Control and Estimation of Systems with Input Output Delays

Lecture Notes in Control and Information Sciences Editors: M. Thoma, M. Morari

355

Huanshui Zhang, Lihua Xie

Control and Estimation of Systems with Input/Output Delays

ABC

Series Advisory Board F. Allgöwer, P. Fleming, P. Kokotovic, A.B. Kurzhanski, H. Kwakernaak, A. Rantzer, J.N. Tsitsiklis

Authors Professor Lihua Xie School of EEE, BLK S2 Nanyang Technological University Nanyang Ave. Singapore 639798 Sinagapore Email: [email protected]

Professor Huanshui Zhang School of Control Science and Engineering Shandong University Jinan P.R. China Email: [email protected]

Library of Congress Control Number: 2007922362 ISSN print edition: 0170-8643 ISSN electronic edition: 1610-7411 ISBN-10 3-540-71118-X Springer Berlin Heidelberg New York ISBN-13 978-3-540-71118-6 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and SPS using a Springer LATEX macro package Printed on acid-free paper

SPIN: 11903901

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543210

Preface

Time delay systems exist in many engineering fields such as transportation, communication, process engineering and more recently networked control systems. In recent years, time delay systems have attracted recurring interests from research community. Much of the research work has been focused on stability analysis and stabilization of time delay systems using the so-called LyapunovKrasovskii functionals and linear matrix inequality (LMI) approach. While the LMI approach does provide an efficient tool for handling systems with delays in state and/or inputs, the LMI based results are mostly only sufficient and only numerical solutions are available. For systems with known single input delay, there have been rather elegant analytical solutions to various problems such as optimal tracking, linear quadratic regulation and H∞ control. We note that discrete-time systems with delays can usually be converted into delay free systems via system augmentation, however, the augmentation approach leads to much higher computational costs, especially for systems of higher state dimension and large delays. For continuous-time systems, time delay problems can in principle be treated by the infinite-dimensional system theory which, however, leads to solutions in terms of Riccati type partial differential equations or operator Riccati equations which are difficult to understand and compute. Some attempts have been made in recent years to derive explicit and efficient solutions for systems with input/output (i/o) delays. These include the study on the H∞ control of systems with multiple input delays based on the stable eigenspace of a Hamlitonian matrix [46]. It is worth noting that checking the existence of the stable eigenspace and finding the minimal root of the transcendent equation required for the controller design may be computationally expensive. Another approach is to split a multiple delay problem into a nested sequence of elementary problems which are then solved based on J-spectral factorizations [62]. In this monograph, our aim is to present simple analytical solutions to control and estimation problems for systems with multiple i/o delays via elementary tools such as projections. We propose a re-organized innovation analysis approach which allows us to convert many complicated delay problems into delay

VI

Preface

free ones. In particular, for linear quadratic regulation of systems with multiple input delays, the approach enables us to establish a duality between the LQR problem and a smoothing problem for a delay free system. The duality contains the well known duality between the LQR of a delay free system and Kalman filtering as a special case and allows us to derive an analytical solution via simple projections. We also consider the dual problem, i.e. the Kalman filtering for systems with multiple delayed measurements. Again, the re-organized innovation analysis turns out to be a powerful tool in deriving an estimator. A separation principle will be established for the linear quadratic Gaussian control of systems with multiple input and output delays. The re-organized innovation approach is further applied to solve the least mean square error estimation for systems with multiple state and measurement delays and the H∞ control and estimation problems for systems with i/o delays in this monograph. We would like to acknowledge the collaborations with Professors Guangren Duan, Yeng Chai Soh and David Zhang on some of the research works reported in the monograph and Mr Jun Xu and Mr Jun Lin for their help in some simulation examples.

Huanshui Zhang Lihua Xie

Symbols and Acronyms

i/o:

input/output.

LQG:

linear quadratic Gaussian.

LQR:

linear quadratic regulation.

PDE:

partial differential equation.

RDE:

Riccati difference (differential) equation.

col{X1 , · · · , Xn }:

the column vector formed by vectors X1 , · · · , Xn .

Rn :

n-dimensional real Euclidean space.

Rn×m :

set of n × m real matrices.

In :

n × n identity matrix.

diag{A1 , A2 , · · · , An }: block diagonal matrix with Aj ( not necessarily square) on the diagonal. 

X:

transpose of matrix X.

P ≥ 0:

symmetric positive semidefinite matrix P ∈ Rn×n .

P > 0:

symmetric positive definite matrix P ∈ Rn×n .

P −1 :

the inverse of the matrix P .

X, Y  :

inner product of vectors X and Y .

E:

mathematical expectation.

L{y1 , · · · , yn }:

linear space spanned by y1 , · · · , yn .



=:

definition.

dim(x):

the dimension of the vector x.

P roj:

projection

VIII

Symbols and Acronyms

M D: δij : e2 :

the total number of Multiplications and Divisions. ⎧ ⎨ 1, i = j, δij = ⎩ 0, i = j. 2 -norm of a discrete-time signal {e(i)},  ∞  2 i.e., e(i) . i=0

2 [0, N ]:

space of square summable vector sequences on [0, N ] with values on Rn .

L2 [0, tf ]:

space of square integrable vector functions on [0, tf ] with values on Rn .

Contents

1.

Krein Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Definition of Krein Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Projections in Krein Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Kalman Filtering Formulation in Krein Spaces . . . . . . . . . . . . . . . 1.4 Two Basic Problems of Quadratic Forms in Krein Spaces . . . . . . 1.4.1 Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4 5 5 6 6

2.

Optimal Estimation for Systems with Measurement Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Single Measurement Delay Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Re-organized Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Re-organized Innovation Sequence . . . . . . . . . . . . . . . . . . . . 2.2.3 Riccati Difference Equation . . . . . . . . . . . . . . . . . . . . . . . . . . ˆ (t | t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Optimal Estimate x 2.2.5 Computational Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Multiple Measurement Delays Case . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Re-organized Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Re-organized Innovation Sequence . . . . . . . . . . . . . . . . . . . . 2.3.3 Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˆ (t | t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Optimal Estimate x 2.3.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 7 9 11 12 13 15 17 18 19 20 22 24 26

3.

Optimal Control for Systems with Input/Output Delays . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Linear Quadratic Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Duality Between Linear Quadratic Regulation and Smoothing Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Solution to Linear Quadratic Regulation . . . . . . . . . . . . . .

27 27 28 29 34

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4.

5.

6.

Contents

3.3 Output Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 44 50

H∞ Estimation for Discrete-Time Systems with Measurement Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 H∞ Fixed-Lag Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 An Equivalent H2 Estimation Problem in Krein Space . . 4.2.2 Re-organized Innovation Sequence . . . . . . . . . . . . . . . . . . . . 4.2.3 Calculation of the Innovation Covariance . . . . . . . . . . . . . . 4.2.4 H∞ Fixed-Lag Smoother . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Computational Cost Comparison and Example . . . . . . . . . 4.2.6 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 H∞ d-Step Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 An Equivalent H2 Problem in Krein Space . . . . . . . . . . . . 4.3.2 Re-organized Innovation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Calculation of the Innovation Covariance . . . . . . . . . . . . . . 4.3.4 H∞ d-Step Predictor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 H∞ Filtering for Systems with Measurement Delay . . . . . . . . . . . 4.4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 An Equivalent Problem in Krein Space . . . . . . . . . . . . . . . . 4.4.3 Re-organized Innovation Sequence . . . . . . . . . . . . . . . . . . . . 4.4.4 Calculation of the Innovation Covariance Qw (t) . . . . . . . . 4.4.5 H∞ Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 54 55 58 59 63 67 69 69 70 73 74 76 77 77 78 80 82 84 85

H∞ Control for Discrete-Time Systems with Multiple Input Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 H∞ Full-Information Control Problem . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Calculation of v ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Maximizing Solution of JN with Respect to Exogenous Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 H∞ Control for Systems with Preview and Single Input Delay . 5.3.1 H∞ Control with Single Input Delay . . . . . . . . . . . . . . . . . . 5.3.2 H∞ Control with Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 88 89 91 96 104 106 106 108 111 113

Linear Estimation for Continuous-Time Systems with Measurement Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Contents

6.2 Linear Minimum Mean Square Error Estimation for Measurement Delayed Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Re-organized Measurement Sequence . . . . . . . . . . . . . . . . . . 6.2.3 Re-organized Innovation Sequence . . . . . . . . . . . . . . . . . . . . 6.2.4 Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˆ (t | t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Optimal Estimate x 6.2.6 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 H∞ Filtering for Systems with Multiple Delayed Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 An Equivalent Problem in Krein Space . . . . . . . . . . . . . . . . 6.3.3 Re-organized Innovation Sequence . . . . . . . . . . . . . . . . . . . . 6.3.4 Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.6 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 H∞ Fixed-Lag Smoothing for Continuous-Time Systems . . . . . . 6.4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 An Equivalent H2 Problem in Krein Space . . . . . . . . . . . . 6.4.3 Re-organized Innovation Sequence . . . . . . . . . . . . . . . . . . . . 6.4.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.

8.

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116 116 117 118 119 120 121 122 123 124 126 127 129 130 132 132 133 136 137 140 141

H∞ Estimation for Systems with Multiple State and Measurement Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Problem Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 H∞ Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Stochastic System in Krein Space . . . . . . . . . . . . . . . . . . . . 7.3.2 Sufficient and Necessary Condition for the Existence of an H∞ Smoother . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 The Calculation of an H∞ Estimator zˇ(t, d) . . . . . . . . . . . 7.4 H∞ Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

148 149 157 162

Optimal and H∞ Control of Continuous-Time Systems with Input/Output Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Linear Quadratic Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Problem Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Solution to the LQR Problem . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Measurement Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 164 164 165 169 175 178 179

143 143 144 145 146

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8.3.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 H∞ Full-Information Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Calculation of v ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 H∞ Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.5 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

180 185 185 187 190 195 199 203

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

1. Krein Space

In this monograph, linear estimation and control problems for systems with i/o delays are investigated under both the H2 and H∞ performance criteria. The key to our development is re-organized innovation analysis in Krein spaces. The reorganized innovation analysis approach, which forms the basis of the monograph, will be presented in Chapter 2. Krein space theory including innovation analysis and projections which can be found in [30], plays an important role in dealing with the H∞ control and estimation problems as it provides a powerful tool to convert an H∞ problem into an H2 one under an appropriate Krein space. Thus, simple and intuitive techniques such as projections can be applied to derive a desired estimator or controller. For the convenience of discussions, in the rest of the monograph we shall first give a brief introduction to Krein space in this chapter.

1.1 Definition of Krein Spaces We briefly introduce the definition and some basic properties of Krein spaces, focusing only on a number of key results that are needed in our later development. Much of the material of this section and more extensive expositions can be found in the book [30]. Finite-dimensional (often called Minkowski) and infinite-dimensional Krein spaces share many of the properties of Hilbert spaces, but differ in some important ways that we highlight below. Definition 1.1.1. (Krein Spaces) An abstract vector space K, <, > that satisfies the following requirements is called a Krein space. 1. K is a linear space over the field of complex numbers C. 2. There exists a bilinear form < ·, · >∈ C on K such that
1) < y, x >=< x, y > , 2) < ax + by, z >= a < x, z > +b < y, z > for any x, y, z ∈ K, a, b ∈ C, where “  ” denotes complex conjugation. 3. The vector space K admits a direct orthogonal sum decomposition K = K+ ⊕ K− , H. Zhang and L. Xie: Cntrl. and Estim. of Sys. with I/O Delays, LNCIS 355, pp. 1–6, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com


(1.1)

2

1. Krein Space

such that K+ , < ·, · > and K− , − < ·, · > are Hilbert spaces, and x, y = 0,

(1.2)

for any x ∈ K+ and y ∈ K− . Remark 1.1.1. Some key differences between Krein spaces and Hilbert spaces include: • Hilbert spaces satisfy not only the above 1)-3), but also the requirement that x, x > 0 when x = 0. • The fundamental decomposition of K defines two projection operators P+ and P− such that P+ K = K+ and P− K = K− . Therefore, for every x ∈ K, we can write x = x+ + x− , where x± = P± x ∈ K± . Note that for every x ∈ K+ , we have x, x ≥ 0, but the converse is generally not true since x, x ≥ 0 does not necessarily imply that x ∈ K+ . • A vector x ∈ K is said to be positive if x, x > 0, neutral if x, x = 0, or negative if x, x < 0. Correspondingly, a subspace M ⊂ K can be positive, neutral, or negative, if all its elements are so, respectively. We now focus on linear subspaces of K. We shall define L{y0 , · · · , yN } as the linear subspace of K spanned by the elements {y0 , y1 , · · · , yN } in K. The Gramian of the collection of the elements {y0 , · · · , yN } is defined as the (N + 1) × (N + 1) (block) matrix 

Ry = [yi , yj ]i,j=0,···,N .

(1.3)




The reflexivity property, yi , yj  = yj , yi  , shows that the Gramian is a Hermitian matrix. Denote the column vector formed by the vectors y0 , y1 , · · · , yN as y = col{y0 , y1 , · · · , yN }. Then the Gramian matrix of the above vector is 

Ry = y, y . Similar to the case in Hilbert space, y is called the “random variables” and their Gramian as the “covariance matrix”, i.e.,  
  
Ry = E yi yj = E yy , where E(·) denotes the mathematical expectation.

1.2 Projections in Krein Spaces

3

In a similar way, if we have two sets of elements {z0 , · · · , zM } and {y0 , · · · , yN }, we shall write z={z0 , z1 , · · · , zM } and y={y0 , y1 , · · · , yN }, and introduce the (M + 1) × (N + 1) cross-Gramian matrix 

Rzy = [zi , yj ]i=0,···,M,j=0,···,N = z, y . Note that




Rzy = Ryz .

1.2 Projections in Krein Spaces We now discuss projections in Krein spaces. Definition 1.2.1. (Projections in Krein Spaces) Given the element z in K, and the elements {y0 , · · · , yN } also in K, we define 

ˆ z to be the projection of z onto L{y} = L{y0 , · · · , yN } if z=ˆ z + z˜,

(1.4)

where zˆ ∈ L{y} and z˜ satisfies the orthogonality condition ˜ z ⊥ L{y}, i.e., ˜ z, yi  = 0 for i = 0, 1, · · · , N. In Hilbert spaces, projections always exist and are unique. However, in Krein spaces, this is not always the case. Indeed, we have the following result Lemma 1.2.1. (Existence and Uniqueness of Projections) In the Hilbert space setting: 1. If the Gramian matrix Ry = y, y is nonsingular, then the projection of z onto L{y} exists, is unique, and is given by −1 ˆ z = z, y y, y y = Rzy Ry−1 y.

(1.5)

2. If the Gramian matrix Ry = y, y is singular, then either 1) R(Ryz ) ⊆ R(Ry ) (where R(A) denotes the column range space of the matrix
A). Here the projection zˆ exists but is not unique. In fact, zˆ = k0 y, where k0 is any solution to the linear matrix equation Ry k0 = Ryz ,

(1.6)

or z does not exist. 2) R(Ryz ) ∈ R(Ry ). Here the projection ˆ Since existence and uniqueness will be important for all future results, we shall make the standing assumption that the Gramian Ry is nonsingular. As is well known, the singularity of Ry implies that y0 , y1 , · · · , yN are linearly dependent, i.e.,


det(Ry )= 0 ⇔ k y = 0 for some nonzero vector k ∈ C N +1 .

4

1. Krein Space

In the Krein space setting, all we can deduce from the singularity of Ry is that there exists a linear combination of {y0 , y1 , · · · , yN } that is orthogonal to every vector in L{y}, i.e., that L{y} contains an isotropic vector. This follows by noting that for any complex matrix k1 and for any k in the null space of Ry , we have




k1 Ry k = k1 y, k y = 0,





which shows that the linear combination k y is orthogonal to k1 y for every k1 ,
i.e., k y is an isotropic vector in L{y}.

1.3 Kalman Filtering Formulation in Krein Spaces Given the state-space model in Krein space x(t + 1) = Φt x(t) + Γt u(t), 0 ≤ t ≤ N, y(t) = Ht x(t) + v(t),

(1.7) (1.8)

and ⎤ ⎡ ⎤ ⎡ ⎡ u(t) u(s) Q(t)δts ⎣ v(t) ⎦ , ⎣ v(s) ⎦ = ⎣ 0 x0 x0 0

⎤ 0 0 ⎦ Π0

0 R(t)δts 0

(1.9)

where x0 is the initial state of the system. Now we shall show that the state-space representation allows us to efficiently compute the innovations by an immediate extension of the standard Kalman filter. Theorem 1.3.1. (Kalman Filter in Krein Space) Consider the Krein statespace representation (1.7)-(1.9). Assume that Ry = [< y(t), y(s) >] is strongly regular1 . Then the innovation can be computed via the formulae ˆ (t), 0 ≤ t ≤ N, w(t) = y(t) − Ht x ˆ (t + 1) = Φt x ˆ (t) + Kp,t (y(t) − Ht x ˆ (t)), x ˆ 0 = 0, x


Kp,t = Φt P (t)Ht [Rw (t)]

−1

,

(1.10) (1.11) (1.12)

where x ˆ(t) is the one-step prediction estimate of x(t), x ˆ0 is the initial estimate,


Rw (t) =< w(t), w(t) >= R(t) + Ht P (t)Ht ,

(1.13)

and P (t) can be recursively computed via the Riccati recursion








P (t + 1) = Φt P (t)Φt − Kp,t Rw (t)Kp,t + Γt Q(t)Γt , P (0) = Π0 . 1

(1.14)

Ry is said to be strongly regular if Ry is nonsingular and all its leading submatrices are nonsingular.

1.4 Two Basic Problems of Quadratic Forms in Krein Spaces

5

1.4 Two Basic Problems of Quadratic Forms in Krein Spaces 1.4.1

Problem 1

Consider the minimization of the following quadratic function over z    −1   z Rz Rzy z J(z, y) = , y y Ryz Ry where the central matrix is the inverse of the Gramian matrix       z z Rz Rzy , = . y y Ryz Ry

(1.15)

(1.16)

It is known that the above minimization problem is related to the H∞ estimation [30]. Now we give the solution in the following theorem. Theorem 1.4.1. [30] Suppose that both Ry and (1.16) are nonsingular. Then 1. The stationary point zˆ of J(z, y) over z is given by zˆ = Rzy Ry−1 y. 2. The value of J(z, y) at the stationary point is


J(ˆ z , y) = y Ry−1 y. 3. zˆ yields a unique minimum of J(z, y) if and only if Rz − Rzy Ry−1 Ryz > 0. Proof: We note that  −1   I 0 Rz Rzy Rz − Rzy Ry−1 Ryz = −1 −Ry Ryz I 0 Ryz Ry   −1 I −Rzy Ry . × 0 I

0 Ry

−1

(1.17)

Hence, we can write J(z, y) as 

Rz − Rzy Ry−1 Ryz J(z, y) = [ z − y y ] 0   −1 z − Rzy Ry y × . y





Ry−1 Ryz




0 Ry

−1

(1.18)

It now follows by differentiation that the stationary point of J(z, y) is equal to
z , y) = y Ry−1 y. From (1.18), it follows that zˆ yields zˆ = Rzy Ry−1 y, and that J(ˆ a unique minimum of J(z, y) if and only if Rz − Rzy Ry−1 Ryz > 0. It should be pointed out that the above minimization problem is related to estimator design, and the presented results in the theorem will be used throughout the monograph.

6

1. Krein Space

1.4.2

Problem 2

Consider the minimization of the following quadratic function over u      x0 Rxc0 x0 Rxc0 yc J(x0 , u) = . u Ryc xc0 Ryc u

(1.19)

We have the following results. Theorem 1.4.2. [30] Suppose Ryc and (1.19) are nonsingular. 1. The stationary point u ˆ of J(x0 , u) over u is given by u ˆ = −Ry−1 c Ry c xc x0 . 0 2. The value of J(x0 , u) at the stationary point is 
 ˆ) = x0 Rxc0 − Rxc0 yc Ry−1 J(x0 , u c Ry c xc x0 . 0 3. uˆ yields an unique minimum of J(x0 , u) if and only if Ryc > 0. Proof: Observe that     Rxc0 I Rxc0 yc Rxc0 yc Rxc0 − Rxc0 yc Ry−1 c Ry c xc 0 = 0 Ryc xc0 Ryc 0 I   I 0 × , I Ry−1 c Rxc y c 0 so we can write J(x0 , u) as





 J(x0 , u) = [ x0 x0 Rxc0 yc Ry−1 c + u ]   x0 × . Ry−1 c Ry c xc x0 + u 0



Rxc0 − Rxc0 yc Ry−1 c Ry c xc 0 0

0 Ryc



(1.20)

0 Ryc



(1.21)

The desired result then follows. It should be pointed out that the above minimization problem is related to controller design, and the presented results in the theorem will be used later.

1.5 Conclusion In this chapter we have introduced Krein spaces and some basic properties of Krein spaces. Although Krein spaces and Hilbert spaces share many characteristics, they differ in a number of ways, especially the special features of the inner product in a Krein space make it applicable to H∞ problems. The related concepts such as projections and Kalman filtering formulation in Krein spaces have been given which will be applied as important tools to the estimation and control problems to be studied in this monograph. We have also presented two minimization problems associated with two quadratic forms.

2. Optimal Estimation for Systems with Measurement Delays

This chapter studies the optimal estimation for systems with instantaneous and delayed measurements. A new approach termed as re-organized innovation analysis is proposed to handle delayed measurements. The re-organized innovation will play an important role not only in this chapter but also in the rest of the monograph.

2.1 Introduction In this chapter we investigate the minimum mean square error (MMSE) estimation problem for systems with instantaneous and delayed measurements. Such problem has important applications in many engineering fields such as communications and sensor fusion [42]. As is well known, the optimal estimation problem for discrete-time systems with known delays can be approached by using system augmentation in conjunction with standard Kalman filtering [3]. However, the augmentation method generally leads to a much expensive computational cost, especially when the delays are large. Our aim here is to present a simple Kalman filtering solution to such problem by adopting the so-called re-organized innovation approach developed in our work [102][107]. It will be shown that the optimal estimator can be computed in terms of the same number of Riccati difference equations (RDEs) (with order of the system ignoring the delays) as that of the measurement channels. The proposed approach in this chapter forms the basis for solving other related problems that are to be studied in the rest of the monograph.

2.2 Single Measurement Delay Case In this section we shall study the Kalman filtering problem for time-varying systems with measurement delays. We consider the system with instantaneous and single delayed measurements described by x(t + 1) = Φx(t) + Γ e(t), H. Zhang and L. Xie: Cntrl. and Estim. of Sys. with I/O Delays, LNCIS 355, pp. 7–26, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com


(2.1)

8

2. Optimal Estimation for Systems with Measurement Delays

y(0) (t) = H(0) x(t) + v(0) (t), y(1) (t) = H(1) x(t − d) + v(1) (t),

(2.2) (2.3)

where x(t) ∈ Rn is the state, e(t) ∈ Rr is the input noise, y(0) (t) ∈ Rp0 and y(1) (t) ∈ Rp1 are respectively the instantaneous and delayed measurements, v(0) (t) ∈ Rp0 and v(1) (t) ∈ Rp1 are the measurement noises. d is the measurement delay which is a known integer. The initial state x(0) and e(t), v(0) (t) and v(1) (t) are mutually uncorrelated white noises with zero means  and known co-

 (s) = variance matrices E [x(0)x (0)] = P0 , E [e(t)e (s)] = Qe δts , E v(0) (t)v(0)   Qv(0) δts and E v(1) (t)v(1) (s) = Qv(1) δts , respectively. In the above, E(·) denotes the mathematical expectation and ⎧ ⎨ 1, i = j, δij = ⎩ 0, i = j.

With the delayed measurement in (2.3), the system (2.1)-(2.3) is not in a standard form to which the standard Kalman filtering is applicable. Let y(t) denote the observation of the state x(t) of the system (2.1)-(2.3): ⎧ ⎪ ⎨ y(0) (t), 0≤t


· · · {H(1) x(t − d)} ] ,

(2.5)

we obtain an augmented state-space model: xa (t + 1) = Φa xa (t) + Γa e(t), y(t) = HLa (t)xa (t) + va (t), where

⎧ ⎪ ⎨ v(0) (t), 0≤t
0 0 Ip1

··· 0 0 .. .

0 0 0

0

0

Ip1

⎡ ⎤ 0⎤ Γ 0⎥ ⎢0⎥ ⎥ ⎢ ⎥ 0 ⎥ , Γa = ⎢ 0 ⎥ , ⎥ ⎢ . ⎥ ⎦ ⎣ .. ⎦ 0 0

(2.6) (2.7)

(2.8)

2.2 Single Measurement Delay Case

⎧ ⎪ ⎨ Ha , 0≤t
9

(2.9)

with Ha = [ H(0) 0 · · · 0 ] , La = [ 0 · · · 0 Ip1 ] .

(2.10)

ˆ (t | t) is given by Then the optimal estimate x ˆ (t | t) = [ In x

ˆ a (t | t), ··· 0]x

(2.11)

ˆ a (t | t) is the optimal estimate of the above augmented system which can where x be obtained by the standard Kalman filter. However, the augmentation generally leads to a higher system dimension and thus higher computational cost. In this section, we shall propose a new method for the optimal filter design without resorting to the system augmentation. Our approach is based on projection and re-organized innovation analysis. It is shown that the proposed approach is computationally attractive as compared with the augmentation approach. 2.2.1

Re-organized Measurements

As is well known, given the measurement sequence {y(s)}ts=0 , the optimal state ˆ (t | t) is the projection of x(t) onto the linear space spanned by the estimator x measurement sequence, denoted by [3, 38]      y(0) y(d − 1) L ,···, , y(d), · · · , y(t) . 0 0 Observe from (2.4) that for 0 ≤ t < d,     L {y(s)}ts=0 = L {y(0) (s)}ts=0 ,

(2.12)

ˆ (t | t) is the standard Kalman filter associated with and the optimal estimator x system (2.1)-(2.2). Thus, in the following we shall focus on the case when t ≥ d. In this situation, the linear space      y(0) y(d − 1) L ,···, , y(d), · · · , y(t) 0 0 is equivalent to ˜ 1 (t − d + 1), · · · , y ˜ 1 (t)} , L {y2 (0), y2 (1), · · · , y2 (t − d); y

(2.13)

˜ 1 (·) are defined as where y2 (·) and y   y(0) (s) y2 (s) = , 0 ≤ s ≤ t − d, y(1) (s + d)   y1 (s) ˜ 1 (s) = y t − d < s ≤ t. , y1 (s) = y(0) (s), 0

(2.14) (2.15)

10

2. Optimal Estimation for Systems with Measurement Delays

Remark 2.2.2. Since the sequence ˜ 1 (t − d + 1), · · · , y ˜ 1 (t)} {y2 (0), y2 (1), · · · , y2 (t − d); y

(2.16)

contains the same information about the system as the re-organized sequence {y2 (0), y2 (1), · · · , y2 (t − d); y1 (t − d + 1), · · · , y1 (t)} ,

(2.17)

throughout the monograph for the simplicity of notation we use (2.17) to denote the linear space spanned by the former although we note that the elements in the latter sequence have different vector dimensions. Similar notation will be applied to linear spaces spanned by innovation sequences to be discussed later. It is easy to know that y2 (s) and y1 (s) satisfy y2 (s) = H2 x(s) + v2 (s),

(2.18)

y1 (s) = H1 x(s) + v1 (s),

(2.19)

where

 H2 =

and

 H(0) , H1 = H(0) H(1)

(2.20)



 v(0) (s) v2 (s) = , v1 (s) = v(0) (s). v(1) (s + d)

(2.21)

Obviously, v2 (s) and v1 (s)  are white noises of zero means and covariance matriQv(0) 0 ces Qv2 = and Qv1 = Qv(0) , respectively. Note that the system 0 Qv(1) (2.1) with measurement (2.18) or (2.19) forms a state-space representation associated with the standard Kalman filtering. The new measurement sequence, {y2 (0), y2 (1), · · · , y2 (t − d); y1 (t − d + 1), · · · , y1 (t)}, is named as a re-organized measurement sequence of {{y(s)}ts=0 }. Throughout the section, we denote t1 = t − d and define the following Definition 2.2.1. Given t > d, the estimator xˆ(s, 2), s ≤ t1 + 1 = t − d + 1 is the optimal estimate of x(s) given the observation sequence: {y2 (0), · · · , y2 (s − 1)}. Similarly, the estimator x ˆ(s, 1), s > t1 + 1 is the optimal estimate of x(s) given the observation sequence: {y2 (0), · · · , y2 (t1 ); y1 (t1 + 1), · · · , y1 (s − 1)}. For s = t1 + 1, xˆ(s, 1) is the optimal estimate of x(s) given the observation {y2 (0), · · · , y2 (t1 )}. Note that x ˆ(t1 + 1, 2) = x ˆ(t1 + 1, 1).

2.2 Single Measurement Delay Case

2.2.2

11

Re-organized Innovation Sequence

We now introduce the re-organized innovation associated with the re-organized measurements introduced in the last subsection. Define: 

ˆ 1 (s, 1), s > t1 , w(s, 1) = y1 (s) − y 

ˆ 2 (s, 2), w(s, 2) = y2 (s) − y

ˆ 2 (0, 2) = 0, 0 ≤ s ≤ t1 , y

(2.22) (2.23)

ˆ 1 (s, 1) with s > t1 +1 is the projection of y1 (s) where, similar to Definition 2.2.1, y onto the linear space formed by {y2 (0), · · · , y2 (t1 ); y1 (t1 + 1), · · · , y1 (s − 1)} ˆ 2 (s, 2) with s ≤ t1 + 1 is the projection of y2 (s) onto the linear space and y formed by {y2 (0), · · · , y2 (s − 1)}. We then have the following relationships ˜ (s, 1) + v1 (s), w(s, 1) = H1 x ˜ (s, 2) + v2 (s), w(s, 2) = H2 x

(2.24)

˜ (s, 1) = x(s) − x ˆ (s, 1), x ˜ (s, 2) = x(s) − x ˆ (s, 2), x

(2.26) (2.27)

(2.25)

where

ˆ (s, 1) and x ˆ (s, 2) are as in Definition 2.2.1. It is clear that x ˜ (s + 1, 1) = and x ˜ (s + 1, 2) when s = t1 . The following lemma shows that {w(·, ·)} is in fact an x innovation sequence. Lemma 2.2.1. {w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(t, 1)} is the innovation sequence which spans the same linear space as that of (2.13) or equivalently L {y(0), · · · , y(t)}. Proof: First, it is readily seen from (2.23) that w(s, 2), s ≤ t1 ( or w(s, 1), s > t1 ) is a linear combination of the observation sequence {y2 (0), · · · , y2 (s)} (or {y2 (0), · · · , y2 (t1 ); y1 (t1 + 1), · · · , y1 (s)}). Conversely, y2 (s), s ≤ t1 , (or y1 (s), s > t1 ) can be given in terms of a linear combination of {w(0, 2), · · · , w(s, 2)} ( or {w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(s, 1)}). Thus, {w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(t, 1)} spans the same linear space as L {y2 (0), · · · , y2 (t1 ); y1 (t1 + 1), · · · , y1 (t)} or equivalently L {y(0), · · · , y(t)}. Next, we show that w(·, ·) is an uncorrelated sequence. In fact, for any s > t1 and τ ≤ t1 , from (2.24) we have ˜ (s, 1)w (τ, 2)] + E [v2 (s)w (τ, 2)] . E [w(s, 1)w (τ, 2)] = E [H1 x

(2.28)

˜ (s, 1) is the state prediction error, it folNote that E [v2 (s)w (τ, 2)] = 0. Since x lows that E [˜ x(s, 1)w (τ, 2)] = 0, and thus E [w(s, 1)w (τ, 2)] = 0, which implies that w(τ, 2) (τ ≤ t1 ) is uncorrelated with w(s, 1) (s > t1 ). Similarly, it can be verified that w(s, 2) is uncorrelated with w(τ, 2) and w(s, 1) is uncorrelated with w(τ, 1) for s = τ . Hence, {w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(t, 1)} is an innovation sequence. ∇

12

2. Optimal Estimation for Systems with Measurement Delays

The white noise sequence, {w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(t, 1)} is termed as re-organized innovation sequence associated with the measurement sequence {y2 (0), · · · , y2 (t1 ); y1 (t1 + 1), · · · , y1 (t)}. Similarly, for any s > t1 , the sequence {w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(s, 1)} is also termed as re-organized innovation sequence associated with the measurement sequence {y2 (0), · · · , y2 (t1 ); y1 (t1 + 1), · · · , y1 (s)}. The re-organized innovation sequence will play a key role in deriving the optimal estimator in the later subsections. 2.2.3

Riccati Difference Equation

For a given t > d, denote 

P2 (s) = E [˜ x(s, 2)˜ x (s, 2)] , 0 ≤ s ≤ t1 , 

P1 (s) = E [˜ x(s, 1)˜ x (s, 1)] , s > t1 .

(2.29) (2.30)

˜ (t1 + 1, 2), it is obvious ˜ (t1 + 1, 1) = x We note that in view of the fact that x that P1 (t1 + 1) = P2 (t1 + 1). Now, it follows from (2.24) and (2.25) that the innovation covariance matrix of w(·, ·) is given by 

Qw (s, 2) = E [w(s, 2)w (s, 2)] = H2 P2 (s)H2 + Qv2 , 0 ≤ s ≤ t1 (2.31) and 

Qw (s, 1) = E [w(s, 1)w (s, 1)] = H1 P1 (s)H1 + Qv1 , s > t1 . (2.32) We have the following results Theorem 2.2.1. For a given t > d, the covariance matrices P2 (·) and P1 (·) can be calculated as follows. – P2 (s), 0 < s ≤ t1 , is calculated by the following standard RDE: P2 (s + 1) = ΦP2 (s)Φ − K2 (s)Qw (s, 2)K2 (s) + Γ Qe Γ  , P2 (0) = P0 ,

(2.33)

where Qw (s, 2) is as in (2.31) and K2 (s) = ΦP2 (s)H2 Q−1 w (s, 2).

(2.34)

– P1 (s), t1 + 1 < s ≤ t, is given by P1 (s + 1) = ΦP1 (s)Φ − K1 (s)Qw (s, 1)K1 (s) + Γ Qe Γ  , P1 (t1 + 1) = P2 (t1 + 1),

(2.35)

where Qw (s, 1) is as in (2.32) and K1 (s) = ΦP1 (s)H1 Q−1 w (s, 1).

(2.36)

2.2 Single Measurement Delay Case

13

Proof: It is obvious that P2 (s + 1) is the covariance matrix of the one step ahead prediction error of the state x(s + 1) associated with the system (2.1) and (2.18). Thus following the standard Kalman filtering theory, P2 (s + 1) satisfies the RDE(2.33). ˆ (s + 1, 1) (s > t1 ) is the projection of the state On the other hand, note that x x(s + 1) onto the linear space L {w(0, 2), · · · , w(t1 , 2), w(t1 + 1, 1), · · · , w(s, 1)} . ˆ (s + 1, 1) can be calculated by using Since w(·, ·) is a white noise, the estimator x the projection formula as ˆ (s + 1, 1) x = P roj {x(s + 1 | w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(s − 1, 1)} +P roj {x(s + 1 | w(s, 1)} = Φˆ x(s, 1) + ΦE [x(s)˜ x (s, 1)] H1 Q−1 w (s, 1)w(s, 1) = Φˆ x(s, 1) + ΦP1 (s)H1 Q−1 w (s, 1)w(s, 1).

(2.37)

It is readily obtained from (2.1) and (2.37) that ˜ (s + 1, 1) = x(s + 1) − x ˆ (s + 1, 1) x = Φ˜ x(s, 1) + Γ e(s) − ΦP1 (s)H1 Q−1 w (s, 1)w(s, 1). (2.38) ˜ (s+ 1, 1) is uncorrelated with w(s, 1) and so is x ˜ (s, 1) with e(s), it follows Since x from the above equation that    P1 (s + 1) + ΦP1 (s)H1 Q−1 w (s, 1)H1 P1 (s)Φ = ΦP1 (s)Φ + Γ Qe Γ ,

∇

which is (2.35). 2.2.4

Optimal Estimate x ˆ(t | t)

In this section we shall give a solution to the optimal filtering problem. Based on the discussion in the previous subsection, the following results are obtained by applying the re-organized innovation sequence. Theorem 2.2.2. Consider the system (2.1)-(2.3). Given d > 0, the optimal ˆ (t | t) is given by filter x ˆ (t, 1)] , ˆ (t | t) = x ˆ (t, 1) + P1 (t)H1 Q−1 x w (t, 1) [y1 (t) − H1 x

(2.39)

ˆ (t, 1) is calculated recursively as where x ˆ (s + 1, 1) = Φ1 (s)ˆ x x(s, 1) + K1 (s)y1 (s), t1 + 1 ≤ s < t,

(2.40)

K1 (s) is as in (2.36) and Φ1 (s) = Φ − K1 (s)H1 ,

(2.41)

14

2. Optimal Estimation for Systems with Measurement Delays

while Qw (s, 1) is as in (2.32) and P1 (s) is computed by (2.35). The initial value ˆ (t1 + 1, 1) = x ˆ (t1 + 1, 2), and x ˆ (t1 + 1, 2) is calculated by the following Kalman x filtering ˆ (s + 1, 2) = Φ2 (s)ˆ ˆ (0, 2) = 0, x x(s, 2) + K2 (s)y2 (s), 0 ≤ s ≤ t1 , x (2.42) where K2 (s) is as in (2.34) and Φ2 (s) = Φ − K2 (s)H2 .

(2.43)

ˆ (t | t) is the projection of the state x(t) onto Proof: By applying Lemma 2.2.1, x the linear space L{w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(t, 1)}. Since w(·, ·) is ˆ (t | t) is calculated by using the projection formula as a white noise, the filter x ˆ (t | t) x = P roj {x(t) | w(0, 2), · · · , w(t1 , 2), w(t1 + 1, 1), · · · , w(t − 1, 1)} +P roj {x(t) | w(t, 1)} ˆ (t, 1) + E [x(t)w (t, 1)] Q−1 =x w (t, 1)w(t, 1) ˆ (t, 1)] , ˆ (t, 1) + P1 (t)H1 Q−1 (2.44) (t, 1) [y1 (t) − H1 x =x w ˆ (s+1, 1) (s > t1 ) is the projection which is (2.39). Similarly, from Lemma 2.2.1, x of the state x(s + 1) onto the linear space L{w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(s, 1)}. Thus, it follows from the projection formula that ˆ (s + 1, 1) x = P roj {x(s + 1) | w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(s, 1)} = Φˆ x(s, 1) + Γ P roj {e(s) | w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(s, 1)} . (2.45) Noting that e(s) is uncorrelated with the innovation sequence, {w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(s, 1)}, we have ˆ (s + 1, 1) = Φˆ x x(s, 1) + ΦE [x(s)w (s, 1)] Q−1 w (s, 1)w(s, 1)


ˆ (s, 1)] = Φˆ x(s, 1) + ΦP1 (s)H1 Q−1 w (s, 1) [y1 (s) − H1 x = Φ1 (s)ˆ x(s, 1) + K1 (s)y1 (s),

(2.46)

ˆ (t1 +1, 1) = x ˆ (t1 +1, 2), which is the which is (2.40). The initial value of (2.40) is x standard Kalman filter of (2.1) and (2.18), and is obviously given by (2.42). ∇

2.2 Single Measurement Delay Case

15

Remark 2.2.3. For k ≥ m, denote Φ1 (k, m) = Φ1 (k − 1) · · · Φ1 (m), Φ1 (m, m) = In ,

(2.47)

where Φ1 (s) is as in (2.41). Then the recursion of (2.40) can be easily rewritten as ˆ (t, 1) = Φ1 (t, t1 + 1)ˆ x x(t1 + 1, 2) +

t−1 

Φ1 (t, s + 1)K1 (s)y1 (s). (2.48)

s=t1 +1

ˆ (t, 1) can be obtained with the From (2.48), it is clear that the estimator x ˆ (t1 + 1, 2), which is given by (2.42). initial value of x Remark 2.2.4. The Kalman filtering solution for system (2.1)-(2.3) with delayed measurement has been given by applying the re-organized innovation analysis. Different from the standard Kalman filtering approach, our approach consists of two parts. The first part is given by (2.40) and (2.35), which is the Kalman filtering for the system (2.1) and (2.19). The second part is given by (2.42) and (2.33), which is the Kalman filtering for the system (2.1) and (2.18). Observe that the solution only relies on two RDEs of dimension n × n. This is in comparison with the traditional augmentation method where one Riccati equation of dimension (n + d × p1 ) × (n + d × p1 ) is involved. In the following subsection, we shall demonstrate that the proposed method indeed possesses computational advantages over the latter. Remark 2.2.5. The above re-organized innovation analysis in Hilbert space can be extended to Krein space to address the more complicated H∞ fixed-lag smoothing and other H∞ estimation problems for time-delay systems which will be demonstrated in the later chapters. 2.2.5

Computational Cost

W shall now compare the computational cost of the presented approach and the system augmentation method. As additions are much faster than multiplications and divisions, it is the number of multiplications and divisions that is used as the operation count. Let M D denote the number of multiplications and divisions. First, note that the algorithm by Theorem 2.2.2 can be summarized as 1. Compute matrix P2 (t1 ) using the RDE (2.33). 2. Compute P1 (s) for t1 + 1 ≤ s < t using (2.35). ˆ (t | t) using (2.39)-(2.42). 3. Compute x ˆ (t | t) in one It is easy to know that the total M D count for obtaining x iteration, denoted as M Dnew , is given by ! M Dnew = 3n3 + (3p0 + r)n2 + 2p20 n + p30 d + (6p1 + 1)n2 ! + 2(p0 + p1 )2 − 2p20 + 4p0 p1 + 2p21 + p0 + p1 n + 2(p0 + p1 )3 + (p0 + p1 )2 − 2p30 .

(2.49)

16

2. Optimal Estimation for Systems with Measurement Delays

On the other hand, recall the Kalman filtering for the augmented state-space ˆ a (t | t) is computed by model (2.5)-(2.10). The optimal filter x ˆ a (t + 1 | t + 1) = Φa x ˆ a (t | t) + Pa (t)HL a (t)Q−1 x w (t)    y(0) (t) ˆ a (t | t) , × − Ha (t + 1)Φa x y(1) (t)

(2.50)

where the matrix Pa (t) satisfies the following RDE   Pa (t + 1) = Φa Pa (t)Φa − Φa Pa (t)HL a (t)Q−1 w (t)HLa (t)Pa (t)Φa + Γa Qe Γa , (2.51)   Qv(0) 0  with Qw (t) = HLa (t)Pa (t)HLa (t) + Qvs and Qvs = Qv2 = . In 0 Qv(1) view of the special structure of the matrices Φa , Γa , HLa (t), the calculation burden for the RDE (2.51) can be reduced. We partition Pa (t) as

Pa (t) = {Pa,ij (t), 1 ≤ i ≤ d + 1, 1 ≤ j ≤ d + 1}, where Pa,11 (t) and Pa,ii (t), i > 1 are of the dimensions n × n and p1 × p1 , respectively. The RDE (2.51) can be simplified as [91]:   −1  Πa (t) = Pa (t) − Pa,1 (t)H(0) [Qv(0) + H(0) Pa,11 (t)H(0) ] H(0) Pa,1 (t),  (t) Σa (t) = Πa (t) − Πa,d+1 (t)[Qv(1) + Πa,(d+1)(d+1) (t)]−1 Πa,d+1

Pa (t + 1) = Γa Qe Γa + Φa Σa (t)Φa ,

(2.52)

where Pa,i (t) and Πa,i (t) represent the ith column blocks of Pa (t) and Πa (t), respectively. Suppose that Σa (t) is partitioned similarly to Pa (t) and Πa (t). By taking into account the structure of the matrices Φa and Γa , (2.52) can be rewritten as Pa (t + 1) = ΦΣa,11 (t)Φ
+ Γ Qe Γ
⎢ H(1) Σa,11 (t)Φ
⎢ ⎢ Σa,21 (t)Φ
⎢ ⎢ .. ⎣ . Σa,d1 (t)Φ
⎡


ΦΣa,11 (t)H(1)
H(1) Σa,11 (t)H(1)
Σa,21 (t)H(1) .. .
Σa,d1 (t)H(1)

ΦΣa,12 (t) H(1) Σa,12 (t) Σa,22 (t) .. . Σa,d2 (t)

··· ··· ··· ··· ···

⎤ ΦΣa,1d (t) H(1) Σa,1d (t) ⎥ ⎥ Σa,2d (t) ⎥ , ⎥ ⎥ .. ⎦ . Σa,dd (t) (2.53)

and the optimal filter (2.50) is then ˆ a (t + 1 | t + 1) x ⎡ Φˆ x (t | t) ⎤

⎡  a1 Pa,11 (t)H(0) ˆ a1 (t | t) ⎥ ⎢ P ⎢ H(1) x  a,21 (t)H(0) ⎢ ⎥ ⎢ ˆ x (t | t) ⎢ ⎥ a2 =⎢ .. ⎥+⎢ .. ⎣ ⎦ ⎣ . .  (t)H(0) P a,(d+1)1 ˆ ad (t | t) x     y(0) (t) xa1 (t | t) H(0) Φˆ × − , ˆ ad (t | t) x y(1) (t)

⎤ Pa,1(d+1) (t) Pa,2(d+1) (t) ⎥ ⎥ −1 ⎥ Qw (t) .. ⎦ . Pa,(d+1)(d+1) (t) (2.54)

2.3 Multiple Measurement Delays Case

17

ˆ a (t | t) = [ x ˆ a1 (t | t) x ˆ a2 (t | t) · · · x ˆ a d+1 (t | t) ] . Let M Daug denote where x ˆ a (t + 1 | t + 1) by (2.52), (2.53) and (2.54). the operation number of calculating x We have M Daug = p21 (n + p1 )d2 + 4p1 n2 + (4p21 + 3p0 p1 )n ! +(p20 + p0 + p21 + p1 )p1 d + 4n3 + (4p0 + 3p1 + 1)n2 +(2p20 + 2p0 − p21 + 2p1 )n + p30 + p20 + p31 .

(2.55)

From (2.49) and (2.55), it is clear that M Daug is of magnitude O(d2 ) whereas M Dnew is linear in d. Thus, when the delay d is sufficiently large, it is easy to know that M Daug > M Dnew . Moreover, the larger the d, the larger the ratio MDaug MDnew . To see this, we consider one example. Example 2.2.1. Consider the system (2.1)-(2.3 ) with n = 3, p0 = 1, r = 1 and p1 = 3. The M D numbers of the proposed approach and the system augmentation approach are compared in Table 2.1 for various values of d.

Table 2.1. Comparison of the Computational Costs d M Dnew M Daug M Daug M Dnew

1 629 605 0.9618

2 753 1052 1.3971

3 877 1607 1.8324

6 1249 3920 3.1385

12 1993 11462 5.7511

2.3 Multiple Measurement Delays Case In this section we shall extend the study of the last section to consider systems with multiple delayed measurements. We consider the linear discrete-time system x(t + 1) = Φx(t) + Γ e(t),

(2.56)

where x(t) ∈ Rn is the state and e(t) ∈ Rr is the system noise. The state x(t) is observed by l + 1 different channels with delays as described by y(i) (t) = H(i) x(t − di ) + v(i) (t), i = 0, 1, · · · , l,

(2.57)

where, without loss of generality, the time delays di , i = 0, 1, · · · , l are assumed to be in a strictly increasing order: 0 = d0 < d1 < · · · < dl , y(i) (t) ∈ Rpi is the ith delayed measurement, and v(i) (t) ∈ Rpi is the measurement noise. The initial state x(0) and the noises e(t) and v(i) (t), i = 0, 1, · · · , l are mutually uncorrelated white noises with zero means and covariance matrices as E [x(0)x (0)] = P0 ,   (j) = Qv(i) δkj , respectively. E [e(k)e (j)] = Qe δkj , and E v(i) (k)v(i)

18

2. Optimal Estimation for Systems with Measurement Delays

Observe from (2.57) that y(i) (t) is in fact an observation of the state x(t − di ) at time t, with delay di . Let y(t) denote all the observations of the system (2.56)-(2.57) at time t, then we have ⎡ ⎤ y(0) (t) ⎢ ⎥ .. y(t) = ⎣ (2.58) ⎦ , di−1 ≤ t < di , . y(i−1) (t) and for t ≥ dl , ⎤ y(0) (t) ⎥ ⎢ y(t) = ⎣ ... ⎦ . y(l) (t) ⎡

(2.59)

The linear optimal estimation problem can be stated as: Given the observation ˆ (t | t) sequence {{y(s)}ts=0 }, find a linear least mean square error estimator x of x(t). Since the measurement y(t) is associated with states at different time instants due to the delays, the standard Kalman filtering is not applicable to the estimation problem. Similar to the single delayed measurement case, one may convert the problem into a standard Kalman filtering estimation by augmenting the state. However, the computation cost of the approach may be very high due to a much increased state dimension of the augmented system [3]. In this section, we shall extend the re-organized innovation approach of the last section to give a simpler derivation and solution for the optimal estimation problem associated with systems of multiple delayed measurements. Throughout the section we denote that 

ti = t − di , i = 0, 1, · · · , l and assume that t ≥ dl for the convenience of discussions. 2.3.1

Re-organized Measurements

In this subsection, the instantaneous and l-delayed measurements will be reorganized as delay free measurements so that the Kalman filtering is applicable. As is well known, given the measurement sequence {y(s)}ts=0 , the optiˆ (t | t) is the projection of x(t) onto the linear space mal state estimator x L {{y(s)}ts=0 } [3, 38]. Note that the linear space L {{y(s)}ts=0 } is equivalent to the following linear space L {yl+1 (0), · · · , yl+1 (tl ); · · · ; yi (ti + 1), · · · , yi (ti−1 ); · · · ; y1 (t1 + 1), · · · , y1 (t)} ,

(2.60)

2.3 Multiple Measurement Delays Case

19

where ⎡

⎤ y(0) (s) ⎢ ⎥ .. yi (s) = ⎣ ⎦. . y(i−1) (s + di−1 )

(2.61)

It is clear that yi (s) satisfies yi (t) = Hi x(t) + vi (t),

i = 1, · · · , l + 1,

(2.62)

with ⎡

⎡ ⎤ ⎤ H(0) v(0) (t) ⎢ ⎢ ⎥ ⎥ .. .. Hi = ⎣ ⎦ , vi (t) = ⎣ ⎦. . . v(i−1) (t + di−1 ) H(i−1) (t + di−1 )

(2.63)

It is easy to know that vi (t) is a white noise of zero mean and covariance matrix Qvi = diag{Qv(0) , · · · , Qv(i−1) }, i = 1, 2, · · · , l + 1.

(2.64)

Note that the measurements in (2.62), termed as re-organized measurements of {{y(s)}ts=0 }, are no longer with any delay. 2.3.2

Re-organized Innovation Sequence

In this subsection we shall define the innovation associated with the re-organized measurements (2.60). First, we introduce a similar definition of projection as in Definition 2.2.1. Definition 2.3.1. The estimator x ˆ(s, i) for ti + 1 < s ≤ ti−1 is the optimal estimation of x(s) given the observation sequence: {yl+1 (0), · · · , yl+1 (tl );

· · · ; yi (ti + 1), · · · , yi (s − 1)}.

(2.65)

For s = ti + 1, x ˆ(s, i) is the optimal estimation of x(s) given the observation sequence: {yl+1 (0), · · · , yl+1 (tl );

· · · ; yi+1 (ti+1 + 1), · · · , yi+1 (ti )}.

(2.66)

From the above definition we can introduce the following stochastic sequence. 

ˆ i (s, i), wi (s, i) = yi (s) − y

(2.67)

ˆ i (s, i) is the optimal estimation of yi (s) given the observawhere for s > ti + 1, y ˆ i (s, i) is the optimal estimation of yi (s) tion sequence of (2.65) and for s = ti +1, y given the observation sequence (2.66). For i = l + 1, it is clear that wl+1 (s, l + 1)

20

2. Optimal Estimation for Systems with Measurement Delays

is the standard Kalman filtering innovation sequence for the system (2.56) and (2.62) for i = l + 1. In view of (2.62), it follows that ˜ (s, i) + vi (s), i = 1, · · · , l + 1, wi (s, i) = Hi x

(2.68)

where ˜ (s, i) = x(s) − x ˆ (s, i), x

i = 1, · · · , l + 1

(2.69)

is the one step ahead prediction error of the state x(s) based on the observations (2.65) or (2.66). The following lemma shows that wi (s, i), i = 1, · · · , l + 1 form an innovation sequence associated with the re-organized observations (2.60). Lemma 2.3.1 {wl+1 (0, l + 1), · · · , wl+1 (tl , l + 1); · · · ; wi (ti + 1, i), · · · , wi (ti−1 , i); · · · ; w1 (t1 + 1, 1), · · · , w1 (t, 1)}

(2.70)

is an innovation sequence which spans the same linear space as: L {yl+1 (0), · · · , yl+1 (tl ); · · · ; yi (ti + 1), · · · , yi (ti−1 ); · · · ; y1 (t1 + 1) · · · , y1 (t)} , or equivalently L{y(0), · · · y(t)}. Proof: The proof is very similar to the single delay case of the last section.

∇

The white noise sequence, {wl+1 (0), · · · , wl+1 (tl ); · · · ; wi (ti + 1), · · · , wi (ti−1 ); · · · ; w1 (t1 + 1), · · · , w1 (t)} is termed as re-organized innovation sequence associated with the measurement sequence {y(0), · · · y(t)}. 2.3.3

Riccati Equation

Let 

˜  (s, i)] , i = l + 1, · · · , 1 Pi (s) = E [˜ x(s, i) x

(2.71)

be the covariance matrix of the state estimation error. For delay free systems, it is well known that the covariance matrix of the state filtering error satisfies a Riccati equation. Similarly, we shall show that the covariance matrix Pi (s) defined in (2.71) obeys certain Riccati equation.

2.3 Multiple Measurement Delays Case

21

Theorem 2.3.1. For a given t > dl , the matrix of Pl+1 (tl + 1) is calculated as  Pl+1 (tl + 1) = ΦPl+1 (tl )Φ − Kl+1 (tl )Qw (tl , l + 1)Kl+1 (tl ) + Γ Qe Γ  , (2.72) Pl+1 (0) = E[x(0)x (0)] = P0 ,

where  Kl+1 (tl ) = ΦPl+1 (tl )Hl+1 Q−1 w (tl , l + 1),  + Qvl+1 . Qw (tl , l + 1) = Hl+1 Pl+1 (tl )Hl+1

(2.73) (2.74)

With the calculated Pl+1 (tl + 1), the matrices of Pi (s) for i = l, · · · , 1 and ti < s ≤ ti + di − di−1 = ti−1 are calculated recursively as Pi (s + 1) = ΦPi (s)Φ − Ki (s)Qw (s, i)Ki (s) + Γ Qe Γ  , Pi (ti + 1) = Pi+1 (ti + 1), i = l, · · · , 1,

(2.75)

where Ki (s) = ΦPi (s)Hi Q−1 w (s, i), 

Qw (s, i) = E [wi (s, i)wi (s, i)] = Hi Pi (s)Hi + Qvi .

(2.76) (2.77)

ˆ (s + 1, i) is the projection of the state x(s + 1) onto the linear Proof: Note that x space L{wl+1 (0, l + 1), · · · , wl+1 (tl , l + 1); · · · ; wi (ti + 1, i), · · · , wi (s, i)}. Since ˆ (s + 1, i) is calculated by using the projection w is a white noise, the estimator x formula as ˆ (s + 1, i) x = P roj {x(s + 1) | wl+1 (0, l + 1), · · · , wl+1 (tl , l + 1); · · · ; wi (ti + 1, i), · · · , wi (s − 1, i)} +P roj{x(s + 1) | wi (s, i)} = Φˆ x(s, i) + ΦE [x(s)x (s, i)] Hi Q−1 w (s, i)wi (s, i)  −1 = Φˆ x(s, i) + ΦPi (s)Hi Qw (s, i)wi (s, i).

(2.78)

It is easily obtained from (2.56) and (2.78) that ˜ (s + 1, i) = x(s + 1) − x ˆ (s + 1, i) x = Φ˜ x(s, i) + Γ e(s) − ΦPi (s)Hi Q−1 w (s, i)wi (s, i).

(2.79)

˜ (s, i) with e(s), it ˜ (s + 1, i) is uncorrelated with wi (s, i) and so is x Since x follows from (2.79) that    Pi (s + 1) + ΦPi (s)Hi Q−1 w (s, i)Hi Pi (s)Φ = ΦPi (s)Φ + Γ Qe Γ ,

which is (2.75). Similarly, we can prove (2.72).

(2.80) ∇

22

2. Optimal Estimation for Systems with Measurement Delays

2.3.4

Optimal Estimate x ˆ(t | t)

In this subsection we shall give a solution to the optimal filtering problem by applying the re-organized innovation sequence and the Riccati equations obtained in the last subsection. ˆ (t | t) Theorem 2.3.2. Consider the system (2.56)-(2.57). The optimal filter x is given by ! ˆ (t | t) = In − P1 (t)H1 Q−1 ˆ (t, 1) + P1 (t)H1 Q−1 x w (t, 1)H1 x w (t, 1)y1 (t),(2.81) ˆ (t, 1) is computed through where Qw (t, 1) = H1 P1 (t)H1 +Qv1 and the estimator x the following steps ˆ (tl + 1, l + 1) with the following standard Kalman filtering – Step 1: Calculate x ˆ (tl + 1, l + 1) = Φl+1 (tl )ˆ x x(tl , l + 1) + Kl+1 (tl )yl+1 (tl ), ˆ (0, l + 1) = 0, x

(2.82)

 where Φl+1 (tl ) = Φ − Kl+1 (tl )Hl+1 , Kl+1 (tl ) = ΦPl+1 (tl )Hl+1 Q−1 w (tl , l + 1), and Pl+1 (tl ) is computed by (2.72), with Pl+1 (0) = P0 . ˆ (t, 1) = x ˆ (t0 , 1) is calculated by the following backward itera– Step 2: Next, x ˆ (tl + 1, l + 1): tion with the initial condition x

ˆ (ti−1 , i) = Φi (ti−1 , ti + 1)ˆ x x(ti + 1, i + 1) ti−1 −1

+



Φi (ti−1 , s + 1)Ki (s)yi (s), i = l, l − 1, · · · , 1

s=ti +1

(2.83) where for k ≥ m, Φi (k, m) = Φi (k − 1) · · · Φi (m), Φi (m, m) = In , Ki (s) = ΦPi (s)Hi Q−1 w (s, i),

(2.84) (2.85)

while Φi (k) = Φ − Ki (k)Hi , Qw (s, i) = Hi Pi (s)Hi + Qvi ,

(2.86) (2.87)

and Pi (s) is calculated by (2.75). ˆ (t | t) is the projection of the state x(t) onto Proof: By applying Lemma 2.3.1, x ˆ (t | t) is calculated the linear space of (2.70). Since w is a white noise, the filter x by using the projection formula as ˆ (t | t) = P roj {x(t) | wl+1 (0, l + 1), · · · , wl+1 (tl , l + 1); x · · · ; w1 (t1 + 1, 1), · · · , w1 (t − 1, 1)} + P roj{x(t) | w1 (t, 1)}. ˆ (t, 1) + E [x(t)w1 (t, 1)] Q−1 =x w (t, 1)w1 (t, 1)  −1 ˆ (t, 1)] ˆ (t, 1) + P1 (t)H1 Qw (t, 1) [y1 (t) − H1 x =x !  −1 ˆ = In −P1 (t)H1 Q−1 x (t, 1) + P (t, 1)H 1 1 (t)H1 Qw (t, 1)y1 (t), (2.88) w

2.3 Multiple Measurement Delays Case

23

ˆ (s + 1, i) is the projection of the state which is (2.81). From Lemma 2.3.1, x x(s + 1) onto the linear space L{wl+1 (0, l + 1), · · · , wl+1 (tl , l + 1); · · · ; wi (ti + 1, i), · · · , wi (s, i)}, it follows from the projection formula that ˆ (s + 1, i) x = P roj {x(s + 1) | wl+1 (0, l + 1), · · · , wl+1 (tl , l + 1); · · · ; wi (ti + 1, i), · · · , wi (s, i)} = Φˆ x(s, i) + P roj{x(s + 1) | wi (s, i)} + Γ P roj {e(s) | wl+1 (0, l + 1), · · · , (2.89) wl+1 (tl , l + 1); · · · ; wi (ti + 1, i), · · · , wi (s, i)} . Noting that e(s) is uncorrelated with the innovation {wl+1 (0, l + 1), · · · , wl+1 (tl , l + 1), · · · ; wi (ti + 1, i), · · · , wi (s, i)}, we have ˆ (s + 1, i) = Φˆ x x(s, i) + ΦE [x(s)wi (s, i)] Q−1 w (s, i)wi (s)  −1 ˆ (s, i)] , = Φˆ x(s, i) + ΦPi (s)Hi Qw (s, i) [yi (s) − Hi x which can be rewritten as ˆ (s + 1, i) = Φi (s)ˆ x x(s, i) + Ki (s)yi (s),

(2.90)

ˆ (ti + 1, i) = x ˆ (ti + 1, i + 1). For each i, it follows from with the initial condition x (2.90) that ˆ (ti−1 , i) x = Φi (ti−1 − 1)ˆ x(ti−1 − 1, i) + Ki (ti−1 − 1)yi (ti−1 − 1) x(ti−1 − 2, i) = Φi (ti−1 − 1)Φi (ti−1 − 2)ˆ +Φi (ti−1 − 1)Ki (ti−1 − 2)yi (ti−1 − 2) + Ki (ti−1 − 1)yi (ti−1 − 1) = ··· = Φi (ti−1 − 1)Φi (ti−1 − 2) · · · Φi (ti + 1)ˆ x(ti + 1, i) + ti−1 −1



Φi (ti−1 − 1)Φi (ti−1 − 2) · · · Φi (s + 1)Ki (s)yi (s)

s=ti +1 ti−1 −1

= Φi (ti−1 , ti + 1)ˆ x(ti + 1, i) +



Φi (ti−1 , s + 1)Ki (s)yi (s).

(2.91)

s=ti +1

The proof is completed.

∇

Remark 2.3.1. The Kalman filtering solution for system (2.56)-(2.57) has been derived by applying the re-organized innovation sequence. Different from the standard Kalman filtering approach, the computation procedure at a given time instant t is summarized as: ˆ (tl + 1, l + 1) by (2.72) and (2.82) with initial – Calculate Pl+1 (tl + 1) and x ˆ (tl , l + 1), respectively. values Pl+1 (tl ) and x

24

2. Optimal Estimation for Systems with Measurement Delays

ˆ (ti−1 , i) by the backward recursive iteration (2.83) for i = l, · · · , 1 – Calculate x ˆ (t, 1) = x ˆ (t0 , 1). and set x ˆ (t | t) using (2.81). – Finally, compute x Observe that the above solution relies on l + 1 Riccati recursions of dimension n× n. 2.3.5

Numerical Example

In this subsection, we present one numerical example to illustrate the computation procedure of the proposed Kalman filtering. Consider the system (2.56)(2.57) with l = 2, d1 = 20, d2 = 40 and     0.8 0 0.6 Φ= , Γ = , 0.9 0.5 0.5 H(0) = [ 1

2 ] , H(1) = [ 2

0.5 ] , H(2) = [ 3 1 ] .

The initial state x(0), the noises e(t), v(0) (t), v(1) (t) and v(2) (t) are with zero means and unity covariance matrices, i.e., P0 = I2 , Qe = 1, Qv(0) = Qv(1) = Qv(2) = 1. Then, it is easy to know that ⎡ ⎤   1 0 0 1 0 Qv1 = 1, Qv2 = , Qv3 = ⎣ 0 1 0 ⎦ , 0 1 0 0 1 ⎡ ⎤   1 2 1 2 , H3 = ⎣ 2 0.5 ⎦ , H1 = [ 1 2 ] , H2 = 2 0.5 3 1 ⎡ ⎤   y(0) (s) y(0) (s) y1 (s) = y(0) (s), y2 (s) = , y3 (s) = ⎣ y(1) (s + 20) ⎦ . y(1) (s + 20) y(2) (s + 40) – For 0 ≤ t < d1 = 20, only one channel measurement is available and is ˆ (t | t) is computed by delay-free, the optimal estimator x ! ˆ (t | t) = In − P1 (t)H1 Q−1 ˆ (t, 1) + P1 (t)H1 Q−1 x w (t, 1)H1 x w (t, 1)y1 (t), (2.92) ˆ (t, 1) is the standard Kalman filter which is given by where x ˆ (t + 1, 1) = Φ1 (t)ˆ ˆ (0, 1) = 0, x x(t, 1) + K1 (t)y1 (t), x

(2.93)

while Φ1 (t) = Φ − K1 (t)H1 , K1 (t) = ΦP1 (t)H1 Q−1 w (t, 1), Qw (t, 1) = H1 P1 (t)H1 + 1,

(2.94) (2.95) (2.96)

and P1 (t) is the solution to the following Riccati equation P1 (t + 1) = ΦP1 (t)Φ −K1 (t)Qw (t, 1)K1 (t)+Γ Γ  , P1 (0) = P0 . (2.97)

2.3 Multiple Measurement Delays Case

25

– For 20 = d1 ≤ t < 40, there are two channel measurements available, and ˆ (t | t) is given by (2.81) as the optimal estimator x ! ˆ (t, 1) + P1 (t)H1 Q−1 ˆ (t | t) = In − P1 (t)H1 Q−1 x w (t, 1)H1 x w (t, 1)y1 (t), (2.98) ˆ (t, 1) is computed in the following steps: where x ˆ (t1 + 1, 2) (t1 = t − d1 = t − 20) by the Kalman filtering (2.82) i) Calculate x with l = 1 as ˆ (0, 2) = 0, ˆ (t1 + 1, 2) = Φ2 (t1 )ˆ x(t1 , 2) + K2 (t1 )y2 (t1 ), x x

(2.99)

where Φ2 (t1 ) = Φ − K2 (t1 )H2 , K2 (t1 ) = ΦP2 (t1 )H2 Q−1 w (t1 , 2) and P2 (t1 ) is computed by P2 (t1 + 1) = ΦP2 (t1 )Φ − K2 (t1 )Qw (t1 , 2)K2 (t1 ) + Γ Γ  , P2 (0) = P0 . (2.100) ˆ (t, 1) in (2.98) is then calculated by (2.83) with i = 1 as ii) x ˆ (t, 1) = Φ1 (t, t1 + 1)ˆ x x(t1 + 1, 2) +

t−1 

Φ1 (t, s + 1)K1 (s)y1 (s),

s=t1 +1

(2.101) ˆ (t1 + 1, 2) is given by (2.99), where x Φ1 (t, s + 1) = Φ1 (t − 1) · · · Φ1 (s + 1),

(2.102)

Φ1 (s), K1 (s) and Qw (s, 1) are respectively as in (2.94)-(2.96), and P1 (s) is the solution to P1 (s + 1) = ΦP1 (s)Φ − K1 (s)Qw (s, 1)K1 (s) + Γ Γ  , P1 (t1 + 1) = P2 (t1 + 1).

(2.103)

– For t ≥ 40, the optimal estimator x(t | t) is given from (2.81) as ! ˆ (t | t) = In − P1 (t)H1 Q−1 ˆ (t, 1) + P1 (t)H1 Q−1 x w (t, 1)H1 x w (t, 1)y1 (t), (2.104) ˆ (t, 1) is computed in the following where x ˆ (t2 + 1, 3) (t2 = t − d2 = t − 40) with the following standard i) Calculate x Kalman filtering (2.82) with l = 2 as ˆ (t2 + 1, 3) = Φ3 (t2 )ˆ ˆ (0, 3) = 0, (2.105) x x(t2 , 3) + K3 (t2 )y3 (t2 ), x

26

2. Optimal Estimation for Systems with Measurement Delays

where Φ3 (t2 ) = Φ − K3 (t2 )H3 , K3 (t2 ) = ΦP3 (t2 )H3 Q−1 w (t2 , 3) and P3 (t2 ) is computed by P3 (t2 + 1) = ΦP3 (t2 )Φ − K3 (t2 )Qw (t2 , 3)K3 (t2 ) + Γ Γ  , (2.106) P3 (0) = E[x(0)x (0)] = P0 . ˆ (t1 + 1, 2) is calculated by (2.83) with i = 2 ii) The estimator x ˆ (t1 + 1, 2) = Φ2 (t1 + 1, t2 + 1)ˆ x(t2 + 1, 3) x t1  + Φ2 (t1 + 1, s + 1)K2 (s)y2 (s),

(2.107)

s=t2 +1

ˆ (t2 + 1, 3) is obtained from (2.105), where x Φ2 (t1 + 1, s + 1) = Φ2 (t1 ) · · · Φ2 (s + 1), Φ2 (s) = Φ − K2 (s)H2 , K2 (s) = ΦP2 (s)H2 Q−1 w (s, 2), Qw (s, 2) = H2 P2 (s)H2 + I2 , and P2 (s) is calculated by P2 (s + 1) = ΦP2 (s)Φ − K2 (s)Qw (s, 2)K2 (s) + Γ Γ  , P2 (t2 + 1) = P3 (t2 + 1). ˆ (t, 1) of (2.81) is then calculated by (2.83) with i = 1 as iii) x ˆ (t, 1) = Φ1 (t, t1 + 1)ˆ x x(t1 + 1, 2) +

t−1 

Φ1 (t, s + 1)K1 (s)y1 (s),

s=t1 +1

(2.108) ˆ (ti + 1, 2) where Φ1 (t, s+ 1) is as in (2.102) and K1 (s) is as in (2.95), and x is calculated by (2.107).

2.4 Conclusion In this chapter we have studied the optimal filtering for systems with instantaneous and single or multiple delayed measurements. By applying the so-called re-organized innovation approach [102], a simple solution has been derived. It includes solving a number of Kalman filters with the same dimension as the original system. As compared with the system augmentation approach, the presented approach is much more computationally attractive, especially when the delays are large. The proposed results in this section will be useful in applications such as sensor fusion [42]. It will also be useful in solving the H∞ fixed-lag smoothing and multiple-step ahead prediction as demonstrated in later chapters.

3. Optimal Control for Systems with Input/Output Delays

In this chapter we study both the LQR control for systems with multiple input delays and the LQG control for systems with multiple i/o delays. For the LQR problem, an analytical solution in terms of a standard RDE is presented whereas the LQG control involves two RDEs, one for the optimal filtering and the other for the state feedback control. The key to our approach is a duality between the LQR control and a smoothing estimation which extends the well known duality between the LQR for delay free systems and the Kalman filtering.

3.1 Introduction We present in this chapter a complete solution to the LQR problem for linear discrete-time systems with multiple input delays. The study of discrete-time delay systems has gained momentum in recent years due to applications in emerging fields such as networked control and network congestion control [96, 82, 2]. For discrete-time systems with delays, one might tend to consider augmenting the system and convert a delay problem into a delay free problem. While it is possible to do so, the augmentation approach, however, generally results in higher state dimension and thus high computational cost, especially when the system under investigation involves multiple delays and the delays are large. Further, in the state feedback case, the augmentation approach generally results in a static output feedback control problem which is non-convex; see the work of [96]. This has motivated researchers in seeking more efficient methods for control of systems with i/o delays. We note that the optimal tracking problem for discrete-time systems with single input delay has been studied in [75]. Our aim in this chapter is to give an intuitive and simple derivation and solution to the LQR problem for systems with multiple input delays. We present an approach based on a duality principle and standard smoothing estimation. We shall first establish a duality between the LQR problem for systems with multiple input delays and a smoothing problem for a backward stochastic delay free system, which extends the well known duality between the LQR of delay free H. Zhang and L. Xie: Cntrl. and Estim. of Sys. with I/O Delays, LNCIS 355, pp. 27–51, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com


28

3. Optimal Control for Systems with Input/Output Delays

systems and the Kalman filtering. With the established duality, the complicated LQR problem for systems with multiple input delays is converted to a smoothing estimation problem for the backward delay free system and a simple solution based on a single RDE of the same order as the original plant (ignoring the delays) is obtained via standard projection in linear space. In this chapter, we shall also consider the LQG problem for systems with multiple i/o delays. By invoking the separation principle and the Kalman filtering for systems with multiple measurement delays studied in the last chapter, a solution to the LQG problem is easily obtained.

3.2 Linear Quadratic Regulation We consider the following linear discrete-time system with multiple input delays x(t + 1) = Φx(t) +

l 

Γ(i) ui (t − hi ), l ≥ 1,

(3.1)

i=0

where x(t) ∈ Rn and ui (t) ∈ Rmi , i = 0, 1, · · · , l represent the state and the control inputs, respectively. Although Φ and Γ(i) , i = 0, 1, · · · , l are allowed to be time-varying, for the sake of simplicity of notations, we confine them to be constant matrices. We assume, without loss of generality, that the delays are in an increasing order: 0 = h0 < h1 < · · · < hl and the control inputs ui , i = 0, 1, · · · , l have the same dimension, i.e., m0 = m1 = · · · = ml = m. Consider the following quadratic performance index for the system (3.1): JN = xN +1 P xN +1 +

−hi l N  i=0 t=0

ui (t)R(i) ui (t) +

N 

x (t)Qx(t),

(3.2)

t=0

where N > hl is an integer, xN +1 is the terminal state, i.e., xN +1 = x(N + 1), P = P  ≥ 0 is the penalty matrix for the terminal state, the matrices R(i) , i = 0, 1, · · · , l, are positive definite and the matrix Q is non-negative definite. The LQR problem is stated as: find the control inputs: ui (t) = Fi (x(t), uj (s), j = 0, 1, · · · , l; −hj ≤ s < t), 0 ≤ t ≤ N − hi , i = 0, 1, · · · , l such that the cost function JN of (3.2) is minimized. We note that in the absence of input delays, the solution to the LQR problem is well known and is related to one backward RDE. In the case of single input delay, the optimal tracking problem has been studied in [75] and a solution is also given in terms of one backward RDE. Here, we aim to give a similar solution for systems with multiple input delays. It is worth pointing out that the presence of delays in multiple input channels makes the control problem much more challenging due to interactions among various input channels.

3.2 Linear Quadratic Regulation

3.2.1

29

Duality Between Linear Quadratic Regulation and Smoothing Estimation

In this section, we shall convert the LQR problem into an optimization problem in a linear space for an associated stochastic system and establish a duality between the LQR problem and a smoothing problem. The duality will allow us to derive a solution for the LQR problem via standard smoothing estimation in the next section. First, the system (3.1) can be rewritten as ⎧ ⎨ Φx(t) + Γ u(t) + u˜(t), h ≤ t < h , t i i+1 (3.3) x(t + 1) = ⎩ Φx(t) + Γ u(t), t≥h, t

where

l

⎧⎡ ⎤ u0 (t − h0 ) ⎪ ⎪ ⎪ ⎢ ⎪ ⎥ .. ⎪ ⎪ ⎣ ⎦ , hi ≤ t < hi+1 , . ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ui (t − hi ) u(t) =

u ˜(t) =

⎪ ⎤ ⎡ ⎪ ⎪ u0 (t − h0 ) ⎪ ⎪ ⎪ ⎪ ⎥ ⎢ .. ⎪ ⎪ ⎦ , t ≥ hl , ⎣ . ⎪ ⎪ ⎩ ul (t − hl ) ⎧ l  ⎪ ⎨ Γ(j) uj (t − hj ), hi ≤ t < hi+1 , j=i+1

⎪ ⎩ 0, ⎧ ⎨[Γ (0) Γt = ⎩[Γ

(3.4)

(3.5)

t ≥ hl , · · · Γ(i) ] , hi ≤ t < hi+1 ,

(3.6)

· · · Γ(l) ] , t ≥ hl .

(0)

Using the above notations, the cost function (3.2) can be rewritten as JN = x (N + 1)P x(N + 1) +

N 

u (t)Rt u(t) +

t=0

where 

Rt =

N 

x (t)Qx(t),

⎧ ⎨ diag{R

· · · , R(i) }, hi ≤ t < hi+1 ,

⎩ diag{R

· · · , R(l) }, t ≥ hl .

(0) , (0) ,

(3.7)

t=0

(3.8)

It should be noted that while the optimal control of the system (3.3) associated with the cost (3.7) seems to be a standard LQR problem, its direct LQR solution does not lead to a causal control law for the system (3.1) in view of u(t) of the form (3.4).

30

3. Optimal Control for Systems with Input/Output Delays

Now we define the following backward stochastic state-space model associated with (3.3) and performance index (3.7): x(t) = Φ x(t + 1) + q(t), y(t) =

Γt x(t

(3.9)

+ 1) + v(t), t = 0, · · · , N,

(3.10)

where x(t) is the state and y(t) is the measurement output. The initial state x(N + 1), q(t) and v(t) are white noises with zero means and covariance matrices x(N + 1), x(N + 1) = P , q(t), q(s) = Qδt,s and v(t), v(s) = Rt δt,s , respectively. It can be seen that the dimensions of q(t) and y(t) are respectively dim{q(t)} = n × 1 and ⎧ ⎨ (i + 1)m × 1, h ≤ t < h , i i+1 dim {y(t)} = (3.11) ⎩ (l + 1)m × 1, t ≥ h , l

and v(t) has the same dimension as y(t). Introduce the column vectors:

⎫  ⎪ x = col{x(0), · · · , x(N )}, ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ u = col{u(0), · · · , u(N )}, ⎪ ⎪ ⎬  y = col{y(0), · · · , y(N )}, ⎪ ⎪ ⎪  ⎪ q = col{q(0), · · · , q(N )}, ⎪ ⎪ ⎪ ⎪ ⎪  v = col{v(0), · · · , v(N )}. ⎭

(3.12)

In the above, x(i), u(i), i = 0, 1, · · · , N are associated with the system (3.3) and y(i), q(i) and v(i) with the backward stochastic system (3.9)-(3.10). Then we have the following result. Lemma 3.2.1. By making use of the stochastic state-space model (3.9)-(3.10), the cost function JN of (3.2) can be put in the following quadratic form     ξ ξ JN = Π , (3.13) u u where ξ = [ x (0) u ˜ (0) · · · u ˜ (hl − 1) ] ,        Rx0 Rx0 y x0 x0 = Π= , , y y Ryx0 Ry x0 = [ x (0)

x (1)

···



x (hl ) ] ,

(3.14) (3.15) (3.16)

with u ˜(i), i = 0, 1, · · · , hl − 1 as defined in (3.5), x0 , x0  = Rx0 , x0 , y = Rx0 y , and y, y = Ry . Proof: By applying (3.3) repeatedly, it is easy to know that the terminal state x(N + 1) and x of (3.12) can be given in terms of the initial state x(0), u˜, and u of (3.12) as

3.2 Linear Quadratic Regulation h l −1

x(N + 1) = ΨN x(0) +

ΨN −i−1 u˜(i) + Cu,

31

(3.17)

i=0

x = ON x(0) +

h l −1

ON −i−1 u ˜(i) + Bu,

(3.18)

i=0

where u ˜(i) is as in (3.5) and ΨN −i−1 = ΦN −i , C = [ ΦN Γ0 , ΦN −1 Γ1 , · · · , ΓN ] , ⎡ ⎤ In ⎢ Φ ⎥ ⎥ ON = ⎢ ⎣ ... ⎦ , ⎡

ON −i−1

ΦN

⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

0 In Φ .. .

ΦN −i−1 0 0 ⎢ Γ0 ⎢ ΦΓ Γ ⎢ 0 1 B=⎢ .. ⎣ ... .

(3.22)

⎤

⎡

∆1 while

(3.21)

⎤

0 .. .

⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣

(3.19) (3.20)

∆2

⎥ ⎥ ⎥, ⎥ ⎦

..

. · · · ΓN −1

(3.23)

∆1 = ΦN −1 Γ0 , ∆2 = ΦN −2 Γ1 .

Next, by using (3.17)-(3.18), the cost function, JN , can be rewritten as the following quadratic form of x(0), u and u ˜: % & h l −1 ΨN −i−1 u˜(i) + Cζ P JN = ΨN x(0) + %

i=0

× ΨN x(0) + % + ON x(0) + % × ON x(0) +

h l −1

& ΨN −i−1 u ˜(i) + Cu

i=0 h l −1 i=0 h l −1 i=0

& ON −i−1 u ˜(i) + Bu

Q &

ON −i−1 u ˜(i) + Bu + u Ru,

(3.24)

32

3. Optimal Control for Systems with Input/Output Delays

where R = diag{R0 , · · · , RN }, N + 1 blocks ' () * Q = diag{ Q, · · · , Q }.

(3.25) (3.26)

We now show that JN can be rewritten as in (3.13). Note that it follows from (3.24) that 



JN = [Ψ ξ + Cu] P [Ψ ξ + Cu] + [Oξ + Bu] Q [Oξ + Bu] + u Ru,       ξ Ψ P Ψ + O QO Ψ  P C + O QB ξ = , u C  P Ψ + B  QO C  P C + B  QB + R u (3.27) where ξ is as in (3.14) and Ψ = [ ΨN

· · · ΨN −i−1

O = [ ON

· · · ON −i−1

· · · ΨN −hl ] , · · · ON −hl ] .

On the other hand, from (3.9), it is easy to know that   x(i) = ΨN −i x(N + 1) + ON −i q, i = 0, · · · , hl ,

(3.28)

where ΨN −i and ON −i are given in (3.19) and (3.22), respectively, and q is defined in (3.12). Putting together (3.28) for i = 0, 1, · · · , hl yields x0 = Ψ  x(N + 1) + O q,

(3.29)

where x0 is defined in (3.16). Similarly, from (3.10), it follows that y = C  x(N + 1) + B  q + v,

(3.30)

where y is defined in (3.12). Combining (3.29) with (3.30) yields         O 0 Ψ x0 x(N + 1) + q + . = v y C B Thus we obtain that      x0 x0 Π= , y y    Ψ P Ψ + O QO Ψ  P C + O QB = . C  P Ψ + B  QO C  P C + B  QB + R Hence, the required result follows from (3.27) and (3.31).

(3.31) ∇

Since the weighting matrices R(i) , i = 0, 1, · · · , l in (3.2) are positive definite, R > 0 and Ry > 0. Thus, by completing the squares for (3.13), we have the following result.

3.2 Linear Quadratic Regulation

33

Lemma 3.2.2. JN can further be rewritten as JN = ξ  Pξ + (u − u∗ ) Ry (u − u∗ ),

(3.32)

where ∗

u =

−Ry−1 Ryx(0) x(0)

−

hl 

Ry−1 Ryx(i) u ˜(i − 1),

(3.33)

i=1

ˆ 0 , x0 − x ˆ 0 , P = x0 − x

(3.34)

ˆ 0 is the projection of x0 onto the linear space L{y(0), · · · , y(N )}. The and x minimizing solution of JN with respect to control input ui (t) is given by

u∗i (t)

i + 1 blocks () * ' = [0 · · · 0 Im ] u∗ (t + hi ),

where, in view of (3.12), u∗ (t + hi ) is the (t + hi + 1)-th block of u∗ in (3.33). Proof: In view of Lemma 3.2.1, the minimizing solution of JN with respect to control input ui (t), i = 0, 1, · · · , l, is readily given by [30] u∗ = −Ry−1 Ryx0 ξ and JN can be written as in (3.32). Further, since Ryx0 =

⎡ x(0) ⎢ x(1) y, ⎢ ⎣ .. .

⎤  ⎥ ⎥ = [ Ryx(0) ⎦

Ryx(1)

· · · Ryx(hl ) ] ,

x(hl ) where Ryx(i) = y, x(i) , i = 0, 1, · · · , hl , the minimizing solution of (3.33) thus follows.

∇

Observe from linear estimation theory [38] that Ry−1 Ryx(i) is the transpose of the gain matrix of the optimal smoothing (or filtering when i = 0) estimate ˆ (i | 0) of the backward system (3.9)-(3.10) which is the projection of the of x state x(i) onto the linear space L{y(0), · · · , y(N )}. Thus, we have converted the LQR problem for the delay system (3.1) into the optimal smoothing problem for the associated system (3.9)-(3.10). In another words, in order to calculate the ˆ (i | 0) optimal controller u∗ , we just need to calculate the smoothing gain of x associated with the backward system (3.9)-(3.10). Therefore, we have established a duality between the LQR of the delay system (3.1) and a smoothing problem for an associated delay free system.

34

3. Optimal Control for Systems with Input/Output Delays

Remark 3.2.1. Observe that when the system (3.1) is delay free, i.e. hi = 0, i = 1, 2, · · · , l, (3.33) becomes u∗ = −Ry−1 Ryx(0) x(0), where Ry−1 Ryx(0) is the transpose of the optimal filtering gain matrix, which is the well known duality between the LQR problem and the Kalman filtering and has played an important role in linear systems theory. Thus, the established duality between the LQR for systems with input delays and the optimal smoothing contains the duality between the LQR problem for delay free systems and the Kalman filtering as a special case. It is expected that the duality will play a significant role in control design for systems with i/o delays. By using the duality of (3.33), we shall present a solution to the LQR problem for systems with multiple input delays via the standard projection in linear space in the next section. 3.2.2

Solution to Linear Quadratic Regulation

In view of the result of the previous section, to give a solution to the LQR problem, we need to calculate the gain matrix Rx(i)y Ry−1 of the smoothing problem associated with the stochastic backward system (3.9)-(3.10). First, define the RDE associated with the Kalman filtering of the backward stochastic system (3.9)-(3.10): Pj = Φ Pj+1 Φ + Q − Kj Mj Kj , j = N, N − 1, · · · , 0,

(3.35)

where the initial condition is given by PN +1 = P with P the penalty matrix of (3.2), Kj = Φ Pj+1 Γj Mj−1 , Mj = Rj +

Γj Pj+1 Γj ,

(3.36) (3.37)

and Γj is defined in (3.6). It follows from the standard Kalman filtering that the optimal filtering estiˆ (i | i), of the backward system (3.9)-(3.10) is given by [30]: mate, x ˆ (i | i) = x

N 

Φi,k Kk y(k),

(3.38)

k=i

where Φj,m = Φj · · · Φm−1 , m ≥ j, Φm,m = In , j = i, · · · , N − 1. Φj = Φ − Kj Γj ,

(3.39) (3.40)

3.2 Linear Quadratic Regulation

35

Also, by applying standard projection, the smoothing estimate is given below. Lemma 3.2.3. Consider the stochastic backward state space model (3.9)-(3.10), ˆ (i | 0) is given by the optimal smoothing estimate x ˆ (i | 0) = x

i−1 

Si (k)y(k) +

k=0

where

N 

Fi (k)y(k)

⎫  Si (k) = Pi Φk+1,i Γk Mk−1 − Φk,i G(k)Kk , 0 ≤ k < i, ⎬ F (k) = [I − P G(i)] Φ K , i ≤ k ≤ N, ⎭ i

n

i

(3.41)

k=i

i,k

(3.42)

k

and G(k) =

k 

−1  Φj,k Γj−1 Mj−1 Γj−1 Φj,k ,

(3.43)

j=1

while Φj,m is given in (3.39). In the above, Pi is the solution to the RDE (3.35) and Mj is given by (3.37). ˆ (i | 0) is the projection of x(i) onto the linear space of Proof: Note that x L{y(0), · · · , y(N )}. By applying the projection formula, we have [30] ˆ (i | 0) = x ˆ (i | i) + x

i 

−1 x(i), w(j − 1)Mj−1 w(j − 1)

j=1

ˆ (i | i) + =x

i 

! −1  ˆ (j | j) , ˜ (j)Γj−1 Mj−1 x y(j − 1) − Γj−1 x(i), x

j=1

(3.44) ˜ (j | j) = x(j) − x ˆ (j | j) is the filtering error, w(j − 1) = y(j − 1) − where x  ˆ (j | j) is the innovation and Mj−1 is the covariance matrix of w(j − 1) Γj−1 x ˜ (j | j) for j < i. which can be computed using (3.37). Now we calculate x(i), x In consideration of (3.9)-(3.10) and by applying the projection lemma, it is easy to know that ˜ (j + 1 | j + 1) + q(j) − Φ Pj+1 Γj Mj−1 v(j), ˜ (j | j) = Φj x x where Φj is as in (3.40). In view of the fact that q(j) and v(j) are independent of x(i) for j < i, it follows that ˜ (j | j) = Pi Φi−1 · · · Φj = Pi Φj,i , x(i), x

(3.45)

where Φj,i is given by (3.39). By taking into account (3.38) and (3.45), it follows from (3.44) that

36

3. Optimal Control for Systems with Input/Output Delays

ˆ (i | 0) = x ˆ (i | i) − x

i 

−1  ˆ (j | j) x Pi Φj,i Γj−1 Mj−1 Γj−1

j=1

+

i 

−1 Pi Φj,i Γj−1 Mj−1 y(j − 1)

j=1

=

N 

Φi,k Kk y(k) +

i−1 

Pi Φj+1,i Γj Mj−1 y(j)

j=0

k=i

−

i 

⎡ ⎤ N  −1  ⎣ Pi Φj,i Γj−1 Mj−1 Γj−1 Φj,k Kk y(k)⎦

j=1

=

N 

k=j

Φi,k Kk y(k) +

i−1 

Pi Φj+1,i Γj Mj−1 y(j)

j=0

k=i

−

k N  

−1  Pi Φj,i Γj−1 Mj−1 Γj−1 Φj,k Kk y(k)

k=i j=1

−

k i−1  

−1  Pi Φj,i Γj−1 Mj−1 Γj−1 Φj,k Kk y(k)

k=1 j=1

=

N 

[In − Pi G(i)] Φi,k Kk y(k)

k=i

+

i−1 

! Pi Φk+1,i Γk Mk−1 − Φk,i G(k)Kk y(k) + Pi Φ1,i Γ0 M0−1 y(0),

k=1

∇

where G(i) is given by (3.43). We complete the proof of the lemma.

Therefore, by the duality established in the previous section, the solution to the LQR problem for the discrete-time system (3.1) is given in the following theorem. Theorem 3.2.1. Consider the system (3.1) and the associated cost (3.2). The optimal controller u∗i (t), i = 0, · · · , l; t = 0, · · · , N − hi that minimizes the cost is given by

u∗i (t)

i + 1 blocks ' () * = [0 · · · 0 Im ] u∗ (t + hi ),

(3.46)

where for t < hl , ∗



u (t) = − [F0 (t)] x(0) −

t  s=1



[Fs (t)] u ˜(s − 1) −

hl 



[Ss (t)] u ˜(s − 1),

s=t+1

(3.47)

3.2 Linear Quadratic Regulation

37

and for t ≥ hl , 

u∗ (t) = − [F0 (t)] x(0) −

hl 



[Fs (t)] u˜(s − 1),

(3.48)

s=1

while u ˜(·) is as in (3.5) and Ss (t) and Fs (t) in (3.42). Proof: From Lemmas 3.2.2 and 3.2.3, we observe that Ry−1 Ryx(0) = [F0 (0), · · · , F0 (N )] , Ry−1 Ryx(s) = [Ss (0), · · · , Ss (s − 1); Fs (s), · · · , Fs (N )] , s = 1, 2, · · · , hl , where Fs (t) and Ss (t) are as in (3.42). In view of (3.33) and (3.4), it is easy to know that u∗ (t) is given by (3.47)-(3.48) and u∗i (t) by (3.46). Thus the proof is completed. ∇ Note, however, that for t > 0, the optimal controller u∗i (t) of (3.46) is given in terms of the initial state x(0) and the past control inputs and is an open-loop control in nature. Our aim is to find the optimal controller of ui (t) in terms of the current state x(t). This problem can be addressed by shifting the time interval from [0, hl ] to [τ, τ + hl ]. Note that for any given τ ≥ 0, the system (3.1) and the cost (3.2) can be rewritten respectively as ⎧ ⎨ Φx(t + τ ) + Γ uτ (t) + u ˜τ (t), hi ≤ t < hi+1 , t x(t + τ + 1) = ⎩ Φx(t + τ ) + Γ uτ (t), t≥h, t

l

(3.49) τ + JN = JN

l τ −1  

ui (t)R(i) ui (t) +

i=0 t=0

τ 

x (t)Qx(t),

(3.50)

t=1

where τ = x (N + 1)P x(N + 1) + JN

N −τ 

[uτ (t)] Rt uτ (t)

t=0

+

N −τ 

x (t + τ )Qx(t + τ ),

(3.51)

t=1

while



uτ (t) =

⎧⎡ ⎤ u0 (t − h0 + τ ) ⎪ ⎪ ⎪ ⎢ ⎪ ⎥ .. ⎪ ⎪ ⎣ ⎦ , hi ≤ t < hi+1 , . ⎪ ⎪ ⎪ ⎪ (t − h + τ ) u ⎪ i i ⎨ ⎪ ⎤ ⎡ ⎪ ⎪ u0 (t − h0 + τ ) ⎪ ⎪ ⎪ ⎪ ⎥ ⎢ .. ⎪ ⎪ ⎦ , t ≥ hl , ⎣ . ⎪ ⎪ ⎩ ul (t − hl + τ )

(3.52)

38

3. Optimal Control for Systems with Input/Output Delays



u ˜τ (t) =

⎧ ⎨ l

j=i+1

Γ(j) uj (t − hj + τ ), hi ≤ t < hi+1 , t + τ ≤ N,

⎩ 0,

(3.53)

otherwise.

Remark 3.2.2. It is easy to see that the above uτ (t) and u ˜τ (t) are related to u(t) and u˜(t) of (3.5) as follows: – For t ≥ hl , uτ (t) ≡ u(t + τ ) and u ˜τ (t) ≡ u ˜(t + τ ). – For τ = 0 and any t, uτ (t) ≡ u(t + τ ) and u ˜τ (t) ≡ u ˜(t + τ ). ˜τ (t) = u ˜(t + τ ). – For t < hl and τ = 0, uτ (t) = u(t + τ ) and u Let the matrix Pjτ (j = N − τ + 1, · · · , 0) obey the following backward RDE: τ Φ + Q − Kjτ Mjτ (Kjτ ) , PNτ −τ +1 = PN +1 = P, Pjτ = Φ Pj+1

(3.54)

τ Γj (Mjτ )−1 , Kjτ = Φ Pj+1

(3.55)

where Mjτ

= Rj +

τ Γj Pj+1 Γj .

(3.56)

Remark 3.2.3. When j ≥ hl , it is not difficult to know that Pjτ = Pτ +j , where Pτ +j is the solution to Riccati equation (3.35). Thus, we just need to calculate Pjτ for j = hl − 1, hl − 2, · · · , 0 using (3.54) with the initial condition of Phτl = Pτ +hl . Similar to (3.39)-(3.40), denote that 

Φτj = Φ − Kjτ Γj ,

(3.57)



Φτj,m = Φτj · · · Φτm−1 , m ≥ j,

(3.58)

with Φτj,j = In . By using a similar line of arguments as for the case of τ = 0 in the above, we have the following result. Lemma 3.2.4. Consider the system (3.49) and the associated cost (3.51). The optimal control associated with uτi (t), i = 0, · · · , l, 0 ≤ t ≤ N − τ − hi that τ minimizes JN , denoted by uτi ∗ (t), is given by uτi ∗ (t)

i + 1 blocks ' () * = [0 · · · 0 Im ] uτ ∗ (t + hi ),

(3.59)

where for t < hl , 

uτ ∗ (t) = − [F0τ (t)] x(τ ) −

t 

hl 



[Fsτ (t)] u˜τ (s − 1) −

s=1



[Ssτ (t)] u˜τ (s − 1),

s=t+1

(3.60) and for t ≥ hl , 

uτ ∗ (t) = − [F0τ (t)] x(τ ) −

hl  s=1



[Fsτ (t)] u˜τ (s − 1),

(3.61)

3.2 Linear Quadratic Regulation

39

while Ssτ (·) and Fsτ (·) are given by

⎫ ! Ssτ (t) = Psτ (Φτt+1,s ) Γt (Mtτ )−1 − (Φτt,s ) Gτ (t)Ktτ , 0 < t < s, ⎬ F τ (t) = [I − P τ Gτ (s)] Φτ K τ , s ≤ t ≤ N, ⎭ s

n

s

s,t

t

(3.62) and Gτ (t) =

t 

τ  (Φτj,t ) Γj−1 (Mj−1 )−1 Γj−1 Φτj,t .

(3.63)

j=1

∇

Proof: The proof is similar to that of Theorem 3.2.1.

Remark 3.2.4. Observe that u∗i (t) is the optimal controller associated with the cost (3.2) while uτi ∗ (t) is the optimal controller associated with the cost (3.51). Furthermore, u∗i (t) is given in terms of initial state x(0) while uτi ∗ (t) in terms of state x(τ ). Corollary 3.2.1. For t = 0, the controller uτi ∗ (0) associated with the cost (3.51) is given by uτi ∗ (0)

i + 1 blocks ' () * = [0 · · · 0 Im ] uτ ∗ (hi ), i = 0, 1, · · · , l,

(3.64)

where 

uτ ∗ (hi ) = − [F0τ (hi )] x(τ ) −

hi 



[Fsτ (hi )] u ˜τ (s − 1)

s=1

−

hl 



[Ssτ (hi )] u ˜τ (s − 1),

(3.65)

s=hi +1

and u˜τ (·) is defined in (3.53). It is clear that uτi ∗ (0) is given in terms of the current state x(τ ) and control inputs ui (t), τ − hi ≤ t ≤ τ − 1, i = 1, · · · , l. ˜0 (t) ≡ u ˜(t) and the RDE (3.54) Remark 3.2.5. Since for τ = 0, u0 (t) ≡ u(t), u reduces to the RDE (3.35), thus uτi ∗ (0) for τ = 0 is the same as u∗i (0) of (3.46). Note that u∗i (τ ) is given in terms of the initial state x(0) while uτi ∗ (0) is given in terms of current state x(τ ). The following result follows similarly from the well known dynamic programming. Lemma 3.2.5. If ui (t) = u∗i (t) for t = 0, · · · , τ − 1; i = 0, · · · , l, then u∗i (τ ) ≡ uτi ∗ (0) |uj (t)=u∗j (t)(

0≤t<τ ; 0≤j≤l) ,

i = 0, · · · , l,

where u∗i (τ ) (τ = 0, · · · , N ) is given by (3.46) and uτi ∗ (0) by (3.64).

(3.66)

40

3. Optimal Control for Systems with Input/Output Delays

Proof: By substituting ui (t) = u∗i (t) for 0 ≤ t < τ ; 0 ≤ i ≤ l into (3.32) and applying some algebraic manipulations, it is not difficult to know that JN |ui (t)=u∗i (t)( 

= ξ Pξ + = ξ  Pξ +

0≤t<τ ; 0≤i≤l) ∗  (u − u ) Ry (u − u∗ ) |ui (t)=u∗i (t)( 0≤t<τ ; 0≤i≤l) , ⎤ ⎡ ⎡ uτ (0) − uτ (0) − uτ∗ (0) ⎥ τ⎢ ⎢ .. .. ⎦ Ry ⎣ ⎣ . . uτ (N − τ ) − uτ∗ (N − τ ) uτ (N − τ ) −

uτ∗ (0)

⎤ ⎥ ⎦,

uτ∗ (N − τ ) (3.67)

where uτ (·) is as in (3.52), and uτ∗ (·), different from uτ ∗ (·), is obtained from uτ (·) with ui (·) replaced by u∗i (·) for i = 0, · · · l, respectively, where u∗i (·) is as in (3.46). Similarly, substituting ui (t) = u∗i (t) for 0 ≤ t < τ ; 0 ≤ i ≤ l into (3.50) yields JN |ui (t)=u∗i (t)( 0≤t<τ ; 0≤i≤l) & % l τ −1 τ     τ = ui (t) R(i) ui (t) + x (t)Qx(t) + JN |ui (t)=u∗i (t) % =

i=0 t=0

t=1

l τ −1 

τ 



ui (t) R(i) ui (t) +

i=0 t=0

& 

x (t)Qx(t) +

ξτ Pτ ξτ

|ui (t)=u∗i (t)

t=1

⎤ uτ (0) − uτ ∗ (0) |ui (t)=u∗i (t)( 0≤t<τ ; 0≤i≤l) ⎥ ⎢ .. +⎣ ⎦ Ryτ . τ τ∗ u (N − τ ) − u (N − τ ) |ui (t)=u∗i (t)( 0≤t<τ ; 0≤i≤l) ⎤ ⎡ uτ (0) − uτ ∗ (0) |ui (t)=u∗i (t)( 0≤t<τ ; 0≤i≤l) ⎥ ⎢ .. ×⎣ ⎦. . uτ (N − τ ) − uτ ∗ (N − τ ) |ui (t)=u∗i (t)( 0≤t<τ ; 0≤i≤l) ⎡

(3.68)

By comparing the terms associated with uτ (s) (0 ≤ s ≤ N − τ ) in (3.67) and those in (3.68), it follows directly that uτ∗ (s) = uτ ∗ (s) |ui (t)=u∗i (t)(

0≤t<τ ; 0≤i≤l) ,

0 ≤ s ≤ N − τ,

or equivalently, u∗i (τ + s) = uτi ∗ (s) |ui (t)=u∗i (t)(

0≤t<τ ; 0≤i≤l) ,

for 0 ≤ s ≤ N − τ − hi and 0 ≤ i ≤ l. The desired result follows by setting s = 0. ∇ Now the main result of this section is summarized below. Theorem 3.2.2. Consider the delay system (3.1) and its cost (3.2). The optimal LQR control ui (τ ), τ = 0, · · · , N − hi , i = 0, · · · , l that minimizes (3.2), can be given by

3.3 Output Feedback Control

u∗i (τ )

41

i + 1 blocks + hi () * '  = − [0 · · · 0 Im ] × [F0τ (hi )] x(τ ) + [Fsτ (hi )] u ˜τ ∗ (s − 1) +

hl 

,

s=1



[Ssτ (hi )] u ˜τ ∗ (s − 1) ,

(3.69)

s=hi +1

where u ˜τ ∗ (·) is given in (3.53) with uj (·) replaced by u∗j (·) for j = 1, · · · , l and τ Fs (·) and Ssτ (·) are defined in (3.62). Proof: From Lemma 3.2.5, we have u∗i (τ ) ≡ uτi ∗ (0) |uj (t)=u∗j (t)(

0≤t<τ ; 0≤j≤l) ,

i = 0, · · · , l.

By substituting uj (t) = u∗j (t)( 0 ≤ t < τ ; 0 ≤ j ≤ l) into (3.64)-(3.65), (3.69) follows directly. ∇ Remark 3.2.6. Observe from (3.62) that for τ = N, N − 1, · · · , 0, to compute the control law (3.69) it involves calculating Psτ , s = hl − 1, · · · , 1, 0. Observe that Psτ , s = hl − 1, · · · , 1, 0 can be computed using (3.54) with the initial condition Phτl = Phl +τ for τ ≤ N − hl and PNτ −τ +1 = P for N − hl < τ ≤ N . Note that (3.54) has the same order as the original plant (ignoring the delays). Thus, our approach has much computational advantage over methods such as the system augmentation. Remark 3.2.7. For delay free systems, i.e., h1 = · · · = hl = 0, it is obvious that the optimal controller (3.69) becomes: ! u∗0 (τ ) = −[F0τ (0)] x(τ ) = − Φ Pτ +1 Γτ Mτ−1 x(τ ),

(3.70)

where Pτ +1 obeys the standard RDE (3.35). Thus the optimal controller u∗0 (τ ) of (3.70) is the same as the well-known LQR solution [30].

3.3 Output Feedback Control In the previous section, we have studied the LQR problem where it is assumed that the system is not subject to noise input and the state is available for feedback. In practice, however, quite often we encounter systems which are not noise free and only noise corrupted measurement instead of the state is available for feedback. This motivates our investigation of the linear LQG control problem. We consider the following discrete linear system with i/o delays: x(t + 1) = Φx(t) +

l 

Γ(i) ui (t − hi ) + Γ e(t),

(3.71)

i=0

y(i) (t) = H(i) x(t − di ) + v(i) (t), i = 0, · · · , l, l ≥ 1,

(3.72)

42

3. Optimal Control for Systems with Input/Output Delays

where x(t) ∈ Rn , ui (t) ∈ Rmi and y(i) (t) ∈ Rpi represent respectively the state, the ith control input and the ith measurement output, and e and v(i) are the process and measurement noises, respectively. Although the study in this section allows that the matrices Φ, Γ(i) , Γ and H(i) are time-varying, for the sake of notation simplification, we confine them to be constant matrices. Similar to the case of the LQR, without loss of generality, we assume that the delays are in an increasing order: 0 = h0 < h1 < · · · < hl , 0 = d0 < d1 < · · · < dl and the control inputs ui , i = 0, 1, · · · , l have the same dimension, i.e., m0 = m1 = · · · = ml = m. The initial state x(0) and the noises e(t) and v(i) (t), i = 0, 1, · · · , l are assumed to be mutually uncorrelated Gaussian noises with zero means and satisfy ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ x(0) Π0 0 0 x(0) ⎦, ⎣ e(t) ⎦ , ⎣ e(s) ⎦ = ⎣ 0 Qe δts (3.73) 0 v(i) (s) v(i) (t) 0 0 Qv(i) δt,s where Π0 , Qe and Qv(i) are given constant matrices with Π0 ≥ 0, Qe ≥ 0 and Qv(i) > 0. To ensure that the regulated signal x(t) and the control input u(t) are simultaneously small, we introduce the cost function JN = xN +1 P xN +1 +

−hi l N 

ui (t)Ri ui (t) +

i=0 t=0

N 

x (t)Qx(t),

(3.74)

t=0

where N > hl is an integer, xN +1 is the terminal state, i.e., xN +1 = x(N + 1), P = P  ≥ 0 is the penalty matrix for the terminal state, the matrices Ri , i = 0, 1, · · · , l, are positive definite and the matrix Q is non-negative definite. Denote ⎧ ⎨ col{y (t), · · · , y  (0) (i−1) (t)}, di−1 ≤ t < di , y(t) = (3.75) ⎩ col{y (t), · · · , y (t)}, t ≥ dl . (0)

(l)

The optimal LQG control problem can be stated as: find the control input ui (t) = Fi (y(0), · · · , y(t)), i = 0, 1, · · · , l such that the cost function E(JN ) is minimized, where JN is defined in (3.74) and E is the mathematical expectation taken with respect to the random noises x(0), e and v(i) , i = 0, 1, · · · , 1. To derive a solution to the optimal LQG control, we first denote the filtering error: ˜ (t | t) = x(t) − x ˆ (t | t), x

(3.76)

ˆ (t | t) is given by the Kalman filter in Chapter 2. It is well known that where x ˜ (t | t) is uncorrelated with x ˆ (t | t). By some simple algebra, we can obtain that x 0 1 E(JN ) = E(JN ) + E(JN ),

(3.77)

3.3 Output Feedback Control

43

where 0 JN

=

ˆ N +1|N +1 P x ˆ N +1|N +1 x

+

−hi l N 

ui (t)Ri ui (t)

i=0 t=0

+

N 

ˆ  (t | t)Qˆ x x(t | t),

(3.78)

t=0 1 ˜ N +1 P x ˜ N +1 + =x JN

N 

˜  (t)Q˜ x(t), x

(3.79)

t=0

ˆ N +1|N +1 is the optimal estimate of x(N + 1) and x ˜ N +1 = x(N + 1) − and x ˆ N +1|N +1 . x 1 Note that JN is not dependent on the control input ui , i = 0, 1, · · · , l. Thus, ˆ (t | t), seek the the LQG problem becomes that given the optimal estimate x 0 ) is minimized. Like the standard control input ui , i = 0, 1, · · · , l such that E(JN LQG control for delay free systems, the optimal control solution can be obtained by the well known separation principle. That is, the optimal LQG control for systems with i/o delays can be obtained by combining the Kalman filtering for systems with delayed measurements as studied in Chapter 2 and the solution to the LQR problem for systems with delayed inputs in the last section. To this end, denote for i = l + 1, l − 1, · · · , 1, ⎡ ⎤ y(0) (s) ⎢ ⎥ .. yi (s) = ⎣ (3.80) ⎦, . y(i−1) (s + di−1 ) ⎤ ⎡ H(0) ⎥ ⎢ .. Hi = ⎣ (3.81) ⎦, . H(i−1) (t + di−1 ) (3.82) Qvi = diag{Qv(0) , · · · , Qv(i−1) }, and introduce the RDE:   Pi (s + 1) = ΦPi (s)Φ − ΦPi (s)Hi Q−1 w (s, i)Hi Pi (s)Φ + Γ Qe Γ , Pi (ti + 1) = Pi+1 (ti + 1), i = l, l − 1, · · · , 0,

(3.83)

where Qw (s, i) = Hi Pi (s)Hi + Qvi , ti = t − di , and Pl+1 (tl ) is computed by   Pl+1 (tl + 1) = ΦPl+1 (tl )Φ − ΦPl+1 (tl )Hl+1 Q−1 w (tl , l + 1)Hl+1 Pl+1 (tl )Φ + Γ Qe Γ  , Pl+1 (0) = Π0 . (3.84)

It follows from Chapter 2 that the Kalman filtering estimate of the system (3.71)-(3.72) can be given by ! ˆ (t | t) = In − P1 (t)H1 Q−1 ˆ (t, 1) + P1 (t)H1 Q−1 x w (t, 1)H1 x w (t, 1)y1 (t), (3.85)

44

3. Optimal Control for Systems with Input/Output Delays

ˆ (t, 1) is computed by where x ˆ (ti−1 + 1, i) = Φi (ti−1 + 1, ti + 1)ˆ x x(ti + 1, i + 1) 

ti−1

+

˜ (s)] , Φi (ti−1 + 1, s + 1) [Ki (s)yi (s) + u

s=ti +1

i = l, l − 1, · · · , 1,

(3.86)

with u ˜(s) defined in (3.5), Φi (k, m) = Φi (k − 1) · · · Φi (m), k ≥ m, Φi (m, m) = In , Φi (s) = Φ − Ki (s)Hi , Ki (s) = ΦPi (s)Hi Q−1 w (s, i), and Pi (s) being the solution of the RDE (3.83). In the above, the initial condition x ˆ(tl + 1, l + 1) is calculated by the following standard Kalman filter ˜ (tl ) + Kl+1 (tl )yl+1 (tl ), ˆ (tl + 1, l + 1) = Φl+1 (tl )ˆ x(tl , l + 1) + u x ˆ (0, l + 1) = 0, x

(3.87)

 where Φl+1 (tl ) = Φ − Kl+1 (tl )Hl+1 and Kl+1 (tl ) = ΦPl+1 (tl )Hl+1 Q−1 w (tl , l + 1).

Theorem 3.3.1. Consider the system (3.71)-(3.72). The optimal LQG control ui (τ ), i = 0, 1, · · · , l that minimizes E(JN ) is given by u∗i (τ )

i + 1 blocks ' () * = − [0 · · · 0 Im ] × ⎛ ⎝[F0τ (hi )] x ˆ (τ | τ ) +

hi 

! τ ∗ ˜ (j − 1) Fjτ (hi ) u

j=1

+

hl 

⎞ !  τ∗ ˜ (j − 1)⎠ , Sjτ (hi ) u

(3.88)

j=hi +1

˜ τ ∗ (·) in (3.53) with ui (·) replaced where Ssτ (·) and Fsτ (·) are given as in (3.62), u ∗ ˆ (τ | τ ) is given by (3.86). by ui (·) for i = 0, · · · , d, and x

3.4 Examples Example 3.4.1. Consider the linear discrete-time system (3.1) with Φ ≡ 1, l = 1, Γ(0) = 1, Γ(1) = 1, h1 = 2 and cost function (3.2) with N = 20, P = 1, R(i) = 1, i = 0, 1 and Q = 1. Following the results in Theorem 3.2.2, we shall compute the optimal controllers u0 (t), t = 0, 1, · · · , 20 and u1 (t), t = 0, 1, 2, · · · , 18.

3.4 Examples

45

– Calculate u0 (20). In this case, τ = N = 20, u ˜20 (0) = u1 (18), Γ020 = 1 and 20 R0 = 1. The optimal controller is obtained from (3.69) as u∗0 (20) = −F020 (0)x(20) − S120 (0)u1 (18). With P120 = P21 = P = 1, the RDE (3.54) gives the solution P020 = 1.5. Then, from (3.62), we have S120 (0) = 0.5 and F020 (0) = 0.5. Therefore, the optimal controller is given by u∗0 (20) = −0.5[x(20) + u1 (18)]. ˜19 (0) = u1 (17), u ˜19 (1) = u1 (18), – Calculate u0 (19). In this case, τ = 19, u 19 19 19 19 Γ0 = 1, Γ1 = 1, R0 = 1, and R1 = 1. The optimal controller is obtained from (3.69) as u19 (0) − S219 (0)˜ u19 (1). u0 (19) = −F019 (0)x(19) − S119 (0)˜ The RDE (3.54) with P119 = P020 = 1.5 gives P020 = 1.6. Then, from (3.62), we have S119 (0) = 0.6, S219 (0) = 0.2 and F019 (0) = 0.6. Therefore, the optimal controller is given by u∗0 (19) = −0.6x(19) − 0.6u1(17) − 0.2u1(18). – Calculate u0 (18) and u1 (18). In this case, τ = 18, u ˜18 (0) = u1 (16), 18 18 18 18 18 18 ˜ (2) u˜ (1) = u1 (18), u  = 0; Γ0 = Γ1 = 1, Γ2 = [ 1 1 ] ; R0 = 1 0 . The optimal controller is obtained from (3.69) as R118 = 1, R218 = 0 1 u∗0 (18) = −F018 (0)x(3) − S118 (0)u1 (16) − S218 (0)u1 (17),  ! ! u∗1 (18) = −[0 1] × F018 (2) x(18) + F118 (2) u1 (16)  ! + F218 (2) u1 (17) Solving the RDE (3.54) with P318 = P21 = 1, we obtain that P218 = 1.3335, P118 = 1.5714, P018 = 1.6111. Then, from (3.62), we have S218 (0) = 0.2222, S118 (0) = 0.6111, F218 (2) = [0.1111, 0.1111], F118 (2) = [0.0555, 0.0555], F018 (2) = [0.0555, 0.0555], F03 (0) = 0.6111. Therefore the optimal controller is given by u∗0 (18) = −0.6111x(18) − 0.6111u1(16) − 0.2222u1(17), u∗1 (18) = −0.0555x(18) − 0.0555u1(16) − 0.1111u1(17). The optimal control u0 (t), u1 (t), t = 0, 1, 2, · · · , 17 can be computed similarly. And the optimal state trajectory is shown in Figure 3.1 where the convergence of the state to the origin is clearly seen.

46

3. Optimal Control for Systems with Input/Output Delays

1.2

1

0.8

0.6

0.4

0.2

0

−0.2

0

2

4

6

8

10

12

14

16

18

20

Fig. 3.1. Optimal state trajectory of the system (horizontal axis: time t; vertical axis: the state x(t))

Example 3.4.2. (Application in ATM congestion control) The mathematical model of congestion control is taken from [1]. The ABR source is the only traffic class which responds to feedback information for the node for rate adjustment to prevent network congestion and to maintain quality of service (QoS) to all connections. The feedback information is the available transmission capacity (bandwidth) and queue level at the bottleneck node. Since the available node capacity for the ABR source changes over time in an unpredictable way due to the higher priority sources, the CBR (constant bit rate) and VBR (variable bit rate) source rates are represented as interferences. Let ξ denote the higher priority source (interference) which is modelled as a stable ARMA process [1]. Such a formulation allows for long-range correlated traffic. Let q(t) be the queue length at the bottleneck and µ(t) the effective service rate available for the traffic of the given source in that link at the beginning of the t-th time slot. The queue length equation is given by q(t + 1) = q(t) +

l 

ri (t) − µ(t)

i=1

≡ q(t) +

l 

vi (t − hi ) − µ(t)

(3.89)

i=1

µ(t) = µ + ξ(t) τ  ξ(t) = ci ξ(t − i) + η(t − 1) i=1

(3.90) (3.91)

3.4 Examples

47

where µ is the constant nominal service rate, ri (t) and vi (t − hi ) are respectively the input rate of the ith source and the calculated ith source rate at switch, {ci , i = 1, 2, · · · , τ } are known parameters. η(t) is a zero-mean i.i.d. Gaussian sequence with variance ρ2 . hi is the round trip delay of the ith source consisting of two path delays, one is the return path delay and the other is forward path delay. On the return path RM cells travel from the switch to the source. On the forward path the user data travels from the source through the congested switch. We should note that round trip delay in transmission is one reason for the disagreement between the switch input and output. The congestion control problem is to find source rate vi such that E {J(q(0), vi (t − hi ), µ(t))} is minimized, where J(q(0), vi (t − hi ), µ(t)) =

N 

1 [q(t) − q)] + 2

t=1

l 

(3.92) 2 λ [ri (t) − ai µ] 2

2

(3.93)

i=1

where q is the target queue length and λ is a weighting factor. It is clear that the objective is to make the queue buffer close to the desired level while the difference between the source rate and the nominal service rate should not be too large. The above criterion combines the performance of queue length and accumulation of the difference between switch input and output. We adopt a second order model for the higher priority source, i.e. τ = 2. To formulate the above congestion control as the LQG control problem for systems with multiple inputs studied in the last section, define ⎡ ⎤ q(t) − q ⎢ ⎥ x(t) = ⎣ ξ(t − 1) ⎦ ξ(t) ui (t − hi ) e(t)

= vi (t − hi ) − ai µ = η(t).

Here ai is the weight for different source rates and

l 

ai = 1.

i=1

The system (3.89)-(3.91) can be put into the form of (3.71) with ⎡ ⎤ ⎡ ⎤ 1 0 −1 0 ⎢ ⎥ ⎢ ⎥ Φ = ⎣ 0 0 1 ⎦, Γ = ⎣ 0 ⎦, 0 c2 c1 1 ⎡ ⎤ 1 ⎢ ⎥ Γ(0) = 0, Γ(i) = ⎣ 0 ⎦ , i = 1, 2, · · · , l. 0 Define the output equation of the ATM system as y(0) (t) =

H(0) x(t) + v(0) (t)

(3.94)

48

3. Optimal Control for Systems with Input/Output Delays

where

 H(0) =

1 0 0 0

0 1



and v(0) is a white noise of zero mean and covariance matrix Qv(0) . The cost function (3.93) can be described by (3.74) with ⎡ ⎤ 1 0 0 ⎢ ⎥ P = 0, Ri = λ2 , Q = ⎣ 0 0 0 ⎦ . 0 0 0 We adopt the similar parameters as given by [82]: – – – – –

The The The The The

bandwidth available for ABR traffic b0 = 1500 cells/s. maximum rate R1,max = 2b0 = 3000 cells/s. buffer length q¯ = 10000 cells/s. buffer set point q = (1/2)ymax = 5000 cells. controller cycle time T = 1 ms.

Here, the delays are fixed. We assume that the link capacity is a 2nd order auto-regressive (AR) process with parameters c1 = c2 = 0.4 and the Gaussian white noise process η(t)’s variance is equal to 1 [82]. We assume that there are 10 sources (i.e. l = 10) with round trip delay from 1 to 10, respectively and N = 100. The weighting between the queue length and the transmission rate is λ = 1 and ai = 1/l(i = 1, 2, · · · , l) are the source sharing. When information of q(t) and ξ(t) are available for feedback, simulation result is in Figure 3.2 where the vertical axis is queue length q(t). The initial queue length of the congested switch is set to be 5100. From the graph we can see that the queue length quickly converges to the target queue length. We now consider a more realistic situation where the round trip delay is time varying. Fix the forward trip delay as 1 but the backward delay is time varying. From the analysis of [82], we know that 1 ≤ hi (t+ 1) ≤ hi (t)+ 1. Here we assume that hi (t) varies around its nominal value hi , i.e., hi (t) − hi = −1, 0 or 1 with probability P (hi (t) = hi ) = 0.50, P (hi (t) = hi + 1) = P (hi (t) = hi − 1) = 0.25. It can be seen from Figure 3.3 that the performance does not change much. On the other hand, the sharing of sources may be different according to the importance of source or the bandwidth of different links. Let the source share be a = [0.0740 0.0386 0.1139 0.1510 0.1384 0.1385 0.1273 0.1485 0.0341 0.0357] and the other parameters remain the same. The switch performance is shown in Figure 3.4 where we can see that the system again possesses very good performance.

3.4 Examples

49

5100

5050

q

t

5000

4950

0

10

20

30

40

50

60

70

80

90

100

t (sample time)

Fig. 3.2. Queue length response

5100

5050

5000

q

t 4950

4900

4850

0

10

20

30

40

50

60

70

80

90

100

t (sample time)

Fig. 3.3. Queue length response

We further examine how the weighting λ affects the congestion control performance. To this end, we set λ = 0.2, which means that we care more about the convergence speed of queue length to the target level. Figure 3.5 verifies that the queue length responds faster, however the system exhibits more oscillations, which is understandable.

50

3. Optimal Control for Systems with Input/Output Delays 5120

5100

5080

5060

qt

5040

5020

5000

4980

4960

0

10

20

30

40

50

60

70

80

90

100

t (sample time)

Fig. 3.4. Queue length response 5100

5080

5060

q

t 5040

5020

5000

4980

0

10

20

30

40

50

60

70

80

90

100

t (sample time)

Fig. 3.5. Queue length response

We also consider the case where there exists output noise and set the covariance of the output measurement covariance Qv(0) = I2 . In this case, the response is shown in Figure 3.6 where one can see that the response becomes worse as compared to Figure 3.2.

3.5 Conclusion In this chapter we have studied the finite-horizon LQR problem for discretetime systems with multiple input delays. An explicit optimal controller is given

3.5 Conclusion

51

5100

5050

qt

5000

4950

0

10

20

30

40

50

60

70

80

90

100

t (sample time)

Fig. 3.6. Queue length response

in terms of the solution of one RDE. The RDE has the same dimension as the original plant. Our solutions have significant computational advantage over traditional approaches such as the system augmentation [19]. It should also be highlighted that our derivation of controller is based on simple duality arguments and standard projection in linear estimation, which is in contrast with traditional dynamic programming [19]. We have also solved the LQG control problem for systems with multiple i/o delays and the solution involves two RDEs, one for the Kalman filtering and the other for the state feedback control. The presented results have been successfully applied to congestion control problem in the end of the chapter.

4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

In this chapter, we shall study the H∞ fixed-lag smoothing and H∞ multiple-stepahead prediction for delay free systems and the H∞ filtering problem for systems with measurement delays. By introducing certain Krein space, it is shown that the three problems can be dealt with in a unified form as the H2 filtering problems for the systems with measurement delays. The presented re-organized innovation analysis approach in Chapter 2 is then applied to derive the estimators.

4.1 Introduction H∞ estimation has attracted a recurring interest since the 1980s. An H∞ estimator is such that the energy gain between the input noises (including process and measurement noises) and the estimation error is bounded by a prescribed level [71, 28, 79] and is applicable to situations where no information on statistics of input noises is available. So far, the basic problem of standard H∞ filtering problem has been well studied. However, problems such as H∞ fixed-lag smoothing, multiple-step ahead prediction and filtering for time delay systems are known to be challenging and deserve further investigations. It is worth noting that in the discrete-time case, the problems of H∞ fixed-lag smoothing, prediction and filtering for systems with measurement delays can be solved via state augmentation [91]. This approach, however, may be computationally expensive when the dimension of the signal to be estimated is high and/or the smoothing lag is large. To obtain more efficient H∞ estimation algorithms, some attempts have been made in recent years; see, e.g., [27, 16, 17, 81]. The purpose of this chapter is to study the H∞ estimation problem, including filtering, multi-step prediction and fixed-lag smoothing, for systems with measurement delays via the re-organized innovation analysis approach in Krein space. The estimators are to be calculated without resorting to the state augmentation. A unified approach is provided. More specifically, by identifying an associated stochastic system in Krein space, each of the three problems is shown to be equivalent to an H2 estimation problem for the associated system with instantaneous and delayed measurements in Krein space. H. Zhang and L. Xie: Cntrl. and Estim. of Sys. with I/O Delays, LNCIS 355, pp. 53–85, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com


54

4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

This chapter is organized as follows. Section 4.2 presents a solution to the H∞ fixed-lag smoothing problem. The H∞ multiple-step-ahead prediction problem is addressed in Section 4.3. The H∞ filtering for systems with measurement delays is studied in Section 4.4.

4.2 H∞ Fixed-Lag Smoothing In this section we consider the H∞ fixed-lag smoothing problem for delay free systems. It shall be shown that the problem is in fact equivalent to an estimation problem for a measurement delayed system in Krein space. We consider the following linear discrete-time system x(t + 1) = Φx(t) + Γ e(t), y(t) = Hx(t) + v(t)

(4.1) (4.2)

x(0) = x0

z(t) = Lx(t)

(4.3)

where x(t) ∈ Rn , e(t) ∈ Rr , y(t) ∈ Rp , v(t) ∈ Rp and z(t) ∈ Rq represent the state, input noise, measurement output, measurement noise and the signal to be estimated, respectively. It is assumed that the input and measurement noises are deterministic signals and are from 2 [0, N ] where N is the time-horizon of the smoothing problem under investigation. The H∞ fixed-lag smoothing problem under investigation is stated as follows: Given a scalar γ > 0, an integer d > 0 and the observation {y(j)}tj=0 , find an estimate zˇ(t − d | t) of z(t − d), if it exists, such that the following inequality is satisfied: N 

sup (x0 ,u,v)=0



[ˇ z (t − d | t) − z(t − d)] [ˇ z (t − d | t) − z(t − d)]

t=d

x0 P0−1 x0

+

N −1 t=0

e (t)e(t)

+

N 

< γ2

(4.4)

v  (t)v(t)

t=0

where P0 is a given positive definite matrix which reflects the relative uncertainty of the initial state to the input and measurement noises. Remark 4.2.1. In the performance criterion (4.4), we have assumed that the initial estimate x ˆ(0|d) is zero and P0 reflects how accurate the estimate is. It is worth pointing out that the above criterion can be easily modified to accommodate the situation where the initial estimate is not zero. Remark 4.2.2. Note that the H∞ fixed-lag smoothing has been addressed in [91] through system augmentation and standard H∞ filtering. The approach is easy to understand but suffers from heavy computational requirement, especially for the case of large smoothing lag and high dimension of the signal to be estimated. Very recently, [17, 16] have presented a J-factorization approach to the steadystate H∞ fixed-lag smoothing without resorting to system augmentation. In the present section, we shall discuss the finite horizon H∞ fixed-lag smoothing for systems using an innovation analysis approach in Krein space.

4.2 H∞ Fixed-Lag Smoothing

4.2.1

55

An Equivalent H2 Estimation Problem in Krein Space

In this section we shall demonstrate that the deterministic H∞ fixed-lag smoothing problem can be converted to an innovation analysis for an associated stochastic system in Krein space. First, in view of (4.4) we define 

Jd,N =

x0 P0−1 x0

+

N −1  t=0



e (t)e(t) +

N 



v (t)v(t) − γ

−2

N 

t=0

vz (t)vz (t),

t=d

(4.5) where vz (t) = zˇ(t − d | t) − Lx(t − d), t ≥ d.

(4.6)

The H∞ fixed-lag smoothing problem is apparently a minimax optimization problem of the above cost function, that is, Jd,N has a minimum over {x(0), e} and the smoother is such that the minimum is positive. To convert the optimization into an innovation analysis problem, in view of (4.1)-(4.2) and (4.6), we introduce the following stochastic system x(t + 1) = Φx(t) + Γ e(t), y(t) = Hx(t) + v(t), zˇ(t − d | t) = Lx(t − d) + vz (t), t ≥ d,

(4.7) (4.8) (4.9)

where e(t), v(t) and vz (t) are assumed to be uncorrelated white noises, with x(0), x(0) = P0 , e(t), e(s) = Qe δts , v(t), v(s) = Qv δts , vz (t), vz (s) = Qvz δts , while Qe = Ir , Qv = Ip , Qvz = −γ 2 Iq .

(4.10)

ˇ(t − d | t) are respectively the observations for In the above system, y(t) and z x(t) and x(t − d) at the time instant t. Since vz (t) is of negative covariance, the above stochastic system should be considered in Krein space [30] rather than Hilbert space. Remark 4.2.3. Note that the elements in (4.1)-(4.3), which are denoted in normal letters, are from Euclidean space while the elements in (4.7)-(4.9), denoted by bold face letters, are from Krein space. They satisfy the same constraints. Combining (4.8) with (4.9) yields, ⎧ ⎪ ⎨ Hx(t) + vo (t), 0≤t
(4.11)

56

4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

where

⎧ ⎪ ⎨ y(t), 0≤t
(4.12)

(4.13)

Note that yo (t) is an observation at time t. It is easy to know that vo (t) is a white noise with vo (t), vo (s) = Qvo (t)δts , where ⎧ ⎪ ⎨ Ip , 0≤t
(4.15)

where y(t), in normal face, is from the measurement system (4.2) and zˇ(t − d | t) is from (4.3). Now let yo be the collection of y0 (t), yo = col {yo (0), · · · , yo (N )} ,

(4.16)

and yo be the collection of the Krein space measurements of system (4.7)-(4.9), i.e., yo = col {yo (0), · · · yo (N )} .

(4.17)

From (4.10) we can show that Jd,N of (4.5) can be re-written as (see also [104]) ⎤ ⎡ ⎤ ⎡ ⎤−1 ⎡ ⎤ x(0) x(0) x(0) x(0) ⎣ e ⎦, Jd,N (x(0), e; yo ) = ⎣ e ⎦ ⎣ e ⎦ , ⎣ e ⎦ yo yo yo yo

(4.18)

e = col{e(0), e(1), · · · , e(N − 1)}, e = col{e(0), e(1), · · · , e(N − 1)},

(4.19) (4.20)

⎡

where

with e(·) from system (4.1) and e(·) from Krein space system (4.7). From (4.18), it is clear that the H∞ fixed-lag smoothing problem is equivalent to that Jd,N has a minimum over {x(0), e} and the smoother is such that

4.2 H∞ Fixed-Lag Smoothing

57

the minimum is positive. To derive conditions under which Jd,N (x(0), e; yo ) is minimum over {x(0), e}, we introduce the innovation associated with the measurements yo (t). Like in standard Kalman filtering, the innovation sequence associated with observation yo (t) is defined as 

ˆ o (t | t − 1), wo (t) = yo (t) − y

(4.21)

ˆ o (t | t − 1) is the projection of yo (t) onto where y L{yo (0), · · · , yo (t − 1)}.

(4.22)

For t < d, the innovation wo (t) is given from (4.11) as ˆ (t | t − 1), wo (t) = y(t) − H x

(4.23)

ˆ (t | t−1) is the Hilbert space projection of x(t) onto L{y(0), · · · , y(t−1)}. where x For t ≥ d, the innovation wo (t) is given from (4.11) as    ˆ (t | t − 1) H 0 x + vo (t), (4.24) wo (t) = yo (t) − ˆ (t − d | t − 1) 0 L x ˆ (t | t − 1) and x ˆ (t − d | t − 1) are respectively the Krein-space projection where x of x(t) and x(t − d) onto (4.22). Denote 

Qwo (t) = wo (t), wo (t).

(4.25)

Qwo (t) is termed as the original innovation covariance matrix which plays an important role for checking the existence of the H∞ fixed-lag smoother. It is clear that for t < d, Qwo (t) > 0.

(4.26)

With the innovation and covariance matrix, we have the following results [31, 104], Lemma 4.2.1. 1) An H∞ estimator zˇ(t − d | t) that achieves (4.4) exists if and only if Qwo (t) and Qvo (t) for 0 ≤ t ≤ N have the same inertia. 0 2)The minimum of Jd,N , if exists, can be given in terms of the innovation wo (t) as 0 = Jd,N

N 

wo (t)Q−1 wo (t)wo (t)

(4.27)

t=0

Now our aims are twofold: Firstly, calculate the covariance matrix of Qwo (t) so that we can check if an H∞ estimator zˇ(t − d | t) that achieves (4.4) exists. 0 Secondly, seck an estimator, if it exists, such that Jd,N of (4.27) is positive. Note that the calculation of Qwo (t) and the minimization of (4.18) subject to the stochastic system (4.7)-(4.9) can be approached by the Krein space Kalman filtering for the system (4.7) and (4.11)[31, 104]. However, unlike the work in [31, 104], (4.11) involves both the instantaneous and delayed measurements to which the Kalman filtering is not directly applicable. We shall provide a reorganized innovation analysis approach for such a problem.

58

4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

4.2.2

Re-organized Innovation Sequence

First, the observations up to time t are collected as {yo (0), · · · , yo (t)}, which span the following linear space L{yo (0), · · · , yo (t)}.

(4.28)

Note that since yo (i), i = 0, 1, · · · , t have different dimensions, the linear space (4.28) is as explained in Remark 2.2.2. For t ≥ d, note the definition of yo in (4.12), the linear space (4.28) can be re-organized as   (4.29) L {yf (s)}t−d s=0 , y(t − d + 1), · · · , y(t) , where 



yf (s) =

 y(s) . zˇ(s | s + d)

(4.30)

Then, the new measurements of y(s) and yf (s) in (4.29) are no longer with time delay, where the former satisfies (4.8) while the latter satisfies:   H yf (s) = x(s) + vf (s), s = 0, 1, · · · , t − d, (4.31) L where



v(s) vf (s) = vz (s + d)

 (4.32) 

 0 Ip is a white noise with vf (s), vf (j) = Qvf δij and Qvf = . Thus, 0 −γ 2 Iq (4.7) and (4.31) form a state space representation to which the Kalman filtering in Krein space can be applied. Now we introduce the innovation associated with the new reorganized measurements as 

ˆ (s), w(s) = y(s) − y

(4.33)

ˆ (s) for s > t − d + 1 is the projection of y(s) onto the linear space where y L{yf (0), · · · , yf (t − d); y(t − d + 1), · · · , y(s − 1)},

(4.34)

ˆ (s) is the projection of y(s) onto the linear space and for s = t − d + 1, y L{yf (0), · · · , yf (s − 1).

(4.35)

Similarly, we define that 

ˆ f (s), t ≤ t − d, wf (s) = yf (s) − y ˆ f (s) is the projection of yf (s) onto the linear space where y

(4.36)

4.2 H∞ Fixed-Lag Smoothing

L{yf (0), · · · , yf (s − 1).

59

(4.37)

Recall the discussion as in Section 2.2, it is easy to know that {wf (0), · · · , wf (t − d), w(t − d + 1), · · · , w(t)}

(4.38)

is a white noise sequence and spans the same linear space as L {yf (0), · · · , yf (t − d), y(t − d + 1), · · · , y(t)}

(4.39)

or equivalently L {yo (0), · · · , yo (t)}. Different from the innovation wo (t), w(s) and wf (s) are termed as reorganized innovations. 4.2.3

Calculation of the Innovation Covariance

In this subsection, we shall calculate the covariance matrix Qwo (t) of the innovation wo (t) by using the re-organized innovations defined in the last subsection. To this end, we shall introduce two standard Riccati equations associated with the reorganized innovation sequence. Firstly, by taking into consideration (4.8) and (4.31), it follows from (4.33)(4.36) that ˜ (s, 1) + v(s), t − d + 1 ≤ s ≤ t, w(s) = H x   H ˜ (s, 2) + vf (s), 0 ≤ s ≤ t − d, x wf (s) = L

(4.40)

˜ (s, 1) = x(s) − x ˆ (s, 1), x ˜ (s, 2) = x(s) − x ˆ (s, 2), x

(4.42)

(4.41)

where

(4.43)

ˆ (s, 1) and x ˆ (s, 2) (s = t + 1) the projections of x(s) onto the linear spaces with x   L {yf (s)}t−d (4.44) s=0 , y(t − d + 1), · · · , y(s − 1) , and L{yf (0), · · · , yf (s − 1),

(4.45)

˜ (t − d + 1, 1) = x ˜ (t − d + 1, 2). respectively. It is obvious that x As in the standard Kalman filtering, we define the one-step ahead prediction error covariance matrices of the state as 

˜ (s, 2), 0 ≤ s ≤ t − d + 1, P2 (s) = ˜ x(s, 2), x 

˜ (s, 1), s ≥ t − d + 1, x(s, 1), x P1 (s) = ˜

(4.46) (4.47)

˜ (s, 2) and x ˜ (s, 2) are as in (4.42)-(4.43). Then, P1 (s) and P2 (s) are where x computed by Riccati equations given in the lemma below.

60

4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

Lemma 4.2.2. The matrix P2 (s) (0 ≤ s ≤ t − d + 1) obeys the following Riccati equation associated with the system (4.7) and (4.31) as P2 (s + 1) = ΦP2 (s)Φ − K2 (s)Qwf (s)K2 (s) + Γ Γ  , P2 (0) = P0 , (4.48) where 

H K2 (s) = ΦP2 (s) L



Q−1 wf (s),

(4.49)

while Qwf (s) is given by 

    H I H + p Qwf (s) = P (s, 2) 0 L L

 0 . −γ 2 Iq

The matrix P1 (s) (t − d + 1 ≤ s ≤ t) obeys the following Riccati equation associated with the system (4.7)-(4.8) as P1 (s + 1) = ΦP1 (s)Φ − K1 (s)Qw (s)K1 (s) + Γ Γ  , P1 (t − d + 1) = P2 (t − d + 1),

(4.50)

where K1 (s) = ΦP1 (s)H  Q−1 w (s), and

(4.51)

Qw (s) = HP1 (s)H  + Ip .

Proof: The proof is similar to the case of the optimal H2 estimation discussed in Chapter 2. ∇ Remark 4.2.4. We have presented two different Riccati equations (4.48) and (4.50). It should be noted that the terminal solution of the Riccati equation (4.48) is the initial condition of the Riccati equation of (4.50). Corollary 4.2.1. Let 

˜ (s + i, 1), i ≥ 0, s ≥ t − d + 1. R(s, i) = x(s), x

(4.52)

Then, R(s, i) obeys the following difference equation: R(s, i + 1) = R(s, i)A (s, i), R(s, 0) = P1 (s),

(4.53)

  A(s, i) = Φ In − P1 (s + i)H  Q−1 w (s + i)H .

(4.54)

where

4.2 H∞ Fixed-Lag Smoothing

61

Proof: Firstly, by applying Kalman filtering, it follows that ˜ (s + i + 1, 1) = Φ˜ x x(s + i, 1) + Γ e(s + i) −ΦP1 (s + i)H  Q−1 w (s + i)w(s + i).

(4.55)

In view of (4.40), it follows directly from (4.55) that ˜ (s + i + 1, 1) = A(s, i)˜ x x(t + i, 1) + Γ e(s + i) −ΦP1 (s + i)H  Q−1 w (s + i)v(s + i).

(4.56)

Note that x(s) is uncorrelated with the white noises e(s + i) and v(s + i). It follows that ˜ (s + i + 1, 1) = R(s, i)A (s, i), R(s, 0) = P1 (s), (4.57) R(s, i + 1) = x(s), x ∇

which is (4.53).

The computation of the original innovation covariance matrix can now be stated in the theorem below. Theorem 4.2.1. For t ≥ d − 1, Qwo (t + 1) of (4.25) is given by   HP1 (t + 1)H  + Ip HR(t − d + 1, d) L Qwo (t + 1) = , LR(t − d + 1, d)H  LP(t − d)L − γ 2 Iq

(4.58)

where P(t − d) = P2 (t − d + 1) −

d 

R(t − d + 1, i − 1)H  Q−1 w (t − d + i)

i=1

×HR (t − d + 1, i − 1),

(4.59)

with P2 (t − d) and P1 (t + 1) computed by (4.48) and (4.50), respectively, R(t − d + 1, i) by (4.53) and Qw (t − d + i) given by Qw (t − d + i) = HP1 (t − d + i)H  + Ip . Proof: For t > d − 1, note (4.12) and (4.21), the innovation wo (t + 1) is given as wo (t + 1)      ˆ (t + 1 | t) H 0 x(t + 1) − x v(t + 1) 0 = + , ˆ (t − d + 1 | t) 0 L x(t − d + 1) − x 0 vz (t + 1) ˆ (t+1 | t) and x ˆ (t−d+1 | t) are the projections of x(t+1) and x(t−d+1) where x onto the linear space L {yo (0), yo (1), · · · , yo (t)}, respectively. By applying the re-organized innovation, L {yo (0), yo (1), · · · , yo (t)} = L {wf (0), · · · , wf (t − d), w(t − d + 1), · · · , w(t)} ,

62

4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

it follows that ˆ (t + 1 | t) = x ˆ (t + 1, 1), x ˆ (t − d + 1 | t) is the projections of x(t − d + 1) onto the linear space and x L {wf (0), · · · , wf (t − d), w(t − d + 1), · · · , w(t)} . Thus, we have 

H wo (t + 1) = 0

0 L



   ˜ (t + 1, 1) x v(t + 1) 0 , + ζ(t) 0 vz (t + 1)

(4.60)

˜ (t + 1, 1) is as in (4.42), i.e., x ˜ (t + 1, 1) = x(t + 1) − x ˆ (t + 1, 1) and where x ˆ (t − d + 1 | t). ζ(t) = x(t − d + 1) − x

(4.61)

By using the projection formula and the re-organized innovation, it follows that ζ(t) = x(t − d + 1) − P roj{x(t − d + 1) | wf (0), · · · , wf (t − d), w(t − d + 1), · · · , w(t)} ˆ (t − d + 1, 2) − = x(t − d + 1) − x

d 

x(t − d + 1), w(t − d + i)

i=1

×Q−1 w (t − d + i)w(t − d + i) ˜ (t − d + 1, 2) − =x

d 

˜ (t − d + i, 1)H  x(t − d + 1), x

i=1

×Q−1 w (t − d + i)w(t − d + i) ˜ (t − d + 1, 2) − =x

d 

R(t − d + 1, i − 1)H 

i=1

×Q−1 w (t

− d + i)w(t − d + i).

(4.62)

From (4.60), the innovation covariance matrix Qwo (t + 1) is given as Qwo (t + 1) = wo (t + 1), wo (t + 1)    ˜ (t + 1, 1) ˜ H 0 ˜ x(t + 1, 1), x x(t + 1, 1), ζ(t) H = ˜ (t + 1, 1) 0 L ζ(t), x ζ(t), ζ(t) 0   I 0 + p . 0 −γ 2 Iq

0 L



(4.63)

Observe that ˜ (t + 1, 1) = P1 (t + 1). ˜ x(t + 1, 1), x

(4.64)

4.2 H∞ Fixed-Lag Smoothing

63

˜ (t + 1, 1) is uncorrelated with w(t − d + i, 1) By considering the fact that x (i = 1, · · · , d), it follows from (4.62) that ˜ (t + 1, 1) = ˜ ˜ (t + 1, 1) ζ(t), x x(t − d + 1, 2), x ˜ (t + 1, 1) = x(t − d + 1), x = R(t − d + 1, d).

(4.65)

Also, from (4.62), ζ(t) is uncorrelated with w(t − d + i) (i = 1, · · · , d), thus we have ζ(t), ζ(t) +

d 

 R(t − d + 1, i − 1)H  Q−1 w (t − d + i)HR (t − d + 1, i − 1)

i=1

= P2 (t − d + 1),

(4.66)

which yields (4.59) directly. Thus, from (4.63)-(4.66), the innovation covariance matrix Qwo (t) for t > d is given by (4.58).∇ 4.2.4

H∞ Fixed-Lag Smoother

In the previous section we have presented preliminary results on the innovation analysis in Krein space. In this section, we shall give our main result on the H∞ fixed-lag smoothing. Theorem 4.2.2. Consider the system (4.1)-(4.3) and the associated performance criterion (4.4). Suppose the recursions (4.48) and (4.50) have bounded solutions. Then for a given scalar γ > 0 and an integer d > 0, an H∞ smoother zˇ(t − d + 1 | t + 1) that achieves (4.4) exists  if and only if, for each t = d − 1, · · · , Ip 0 N − 1, Qwo (t + 1) and Qvo (t + 1) = have the same inertia1 , 0 −γ 2 Iq where Qwo (t + 1) is calculated by (4.58). In this situation, the central smoother is given by zˇ(t − d + 1 | t + 1) = Lˆ x(t − d + 1 | t + 1, t − d),

(4.67)

where x ˆ(t−d+1 | t+1, t−d) is defined as the Krein space projection of x(t−d+1) onto the linear space L{yf (0), · · · , yf (t − d); y(t − d + 1), · · · , y(t + 1)}, which can be computed by xˆ(t − d + 1 | t + 1, t − d) = xˆ(t − d + 1, 2) +

d+1 

R(t − d + 1, i − 1)H 

i=1

×Q−1 ˆ(t − d + i, 1)] , w (t − d + i) [y(t − d + i) − H x 1

(4.68)

The inertia of a matrix is the numbers of positive and negative eigenvalues of the matrix.

64

4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

while x ˆ(s + 1, 1) (s > t − d) in (4.68) is calculated recursively as x ˆ(s + 1, 1) = Φˆ x(s, 1) + K1 (s) [y(s) − H x ˆ(s, 1)] , x ˆ(t − d + 1, 1) = x ˆ(t − d + 1, 2),

(4.69)

ˆ(t − d + 1, 2) in (4.68) and with K1 (s) computed by (4.51) and the initial value x (4.69) given by the Kalman recursion:     H x ˆ(t − d + 1, 2) = Φˆ x(t − d, 2) + K2 (t − d) yf (t − d) − x ˆ(t − d, 2) , L x ˆ(0, 2) = 0. (4.70)   y(t − d) where K2 (t − d) is as in (4.49) and yf (t − d) = . zˇ(t − d | t) Proof: First, in view of (4.18), the H∞ estimator zˇ(t − d + 1 | t + 1) (t ≥ −1) 0 with respect that achieves (4.4) exists if and only if Jd,N has a minimum Jd,N 0 to (x(0), e) and zˇ(t−d+1 | t+1) can be chosen such that Jd,N > 0, where the 0 can be given in terms of the original innovation wo (t + 1) as minimum Jd,N [31, 104] 0 = Jd,N

N −1 

wo (t + 1)Q−1 wo (t + 1)wo (t + 1).

(4.71)

t=−1

Recall that Jd,N has a minimum over {x(0); e} iff Qwo (t + 1) and Qvo (t + 1) for −1 ≤ t ≤ N − 1 have the same inertia [31, 104]. Note that for −1 ≤ t < d − 1, Qvo (t + 1) > 0 and Qwo (t + 1) > 0. Thus, an H∞ estimator zˇ(t − d + 1 | t + 1) exists iff Qwo (t + 1) and Qvo (t + 1) for d − 1 ≤ t ≤ N − 1 have the same inertia. Next, we prove (4.67) based on (4.27). In light of (4.58), Qwo (t + 1) (t ≥ d − 1) has the following factorization   Q11 (t) Q12 (t) Qwo (t + 1) = Q21 (t) Q22 (t)     Ip Q11 (t) 0 0 Ip Q−1 (t)Q12 (t) 11 = , Q21 (t)Q−1 0 ∆(t) 0 Iq 11 (t) Iq (4.72) where Q11 (t) = HP1 (t + 1)H  + Ip , Q21 (t) = LR(t − d + 1, d)H  ; Q12 (t) = Q21 (t), Q22 (t) = LP(t − d)L − γ 2 Iq ,

(4.73)

and ∆(t) = Q22 (t) − Q21 (t)Q−1 11 (t)Q12 (t).

(4.74)

4.2 H∞ Fixed-Lag Smoothing

65

Note that for −1 ≤ t < d − 2, wo (t + 1) is as wo (t + 1) ≡ woy (t + 1) = y(t + 1) − yˆ(t + 1 | t),

(4.75)

and for t ≥ d − 1, wo (t + 1) is as       woy (t + 1) y(t + 1) yˆ(t + 1 | t) wo (t + 1) ≡ = − woz (t + 1) zˇ(t − d + 1 | t + 1) zˆ(t − d + 1 | t) (4.76) where zˆ(t−d+1 | t) and yˆ(t−d+1 | t) are obtained from the projections of ˇz(t−d) and y(t − d + 1) onto the Krein space L{yo (0), · · · , yo (t)} = L{yf (0), · · · , yf (t − d); y(t − d + 1), · · · , y(t)}, respectively. 0 is equivalently written By applying the factorization (4.72), the minimum Jd,N as 0 = Jd,N

d−2 

wo (t + 1)Q−1 wo (t + 1)wo (t + 1)

t=−1

+

N −1 

wo (t + 1)Q−1 wo (t + 1)wo (t + 1)

t=d−1

=

d−1 

wo (t)Q−1 wo (t)wo (t) +

t=0

N −1 

[y(t + 1) − yˆ(t + 1 | t)] Q−1 11 (t)

t=d−1

×[y(t + 1) − yˆ(t + 1 | t)] +

N −1 

zy (t + 1)∆−1 (t)zy (t + 1), (4.77)

t=d−1

with zy (t + 1) = zˇ(t − d + 1 | t + 1) − zˆ(t − d + 1 | t) − Q21 (t)Q−1 11 (t) × [y(t + 1) − yˆ(t + 1 | t)] .

(4.78)

It is not difficult to show that if Qwo (t + 1) and Qvo (t + 1) have the same inertia, then Q11 (t) > 0. Thanks to the fact that Qwo (t + 1) and Qvo (t + 1) have the same inertia, from (4.72) and (4.14), Q11 (t) > 0 implies ∆(t) < 0. We note that 0 any choice of estimator that renders Jd,N > 0 is an acceptable one, and that Qwo (t + 1) > 0 for −1 ≤ t ≤ d − 2. Thus the estimator can be obtained from (4.77) by setting zy (t + 1) = 0,

(4.79)

which implies that zˇ(t − d + 1 | t + 1) = zˆ(t − d + 1 | t) + Q21 (t)Q−1 ˆ(t + 1 | t)] 11 (t) [y(t + 1) − y −1 = zˆ(t − d + 1 | t) + Q21 (t)Q11 (t) × [y(t + 1) − yˆ(t + 1 | t, t − d)] .

(4.80)

66

4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

Furthermore, by considering that 3 4 Q11 (t) = woy (t + 1), woy (t + 1) = Qwy (t + 1), 3 4 Q21 (t) = woz (t + 1), woy (t + 1) 4 3 = ˇ z(t − d + 1 | t + 1), woy (t + 1) , and that zˆ(t − d + 1 | t) is obtained from the projection of zˇ(t − d + 1) onto the Krein space L{yf (0), · · · , yf (t − d); y(t − d + 1), · · · , y(t)}, it follows from (4.80) that zˇ(t − d + 1 | t + 1) = zˆ(t − d + 1 | t + 1, t − d),

(4.81)

where zˆ(t − d + 1 | t + 1, t − d) is obtained from the projection of zˇ(t − d + 1) onto L{yf (0), · · · , yf (t − d); y(t − d + 1), · · · , y(t + 1)}. Thus zˇ(t − d + 1 | t + 1) = Lˆ x(t − d + 1 | t + 1, t − d), which is (4.67). Finally we prove that the projection x ˆ(t − d + 1 | t + 1, t − d) is calculated by (4.68). Note that x ˆ(t − d + 1 | t + 1, t − d) is obtained from the projection of the state x(t − d + 1) onto L{wf (0), · · · , wf (t − d); w(t − d + 1), · · · , w(t + 1)}. Since w is a white noise, the estimator x ˆ(t − d + 1 | t + 1, t − d) is calculated by using the projection formula as ˆ (t − d + 1 | t + 1, t − d) x = P roj {x(t − d + 1) | wf (0), · · · , wf (t − d)} + P roj {x(t − d + 1) | w(t − d + 1), · · · , w(t + 1)} ˆ (t − d + 1, 2) + =x

d+1 

x(t − d + 1), w(t − d + i)

i=1

×Q−1 w (t − d + i)w(t − d + i) ˆ (t − d + 1, 2) + =x

d+1 

R(t − d + 1, i − 1) ×

i=1

ˆ (t − d + i, 1)] , H  Q−1 w (t − d + i) [y(t − d + i) − H x

(4.82)

which is (4.68). Similarly, by applying the projection formula and the re-organized innovation sequence, for s > t − d, it follows that ˆ (s + 1, 1) = P roj {x(s + 1) | wf (0), · · · , wf (t − d); w(t − d + 1), x · · · , w(s)} = P roj {x(s + 1) | wf (0), · · · , wf (t − d); w(t − d + 1), · · · , w(s − 1)} + x(s + 1), w(s) Q−1 w (s)w(s)  −1 ˆ (s, 1)] = Φˆ x(s, 1) + ΦP1 (s)H Qw (s) [y(s) − H x which is (4.69). Similarly, we derive (4.70) immediately.

(4.83) ∇

4.2 H∞ Fixed-Lag Smoothing

67

Remark 4.2.5. In Theorem 4.2.2, we have presented a solution to the H∞ fixed-lag smoothing. Unlike [91] where the problem is converted to the H∞ filtering for an augmented system, our solution is given in terms of two RDEs of the same dimension as that of the original plant and can be considered as the H∞ counterpart of the forward and backward algorithm of the H2 fixed-lag smoothing [65]. 4.2.5

Computational Cost Comparison and Example

We now analyze the computational cost of the H∞ fixed-lag smoothing algorithm presented in Theorem 4.2.2, in comparison with the state augmentation method in [91]. To this end, we recapture the procedures for computing an H∞ smoother using the above two methods. From Theorem 4.2.2 of this section, the procedure for computing an H∞ fixed-lag smoother zˇ(t − d + 1 | t + 1) is as follows – First, compute the matrix P2 (t − d + 1) using the RDE (4.48), with the value of P2 (t − d) obtained from the last iteration as the initial value. – Calculate P1 (s) (t − d + 1 ≤ s ≤ t) recursively using (4.50) with the initial condition P1 (t − d + 1) = P2 (t − d + 1) obtained in the last step. – Calculate innovation covariance Qwo (t + 1) using Theorem 4.2.1 and then check the existence condition of an H∞ smoother. – Compute the H∞ smoother zˇ(t − d | t) using Theorem 4.2.2. Recall the result in [91]. The smoother is computed with the help of the following augmented model xa (t + 1) = Φa xa (t) + Γa e(t), y(t) = Ha xa (t) + v(t), zˇ(t − d | t) = Lx(t − d) = La (t)xa (t),

(4.84) (4.85) (4.86)

where 

xa (t) = [ x (t) z  (t − 1) · · · z  (t − d) ] , Ha = [ H 0 · · · 0 ] , La (t) = [ 0

(4.87) (4.88)

· · · 0, Iq ] f or t > d, and La (t) = 0 f or t ≤ d,

⎡

Φ ⎢L ⎢ Iq Φa = ⎢ ⎣

⎤

··· Iq

(4.89) ⎡

⎤

Γ ⎥ ⎢ 0⎥ ⎥ .. ⎥ . ⎥ , Γa = ⎢ ⎣ ⎦ . ⎦ 0 0

(4.90)

Then the H∞ fixed-lag smoother is equivalent to the standard H∞ filtering problem of the above augmented system (4.84)-(4.86), which has been studied in [91]. Computational Costs. As additions are much faster than multiplications and divisions, it is the number of multiplications and divisions, counted together,

68

4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

that is used as the operation count. Let M D(t) denote the number of multiplications and divisions at time instant t. A. Re-organized innovation approach It follows that the operation count at time instant t with the new approach, denoted as M Dnew (t) (the multiplication is from right to left), is given as M Dnew (t) = (d + 1)(p + n)2 + d(2n3 + n2 + pn + p2 ) +d(3n2 + 2pn + p2 ) + d(3n3 + 3pn2 + rn2 + 2p2 n + p3 ) +3n2 + 2(p + q)n + (p + q)2 + 3n3 +3(p + q)n2 + rn2 + 2(p + q)2 n + (p + q)3 + pn.

(4.91)

B. Augmentation approach Note that the augmented system (4.84)-(4.86) is of dimension of n + dp, the MD count for computing an smoother by using the algorithm of [91] in each iteration, denoted by M Daug (t), is given by M Daug (t) = 2(n + dq)3 + (n + q + r)(n + dq)2 + (2pn + p2 + q 2 )(n + dq) +pn2 + p2 n + p3 + q 3 + ! d (2p + 4)n2 + (2p2 + 4p + q)n + 2p3 + 2p2 + qn. (4.92) It is clear that the order of d in M Dnew (t) is 1 and the order in M Daug (t) is 3. That is, when the smoothing lag d is sufficiently large, it is not difficult to see that M Daug (t)  M Dnew (t). This can also be seen from the examples below. Example 4.2.1. Consider the system (4.1)-(4.3) with n = 3, p = 1, q = 3 and r = 1. We investigate the relationship between the lag d and the MD number. Table 4.1. Comparison of the Computational Costs d M Daug (t) M Dnew (t)

1 1022 691

2 2631 932

3 5392 1173

4 9629 1414

5 15666 1655

10 84191 2860

20 542541 5270

Assume that n, m and r are as in the above, but p = 2, the MD number in relation to the lag d is given in Table 4.2. Table 4.2. Comparison of the Computational Costs d M Daug (t) M Dnew (t)

1 639 569

2 1395 810

3 2559 1051

4 4227 1292

5 6495 1533

10 30195 2738

20 177795 5148

4.3 H∞ d-Step Prediction

69

Clearly, the presented new approach is more efficient than the augmented model approach, especially when the lag d is large. 4.2.6

Simulation Example

Example 4.2.2. Consider the discrete-time system (4.1)-(4.3) with ⎡ ⎤ ⎡ ⎤ 0.8 −0.2 0 0 0 Φ=⎣ 0 0.3 0.5 ⎦ , Γ = ⎣ 0.5 0 ⎦ 0 0 0.9 1 0 ⎡ ⎤   0.1 0 0 0.5 0.8 0 H = [ 1 −0.8 0.6 ] , L = , P0 = ⎣ 0 0.1 0 ⎦ . 0 0 0.6 0 0 0.1 We shall investigate the relationship between the lag d and the achievable optimal γ for all 0 ≤ t ≤ N where N = 20. By applying the method presented in this section, the results are shown in Table 4.3. Table 4.3. The Relationship between the lag d and γ d γopt

0 2.1

1 1.76

2 1.49

3 1.35

4 1.26

5 1.25

By using the state augmentation approach [91], the same relationship between d and γ is obtained as in the above table.

4.3 H∞ d-Step Prediction In the above section, we have studied the H∞ fixed-lag smoothing problem. In this section we shall investigate the H∞ multiple-step-ahead prediction problem. Consider the following linear system x(t + 1) = Φx(t) + Γ e(t), y(t) = Hx(t) + v(t) z(t) = Lx(t)

x(0) = x0

(4.93) (4.94) (4.95)

where x(t) ∈ Rn , e(t) ∈ Rr , y(t) ∈ Rp , v(t) ∈ Rp and z(t) ∈ Rq represent the state, input noise, measurement output, measurement noise and the signal to be estimated, respectively. It is assumed that the input and measurement noises are deterministic signals and are from 2 [0, N ] where N is the time-horizon of the prediction problem under investigation.

70

4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

The H∞ d-step prediction problem under investigation is stated as follows: Given a scalar γ > 0, an integer d > 0 and the observation {y(j)}tj=0 , find an estimate zˇ(t + d | t) of z(t + d), if it exists, such that the following inequality is satisfied: N 

sup (x0 ,u,v)=0

[ˇ z (t + d | t) − z(t + d)] [ˇ z (t + d | t) − z(t + d)]

t=−d

x0 P0−1 x0

+

N +d−1 

e (t)e(t) +

t=0

N 

< γ 2 (4.96) v  (t)v(t)

t=0

where P0 is a given positive definite matrix which reflects the relative uncertainty of the initial state to the input and measurement noises. 4.3.1

An Equivalent H2 Problem in Krein Space

Similar to the discussion for the H∞ fixed-lag smoothing, we define 

Jd,N = x0 P0−1 x0 +

N +d−1

e (t)e(t) +

t=0

−γ −2

N 

N 

v  (t)v(t)

t=0

vz (t + d)vz (t + d),

(4.97)

t=−d

where vz (t + d) = zˇ(t + d | t) − Lx(t + d), t ≥ −d.

(4.98)

Introduce the following stochastic system x(t + 1) = Φx(t) + Γ e(t), y(t) = Hx(t) + v(t), ˇ z(t + d | t) = Lx(t + d) + vz (t + d), t ≥ −d,

(4.99) (4.100) (4.101)

where e(t), v(t) and vz (t) are assumed to be uncorrelated white noises, with x(0), x(0) = P0 , e(t), e(s) = Qe δts , v(t), v(s) = Qv δts , vz (t), vz (s) = Qvz δts , while Qe = Ir , Qv = Ip , Qvz = −γ 2 Iq .

(4.102)

ˇ(t + d | t) are respectively the observations for In the above system, y(t) and z x(t) and x(t + d) at the time instant t. Since vz (t) is of negative covariance, the above stochastic system should be considered in Krein space [30] rather than Hilbert space.

4.3 H∞ d-Step Prediction

71

Let yo (t) be the measurement at time t and vo (t) be associated measurement noise at time t, then we have ⎧ ⎪ ⎨ zˇ(t + d | t), −d ≤ t < 0   yo (t) = (4.103) y(t) ⎪ ,t≥0 ⎩ ˇ z(t + d | t) ⎧ ⎪ ⎨ vz (t + d), −d ≤ t < 0   vo (t) = (4.104) v(t) ⎪ , t ≥ 0. ⎩ vz (t + d) Combining (4.100) with (4.101) yields, ⎧ ⎪ ⎨ Lx(t + d) + vo (t), −d ≤ t < 0    yo (t) = H 0 x(t) ⎪ + vo (t), t ≥ 0. ⎩ 0 L x(t + d)

(4.105)

It is easy to know that vo (t) is a white noise with vo (t), vo (s) = Qvo (t)δts , where ⎧ ⎪ ⎨ −γ 2 Iq , −d ≤ t < 0   Qvo (t) = (4.106) 0 I ⎪ , t ≥ 0. ⎩ p 2 0 −γ Iq Let yo be the collection of the measurements of system (4.93)-(4.94) up to time N , i.e., yo = col {yo (−d), · · · , yo (N )} ,

(4.107)

and yo be the collection of the Krein space measurements of system (4.99)(4.101) up to time N , i.e., yo = col {yo (−d), · · · , yo (N )} .

(4.108)

With the Krein space state-space model of (4.99)-(4.101), we can show that Jd,N of (4.97) can be rewritten as (see also [104]) ⎤ ⎡ ⎤ ⎡ ⎤−1 ⎡ ⎤ x(0) x(0) x(0) x(0) ⎣ e ⎦ , (4.109) Jd,N (x(0), e; yo ) = ⎣ e ⎦ ⎣ e ⎦ , ⎣ e ⎦ yo yo yo yo ⎡

where e = col{e(0), e(1), · · · , e(N + d − 1)},

(4.110)

e = col{e(0), e(1), · · · , e(N + d − 1)},

(4.111)

with e(·) from system (4.93) and e(·) from the Krein space system (4.99).

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4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

From (4.109), it is clear that the H∞ prediction problem is equivalent to that Jd,N has a minimum over {x(0), e} and the predictor is such that the minimum is positive. Now the problem is to derive conditions under which Jd,N (x(0), e; yo ) is minimum over {x(0), e}, and find a predictor such that the minimum of Jd,N (x(0), e; yo ) is positive. Similar to the H∞ fixed-lag smoothing discussed in the last section, we introduce the innovation associated with the measurements yo (t). 

ˆ o (t | t − 1), wo (t) = yo (t) − y

(4.112)

ˆ o (t | t − 1) is the projection of yo (t) onto where y L{yo (−d), · · · , yo (0), · · · , yo (t − 1)}.

(4.113)

For −d ≤ t < 0, the innovation wo (t) is given from (4.105) as wo (t) = ˇ z(t + d | t) − Lˆ x(t + d | t − 1),

(4.114)

ˆ (t + d | t − 1) is the projection of x(t + d) onto L{y(−d), · · · , y(t − 1)}. where x For t ≥ 0, the innovation wo (t) is given from (4.105) as 

H wo (t) = yo (t) − 0

0 L



 ˆ (t | t − 1) x + vo (t), ˆ (t + d | t − 1) x

(4.115)

ˆ (t | t − 1) and x ˆ (t + d | t − 1) are respectively the projections of x(t) and where x x(t + d) onto (4.113). Furthermore, let 

Qwo (t) = wo (t), wo (t).

(4.116)

Qwo (t) is the covariance matrix of the innovation wo (t). Note that Qvo (t) < 0 for −d ≤ t < 0, it is easy to know that Qwo (t) < 0,

− d ≤ t < 0.

(4.117)

With the innovation and covariance matrix, we have the following results [31, 104], Lemma 4.3.1. 1) The H∞ estimator zˇ(t + d | t) that achieves (4.96) exists if and only if Qwo (t) and Qvo (t) for −d ≤ t ≤ N have the same inertia. 0 , if exists, can be given in terms of the innovation 2)The minimum of Jd,N wo (t) as 0 Jd,N =

N  t=−d

wo (t)Q−1 wo (t)wo (t).

(4.118)

4.3 H∞ d-Step Prediction

4.3.2

73

Re-organized Innovation

Similar to the line of arguments for fixed-lag smoothing, we shall apply the re-organized innovation analysis approach to derive the main results. First, the measurements up to time t is denoted by {yo (−d), · · · yo (t)},

(4.119)

which, for t ≥ 0, can be equivalently re-organized as   {yf (s)}ts=0 , zˇ(t + 1 | t − d + 1), · · · , ˇz(t + d | t) , where



y(s) yf (s) = ˇ z(s | s − d) satisfies:

 yf (s) =

 (4.121)

 H x(s) + vf (s), s = 0, 1, · · · , t, L

with



v(s) vf (s) = vz (s)

(4.120)

(4.122)

 (4.123) 

 Ip 0 . 0 −γ 2 Iq It is now clear that the measurements in (4.120) are no longer with time delays. Secondly, we introduce the innovation associated with the reorganized measurements (4.120). Given time instant t and t + 1 ≤ s ≤ t + d, let

a white noise satisfying that vf (s), vf (t) = Qvf δst and Qvf =



w(s) = zˇ(s | s − d) − zˆ(s),

(4.124)

where ˆ z(s) for s > t + 1 is the projection of zˇ(s | s − d) onto the linear space   z(t + 1 | t − d + 1), · · · , zˇ(s − 1 | s − d − 1) , (4.125) L {yf (s)}ts=0 , ˇ and for s = t + 1, ˆ z(s) is the projection of ˇ z(s | s − d) onto the linear space L{yf (0), · · · , yf (t)}.

(4.126)

Similarly, we define that 

ˆ f (s), 0 ≤ s ≤ t, wf (s) = yf (s) − y

(4.127)

ˆ f (s) is the projection of yf (s) onto the linear space where y L{yf (0), · · · , yf (s − 1)}.

(4.128)

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4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

Recall the discussion in Section 2.2, it is easy to know that {wf (0), · · · , wf (t), w(t + 1), · · · , w(t + d)}

(4.129)

is a white noise sequence and spans the same linear space as (4.120) or equivalently (4.119). Different from the innovation wo (s) (−d ≤ s ≤ t), w(s) (0 ≤ s ≤ t) and wf (s) (t + 1 ≤ s ≤ t + d) are termed as reorganized innovations. 4.3.3

Calculation of the Innovation Covariance

In this subsection, we shall calculate the covariance matrix Qwo (t) of the innovation wo (t) using the re-organized innovation. To compute the covariance matrix, we first derive the Riccati equation associated with the reorganized innovation. By considering (4.100) and (4.122), it follows from (4.124)-(4.127) that w(s) = L˜ x(s, 1) + vz (s), t + 1 ≤ s ≤ t + d,   H ˜ (s, 2) + vf (s), 0 ≤ s ≤ t, x wf (s) = L

(4.130) (4.131)

where ˜ (s, 1) = x(s) − x ˆ (s, 1), x ˜ ˆ x(s, 2) = x(s) − x(s, 2).

(4.132) (4.133)

ˆ (s, 1), s > t + 1 is the projection of x(s) onto the linear space In the above, x   L {yf (s)}ts=0 , zˇ(t + 1 | t − d + 1), · · · , ˇz(s − 1 | s − d − 1) (4.134) ˆ (s, 2) (s ≤ t + 1) the projection of x(s) onto the linear space and x L{yf (0), · · · , yf (s − 1)}.

(4.135)

ˆ (s, 1) is the projection of x(t + 1) onto the linear space For s = t + 1, x L{yf (0), · · · , yf (t)}.

(4.136)

˜ (t + 1, 1) = x ˜ (t + 1, 2). Obviously, x Let 

˜ (s, 2), 0 ≤ s ≤ t + 1, P2 (s) = ˜ x(s, 2), x 

˜ (s, 1), t + 1 ≤ s ≤ t + d. P1 (s) = ˜ x(s, 1), x

(4.137) (4.138)

4.3 H∞ d-Step Prediction

75

Then, we have the following Lemma Lemma 4.3.2. Given time instant t, the matrix P2 (s) for 0 ≤ s ≤ t + 1 obeys the following Riccati equation associated with the system (4.99) and (4.122): P2 (s + 1) = ΦP2 (s)Φ − K2 (s)Qwf (s)K2 (s) + Γ Γ  , P2 (0) = P0 , (4.139) where  K2 (s) = ΦP2 (s)

H L



Q−1 wf (s),

(4.140)

while Qwf (s) is given by  Qwf (s) =

    H H I + p P2 (s) 0 L L

 0 . −γ 2 Iq

The matrix P1 (s) for t + 1 ≤ s < t + d obeys the following Riccati equation associated with the system (4.99)-(4.100): P1 (s + 1) = ΦP1 (s)Φ − K1 (s)Qw (s)K1 (s) + Γ Γ  , P1 (t + 1) = P2 (t + 1),

(4.141)

where K1 (s) = ΦP1 (s)L Q−1 w (s),

(4.142)

Qw (s) = LP1 (s)L − γ 2 Iq .

(4.143)

and

Proof: The proof is similar to the case of the optimal H2 estimation discussed in Chapter 2. ∇ It should be noted that the terminal solution of the Riccati equation (4.139) is the initial condition of the Riccati equation (4.141). Denote 

˜ (s + i, 1), i ≥ 0, s ≥ t + 1. R(s, i) = x(s), x

(4.144)

By a similar line of discussions as for the fixed-lag smoothing, it is easy to know that R(s, i) satisfies the following difference equation: R(s, i + 1) = R(s, i)A (s, i), R(s, 0) = P1 (s),

(4.145)

  A(s, i) = Φ In − P1 (s + i)L Q−1 w (s + i)L .

(4.146)

where

Now we are in the position to give the computation of innovation covariance matrix of wo (t).

76

4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

Theorem 4.3.1. For t + 1 ≥ 0, Qwo (t + 1) is given by   HR(t + 1, d)L HP(t)H  + Ip Qws (t + 1) = , LR (t + 1, d)H  P1 (t + d + 1) − γ 2 Iq

(4.147)

where P(t) = P2 (t + 1) −

d 

 R(t + 1, i − 1)L Q−1 w (t + i)LR (t + 1, i − 1),(4.148)

i=1

while R(t + 1, i) is computed by (4.145) and Qw (t + i) = LP1 (t + i)L − γ 2 Iq . Proof: The proof is similar to the case for the H∞ fixed-lag smoothing in the last section. ∇ 4.3.4

H∞ d-Step Predictor

Having presented the Riccati equations and calculated the covariance matrix of innovation in the last subsection, we are now in the position to give the main result of the H∞ d-step prediction. Theorem 4.3.2. Consider the system (4.93)-(4.95) and the associated performance criterion (4.96). Then for a given scalar γ > 0 and d > 0, an H∞ predictor zˇ(t+d | t) that achieves (4.96) exists if and only if, for each t = −1, · · · , N −1,  0 Ip have the same inertia, where Qwo (t+1) Qwo (t+1) and Qvo (t+1) = 0 −γ 2 Iq is calculated by (4.147). In this situation, the central predictor is given by zˇ(t + d | t) = Lˆ x(t + d, 1),

(4.149)

where x ˆ(t + d, 1) is obtained from the Krein space projection of x(t + d) onto the z(t + 1 | t − d + 1), · · · , ˇz(t + d − 1 | t − 1)}, linear space of L{yf (0), · · · , yf (t); ˇ which can be computed recursively as z (s | s − d) − Lˆ x(s, 1)] , x ˆ(s + 1, 1) = Φˆ x(s, 1) + K1 (s) [ˇ s = t + 1, · · · , t + d − 1, xˆ(t + 1, 1) = xˆ(t + 1, 2). (4.150) ˆ(t + 1, 2) in (4.150) In the above, K1 (s) is as in (4.142) and the initial value x is given by the Kalman recursion:     H x ˆ(t + 1, 2) = Φˆ x(t, 2)+K2(t) yf (t)− x ˆ(t, 2) , x ˆ(0, 2) = 0, (4.151) L   y(t) where K2 (t) is as in (4.140) and yf (t) = . zˇ(t | t − d) Proof: The proof is similar to the case for the H∞ fixed-lag smoothing in the last section. ∇

4.4 H∞ Filtering for Systems with Measurement Delay

77

4.4 H∞ Filtering for Systems with Measurement Delay In this section we study the H∞ filtering for measurement delayed systems. By identifying an associated stochastic system in Krein space, the H∞ filtering is shown to be equivalent to the Krein space Kalman filtering with measurement delay. Necessary and sufficient existence conditions of the H∞ filtering are given in terms of the innovation covariance matrix of the identified stochastic system. 4.4.1

Problem Statement

We consider the following linear system for the H∞ filtering problem. x(t + 1) = Φx(t) + Γ e(t), x(0) = x0 , z(t) = Lx(t),

(4.152) (4.153)

where x(t) ∈ Rn , e(t) ∈ Rr , and z(t) ∈ Rq represent the state, input noise, and the signal to be estimated, respectively. Φ, Γ , L are bounded matrices with dimensions of n × n, n × r and q × n, respecively. Assume that the state x(t) is observed by two channels described by y(0) (t) = H(0) x(t) + v(0) (t), y(1) (t) = H(1) x(t − d) + v(1) (t),

(4.154) (4.155)

where H(i) (i = 0, 1) is of dimension pi × n. In (4.154)-(4.155), y(0) (t) is a measurement without delay whereas y(1) (t) ∈ Rp1 is a delayed measurement and v(i) (t) ∈ Rpi , i = 0, 1 are measurement noises. It is assumed that the input noise e is from L2 [0, N ] and measurement noises v(0) and v(1) respectively from L2 [0, N ] and L2 [d, N ], where N > 0 is the time-horizon of the filtering problem under consideration. Let y(t) denote the observation of the system (4.154)-(4.155) at time t and v(t) the related observation noise at time t, then we have ⎧ ⎨ y (t), 0 ≤ t < d, (0) y(t) = (4.156)   ⎩ col y (t), y (t) , t ≥ d (0)

v(t) =

⎧ ⎨v

(1)

0 ≤ t < d,   ⎩ col v (t), v (t) , t ≥ d. (0) (1) (0) (t),

(4.157)

The H∞ filtering problem is stated as: Given a scalar γ > 0 and the observation {y(s), 0 ≤ s ≤ t}, find a filtering estimate zˇ(t | t) of z(t), if it exists, such that the following inequality is satisfied:
N z (t | t) − z(t)] [ˇ z (t | t) − z(t)] t=0 [ˇ < γ 2 , (4.158) sup N −1
N
 P −1 x + x e (t)e(t) + v (t)v(t) (x0 ,u,vd )=0 0 0 0 t=0 t=0 where P0 is a given positive definite matrix which reflects the relative uncertainty of the initial state to the input and measurement noises.

78

4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

4.4.2

An Equivalent Problem in Krein Space

First, in view of (4.158) we define 

Jd,N = x0 P0−1 x0 +

N −1 

N 

t=0

t=0

e (t)e(t)+

v  (t)v(t)−γ −2

N 

vz (t)vz (t), (4.159)

t=d

where vz (t) = zˇ(t | t) − Lx(t).

(4.160)

The H∞ filtering problem is equivalently stated as: 1) Jd,N has a minimum over {x0 , e}, and 2) the filter zˇ(t | t) is such that the minimum is positive. We investigate the above minimization problem by introducing a Krein space model as discussed in the last section. Define the following stochastic system associated with (4.152)-(4.155): x(t + 1) = Φx(t) + Γ e(t), y(0) (t) = H(0) x(t) + v(0) (t), y(1) (t) = H(1) x(t − d) + v(1) (t).

(4.161) (4.162) (4.163)

We also introduce a ‘fictitious’ observation system: ˇ z(t | t) = Lx(t) + vz (t), t ≥ 0,

(4.164)

where zˇ(t | t) ∈ Rq implies the observation of the state x(t) at time t. In the above, the matrices Φ, Γ , H(i) and L are the same as in (4.152)-(4.155). Note that x(·), e(·), yi (·), vi (·), ˇ z(·) and vz (·) , in bold faces, are Krein space elements. Assumption 4.4.1. The initial state x(0) and the noises e(t), v(i) (t) (i = 0, 1) and vz (t) are mutually uncorrelated white noises with zero means and known covariance matrices as P0 , Qe = Ir , Qv(i) = Ipi and Qvz = −γ 2 Iq , respectively. Combining (4.162) with (4.164) yields, ¯ ¯ (t) = Hx(t) ¯ (t), y +v where

(4.165)



 y(0) (t) , ˇ z(t | t)   v(0) (t) ¯ (t) = v , vz (t)   ¯ = H(0) . H L

¯ (t) = y

(4.166) (4.167) (4.168)

¯ is as The covariance of v Qv¯ = diag{Qv(0) , Qvz } = diag{Ip0 , −γ 2 Iq }.

(4.169)

4.4 H∞ Filtering for Systems with Measurement Delay

79

Let y(t) be the measurement at time t for system (4.163) and (4.165), i.e., ⎧ ⎪ ⎨y ¯ (t), 0≤t
⎧ ⎪ ¯ ⎨ Hx(t) ¯ (t), +v 0≤t
where

⎧ ⎪ ⎨v ¯ (t), 0≤t
Note that v(t) is a white noise with v(t), v(s) = Qv (t)δts , where ⎧  Ip0 0 ⎪ ⎪ , 0≤t
(4.171)

(4.172)

(4.173)

Similarly, denote ⎧ ⎪ ⎨ y¯(t), 0≤t
(4.174)

(4.175)

Now let us introduce the innovation sequence associated with observation y(t): 

ˆ (t | t − 1), w(t) = y(t) − y

(4.176)

ˆ (t | t − 1) is the Krein space projection of y(t) onto where y L{y(0), · · · , y(t − 1)}.

(4.177)

For t < d, the innovation w(t) is given from (4.165) as ¯x ¯ (t) − H ˆ (t | t − 1), w(t) = y

(4.178)

80

4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

ˆ (t | t − 1) is the projections of x(t) onto L{¯ ¯ (t − 1)}. For t ≥ d, where x y(0), · · · , y the innovation w(t) is given from (4.171) as    ˆ (t − d | t − 1) H(1) 0 x w(t) = y(t) − , (4.179) ¯ ˆ (t | t − 1) 0 H x ˆ (t | t − 1) and x ˆ (t − d | t − 1) are respectively the projection of x(t) and where x x(t − d) onto (4.177). Denote 

Qw (t) = w(t), w(t).

(4.180)

Qw (t) is the original innovation covariance matrix which plays an important role for checking the existence of an H∞ filter. Now we have the following results [31, 104]. Lemma 4.4.1. 1) An H∞ estimator zˇ(t | t) that achieves (4.158) exists if and only if Qw (t) and Qv (t), 0 ≤ t ≤ N , have the same inertia. 0 , if exists, can be given in terms of the innovation 2)The minimum of Jd,N w(t) as 0 Jd,N =

N 

w (t)Q−1 w (t)w(t).

(4.181)

t=0

In the following, we shall firstly calculate the covariance matrix Qw (t) so that we can check if an H∞ estimator zˇ(t | t) that achieves (4.158) exists. Further, 0 seck an estimator zˇ(t | t), if exists, such that Jd,N of (4.181) is positive. 4.4.3

Re-organized Innovation Sequence

First, the observations up to time t are collected as {y(0), · · · , y(t)}, which span the following linear space L{y(0), · · · y(t)}. For t < d, the linear space (4.182) is   L {¯ y(s)}ts=0 .

(4.182)

(4.183)

For t ≥ d, note the definition of y in (4.170), the linear space (4.182) can be re-organized equivalently as   ¯ (t − d + 1), · · · , y ¯ (t) , (4.184) L {yf (s)}t−d s=0 , y where



 y(1) (s + d) yf (s) = . ¯ (s) y 

(4.185)

4.4 H∞ Filtering for Systems with Measurement Delay

81

¯ (s) and yf (s) in (4.184) are no longer with time Then, the measurements of y delay, where the former satisfies (4.165) while the latter satisfies:  yf (s) =

 H(1) ¯ x(s) + vf (s), s = 0, 1, · · · , t − d, H

(4.186)

where 

v(1) (s + d) vf (s) = ¯ (s) v

 (4.187)

 0 Ip0 +p1 . 0 −γ 2 Iq Thus, (4.161) and (4.186) form a state space representation to which the Kalman filtering in Krein space can be applied. Now we define the innovation associated with the re-organized measurements. ¯ (t − d + 1), · · · , y ¯ (t), the innovation is Associated with the measurements y 

is a white noise with vf (s), vf (j) = Qvf δij and Qvf =



ˆ ¯ ¯ (s), ¯ (s) − y w(s) =y

(4.188)

ˆ ¯ (s) is the projection of y ¯ (s) onto the linear space where y ¯ (t − d + 1), · · · , y ¯ (s − 1)}. L{yf (0), · · · , yf (t − d); y

(4.189)

Associated with measurement yf (s), the innovation is 

ˆ f (s), wf (s) = yf (s) − y

(4.190)

ˆ f (s) is the projection of yf (s) onto the linear space where y L{yf (0), · · · , yf (s − 1)}.

(4.191)

¯ − d + 1), · · · , w(t)} ¯ {wf (0), · · · , wf (t − d), w(t ,

(4.192)

It is easy to know that

is a white noise sequence and spans the same linear space as ¯ (t − d + 1), · · · , y ¯ (t)} L {yf (0), · · · , yf (t − d), y

(4.193)

or equivalently L {y(0), · · · , y(t)}. ¯ Different from the innovation w(t) defined in the last subsection, w(s) and wf (s) are termed as reorganized innovation. By applying the standard Kalman filtering, the re-organized innovation can be calculated.

82

4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

4.4.4

Calculation of the Innovation Covariance Qw (t)

In this subsection, we shall calculate the covariance matrix Qw (t) of the innovation w(t) by using the re-organized innovation defined in the last subsection. Firstly, from (4.161) and (4.186), it follows that ¯x ¯ ˜ (s, 1) + v ¯ (s), t − d + 1 ≤ s ≤ t, w(s) =H   H(1) ˜ (s, 2) + vf (s), 0 ≤ s ≤ t − d, wf (s) = x ¯ H

(4.194) (4.195)

where ˜ (s, 1) = x(s) − x ˆ (s, 1), x ˜ (s, 2) = x(s) − x ˆ (s, 2), x

(4.196) (4.197)

ˆ (s, 1) is the projection of x(s) onto the linear space while x   ¯ (t − d + 1), · · · , y ¯ (s − 1) , L {yf (s)}t−d s=0 , y

(4.198)

ˆ (s, 2) (s = t + 1) is the projection of x(s) onto the linear space and x L{yf (0), · · · , yf (s − 1).

(4.199)

˜ (t − d + 1, 1) = x ˜ (t − d + 1, 2). Obviously, x Then, as in the standard Kalman filtering, we define the one-step ahead prediction error covariance matrices of the state as 

˜ (s, 2), 0 ≤ s ≤ t − d + 1, x(s, 2), x P2 (s) = ˜ 

˜ (s, 1), s ≥ t − d + 1, P1 (s) = ˜ x(s, 1), x

(4.200) (4.201)

˜ (s, 2) and x ˜ (s, 2) are as in (4.196)-(4.197). Then, P1 (s) and P2 (s) are where x computed by the Riccati equation given in the lemma below. Lemma 4.4.2. The matrix P2 (s) (0 ≤ s ≤ t − d + 1) obeys the following Riccati equation associated with the system (4.161) and (4.186): P2 (s + 1) = ΦP2 (s)Φ − K2 (s)Qwf (s)K2 (s) + Γ Γ  , P2 (0) = P0 , (4.202) where



 H(1) K2 (s) = ΦP2 (s) Q−1 wf (s), ¯ H      H(1) H(1) I (s) + p0 +p1 P Qwf (s) = 2 ¯ ¯ H H 0

(4.203)  0 . −γ 2 Iq

(4.204)

The matrix P1 (s) (t − d + 1 ≤ s ≤ t) obeys the following Riccati equation associated with the system (4.161) and (4.165): P1 (s + 1) = ΦP1 (s)Φ − K1 (s)Qw¯ (s)K1 (s) + Γ Γ  , P1 (t − d + 1) = P2 (t − d + 1),

(4.205)

4.4 H∞ Filtering for Systems with Measurement Delay

83

where ¯  Q−1 K1 (s) = ΦP1 (s)H w ¯ (s),  ¯ ¯ Qw¯ (s) = HP1 (s)H + Qv¯ .

(4.206) (4.207)

Proof: The proof is similar to the case of the optimal H2 estimation discussed in Chapter 2. ∇ Now we present the computation for the original innovation covariance matrix Qw (t) in the theorem below. Theorem 4.4.1. 1) For 0 ≤ t < d − 1, Qw (t + 1) of (4.180) is given by ¯ 1 (t + 1)H ¯  + Qv¯ , Qw (t + 1) = HP

(4.208)

where P1 (t + 1) is calculated by (4.205) with initial condition P1 (0) = P0 . 2) For t ≥ d − 1, Qw (t + 1) of (4.180) is given by   ¯  H(1) P(t − d)H(1) + Ip1 H(1) R(t − d + 1, d)H Qw (t + 1) = ¯  (t − d + 1, d)H  ¯ 1 (t + 1)H ¯  + Qv¯ , (4.209) HR HP (1) where P(t − d) = P2 (t − d + 1) −

d 

¯  Q−1 (t − d + i) R(t − d + 1, i − 1)H w ¯

i=1

¯  (t − d + 1, i − 1), ×HR

(4.210)

and R(t − d + 1, d) is calculated as in the following recursion   ¯  Q−1 (s + i)H ¯ , R(s, 0) = P1 (s), R(s, i + 1) = R(s, i)Φ In − P1 (s + i)H w ¯ (4.211) while Qw¯ (·) is as (4.207). In the above, P2 (s) (0 ≤ s ≤ t − d + 1) and P1 (s) (t − d + 1 ≤ s ≤ t) are solutions to the Riccati equations (4.202) and (4.205), respectively. Proof: The proof is similar to the case for the H∞ fixed-lag smoothing in the last section and thus omitted. ∇ 4.4.5

H∞ Filtering

In this section, we shall give our main result on the H∞ filtering based on the discussion in the above. Theorem 4.4.2. Consider the system (4.152)-(4.155) and the associated performance criterion (4.158). Then for a given scalar γ > 0 and d > 0, an H∞ filter zˇ(t+1 | t+1) that achieves (4.158) exists if and only if, for each t = 0, · · · , N −1,

84

4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

Qw (t + 1) and Qv (t + 1) have the same inertia, where Qw (t + 1) is calculated by (4.208) or (4.209). In this situation, a suitable H∞ filter is given by zˇ(t − d + 1 | t + 1) = Lˆ x(t − d + 1 | t + 1, t − d),

(4.212)

where x ˆ(t−d+1 | t+1, t−d) is defined as the Krein space projection of x(t−d+1) onto the linear space ¯ (t − d + 1), · · · , y ¯ (t), y(0) (t + 1)}. L{yf (0), · · · , yf (t − d + 1); y The above projection can be computed by x ˆ(t − d + 1 | t + 1, t − d) = x ˆ(t − d + 1, 2) +

d 

¯ R(t − d + 1, i − 1)H

i=1

×Q−1 w ¯ (t

! ¯ xˆ(t − d + i, 1) + − d + i) y¯(t − d + i) − H

!  R0 (t − d + 1, d)H(0) Q−1 ˆ(t + 1, 1) , w0 (t + 1) y(0) (t + 1) − H(0) x

(4.213)

where Qw0 (t + 1) = H(0) P1 (t + 1)H(0) + Ip0 , (4.214)    −1 ¯ ¯ R0 (t − d + 1, d) = R(t−d+1, d−1)Φ In −P1 (t)H Qw¯ (t)H , (4.215) while R(t − d + 1, ·) is calculated by (4.211). In the above, x ˆ(s + 1, 1) (s > t − d) in (4.213) is calculated recursively as ! ¯x x ˆ(s + 1, 1) = Φˆ x(s, 1) + K1 (s) y¯(s) − H ˆ(s, 1) , x ˆ(t − d + 1, 1) = xˆ(t − d + 1, 2), (4.216) where K1 (s) is as in (4.206) and the initial condition xˆ(t − d + 1, 2) in (4.213) and (4.216) is given by the Kalman recursion: xˆ(t − d + 1, 2)

    H(1) = Φˆ x(t − d, 2) + K2 (t − d) yf (t − d) − x ˆ (t − d, 2) , ¯ H xˆ(0, 2) = 0, (4.217)   y (t − d) where K2 (t − d) is as in (4.203) and yf (t − d) = (1) . y¯(t − d) Proof: The proof is similar to the case of the H∞ fixed-lag smoothing, which is omitted here. ∇ Remark 4.4.1. In this section we have presented the H∞ filtering for single measurement delay systems. The more general case with multiple measurement delays can be dealt with similarly by the same line of discussions as for the single measurement delay case in the above.

4.5 Conclusion

85

4.5 Conclusion In this chapter, we have studied the H∞ fixed-lag smoothing, multiple-step prediction and H∞ filtering for systems with delayed measurement. It has been clearly shown that the above three different problems can be solved in a unified form. That is, by introducing the stochastic systems in an indefinite space (Krein space), the problems are transformed into the H2 estimation with measurement delays, which are then solved by applying the so-called re-organized innovation analysis approach developed in Chapter 2.

5. H∞ Control for Discrete-Time Systems with Multiple Input Delays

In this chapter, we study the H∞ full-information control problem for linear discrete-time systems with multiple input delays. The problem is first written equivalently as an optimization problem associated with an indefinite quadratic form. Then the approach developed for the LQR control for multiple input delayed systems in the last chapter is extended to derive existence conditions and an H∞ controller.

5.1 Introduction We have presented a complete analytical solution to the LQR problem for systems with multiple input delays and the LQG problem for systems with multiple input and output delays in Section 3.2 and Section 3.3, respectively. Note in the LQG control, the statistics of the system noise and measurement noise are assumed to be known. However, this assumption does not generally hold in the real world. In this chapter we shall consider the H∞ full-information control where the input delay system is subject to energy bounded exogenous noise and measurement noise. In the discrete-time context, control problems for systems with input delays have received renewed interests due to applications in areas such as network congestion control and networked control systems; see, e.g. [2, 35, 55, 96]. As in the LQR case, the H∞ control problem may be treated by state augmentation. However, state augmentation can lead to much more expensive computation especially when the system under investigation involves multiple delays and the delays are large. On the other hand, in the state feedback case, the augmentation approach generally leads to a static output feedback control problem which is non-convex [96]. In this chapter, we propose a new approach to the H∞ full-information control problem. By converting the problem into an optimization problem in Krein space, the H∞ control problem is shown to be a dual problem of H2 smoothing for an associated system. The latter can then be solved via an innovation analysis in Krein space. Thus, the duality enables us to address the complicated multiple input delays in a simple manner. Our solvability conditions rely on the existence H. Zhang and L. Xie: Cntrl. and Estim. of Sys. with I/O Delays, LNCIS 355, pp. 87–113, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com


88

5. H∞ Control for Discrete-Time Systems with Multiple Input Delays

of bounded solutions to two Riccati difference equations (RDEs) of the same order as the original plant (ignoring the delays). An explicit controller is given in terms of the solutions of the two RDEs. As the special cases of the fullinformation control problem, solutions to the H∞ control of systems with single input delay and H∞ control with preview are obtained. Note that the H∞ control with preview has been solved in [89] using a different approach. The present solution to the H∞ control of the multiple input delays problem can be viewed as the discrete-time counterpart of [44].

5.2 H∞ Full-Information Control Problem We consider the following discrete linear system for the H∞ control problem. x(t + 1) = Φx(t) +

l 

G(i) ei (t − hi ) +

i=0

⎤ Cx(t) ⎢ D(0) u0 (t − h0 ) ⎥ ⎥ ⎢ ⎥ ⎢ z(t) = ⎢ D(1) u1 (t − h1 ) ⎥ ⎥ ⎢ .. ⎦ ⎣ . D(l) ul (t − hl ) ⎡

l 

B(i) ui (t − hi ),

(5.1)

i=0

(5.2)

where x(t) ∈ Rn , ei (t) ∈ Rri , ui (t) ∈ Rmi , and z(t) ∈ Rq represent the state, the exogenous input, the control input and the controlled signal, respectively. Φ, B(i) , G(i) , C, and D(i) are bounded time-varying matrices. It is assumed that the exogenous inputs are deterministic signals and are from 2 [0, N ] where N is the time-horizon of the control problem under investigation. Without loss of generality, we assume that the delays are in a strictly increasing order: 0 = h0 < h1 < · · · < hl and the control inputs ui , i = 0, 1, · · · , l and the exogenous inputs ei , i = 0, 1, · · · , l respectively have the same dimension, i.e., m0 = m1 = · · · = ml = m and r0 = r1 = · · · = rl = r. We also assume that for t < 0,  D(i) > 0, i = 0, 1, · · · , l. ei (t) = 0, ui (t) = 0, i = 0, 1, · · · , l, and D(i) The H∞ full-information control under investigation is stated as follows: for a given positive scalar γ, find a finite-horizon full-information control strategy ui (t) = Fi (x(t), (ej (τ ), uj (τ )) |0≤j≤l,0≤τ
i = 0, 1, · · · , l

such that sup {x(0),ej (t)|0≤j≤l,0≤t
J(x(0), ej (t), uj (t)) < γ 2

(5.3)

where J(x(0), ej (t), uj (t)) =

z2[0,N ] x (0)Π −1 x(0) + e2[0,N ]

,

(5.4)

5.2 H∞ Full-Information Control Problem

89



e(t) = [e0 (t − h0 ) · · · el (t − hl )] , and Π is a given positive definite matrix which reflects the uncertainty of the initial state relative to the energy of the exogenous inputs. 5.2.1

Preliminaries

In this section, we shall convert the H∞ full-information control problem into an optimization problem in Krein space for an associated stochastic model and derive conditions under which the optimizing solution exists and give an explicit formula to compute the optimizing solution. Considering the performance index (5.3), we define 

∞ = x (0)Π −1 x(0) − γ −2 JN , JN

(5.5)

where JN = z2[0,N ] − γ 2 e2[0,N ] =

−hi l N 

vi (t)R(i) vi (t)

i=0 t=0

+

N 

x (t)Qx(t),

(5.6)

Q = C  C,

(5.7)

t=1

with  D(i) , − γ 2 Ir }, R(i) = diag{D(i)   ui (t) vi (t) = . ei (t)

(5.8)

It is clear that an H∞ controller ui (t) achieves (5.3) if and only if it satisfies ∞ that JN of (5.5) is positive for all non-zero {x(0); ei (t), 0 ≤ t ≤ N −hi , 0 ≤ i ≤ l}. Denote ⎧⎡ ⎤ ⎪ v0 (t − h0 ) ⎪ ⎪ ⎢ ⎥ ⎪ .. ⎪ ⎪ ⎣ ⎦ , hi ≤ t < hi+1 , . ⎪ ⎪ ⎨  v (t − hi ) ⎤ v(t) = ⎡ i (5.9) v ⎪ 0 (t − h0 ) ⎪ ⎪⎢ ⎪ ⎥ .. ⎪ ⎪ ⎦ , t ≥ hl . ⎪⎣ ⎪ ⎩ vl (t − hl ) ⎧ l  ⎪ ⎨ Γ(j) vj (t − hj ), hi ≤ t < hi+1 ,  v˜(t) = j=i+1 (5.10) ⎪ ⎩ 0, t ≥ hl ⎧ ⎨[Γ · · · Γ(i) ] , hi ≤ t < hi+1 (0)  (5.11) Γt = ⎩[Γ · · · Γ(l) ] , t ≥ hl (0) ⎧ ⎨ diag{R , · · · , R }, h ≤ t < h , i i+1  (0) (i) Rt = (5.12) ⎩ diag{R , · · · , R }, t ≥ hl (0)

(l)

90

5. H∞ Control for Discrete-Time Systems with Multiple Input Delays

where Γ(i) = [ B(i)

G(i) ] .

(5.13)

Note that v(t) and v˜(t) above are vectors involving both the control inputs and the exogenous inputs. Using the above notations and taking into account that ui (t) = 0, ei (t) = 0, i = 0, 1, · · · , l for t < 0, system (5.1) can be rewritten as ⎧ ⎨ Φx(t) + Γ v(t) + v˜(t), h ≤ t < h , t i i+1 x(t + 1) = (5.14) ⎩ Φx(t) + Γ v(t), t≥h t

l

and the cost function (5.6) can be rewritten as JN =

N 

v  (t)Rt v(t) +

N 

t=0

x (t)Qx(t).

(5.15)

t=0

Now we define the following backward stochastic state-space model associated with the system (5.14) and the cost (5.15): x(t) = Φ x(t + 1) + u(t), y(t) = Γt x(t + 1) + v(t),

x(N + 1) = 0

(5.16) (5.17)

where t = 0, 1, · · · , N and u(t) and v(t) are white noises of zero means and satisfy u(t), u(s) = Qδt,s and v(t), v(s) = Rt δt,s , respectively. It can be seen that the dimensions of u(t), y(t) and v(t) are as dim{u(t)} = n × 1, ⎧ ⎨ (i + 1)(m + r) × 1, h ≤ t < h , i i+1 (5.18) dim {y(t)} = ⎩ (l + 1)(m + r) × 1, t ≥ h l

and v(t) has the same dimension as y(t). Remark 5.2.1. It should be pointed out that the notations x(t), u(t) and v(t), in bold faces, are completely different from the normal face notations x(t), u(t) and v(t) in (5.1)-(5.3). The former are Krein space elements with zero means and certain covariances while the latter are deterministic. Observe from (5.7) that covariance Rt is indefinite, which is allowed as discussed in Chapter 2. Denote 

y = col{y(0), · · · , y(N )}, 

(5.19) 

x0 = [ x (0) x (1) · · · x (hl ) ] ,

(5.20)



v = col{v(0), · · · , v(N )}, 

(5.21) 

ξ = [ x (0) v˜ (0) · · · v˜ (hl − 1) ] .

(5.22)

We have the following result. 

Lemma 5.2.1. Assume that Ry = y, y is nonsingular. By making use of the stochastic state-space model (5.16)-(5.17), the cost function JN of (5.6) can be rewritten as the following quadratic form:

5.2 H∞ Full-Information Control Problem

JN = ξ  Pξ + (v − v ∗ ) Ry (v − v ∗ ),

91

(5.23)

where ∗

v =

−Ry−1 Ryx(0) x(0)

−

hl 

Ry−1 Ryx(i) v˜(i − 1),

(5.24)

i=1

ˆ 0 , x0 − x ˆ 0 , Ry = y, y, Ryx(i) = y, x(i), P = x0 − x

(5.25)

ˆ 0 is the projection of x0 onto the linear space L{y(0), · · · , y(N )}. and x Proof: The proof is similar to the case of H2 optimal control discussed in Chapter 3. ∇ In Lemma 5.2.1, we require that Ry be invertible. Observe that Ry consists of the covariance matrices of y(t), t = 0, 1, · · · , N of the backward system (5.16)(5.17), the condition for its invertibility follows from standard H∞ estimation arguments of [30] and shall become clear later. On the other hand, from linear estimation theory [30], it is clear that Ry−1 Ryx(i) is the transpose of the gain matrix of the smoothing (or filtering ˆ (i | 0) of the backward system (5.16)-(5.17) which is when i = 0) estimate of x the projection of the state x(i) onto the linear space L{y(0), · · · , y(N )}. This enables us to obtain v ∗ via a smoothing approach, which is given below. 5.2.2

Calculation of v ∗

First, we define the following backward Riccati difference equation (RDE) associated with Krein state-space system (5.16)-(5.17): Pj = Φ Pj+1 Φ + Q − Kj Mj Kj ,

PN +1 = 0,

(5.26)

where Kj = Φ Pj+1 Γj Mj−1 , Mj = Rj +

Γj Pj+1 Γj .

(5.27) (5.28)

Decompose v ∗ of (5.24) in a similar way as (5.21), i.e., v ∗ = col{v ∗ (0), · · · , v ∗ (N )}, and decompose v ∗ (t) as in (5.9), i.e., ⎧⎡ ∗ ⎤ v0 (t − h0 ) ⎪ ⎪ ⎪ ⎢ ⎪ ⎥ .. ⎪ ⎪ ⎣ ⎦ , hi ≤ t < hi+1 , . ⎪ ⎪ ⎨ ∗  v (t − hi ) ⎤ v ∗ (t) = ⎡ ∗i v ⎪ 0 (t − h0 ) ⎪ ⎪ ⎪ ⎥ ⎢ .. ⎪ ⎪ ⎦ , t ≥ hl . ⎣ . ⎪ ⎪ ⎩ vl∗ (t − hl ) Then, the element vi∗ (t) of (5.30) can be computed as follows.

(5.29)

(5.30)

92

5. H∞ Control for Discrete-Time Systems with Multiple Input Delays

Theorem 5.2.1. Consider the system (5.1)-(5.2) and the associated cost (5.6). Suppose the RDE (5.26) has a bounded solution Pj , j = N, N − 1, · · · , 0. Then vi∗ (t) is calculated by vi∗ (t)

i + 1 blocks ' () * = − [0 · · · 0 Im+r ] v ∗ (t + hi ),

(5.31)

where for t < hl , 

v ∗ (t) = − [F0 (t)] x(0) −

t 



[Fk (t)] v˜(k − 1)

k=1 hl 

−

[Sk (t)] v˜(k − 1),

(5.32)

k=t+1

and for t ≥ hl , 

v ∗ (t) = − [F0 (t)] x(0) −

hl 



[Fk (t)] v˜(k − 1),

(5.33)

k=1

while v˜(·) is as in (5.10) and Sk (t) and Fk (t) are calculated by ! Sk (t) = Pk Φt+1,k Γt Mt−1 − Φt,k G(t)Kt , 0 ≤ t ≤ k − 1, Fk (t) = [In − Pk G(k)] Φk,t Kt ,

k ≤ t ≤ N,

(5.34)

with G(k) =

k 

−1  Φj,k Γj−1 Mj−1 Γj−1 Φj,k ,

(5.35)

j=1

and Φj,m = Φj · · · Φm−1 , m ≥ j, Φm,m = I.

(5.36)

In the above, Pj is the solution to the RDE (5.26) and Φj = Φ − Kj Γj ,

(5.37)

Note that for any τ ≥ 0, vi∗ (τ ) is given in terms of the initial state x(0). In order to obtain a control law vi∗ (τ ) that is dependent on the current state x(τ ), we just shift the time interval [0, hl ] to [τ, hl + τ ], and adopt a similar line of discussions as in the above. To this end, rewrite the system (5.1)-(5.2) as ⎧ ⎨ Φx(t + τ ) + Γ v τ (t) + v˜τ (t), h ≤ t < h , t i i+1 x(t + τ + 1) = ⎩ Φx(t + τ ) + Γ v τ (t), t≥h t

l

(5.38)

5.2 H∞ Full-Information Control Problem

where Γt is as in (5.11) and ⎧⎡ ⎤ v0 (t + τ − h0 ) ⎪ ⎪ ⎪ ⎪⎢ ⎥ .. ⎪ ⎪ ⎣ ⎦ , hi ≤ t < hi+1 , . ⎪ ⎪ ⎨  v (t + τ − hi ) ⎤ v τ (t) = ⎡ i v ⎪ 0 (t + τ − h0 ) ⎪ ⎪ ⎪ ⎥ ⎢ .. ⎪ ⎪ ⎦ , t ≥ hl ⎣ . ⎪ ⎪ ⎩ vl (t + τ − hl ) ⎧  ⎨ l  j=i+1 Γ(j) vj (t + τ − hj ), hi ≤ t < hi+1 , v˜τ (t) = ⎩ 0, t≥h.

93

(5.39)

(5.40)

l

(5.41) Similarly, we rewrite the cost function as τ + JN = JN

l τ −1  

vi (t)R(i) vi (t) +

τ 

x (t)Qx(t),

(5.42)

x (t + τ )Qx(t + τ )

(5.43)

i=0 t=0

t=1

where τ JN =

N −τ 

v τ (t) Rt v τ (t) +

t=0

N −τ  t=1

with Rt as given in (5.12). Define the following stochastic state-space model associated with (5.38) and (5.43): xτ (t) = Φ xτ (t + 1) + uτ (t), xτ (N − τ + 1) = x(N + 1), yτ (t) = Γt xτ (t + 1) + vτ (t), t = N − τ, · · · , 0,

(5.44) (5.45)

where Γt is as in (5.11), uτ (t) and vτ (t) are white noises with zero mean and covariance of Q and Rt , respectively. Associated with the system (5.44)-(5.45), define the Riccati equation as τ Φ + Q − Kjτ Mjτ Kjτ , Pjτ = Φ Pj+1

PNτ −τ +1 = PN +1 = 0,

(5.46)

where τ Kjτ = Φ Pj+1 Γj (Mjτ )−1 ,

Mjτ

= Rj +

τ Γj Pj+1 Γj .

(5.47) (5.48)

Remark 5.2.2. When j ≥ hl , it is not difficult to know that Pjτ = Pτ +j , where Pτ +j is the solution to Riccati equation (5.26). Thus, in order to calculate the Riccati equation (5.46), we just need to calculate Pjτ for j = hl − 1, hl − 2, · · · , 0 by (5.46) with the initial condition of Phτl = Pτ +hl .

94

5. H∞ Control for Discrete-Time Systems with Multiple Input Delays

Denote that yτ = col{yτ (0), · · · , yτ (N − τ )}.

(5.49)

With a similar line of discussion as in the above, by completing the square for τ JN , it follows that τ JN = ξ τ  P τ ξ τ + (v τ − v τ ∗ ) Ryτ (v τ − v τ ∗ ) ,

(5.50)

where Ryτ = yτ , yτ , v τ = col{v τ (0), · · · , v τ (N − τ )},

(5.51)

and P τ and ξ τ have a similar definition as in (5.23). Decomposing v τ ∗ in the same form as v τ : v τ ∗ = col{v τ ∗(0), · · · v τ ∗ (N − τ )},

(5.52)

and decompose v τ ∗ (t) as

⎧ ⎡ τ∗ ⎤ v0 (t − h0 ) ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ .. ⎪ ⎪ ⎦ , hi ≤ t < hi+1 , . ⎪⎣ ⎪ ⎨ τ ∗  v (t − hi ) ⎤ v τ ∗ (t) = ⎡ τi ∗ ⎪ v0 (t − h0 ) ⎪ ⎪ ⎪ ⎥ .. ⎪⎢ ⎪ ⎦ , t ≥ hl . ⎣ . ⎪ ⎪ ⎩ τ∗ vl (t − hl )

(5.53)

Similar to the discussion as in the above for the case of τ = 0, it is easy to know that viτ ∗ (t) for t = 0 is given by i + 1 blocks ' () * viτ ∗ (0) = − [0 · · · 0 Im+r ] v τ ∗ (hi ),

(5.54)

where v τ ∗ (hi ) = − [F0τ (hi )] x(τ ) −

hi 

[Fkτ (hi )] v˜τ (k − 1)

k=1

−

hl 



[Skτ (hi )] v˜τ (k − 1).

(5.55)

k=hi +1

In the above, v˜τ (k) is as in (5.40) while Skτ (·) and Fkτ (·) are given by ! Skτ (hi ) = Pkτ (Φτhi +1,k ) Γhi (Mhτi )−1 − (Φτhi ,k ) Gτ (hi )Khτi , Fkτ (hi ) = [In − Pkτ Gτ (k)] Φτk,hi Khτi ,

(5.56)

and Gτ (hi ) =

hi  j=1

τ  (Φτj,hi ) Γj−1 (Mj−1 )−1 Γj−1 Φτj,hi ,

(5.57)

5.2 H∞ Full-Information Control Problem

Φτj,t = Φτj · · · Φτt−1 , t ≥ j,

Φτj,j = I,

95

(5.58)

with Φτj = Φ − Kjτ Γj .

(5.59)

Note that Φτs · · · Φτt = I for t < s. The following theorem shows the relationship between vi∗ (τ ) in Theorem 5.2.1 and viτ ∗ (0) of (5.54). Theorem 5.2.2. Given a scalar γ, suppose the RDE (5.26) has a bounded solution Pt , t = N, N − 1, · · · , 0 and for any τ > 0, the RDE (5.46) admits a bounded solution Pjτ , j = min{hl − 1, N − τ }, · · · , 1, 0. If vi (t) = vi∗ (t) for t = 0, · · · , τ − 1; i = 0, · · · , l, then vi∗ (τ ) ≡ viτ ∗ (0) |vj (t)=vj∗ (t)(

0≤t<τ ; 0≤j≤l)

i + 1 blocks ' () * = − [0 · · · 0 Im+r ] × + hi  [F0τ (hi )] x(τ ) − [Fkτ (hi )] v˜τ ∗ (k − 1) k=1

−

hl 

 [Skτ (hi )]

τ∗

,

v˜ (k − 1) ,

(5.60)

k=hi +1

where v˜τ ∗ (·) is as in (5.40) with vi (·) replaced by vi∗ (·) for i = 0, · · · , l, Fkτ (·) and Skτ (·) are given by (5.56). Proof: The proof is similar to that of Lemma 3.2.5.

∇

The above theorem shows that v ∗ (τ ) is now given in terms of the current state x(τ ). Remark 5.2.3. It is readily known that Pjτ = Pj+τ when j ≥ hl . Note that the latter has been computed from (5.26). Hence, for a given τ > 0 and N −τ +1 ≥ hl , we only need to recursively compute Pjτ , j = hl − 1, hl − 2, · · · , 0 using (5.46) with Phτl = Pτ +hl . For the case when N − τ + 1 < hl , we compute Pjτ using (5.46) with the terminal condition PNτ −τ +1 = PN +1 = 0. The above results give an explicit calculation of the quadratic form (5.23). Note, however, that the exogenous inputs play a contradictory role with the control inputs, namely the former aims to maximize the cost whereas the latter minimize the cost. Since the second term of (5.23) is not definite and v involves both the control inputs and the exogenous inputs, it is not clear from (5.23) that under what conditions a minimax solution exists. Thus, some further simplification of (5.23) is needed which is given below.

96

5. H∞ Control for Discrete-Time Systems with Multiple Input Delays

5.2.3

Maximizing Solution of JN with Respect to Exogenous Inputs

In this subsection, we shall discuss conditions under which a maximizing solution of JN with respect to the exogenous inputs exists. To this end, we recall the Krein space stochastic model (5.16)-(5.17) and decompose the observation y(t) and the noise v(t) as follows: ⎧ ⎨ col{y (t), · · · , y (t)}, h ≤ t < h , 0 i i i+1 (5.61) y(t) = ⎩ col{y0 (t), · · · , yl (t)}, t ≥ hl ⎧ ⎨ col{v (t), · · · , v (t)}, h ≤ t < h , 0 i i i+1 v(t) = (5.62) ⎩ col{v0 (t), · · · , vl (t)}, t ≥ hl where yi (t) ∈ Rm and vi (t) ∈ Rm , i = 0, · · · , l, satisfy  x(t + 1) + vi (t), 0 ≤ t ≤ N − hi , yi (t) = Γ(i)

(5.63)

and vi (t), vi (s) = R(i) δt,s . In view of the input delays of the original system, we re-organize (5.61)-(5.62) as follows: ⎧ ⎨ col{y (t + h ), · · · , y (t + h )}, 0 ≤ t ≤ N − h , 0 0 l l l  ¯ (t) = y ⎩ col{y (t + h ), · · · , y (t + h )}, N − h
0

i

i

i+1

i

(5.64) and

⎧ ⎨ col{v0 (t + h0 ), · · · , vl (t + hl )}, 0 ≤ t ≤ N − hl , ¯ (t) = v ⎩ col{v (t + h ), · · · , v (t + h )}, N − h 0 0 i i i+1 < t ≤ N − hi . 

(5.65) ¯ (t) obeys It is easy to know that for 0 ≤ t ≤ N − hl , y ⎡  ⎤ Γ(0) x(t + h0 + 1) ⎢ ⎥ .. ¯ (t) = ⎣ ¯ (t), y ⎦+v .  Γ(l) x(t + hl + 1) and for N − hi+1 < t ≤ N − hi , ⎡  ⎤ Γ(0) x(t + h0 + 1) ⎢ ⎥ .. ¯ (t) = ⎣ ¯ (t). y ⎦+v .  Γ(i) x(t + hi + 1)

(5.66)

(5.67)

Further, it is not difficult to verify that the linear space generated by ¯ (N )} is the same as the one generated by {y(0), · · · , y(N )}. {¯ y(0), · · · , y

5.2 H∞ Full-Information Control Problem

97

We denote that 

v¯ = col{¯ v (0), · · · , v¯(N )},

(5.68)



¯ = col{¯ ¯ (N )}, y y(0), · · · , y

(5.69)

¯ (·), in bold face, is as in (5.64) and v¯(t), in normal face, is as where y ⎧ ⎨ col{v (t), · · · , v (t)}, 0 ≤ t ≤ N − h , 0 l l  v¯(t) = ⎩ col{v (t), · · · , v (t)}, N − h
i

i+1

i

(5.70) with vi (t), in normal face, as defined in (5.8). Then we have the following result. Lemma 5.2.2. Assuming that Ry is invertible, JN of (5.23) can be rewritten as: v − v¯∗ ) Ry¯ (¯ v − v¯∗ ), JN = ξ  Pξ + (¯

(5.71)

¯  and v¯∗ is obtained from v¯ with vi (t) replaced by vi∗ (t) for y, y where Ry¯ = ¯ i = 0, · · · , l and t = 0, · · · , N − hi . Proof: Recall Lemma 5.2.1, JN = ξ  Pξ + (v − v ∗ ) Ry (v − v ∗ ).

(5.72) ∗ 

∗

Then the proof is straightforward by verifying that (v − v ) Ry (v − v ) is equal v − v¯∗ ). ∇ to (¯ v − v¯∗ ) Ry¯ (¯ ¯ Now we introduce the innovation sequence w(t) associated with the new obser¯ (t) as vation y ˆ ˆ ¯ (N | N + 1) = 0, ¯ (t | t + 1), y ¯ ¯ (t) − y w(t) =y

(5.73)

ˆ ¯ (t | t + 1) is the projection of y ¯ (t) onto the linear space L{¯ y(t + where y ¯ (N )} = L{w(t ¯ + 1), · · · , w(N ¯ )}. From (5.73), it is easy to observe 1), · · · , y ¯ (N )} = F2 × col{w(0), ¯ ¯ )}, col{¯ y(0), · · · , y · · · , w(N where

⎡I ⎢0 F2 = ⎢ ⎣ .. . 0

∆0,1 I .. .

∆0,2 ∆1,2 .. .

0

0

· · · ∆0,N ⎤ · · · ∆1,N ⎥ .. ⎥ ⎦ ··· . ··· I

(5.74)

(5.75)

¯ with ∆i,j denoting entry that is immaterial. Since w(t) is a mutually uncorrelated white noise, it follows that ¯ y, y Ry¯ = ¯ ¯ 0, · · · M ¯N} × F, = F2 × diag{M 2

(5.76)

¯ t = w(t), ¯ ¯ ¯ w(t) is the covariance matrix of the innovation w(t), which where M can be calculated using the lemma below.

98

5. H∞ Control for Discrete-Time Systems with Multiple Input Delays

Lemma 5.2.3. Assume that the RDE (5.46) exists a bounded solution Pjτ for any τ > 0 and j = min{hl − 1, N − τ }, · · · , 1, 0. Then, ¯ τ is given by 1. For τ ≤ N − hl , the covariance matrix M ! ¯ τ = diag{Γ  , · · · , Γ  } P¯τ +1 (i, j) M diag{Γ(0) , · · · , Γ(l) } (0) (l) (l+1)×(l+1) +diag{R(0), · · · , R(l) }, (5.77) where P¯τ (i, j) = P¯τ (j, i), and for i ≥ j, P¯τ (i, j) is given by  P¯τ (i, j) = Phτi (Φτhj ,hi ) I − Gτ (hj )Phτj ;

(5.78)

¯ τ is given by 2. For N − hk+1 < τ ≤ N − hk , the covariance matrix M ! ¯ τ = diag{Γ  , · · · , Γ  } P¯τ +1 (i, j) M diag{Γ(0) , · · · , Γ(k) } (0) (k) (k+1)×(k+1) +diag{R(0) , · · · , R(k) },

(5.79)

where P¯τ (i, j) is calculated by (5.78). ¯ τ −1 . Firstly, we consider Proof: For the convenience of discussion, we calculate M the case of τ − 1 ≤ N − hl . Note that ¯ τ −1 = w(τ ¯ − 1), w(τ ¯ − 1), M

(5.80)

where ˆ ¯ (τ − 1 | τ ) ¯ − 1) = y ¯ (τ − 1) − y w(τ ⎡  ⎤ ⎤ ⎡ ˆ (τ + h0 | τ )} Γ(0) {x(τ + h0 ) − x v0 (τ + h0 − 1) ⎢ ⎥ ⎥ ⎢ .. .. =⎣ ⎦, ⎦+⎣ . .  vl (τ + hl − 1) ˆ (τ + hl | τ )} Γ(l) {x(τ + hl ) − x (5.81) ˆ (τ + hj | τ ) (j = 0, · · · , l) is the projection of x(τ + hj ) onto the linear where x ¯ (N )}. In view of (5.64), it is readily known that the linear space L{¯ y(τ ), · · · , y ¯ (N )} is the same as L{yτ (0), · · · , yτ (N − τ )} or the same space of L{¯ y(τ ), · · · , y as the linear space L{wτ (0), · · · , wτ (N − τ )}, where yτ (t) is as in (5.45) and wτ (t) is the associated innovation given by ˆ τ (t | t + 1), y ˆ τ (N − τ | N + 1 − τ ) = 0, wτ (t) = yτ (t) − y

(5.82)

ˆ τ (t | t + 1) is the projection of yτ (t) onto the linear space L{yτ (t + while y ˆ (τ + hj | τ ) (j = 0, · · · , l) is the projection of x(τ + hj ) 1), · · · , yτ (N − τ )}. Thus x onto the linear space L{wτ (0), · · · , wτ (N − τ )} and is given by hj −1

ˆ (τ + hj | τ ) = x ˆ (τ + hj | τ + hj ) + x

 s=0

x(τ + hj ), wτ (s)(Msτ )−1 wτ (s)

5.2 H∞ Full-Information Control Problem

99

hj −1

ˆ (τ + hj | τ + hj ) + =x



˜ τ (s + 1 | s + 1) x(τ + hj ), x

s=0

˜ τ (s + 1 | s + 1) + vτ (s)] , ×Γs (Msτ )−1 [Γs x

(5.83)

where Msτ , the innovation covariance of wτ (s), is as in (5.48), while ˜ τ (s + 1 | s + 1) = x(τ + s + 1) − x ˆ (τ + s + 1 | τ + s + 1), x ˆ (τ + s + 1 | τ + s + 1) is the projection of x(τ + s + 1) onto the linear and x ¯ + s + 1), · · · , w(N ¯ )}. It follows space L{wτ (s + 1), · · · , wτ (N − τ )} = L{w(τ from (5.83) that ˆ (τ + hj | τ ) x(τ + hj ) − x hj −1

˜ τ (hj | hj ) − =x Γs (Msτ )−1



˜ τ (s + 1 | s + 1) × x(τ + hj ), x

s=0  τ ˜ (s [Γs x

+ 1 | s + 1) + vτ (s)] .

(5.84)

On the other hand, note that τ ˜ τ (s | s) = Φτs x ˜ τ (s + 1 | s + 1) + L u(τ + s) − Φ Ps+1 x Γs (Msτ )−1 vτ (s), (5.85) τ satisfies the RDE (5.46). Note the fact where Φτs is defined in (5.59) and Ps+1 τ τ that u (s) and v (s) are independent of x(τ + t) for s < t, it follows from (5.85) that

˜ τ (hj | hj ) = Phτi (Φτhi −1 ) · · · (Φτhj ) , x(τ + hi ), x ˜ (s + 1 | s + 1) = x(τ + hj ), x τ

˜ (s + 1 | s + 1) = x(τ + hi ), x τ

Phτj (Φτhj −1 ) · · · (Φτs+1 ) , Phτi (Φτhi −1 ) · · · (Φτs+1 ) .

(5.86) (5.87) (5.88)

 ˆ (τ + hi | τ ), x(τ + Therefore, for i ≥ j, the matrix P¯τ (i, j) = x(τ + hi ) − x ˆ (τ + hj | τ ) = x(τ + hi ), x(τ + hj ) − x ˆ (τ + hj | τ ) is calculated by hj ) − x hj −1

P¯τ (i, j) = Phτi (Φτhi −1 ) · · · (Φτhj ) −



Phτi (Φτhi −1 ) · · · (Φτs+1 ) Γs (Msτ )−1

s=0

5 6 ×(Γs ) Phτj (Φτhj −1 ) · · · (Φτs+1 )  = Phτi (Φτhj ,hi ) I − Gτ (hj )Phτj . 

In view of (5.81), (5.77) follows directly from (5.80).

(5.89)

100

5. H∞ Control for Discrete-Time Systems with Multiple Input Delays

The case of N − hi+1 ≤ τ − 1 < N − hi can be proved in a similar way.

∇

¯ With the innovation of w(t), it follows that Theorem 5.2.3. The linear quadratic form JN of (5.23), if exists, can be rewritten as JN = ξ  Pξ +

N 

 ¯ {¯ v (τ ) − v¯∗ (τ )} M v (τ ) − v¯∗ (τ )} , τ {¯

(5.90)

τ =0

where v¯∗ (τ ) is obtained from v¯(τ ) with the replacement of vi (τ ) by vi∗ (τ) of (5.60). Proof: From (5.71) and (5.76), it follows that v − v¯∗ ) Ry¯ (¯ v − v¯∗ ) JN = ξ  Pξ + (¯ ¯ 0, · · · M ¯ N } × F2 (¯ = ξ  Pξ + (¯ v − v¯∗ ) F2 × diag{M v − v¯∗ ) 2 1 N τ −1     ∗ ∗ ¯τ × ∆s,τ × [¯ v (s) − v¯ (s)] + v¯(τ ) − v¯ (τ ) M = ξ Pξ + 1τ −1 

τ =0

∆s,τ

s=0

2 ∗

∗

× [¯ v (s) − v¯ (s)] + v¯(τ ) − v¯ (τ ) .

(5.91)

s=0

Similar to the discussion in [30], we have τ −1

∆s,τ × [¯ v (s) − v¯∗ (s)] + v¯(τ ) − v¯∗ (τ ) = v¯(τ ) − v¯∗ (τ ).

s=0

∇

Thus we derive (5.90) directly from (5.91).

To find conditions for the existence of a maximizing solution of JN with respect to {ei (t), 0 ≤ i ≤ l; 0 ≤ t ≤ N − hi }, we shall separate the control inputs from the disturbance inputs in v. To this end, denote    u ¯(t) v¯a (t) = , (5.92) e¯(t) with

⎧ ⎨ col{u (t), · · · , u (t)}, 0 ≤ t ≤ N − h , 0 l l  u ¯(t) = ⎩ col{u (t), · · · , u (t)}, N − h 0 i i+1 < t ≤ N − hi ⎧ ⎨ col{e0 (t), · · · , el (t)}, 0 ≤ t ≤ N − hl ,  e¯(t) = ⎩ col{e (t), · · · , e (t)}, N − h
i

i+1

(5.93)

i

Lemma 5.2.4. The linear quadratic form JN of (5.90), if exists, can be further rewritten as

5.2 H∞ Full-Information Control Problem

JN = x (0)P0 x(0) +

N 

 ˜ [¯ va (τ ) − v¯a∗ (τ )] M va (τ ) − v¯a∗ (τ )] , τ [¯

101

(5.94)

τ =0

where 

– v¯a∗ (τ ) is obtained from v¯a (τ ) with ei (τ ) and ui (τ ) replaced by e∗i (τ) = [0, Ir ]vi∗ (τ) 

and u∗i (τ ) = [Im , 0]vi∗ (τ ), respectively, while vi∗ (τ ) is given by (5.60). ˜ t is calculated by – The matrix M – For t ≤ N − hl ! ˜ t = Θl P¯t+1 (i, j) M Θ (l+1)×(l+1) l   +diag{D(0) D(0) , · · · , D(l) D(l) ; −γ 2 Ir , · · · , −γ 2 Ir }.

(5.95)

– For N − hi+1 < t ≤ N − hi ! ˜ t = Θ P¯t+1 (i, j) M Θ i (i+1)×(i+1) i   +diag{D(0) D(0) , · · · , D(i) D(i) ; −γ 2 Ir , · · · , −γ 2 Ir }.

(5.96) In the above, Θi = diag{B(0) , · · · , B(i) ; G(0) , · · · , G(i) } and P¯t+1 (i, j) is given by (5.78). Proof: In the light of (5.70), for 0 ≤ t < N − hl , ⎤ ⎡ u0 (t) ⎤ ⎡ v0 (t) ⎢ e0 (t) ⎥ ⎥ ⎥ ⎢ ⎢ . ⎥ v¯(t) = ⎣ ... ⎦ = ⎢ ⎢  ..  ⎥ , ⎣ u (t) ⎦ vl (t) l el (t)

(5.97)

and for N − hi+1 < t ≤ N − hi , ⎤ u0 (t) v0 (t) ⎢ e0 (t) ⎥ ⎥ ⎢ .. ⎥ ⎢ . ⎥ v¯(t) = ⎣ . ⎦ = ⎢ ⎢  ..  ⎥ . ⎣ u (t) ⎦ vi (t) i ei (t) ⎡

⎤

⎡

(5.98)

Using the notations as in (5.92)-(5.93), it is easy to know v¯a = Tt v¯, where v¯ is as (5.68) and v¯a is given by 

va (0), · · · , v¯a (N )}, v¯a = col{¯

(5.99)

102

5. H∞ Control for Discrete-Time Systems with Multiple Input Delays

and for 0 ≤ t ≤ N − hl , Tt is of the form: '⎡

Im ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ .. ⎢ . ⎢ ⎢ 0 ⎢ ⎢ Tt = ⎢ ⎢··· ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ . ⎣ .. 0

2(l+1) blocks () 0 0 0 ··· 0 0 0 ··· 0 Im 0 · · · .. .

0 0 0

0 Im 0

0 0 0

0 0 0 .. .

0

0

0

0

0

···

Im

··· Ir 0 0

··· 0 0 0

··· 0 Ir 0 .. .

··· 0 0 0

··· 0 0 Ir

··· ··· ··· ···

··· 0 0 0

0

0

0

0

0

···

0

Ir

2(i+1) blocks () 0 0 0 ··· 0 0 0 ··· 0 Im 0 · · · .. .

0 0 0

0 0 0 .. .

⎤*

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥, ···⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ .. ⎥ . ⎦

and for N − hi+1 < t ≤ N − hi , '⎡

Im ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ .. ⎢ . ⎢ ⎢ 0 ⎢ ⎢ Tt = ⎢ ⎢··· ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ . ⎣ .. 0

0 0 0

0 Im 0

0

0

0

0

0

···

Im

··· Ir 0 0

··· 0 0 0

··· 0 Ir 0 .. .

··· 0 0 0

··· 0 0 Ir

··· ··· ··· ···

··· 0 0 0

0

0

0

0

0

···

0

⎤*

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥, ···⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ .. ⎥ . ⎦ Ir

while Im and Ir are respectively the unitary matrices with the dimension of m × m and r × r. Note that Tt = Tt−1 , it follows from (5.90) that JN = ξ  Pξ +

N 

˜ τ [¯ [¯ vr (τ ) − v¯r∗ (τ )] M vr (τ ) − v¯r∗ (τ )] ,

(5.100)

τ =0

where ¯ t T −1 . ˜ t = [T  ]−1 M M t t

(5.101)

˜ t . By considering (5.13), for t ≤ N − hl , it follows from Now we calculate M Lemma 5.2.3 that ¯ t Tt−1 ˜ t = [Tt ]−1 M M !  P¯t+1 (i, j) (l+1)×(l+1) Θl,t = Θl,t +diag{R0,t , · · · , Rl,t ; −γ 2 Ir , · · · , −γ 2 Ir },

(5.102)

5.2 H∞ Full-Information Control Problem

103

and for N − hi+1 < t ≤ N − hi , we have !  ˜ t = Θi,t P¯t+1 (i, j) (i+1)×(i+1) Θi,t M +diag{R0,t , · · · , Ri,t ; −γ 2 Ir , · · · , −γ 2 Ir },

(5.103)

where P¯t+1 (i, j) is given by (5.78). Since vi (t) = 0 for t < 0, then (5.94) follows from (5.100), which completes the proof of the lemma. ∇ ˜ t is invertible, we can apply the Next, for the case when the (2, 2)-block of M ˜ LDU factorization to the covariance matrix Mt :     ˜ 1,1 (t) M ˜ 1,2 (t) ˜ −1 (t) ˜ 1,2 (t)M M I M 2,2 ˜ Mt ≡ ˜ ˜ 2,2 (t) = 0 I M2,1 (t) M    I 0 ∆(t) 0 (5.104) × ˜ −1 (t)M ˜ 2,1 (t) I , ˜ 2,2 (t) M 0 M 2,2 where ˜ 1,1 (t) − M ˜ 1,2 (t)M ˜ −1 (t)M ˜ 2,1 (t). ∆(t) = M 2,2

(5.105)

Then we have the following result. Lemma 5.2.5. Consider the system (5.1)-(5.2) and the performance criterion (5.3). For a given scalar γ, suppose the RDE (5.26) has a bounded solution Pt , t = N, N − 1, · · · , 0 and for any τ > 0, the RDE (5.46) admits a bounded solution Pjτ , j = min{hl − 1, N − τ }, · · · , 1, 0. Then a maximizing solution of JN with respect to {ei (t), 0 ≤ i ≤ l; 0 ≤ t ≤ N − hi } exists if and only if ˜ 2,2 (t) < 0, M

(5.106)

˜ 2,2 (t) is the (2, 2)-block of M ˜ t which is given by (5.95) for t ≤ N − hl where M or (5.96) for N − hi+1 < t ≤ N − hi . In this case, the maximum value of JN with respect to the exogenous inputs is given by JN,max = x (0)P0 x(0) +

N 

[¯ u(τ ) − u ¯∗ (τ )] ∆(τ )[¯ u(τ ) − u ¯∗ (τ )]. (5.107)

τ =0

Proof: By applying the LDU factorization, JN of (5.94) can be further written as JN = x (0)P0 x(0) +

N 

[¯ u(τ ) − u ¯∗ (τ )] ∆(τ )[¯ u(τ ) − u ¯∗ (τ )] +

τ =0 N 

˜ 2,2 (t)[¯ [¯ e(τ ) − e˜∗ (τ )]T M e(τ ) − e˜∗ (τ )],

(5.108)

t=0

where



    u ¯(τ ) − u ¯∗ (τ ) I 0 u¯(τ ) − u ¯∗ (τ ) = ˜ −1 , ˜ 2,1 (t) I M2,2 (t)M e¯(τ ) − e˜∗ (τ ) e¯(τ ) − e¯∗ (τ )

(5.109)

104

5. H∞ Control for Discrete-Time Systems with Multiple Input Delays

and u¯∗ (τ ) is obtained from u ¯(τ ) with ui (τ ) replaced by u∗i (τ ) for i = 0, · · · , l, ∗ and e¯ (τ ) is obtained from e¯(τ ) with ei (τ ) replaced by e∗i (τ ) for i = 0, · · · , l. It is then obvious that JN exists a maximizing solution with respect to {ei (t), 0 ≤ i ≤ l; 0 ≤ t ≤ N − hi }, if and only if ˜ 2,2 (t) < 0. M

(5.110) ∇

In this case the maximum value of JN is given by (5.107). 5.2.4

Main Results

In this section we shall present a solution to the H∞ full-information control. Theorem 5.2.4. Consider the system (5.1)-(5.2) and the performance (5.3). For the given γ > 0, suppose the RDE (5.26) has a bounded solution Pt , t = N, N − 1, · · · , 1, 0 and for any τ > 0, the RDE (5.46) admits a bounded solution Pjτ , j = min{hl , N − τ + 1}, · · · , 1, 0. Then an H∞ controller that solves the H∞ full-information control problem exists if and only if Π −1 − γ −2 P0 > 0, ˜ 2,2 (t) < 0, M

(5.111) (5.112)

˜ 2,2 (t) is the (2, 2)-block of where P0 is the terminal value of Pt of (5.26), and M ˜ Mt which is given by (5.95) for t ≤ N − hl or (5.96) for N − hi+1 < t ≤ N − hi . In this case, a suitable H∞ controller (central controller) u∗i (τ ) is given by u∗i (τ ) = [Im , 0]vi∗ (τ ),

(5.113)

where vi∗ (τ ), i = 0, 1, · · · , l, is calculated by vi∗ (τ )

i + 1 blocks () * ' = [0 · · · 0 Im+r ] × 1 

− [F0τ (hi )] x(τ ) −

hi  k=1

−

hl 



[Fkτ (hi )] v˜τ ∗ (k − 1)



2

[Skτ (hi )] v˜τ ∗ (k − 1) ,

(5.114)

k=hi +1

while Skτ (·) and Fkτ (·) are given in (5.56), and 

v˜τ ∗ (t) =

⎧ l  ⎪ ⎨ Γ(j) vj∗ (t + τ − hj ), hi ≤ t < hi+1 , i = 0, 1, · · · , l − 1 j=i+1

⎪ ⎩ 0,

t ≥ hl . (5.115)

5.2 H∞ Full-Information Control Problem

105

Proof: Substituting (5.94) into (5.5) and using (5.104) yields ∞ JN = x (0)Π −1 x(0) − γ −2 JN

= x (0)[Π −1 − γ −2 P0 ]x(0) −γ −2

N 

[¯ u(τ ) − u¯∗ (τ )]T ∆(τ )[¯ u(τ ) − u¯∗ (τ )]

τ =0

−γ −2

N 

˜ 2,2 (t)[¯ [¯ e(τ ) − e˜∗ (τ )]T M e(τ ) − e˜∗ (τ )],

(5.116)

t=0

where

  =

u¯(τ ) − u ¯∗ (τ ) e¯(τ ) − e˜∗ (τ )



I ˜ −1 (t)M ˜ 2,1 (t) M 2,2

0 I



u ¯(τ ) − u¯∗ (τ ) e¯(τ ) − e¯∗ (τ )

 (5.117)

while u¯∗ (τ ) is obtained from u¯(τ ) with ui (τ ) replaced by u∗i (τ ) for i = 0, · · · , l, and e¯∗ (τ ) is obtained from e¯(τ ) with ei (τ ) replaced by e∗i (τ ) for i = 0, · · · , l. Recall the discussion in [30] (Theorem 9.5.1), an H∞ control input u(τ ) that ∞ > 0 exists if and only if achieves JN Π −1 − γ −2 P0 > 0, ˜ 2,2 (t) < 0. M

(5.118) (5.119)

In view of (5.116), the suitable controller can be chosen such that u ¯(τ ) = u ¯∗ (τ ).

(5.120)

Therefore, the controller is u∗i (τ ), which is given by (5.113) and (5.114).

∇

Remark 5.2.4. From Theorem 5.2.4, the solution to the H∞ control of systems with multiple input delays requires solving a standard RDE (5.26). In addition, for every τ > 0, the RDE (5.46) is to be computed for Pjτ , j = min{hl − 1, N − τ }, · · · , 1, 0. Both the RDEs (5.26) and (5.46) have the same order as the original system (ignoring the delays). The result shares some similarity with the H∞ fixed-lag smoothing in [37, 111, 106] where an H∞ -type of RDE together with some Riccati recursions related to the length of the lag are to be solved. The solution has clear computational advantage over the traditional approach of state augmentation even if the latter can be applied to address our problem as the latter usually leads to higher system dimension. It is worthy pointing out that it is not clear at this stage if the existence of bounded solution of (5.46) is also necessary for the solvability of the H∞ control problem. Remark 5.2.5. In the delay free case, i.e., h1 = · · · = hl = 0, it is obvious that the H∞ controller (5.113)-(5.114) reduces to: u∗0 (τ ) = [Im , 0]v0∗ (τ ) = −[Im , 0] × [F0τ (0)] x(τ ) = −[Im , 0] × Kτ x(τ ),

(5.121)

106

5. H∞ Control for Discrete-Time Systems with Multiple Input Delays

where Kτ is as in (5.27). And the existence condition becomes Π −1 − γ −2 P0 > 0, M2,2 (t) < 0,

(5.122) (5.123)

where M2,2 (t) is the (2, 2)-block of Mt which is given by (5.28). Thus the obtained result is the same as the well-known full-information control solution [30].

5.3 H∞ Control for Systems with Preview and Single Input Delay 5.3.1

H∞ Control with Single Input Delay

The linear single input delay system is described by x(t + 1) = Φx(t) + Ge(t) + Bu(t − h) ⎡ ⎤ Cx(t) ⎦. z(t) = ⎣ Du(t − h)

(5.124) (5.125)

The H∞ control with single input delay is stated as follows: for a given positive scalar γ, find a finite-horizon full-information control strategy u(t) = F (x(t), u(τ ) |t−h≤τ
sup

 −1 x(0) + e2 {x(0),e(t)|0≤t
< γ2

(5.126)

where Π is a given positive definite matrix which reflects the uncertainty of the initial state relative to the energy of the exogenous inputs. Similar to the discussion in the last section, associated with the system (5.124)-(5.125) and the cost function (5.126), we introduce the following notations: 











R(1) = D D and Q = C  C, v0 (t) = e(t), v1 (t) = u(t), Γ(0) = G, Γ(1) = B, and ⎧ ⎪ ⎨ v0 (t + τ ), 0 ≤ t < h,    τ v (t) = (5.127) v0 (t + τ ) ⎪ ,t≥h ⎩ v1 (t + τ − h) ⎧ ⎨ Bv (t + τ − h), 0 ≤ t < h, 1  (5.128) v˜τ (t) = ⎩ 0, t≥h ⎧ ⎨Γ , 0≤t
(1)

5.3 H∞ Control for Systems with Preview and Single Input Delay

107

⎧ ⎨ −γ 2 I , 0 ≤ t < h, r  Rt = ⎩ diag{−γ 2I , R }, t ≥ h. r (1) (5.130) Define the following Riccati equation: Pj = Φ Pj+1 Φ + Q − Kj Mj Kj ,

j = N, N − 1, · · · , 0,

(5.131)

where PN +1 = 0 and Kj = Φ Pj+1 Γj Mj−1 , Mj = Rj +

(5.132)

Γj Pj+1 Γj .

Also, define the following Riccati equation:   τ Pjτ = Φ Pj+1 Φ + Q − Kjτ Mjτ Kjτ ,

(5.133)

j = h − 1, · · · , 0,

(5.134)

where Phτ = Pτ +h which is computed by (5.131) and τ Kjτ = Φ Pj+1 Γj (Mjτ )−1 , τ τ Γj . Mj = Rj + Γj Pj+1

(5.135) (5.136)

Remark 5.3.1. When N + 1 < τ + h ≤ N + h, i.e., N + 1 − h < τ ≤ N , the Riccati equation (5.134) becomes   τ Pjτ = Φ Pj+1 Φ + Q − Kjτ Mjτ Kjτ ,

j = N − τ, · · · , 0,

(5.137)

with the initial value PNτ −τ +1 = PN +1 = 0. Following a similar discussion as in the last section, we have Theorem 5.3.1. Consider the system (5.124)-(5.125). Given a scalar γ > 0, suppose the RDE (5.131) has a bounded solution Pt , t = N, N − 1, · · · , 0 and for any τ > 0, the RDE (5.134) for τ ≤ N + 1 − h or (5.137) for N + 1 − h < τ ≤ N admits a bounded solution Pjτ , j = min{h − 1, N − τ }, · · · , 0. Then, there exists an H∞ controller that solves the H∞ control problem if and only if Π −1 − γ −2 P0 > 0, ¯ 1,1 (t) < 0, M

(5.138) (5.139)

¯ 1,1 (t) is where P0 is the terminal value of Pτ of (5.131), and for t ≤ N − h, M ¯ the (1,1)-block of matrix Mt which is given by ! ¯ t = diag{G , B  } P¯t+1 (i, j) M diag{G, B} 2×2 +diag{−γ 2Ir , R(1) }

(5.140)

108

5. H∞ Control for Discrete-Time Systems with Multiple Input Delays

and for N − h < t ≤ N , ¯ 1,1 (t) = M ¯ t = G P¯t+1 (0, 0)G − γ 2 Ir , M (5.141) where P¯t+1 (i, j) is given by P¯τ (0, 0) = P0τ , P¯τ (1, 0) = P¯τ (0, 1) = Phτ (Φτ0,h ) , P¯τ (1, 1) = Phτ [I − Gτ (h)Phτ ] .

(5.142) (5.143) (5.144)

In this situation, a suitable H∞ controller is given by + , h    ∗ τ τ ∗ u (τ ) = −[0, Im ] × [F0 (h)] x(τ ) + [Fk (h)] Gu (τ + k − h − 1) , k=1

(5.145) where Fkτ (h) is as Fkτ (h) = [In − Pkτ Gτ (k)] Φτk,h Khτ ,

(5.146)

and Gτ (h) =

h 

τ  (Φτj,h ) Γj−1 (Mj−1 )−1 Γj−1 Φτj,h ,

(5.147)

Φτj,t = Φτj · · · Φτt−1 , t ≥ j,

(5.148)

j=1

Φτj,j = I,

while Φτs · · · Φτt = I for t < s, and Φτj = Φ − Kjτ Γj . 5.3.2

(5.149)

H∞ Control with Preview

We consider the following system for the H∞ control with preview: x(t + 1) = Φx(t) + Ge(t − h) + Bu(t), ⎤ ⎡ Cx(t) ⎦. z(t) = ⎣ Du(t)

(5.150) (5.151)

The H∞ control with preview is stated as follows: for a given positive scalar γ, find a finite-horizon full-information control strategy u(t) = F (x(t), e(τ ) |t−h≤τ
5.3 H∞ Control for Systems with Preview and Single Input Delay

109

such that z2[0,N ]

sup {x(0),e(t)|0≤t
x (0)Π −1 x(0) + e2[0,N −h]

< γ2,

(5.152)

where Π is a given positive definite matrix which reflects the uncertainty of the initial state relative to the energy of the exogenous inputs. Note that infinite horizon case of the above problem has been addressed in [89]. Similar to the discussion as in the last section, associated with the system (5.150)-(5.151) and the cost function (5.152), we introduce the following notations, ⎧ ⎪ ⎨ u(t + τ ), 0 ≤ t < h,    τ v (t) = (5.153) u(t + τ ) ⎪ ,t≥h ⎩ e(t + τ − h) ⎧ ⎨ Ge(t + τ − h), 0 ≤ t < h,  (5.154) v˜τ (t) = ⎩ 0, t≥h ⎧ ⎨ B, 0≤t
1

We introduce the following Riccati equation: Pj = Φ Pj+1 Φ + Q − Kj Mj Kj ,

j = N, N − 1, · · · , 0,

(5.157)

where PN +1 = 0 and Kj = Φ Pj+1 Γj Mj−1 , Mj = Rj +

(5.158)

Γj Pj+1 Γj .

Also, define the following Riccati equation:   τ Φ + Q − Kjτ Mjτ Kjτ , Pjτ = Φ Pj+1

(5.159)

j = h − 1, · · · , 0,

(5.160)

where Phτ = Pτ +h which is computed by (5.157), and τ Γj (Mjτ )−1 , Kjτ = Φ Pj+1

Mjτ

= Rj +

τ Γj Pj+1 Γj .

(5.161) (5.162)

110

5. H∞ Control for Discrete-Time Systems with Multiple Input Delays

Remark 5.3.2. When N + 1 < τ + h ≤ N + h, i.e., N + 1 − h < τ ≤ N , the Riccati equation (5.160) becomes   τ Φ + Q − Kjτ Mjτ Kjτ , j = N − τ, · · · , 0, (5.163) Pjτ = Φ Pj+1 with the initial condition PNτ −τ +1 = PN +1 = 0. Then, the following result follows. Theorem 5.3.2. Consider the system (5.150)-(5.151). For a given scalar γ > 0, suppose the RDE (5.157) has a bounded solution Pt , t = N, · · · , 0 and for any τ > 0, the RDE (5.160) for τ ≤ N + 1 − h or (5.163) for N + 1 − h < τ ≤ N admits a bounded solution Pjτ , j = min{h − 1, N − τ }, · · · , 0. Then there exists an H∞ controller solving the H∞ control problem with preview if and only if Π −1 − γ −2 P0 > 0, ¯ 2,2 (t) < 0, M

(5.164)

0≤t≤N −h (5.165) ¯ where P0 is the terminal value of Pτ of (5.157), and M2,2 (t) is the (2,2)-block ¯ t which, for t ≤ N − h, is given by of matrix M ! ¯ t = diag{B  , G } P¯t+1 (i, j) M diag{B, G} 2×2



+diag{D D, −γ I}, ¯ where Pt+1 (i, j) is computed by P¯τ (0, 0) = P τ , 2

0

P¯τ (1, 0) = P¯τ (0, 1) = Phτ (Φτ0,h ) , P¯τ (1, 1) = Phτ [I − Gτ (h)Phτ ] .

(5.166)

(5.167) (5.168) (5.169)

In this situation, a suitable H∞ controller is given by + , h    u∗0 (τ ) = − [F0τ (0)] x(τ ) + [Skτ (0)] Ge∗ (τ + k − h − 1) , k=1

(5.170) where

F0τ (0)

=

K0τ

Skτ (h)

and

! = Pkτ (Φτh+1,k ) Γh (Mhτ )−1 − (Φτh,k ) Gτ (h)Khτ ,

(5.171)

with Gτ (h) =

h 

τ  (Φτj,h ) Γj−1 (Mj−1 )−1 Γj−1 Φτj,h ,

(5.172)

j=1

Φτj,j = I,

Φτj,t = Φτj · · · Φτt−1 , t ≥ j

(5.173)

and Φτj = Φ − Kjτ Γj . Note that

Φτs

· · · Φτt

= I for t < s.

(5.174)

5.4 An Example

111

5.4 An Example Consider the mathematical model of congestion control taken from [1]: qt+1 = qt +

d 

v¯i,t−hi − µt

(5.175)

i=1

µt = µ + ξt τ  ξt = li ξt−i + ηt−1

(5.176) (5.177)

i=1

where qt is the queue length at the bottleneck, ξt is the higher priority source (interference) which is modelled as a stable auto-regressive (AR) process as in (5.177) with ηt the noise input. µt denotes the effective service rate available for the traffic of the given source, µ is the constant nominal service rate, and v¯i,t−hi is the input rate of the i-th source. hi is the round trip delay of the i-th source consisting of two path delays, one is the return path (from the switch to the source) delay and the other is forward path (from the source through the congested switch) delay. We note that round trip delay in transmission is one reason for the disagreement between the switch input and output. For a prescribed γ > 0, the H∞ congestion control problem is to find source rate v¯ such that sup {q0 ,¯ vi,t−hi |0≤i≤d,0≤k≤N −hi }

J(q0 , v¯i,t−hi , µt ) < γ 2 ,

(5.178)

for any non-zero η ∈ 2 [0, N ], where J(q0 , v¯i,t−hi , µt )   N d   λ2 (¯ vi,t−hi − ai µ)2 (qt − q¯)2 + =

t=1

i=1 N  t=1

(5.179) ηt2

with q¯ the target queue length and λ a weighting factor. ai satisfying

d 

ai = 1 is

i=1

the weight for different source rates which determines the allocation of bandwidth for each channel. The objective is to make the queue buffer close to the desired level while the difference between the source rate and the nominal service rate should not be too large. We adopt a second order AR model for the higher priority source, i.e. τ = 2 [82]. Denote ⎛ ⎞ qt − q¯ x(t) = ⎝ ξt−1 ⎠ , ξt

112

5. H∞ Control for Discrete-Time Systems with Multiple Input Delays

ui (t − hi ) = v¯i,t−hi − ai µ, ⎛ q − q¯ ⎞ t

⎜ λu1 (t − h1 ) ⎟ ⎟, z(t) = ⎜ .. ⎝ ⎠ . λud (t − hd ) e(t) = ηt .

The system (5.175)-(5.177) can be put into the form of (5.1)-(5.2) with ⎞ ⎛ ⎛ ⎞ 1 0 −1 0 Φt = ⎝ 0 0 1 ⎠ , G0,t = ⎝ 0 ⎠ , 0 l2 l1 1 Gi,t = 0 (i > 0), B0,t = 0, ⎛ ⎞ 1 Bi,t = ⎝ 0 ⎠ (i = 1, · · · , d), Ct = 1, 0 and the cost function (5.6) with ⎛ ⎞ 1 0 0  Di,t = λ2 . Qt = ⎝ 0 0 0 ⎠ , Di,t 0 0 0 The initial state is assumed to be known. We adopt the similar parameters as given by [82]: The buffer length ymax = 10000 cells/s The buffer set point q¯ = (1/2)ymax = 5000 cells The controller cycle time T = 1 ms. Assume that there are 10 sources with round trip delays from 1 to 10, respectively, i.e., d = 10, hi = i, i = 1, 2, · · · , 10. We also assume that l1 = l2 = 0.4 and the Gaussian white noise process ηt ’s variance is equal to 1 [82]. The time horizon N is 100 and we set γ = 15. The weighting between the queue length and the transmission rate is λ = 1 and ai = 1/d(i = 1, 2, · · · , d) are the source sharing. By applying Theorem 5.2.4, we design an H∞ controller. Simulation result for the designed controller is shown in Figure 5.1 where the vertical axis is the queue length qt . The initial queue length of the congested switch is set to be 5100. From the graph we can see that the queue length quickly converges to the target queue length. As mentioned in [1], the congestion control problem considered in this example can be studied with the state augmentation and standard H∞ control theory. In what follows, we shall show that the augmentation approach will lead to more expensive computation than the one of the presented approach, especially when the delays are large. In fact, if we apply the state augmentation approach to the problem, a RDE with dimension of hd + τ + 1 [1] will be solved where hd is the maximum delay in (5.175) and τ is the order of the AR process of (5.176), and thus the operation number (the total number of multiplication and division) can

5.5 Conclusion

113

be roughly estimated as of O((hd + τ + 1)3 ) (note that the computation cost of controller design is mainly boiled down to the solving of the RDE) for each iteration step. While if we apply the presented approach in this paper, we require to solve the two RDEs with dimension of τ + 1, in which the computation cost can be roughly computed as O((τ + 1)3 + hd (τ + 1)3 ). It is easy to know that (hd +τ +1)3 >> (1+hd )(τ +1)3 when the delay hd is large enough. In the example, we have hd = 10, τ = 2 and thus (hd +τ +1)3 = 2197 where (1+hd)(τ +1)3 = 297. In other words, the augmentation approach requires a much larger operation number (the sum of multiplications and divisions) than the presented approach. On the other hand, it should be noted that the augmentation approach generally leads to a static output feedback control problem of non-convex [96]. 5120

5100

5080

5060

5040

qt 5020

5000

4980

4960

4940

0

10

20

30

40

50

60

70

80

90

100

t (sample time)

Fig. 5.1. Queue length response

5.5 Conclusion We have presented a simple solution to the H∞ control of systems with multiple input delays. We solved the problem by converting it to an optimization associated with a stochastic system in Krein space and thus establishing a duality between the H∞ full-information control and an H∞ smoothing problem. The duality enables us to address the full-information control problem via a simple approach. Our solvability condition and the construction of the controller are given in terms of the solutions of two Riccati difference equations of the same order as the original plant, which is advantageous over methods such as system augmentation. Our result assumes the existence of bounded solutions of two Riccati difference equations whose necessity deserves further studies. While this chapter only deals with systems with input delays via state feedback, the presented approach together with the existing studies on H∞ estimation for systems with delayed measurements [109] would make it possible to solve the H∞ control of systems with multiple i/o delays via dynamic output feedback.

6. Linear Estimation for Continuous-Time Systems with Measurement Delays

In this chapter, we shall study the optimal estimation problems for continuoustime systems with measurement delays under both the minimum variance and H∞ performance criteria. We also investigate a related H∞ fixed-lag smoothing problem. The key technique that will be applied in this chapter is the reorganized innovation analysis approach in both Hilbert spaces and Krein spaces.

6.1 Introduction This chapter is concerned with the H2 and H∞ estimation problems for continuous-time systems with measurement delays. This problem may be approached by a partial differential equation approach or an infinite dimensional system theory approach. However, these approaches lead to solutions either in terms of partial Riccati differential equations or operator Riccati equations which are difficult to understand and implement in practice [4, 13, 12, 52, 54]. In order to obtain an effective approach to the problems, some attempts have been made. In [72], the H∞ estimation for systems with measurement delay is studied and the derived filter is given in terms of one H∞ Riccati equation and one H2 Riccati equation. Notice that only single measurement delay was considered in [72]. Our aim here is to present a simple approach for the problems in more general case without involving Riccati Partial Differential Equation (PDE) or operator Riccati equation. First, we study the linear minimum mean square error (H2 ) estimation for systems with multiple delayed measurements. By applying the re-organized innovation analysis, an estimator is derived in a simple manner. The estimator is designed by performing two standard differential Riccati equations. Secondly, we shall investigate the H∞ filtering problem for systems with measurement delays. By introducing an appropriate dual stochastic system in Krein space, it will be shown that the calculation of the central H∞ estimator is the same as that of the H2 estimator associated with the dual system. Thirdly, we shall study the complicated H∞ fixed-lag smoothing problem which has been an open problem for many years. By defining a dual stochastic system H. Zhang and L. Xie: Cntrl. and Estim. of Sys. with I/O Delays, LNCIS 355, pp. 115–141, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com


116

6. Linear Estimation for Continuous-Time Systems with Measurement Delays

in Krein space, the H∞ fixed-lag smoothing problem is converted into an H2 estimation problem for an associated system with instantaneous and delayed measurements. The existence condition and estimator are derived by using the re-organized innovation analysis.

6.2 Linear Minimum Mean Square Error Estimation for Measurement Delayed Systems In this section we study the optimal estimation problem for systems with multiple measurement delays. The standard Kalman filtering formulation is not applicable to such problem due to the delays in measurements. We shall extend the reorganized innovation analysis developed in the previous chapters to continuoustime systems. 6.2.1

Problem Statement

We consider the linear time-invariant system described by ˙ x(t) = Φx(t) + Γ e(t),

(6.1)

where x(t) ∈ Rn is the state, and e(t) ∈ Rr is the system noise. The state is observed by l + 1 channels with delays given by y(i) (t) = H(i) x(t − di ) + v(i) (t), i = 0, 1, · · · , l,

(6.2)

where y(i) (t) ∈ Rpi is the delayed measurement, v(i) (t) ∈ Rpi is the measurement noise. The initial state x(0), and noises e(t) and v(i) (t), i = 0, 1, · · · , l, are uncorrelated white noises with zero means and covariance matrices as E [x(0)x (0)] =  x(0), x(0) = P0 , E [e(t)e (s)] = Qe δt−s , and E v(i) (t)v(i) (s) = Qv(i) δt−s ,

respectively, where E(·) denotes the expectation. Without loss of generality, the time delays di are assumed to be in a strictly increasing order: 0 = d0 < d1 < · · · < dl . In (6.2), y(i) (t) means the observation of the state x(t − di ) at time t, with delay di . Let y(t) be the observation of the system (6.2) at time t, then we have ⎤ ⎡ y(0) (t) ⎥ ⎢ .. (6.3) y(t) = ⎣ ⎦ , di−1 ≤ t < di , . y(i−1) (t) and for t ≥ dl ,

⎡

⎤ y(0) (t) ⎢ ⎥ y(t) = ⎣ ... ⎦ . y(l) (t)

(6.4)

The problem to be studied is stated as: Given the observation {y(s)|0≤s≤t }, ˆ (t | t) of x(t). find a linear least mean square error estimator x

6.2 Linear Minimum Mean Square Error Estimation

117

Remark 6.2.1. For the convenience of discussion, we shall assume that the time instant t ≥ dl throughout the section. When the measurement (6.2) is delay free, i.e., di = 0 for 1 ≤ i ≤ l, the above problem can be solved by using the standard Kalman filtering formulation. However, in the case when the system (6.2) is of measurement delays, the standard Kalman filtering is not applicable. One possible way to such problem is the PDE approach [13, 52], which however leads to difficult computation. Our aim is to present an analytical solution in terms of Riccati Differential Equations (RDEs). The key is the re-organization of the measurement and innovation sequences. Throughout the section we denote that 

ti = t − di , i = 0, 1, · · · , l. 6.2.2

Re-organized Measurement Sequence

In this subsection, the instantaneous and l-delayed measurements will be reorganized as delay free measurements so that the Kalman filtering is applicable. As is well known, given the measurement sequence {y(s)|0≤s≤t }, the optiˆ (t | t) is the projection of x(t) onto the linear space mal state estimator x L {y(s)|0≤s≤t } [38, 3]. Note that the linear space L {y(s)|0≤s≤t } is equivalent to the following linear space (see Remark 2.2.2 for explanation): L{yl+1 (s)|0≤s≤tl ; · · · ; yi (s)|ti <s≤ti−1 ; · · · ; y1 (s)|t1 <s≤t }, (6.5) where

⎡

⎤ y(0) (s) ⎢ ⎥ .. yi (s) = ⎣ ⎦ . y(i−1) (s + di−1 )

(6.6)

is a new measurement obtained by re-organizing the delayed measurement y(t). It is clear that the new measurement yi (t) satisfies that yi (t) = Hi x(t) + vi (t), where

i = 1, · · · , l + 1,

⎤ ⎡ ⎤ H(0) v(0) (t) ⎥ ⎢ ⎢ ⎥ .. Hi = ⎣ ... ⎦ , vi (t) = ⎣ ⎦, . H(i−1) v(i−1) (t + di−1 )

(6.7)

⎡

(6.8)

and vi (t) is a white noise of zero mean and covariance matrix Qvi = diag{Qv(0) , · · · , Qv(i−1) }, i = 1, · · · , l + 1.

(6.9)

Note that the measurement sequence in (6.5) are no longer with any delay, which is termed as the re-organized measurement sequence of {y(s)|0≤s≤t }. ˆ (t | t) becomes the projection of x(t) onto the linear The optimal estimate x space (6.5).

118

6. Linear Estimation for Continuous-Time Systems with Measurement Delays

6.2.3

Re-organized Innovation Sequence

In this subsection we shall define the innovation associated with the re-organized measurement sequence (6.5). To this end, we first define the projection in the linear space spanned by the re-organized measurement sequence. Definition 6.2.1. Given the re-organized observation yi (s) with ti ≤ s ≤ ti−1 . ˆ i (s) denotes the projection of yi (s) onto the linear space: For s > ti , y L{yl+1 (τ )|0≤τ ≤tl ;

· · · ; yk (τ )|tk <τ ≤tk−1 ;

· · · ; yi (τ )|ti <τ <s }.

(6.10)

ˆ i (s) denotes the projection of yi (s) onto the linear space: For s = ti , y L{yl+1 (τ )|0≤τ ≤tl ;

· · · ; yk (τ )|tk <τ ≤tk−1 ;

· · · ; yi+1 (τ )|ti+1 <τ
From the above definition we can introduce the following stochastic sequence. 

ˆ i (s), wi (s) = yi (s) − y

(6.12)

ˆ i (s) is defined as in the above. The sequence wi (·) is the prediction error where y of the re-organized measurement yi (s). For i = l + 1, it is clear that wl+1 (s) is the standard Kalman filtering innovation sequence for the system (6.1) and (6.7) for i = l + 1. In view of (6.7), it follows that ˜ (s, i) + vi (s), i = 1, · · · , l + 1, wi (s) = Hi x

(6.13)

where ˜ (s, i) = x(s) − x ˆ (s, i), x

i = 1, · · · , l + 1,

(6.14)

ˆ (s, i) has a similar definition as y ˆ i (s), i.e., for s > ti , x ˆ (s, i) is the prowhile x ˆ (s, i) denotes the jection of x(s) onto the linear space of (6.10) and for s = ti , x projection of x(s) onto the linear space of (6.11). The following lemma shows that the sequence {w} is the innovation associated with the re-organized observation (6.5). Lemma 6.2.1. The stochastic sequence {wl+1 (s)|0≤s≤tl ; · · · ; wi (s)|ti <s≤ti−1 ;

· · · ; w1 (s)|t1 <s≤t }

(6.15)

is the innovation sequence which spans the same linear space as L{yl+1 (s)|0≤s≤tl ;

· · · ; yi (s)|ti <s≤ti−1 ;

· · · ; y1 (s)|t1 <s≤t },

or equivalently L{y(s)|0≤s≤t }. Proof: To simplify the discussion, we consider the case of single measurement delay system, i.e., l = 1. The case of multiple measurement delays can be discussed in a similar way. In the case of single measurement delay, it is readily

6.2 Linear Minimum Mean Square Error Estimation

119

seen from (6.12) that w2 (s) for s ≤ t1 ( or w1 (s), s > t1 ) is a linear combination of the observations {y2 (τ ) |0≤τ ≤s } (or {y2 (τ ) |0≤τ ≤t1 ; y1 (τ ) |t1 <τ ≤s }). Conversely, y2 (s), s ≤ t1 , (or y1 (s), s > t1 ) can be given in terms of a linear combination of w2 (τ )|0≤τ <s ( or {w2 (τ )|0≤τ ≤t1 ; w1 (τ )|t1 <τ <s }). Thus, {w2 (τ )|0≤τ ≤t1 ; w1 (τ )|t1 <τ ≤t } spans the same linear space as L {y2 (τ )|0≤τ ≤t1 ; y1 (τ )|t1 <τ ≤t } or equivalently L {y(τ )|0≤τ ≤t }. Next, we show that wi (·)(i = 1, 2) is an uncorrelated sequence. In fact, for any s > t1 and τ ≤ t1 where t = t − d1 , it follows from (6.13) that x(s, 1)w2 (τ )] + E [v2 (s)w2 (τ )] . E [w1 (s)w2 (τ )] = H1 E [˜

(6.16)

˜ (s, 1) is the state prediction error, it folNote that E [v2 (s)w2 (τ )] = 0. Since x lows that E [˜ x(s, 1)w2 (τ )] = 0, and thus E [w1 (s)w2 (τ )] = 0, which implies that w2 (τ ) (τ ≤ t1 ) is uncorrelated with w1 (s) (s > t1 ). Similarly, it can be verified that w2 (s) is uncorrelated with w2 (τ ) for s = τ and w1 (s0 ) is uncorrelated with w1 (τ0 ) for s0 = τ0 . Hence, {w2 (τ )|0≤τ ≤t1 ; w1 (τ )|t1 <τ ≤t } is an innovation sequence. This completes the proof of the lemma. ∇ The white noise sequence {wl+1 (s)|0≤s≤tl ; · · · ; wi (s)|ti <s≤ti−1 ; · · · ; w1 (s)|t1 <s≤t } is termed as the re-organized innovation sequence associated with the measurement sequence {y(s)|0≤s≤t }. 6.2.4

Riccati Equation

Let 

˜  (s, i)] , i = l + 1, · · · , 1 Pi (s) = E [˜ x(s, i) x

(6.17)

be the covariance matrix of the state estimation error. For a delay free system, it is well known that the covariance matrix of state filtering error satisfies a Riccati equation. Similarly, we shall show that the covariance matrix Pi (s) defined in (6.17) obeys certain Riccati equations. Theorem 6.2.1. Given time instant t, the matrix of Pl+1 (tl ) can be calculated as
 P˙l+1 (s) = ΦPl+1 (s) + Pl+1 (s)Φ − Pl+1 (s)Hl+1 Q−1 vl+1 Hl+1 Pl+1 (s)


+ Γ Qe Γ , Pl+1 (0) = P0 .

(6.18)

With Pl+1 (tl ), the matrices Pi (s), i = l, · · · , 1 and ti ≤ s ≤ ti + di − di−1 = ti−1 , can then be calculated as

P˙ i (s) = ΦPi (s) + Pi (s)Φ − Pi (s)Hi Q−1 vi Hi Pi (s) + Γ Qe Γ ,

Pi (ti ) = Pi+1 (ti ), i = l, · · · , 1.

(6.19)

In (6.19), the terminal value at step i is used as the initial value for the step i = i − 1, i.e., Pi (ti−1 ) ≡ Pi−1 (ti−1 ).

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6. Linear Estimation for Continuous-Time Systems with Measurement Delays

ˆ (s, i) is the projection of the state x(s) onto the linear space Proof: Note that x spanned by L{wl+1 (τ )|0≤τ ≤tl , · · · , wi (τ )|ti <τ <s }. Since w is a white noise, ˆ (s, i) is calculated by using the projection formula as the estimator x 9 tl ˆ (s, i) = x x(s), wl+1 (τ )Q−1 vl+1 wl+1 (τ )dτ 0

+

9 l 

k=i+1 s

9 +

ti

tk−1

tk

x(s), wk (τ )Q−1 vk wk (τ )dτ

x(s), wi (τ )Q−1 vi wi (τ )dτ.

(6.20)

By differentiating both sides of (6.20) with respect to s, we have ˆ˙ (s, i) = Φˆ x x(s, i) + x(s), wi (s)Q−1 vi wi (s) = Φˆ x(s, i) + Pi (s)Hi Q−1 vi wi (s),

(6.21)



ˆ (s, i) obeys the ˆ (ti , i + 1). Therefore, Σi (s) = ˆ x(s, i), x with initial condition x equation
Σ˙ i (s) = ΦΣi (s) + Σi (s)Φ + Pi (s)Hi Q−1 vi Hi Pi (s).

(6.22)



It is also clear that Π(s) = x(s), x(s) satisfies the linear differential equation

˙ Π(s) = ΦΠ(s) + Π(s)Φ + Γ Qe Γ .

(6.23)

ˆ (s, i) + x ˜ (s, i), we have that Π(s) = In view of the decomposition x(s) = x Σi (s) + Pi (s). Then, (6.19) and (6.18) follow and thus the proof is completed.∇ 6.2.5

Optimal Estimate x ˆ(t | t)

In this subsection we shall give a solution to the optimal filtering problem by applying the re-organized innovation sequence and the Riccati equations obtained in the last subsection. ˆ (t | t) is Theorem 6.2.2. Consider the system (6.1)-(6.2). The optimal filter x given by ˆ (t | t) = x ˆ (t, 1), x

(6.24)

ˆ (t, 1) is computed in the following steps where x ˆ (tl , l + 1) with the following standard Kalman filtering – Step 1: Calculate x ˆ˙ (s, l + 1) = Φl+1 (s)ˆ x x(s, l + 1) + Kl+1 (s)yl+1 (s), ˆ (0, l + 1) = 0, x

(6.25)

 where Φl+1 (s) = Φ−Kl+1 (s)Hl+1 , Kl+1 (s) = ΦPl+1 (s)Hl+1 Q−1 vl+1 and Pl+1 (s) is computed by (6.18), with Pl+1 (0) = P0 .

6.2 Linear Minimum Mean Square Error Estimation

121

ˆ (tl , l + 1) calculated in the above, x ˆ (t, 1) = – Step 2: With the initial value x ˆ (t0 , 1) is calculated by the following backward iteration: x 9 ti−1 ˆ (ti−1 , i) = Φi (ti−1 , ti )ˆ x x(ti , i) + Φi (ti−1 , s)Ki (s)yi (s)ds, ti

ˆ (ti , i) = x ˆ (ti , i + 1), ti = t − di , x

i = l, · · · , 1;

(6.26) where Φi (t, τ ) is the transition matrix of Φi (t), Φi (s) = Φ − Ki (s)Hi , Ki (s) =

ΦPi (s)Hi Q−1 vi ,

(6.27) (6.28)

and Pi (s) is calculated by (6.19). Proof: From (6.12) and (6.21), we can conclude (6.25). From (6.21), it follows that ˆ˙ (s, i) = Φi (s)ˆ x x(s, i) + Ki (s)yi (s), ˆ (ti , i) = x ˆ (ti , i + 1), ti ≤ s ≤ ti−1 , i = l, l − 1, · · · , 1, x (6.29) where Φi (s) and Ki (s) are respectively given in (6.27) and (6.28). Then (6.26) follows directly from (6.29). ∇ Remark 6.2.2. The Kalman filtering solution for measurement delayed system (6.1)-(6.2) is given by applying the re-organized innovation sequence. Different from the standard Kalman filtering approach, the computation procedure for a given time instant t is summarized as ˆ (tl , l + 1) by (6.18) and (6.25) with initial values – Calculate Pl+1 (tl ) and x ˆ (0, l + 1) = 0, respectively. Pl+1 (0) = P0 and x ˆ (tl , l + 1) as the initial values, the estimator x ˆ (t, 1) is – With Pl+1 (tl ) and x calculated by the backward recursion of (6.26) when i = 1. Observe that the solution only relies on Riccati equations of order n. 6.2.6

Numerical Example

In this subsection we give one example to demonstrate the calculation procedure and the validity of the proposed approach. Consider the system (6.1)-(6.2) with l = 2, d1 = 2 sec., d2 = 3 sec. and     −1 0 1 Φ= , Γ = , H(0) = [ 1 2 ] , H(1) = [ 1 0.5 ] , H(2) = [ 3 2 ] . 1 −2 1 Assume that e(t), v(0) (t), v(1) (t) and v(2) (t) are uncorrelated white noises with zero means and unity covariance matrices. Our aim is to calculate the optimal

122

6. Linear Estimation for Continuous-Time Systems with Measurement Delays

ˆ (t | t) of the signal x(t) based on measurements {y(s) |0≤s≤t }, where estimate x y(s) is defined in (6.3)-(6.4). We consider t ≥ d2 = 3. To compute the filter, we re-organize the linear space L {y(s) |0≤s≤t } equivalently as L {y3 (s) |0≤s≤t2 ; y2 (s) |t2 <s≤t1 ; y1 (s) |t1 <s≤t } ,

(6.30)

where t2 = t − 3, t1 = t − 2, and the new measurements yi (s) (i = 1, 2, 3) are given by y3 (s) = [ y(0) (s) y2 (s) = [ y(0) (s)



y(1) (s + 2) y(2) (s + 3) ] ,  y(1) (s + 2) ] ,

y1 (s) = y(0) (s) which satisfy yi (t) = Hi x(t) + vi (t), where

(6.31)

⎤ 1 2 1 2 2 ] , H2 = , H3 = ⎣ 2 0.5 ⎦ , 1 0.5 3 2 

H1 = [ 1

i = 1, 2, 3,



⎡

and vi (t) (i = 1, 2, 3) are white noises with zero means and unity covariances Qvi = Ii where Ii is with dimension of i × i. ˆ (t | t) can be summarized as The computation procedure of the optimal filter x – Step 1: Compute P3 (s) by (6.18) with l = 2 and initial value P3 (0) = 0. ˆ (s, 3) by (6.25) with l = 2 and initial value x ˆ (0, 3) = 0. Compute x – Step 2: With the initial values P2 (t2 ) ≡ P3 (t2 ) obtained in Step 1, compute ˆ (t1 , 2) is calculated by (6.26) P2 (s) (t2 ≤ s ≤ t1 ) by (6.19) with i = 2. Then x ˆ (t2 , 3) obtained in Step 1. ˆ (t2 , 2) ≡ x with i = 2 and initial value x ˆ (t1 , 1) ≡ x ˆ (t1 , 2), P1 (s) (t1 ≤ s ≤ t) and – Step 3: With P1 (t1 ) ≡ P2 (t1 ) and x ˆ (t0 , 1) can be given by (6.19) and (6.26), respectively. Then we obtain the x ˆ (t | t) = x ˆ (t0 , 1). optimal filter as x

6.3 H∞ Filtering for Systems with Multiple Delayed Measurements In this section we study the H∞ filtering for measurement delayed systems. By identifying a Krein space stochastic system associated with the original system and the performance criterion under consideration, the calculation of the central H∞ filter is similar to that of an H2 filter with measurement delays. A necessary and sufficient existence condition for an H∞ filter is related to the inertia of the innovation covariance matrix of the identified stochastic system.

6.3 H∞ Filtering for Systems with Multiple Delayed Measurements

6.3.1

123

Problem Statement

We consider the following linear system for the H∞ filtering problem. x(t) ˙ = Φx(t) + Γ e(t), z(t) = Lx(t),

(6.32) (6.33)

where x(t) ∈ Rn , e(t) ∈ Rr , and z(t) ∈ Rq represent the state, input noise, and the signal to be estimated, respectively. Φ, Γ , L are bounded matrices with dimensions of n × n, n × r and q × n, respecively. Assume that the state x(t) is observed by different systems with delays described by y(i) (t) = H(i) x(ti ) + v(i) (t), i = 0, 1, · · · , l,

(6.34)

ti = t − di ,

(6.35)

where y(i) ∈ Rpi ,

and H(i) (i = 0, 1, · · · , l) are observation matrices with dimensions of pi × n. Withought loss of generality, we assume that the time delays are in a strictly increasing order: 0 = d0 < d1 < · · · < dl . In (6.34), y(i) (t) ∈ Rpi , i ≥ 1, are the delayed measurements and v(i) (t) ∈ Rpi are the measurement noises. It is assumed that the input noise e is from L2 [0, T ] and measurement noises v(i) from L2 [di , T ], where T > 0 is the time-horizon of the filtering problem under consideration. Let y(t) and v(t) respectively denote the observation of the system (6.34) and the observation noise at time t, then we have ⎧  ⎨ col y (t), · · · , y (0) (i−1) (t) , di−1 ≤ t < di , (6.36) y(t) = ⎩ col y (t), · · · , y (t) , t ≥ d (0)

l

(l)

⎧  ⎨ v (t), · · · , v (0) (i−1) (t) , di−1 ≤ t < di , v(t) =  ⎩ v (t), · · · , v (t) , t ≥ d . l (0) (l)

(6.37)

The H∞ filtering problem for the system is stated as: Given a scalar γ > 0 and the observation {y(s), 0 ≤ s ≤ t}, find a filtering estimate zˇ(t | t) of z(t), if it exists, such that the following inequality is satisfied: :T

sup (x0 ,u,vd )=0

z (t | 0 [ˇ −1  x (0)Π0 x(0) 2

<γ ,




t) − z(t)] [ˇ z (t | t) − z(t)] dt :T
:T + 0 e (t)e(t)dt + 0 v
(t)v(t)dt (6.38)

where Π0 is a given positive definite matrix which reflects the relative uncertainty of the initial state.

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6. Linear Estimation for Continuous-Time Systems with Measurement Delays

Remark 6.3.1. It should be noted that the H∞ filtering of linear systems with single delayed measurement has been studied in [74, 23] where only sufficient solution condition has been given. However, the case of the multiple delayed measurements is much more complicated and remains to be investigated. Our aim in this chapter is to give a necessary and sufficient condition for the finite horizon H∞ filtering. 6.3.2

An Equivalent Problem in Krein Space

We start our work by studying the following stochastic system associated with (6.32)-(6.34) in Krein space: ˙ x(t) = Φx(t) + Γ e(t),

(6.39)

y(i) (t) = H(i) x(ti ) + v(i) (t), i = 0, 1, · · · , l.

(6.40)

We also introduce a ‘fictitious’ observation for the system: ˇ z(t | t) = Lx(t) + vz (t), t ≥ 0,

(6.41)

where zˇ(t | t) ∈ Rq implies the observation of the state x(t) at time t. In the above the matrices Φ, Γ , H(i) and L have the same definition as in (6.32)-(6.34). Note that x(.), e(.), yi (.), vi (.), zˇ(.) and vz (.) , in bold faces, are Krein space elements. Assumption 6.3.1. The initial state x(0) and the noises e(t), vi (t) (i = 1, · · · , l) and vz (t) are mutually uncorrelated white noises with zero means and known covariance matrices Π0 , Qe = Ir , Qv(i) = Ipi and Qvz = −γ 2 Iq , respectively. Combining (6.40) for i = 0 with (6.41) yields, ¯ (0) x(t) + v ¯ (0) (t) = H ¯ (0) (t), y where

(6.42)



 y0 (t) ¯ (0) (t) = y , zˇ(t | t)   v(0) (t) ¯ (0) (t) = v , vz (t)   ¯ (0) = H(0) . H L

(6.43) (6.44) (6.45)

¯ (0) is of the form: The covariance of v Qv¯ = diag{Qv(0) , Qvz } = diag{Ip0 , −γ 2 Iq }. Then the Krein-space measurement at time t is given by ⎧  ⎨ col y ¯ (0) (t), y(1) (t), · · · , y(i−1) (t) , di−1 ≤ t < di , ¯ (t) = y  ⎩ col y ¯ (t), y (t), · · · , y (t) , t ≥ dl . (0)

(1)

(l)

(6.46)

(6.47)

6.3 H∞ Filtering for Systems with Multiple Delayed Measurements

125

¯ (t) is different from the Hilbert Note that the Krein space measurement y measurement y(t) defined in (6.36). The linear space generated by the Krein space measurements up to time t is denoted as L {¯ y(τ ), 0 ≤ τ ≤ t} .

(6.48)

Having introduced the stochastic system in Krein space, it is easy to show that the H∞ filtering problem to be addressed is equivalent to an H2 estimation problem. Lemma 6.3.1. Consider the system (6.32)-(6.34) together with the performance criterion (6.38) and the associated stochastic system (6.39)-(6.41).Then, for a given scalar γ > 0, there exists an estimator zˇ(t | t)(0 ≤ t ≤ T ) that achieves (6.38) if and only if x ˆ(ti | t) (di ≤ t ≤ T ) for i = 0, 1, · · · , l exist, where x ˆ(ti | t) is obtained from the projection of x(ti ) onto the linear space of (6.48). Furthermore, if the above conditions are satisfied, a suitable H∞ filter is given by zˇ(t | t) = Lˆ x(t | t),

(6.49)

where x ˆ(t | t) is obtained from the projection of x(t) onto the linear space of (6.48). Proof: In view of (6.38), it is not difficult to observe that there exists an estomator zˇ(t | t) that achieves (6.38) if and only if 

JT = x (0)Π0−1 x(0) + −γ −2

9

T




0

9

T

9

e (s)e(s)ds +

T




v (s)v(s)ds 0




vz (s)vz (s)ds

(6.50)

0

has a minimum JTm over {x(0); e(s), 0 ≤ s ≤ T } and an estimator can be chosen such that the minimum is positive. Recall the discussion in [30], [111]-[109], JT has a minimum if and only if the innovation of the Krein-space measurements ¯ (t) (0 ≤ t ≤ T ), denoted as w(t), ¯ y exists. Note that the innovation associated ¯ (t) is given by with y ˆ ¯ (t | t) ¯ ¯ (t) − y w(t) =y ⎧   ⎪ ¯ (0) x ⎪ ¯ (0) (t) − H ˆ (t | t), · · · , y(i−1) (t) − H(i−1) x ˆ (ti−1 | t) col y ⎪ ⎪ ⎪ ⎪ ⎨ di−1 ≤ t < di , =   ⎪ ¯ (0) x ⎪ ˆ (t | t), · · · , y(l) (t) − H(l) x ˆ (tl | t) , ¯ (0) (t) − H col y ⎪ ⎪ ⎪ ⎪ ⎩ t ≥ dl . (6.51)

126

6. Linear Estimation for Continuous-Time Systems with Measurement Delays

Therefore, there exists an estimator zˇ(t | t)(0 ≤ t ≤ T ) that achieves (6.38) if and only if x ˆ(ti | t) (di ≤ t ≤ T ) for i = 0, 1, · · · , l exist. Furthermore, if the innovation exists, the minimum of JT is given by 9 T JTm = w ¯ (s)Qw¯ (s)w(s)ds, ¯ (6.52) 0

¯ and is calculated as where Qw¯ (s) is the covariance of the innovation w(s), ⎧ ⎨ diag{Q , I , · · · , I v ¯ p1 pi−1 }, di−1 ≤ t < di , Qw¯ (s) = (6.53) ⎩ diag{Q , I , · · · , I }, t ≥ d , v ¯

p1

pl

l

where Qv¯ is as in (6.46). Thus it follows from (6.51)-(6.52) that 9 T ! ! y(0) (s) − H(0) x JTm = ˆ(s | s) y(0) (s) − H(0) x ˆ(s | s) ds 0

−γ +

l 9  k=1

9

T

2

[ˇ z (s | s) − Lˆ x(s | s)] [ˇ z (s | s) − Lˆ x(s | s)] ds

0 T

! ! ˆ(s − dk | s) y(k) (s) − H(k) x ˆ(s − dk | s) ds. y(k) (s) − H(k) x

dk

(6.54) To achieve JTm > 0, a natural choice is zˇ(t | t) = Lˆ x(t | t), which is (6.49). Thus the lemma is established.

∇

As shown in the above Lemma, in order to obtain the H∞ filter, we need to check the existence of the projection xˆ(t − di | t) and calculate x ˆ(t | t), if it exists. Note the observation equation involves the time delays, so the problem is much more complicated than the standard H∞ filtering. To overcome the difficulties, we shall apply the re-organized innovation approach developed earlier. 6.3.3

Re-organized Innovation Sequence

Lemma 6.3.2. The linear space of (6.48) is equivalent to L{yl+1 (τ )|0≤τ ≤tl ;

· · · ; yi (τ )|ti <τ ≤ti−1 ; · · · ; y1 (τ )|t1 <τ ≤t },

(6.55)

¯ (0) (τ ) and for i > 1 where y1 (τ ) = y ⎡

⎤ ¯ (0) (τ ) y ⎢ y(1) (τ + d1 ) ⎥ ⎢ ⎥ yi (τ ) = ⎢ ⎥, .. ⎣ ⎦ . y(i−1) (τ + di−1 )

(6.56)

6.3 H∞ Filtering for Systems with Multiple Delayed Measurements

127

satisfies that yi (τ ) = Hi x(τ ) + vi (τ ),

i = 1, · · · , l + 1,

¯ (0) , v1 (τ ) = v ¯ (0) (τ ) and for i > 1 with H1 = H ⎡ ¯ ⎡ ⎤ ⎤ ¯ (0) (τ ) H(0) v ⎢ H(1) ⎥ ⎢ v(1) (τ + d1 ) ⎥ ⎢ ⎢ ⎥ ⎥ Hi = ⎢ . ⎥ , vi (τ ) = ⎢ ⎥. .. ⎣ .. ⎦ ⎣ ⎦ . v(i−1) (τ + di−1 ) H(i−1)

(6.57)

(6.58)

Moreover, vi (τ ) is a white noise of zero mean and covariance matrix Qv1 = Qv¯(0) and for i > 1 Qvi = diag{Qv¯(0) , Qv(1) , · · · , Qv(i−1) }. Proof: It is straightforward by re-organizing the delayed observations.

(6.59) ∇

Now we introduce the innovation sequence associated with the re-organized measurements. Definition 6.3.1. For i = l + 1, · · · , 1 and ti < τ ≤ ti−1 , let 

ˆ i (τ ), wi (τ ) = yi (τ ) − y

(6.60)

ˆ i (τ ) is the projection of yi (τ ) onto the following linear where for ti < s ≤ ti−1 , y space L{yl+1 (τ )|0≤τ ≤tl ;

· · · ; yk (τ )|tk <τ ≤tk−1 ; · · · ; yi (τ )|ti <τ <s },

(6.61)

ˆ i (τ ) is the projection of yi (τ ) onto the linear space of and for s = ti , y L{yl+1 (τ )|0≤τ ≤tl ;

· · · ; yk (τ )|tk <τ ≤tk−1 ; · · · ; yi+1 (τ )|ti+1 <τ
(6.62)

Based on the above definition, it is clear [111, 102] that the sequence wi (.) is mutually uncorrelated and {wl+1 (τ )|0≤τ ≤tl+1 , · · · , wi (τ )|ti <τ ≤di−1 ; · · · , w1 (τ )|t1 <τ ≤t },

(6.63)

spans the same linear space as (6.48). Since wi (τ ) is obtained with the reorganized observations, so it is termed as re-organized innovation. 6.3.4

Riccati Equation

In this subsection we compute the covariance matrix of the re-organized innovation sequence defined in the above subsection. From the definition of the innovation sequence, it is easy to know that ˜ (τ, i) + vi (τ ), ti < τ ≤ ti−1 , i = l + 1, · · · , 1, wi (τ ) = Hi x

(6.64)

128

6. Linear Estimation for Continuous-Time Systems with Measurement Delays

where ˜ (τ, i) = x(τ ) − x ˆ (τ, i) x

(6.65)

is the filtering error and its covariance matrix is defined as 

˜ (τ, i) ] , ti < τ ≤ ti−1 . Pi (τ ) = E [˜ x(τ, i) x

(6.66)

The following theorem shows that Pi (τ ) satisfies one standard Riccati equation. Theorem 6.3.1. Given the time instant t, the cross-covariance matrices Pi (τ ) (ti < τ ≤ ti−1 ) for i = l + 1, · · · , 1 can be calculated by:




P˙ i (τ ) = ΦPi (τ ) + Pi (τ )Φ − Ki (τ )Qvi [Ki (τ )] + Γ Γ , Pi (ti ) = Pi+1 (ti ),

(6.67) where Ki (τ ) = Pi (τ )Hi Q−1 vi ,

(6.68)

while the initial value of Pi (ti ) is the terminal value of Pi+1 (ti ) calculated in step i + 1, and the initial value of Pl+1 (0) is set as Π0 . ˆ (τ, i) is the projection of x(τ ) onto the linear space Proof: Note that for τ > ti , x L{wl+1 (s)|0≤s≤tl , · · · , wi (s)|ti <s<τ } and can be obtained by applying the projection formula as 9 tl ˆ (τ, i) = x x(τ ), wl+1 (s)Q−1 vl+1 wl+1 (s)ds 0

+

9 l 

k=i+1 9 τ

+ ti

tk−1

tk

x(τ ), wk (s)Q−1 vk wk (s)ds

x(τ ), wi (s)Q−1 vi wi (s)ds.

(6.69)

By differentiating both sides of (6.69) with respect to τ , we have dˆ x(τ, i) = Φˆ x(τ, i) + x(τ ), wi (τ )Q−1 vi wi (τ ) dτ = Φˆ x(τ, i) + Ki (τ )wi (τ ),

(6.70)



ˆ (τ, i) obeys the where Ki (τ ) is given by (6.68). Therefore, Σi (τ ) = ˆ x(τ, i), x equation

d Σi (τ ) = ΦΣi (τ ) + Σi (τ )Φ + Ki (τ )Qvi [Ki (τ )] . dτ

(6.71)

6.3 H∞ Filtering for Systems with Multiple Delayed Measurements

129



On that other hand, it is clear that Π(τ ) = x(τ ), x(τ ) satisfies the linear differential equation

d Π(τ ) = ΦΠ(τ ) + Π(τ )Φ + Γ Γ . dτ

(6.72)

ˆ (τ, i) + x ˜ (τ, i) and x ˆ (τ, i) is orthogonal to In view of the decomposition x(τ ) = x ˜ (τ, i), we have that Π(τ ) = Σi (τ ) + Pi (τ ). Then, (6.67) follows and thus the x proof is completed. ∇ 6.3.5

Main Results

Theorem 6.3.2. Consider the system (6.32)-(6.34) and the associated performance criterion (6.38). Given a scalar γ > 0, there exists a filter zˇ(t | t) that achieves (6.38) if and only if the matrices Pi (τ ) for ti−1 < τ ≤ ti (i = 1, · · · , l) over 0 ≤ t ≤ T is bounded, where Pi (τ ) is computed from Theorem 6.3.1. If the above conditions are met, a suitable filter zˇ(t | t) is given by zˇ(t | t) = Lˆ x(t, 1),

(6.73)

where xˆ(t, 1) can be computed following the steps below. ˆ (tl , l + 1) by Step 1: With the initial value xˆ(0, l + 1) = 0, compute x x ˆ˙ (s, l + 1) = Φl+1 (s)ˆ x(s, l + 1) + Kl+1 (s)yl+1 (s),

(6.74)

Φl+1 (s) = Φ − Kl+1 (s)Hl+1 ,  Q−1 Kl+1 (s) = Pl+1 (s)Hl+1 vl+1 ,

(6.75)

where

and Pl+1 (s) is computed by (6.67) with the initial value Pl+1 (0) = Π0 . ˆ (tl , l + 1) and Pl+1 (tl ) as the initial values, compute x Step 2: Using x ˆ(ti−1 , i) (i = l, · · · , 1) as ˆ (ti , i) = x ˆ (ti , i + 1), x(s, i) + Ki (s)yi (s), x x ˆ˙ (s, i) = Φi (s)ˆ ti < s ≤ ti−1 ,

(6.76)

where Φi (s) = Φ − Ki (s)Hi , Ki (s) = Pi (s)Hi Q−1 vi ,

(6.77)

while Pi (s) is computed by (6.67). Proof: (6.74) and (6.76) follow directly from Theorem 6.3.1. Now we show the necessity and the sufficiency. By using the re-organized innovation sequence and projection formula, we have

130

6. Linear Estimation for Continuous-Time Systems with Measurement Delays

x ˆ(ti | t) = x ˆ(ti , i + 1) +

1 9  k=i t

9 =x ˆ(ti , i + 1) + +

2 9 

tk−1

tk

x(ti ), wk (τ )Q−1 vk wk (τ )dτ

P1 (ti , τ ) [ H0

−γ 2 L ] w1 (τ )dτ

t1 tk−1

Pk (ti , τ )Hk wk (τ )dτ,

(6.78)

˜ (τ, k). Pk (ti , τ ) = x(ti ), x

(6.79)

k=i

tk

where

Note Pk (ti , τ ) is different from Pi (τ ) defined in the last subsection. It is not difficult to show that Pk (ti , τ ) = Pk (τ ) for τ = ti and for τ > ti , Pk (ti , τ ) satisfy dPk (ti , τ ) = Pk (ti , τ ) [Φ{In − Pk (τ )Hk Hk }] . dτ

(6.80)

Sufficiency: If the matrices Pi (τ ) for ti < τ ≤ ti−1 over 0 ≤ t ≤ T is bounded, ˆ (ti | t) for 0 ≤ i ≤ l exists, which implies it is easy to observe from (6.78) that x that the H∞ filter zˇ(t | t) that achieves (6.38) exists. Necessity: When the H∞ filtering problem is solvable, i.e. there exists an estimate zˇ(t | t) that achieves (6.38). It is not difficult to observe that the H∞ filtering problem for the system (6.32)-(6.33) with observation equation (6.57) exists, which implies that Pl+1 (τ ) is bounded for 0 ≤ τ ≤ T . Now we show that Pi (τ ) for ti < τ ≤ ti−1 (i = 1, · · · , l) over 0 ≤ t ≤ T are bounded. Actually, in view of ˆ(ti | t) are not solvable, viz, zˇ(t | t) that (6.79)-(6.80), if Pi (τ ) is unbounded, x achieves (6.38) does not exist. The proof is thus completed. ∇ Remark 6.3.2. Note that (6.76) can be written as 9 ti−1 x ˆ(ti−1 , i) = Φi (ti−1 , ti )ˆ x(ti , i) + Φi (ti−1 , s)Ki (s)yi (s)ds, ti

i = l, · · · , 1;

ti = t − di , x ˆ(ti , i) = x ˆ(ti , i + 1),

(6.81)

where Φi (t, τ ) is the transition matrix of Φi (t). In the above backward iteration, the initial value is x ˆ(tl , l + 1) which can be computed by (6.74). Thus, we can obtain the terminal estimate x ˆ(t, 1) = x ˆ(t0 , 1) from (6.81) when i = 1. 6.3.6

Numerical Example

Consider the linear time-varying continuous-time system (6.32)-(6.34) with two delays d1 = 0.2sec., d2 = 0.3sec. and     t 1 1 Φ= , Γ = , H(0) = [ 1 0.5 ] , 0 1 1 H(1) = [ 1

2 ] , H(2) (t) = [ 5

2t ] , L = [ 1

1].

(6.82)

6.3 H∞ Filtering for Systems with Multiple Delayed Measurements

131



 0.2 The initial state is x(0) = . We study the H∞ filter problem with γ 2 = 2.5 0.5   1 0 in (6.38). In simulation, we take the time-horizon T = 2sec. and Π0 = 0 1 and sampling period as 0.02sec., the system noise e(t) is generated by e(t) = 10cos(t), the observation noise v0 (t), v1 (t) and v2 (t) are Gaussian random noises with zero means and unit covariance matrices. Recall Lemma 6.3.2, it is easy to know   1 0.5 ¯ H(0) = , (6.83) 1 1   ¯ (0) = 1 0.5 , H1 = H (6.84) 1 1 ⎡ ⎤ 1 0.5 H2 = ⎣ 1 1 ⎦ , (6.85) 1 2 ⎡ ⎤ 1 0.5 1 ⎥ ⎢1 H3 (t) = ⎣ (6.86) ⎦. 1 2 5 2t + 1 Similarly, the covariance matrices of the re-organized observation noises are Qv1 = diag{1, −2.5}, Qv2 = diag{1, −2.5, 1}, Qv3 = diag{1, −2.5, 1, 1}. (6.87)   1 0 Let the initial estimate x ˆ(0, 2 + 1) = 0 with covariance P0 = Π0 = , 0 1 the filter is computed by Theorem 6.3.2. The simulated results are shown in 250

200

150

100

1

50

0

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Fig. 6.1. Tracking performance of the filter

1.6

1.8

2

132

6. Linear Estimation for Continuous-Time Systems with Measurement Delays

Figures 6.1-6.2. In Figure 6.1, ’1’ denotes the actual signal whereas ’2’ is the filtered estimate. It is easy to observe that the filter has a good tracking performance. Figure 6.2 shows the ratios between the energies of the estimation error and the input noises, where line ’1’ is for the H∞ filtering with observation y0 (t) only while ’2’ is for the H∞ filtering with the instantaneous observation y0 (t) and two delayed measurements y1 (t) and y2 (t). It is clearly shown that both line ’1’ and ’2’ are below the given γ 2 = 2.5. Moreover, line ’2’ is lower than the line ’1’ which implies that the more the available observation information about the system, the better performance the filter.

2.5

2

1.5

1

2

1

0.5

0

0

0.2

0.4

0.8

0.6

1

1.2

1.4

1.6

2

1.8

Fig. 6.2. The ratios of energy between the filtering error and the input noises

6.4 H∞ Fixed-Lag Smoothing for Continuous-Time Systems In this section we study the H∞ fixed-lag smoothing problem. It will be shown that the H∞ fixed-lag smoothing is related to the optimal filtering problem with instantaneous and delayed measurements in Krein space. The developed re-organized innovation analysis approach in the last section will be applied to derive the smoothing estimator. 6.4.1

Problem Statement

We consider the following continuous time-varying system: x(t) ˙ = Φx(t) + Γ e(t), y(t) = Hx(t) + v(t),

(6.88) (6.89)

z(t) = Lx(t),

(6.90)

6.4 H∞ Fixed-Lag Smoothing for Continuous-Time Systems

133

where x(t) ∈ Rn , e(t) ∈ Rr , y(t) ∈ Rp , v(t) ∈ Rp and z(t) ∈ Rq represent the state, input noise, measurement output, measurement noise and the signal to be estimated, respectively. It is assumed that the input and measurement noises are deterministic signals and are from L2 [0, T ] where T is the time-horizon of the estimation problem under investigation. The H∞ fixed-lag smoothing problem is stated as: Given scalars γ > 0 and d > 0 and the observation {y(s), 0 ≤ s ≤ t}, find an estimate zˇ(t − d | t) of z(t − d), if exists, such that the following inequality is satisfied: :T d

sup (x0 ,e,v)=0



[ˇ z (t − d | t) − z(t − d)] [ˇ z (t − d | t) − z(t − d)] dt < γ 2 , (6.91) :T :T −1 x (0)P0 x(0) + 0 e (t)e(t)dt + 0 v  (t)v(t)dt

where “  ” stands for matrix transposition and P0 is a given positive definite matrix which reflects the uncertainty of the initial state relative to the input and measurement noises. 6.4.2

An Equivalent H2 Problem in Krein Space

Firstly, we define that 

JT = x (0)P0−1 x(0) + −γ −2

9

9

T

e (t)e(t)dt +

9

0 T

T

v  (t)v(t)dt

0

vz (t)vz (t)dt

(6.92)

d

where vz (t) = zˇ(t − d | t) − Lx(t − d), or equivalently zˇ(t − d | t) = Lx(t − d) + vz (t),

(6.93)

and vz (t) = 0 for t < d. Similar to the discussion in [108], we introduce the following Krein space stochastic system associated with the deterministic system (6.88)-(6.90): ˙ x(t) = Φx(t) + Γ e(t), y(t) = Hx(t) + v(t), ˇ z(t − d | t) = Lx(t − d) + vz (t),

(6.94) (6.95) t ≥ d,

(6.96)

where the initial state x(0), and noises e(t), v(t) and vz (t), in bold faces, are assumed to be mutually uncorrelated white noises of zero means and covariances Qx0 = P0 , Qe = Ir , Qv = Ip and Qvz = −γ 2 Iq , respectively. We shall introduce a re-organized innovation approach. To this end, the observation of the Krein space system at time t is denoted as ⎧ ⎪ ⎨ y(t), 0≤t
134

6. Linear Estimation for Continuous-Time Systems with Measurement Delays

Note that yo (t) is related to x(t) and x(t − d). As in the standard Kalman filtering, the innovation sequence associated with observation yo (t) is defined as 

ˆ o (t), wo (t) = yo (t) − y

(6.98)

ˆ o (t) is the projection of yo (t) onto where y L{yo (τ )|0≤τ
(6.99)

For t < d, the innovation wo (t) is given from (6.98) as ˆ (t), wo (t) = y(t) − H x

(6.100)

ˆ (t) is the Hilbert space projection of x(t) onto L{y(τ )|0≤τ
˜ (t), M (t, t) = x(t), x 

˜ (s), t − d ≤ s ≤ t, M (t − d, s) = x(t − d), x

(6.102) (6.103)

˜ (s) = x(s) − x ˆ (s), and x ˆ (s) is the Krein space projection of x(s) onto where x L{yo (τ )|0≤τ <s }, while yo (τ ) is as in (6.97). Now we have the following results Theorem 6.4.1. Consider the system (6.88)-(6.90) and the associated performance criterion (6.91). Given scalars γ > 0 and d > 0, the finite horizon H∞ fixed-lag smoothing problem is solvable if and only if the matrices M (t, t) and M (t − d, s) as defined in (6.102)-(6.103) are bounded for t − d ≤ s ≤ t over d ≤ t ≤ T . In this case, a suitable H∞ smoother zˇ(t − d | t) is the projection of the Krein space element z(t−d) onto the linear space spanned by the Krein-space observation {yo (τ )|0 ≤ τ < t}, which is given by zˇ(t − d | t) = Lˆ x(t − d | t),

(6.104)

where x ˆ(t − d | t) is obtained from the projection of Krein space state x(t − d) onto linear space L{yo (τ )|0≤τ
or equivalently,

9

t

x ˆ(t − d | t) = xˆ0 (t − d | t) + t−d

M (t − d, s)H  [y(s) − H x ˆ(s)] ds, (6.106)

6.4 H∞ Fixed-Lag Smoothing for Continuous-Time Systems

: t−d



while x ˆ0 (t − d | t) =

135

M (t − d, s)H  [y(s) − H x ˆ(s)] ds satisfies

0

x ˆ˙ 0 (t − d | t) = Φˆ x0 (t − d | t) + M (t − d, t − d)H  [y(t − d) − H x ˆ(t − d)] , (6.107) x ˆ0 (0 | d) = 0. The estimator xˆ(s), t − d ≤ s ≤ t, in the above is calculated by x ˆ˙ (s) = {Φ − M (s, s)H  H} xˆ(s) + M (s, s)H  y(s).

(6.108)

Proof: In view of (6.92) and the discussion in [30], an estimator zˇ(t − d | t) exists iff the innovation wo (t)(0 ≤ t ≤ T ), defined in (6.98), exists. From (6.101), it is readily known that the existence of wo (t)(0 ≤ t ≤ T ) implies that both xˆ(t) and x ˆ(t − d | t) exist for 0 ≤ t ≤ T . Furthermore, the minimum of JT with respect to {x(0); e(s), 0 ≤ s ≤ t} is given by 9 T wo (s)Qwo (s)−1 wo (s)ds JTmin = 0

9 =

d

[y(s) − yˆ(s)] [y(s) − yˆ(s)] +   9 T y(s) − H x ˆ(s) Ip z ˇ (s − d | s) − Lˆ x (s − d | s) 0 d   y(s) − H x ˆ(s) × ds. zˇ(s − d | s) − Lˆ x(s − d | s) 0

0



−γ −2 Iq (6.109)

An estimator zˇ(s − d | s) is to be chosen such that JTmin > 0. One natural choice is that zˇ(s − d | s) = Lˆ x(s − d | s),

(6.110)

where x ˆ(s−d | s) is the projection of x(s−d) on the linear space L{yo (τ )|0≤τ <s }. Now we calculate x ˆ(s− d | s) and xˆ(s). Applying the projection formula, we have that 9 s x(s), wo (τ )Q−1 x ˆ(s − d | s) = wo (τ )wo (τ )dτ 0

9

d

=−

x(s), wo (τ )[y(τ ) − yˆ(τ )]dτ +    9 s I y(s) − H x ˆ(s) 0 x(s), wo (τ ) p dτ. zˇ(s − d | s) − Lˆ x(s − d | s) 0 −γ −2 Iq d 0

(6.111) In view of (6.110), we have

9

s

x(s), wo (τ )[y(τ ) − yˆ(τ )]dτ

x ˆ(s − d | s) = 0

9 =

0

s

M (t − d, s)H  [y(s) − H x ˆ(s)] ds.

(6.112)

136

6. Linear Estimation for Continuous-Time Systems with Measurement Delays

Similarly, we can prove (6.108). Since an estimator zˇ(t − d | t) exists iff both x ˆ(t) and x ˆ(t − d | t) exist for 0 ≤ t ≤ T . In view of (6.105) and (6.108), it is readily known that the finite horizon H∞ fixed-lag smoothing problem is solvable if and only if the matrices M (t, t) and M (t − d, s) as defined in (6.102)-(6.103) are bounded for t − d ≤ s ≤ t over d ≤ t ≤ T . The proof of the theorem is completed. ∇ Remark 6.4.1. Note that in [108] the matrices M (t, t) and M (t − d, s) (t − d ≤ s ≤ t) are related to a Riccati type of PDE together with two boundary conditions governed by one PDE and one RDE, which poses a significant numerical difficulty. In the following we shall propose a RDE approach for calculating a smoother. 6.4.3

Re-organized Innovation Sequence

First we consider the linear space L{yo (τ ), 0 ≤ τ < s} from the Krein space system described by (6.94)-(6.96). In view of (6.97), we have L{yo (τ )|0≤τ <s } = L{y2 (τ )|0≤τ <s−d ; y1 (τ )|s−d≤τ <s }, where 



y2 (τ ) =

 y(τ ) = H2 x(τ ) + v2 (τ ), ˇ z(τ | τ + d)



y1 (τ ) = y(τ ) = H1 x(τ ) + v1 (τ ), with

 H2 =

   H v(τ ) , v2 (τ ) = , L vz (τ + d)

H1 = H,

v1 (τ ) = v(τ ).

(6.113)

(6.114) (6.115)

(6.116) (6.117)

In the above, v2 (τ ) and v1 (τ ) are Krein space noises with zero means and covariances of Qv2 = diag{Ip , −γ 2 Iq } and Qv1 = Ip , respectively. Introduce the following sequence 

ˆ 2 (θ | θ, θ), w(θ, θ) = y2 (θ) − y 

ˆ 1 (s | s, θ), s > θ, w(s, θ) = y1 (s) − y

(6.118) (6.119)

ˆ 2 (r | θ, θ) is the projection of y2 (θ) onto the linear space spanned by where y re-organized measurements L{y2 (τ )|0≤τ <θ }, ˆ 1 (s | s, θ) is the projection of y1 (s) onto the linear space spanned by and y re-organized measurements L{y2 (τ )|0≤τ <θ ; y1 (τ )|θ≤τ <s }.

6.4 H∞ Fixed-Lag Smoothing for Continuous-Time Systems

137

It is easy to observe the relationships ˜ (θ, θ) + v2 (θ), w(θ, θ) = H2 x ˜ (s, θ) + v1 (s), s > θ, w(s, θ) = H1 x

(6.120) (6.121)

where 

˜ (s, θ) = x(s) − x ˆ (s | s, θ), x

(6.122)

ˆ (s | s, θ) is defined similarly to y ˆ 1 (s | s, θ), i.e. it is the projection of x(s) and x onto L {y2 (τ )|0≤τ <θ ; y1 (τ )|θ≤τ <s } . It can be shown [111] that the stochastic process {w(τ, τ )|0≤τ
(6.123)

is a mutually uncorrelated white noise and spans the same linear space as L{y2 (τ )|0≤τ
(6.124)

(6.123) is termed as a re-organized innovation sequence associated with (6.124) which will play an important role in deriving a simple algorithm for computing matrices M (t, t) and M (t − d, s). 6.4.4

Main Results

Denote 

τ ˜ (s, θ), s ≥ θ, Ps,θ = x(τ ), x

(6.125)

˜ (s, θ) is as in (6.122). Note that x ˜ (t, t − d) = x ˜ (t) and x ˜ (s, s − d) = x ˜ (s). where x Thus, from definition (6.102)-(6.103) and (6.125), it follows that t ≡ M (t, t), Pt,t−d

(6.126)

t−d Ps,s−d

(6.127)

≡ M (t − d, s), t − d ≤ s ≤ t.

We now present the main result of this chapter which provides a way of computing M (t, t) and M (t − d, s) in terms of Riccati differential equations. Theorem 6.4.2. Consider the system (6.94)-(6.96). The following results hold: 

θ 1. P (θ) = Pθ,θ satisfies

  dP (θ) = ΦP (θ) + P (θ)Φ − P (θ) H  H − γ −2 L L P (θ) + Γ Γ  , (6.128) dθ with initial value P (0) = P0 .

138

6. Linear Estimation for Continuous-Time Systems with Measurement Delays

θ 2. Pθ,t−d (t − d ≤ θ ≤ t) satisfies the following H2 type of RDE: θ ∂ Pθ,t−d θ θ θ θ = ΦPθ,t−d + Pθ,t−d Φ − Pθ,t−d H  HPθ,t−d + Γ Γ , ∂θ t−d = P (t − d). Pt−d,t−d

(6.129)

t is obtained. When θ = t, M (t, t) = Pt,t−d t−d 3. Pθ,s−d (t−d ≤ θ ≤ s) satisfies the following linear matrix differential equation t−d ∂ Pθ,s−d

∂θ

t−d t−d = Pθ,s−d A (θ, s − d), Pt−d,s−d ,

(6.130)

where   θ H H , A(θ, s − d) = Φ(θ) In − Pθ,s−d

(6.131)

t−d θ when θ = s, M (t − d, s) = Ps,s−d is obtained and Pθ,s−d is given in the last step.

˜ (θ, θ) = ˜ ˜ (θ, θ), where Proof: It is easily seen that P (θ) = x(θ), x x(θ, θ), x ˜ (θ, θ) = x(θ) − x ˆ (θ | θ, θ) and x ˆ (θ | θ, θ) is the projection of x(θ) onto x L {y2 (τ )|0≤τ <θ } ,

(6.132)

where y2 (τ ) is as (6.114). By applying the projection formula, it follows that 9

θ

ˆ (θ | θ, θ) = x

x(θ), w(τ, τ )Q−1 w (τ, τ )w(τ, τ )dτ

0

9 =

0

θ

x(θ), w(τ, τ )Q−1 v2 w(τ, τ )dτ.

(6.133)

By differentiating both sides of the above equation with respect to θ, ˆ (θ | θ, θ) ∂x = Φˆ x(θ | θ, θ) + x(θ), w(θ, θ)Q−1 v2 w(θ, θ) ∂θ = Φˆ x(θ | θ, θ) + P (θ)H2 Q−1 v2 w(θ, θ).

(6.134)



θ ˆ (θ | θ, θ) obeys the equation Therefore, Σθ,θ = ˆ x(θ | θ, θ), x θ ! θ ∂ Σθ,θ θ θ θ = ΦΣθ,θ + Σθ,θ Φ + Pθ,θ . H  H − γ −2 L L Pθ,θ ∂θ

(6.135)



On the other hand, it is obvious from (6.94) that Π(θ) = x(θ), x(θ) obeys the linear differential equation d Π(θ) = ΦΠ(θ) + Π(θ)Φ + Γ Γ  . dθ

(6.136)

6.4 H∞ Fixed-Lag Smoothing for Continuous-Time Systems

139

ˆ (θ | θ, θ) + e(θ, θ) implies that Π(θ) = Note that the decomposition x(θ) = x θ + P (θ) and hence (6.128) follows. Now we prove (6.129). By using the Σθ,θ innovation theory, we have 9

s−d

ˆ (θ | θ, s − d) = x 9

0 θ

+ s−d

x(θ), w(τ, τ )Q−1 v2 w(τ, τ )dτ

x(θ), w(τ, s − d)Q−1 v1 w(τ, s − d)dτ,

(6.137)



 Ip 0 where Qv2 = and Qv1 = Ip are respectively the covariance matri0 −γ 2 Iq ces of the innovation w(τ, τ ) and w(τ, s − d). By applying a similar discussion as in the proof of (6.128), (6.129) can be easily obtained. Next, it follows from (6.125) that t−d ˜ (θ, s − d), Pθ,s−d = x(t − d), x

(6.138)

where θ ≥ t − d. By applying (6.134) and (6.121), we have ! ˜ (θ, s − d) ∂x θ θ ˜ (θ, s − d) + Γ e(θ) − Pθ,s−d = Φ − Pθ,s−d H H x v1 (θ) ∂θ θ = A(θ, s − d)˜ x(θ, s − d) + Γ e(θ) − Pθ,s−d v1 (θ). (6.139) Then, it follows that t−d ∂ Pθ,s−d

∂θ

˜ (θ, s − d) ∂x  ∂θ t−d t−d = Pθ,s−d A (θ, s − d) + x(t − d), Γ e(θ) − Pθ,s−d v1 (θ). = x(t − d),

(6.140) Since θ ≥ t − d, x(t − d) is uncorrelated with e(θ) and v(θ), (6.130) can be obtained directly from the above equation. This completes the proof of the theorem. ∇ Remark 6.4.2. By taking into account (6.126) and (6.127), it can be seen from Theorem 6.4.1 that the design of H∞ smoother is based on one standard H∞ filtering RDE (6.128) and one H2 RDE (6.129). If the calculated M (t, t) and M (t − d, s) are bounded over t − d ≤ s ≤ t and 0 ≤ t ≤ T , an H∞ fixed-lag smoother exists and can be given by Theorem 6.4.1. It should be noted that if an H∞ filter exists, that is, there exists a bounded solution to the RDE (6.128) over 0 ≤ s ≤ T − d, then an H∞ fixed-lag smoother also exists, i.e. M (t, t) and M (t − d, s) are bounded over t − d ≤ s ≤ t and 0 ≤ t ≤ T . This can be seen from Theorem 6.4.1 and the standard Kalman filtering result. The converse, however, is not true. That is, the existence of bounded M (t, t) and M (t − d, s) does not require the boundedness of P (θ) for 0 ≤ θ ≤ t − d. We shall give an example to illustrate this point in the following section.

140

6. Linear Estimation for Continuous-Time Systems with Measurement Delays

6.4.5

Examples

Consider the linear continuous-time system (6.88)-(6.90) with Φ = 0.5, Γ = √1 , H = L = 1 and P0 = 2 . We shall investigate the H∞ fixed-lag smoothing 3 6 estimation problem for d = 2 and γ = √12.5 . Firstly, the solution to (6.128) with P (0) = P0 = 23 is given by 1 1 P (s) = − + . 3 1 − 1.5s

(6.141)

t−2 t Since M (t, t) = Pt,t−2 and M (t − 2, s) = Ps,s−2 , it follows from (6.129) and (6.130) that

1 , (6.142) c1 (t − 2)e−1.291t + 0.7746   e1.291(t−2) M (t − 2, s) = e0.5727(s−t+2) −0.1455 + , c1 (s − 2) + 0.7746e1.291s (6.143) M (t, t) = −0.1455 +

where c1 (t − 2) is given by c1 (s − 2) = −0.7746e1.291(s−2) +

3 − 4.5(s − 2) e1.291(s−2) . −3.5635 + 0.8453(s − 2) (6.144)

In view of (6.141), it is obvious that P (t) is unbounded for t = 1/1.5 which implies that the H∞ filter zˇ(t | t) is not solvable for t = 1/1.5. On the other hand, from (6.142)-(6.143) we can know that M (t, t) is bounded for 2 ≤ t < 2 + 1/1.5 and M (t − 2, s) is bounded for t − 2 ≤ s ≤ t over 2 ≤ t < 2 + 1/1.5. Further, it can be calculated from (6.142)-(6.143) that M (t, t) = 1.2526 and M (t − 2, s) is bounded for t = 2 + 1/1.5, 1/1.5 ≤ s ≤ 2 + 1/1.5. Thus, from Theorem 6.4.2, the 2-lag smoother zˇ(t | t + 2) exists at t = 1/1.5. Remark 6.4.3. When the approach in [108] is applied, an H∞ smoother for the system in the example is given by zˆ(t | t + 2) = x ˆ(t + 2, 2), where x ˆ(t + 2, 2) is computed from the following PDE ∂x ˆ(t, τ ) ∂ x ˆ(t, τ ) + = P (t, τ, 0) [y(t) − x ˆ(t, 0)] , ∂t ∂τ ∂x ˆ(t, 0) = 0.5ˆ x(t, 0) + P (t, 0, 0) [y(t) − x ˆ(t, 0)] , ∂t where P (t, τ, 0) is the solution to the following Riccati type of PDE and boundary conditions ∂P (t, τ1 , τ2 ) ∂P (t, τ1 , τ2 ) ∂P (t, τ1 , τ2 ) + + ∂t ∂τ1 ∂τ2

6.5 Conclusion

141

= −P (t, τ1 , 0)P (t, 0, τ2 ) + 0.4P (t, τ1 , 2)P (t, 2, τ2 ), (6.145) ∂P (t, τ1 , 0) ∂P (t, τ1 , 0) + ∂t ∂τ1 = 0.5P (t, τ1 , 0) − P (t, τ1 , 0)P (t, 0, 0) + 0.4P (t, τ1 , 2)P (t, 2, 0), (6.146) ∂P (t, 0, 0) = P (t, 0, 0)+1/6−P 2(t, 0, 0)+0.4P (t, 0, 2)P (t, 2, 0), (6.147) ∂t where P (t, 0, s) = P (t, s, 0). It does not seem possible to give an analytical solution to the above PDEs. A solution may be obtained by numerical methods such as the finite element method, which is, however, computationally much more expensive than the present approach.

6.5 Conclusion In this chapter, we have proposed a re-organized innovation analysis approach to the H2 estimation for continuous-time systems with measurement delays. An estimator is designed by employing two standard Riccati differential equations. As an application of the proposed approach, we have studied the H∞ filtering problem for systems with measurement delays and the complicated H∞ fixed-lag smoothing. Necessary and sufficient solutions to the problems have been given in terms of Riccati differential equations. We have shown that the proposed re-organized innovation analysis approach plays key roles in dealing with the estimation problems for systems with delayed measurements and the related problems. In fact, by defining an appropriate stochastic system in Krein space, the H∞ filtering with measurement delays and the H∞ fixed-lag smoothing can be converted into H2 ones for associated systems with measurement delays in Krein space.

7. H∞ Estimation for Systems with Multiple State and Measurement Delays

In this chapter, the H∞ estimation problem for more general systems with multiple state and measurement delays is considered. The H∞ filtering problem is first converted into an H2 estimation associated with a stochastic system with multiple state and measurement delays in Krein space and then an estimator and its necessary and sufficient existence conditions are given in terms of partial differential equations.

7.1 Introduction In this chapter, we investigate the H∞ estimation for systems with delays that appear in both the state and measurement. In the H2 case, the problems have been addressed in [4, 13, 12, 52, 54]. In particular, a solution has been given in [52] in terms of the solution of a partial differential equation with boundary conditions. In the context of H∞ performance, note that delay systems are a class of infinite-dimensional systems and may be approached by the methods of [93, 34] which involve operator Riccati equations. However, the approaches are not easy to understand. Also, inspired by the early work [52] for the H2 estimation of delay systems and the Krein space approach to the H∞ estimation in the previous chapters, we aim to present a partial differential equation approach to the H∞ filtering with both state and measurement delays. By converting the H∞ filtering problem into an indefinite quadratic optimization one, an innovation approach in Krein space [30] is proposed to give a necessary and sufficient condition for the existence of an H∞ smoother in terms of a bounded solution of a Riccati type of partial differential equation. A smoother is then constructed. The case of filtering is in fact a special case of the smoothing. This chapter further demonstrates that the Krein space innovation approach is a powerful tool in dealing with complicated problems for systems with delays not only in i/o but also in state. Our result can be considered as the H∞ counterpart of the H2 result in [52] even though a different derivation method is applied. We further show that the H∞ prediction problem can be approached in the similar way and hence a necessary and sufficient condition is obtained. As special cases, solutions to the H∞ fixedlag smoothing and prediction for systems without delays are also given. H. Zhang and L. Xie: Cntrl. and Estim. of Sys. with I/O Delays, LNCIS 355, pp. 143–162, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com


144

7. H∞ Estimation for Systems with Multiple State and Measurement Delays

7.2 Problem Statements We consider the linear system with multiple time delays described by x(t) ˙ =

k 

Φi x(t − hi ) + Γ e(t),

(7.1)

Hi x(t − hi ) + v(t),

(7.2)

i=0

y(t) =

k  i=0

z(t) = Lx(t),

(7.3)

where x(t) ∈ Rn , e(t) ∈ Rr , y(t) ∈ Rp , v(t) ∈ Rp and z(t) ∈ Rq represent the state, input noise, measurement output, measurement noise and the signal to be estimated, respectively and Φi , Γ, Hi and L are bounded matrices with appropriate dimension. It is assumed that the input and measurement noises are deterministic and are from L2 [0, T ] where T > 0 is the time-horizon. The scalar quantities 0 = h0 < h1 < · · · < hk are known constant time delays of the  system. Denote h = max{h0 , · · · , hk }. The initial condition x(s)(−h ≤ s ≤ 0) is unknown. The general H∞ estimation problem including filtering, prediction and fixedlag smoothing is stated as: – H∞ Fixed-lag Smoothing: Given a desired noise attenuation level γ > 0, the smoothing lag d ≥ 0 and the observation {y(s), 0 ≤ s ≤ t}, find an estimate zˇ(t, d) of z(t − d), if it exists, such that the following inequality is satisfied: sup {x(s)|−h≤s≤0 ,u,v}=0 :T
[ˇ z (t, d) − z(t − d)] [ˇ z (t, d) − z(t − d)] dt d < γ2 :0 : :T
T −1   x (t)Πt x(t)dt + 0 e (t)e(t)dt + 0 v (t)v(t)dt −h

(7.4)

where Πt , t ∈ [−h, 0] is a given positive definite matrix function which reflects the relative uncertainty of the initial state x(t), −h ≤ t ≤ 0 about the origin to the input and measurement noises. We note that when d = 0, the above defines an H∞ filtering problem. – H∞ Prediction: Given the desired noise attenuation level γ > 0, the prediction time lead d > 0 and the observation {y(s), 0 ≤ s ≤ t − d}, find an estimate zˇ(t − d, −d) of z(t), if exists, such that the following inequality is satisfied: sup {x(s)|−h≤s≤0 ,u,v}=0 :T
[ˇ z (t − d, −d) − z(t)] [ˇ z (t − d, −d) − z(t)] dt 0 < γ 2 , (7.5) :0 :T
: T −d
−1  (t)Π x x(t)dt + e (t)e(t)dt + v (t)v(t)dt t −h 0 0 where Πt has the same meaning as in the fixed-lag smoothing.

7.3 H∞ Smoothing

145

Note that the use of notation zˇ(t − d, −d) for the prediction estimate of z(t) = z(t − d − (−d)) follows from the practice in the smoothing where zˇ(t, d) stands for the estimate of z(t − d). Remark 7.2.1. The H∞ filtering and fixed-interval smoothing problems for linear systems without delays have been solved in [71]. Their solutions are in terms of a bounded solution of a Riccati differential equation. For systems with delays, the problem is generally very difficult due to its infinite dimensional nature. There have been many attempts using a finite dimensional approach [74, 23, 68, 72, 67]. However, the solutions in [74, 23] are only sufficient and the results [68, 72, 67] only address systems with delays in the output. Although systems with delays in state may be approached by solutions for infinite dimensional systems, those solutions are given in terms of operator Riccati equations and are not explicit. To our knowledge, there has been no necessary and sufficient condition in explicit form for the H∞ estimation of linear systems with delays in state which appear in both the state and output equations. Our aim here is to provide such a condition using an innovation approach in Krein space [103].

7.3 H∞ Smoothing In this section, we shall first convert the deterministic fixed-lag smoothing problem into a stochastic optimization problem. The latter is then solved using an innovation approach in Krein space. In view of (7.4), we define 9 0 9 T 9 T   −1  JS (T ) = x (s)Πs x(s)ds + e (s)e(s)ds + v  (s)v(s)ds −h

−γ −2

0

9

T

0

vz (s)vz (s)ds,

(7.6)

d

where vz (s) = zˇ(s, d) − z(s − d) = zˇ(s, d) − Lx(s − d).

(7.7)

Let vz (s) = 0 for s < d. Then, JS (T ) can be rewritten as 9 0 9 T  −1 JS (T ) = x (s)Πs x(s)ds + e (s)e(s)ds −h

9

+ 0

T



v(s) vz (s)

 

0

Ip 0

0

−γ −2 Iq



 v(s) ds. vz (s)

(7.8)

It follows from [30] that an estimate zˇ(t, d)(d ≤ t ≤ T ) that achieves (7.4) exists if and only if: 1) the above quadratic function JS (T ) has a minimum JSmin with respect to e(t) (0 ≤ t ≤ T ) and x(t) (−h ≤ t ≤ 0); and 2) zˇ(t, d) (d ≤ t ≤ T ) can be chosen such that the minimum is positive for all y(·).

146

7. H∞ Estimation for Systems with Multiple State and Measurement Delays

Hence, the H∞ smoothing problem then becomes that of deriving the existence condition for the minimum JTmin of JS (T ) with respect to e(t) (0 ≤ t ≤ T ) and x(t) (−h ≤ t ≤ 0) and further find zˇ(t, d) (d ≤ t ≤ T ) (an estimate) to satisfy that JTmin > 0. To solve the latter problem, in the sequel we shall adopt an innovation approach in Krein space. 7.3.1

Stochastic System in Krein Space

In association with system (7.1)-(7.3) and the cost function (7.8), we introduce the stochastic system below. ˙ x(t) =

k 

Φi x(t − hi ) + Γ e(t),

(7.9)

Hi x(t − hi ) + v(t),

(7.10)

i=0

y(t) =

k  i=0

zˇ(t, d) = Lx(t − d) + vz (t), t ≥ d, (7.11) where the initial state x(τ ) (−h ≤ τ ≤ 0) and e(t), v(t) and vz (t) are mutually uncorrelated white noises with zero means and known covariance matrices, i.e., ⎤ ⎡ ⎤ ⎡ x(τ1 )  x(τ ) ⎢ e(t) ⎥ ⎢ e(t1 ) ⎥ ⎦, ⎣ ⎦ ⎣ v(t1 ) v(t) vz (t1 ) vz (t) ⎤ ⎡ Πτ δ(τ − τ1 ) 0 0 0 0 0 0 Qe (t)δ(t − t1 ) ⎥ ⎢ =⎣ ⎦, 0 0 Qv (t)δ(t − t1 ) 0 0 0 0 Qvz (t)δ(t − t1 ) (7.12) where −h ≤ τ, τ1 ≤ 0, Qe (t) = Ir , Qv (t) = Ip for t ≥ 0 and ⎧ ⎪ ⎪ 0, t
7.3 H∞ Smoothing

Denote



yz (t) =

⎧ ⎪ ⎪ y(t), ⎪ ⎪ ⎨

147

0≤t
  ⎪ ⎪ y(t) ⎪ ⎪ ,t≥d ⎩ ˇ z(t, d)

(7.13)

which is the measurement of the stochastic system (7.9)-(7.11) in an indefinite linear space. It follows from (7.13) that for 0 ≤ t < d yz (t) =

k 

Hi x(t − hi ) + v(t),

(7.14)

i=0

and for t ≥ d,

⎡

k 

⎤ Hi x(t − hi ) ⎦

yz (t) = ⎣ i=0 Lx(t − d)



 v(t) + . vz (t)

(7.15)

The measurements up to time t are collected as {yz (s), 0 ≤ s ≤ t}.

(7.16)

ˆz (s, 0) the projection of yz (s) onto Similar to the case in Hilbert space, define y L{yz (r), r < s}. It should be noted that unlike in Hilbert space a projection ˆ z (s, 0) exists, we define the in Krein space may not exist. If the projection y innovation of the observation yz (s) as 

ˆ z (s, 0). wz (s) = yz (s) − y

(7.17)

Note that wz (s) is in fact the prediction error of the observation. Obviously, the linear space L{yz (s), s < t} is equivalent to L{wz (s), s < t} [38]. ˆ (t, d) and x ˆ (t, hi ) of Lemma 7.3.1. Suppose that there exist the projections x x(t − d) and x(t − hi ), respectively, onto the linear space of L{yz (s), s < t}. Then, the innovation wz (t), defined by (7.17), can be given as ⎧ k  ⎪ ⎪ ˆ (t, hi ), y(t) − Hi x 0≤t
Qwz (t) = wz (t), wz (t)

(7.19)

148

7. H∞ Estimation for Systems with Multiple State and Measurement Delays

is given by ⎧ ⎪ ⎪ Ip , ⎪ ⎪ ⎨ Qwz (t) =

 ⎪ ⎪ Ip ⎪ ⎪ ⎩ 0

0≤t
(7.20)

ˆ (t, d) and x ˆ (t, hi ), 0 ≤ i ≤ k, (7.18) follows Proof: With the existence of x directly from (7.14)-(7.15) and the definition (7.17). Further, from (7.18), we have ⎧ k  ⎪ ⎪ ˆ (t, hi )] + v(t), Hi [x(t − hi ) − x 0≤t
(7.22)

Sufficient and Necessary Condition for the Existence of an H∞ Smoother

Based on the above discussion, we have the following result which provides a necessary and sufficient condition for the existence of an H∞ fixed-lag smoother. Theorem 7.3.1. Consider the system (7.1)-(7.3) and the associated performance criterion (7.4). Then, for a given scalar γ > 0, a smoother zˇ(t, d)(d ≤ t ≤ T ) that achieves (7.4) exists if and only if – x ˆ(t, d) exists for 0 ≤ t ≤ T , – x ˆ(t, hi ) exists for 0 ≤ t ≤ T and 0 ≤ i ≤ k, where x ˆ(t, d) and x ˆ(t, hi ) are respectively given from the projections of x(t − d) and x(t − hi ) onto {yz (s), s < t}. In this situation, a suitable H∞ smoother (central estimator) zˇ(t, d) is given by zˇ(t, d) = Lˆ x(t, d).

(7.23)

7.3 H∞ Smoothing

149

Proof: Recall that the quadratic cost JS (T ) in (7.8) has a minimum JSmin with respect to e(t) (0 ≤ t ≤ T ) and x(t) (−h ≤ t ≤ 0) if and only if the innovation wz (t) (0 ≤ t ≤ T ), defined by (7.17), exists (see [30], Chapter 16, pp. 499-529). It is readily known from (7.18) that the existence of wz (t) (0 ≤ t ≤ T ) requires that x ˆ(t, d) for d ≤ t ≤ T and xˆ(t, hi ) for 0 ≤ t ≤ T and 0 ≤ i ≤ k exist. Furthermore, if the innovation wz (s) exists, the minimum of JS (T ) with respect to e(t) (0 ≤ t ≤ T ) and x(τ ) (−h ≤ τ ≤ 0) is given by 9

T

JSmin = 0

9

d

=

wz (t)Q−1 wz (t)wz (t)dt wz (t)wz (t)dt +

0

9

T

wz (t)



d

Ip 0



0 −γ −2 Iq

wz (t)dt, (7.24)

where wz (t) is as defined in (7.18). By using (7.18), it follows that 9

T

JSmin =

% y(t) −

0

−γ −2

9

k 

& % y(t) −

Hi x ˆ(t, hi )

i=0 T

k 

& Hi x ˆ(t, hi ) dt

i=0 

[ˇ z (t, d) − Lˆ x(t, d)] [ˇ z (t, d) − Lˆ x(t, d)] dt.

(7.25)

d

It is then clear that a natural choice of zˇ(t, d) that renders JSmin > 0 is given by zˇ(t, d) = Lˆ x(t, d),

(7.26)

for all d ≤ t ≤ T , where x ˆ(t, d) is given from the projection of x(t − d) onto ∇ {yz (s), s < t}. 7.3.3

The Calculation of an H∞ Estimator zˇ(t, d)

In order to compute an H∞ smoother zˇ(t, d) from (7.23), we need to check ˆ(t, d). To this end, define the if x ˆ(t, d) and x ˆ(t, hi ) exist, and if so compute x cross-covariance matrix of the estimates of x(t − τ1 ) and x(t − τ2 ) as 

ˆ (t, τ1 ), x(t − τ2 ) − x ˆ (t, τ2 ), P (t, τ1 , τ2 ) = x(t − τ1 ) − x

(7.27)

ˆ (t, τi ) (i = 1, 2) is the projection of x(t − τi ) where τ1 ≥ 0, τ2 ≥ 0 and x onto L{yz (s), s < t}), yz (s) is as in (7.13). It is obvious that P (t, τ1 , τ2 ) = P (t, τ2 , τ1 ). We have the following results Theorem 7.3.2. The matrix P (t, τ1 , τ2 ) (τ1 ≥ 0, τ2 ≥ 0), defined as in (7.27), is the solution to the following Riccati type of partial differential equation and boundary conditions

150

7. H∞ Estimation for Systems with Multiple State and Measurement Delays

∂P (t, τ1 , τ2 ) ∂P (t, τ1 , τ2 ) ∂P (t, τ1 , τ2 ) + + ∂t ∂τ1 ∂τ2 k  =− P (t, τ1 , hi )Hi Hj P (t, hj , τ2 ) i,j=0

+γ

−2

P (t, τ1 , d)L LP (t, d, τ2 ),

(7.28)

k  ∂P (t, τ1 , 0) ∂P (t, τ1 , 0) + = P (t, τ1 , hi )Φi ∂t ∂τ1 i=0

−

k 

P (t, τ1 , hi )Hi Hj P (t, hj , 0)

i,j=0

+γ −2 P (t, τ1 , d)L LP (t, d, 0),

(7.29)

k k   ∂P (t, 0, 0) = Φi P (t, hi , 0) + P (t, 0, hi )Φi + Γ Γ  ∂t i=0 i=0

−

k 

P (t, 0, hi )Hi Hj P (t, hj , 0)

i,j=0

+γ −2 P (t, 0, d)L LP (t, d, 0),

(7.30)

where P (t, 0, s) = P (t, s, 0) .

(7.31)

In addition, the initial value P (0, τ1 , τ2 ), 0 ≤ τ1 , τ2 ≤ h is as P (0, τ1 , τ2 ) = x(−τ1 ), x(−τ2 ) = Π−τ1 δ(τ1 − τ2 ).

(7.32)

ˆ (t, τ ) is the projection of x(t − τ ) onto L{yz (s), s < t} or equivProof: Since x alently L{wz (s), s < t}, where wz (s) is the innovation, defined as (7.17), by using the projection formula [20], we have 9 t ˆ (t, τ ) = x x(t − τ ), wz (s)Q−1 (7.33) wz (s)wz (s)ds, 0

where Qwz (s) is as in (7.20). Thus, by considering the orthogonality of the ˆ (t, τ2 ) with the innovation wz (s), 0 ≤ s ≤ t, (7.27) estimation error x(t − τ2 ) − x is further given by ˆ (t, τ2 ) P (t, τ1 , τ2 ) = x(t − τ1 ), x(t − τ2 ) − x(t − τ1 ), x = x(t − τ1 ), x(t − τ2 ) − 9 t  x(t − τ1 ), wz (s)Q−1 wz (s)x(t − τ2 ), wz (s) ds. 0

(7.34)

7.3 H∞ Smoothing

151

Differentiating (7.34) with respect to t yields, ∂P (t, τ1 , τ2 ) ˙ − τ1 ), x(t − τ2 ) − = x(t ∂t 9 t  ˙ − τ1 ), wz (s)Q−1 x(t wz (s)x(t − τ2 ), wz (s) ds 0

˙ − τ2 ) − +x(t − τ1 ), x(t 9 t ˙ − τ2 ), wz (s) ds x(t − τ1 ), wz (s)Q−1 wz (s)x(t 0

 −x(t − τ1 ), wz (t)Q−1 wz (t)x(t − τ2 ), wz (t) .

(7.35)

Similarly, differentiating (7.34) with respect to τ1 and τ2 respectively yields, ∂P (t, τ1 , τ2 ) ˙ − τ1 ), x(t − τ2 ) + = −x(t ∂τ1 9 t  ˙ − τ1 ), wz (s)Q−1 x(t wz (s)x(t − τ2 ), wz (s) ds, (7.36) 0

∂P (t, τ1 , τ2 ) ˙ − τ2 ) + = −x(t − τ1 ), x(t ∂τ2 9 t ˙ − τ2 ), wz (s) ds. (7.37) x(t − τ1 ), wz (s)Q−1 wz (s)x(t 0

Then, the addition of (7.35)-(7.37) yields ∂P (t, τ1 , τ2 ) ∂P (t, τ1 , τ2 ) ∂P (t, τ1 , τ2 ) + + ∂t ∂τ1 ∂τ2  = −x(t − τ1 ), wz (t)Q−1 wz (t)x(t − τ2 ), wz (t) .

(7.38)

Next, by considering the innovation of (7.21), we have x(t − τ1 ), wz (t) ⎧  k  ⎪ ⎪ ˆ (t, hi )] + v(t) , Hi [x(t − hi ) − x 0≤t
152

7. H∞ Estimation for Systems with Multiple State and Measurement Delays

The last equality is due to the fact that v(t) and vz (t) are independent of x(t − τ1 ). By using (7.22), (7.38) is further rewritten as ∂P (t, τ1 , τ2 ) ∂P (t, τ1 , τ2 ) ∂P (t, τ1 , τ2 ) + + ∂t ∂τ1 ∂τ2  ⎧  k   k ⎪   ⎪ − P (t, τ , h )H P (t, τ , h )H ,0≤t
t

 x(t − τ1 ), wz (s)Q−1 wz (s)x(t), wz (s) ds.

(7.40) Differentiation with respect to t and τ1 , respectively, and addition of the results yield ∂P (t, τ1 , 0) ∂P (t, τ1 , 0) ˙ + = x(t − τ1 ), x(t) − ∂t ∂τ1 9 t ˙ x(t − τ1 ), wz (s)Q−1 wz (s) ds wz (s)x(t), 0

 −x(t − τ1 ), wz (t)Q−1 wz (t)x(t), wz (t) .

(7.41) On the other hand, by applying (7.9) and (7.34), we have 9 ˙ − x(t − τ1 ), x(t) 0

=

k 

˙ x(t − τ1 ), wz (s)Q−1 wz (s) ds wz (s)x(t),

[x(t − τ1 ), x(t − hi )−

i=0 t

9 0

=

t

  x(t − τ1 ), wz (s)Q−1 (s)x(t − h ), w (s) ds Φi i z wz

k 

P (t, τ1 , hi )Φi .

i=0

From (7.39), it follows that  x(t − τ1 ), wz (t)Q−1 wz (t)x(t), wz (t)

(7.42)

7.3 H∞ Smoothing

⎧ & % k k ⎪   ⎪   ⎪ P (t, τ1 , hi )Hi P (t, 0, hj )Hj , 0 ≤ t < d ⎪ ⎪ ⎪ i=0 j=0 ⎪ ⎪ ⎪ ⎪ ⎨ & =  % k ⎪  k  ⎪ ⎪   ⎪ P (t, τ1 , hi )Hi P (t, 0, hj )Hj ⎪ ⎪ ⎪ i=0 j=0 ⎪ ⎪ ⎪ ⎩ −γ −2 P (t, τ , d)L [P (t, 0, d)L ] , t ≥ d. 1

153

(7.43)

Thus (7.29) is derived by substituting (7.42) and (7.43) into (7.41). Finally, setting τ1 = τ2 = 0 in (7.34) we have 9

t

P (t, 0, 0) = x(t), x(t) − 0

 x(t), wz (s)Q−1 wz (s)x(t), wz (s) ds.

(7.44) Differentiating (7.44) with respect to t yields ∂P (t, 0, 0) ˙ = x(t), x(t) − ∂t

9

t

0

 ˙ x(t), wz (s)Q−1 wz (s)x(t), wz (s) ds

9

˙ +x(t), x(t) − 0

t

˙ x(t), wz (s)Q−1 wz (s) ds wz (s)x(t),

 −x(t), wz (t)Q−1 wz (t)x(t), wz (t) .

(7.45)

Note also that 9 ˙ x(t), x(t) − =

k 



0

t

 ˙ x(t), wz (s)Q−1 wz (s)x(t), wz (s) ds

9

Φi x(t − hi ), x(t) −

x(t − 0

i=0

=

t

k 

 hi ), wz (s)Q−1 wz (s)x(t), wz (s) ds

Φi P (t, hi , 0).



(7.46)

i=0

Thus (7.30) is obtained directly from (7.45) by using (7.46) and (7.42)-(7.43). ˆ (0, τ1 ) = 0 and x ˆ (0, τ2 ) = 0, it follows from From definition (7.27), note that x (7.12) that P (0, τ1 , τ2 ) = x(−τ1 ), x(−τ2 ) = Π−τ1 δ(τ1 − τ2 ). This completes the proof of Lemma.

(7.47) ∇

Remark 7.3.2. It should be noted that for the filtering case of d = 0, and when the system is with no delays i.e., Φi = 0 and Hi = 0 for i = 1, · · · , k, (7.30) becomes a standard differential Riccati equation for H∞ filtering.

154

7. H∞ Estimation for Systems with Multiple State and Measurement Delays

Based on the solution P (t, τ1 , τ2 ) in Theorem 7.3.2, we shall now give a solution to the fixed-lag smoothing problem. Theorem 7.3.3. Consider the system (7.1)-(7.3) and the associated performance criterion (7.4). Given the desired noise attenuation γ > 0 and the smoothing lag d ≥ 0, the H∞ fixed-lag smoothing problem of (7.4) is solvable if and only if there exists a bounded matrix solution P (t, τ, hi ) (0 ≤ i ≤ k) for 0 ≤ t ≤ T and 0 ≤ τ ≤ hmax , where hmax = max{h, d} = max{h0 , · · · , hk , d}, to the partial differential equations (7.28)-(7.30). In this case, the central estimator zˇ(t, d) is given by zˇ(t, d) = Lˆ x(t, d),

(7.48)

where xˆ(t, d) is computed as % & k  ˆ(t, τ ) ∂x ˆ(t, τ ) ∂ x + = K(t, τ, t) y(t) − Hi x ˆ(t, hi ) , (7.49) ∂t ∂τ i=0 % & k k   ∂x ˆ(t, 0) = Φi x ˆ(t, hi ) + K(t, 0, t) y(t) − Hi x ˆ(t, hi ) , ∂t i=0 i=0 (7.50) where K(t, τ, t) =

k 

P (t, τ, hi )Hi ,

(7.51)

i=0

and initial value xˆ(0, τ ) = 0 for τ ≥ 0. Proof: Note that x ˆ(t, τ ) is obtained from the projection of x(t − τ ) onto L{yz (s), s < t} or equivalently L{wz (s), s < t}, which is given by 9 t x(t − τ ), wz (s)Q−1 (7.52) x ˆ(t, τ ) = wz (s)wz (s)ds. 0

Differentiating (7.52) with respect to t and τ , respectively, yields, 9 t ∂x ˆ(t, τ ) ˙ − τ ), wz (s)Q−1 = x(t wz (s))wz (s)ds + ∂t 0 x(t − τ ), wz (t)Q−1 wz (t)wz (t), 9 t ∂x ˆ(t, τ ) ˙ − τ ), wz (s)Q−1 =− x(t wz (s))wz (s)ds. ∂τ 0

(7.53) (7.54)

The addition of (7.53) and (7.54) yields, ∂x ˆ(t, τ ) ∂ x ˆ(t, τ ) + = x(t − τ ), wz (t)Q−1 wz (t)wz (t). ∂t ∂τ

(7.55)

7.3 H∞ Smoothing

155

In view of (7.18) and noting that zˇ(t, τ ) = Lˆ x(t, τ ), the innovation wz (t) is given by ⎧ k  ⎪ ⎪ y(t) − Hi x ˆ(t, hi ), 0≤t
9

t

−1 ˙ x(t), wz (s)Q−1 wz (s)wz (s)ds + x(t), wz (t)Qwz (t)wz (t)

0 k  i=0

9

t

Φi (t) 0

x(t − hi ), wz (s)Q−1 wz (s)wz (s)ds +

x(t), wz (t)Q−1 wz (t)wz (t) =

k 

Φi (t)ˆ x(t, hi ) + x(t), wz (t)Q−1 wz (t)wz (t).

(7.58)

i=0

In view of (7.39) and (7.56), (7.50) follows directly from (7.58). Recall that a smoother zˇ(t, d)(0 ≤ t ≤ T ) that achieves (7.4) exists if and only if x ˆ(t, d) for d ≤ t ≤ T and xˆ(t, hi ) for 0 ≤ t ≤ T and 0 ≤ i ≤ k exist, which, from (7.49)−(7.50) and (7.51), is equivalent to that there exists bounded matrix solution P (t, τ, hi ) (0 ≤ i ≤ k) for 0 ≤ t ≤ T and 0 ≤ τ ≤ hmax to the partial differential equations (7.28)-(7.30). The result is established. ∇ Remark 7.3.3. Theorem 7.3.3 gives a necessary and sufficient condition for the existence of a H∞ filter (d = 0) or fixed-lag smoother for linear delay systems. The result involves solving a Riccati type of partial differential equation with given boundary conditions and can be considered as the H∞ counterpart of the H2 result in [52]. The solution clearly demonstrates the infinite dimensional nature of the problem. Remark 7.3.4. – It should be noted that (7.50) bears the similarity of the Kalman filtering Actually,  formulation. K(t, 0, t) can be regarded as the gain k matrix while y(t) − i=0 Hi x ˆ(t, hi ) is the innovation.

156

7. H∞ Estimation for Systems with Multiple State and Measurement Delays

– By setting Φi ≡ 0 and Hi ≡ 0, i = 1, 2, · · · , k in (7.1)-(7.3), Theorem 7.3.3 provides a necessary and sufficient condition for the H∞ fixed-lag smoothing problem of systems without delays. – In the case of filtering with no delays, i.e., d = 0, Φi ≡ 0 and Hi ≡ 0, i = 1, 2, · · · , k , it reduces to the result of [71]. Note that in this case, the filter equation (7.50) becomes ∂x ˆ(t, 0) = Φ0 x ˆ(t, 0) + K(t, 0, t) [y(t) − H0 x ˆ(t, 0)] , ∂t

(7.59)

with K(t, 0, t) = P (t, 0, 0)H0 , and the covariance equation (7.30) becomes ∂P (t, 0, 0) = Φ0 P (t, 0, 0) + P (t, 0, 0)Φ0 + Γ Γ  ∂t −P (t, 0, 0)H0 H0 P (t, 0, 0) γ −2 + P (t, 0, 0)L LP (t, 0, 0)

(7.60)

with P (0, 0, 0) = Π0 . Thus the derived filter is the same as the one obtained in [71]. – By setting γ → ∞ in (7.4), zˇ(t, d) becomes an H2 estimator. Note that when γ → ∞, the Riccati type of partial differential equation and boundary conditions (7.28)-(7.30) becomes that ∂P (t, τ1 , τ2 ) ∂P (t, τ1 , τ2 ) ∂P (t, τ1 , τ2 ) + + ∂t ∂τ1 ∂τ2 k  =− P (t, τ1 , hi )Hi Hj P (t, hj , τ2 ),

(7.61)

i,j=0

∂P (t, τ1 , 0) ∂P (t, τ1 , 0)  + = P (t, τ1 , hi )Φi ∂t ∂τ1 i=0 k

−

k 

P (t, τ1 , hi )Hi Hj P (t, hj , 0),

(7.62)

i,j=0 k k   ∂P (t, 0, 0) = Φi P (t, hi , 0) + P (t, 0, hi )Φi ∂t i=0 i=0

+Γ Γ  −

k 

P (t, 0, hi )Hi Hj P (t, hj , 0).

(7.63)

i,j=0

In this case of γ → ∞, the obtained results in this paper are the same as in [52] for the H2 optimal filtering in linear systems with time delays and Q1 (t) = Γ Γ T and Q2 (t) = Ip .

7.4 H∞ Prediction

157

Remark 7.3.5. To compute the optimal H∞ smoother zˇ(t, d), we need the solution to the partial differential equations (7.28)-(7.30)) and (7.49)-(7.50). Note that the partial differential equations (7.28)-(7.30)) and (7.49)-(7.50) are nonlinear. An analytical solution to these equations may not be possible even if it exists. However, they may be solved by numerical methods such as the finite element method [33, 52].

7.4 H∞ Prediction In the previous section we have addressed the H∞ fixed-lag smoothing and filtering problems. In this section, we shall extend the innovation analysis technique to deal with the H∞ prediction problem. In view of (7.5), we define 

JP (T ) =

9

0

−h

x (s)Πs−1 x(s)ds +

T

e (s)e(s)ds +

9

0

9

−γ −2

9

T

T −d

v  (s)v(s)ds

0

vz (s − d)vz (s − d)ds,

(7.64)

0

where 

vz (s − d) = zˇ(s − d, −d) − z(s) = zˇ(s − d, −d) − Lx(s).

(7.65)

Let v(s) = 0 for s < 0, v 0 (s) = v(s − d) and vz0 (s) = vz (s − d). Then, JP (T ) can be rewritten as 9 0 9 T JP (T ) = x (s)Πs−1 x(s)ds + e (s)e(s)ds −h

T



T



0

  Ip v(s − d) 0 ds vz (s − d) 0 −γ −2 Iq 0 9 T 9 0  −1 x (s)Πs x(s)ds + e (s)e(s)ds = 9

+

−h

9

+ 0

v(s − d) vz (s − d)

0

v (s) vz0 (s)

 

 

0

Ip 0

0 −γ −2 Iq



 v 0 (s) ds. vz0 (s)

(7.66)

In view of (7.2) and (7.65), define the following measurement and fictitious measurement: y 0 (t) =

k 

Hi0 x(t − h0i ) + v 0 (t), t ≥ d

(7.67)

i=0

zˇ0 (t, 0) = Lx(t) + vz0 (t), where y 0 (t) = y(t − d), Hi0 = Hi , zˇ0 (t, 0) = zˇ(t − d, −d) and h0i = hi + d.

(7.68)

158

7. H∞ Estimation for Systems with Multiple State and Measurement Delays

Remark 7.4.1. Note that the time delays in observation equation (7.67) is h0i which is different from the one in system (7.1). So the problem considered in this section is different from the one studied in the last section in which the delays in both the system and observation equations are the same as hi . According to the discussion in the last section, zˇ(t − d, −d) = zˇ0 (t, 0)(0 ≤ t ≤ T ) that achieves (7.5) exists if and only if [30]: 1) JP (T ) has a minimum JPmin , with respective e(t) (0 ≤ t ≤ T ) and x(t) (−h ≤ t ≤ 0) and 2) zˇ0 (t, 0) can be chosen such that JPmin is positive for all y 0 (·). Similar to the case as in the last section, we introduce the stochastic system below. ˙ x(t) =

k 

Φi x(t − hi ) + Γ e(t),

(7.69)

i=0

y0 (t) =

k 

Hi0 x(t − h0i ) + v0 (t),

t≥d

(7.70)

i=0

ˇ z0 (t, 0) = Lx(t) + vz0 (t),

(7.71)

where the initial state x(τ ) (−h ≤ τ ≤ 0) and e(t), v0 (t) (t ≥ d) and vz0 (t) are mutually uncorrelated white noises with zero means and known covariance matrices, i.e., ⎤ ⎡ ⎤ ⎡ x(τ1 )  x(τ ) ⎢ e(t) ⎥ ⎢ e(t1 ) ⎥ ⎦ ⎣ 0 ⎦, ⎣ 0 v (t1 ) v (t) vz0 (t) vz0 (t1 ) ⎡ Πτ δ(τ − τ1 ) 0 0 Qe (t)δ(t − t1 ) ⎢ =⎣ 0 0 0 0

⎤ 0 0 0 0 ⎥ ⎦, Qv0 (t)δ(t − t1 ) 0 0 Qvz0 (t)δ(t − t1 ) (7.72)

while −h ≤ τ, τ1 ≤ 0, Qe (t) = Ir , Qvz0 (t) = −γ 2 Iq for t ≥ 0 and ⎧ ⎨ 0, Qv0 (t) =

⎩

t
Ip , t ≥ d.

Define the cross-covariance matrix of the estimates of x(t−τ1 ) and x(t−τ2 ) as P (t, τ1 , τ2 )



ˆ (t, τ1 ), x(t − τ2 ) − x ˆ (t, τ2 ), = x(t − τ1 ) − x

(7.73)

ˆ (t, τi ) (i = 1, 2) is the projection of x(t − τi ) onto where τ1 ≥ 0, τ2 ≥ 0 and x L{yz0 (s), s < t}), where

7.4 H∞ Prediction

⎧ 0 zˇ (t, 0) = Lx(t) + vz0 (t), 0≤t
159

(7.74)



ˆ z0 (t, 0), From (7.74), the innovation of yz0 (s), denoted by wz0 (t) = yz0 (t) − y ˆ z0 (t, 0) is the projection of yz0 (t) onto L{yz0 (s), s < t}, is given where y ⎧ 0 zˇ (t, 0) − Lˆ x(t, 0), 0≤t
Then the innovation covariance matrix of wz0 (t), denoted by Qwz0 (t) = wz0 (t), wz0 (t), is computed by ⎧ −γ 2 Iq , 0≤t
∂P (t, τ1 , τ2 ) ∂P (t, τ1 , τ2 ) + ∂τ1 ∂τ2 k  = − P (t, τ1 , hi + d)H  i (t − d)Hj (t − d)P (t, hj + d, τ2 )

+

i,j=0

+γ −2 P (t, τ1 , 0)L LP (t, 0, τ2 ), ∂P (t, τ1 , 0) ∂t

(7.78)

∂P (t, τ1 , 0)  = P (t, τ1 , hi )Φi ∂τ1 i=0 k

+

−

k 

P (t, τ1 , hi + d)H  i (t − d)Hj (t − d)P (t, hj + d, 0)

i,j=0

+γ −2 P (t, τ1 , 0)L LP (t, 0, 0),

(7.79)

160

7. H∞ Estimation for Systems with Multiple State and Measurement Delays

∂P (t, 0, 0) ∂t

k 

=

Φi P (t, hi , 0) +

i=0

−

k 

P (t, 0, hi )Φi + Γ Γ 

i=0

k 

P (t, 0, hi + d)H  i (t − d)Hj (t − d)P (t, hj + d, 0)

i,j=0

+γ −2 P (t, 0, 0)L LP (t, 0, 0),

(7.80)

where Hi (t − d) = 0 for t < d, Hi (t − d) = Hi for t ≥ d, and P (t, 0, s) = P (t, s, 0) .

(7.81)

In addition, the initial value P (0, τ1 , τ2 ), 0 ≤ τ1 , τ2 ≤ h is as P (0, τ1 , τ2 ) = x(−τ1 ), x(−τ2 ) = Π−τ1 δ(τ1 − τ2 ).

(7.82)

Proof: From the definition of (7.73), by a similar discussion as in last Section, we have P (t, τ1 , τ2 )

ˆ (t, τ2 ) = x(t − τ1 ), x(t − τ2 ) − x(t − τ1 ), x = x(t − τ1 ), x(t − τ2 ) − 9 t 0  x(t − τ1 ), wz0 (s)Q−1 w 0 (s)x(t − τ2 ), wz (s) ds. 0

z

(7.83) By differentiating (7.83) with respect to, t, τ1 and τ2 , respectively and make addition yields ∂P (t, τ1 , τ2 ) ∂t

+ =

∂P (t, τ1 , τ2 ) ∂P (t, τ1 , τ2 ) + ∂τ1 ∂τ2 0  −x(t − τ1 ), wz0 (t)Q−1 w 0 (t)x(t − τ2 ), wz (t) ,

(7.84)

z

Next, by considering the innovation of (7.76), we have x(t − τ1 ), wz0 (t) 4 ⎧ 3 ˆ (t, 0)] + vz0 (t) , 0 ≤ t < d x(t − τ1 ), L [x(t) − x ⎪ ⎪ ⎪ ⎨ ⎡ k ⎤  0    0 = 0 0 ˆ x(t − h H ) − x (t, h ) ⎪ i i i ⎣ ⎦ + v0 (t) ⎪ , t≥d ⎪ ⎩ x(t − τ1 ), i=0 vz (t) ˆ (t, 0)] L [x(t) − x ⎧ 0≤t
By taking into account (7.77) and (7.85), (7.78) follows from (7.84). Similarly we can establish (7.79)-(7.80). ∇ Now we are in the position to present the main result of this section.

7.4 H∞ Prediction

161

Theorem 7.4.2. Consider the system (7.1)-(7.3) and the associated performance criterion (7.5). Given a desired noise attenuation γ > 0, the prediction time lead d > 0, the H∞ prediction problem of (7.5) is solvable if and only if there exists a bounded matrix solution P (t, τ, hi + d) (0 ≤ i ≤ k) for 0 ≤ t ≤ T and 0 ≤ τ ≤ h0max , where h0max = max{h0 + d, · · · , hk + d}, to the partial differential equations (7.78)-(7.80). In this case, a suitable predictor zˇ(t − d, −d) = zˇ0 (t, 0) is given by zˇ0 (t, 0) = Lˆ x(t, 0),

(7.86)

where xˆ(t, 0) is computed from ∂x ˆ(t, τ ) ∂ x ˆ(t, τ ) + ∂t ∂τ

% = K(t, τ, t) y(t − d) −

k 

& Hi x ˆ(t, hi + d) ,

i=0

(7.87) ∂x ˆ(t, 0) ∂t

=

k 

Φi x ˆ(t, hi ) +

i=0

%

K(t, 0, t) y(t − d) −

k 

& Hi x ˆ(t, hi + d) ,

i=0

(7.88) with K(t, τ, t) =

k 

P (t, τ, hi + d)Hi ,

(7.89)

i=0

and initial value xˆ(0, τ ) = 0 τ ≥ 0. Proof: As we discuss in the above, if the innovation wz0 (s) exists, the minimum of JP (T ) with respect to {x(0); u(s), 0 ≤ s ≤ T } is given by 9 T  0 JPmin = wz0 (t)Q−1 (7.90) w 0 (t)wz (t)dt, z

0

where wz0 (t) is as defined in (7.75). By using (7.75), it follows that 9 T ! ! JPmin = −γ −2 x(t, 0) zˇ0 (t, 0) − Lˆ x(t, 0) dt + zˇ0 (t, 0) − Lˆ 9

T

%

0

y (t) − 0

d

k 

& % Hi0 xˆ(t, h0i )

i=0

y (t) − 0

k 

& Hi0 x ˆ(t, h0i )

dt.(7.91)

i=0

It is then clear that a natural choice of zˇ0 (t, 0) that renders JPmin > 0 is given by zˇ0 (t, 0) = Lˆ x(t, 0),

(7.92)

162

7. H∞ Estimation for Systems with Multiple State and Measurement Delays

for all 0 ≤ t ≤ T , where x ˆ(t, 0) is obtained from the projection of x(t) onto L{yz0 (s), s < t} or equivalently L{wz0 (s), s < t}, which is given by 9 x ˆ(t, 0) = 0

t

0 x(t), wz0 (s)Q−1 w 0 (s)wz (s)ds. z

(7.93)

Differentiating (7.93) with respect to t yields, ∂x ˆ(t, 0)  0 = Φi x ˆ(t, hi ) + x(t), wz0 (t)Q−1 wz0 (t)wz (t). ∂t i=0 k

(7.94)

x(t, 0), (7.88) In view of (7.75), (7.77) and (7.85) and note that zˇ0 (t, 0) = Lˆ follows directly from (7.94). On the other hand, note that 9 ˆ (t, τ ) x

= 0

t

0 x(t − τ ), wz0 (s)Q−1 w 0 (s)wz (s)ds, z

(7.95)

By applying a similar discussion, (7.87) is obtained. Recall that a predictor zˇ(t − d, −d)(0 ≤ t ≤ T ) that achieves (7.5) exists if and only if the innovation wz0 (t) for 0 ≤ t ≤ T exists which, from (7.75), is equivalent to that x ˆ(t, 0) and x ˆ(t, h0i ) for 0 ≤ i ≤ k exist. Thus, from (7.87)(7.89), a predictor zˇ(t − d, −d)(0 ≤ t ≤ T ) exists if and only if there exists a bounded matrix solution P (t, τ, h0i ) (0 ≤ i ≤ k) for 0 ≤ t ≤ T and 0 ≤ τ ≤ h0max , to the partial differential equations (7.78)-(7.80). The result is established. ∇

7.5 Conclusion In this chapter we have studied the H∞ estimation problem for linear systems with multiple state delays. A necessary and sufficient condition for the existence of an estimator is obtained in terms of a partial differential equation with boundary conditions. The approach applied in this paper is the innovation analysis in Krein space, which is completely different from the traditional game theoretic theory and operator approach. Note that it is not clear if the latter approaches could be applied to give an explicit solution to the problem due to its complexity. It has been shown that the presented result in this chapter includes the standard H∞ filtering result given in a previous work as a special case. We have also solved the H∞ fixed-lag smoothing problem for linear systems without delays. Due to the duality of the control and filtering, we believe that the presented results can be extended to the H∞ control for time delay systems to give a necessary and sufficient condition.

8. Optimal and H∞ Control of Continuous-Time Systems with Input/Output Delays

In this chapter we study the LQR, LQG and H∞ control for continuous-time systems with multiple i/o delays. Similar to the discrete-time case in Chapter 6, we shall establish a duality between the LQR and a smoothing estimation for an associated backward stochastic system without delay. The duality allows us to give a simple solution to the LQR problem with elementary tools such as projections. We then establish a separation principle and solve the LQG control for systems with multiple i/o delays. Finally the H∞ full-information control with input delays is also solved.

8.1 Introduction For continuous-time systems, time delay problems can in principle be treated by the infinite-dimensional system theory [93, 19]. This approach, however, leads to a solution in terms of a Riccati type partial differential equation or operator Riccati equations which are difficult to understand and compute. Much effort has been made in order to derive an explicit and efficient solution for systems with i/o delays. An earlier treatment of multiple delay systems in [29] involved rather complicated computation and the results are restrictive. [46] discusses a general class of H∞ control problem for systems with multiple input delays and treats the LQR problem as a limiting case of the H∞ control. The solvability condition and analytic solution of [46] are established based on the stable eigenspace of a Hamlitonian matrix. However, checking the existence of the stable eigenspace and finding the minimal root of the transcendent equation required for the controller design may be computationally expensive. By splitting the delay problem into a nested sequence of elementary problems, [62] derives a complete solution to the standard H∞ problem for systems having multiple i/o delays. The solvability and explicit solution depend on a number of J-spectral factorizations. [70] studies the H2 control problem of systems with i/o delays using a frequency/time domain approach where the given solution is actually not analytical and explicit. In this chapter, we focus on the finite horizon control problems under both the H2 and H∞ performances for i/o delayed systems. The establishment of duality, H. Zhang and L. Xie: Cntrl. and Estim. of Sys. with I/O Delays, LNCIS 355, pp. 163–203, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com


164

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

the separation principle, the indefinite positive LQR optimization theory and the Kalman filtering formulation will be the key techniques. These techniques allow us to give a simple derivation and solution to the problems under investigations. The chapter is organized as follows. In Section 8.2, we study the LQR problem for multiple input delay systems. We first establish the duality between the LQR for the multiple input delayed system and a smoothing problem for an associated system without delay. The duality allows us to solve the very complicated LQR problem via elementary tools such as projections. An analytical solution to the LQR problem is then presented in terms of Riccati differential equations. In Section 8.3, the measurement feedback control for i/o delayed systems is studied by establishing a separation principle. In Section 8.4, we investigate the H∞ full-information control problem. The problem is solved with an indefinite LQR approach.

8.2

Linear Quadratic Regulation

In this section we shall study the LQR problem for continuous-time systems with multiple input delays. Recall that for linear delay free systems, the LQR control is dual to the Kalman filtering and the state feedback controller is obtained directly by calculating the Kalman filtering gain matrix. Here we extend this duality for systems with multiple input delays and establish a duality between the LQR for systems with multiple input delays and a smoothing estimation. Our duality contains the duality for delay free systems as a special case. 8.2.1

Problem Statements

Consider the following continuous-time system with multiple input delays: x(t) ˙ = Φx(t) +

l 

Γ(i) ui (t − hi ), l ≥ 1,

(8.1)

i=0

where x(t) ∈ Rn and ui (t) ∈ Rmi represent the state and the control input, respectively. Φ and Γ(i) , i = 0, 1, · · · , l are bounded matrices. Without loss of generality, we assume that the delays are in an increasing order: 0 = h0 < h1 < · · · < hl , and the control input signals have the same dimension, i.e. m0 = m1 = · · · = ml = m. Associated with the system (8.1), we introduce the following quadratic performance cost: 9 tf l 9 tf −hi  ui (t)R(i) ui (t)dt + x (t)Qx(t)dt, (8.2) Jtf = xtf Pf xtf + i=0

0

0

where tf > hl , xtf is the terminal state, Pf = Pf ≥ 0 imposes a penalty on the terminal state, R(i) is a bounded positive definite matrix function and Q is a non-negative definite bounded matrix function. The LQR problem for the system (8.1) is stated as: find control inputs ui (t − hi ), hi ≤ t ≤ tf , i = 0, 1, · · · , hl that minimize the performance cost (8.2).

8.2 Linear Quadratic Regulation

165

Remark 8.2.1. The LQR problem for continuous-time systems with input delays has been investigated in some existing works such as [19, 70]. [19] treated general systems with input and output delays using infinite dimensional system theory where the optimal solution is given in terms of operator Riccati differential equations and differential matrix equations. In [70], the LQR problem for systems with multiple i/o delays in the infinite horizon case has been dealt with using a time-domain approach. 8.2.2

Preliminaries

Similar to the discussion in Chapter 3 for discrete-time systems, we shall convert the LQR problem for continuous-time systems into an optimization problem associated with a stochastic model, which will enable us to solve the LQR problem via a smoothing approach in the next subsection. We first introduce the following notations: ⎧⎡ ⎤ u0 (t − h0 ) ⎪ ⎪ ⎪ ⎢ ⎪ ⎥ .. ⎪ ⎪ ⎣ ⎦ , hi ≤ t < hi+1 , . ⎪ ⎪ ⎨  (t − h ) u i ⎤ u(t) = ⎡ i (8.3) u0 (t − h0 ) ⎪ ⎪ ⎪ ⎪⎢ ⎥ .. ⎪ ⎪ ⎦ , t ≥ hl , ⎣ . ⎪ ⎪ ⎩ ul (t − hl )



u ˜(t) =

⎧ l  ⎪ ⎨ Γ(j) uj (t − hj ), hi ≤ t < hi+1 , j=i+1

⎪ ⎩ 0, ⎧ ⎨[Γ  (0) Γ (t) = ⎩[Γ

· · · Γ(i) ] , hi ≤ t < hi+1 , · · · Γ(l) ] , t ≥ hl ,

(0)



R(t) =

(8.4)

t ≥ hl ,

⎧ ⎨ diag{R

· · · , R(i) }, hi ≤ t < hi+1 ,

⎩ diag{R

· · · , R(l) }, t ≥ hl .

(0) , (0) ,

(8.5)

(8.6)

Then the system (8.1) and the cost (8.2) can be rewritten respectively as ⎧ ⎨ Φx(t) + Γ (t)u(t) + u ˜(t), hi ≤ t < hi+1 , (8.7) x(t) ˙ = ⎩ Φx(t) + Γ (t)u(t), t≥h l

and Jtf = xtf Pf xtf +

9 0

tf

u (t)R(t)u(t)dt +

9 0

tf

x (t)Qx(t)dt.

(8.8)

166

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

Hence, the state x(t) can be given in terms of the initial state x(0) and the control input u as 9 t 9 t Π(t, τ )˜ u(τ )dτ + Π(t, τ )Γ (τ )u(τ )dτ, (8.9) x(t) = Π(t, 0)x(0) + 0

0

where Π(t, τ ) is the transition matrix corresponding to the state matrix Φ and satisfies the matrix differential equation dΠ(t, τ ) = ΦΠ(t, τ ), Π(τ, τ ) = I. dt

(8.10)

Remark 8.2.2. For t ≥ hl , since u ˜(t) = 0, (8.9) can be rewritten as 9

9

hl

Π(t, τ )˜ u(τ )dτ +

x(t) = Π(t, 0)x(0) +

t

Π(t, τ )Γ (τ )u(τ )dτ. (8.11) 0

0

By dropping the explicit dependence on t, we may write the equation (8.9) (or (8.11)) in an operator form: x = Ox(0) + H˜ u + Pu,

(8.12)

where O is the operator that maps x(0) to x(·) according to the rule x(t) = Π(t, 0)x(0), and H and P are the integral operators that map u ˜ and u to x(·), ac: min{t,hl } :t cording the rules x(t) = 0 Ψ (t, τ )˜ u(τ )dτ and x(t) = 0 Ψ (t, τ )Γ (τ )u(τ )dτ , respectively. Let s be the continuously indexed collection of Krein space variables, i.e., s = {s(t); 0 ≤ t ≤ tf }. Motivated by the discrete-time setting, we define 9 tf  s Ms ≡ s (τ )M (τ )s(τ )dτ

(8.13)

(8.14)

0

for any non-negative time-varying bounded matrix M (τ ). Then, the cost function, Jtf , can be rewritten as a quadratic form of x(0), u˜ and u: Jtf = [Ox(0) + H˜ u + Pu] Pf [Ox(0) + H˜ u + Pu] + [Ox(0) + H˜ u + Pu] Q × [Ox(0) + H˜ u + Pu] + u Ru ⎡ ⎤ x(0) ˜  u ] Ω ⎣ u ˜ ⎦ = [ x (0) u u where



Ω11 Ω= Ω21

(8.15)  Ω12 , Ω22

(8.16)

8.2 Linear Quadratic Regulation

with

167



 O Pf O + O QO O Pf H + O QH , H Pf O + H QQ H Pf H + H QH    O Pf P + O QP  = , , Ω21 = Ω12 H Pf P + H QP

Ω11 = Ω12

Ω22 = P  Pf P + P  QP + R.

(8.17)

Next, in association with (8.12), we introduce the following operator form of a stochastic linear model: x(0) = O xtf + O η, x = H xtf + H η,

(8.18) (8.19)

y = P  xtf + P  η + v,

(8.20)

where xtf , η and v in bold face are linear space random vectors satisfying ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ xtf xtf Pf 0 0 ⎣ η ⎦, ⎣ η ⎦ = ⎣ 0 Q 0 ⎦, (8.21) v v 0 0 R with Q and R the diagonal operators of Kernels Qδ(t − τ ) and R(t)δ(t − τ ), respectively. From (8.11), it is easy to observe that the time horizon of x in (8.19) is (0, hl ], i.e. x = {x(t), 0 < t ≤ hl }, whereas y = {y(t), 0 ≤ t ≤ tf }. Then it is straightforward to see that ⎤ ⎡ ⎤ ⎡ x(0) x(0) ⎣ x ⎦, ⎣ x ⎦ = Ω (8.22) y y where Ω is as in (8.16). It is not difficult to see that the system (8.18) and (8.20) is equivalent to the following backward dual state-space model, ˙ − x(t) = Φ x(t) + η(t), y(t) = Γ  (t)x(t) + v(t), with x(t) |t=tf = xtf , and ⎤ ⎡ ⎤ ⎡ ⎡ xtf Pf xtf ⎣ η(t) ⎦ , ⎣ η(τ ) ⎦ = ⎣ 0 v(t) v(τ ) 0

(8.23) (8.24) ⎤ 0 ⎦, 0 R(t)δ(t − τ )

0 Qδ(t − τ ) 0

(8.25)

where dim{η(t)} = n × 1,

⎧ ⎨ (i + 1)m × 1, h ≤ t < h i i+1 dim {y(t)} = ⎩ (l + 1)m × 1, t ≥ h l

and v(t) has the same dimension as y(t).

(8.26)

168

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

Denote 

Ry = y, y,

(8.27)



Ryx = y, x,

(8.28)

where x is as in (8.19) and Ry is an integral operator, determined by its kernel, say Ry (·, ·), which is given as Ry (t, s) = y(t), y(s),

0 ≤ t, s ≤ tf ,

(8.29)

and Ryx is determined by kernel Ryx (·, ·) which is given by Ryx (t, s) = y(t), x(s), 0 ≤ t ≤ tf , 0 ≤ s ≤ hl .

(8.30)

By taking into account (8.22), (8.15) can be rewritten as: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x(0) x(0) x(0) x(0) Jtf = ⎣ u˜ ⎦ ⎣ x ⎦ , ⎣ x ⎦ ⎣ u ˜ ⎦ u y y u      ξ Rx0 Rx0 y ξ = , u u Ryx0 Ry where

 x0 =

(8.31)

   x(0) x(0) , ξ= , x u ˜

(8.32)

u ˜ = {˜ u(t), 0 ≤ t ≤ hl }, u = {u(t), 0 ≤ t ≤ tf },

(8.33) (8.34)

and x0 , x0  = Rx0 , x0 , y = Rx0 y , and y, y = Ry . Then we obtain that [30] Jtf = ξ  Pξ + (u − u∗ ) Ry (u − u∗ ),

(8.35)

u∗ = {u∗ (t), 0 ≤ t ≤ tf } = −Ry−1 Ryx0 ξ, ˆ 0 , x0 − x ˆ 0 , P = x0 − x ˆ 0 = {ˆ x x(s), 0 ≤ s ≤ hl },

(8.36) (8.37)

where

ˆ (s) is the projection of x(s) onto the linear space L{y(r), while x Since    x(0) = [ Ryx(0) Ryx ] , Ryx0 = y, x

(8.38) 0 ≤ r ≤ tf }.

by taking into account (8.36), the minimizing solution is further given as u∗ = −Ry−1 Ryx(0) x(0) − Ry−1 Ryx u ˜.

(8.39)

8.2 Linear Quadratic Regulation

169

It is clear that Ry−1 Ryx(0) is the transpose of the gain matrix of the filtering ˆ (0 | 0) which is the projection of the linear space state x(0) onto the estimate x linear space L{y(t); 0 ≤ t ≤ tf }. Similarly, it is easy to know that Ry−1 Ryx with x = {x(t), 0 < t ≤ hl } is the transpose of the gain matrix of the smoothing ˆ (τ | 0) which is the projection of the linear space state x(τ ) (0 < τ ≤ estimate x ˜(τ ) = 0 for τ ≥ hl , (8.39) hl ) onto the linear space L{y(t); 0 ≤ t ≤ tf }. Since u can be rewritten as 9 hl u∗ = −Ry−1 Ryx(0) x(0) − Ry−1 Ryx(s) u ˜(s)ds 0

= −Ry−1 Ryx(0) x(0) −

l 9  i=1

hi

hi−1

Ry−1 Ryx(s) u ˜(s)ds.

(8.40)

To give a solution to the optimal LQR problem, the key is to compute the filtering gain matrix Rx(0)y Ry−1 and the smoothing gain matrix Rx(τ )y Ry−1 (0 < τ ≤ hl ) which will be derived in the next subsection. 8.2.3

Solution to the LQR Problem

In this subsection we shall give an explicit solution to the LQR problem for the delay system (8.1). ˆ (0 | 0). By applying the standard Firstly, we shall find the filter gain matrix of x Kalman filtering formulation for the backward system (8.23)-(8.24), it follows that ˆ˙ (t | t) = Φ(t)ˆ −x x(t | t) + K(t)y(t),

(8.41)

K(t) = P (t)Γ (t)R(t)−1 , Φ(t) = Φ − K(t)Γ  (t),

(8.42) (8.43)

where

and the matrix P (t) obeys the following backward Riccati equation − P˙ (t) = Φ P (t) + P (t)Φ + Q − K(t)R(t)K(t)

(8.44)

with the terminal condition Pf . Let Ψ (t, τ ) be the transition matrix of −Φ(t), then we have 9 t ˆ (t | t) = − x Ψ (t, τ )K(τ )y(τ )dτ + Ψ (t, tf )ˆ x(tf | tf ), (8.45) tf

ˆ (tf | tf ) = 0. with x ˆ (t | 0). Next, we shall find the smoother x

170

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

Lemma 8.2.1. For the system with backward state-space model (8.23)-(8.24), ˆ (t | 0) is given by the smoother x 9 ˆ (t | 0) = x

9

tf

t

Ψ (t, s)K(s)y(s)ds + P (t)Ψ  (s, t)Γ (s)R−1 (s)y(s)ds − 0 t < 9 t; 9 tf  −1  Ψ (s, r)K(r)y(r)dr ds. P (t)Ψ (s, t)Γ (s)R (s)Γ (s) s

0

(8.46) Proof: By applying the projection formula [38], we have 9

t

ˆ (t | 0) = x ˆ (t | t) + x

x(t), w(s)R−1 (s)w(s)ds

0

9

t

ˆ (t | t) + =x

ˆ (s | s)] ds, ˜ (s | s)Γ (s)R−1 (s) [y(s) − Γ (s) x x(t), x

0

(8.47) ˜ (s) = x(s) − x ˆ (s | s) and w(s) = y(s) − Γ (s) x ˆ (s | s) is the innovation where x and R(s) (same as in (8.6)) is the covariance matrix of w(s). Now we calculate ˜ (s | s) for s < t. Note that from (8.41), x(t), x ˆ˙ (s | s) = Φ x ˆ (s | s) + K(s) [y(s) − Γ (s) x ˆ (s | s)] −x   ˆ (s | s) + K(s) [Γ (s) x ˜ (s | s) + v(s)] . =Φx

(8.48)

By taking into account (8.23) and (8.48) we have ˜˙ (s | s) = Φ(s)˜ −x x(s | s) + η(s) − K(s)v(s), where Φ(s) is as in (8.43). From (8.49), we obtain 9 s 9 ˜ (s | s) = x Ψ (s, r)K(r)v(r)dr − t

(8.49)

s

Ψ (s, r)η(r)dr + Ψ (s, t)˜ x(t | t),

t

(8.50) which implies ˜ (s | s) = P (t)Ψ  (s, t), x(t), x since v(r) and q(r), r < t, are independent of x(t). By taking into account (8.51), it follows from (8.47) that 9 t ˆ (t | 0) = x ˆ (t | t) + x P (t)Ψ  (s, t)Γ (s)R−1 (s)y(s)ds 0 9 t ˆ (s | s)ds. P (t)Ψ  (s, t)Γ (s)R−1 (s)Γ (s) x − 0

(8.51)

(8.52)

8.2 Linear Quadratic Regulation

171

ˆ (tf | tf ) = 0, the filter x ˆ (s | s) On the other hand, in view of the fact that x is obtained from (8.45) as 9 s ˆ (s | s) = − Ψ (s, r)K(r)y(r)dr. (8.53) x tf

Substituting (8.53) into (8.52), (8.46) is obtained directly. This completes the proof of the lemma. ∇ Having given the filter and smoother, we are now in the position to present the optimal controller. Theorem 8.2.1. Consider the system (8.1) and the performance index (8.2). For t ≤ hl − hi , the optimal controller u∗i (t) (i = 0, 1, · · · , l) is given by u∗i (t) = −K(t + hi )x(0) −

9

t+hi

K1 (t + hi , s)˜ u(s)ds h0

9

hl

−

K2 (t + hi , s)˜ u(s)ds,

(8.54)

t+hi

and for t > hl − hi , u∗i (t)

9 = −K(t + hi )x(0) −

hl

K1 (t + hi , s)˜ u(s)ds,

(8.55)

h0

where −1  Γ(i) P (t + hi )Ψ  (0, t + hi ), K(t + hi ) = R(i)

(8.56) ! −1  K1 (t + hi , s) = R(i) Γ(i) P (t + hi )Ψ  (s, t + hi ) In − G0 (s)P (s) , (8.57) ! −1  0 K2 (t + hi , s) = R(i) Γ(i) In − P (t + hi )G (t + hi ) Ψ (t + hi , s)P (s), (8.58) and

9 0

G (s) =

s

Ψ  (r, s)Γ (r)R(r)−1 Γ (r) Ψ (r, s)dr,

(8.59)

0

while Ψ (s, ·) is the transition matrix of −Φ(s) = K(s)Γ  (s)−Φ , K(s) is as given in (8.42) and P (t) is the solution to the Riccati equation (8.44). Proof: Recall from (8.39) that Ry−1 Ryx(0) is the transpose of the gain matrix of ˆ (0 | 0), and Ry−1 Ryx(s) is the transpose of smoother gain of x ˆ (s | 0) the filter x of (8.46). ˆ (s | 0) can be rewritten as Note that the smoother of x 9 tf 9 s ˆ (s | 0) = x Ψ (s, r)K(r)y(r)dr + P (s)Ψ  (r, s)Γ (r)R−1 (r)y(r)dr s

0

172

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

9

s

9

τ

−P (s)



Ψ (r, s)Γ (r)R

0

9

0

tf

−P (s)

9

−1





(r)Γ (r)Ψ (r, τ )dr K(τ )y(τ )dτ  s  −1  Ψ (r, s)Γ (r)R (r)Γ (r)Ψ (r, τ )dr K(τ )y(τ )dτ.

0

s

(8.60) With the duality of the control and smoothing, the optimal controller u∗ (t+hi ) (i = 0, 1, · · · , l) for t ≤ hl − hi can be given directly from (8.46) as u∗ (t + hi ) = −K  (t + hi )Ψ  (0, t + hi )x(0) 9 t+hi −K  (t + hi ) [Ψ  (s, t + hi ) − Σ(s, t + hi )P (s)] u˜(s)ds 9 −

h0 hl

[R−1 (t + hi )Γ  (t + hi )Ψ (t + hi , s) −

t+hi 

¯ t + hi )]P (s)˜ K (t + hi )Σ(s, u(s)ds,

(8.61)

and for t > hl − hi , u∗ (t + hi ) is given by u∗ (t + hi ) = −K  (t + hi )Ψ  (0, t + hi )x(0) 9 hl  ¯ t + hi )P (s)˜ Σ(s, +K (t + hi ) u(s)ds,

(8.62)

h0

where 9 Σ(s, t + hi ) =

s

Ψ  (r, t + hi )Γ (r)R−1 (r)Γ  (r)Ψ (r, s)dr,

(8.63)

0

9 ¯ t + hi ) = Σ(s,

t+hi

Ψ  (r, t + hi )Γ (r)R−1 (r)Γ  (r)Ψ (r, s)dr.

(8.64)

0

Note that Ψ  (r, t + hi ) = Ψ  (s, t + hi )Ψ  (r, s) and Ψ (r, s) = Ψ (r, t + hi )Ψ (t + hi , s), it follows that Σ(s, t + hi ) = Ψ  (s, t + hi )G0 (s), ¯ t + hi ) = G0 (t + hi )Ψ (t + hi , s), Σ(s,

(8.65) (8.66)

where G0 (·) is as (8.59). Further, observe that K  (t + hi ) = R−1 (t + hi )Γ  (t + hi )P (t + hi ), Γ (t + hi ) = [ Γ(0) · · · Γ(i) ] and R(t + hi ) = diag{R(0) , · · · , R(i) }, it follows that i + 1 blocks ' () * −1  [0 · · · 0 Im ] K  (t + hi ) = R(i) Γ(i) P (t + hi ),

(8.67)

i + 1 blocks () * ' −1  Γ(i) . [0 · · · 0 Im ] R−1 (t + hi )Γ  (t + hi ) = R(i)

(8.68)

8.2 Linear Quadratic Regulation

173

i + 1 blocks () * ' Thus, for t ≤ hl − hi , the optimal controller = [0 · · · 0 Im ] ×u∗ (t + hi ) is given by (8.54). For t > hl − hi , the optimal controller follows from (8.62) as (8.55). This completes the proof of the theorem. ∇ u∗i (t)

Note, however, that for τ > 0, the optimal controller u∗i (τ ) of (8.54)-(8.55) is given in terms of the initial state x(0) rather than the current state x(τ ). Similar to the discrete-time case, this problem can be addressed by shifting the time interval from [0, hl ] to [τ, τ + hl ]. To this end, we first introduce the following notations. For any given τ ≥ 0, denote: ⎧⎡ ⎤ u0 (t + τ − h0 ) ⎪ ⎪ ⎪ ⎪⎢ ⎥ .. ⎪ ⎪ ⎣ ⎦ , hi ≤ t < hi+1 , . ⎪ ⎪ ⎨  u (t + τ − hi ) ⎤ uτ (t) = ⎡ i (8.69) u ⎪ 0 (t + τ − h0 ) ⎪ ⎪ ⎪ ⎥ ⎢ .. ⎪ ⎪ ⎦ , t ≥ hl ⎣ . ⎪ ⎪ ⎩ ul (t + τ − hl ) ⎧  ⎨ l  j=i+1 Γ(j) uj (t + τ − hj ), hi ≤ t < hi+1 , t + τ ≤ tf u˜τ (t) = (8.70) ⎩ 0, otherwise. Using the notations of (8.69)-(8.70), for a given τ > 0, the system (8.1) and the cost (8.2) can be rewritten respectively as ⎧ ⎨ Φx(t + τ ) + Γ (t)uτ (t) + u ¯τ (t), hi ≤ t < hi+1 , (8.71) x(t ˙ + τ) = ⎩ Φx(t + τ ) + Γ (t)uτ (t), t≥h, l

where Γ (t) is as in (8.5). The cost function can be rewritten as Jtf = Jtτf +

l 9  i=0

where

0

τ

ui (t)R(i) ui (t)dt +

9

τ

x (t)Qx(t)dt,

(8.72)

0

9 tf −τ Jtτf = x (tf )Pf x(tf ) + [uτ (t)] R(t)uτ (t)dt 0 9 tf −τ  x (t + τ )Qx(t + τ )dt, +

(8.73)

0

with R(t) as given in (8.6). For any given τ ≥ 0, define the following RDE: −

dP τ (s)  = Φ P τ (s) + P τ (s)Φ + Q − K τ (s)R(s) [K τ (s)] , ds

(8.74)

where P τ (tf − τ ) = P and K τ (s) = P τ (s)Γ (s)R(s)−1 .

(8.75)

174

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

By using a similar line of discussions as in Theorem 8.2.1, we obtain the following result. Theorem 8.2.2. Consider the system (8.71) and the associated cost (8.73). For t ≤ hl − hi , the optimal controller uτi ∗ (t) (i = 0, 1, · · · , l) associated with ui (t + τ ) that minimizes (8.73) is given by uτi ∗ (t)

9

t+hi

= −K (t + hi )x(τ ) − τ

K1τ (t + hi , s)˜ uτ (s)ds h0

9

hl

−

K2τ (t + hi , s)˜ uτ (s)ds,

t+hi

(8.76) and for t > hl − hi , uτi ∗ (t)

9

hl

= −K (t + hi )x(τ ) − τ

K1τ (t + hi , s)˜ uτ (s)ds,

(8.77)

h0

where −1  Γ(i) P τ (t + hi )[Ψ τ (0, t + hi )] , K τ (t + hi ) = R(i)

K1τ (t K2τ (t

+ hi , s) = + hi , s) =

and

9 τ

G (s) =

s

(8.78)

−1  R(i) Γ(i) P τ (t + hi )[Ψ τ (s, t + hi )] × [In − Gτ (s)P τ (s)] , −1  R(i) Γ(i) [In − P τ (t + hi )Gτ (t + hi )]

(8.79)

×Ψ τ (t + hi , s)P τ (s).

(8.80)

[Ψ τ (r, s)] Γ (r)R−1 (r)Γ  (r)Ψ τ (r, s)dr.

(8.81)

0

In the above Ψ τ (s, ·) is the transition matrix of − Φτ (s) = K τ (s)Γ  (s) − Φ .

(8.82)

Corollary 8.2.1. The optimal controller uτi ∗ (t), i = 0, 1,· · · , l for t = 0 is given by uτi ∗ (0) = −K τ (hi )x(τ ) − 9 −

9

hi

K1τ (hi , s)˜ uτ (s)ds h0

hl

K2τ (hi , s)˜ uτ (s)ds.

(8.83)

hi

Observe that u∗i (τ ) is the optimal controller associated with the cost (8.2) given in terms of the initial state x(0) while uτi ∗ (0) is the optimal controller associated with the cost (8.72) given in terms of the current state x(τ ).

8.2 Linear Quadratic Regulation

175

Lemma 8.2.2. If ui (t) = u∗i (t) for 0 ≤ t < τ ; i = 0, · · · , l, then u∗i (τ ) ≡ uτi ∗ (0) |uj (t)=u∗j (t)(

0≤t<τ ; 0≤j≤l) ,

i = 0, · · · , l,

(8.84)

where u∗i (τ ) (0 ≤ τ ≤ tf ) is given by (8.54)-(8.55) and uτi ∗ (0) is given by (8.83). Proof: The proof is similar to the discrete-time case in Chapter 3.

∇

Now the main result of this section is summarized below. Theorem 8.2.3. Consider the delay system (8.1) and its cost (8.2). The optimal LQR control ui (τ ), 0 ≤ τ ≤ tf − hi , i = 0, · · · , l that minimizes (8.2), is calculated by 9 hi K1τ (hi , s)˜ uτ ∗ (s)ds u∗i (τ ) = −K τ (hi )x(τ ) − 9 −

h0 hl

K2τ (hi , s)˜ uτ ∗ (s)ds,

(8.85)

hi

while u ˜τ ∗ (·) is as in (8.70) with uj (·) replaced by u∗j (·) for j = 1, · · · , l. Proof: From Lemma 8.2.2, we have u∗i (τ ) ≡ uτi ∗ (0) |uj (t)=u∗j (t)(

0≤t<τ ; 0≤j≤l) ,

i = 0, · · · , l.

Substituting uj (t) = u∗j (t)( 0 ≤ t < τ ; 0 ≤ j ≤ l) into (8.83), (8.85) follows immediately. ∇ Remark 8.2.3. Similar to the discrete-time case, the optimal control law involves computing P τ (s), 0 ≤ s < min{hl , tf − τ } of (8.74) where for tf − τ ≥ hl , the initial condition P τ (hl ) = P (τ + hl ) and for tf − τ < hl , the initial condition P τ (tf − τ ) = Pf . Remark 8.2.4. For delay free systems, i.e., h1 = · · · = hl = 0, it is obvious that the optimal controller (8.85) becomes: −1  −1  Γ(0) P τ (0)[Ψ τ (0, 0)] x(τ ) = −R(0) Γ(0) P τ (0)x(τ ), u∗0 (τ ) = −R(0)

(8.86)

where Ψ τ (0, 0) = I has been used in the second equality. Note that P τ (0) = P (τ ) obeys the standard Riccati equation (8.44). Thus the optimal controller u∗0 (τ ) of (8.86) is the same as the well-known LQR solution [30]. 8.2.4

An Example

The purpose of this section is to demonstrate the computational procedure of the proposed optimal LQR controller design. For simplicity, we only discuss first order systems. High order systems follow the same procedure in designing their optimal controllers. Consider the system (8.1) with Φ ≡ 0, l = 1, Γ(0) ≡ 1, Γ(1) ≡ 1 and cost function (8.2) with R(0) ≡ 1 and R(1) ≡ α−2 . We shall give the optimal controller by applying the results in the last subsection.

176

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

We first investigate the optimal controller u0 (τ ) for tf − h1 < τ ≤ tf . Under this case, ⎧ ⎨ 1, t − h < s ≤ t f 1 f Γ (s) = (8.87) ⎩ 0, otherwise, ⎧ ⎨ 1, t − h < s ≤ t , f 1 f R(s) = (8.88) ⎩ 0, otherwise. Solving the Riccati differential equation (8.74), with P τ (tf −τ ) = Pf , we have that 1 + ce2r Pf − 1 2(τ −tf ) P τ (r) = e , c= . 1 − ce2r Pf + 1 Then we obtain from (8.82) and (8.81) that ce2s − 1 r−s e , ce2r − 1 (ce2s − 1)(e2s − 1) Gτ (s) = . 2(c − 1)e2s

Ψ τ (r, s) =

Note that

⎧ ⎨ u∗ (s + τ − h ), t − τ − h ≤ s < t − τ, 1 f 1 f 1 u ˜τ ∗ (s) = ⎩ 0, otherwise.

(8.89) (8.90)

(8.91)

Therefore, from Theorem 8.2.3, the optimal controller u∗0 (τ ) for tf − h1 ≤ τ ≤ tf is given by 9 h1 1 1 + ce2s ∗ 1+c u∗0 (τ ) = − x(τ ) − u1 (s + τ − h1 )ds. (8.92) 1−c 1−c 0 es For 0 ≤ τ ≤ tf − h1 , ⎧ ⎨ 1,

0 ≤ s < h1 ⎩[1 1], s ≥ h , 1 ⎧ ⎨ 1, 0 ≤ s < h1 , R(s) = ⎩ diag{1, α−2 }, s ≥ h , 1 ⎧ ⎨ u∗ (s + τ − h ), 0 ≤ s < h , 1 1 1 u ˜τ ∗ (s) = ⎩ 0, s≥h . Γ (s) =

(8.93)

(8.94)

(8.95)

1

Note that Riccati equation (8.74) can be solved by two steps: The first step is to obtain P τ (s) for h1 ≤ s ≤ tf − h1 . In this step the Riccati equation (8.74) is with Γ (s) = [ 1 1 ], R(s) = diag{1, α−2}, and initial value P τ (tf − τ ) = Pf . Its solution is given by

8.2 Linear Quadratic Regulation

P τ (s) = P (s) =

c1 e2α0 s + α0 α0 [α0 Pf − 1] , c1 = 2α0 tf , α0 [−c1 e2α0 s + α0 ] e [α0 Pf + 1]

177

(8.96)

√ where α0 = 1 + α2 . In the second step, for 0 ≤ s < h1 , we obtain P τ (s) by solving the Riccati equation (8.74) with Γ (s) = 1, R(s) = 1, and initial value P τ (h1 ) = P (τ + h1 ). The result is P τ (r) =

1 + c2 e2r P (τ + h1 ) − 1 . , c2 = 2h1 1 − c2 e2r e [P (τ + h1 ) + 1]

where P (τ + h1 ) is as (8.96). Then we get Ψ τ (r, s) =

c2 e2s − 1 r−s (c2 e2s − 1)(e2s − 1) τ e , G (s) = . c2 e2r − 1 2(c2 − 1)e2s

Therefore the optimal controllers u0 (τ ) and u1 (τ ) for 0 ≤ τ < tf − h1 are given from Theorem 8.2.3 as 9 h1 1 + c2 e2s ∗ 1 + c2 1 x(τ ) − u1 (s + τ − h1 )ds, (8.97) u∗0 (τ ) = − 1 − c2 1 − c2 0 es + , 9 h1 2h1 ! ∗ ∗ 2 1 + c2 e s −s u1 (τ ) = α e +e u1 (s + τ − h1 )ds . × x(τ ) + 0.5 (c2 − 1)eh1 0 (8.98) Figures 8.1 and 8.2 show the optimal state trajectories for h1 of 0.5 sec. and 0.1 sec., respectively, whereas Figure 8.3 gives a comparison of optimal costs. 1 a=5 a=1 a=2

h1=0.1 0.9

0.8

0.7

state

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

Fig. 8.1. The optimal state trajectories for h1 = 0.1 sec. and α = 1, 2, 5, respectively

178

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays 1 a=1 a=2 a=5

h1=0.5 0.9

0.8

0.7

state

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

Fig. 8.2. The optimal state trajectories for h1 = 0.5 sec. and α = 1, 2, 5, respectively 1 h1=0.1 h1=0.25 h1=0.5

0.9

0.8

0.7

Optimal cost

0.6

0.5

0.4

0.3

0.2

0.1

0

0

2

4

6

8

10

12

14

16

18

20

α

Fig. 8.3. Comparison of the optimal costs for h1 = 0.1, 0.25 and 0.5 sec., respectively

8.3 Measurement Feedback Control In the LQR problem we assumes that the controller is able to access to the state vector x(t). This is often not a very reasonable assumption since in many applications one has only access to a certain measurement signal that is related

8.3 Measurement Feedback Control

179

to the state. Since in these situations the controller can only use the measurement signal, such problem is referred to as measurement feedback control problem. As in delay free systems, the solution to the measurement feedback problem for systems with i/o delays is given by the same state feedback control law where the state is now replaced by its estimate. 8.3.1

Problem Statement

We consider the following continuous-time linear system with multiple i/o delays ˙ x(t) = Φx(t) +

l 

Γ(i) ui (t − hi ) + e(t),

(8.99)

i=0

y(i) (t) = H(i) x(t − di ) + v(i) (t), i = 0, · · · , l, l ≥ 1,

(8.100)

where x(t) ∈ Rn and ui (t) ∈ Rmi represent respectively the state and the control input, e and v are the exogenous input and measurement noise respectively. Φ, Γ(i) and H(i) are bounded matrices. Without loss of generality, we assume that the delays are in an increasing order: 0 = h0 < h1 < · · · < hl and the control inputs ui , i = 0, 1, · · · , l have the same dimension, i.e., m0 = m1 = · · · = ml = m. We assume that the initial value x(0), e and v are zero mean random variables with variances given by ⎤ ⎤ ⎡ ⎤ ⎡ ⎡ 0 0 x(0) Π0 x(0) ⎦. ⎣ e(t) ⎦ , ⎣ e(s) ⎦ = ⎣ 0 Qe δts (8.101) 0 0 0 Qv(i) δt,s v(i) (s) v(i) (t) To ensure that the regulated signal x(t) and control u(t) are simultaneously small, we shall introduce the cost function, 9 tf l 9 tf −hi  Jtf = xtf Pf xtf + ui (t)R(i) ui (t)dt + x (t)Qx(t)dt, i=0

0

0

(8.102) where tf > hd is an integer, xtf is the terminal state, i.e. xtf = x(tf ), Pf = Pf ≥ 0 reflects the penalty on the terminal state, the matrix functions R(i) , i = 0, 1, · · · , l, are positive definite and bounded and the matrix function Q is non-negative definite and bounded. Let y(t) be the observation of system (8.100) at time t, then y(t) is given by ⎧ ⎨ col{y (t), · · · , y  (0) (i−1) (t)}, di−1 ≤ t < di , (8.103) y(t) = ⎩ col{y (t), · · · , y (t)}, t ≥ d . (0)

(l)

l

The measurement feedback control problem can now be stated as follows. Find the input sequences {ui (t) = Fi (y(τ )|0≤τ ≤t } (i = 0, · · · , l) such that the cost function E(Jtf ) of (8.102) is minimized, where E is the mathematical expectation taken with respect to the random noises of e and v.

180

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

8.3.2

Solution

We first present the Kalman filtering formulation for the system (8.99)-(8.100). ˆ (t) is the optimal estimation of state x(t) given the measurements Suppose that x of {y(τ ), 0 ≤ τ < t}. Then, by projection formula, it follows that ˆ˙ (t) = Φˆ x x(t) +

l 

Γ(i) ui (t − hi ) + x(t), w(t)Q−1 w (t)w(t),

(8.104)

i=0

where w(t) is the innovation of the measurement y(t), i.e., ˆ (t), w(t) = y(t) − y ˆ (t) is the optimal linear estimation of y(t) based on the measurement of while y {y(τ ), 0 ≤ τ < t} and Qw (t) is the covariance matrix of the innovation w(t). For the convenience of discussion, denote K(t) = x(t), w(t) Q−1 w (t),

(8.105)

then (8.104) leads to ˆ˙ (t) = Φˆ x x(t) +

l 

Γ(i) ui (t − hi ) + K(t)w(t).

(8.106)

i=0

Note that w(t) is a white noise with zero mean and covariance matrix of Qw (t). Remark 8.3.1. (8.106) has a similar form as the standard Kalaman filtering formulation, however, due to the fact that the measurements are with multiple delays, there is no direct way to calculate the gain matrix of K(t) as in the standard Kalman filtering where the gain matrix is computed by performing one Riccati equation. A. Derivation of the controller Now we derive the optimal measurement feedback controller. Firstly, let ˜ (t) = x(t) − x ˆ (t), x

(8.107)

ˆ (t) is the optimal filter of (8.106). It is obvious that x ˜ (t), the filtering where x ˆ (t). By some simple algebra evaluations, we obtain error, is uncorrelated with x that,   E Jtf = Jt0f + Jt1f , (8.108) where

+ ˆ tf Pf x ˆ tf x

Jt0f

=E

Jt1f

; ˜ tf Pf x ˜ tf + =E x

+

l 9 

tf −hi

i=0 0 9 tf 

ui (t)R(i) ui (t)dt <

˜ (t)Q˜ x x(t)dt .

0

9

tf

, 

ˆ (t)Qˆ x x(t)dt ,

+ 0

(8.109)

8.3 Measurement Feedback Control

181

Note that Jt1f does not contain any control input u, so the problem under investigation is converted to the one of seeking a control input u such that Jt0f is minimized, with the constraint of (8.106). Thus the measurement feedback control becomes a typical LQG problem. In order to give the optimal solution to the above LQG problem, we introduce the following notations for any given τ ≥ 0, ⎧ ⎨ l  j=i+1 Γ(j) uj (t − hj + τ ), hi ≤ t < hi+1 , t + τ ≤ N ˜ τ (t) = u ⎩ 0, otherwise



Γ (t) =

(8.110)

⎧ ⎨[Γ

· · · Γ(i) ] , hi ≤ t < hi+1 ,

(0)

(8.111)

⎩[Γ · · · Γ(l) ] , t ≥ hl , (0) ⎧ ⎨ diag{R , · · · , R }, h ≤ t < h , i i+1  (0) (i) R(t) = ⎩ diag{R , · · · , R }, t ≥ h . l (0) (l)

(8.112)

Furthermore, we define the following Riccati equation dP τ (s) = Φ P τ (s) + P τ (s)Φ + Q − K τ (s)R(s)(K τ (s)) , ds where P τ (tf − τ ) = Pf and −

K τ (s) = P τ (s)Γ (s)R(s)−1 .

(8.113)

(8.114)

Similar to the standard LQG control and following the results of last section (see also [100]), the optimal solution ui (τ ) that minimizes E(Jtf ) is given by 9 t+hi uτi ∗ (t) = −K τ (t + hi )ˆ x(τ ) − K1τ (t + hi , s)˜ uτ (s)ds 9 −

h0 hl

K2τ (t + hi , s)˜ uτ (s)ds,

(8.115)

t+hi

and for t > hl − hi , x(τ ) − uτi ∗ (t) = −K τ (t + hi )ˆ

9

hl

K1τ (t + hi , s)˜ uτ (s)ds,

(8.116)

h0

where −1  K τ (t + hi ) = R(i) Γ(i) P τ (t + hi )[Ψ τ (0, t + hi )] ,

(8.117)

−1  K1τ (t + hi , s) = R(i) Γ(i) P τ (t + hi )[Ψ τ (s, t + hi )]

× [In − Gτ (s)P τ (s)] , K2τ (t

+ hi , s) =

−1  R(i) Γ(i) τ

(8.118)

[In − P (t + hi )G (t + hi )] τ

×Ψ (t + hi , s)P τ (s)

τ

(8.119)

182

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

and 9 Gτ (s) =

s

[Ψ τ (r, s)] Γ (r)[R(r)]−1 Γ  (r)Ψ τ (r, s)dr.

(8.120)

0

In the above, Ψ τ (s, ·) is the transition matrix of − Φτ (s) = K τ (s)Γ  (s) − Φ .

(8.121)

˜ τ ∗ (·) is as in (8.110) with ui (·) replaced by u∗i (·) for i = 0, · · · , l. In addition, u ˆ (t) B. Calculation of the Kalman filter x ˆ (t) in (8.115). It is easy Now the problem is to calculate the Kalman filter x ˆ to know that x(t) is not a standard Kalman filter as the measurements of ˆ (t), we y(τ ), 0 ≤ τ < t are with multiple time delays. In order to calculate x shall apply the re-organized innovation analysis approach. Note that the linear space L {y(s)|0≤s
(8.123)

and ⎡

⎤ y(0) (s) ⎢ ⎥ .. yi (s) = ⎣ ⎦ . y(i−1) (s + di−1 )

(8.124)

is a new measurement. It is clear that the new measurement yi (s) satisfies that yi (s) = Hi x(s) + vi (s),

i = 1, · · · , l + 1,

(8.125)

with ⎤ ⎡ ⎤ H(0) v(0) (s) ⎢ ⎢ . ⎥ ⎥ .. Hi = ⎣ ⎦ , vi (s) = ⎣ .. ⎦ . . H(i−1) (s + di−1 ) v(i−1) ⎡

(8.126)

In the above, vi (s) is a white noise of zero mean and covariance matrix Qvi = diag{Qv(0) , · · · , Qv(i−1) },

i = 1, · · · , l + 1.

(8.127)

8.3 Measurement Feedback Control

183

Now we have the following results, ˆ (t) = Theorem 8.3.1. Consider the system (8.99)-(8.100). The optimal filter x ˆ (t, 1) is calculated by x ˆ (ti−1 , i) = Ψi (ti−1 , ti )ˆ x x(ti , i) + % & 9 ti−1 l  Ψi (ti−1 , s) Ki (s)yi (s) + Γ(i) ui (s − hi ) ds, ti

i=0

i = l, l − 1, · · · , 1;

ˆ (ti , i) = x ˆ (ti , i + 1) ti = t − di , x

(8.128)

where Ψi (t, τ ) is the transition matrix of Φi (t) and Φi (t) = Φ − Ki (t)Hi ,

(8.129)

Pi (t)Hi Q−1 vi ,

(8.130)

Ki (t) = while Pi (t) is computed by

P˙i (t) = ΦPi (t) + Pi (t)Φ + Qe − Ki (t)Qvi Ki (t),

(8.131)

with Pi (ti ) = Pi+1 (ti ). ˆ (tl , l + 1) and Pl+1 (tl ) are calculated by the In the above the intial values x following standard Kalman filter ˆ˙ (s, l + 1) = Φl+1 (s)ˆ ˜ (s) + Kl+1 (s)yl+1 (s), x x(s, l + 1) + u ˆ (0, l + 1) = 0, x

(8.132)

Φl+1 (s) = Φ − Kl+1 (s)Hl+1 ,  Q−1 Kl+1 (s) = Pl+1 (s)Hl+1 vl+1

(8.133)

where

with Pl+1 (s) computed by  P˙l+1 (s) = ΦPl+1 (s) + Pl+1 (s)Φ + Qe − Kl+1 (s)Qvl+1 Kl+1 (s),

Pl+1 (0) = Π0 .

(8.134)

Proof: To start off the proof, we first introduce some notations: ˆi (s) for s > ti denotes the optimal estimate of yi (s) given 1) The estimator y the observation L {yl+1 (τ )|0≤τ ≤tl ; · · · ; yi (τ )|ti <τ <s } .

(8.135)

ˆ i (s) is the optimal estimate of yi (s) given the observation 2) For s = ti , y   (8.136) L yl+1 (τ )|0≤τ ≤tl ; · · · ; yi+1 (τ )|ti+1 <τ
184

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

From the above notations we can introduce the innovation sequence associated with the re-organized measurements: 

ˆ i (s). wi (s) = yi (s) − y

(8.137)

For i = l + 1, it is clear that wl+1 (s) is the standard Kalman filtering innovation sequence for the system (8.99) and (8.125) for i = l + 1. In view of (8.125), it follows that ˜ (s, i) + vi (s), i = 1, · · · , l + 1 wi (s) = Hi x

(8.138)

˜ (s, i) = x(s) − x ˆ (s, i), x

(8.139)

where i = 1, · · · , l + 1

is the one-step ahead prediction error of the state x(s) based on the obserˆ (s, i) for s > ti is the provations (8.135) or (8.136) respectively. Note that x jection of the state x(s) onto the linear space spanned by the innovation {wl+1 (τ )|0≤τ ≤tl ; · · · ; wi (τ )|ti <τ <s }, it follows from the projection formula that ˆ (s, i) = P roj{x(s) | wl+1 (τ )|0≤τ ≤tl ; · · · ; wi (τ )|ti <τ <s } x 9 tl x(s), wl+1 (τ )Q−1 = vl+1 wl+1 (τ )dτ 0

+

9 l 

k=i+1 9 s

+ ti

tk−1

tk

x(s), wk (τ )Q−1 vk wk (τ )dτ

x(s), wi (τ )Q−1 vi wi (τ )dτ.

(8.140)

By differentiating both sides of (8.140) with respect to s, we have ˆ˙ (s, i) = Φˆ x x(s, i) + x(s), wi (s)Q−1 vi wi (s) +

l 

Γ(i) ui (s − hi )

i=0

= Φi (s)ˆ x(s, i) + Ki (s)yi (s) +

l 

Γ(i) ui (s − hi ).

(8.141)

i=0

Then, it follows from (8.141) that ˆ (ti−1 , i) = Ψi (ti−1 , ti )ˆ x x(ti , i) + % & 9 ti−1 l  ˜ i (s)˜ Ψi (ti−1 , s) K yi (s) + Γ(i) ui (s − hi ) ds, ti

i=0

ˆ (ti , i) = x ˆ (ti , i + 1), i = l, · · · , 1, x

(8.142)

ˆ (tl , l + 1) where Ψi (t, τ ) is the transition matrix of Φi (t). With the initial value x ˆ (ti−1 , i) for i = l, l − 1, · · · , 1. Note the and using (8.142), we can compute x

8.4 H∞ Full-Information Control

185

ˆ (t0 , 1) for (8.142) is the same as x ˆ (t, 1), and the initial value terminal value of x ˆ (tl , l + 1) which is the solution of the standard Kalman filter of of (8.142) is x (8.141) with i = l + 1. Now the proof is completed. ∇ The main results are summarized as Theorem 8.3.2. (Optimal Measurement Feedback Controller) Consider the state-space model (8.99)-(8.100). Suppose the controller ui (t) is allowed to be causal linear function of the measurements of {y(τ ), 0 ≤ τ ≤ t}, i.e., ui (t) = Fi (y(τ )|0≤τ ≤t ).

(8.143)

ˆ (t) Then the optimal solution that minimizes (8.102) is given by (8.115), where x is the Kalman filter calculated by Theorem 8.3.1.

8.4 H∞ Full-Information Control In this section, we shall study the H∞ full-information control problem for multiple input delayed systems. By converting the H∞ control problem into a min-max problem of an indefinite quadratic form, we derive the conditions under which an H∞ full-information controller exists and obtain an H∞ controller, following a similar discussion as in Section 8.2. Note that the infinite horizon case of the above problem has been independently studied in some recent papers [44, 62]. [44] has investigated the problem using the operator theory where a controller is designed with a Hamiltonian matrix and the solutions of a number of complicated differential equations while [62] considers the problem with the standard J-spectral factorization approach. 8.4.1

Problem Statement

We consider the following continuous-time linear system for the H∞ control problem. x(t) ˙

= Φx(t) +

l 

B(i) wi (t − hi )

i=0

+ ⎡ ⎢ ⎢ ⎢ z(t) = ⎢ ⎢ ⎣

l 

G(i) ui (t − hi ), l ≥ 1,

i=0

Cx(t) D(0) u0 (t − h0 ) D(1) u1 (t − h1 ) .. .

(8.144)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(8.145)

D(l) ul (t − hl ) where x(t) ∈ Rn , wi (t) ∈ Rri , ui (t) ∈ Rmi , and z(t) ∈ Rq represent the state, the exogenous input, the control input and the controlled signal, respectively.

186

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

Φ, B(i) , G(i) , C, and D(i) , i = 0, 1, · · · , l are bounded matrices. It is assumed that the exogenous inputs are deterministic signals and are from L2 [0, tf ] where tf is the time-horizon of the control problem under investigation. Without loss of generality, we assume that the delays are in a strictly increasing order: 0 = h0 < h1 < · · · < hl and the control inputs ui , i = 0, 1, · · · , l and the exogenous inputs wi , i = 0, 1, · · · , l, respectively, have the same dimension, i.e., m0 = m1 = · · · = ml = m and r0 = r1 = · · · = rl = r. We also assume that for t < 0,  D(i) > 0, i = 0, 1, · · · , l. wi (t) = 0, ui (t) = 0, i = 0, 1, · · · , l, and for t ≥ 0, D(i) The H∞ full-information control under investigation is stated as follows: for a given positive scalar γ, find a finite-horizon full-information control strategy ui (t) = Fi (x(t), (wj (τ ), uj (τ )) |0≤j≤l,

0≤τ
i = 0, 1, · · · , l

such that sup {x(0),wj (t)|0≤j≤l,

0≤t
J(x(0), wj (t), uj (t)) < γ 2

(8.146)

where J(x(0), wj (t), uj (t)) =

z2[0,tf ] x (0)Π −1 x(0) + w2[0,tf ]

,

(8.147)

w(t) = [w0 (t − h0 ) · · · wl (t − hl )] , and Π is a given positive definite matrix which reflects the uncertainty of the initial state relative to the energy of the exogenous inputs. Remark 8.4.1. Note that system (8.145) is encountered in many practical applications such as in network congestion control and wind tunnel or tandem connected processes. – Network congestion control. A network excises control over the best-effort traffic by assigning input rates based on the congestion in the network. The congestion control can be formulated as a feedback control problem for an input delayed system described by (8.145); see [1]. In this case, the action delay hi of system (8.145) consists of downstream and upstream delays, where the former is the delay between the time that the bottleneck node issues its command to the time that it takes for source to receive this command and the latter is the time that it takes for data packets generated by the source to reach the bottleneck node. Moreover, the action delays are different for different sources and hence multiple input delays appear in (8.145). – Control of unilateral delay systems. System (8.145) is directly related to some complicated unilateral delay systems which arise in wind-tunnel or tandem connected processes [24]. In the unilateral delay system, the plant is described exactly as (8.145) with three different delays h0 = 0, h1 = h, h2 = 2h. – Some special cases of system (8.145) have been studied for a number of wellknown problems. When d = 1, Gi,t = 0 for i = 0, 1 and B0,t = 0, the optimal tracking control with single input delay has been studied in [75]. When d = 1, Bi,t = 0 for i = 0, 1 and G0,t = 0, the H∞ control with preview has been studied in a very recent research work [89].

8.4 H∞ Full-Information Control

8.4.2

187

Preliminaries

In this section, we shall convert the H∞ full-information control problem into an optimization problem in Krein space for an associated stochastic model and derive conditions under which the optimizing solution exists and give an explicit formula to compute the optimizing solution. Considering the performance index (8.146), we define Jt∞ f



=

x (0)Π −1 x(0) − γ −2 Jtf ,

(8.148)

where Jtf

= =

z2[0,tf ] − γ 2 w2[0,tf ] 9 l 9 tf −hi   vi (t)R(i) vi (t)dt + i=0

0

tf

x (t)Qx(t)dt,

(8.149)

0

with R(i) vi (t)

 = diag{D(i) D(i) , − γ 2 Ir },   ui (t) = . wi (t)

Q = C C

(8.150) (8.151)

It is clear that an H∞ controller ui (t) achieves (8.146) if and only if it satisfies that Jt∞ of (8.148) is positive for all non-zero {x(0); wi (t), 0 ≤ t ≤ tf − hi , 0 ≤ f i ≤ l}. Denote ⎧ ⎡ ⎤ v0 (t − h0 ) ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ .. ⎪ ⎪ ⎣ ⎦ , hi ≤ t < hi+1 , . ⎪ ⎪ ⎨  ⎡ vi (t − hi ) ⎤ v(t) = (8.152) v0 (t − h0 ) ⎪ ⎪ ⎪ ⎢ ⎪ ⎥ .. ⎪ ⎪ ⎦ , t ≥ hl . ⎪ ⎣ ⎪ ⎩ vl (t − hl ) ⎧ l  ⎨ Γ(j) vj (t − hj ), hi ≤ t < hi+1 ,  v˜(t) = (8.153) ⎩ j=i+1 0, t ≥ hl  [ Γ(0) · · · Γ(i) ] , hi ≤ t < hi+1  (8.154) Γ (t) = [ Γ(0) · · · Γ(l) ] , t ≥ hl   diag{R(0) , · · · , R(i) }, hi ≤ t < hi+1 , R(t) = (8.155) diag{R(0) , · · · , R(l) }, t ≥ hl where Γ(i) = [ B(i)

G(i) ] .

(8.156)

188

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

Note that v(t) and v˜(t) above are vectors involving both the control inputs and the exogenous inputs. Using the above notations, system (8.144) can be rewritten as 1 x(t) ˙ =

Φx(t) + Γ (t)v(t) + v˜(t), hi ≤ t < hi+1 , Φx(t) + Γ (t)v(t), t ≥ hl

(8.157)

and the cost function (8.149) can be rewritten as 9 Jtf

tf

=

v  (t)R(t)v(t)dt +

0

9

tf

x (t)Qx(t)dt.

(8.158)

0

In order to find a controller u∗ (·) such that the indefinite quadratic Jtf is positive, the key problem is to complete square for Jtf associated with v(·). To this end, we define a backward stochastic state-space model in association with system (8.157) and the cost (8.158) as, ˙ − x(t) y(t)

= Φ x(t) + η(t), x(tf ) = 0 = Γ (t) x(t) + v(t),

(8.159) (8.160)

where 0 ≤ t ≤ tf and η(t) and v(t) are white noises of zero means and satisfy η(t), η(s) = Qδt,s and v(t), v(s) = R(t)δt,s , respectively. It can be seen that the dimensions of q(t) and y(t) are respectively dim{η(t)} = n × 1, 1 dim {y(t)} =

(i + 1)(m + r) × 1, hi ≤ t < hi+1 , (l + 1)(m + r) × 1,

t ≥ hl

(8.161)

and v(t) has the same dimension as y(t). Remark 8.4.2. It should be pointed out that in system (8.159)-(8.160) the notations x(t) and v(t), in bold faces, are Krein space elements with zero means and certain covariances. Observe from (8.150) that covariance R(t) is indefinite, which is allowed for Krein space elements [30]. Let y be the continuously indexed collection of Krein space variables [30], i.e. y = {y(t); 0 ≤ t ≤ tf },

(8.162)

where y(t) is from (8.160), and x = {x(t); 0 < t ≤ hl },

(8.163)

where x(t) is from (8.159) and hl is the maximum time delay. Further, mimic to the discrete-time setting, we denote the Gramian operator

8.4 H∞ Full-Information Control 

= y, y,

Ry

189

(8.164)

where Ry is determined by its kernel, Ry (·, ·), which is defined as Ry (t, s) = y(t), y(s),

0 ≤ t, s ≤ tf .

(8.165)

The cross Gramian operator Ryx is determined by its kernel, say Ryx (·, ·), which is given by Ryx (t, s) = y(t), x(s),

0 ≤ t ≤ tf , 0 < s ≤ h l .

(8.166)

Similar to (8.162), we define η and v as η v

= {η(t); 0 ≤ t ≤ tf }, = {v(t); 0 ≤ t ≤ tf }.

Obviously, it follows that      q q Q , = v v 0

 0 , R

(8.167) (8.168)

(8.169)

where Q and R are the diagonal operators with Kernels Qδ(t−τ ) and R(t)δ(t−τ ), respectively. By a similar line of arguments as in Section 8.2, we have the following results Lemma 8.4.1. (8.158) can be rewritten as: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x(0) x(0) x(0) x(0) Jtf = ⎣ v˜ ⎦ ⎣ x ⎦ , ⎣ x ⎦ ⎣ v˜ ⎦ v y y v      Rx0 Rx0 y ξ ξ , = Ryx0 Ry v v where

 x0

=

 x(0) , x

v˜ = {˜ v (t), 0 ≤ t ≤ hd },   x(0) ξ = , v˜

(8.170)

(8.171) (8.172) (8.173)

x0 , x0  = Rx0 , x0 , y = Rx0 y , and y, y = Ry . The terms ξ  Rx0 ξ, ξ  Rx0 y v and v  Ry v in (8.170) are defined as 9 hl 9 hl ξ  Rx0 ξ = ξ  (t)Rx0 (t, s)ξ(s)dtds, (8.174) 0

ξ  Rx0 y v 

v Ry v

9

0

hl

9

tf

= 0

9

0

tf

9

= 0

0

tf

ξ  (t)Rx0 y (t, s)v(s)dtds,

(8.175)

v  (t)Ry (t, s)v(s)dtds.

(8.176)

190

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

Then, by completing the squares, Jtf of (8.170) can be further written as [101] ξ  Pξ + (v − v ∗ ) Ry (v − v ∗ ),

(8.177)

{v ∗ (t), 0 ≤ t ≤ tf } = −Ry−1 Ryx0 ξ, ˆ 0 , x0 − x ˆ 0 , x0 − x & % ˆ (0) x ˆ = {ˆ , x x(s), 0 < s ≤ hl }, ˆ x

(8.178)

=

Jtf where v∗

=

P

=

ˆ0 x

=

ˆ (s) is the projection of x(s) onto the linear space L{y(r), while x Since    x(0) Ryx0 = y, x = [ Ryx(0) Ryx ] ,

(8.179) (8.180) 0 ≤ r ≤ tf }.

(8.181)

by taking into account (8.178), v ∗ is given as v ∗ = −Ry−1 Ryx(0) x(0) − Ry−1 Ryx v˜.

(8.182)

Since v˜(τ ) = 0 for τ > hl , (8.182) can be rewritten as v∗

=

−Ry−1 Ryx(0) x(0) −

9 0

hl

Ry−1 Ryx(s) v˜(s)ds.

(8.183)

In (8.183), it is clear that Ry−1 Ryx(0) is the transpose of the gain matrix of ˆ (0 | 0) which is the projection of the state x(0) onto the the filtering estimate x linear space L{y(t); 0 ≤ t ≤ tf }. Similarly, it is easy to know that Ry−1 Ryx (s) ˆ (s | 0) which is is the transpose of the gain matrix of the smoothing estimate x the projection of the state x(s) onto the linear space L{y(t); 0 ≤ t ≤ tf }. The above establishes a duality between the H∞ full-information control problem for systems with multiple input delays and the H∞ filtering and smoothing problem for the associated system (8.159)-(8.160). The duality relationship of (8.183) allows us to consider the complicated H∞ control problem with input delays in an intuitive way by calculating the gains of smoothing and filtering for the stochastic system (8.159)-(8.160). 8.4.3

Calculation of v ∗

In view of (8.183), in order to calculate v ∗ , we should find the optimal estimate ˆ (t | 0) where t ≥ 0. Assume that the RDE: of x − P˙ (t) =

Φ P (t) + P (t)Φ + Q − K(t)R(t)K  (t),

(8.184)

8.4 H∞ Full-Information Control

191

with the terminal condition P (tf ) = 0 admits a bounded solution P (t), 0 ≤ t ≤ tf , where K(t) = P (t)Γ (t)R−1 (t).

(8.185) (8.186)

Let Ψ¯ (t, τ ) be the transition matrix of −Φ(t), where Φ(t) = Φ − K(t)Γ  (t).

(8.187)

ˆ (t | 0) of the stochastic backward system (8.159)-(8.160) Then the smoother x can be calculated by the lemma below. Lemma 8.4.2. Considering the stochastic backward system (8.159)-(8.160), assume that the RDE (8.184) admits a bounded solution P (t), 0 ≤ t ≤ tf . Then, ˆ (t | 0) is given by the smoother x 9 t 9 tf ˆ (t | 0) = Ψ¯ (t, s)K(s)y(s)ds + P (t)Ψ¯  (s, t)Γ (s)R−1 (s)y(s)ds − x 0 t < 9 tf 9 t;  −1  ¯ ¯ Ψ (s, r)K(r)y(r)dr ds. P (t)Ψ (s, t)Γ (s)R (s)Γ (s) 0

s

(8.188) Proof: The detailed proof can be found in Section 8.2.

∇

Having given the filter and smoother, v ∗ can be given from (8.183) readily. Decompose v ∗ as in (8.178) and v ∗ (t) as in (8.152), i.e. ⎧ ⎡ ∗ ⎤ v0 (t − h0 ) ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ .. ⎪ ⎪ ⎣ ⎦ , hi ≤ t < hi+1 , . ⎪ ⎪ ⎨ ∗  ⎡ v∗i (t − hi ) ⎤ v ∗ (t) = (8.189) v0 (t − h0 ) ⎪ ⎪ ⎪ ⎪ ⎥ ⎢ .. ⎪ ⎪ ⎦ , t ≥ hl . . ⎪ ⎣ ⎪ ⎩ ∗ vl (t − hl ) Then, vi∗ (t) of (8.189) can be computed as follows. Theorem 8.4.1. Consider the system (8.144)-(8.145) and the associated cost (8.149). Suppose the RDE (8.184) has a bounded solution P (t), 0 ≤ t ≤ tf . Then, v ∗ can be given by (8.178), where v ∗ (t) is as in (8.189), and vi∗ (t) for t ≤ hl − hi is given by (i = 0, 1, · · · , l)  −1  vi∗ (t) = −R(i) Γ(i) × P (t + hi )Ψ¯  (0, t + hi )x(0) 9 t+hi ! Ψ¯  (s, t + hi ) In − G0 (s)P (s) v˜(s)ds +P (t + hi ) h0 , 9 h l ! 0 ¯ + v (s)ds , In − P (t + hi )G (t + hi ) Ψ (t + hi , s)P (s)˜ t+hi

(8.190)

192

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

and for tf − hi ≥ t > hl − hi , vi∗ (t)

 −1  −R(i) Γ(i) P (t + hi ) Ψ¯  (0, t + hi )x(0) , 9 h l !  0 ¯ − Ψ (s, t + hi ) In − G (s)P (s) v˜(s)ds ,

=

(8.191)

h0

where

9

s

G0 (s) =

Ψ¯  (r, s)Γ (r)R−1 (r)Γ  (r)Ψ¯ (r, s)dr,

(8.192)

0

while Ψ¯ (s, .) is the transition matrix of −Φ(s) = K(s)Γ  (s) − Φ and K(s) is given in (8.185). ˆ (s | 0) can be rewritten from (8.188) as Proof: Note that the smoother of x 9 tf 9 s ˆ (s | 0) = x Ψ¯ (s, r)K(r)y(r)dr + P (s)Ψ¯  (r, s)Γ (r)R−1 (r)y(r)dr 0 s  9 s 9 τ  −1  ¯ ¯ Ψ (r, s)Γ (r)R (r)Γ (r)Ψ (r, τ )dr K(τ )y(τ )dτ −P (s) 0 0  9 tf 9 s Ψ¯  (r, s)Γ (r)R−1 (r)Γ  (r)Ψ¯ (r, τ )dr K(τ )y(τ )dτ. −P (s) 0

s

(8.193) From (8.183), it follows that v∗

= −Ry−1 Ryx(0) x(0) −

l 9  i=1

hi

hi−1

Ry−1 Ryx(s) v˜(s)ds,

(8.194)

ˆ (0 | 0), and where Ry−1 Ryx(0) is the transpose of the gain matrix of the filter x ˆ (s | 0). By the duality between Ry−1 Ryx(s) is the transpose of smoother gain of x the H∞ control and the smoothing established in the last subsection, v ∗ (t + hi ) (i = 0, 1, · · · , l) for t ≤ hl − hi can be obtained directly from (8.193) as v ∗ (t + hi ) =

−K  (t + hi )Ψ¯  (0, t + hi )x(0) 9 t+hi !  Ψ¯  (s, t + hi ) − Σ(s, t + hi )P (s) v˜(s)ds −K (t + hi ) 9 −

h0 hl

[R−1 (t + hi )Γ  (t + hi )Ψ¯ (t + hi , s)

t+hi 

¯ t + hi )]P (s)˜ −K (t + hi )Σ(s, v (s)ds,

(8.195)

∗

and for tf ≥ t > hl − hi , v (t + hi ) is given by v ∗ (t + hi ) =

−K  (t + hi )Ψ¯  (0, t + hi )x(0) 9 hl ! Ψ¯  (s, t + hi ) − Σ(s, t + hi )P (s) v˜(s)ds, −K  (t + hi ) h0

(8.196)

8.4 H∞ Full-Information Control

where

9 Σ(s, t + hi )

9 ¯ t + hi ) Σ(s,

s

=

Ψ¯  (r, t + hi )Γ (r)R−1 (r)Γ  (r)Ψ¯ (r, s)dr,

193

(8.197)

0 t+hi

=

Ψ¯  (r, t + hi )Γ (r)R(r)−1 Γ (r) Ψ¯ (r, s)dr. (8.198)

0

Note that Ψ¯  (r, t + hi ) = Ψ¯  (s, t + hi )Ψ¯  (r, s) and Ψ¯ (r, s) = Ψ¯ (r, t + hi )Ψ¯ (t + hi , s), it follows that Σ(s, t + hi ) ¯ t + hi ) Σ(s,

= Ψ¯  (s, t + hi )G0 (s), = G0 (t + hi )Ψ¯ (t + hi , s),

(8.199) (8.200)

where G0 (·) is as (8.192). Further, observe that K  (t + hi )

= R−1 (t + hi )Γ  (t + hi )P (t + hi ),

(8.201)

= diag{R(0) , · · · , R(i) }, = [ Γ(0) · · · Γ(i) ] ,

(8.202) (8.203)

i + 1 blocks ' () * −1  [0 · · · 0 Im ] K  (t + hi ) = R(i) Γ(i) P (t + hi ),

(8.204)

i + 1 blocks () * ' −1  Γ(i) . [0 · · · 0 Im ] R−1 (t + hi )Γ (t + hi ) = R(i)

(8.205)

R(t + hi ) Γ (t + hi ) and

i + 1 blocks ' () * Thus, for t ≤ = [0 · · · 0 Im ] ×v ∗ (t+hi ) which is given by (8.190). Similarly, for tf ≥ t > hl − hi , following from (8.196), the optimal controller is given by (8.191). This completes the proof of the theorem. ∇ hl −hi , vi∗ (t)

In Theorem 8.4.1, vi∗ (τ ) is given in terms of the initial state x(0) rather than the current state x(τ ). This problem can be addressed by shifting time interval from [0, hl ] to [τ, τ + hl ]. To this end, we first introduce the following notations. For any given τ ≥ 0, denote: ⎧ ⎡ ⎤ v0 (t + τ − h0 ) ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ .. ⎪ ⎪ ⎣ ⎦ , hi ≤ t < hi+1 , . ⎪ ⎪ ⎨  ⎡ vi (t + τ − hi ) ⎤ v τ (t) = (8.206) v0 (t + τ − h0 ) ⎪ ⎪ ⎪ ⎪ ⎥ ⎢ .. ⎪ ⎪ ⎦ , t ≥ hl ⎣ . ⎪ ⎪ ⎩ vl (t + τ − hl ) 1  l  j=i+1 Γ(j) vj (t + τ − hj ), hi ≤ t < hi+1 , v˜τ (t) = (8.207) 0, t ≥ hl .

194

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

Using the notations of (8.206)-(8.207), for a given τ > 0, the system (8.144) and the cost (8.149) can be rewritten respectively as 1 Φx(t + τ ) + Γ (t)v τ (t) + v¯τ (t), hi ≤ t < hi+1 , (8.208) x(t ˙ + τ) = t ≥ hl Φx(t + τ ) + Γ (t)v τ (t), and Jtf

=

Jtτf +

l 9  i=0

τ

vi (t)R(i) vi (t)dt +

9

0

τ

x (t)Qx(t)dt,

(8.209)

0

where 9 Jtτf

tf −τ

=

τ



9 τ

tf −τ

[v (t)] R(t)v (t)dt + 0

x (t + τ )Qx(t + τ )dt. (8.210)

0

Following a similar discussion and the well known dynamic programming arguments, the H∞ control for system (8.208) associated with the cost index (8.210) is given in the theorem below. Theorem 8.4.2. Given a scalar γ, suppose that the RDE (8.184) has a bounded solution P (t), 0 ≤ t ≤ tf and for any τ > 0, the following RDE with initial value P τ (t) |t=hl = P (τ +hl ) for hl +τ ≤ tf or P τ (t) |t=tf −τ = P (tf ) = 0 for hl +τ > tf admits a bounded solution P τ (t), 0 ≤ t ≤ min{hl , tf − τ }, −

dP τ (t) dt

=



Φ P τ (t) + P τ (t)Φ + Q − Ktτ R(t) (Ktτ )

(8.211)

where Ktτ = P τ (t)Γ (t)R(t)−1 .

(8.212)

Let v ∗ be decomposed as in (8.178), i.e. v ∗ = {v ∗ (t), 0 ≤ t ≤ tf }, where v ∗ (τ ) as in (8.189). Then vi∗ (τ ) in (8.189) is calculated by  −1  vi∗ (τ ) = −R(i) Γ(i) × P τ (hi )[Ψ¯ τ (0, hi )] x(τ )+ 9 hi P τ (hi ) [Ψ¯ τ (s, hi )] {In − Gτ (s)P τ (s)} v˜τ ∗ (s)ds+ h0 , 9 hl τ τ τ τ τ∗ ¯ [In − P (hi )G (hi )] Ψ (hi , s)P (s)˜ v (s)ds , (8.213) hi+1

while v˜τ ∗ (·) is as in (8.207) with vi (·) replaced by vi∗ (·) for i = 0, · · · , l, Σ τ (·, ·) and Gτ (·) is given by 9 s ! Ψ¯ τ (r, s) Γ (r)R−1 (r)Γ  (r)Ψ¯ τ (r, s)dr. Gτ (s) = (8.214) 0

8.4 H∞ Full-Information Control

195

In the above, R(r) and Γ (r) are respectively as in (8.155) and (8.154) and Ψ¯ τ (s, ·) is the transition matrix of −Φτ (s) = K τ (s)Γ  (s) − Φ . The results of Theorems 8.4.1 and 8.4.2 give an explicit calculation of the quadratic form (8.177). Note, however, that the exogenous inputs play a contradictory role with the control inputs, namely the former aims to maximize the cost whereas the latter minimize the cost. Since the second term of (8.177) is not definite and v involves both the control inputs and the exogenous inputs, it is not clear from (8.177) that under what conditions a minimax solution exists. Thus, some further simplification of (8.177) is needed, which will be given in the next section. 8.4.4

H∞ Control

In this subsection, we shall discuss conditions under which a maximizing solution of Jtf with respect to the exogenous inputs exists and then derive a suitable H∞ controller. To this end, we recall the Krein space stochastic model (8.159)-(8.160) and decompose the observation y(t) and the noise v(t) as follows: ⎧ ⎡ ⎤ y0 (t) ⎪ ⎪ ⎪ ⎢ .. ⎥ ⎪ ⎪ ⎪ ⎣ . ⎦ , hi ≤ t < hi+1 , ⎪ ⎪ ⎨ ⎡ yi (t) ⎤ y(t) = (8.215) y0 (t) ⎪ ⎪ ⎪ ⎪ ⎢ .. ⎥ ⎪ ⎪ ⎣ . ⎦ , t ≥ hl ⎪ ⎪ ⎩ yl (t) ⎧ ⎡ ⎤ v0 (t) ⎪ ⎪ ⎪ ⎢ .. ⎥ ⎪ ⎪ ⎪ ⎣ . ⎦ , hi ≤ t < hi+1 , ⎪ ⎪ ⎨ ⎡ vi (t) ⎤ (8.216) v(t) = v0 (t) ⎪ ⎪ ⎪ ⎪ ⎥ ⎢ . ⎪ ⎪ ⎪ ⎣ .. ⎦ , t ≥ hl ⎪ ⎩ vl (t) where yi (t) ∈ Rm+r and vi (t) ∈ Rm , i = 0, · · · , l, satisfy yi (t)

 = Γ(i) x(t) + vi (t), 0 ≤ t ≤ tf − hi ,

(8.217)

and vi (t), vi (s) = R(i) δt,s . In view of the input delays of the original system, we re-organize (8.215)-(8.216) as follows: 1 col{y0 (t + h0 ), · · · , yl (t + hl )}, 0 ≤ t ≤ tf − hl ,  ¯ (t) = y (8.218) col{y0 (t + h0 ), · · · , yi (t + hi )}, tf − hi+1 < t ≤ tf − hi

196

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

and 1 

¯ (t) = v

col{v0 (t + h0 ), · · · , vl (t + hl )}, 0 ≤ t ≤ tf − hl , (8.219) col{v0 (t + h0 ), · · · , vi (t + hi )}, tf − hi+1 < t ≤ tf − hi .

¯ (t) obeys It is easy to know that for 0 ≤ t ≤ tf − hl , y ⎡ ¯ (t) = y

⎤  Γ(0) x(t + h0 ) ⎢ ⎥ .. ¯ (t), ⎣ ⎦+v .  Γ(l) x(t + hl )

(8.220)

and for tf − hi+1 < t ≤ tf − hi , ⎡ ¯ (t) = y

⎤  Γ(0) x(t + h0 ) ⎢ ⎥ .. ¯ (t). ⎣ ⎦+v .  Γ(i) x(t + hi )

¯ v (t), is calculated by ¯ (t), denoted by R The covariance matrix of v 1 diag{R(0) , · · · , R(l) }, 0 ≤ t ≤ tf − hl , ¯ v (t) = R diag{R(0) , · · · , R(i) }, tf − hi+1 < t ≤ tf − hi .

(8.221)

(8.222)

Further, it is not difficult to verify that the linear space generated by {¯ y(t), 0 ≤ t ≤ tf } is the same as the one generated by {y(t), 0 ≤ t ≤ tf }. We denote that v¯ =



{¯ v (t), 0 ≤ t ≤ tf },

(8.223)



{¯ y(t), 0 ≤ t ≤ tf },

(8.224)

¯ y

=

¯ (·) is as in (8.218) and where y 1 col{v0 (t), · · · , vl (t)}, 0 ≤ t ≤ tf − hl ,  v¯(t) = col{v0 (t), · · · , vi (t)}, tf − hi+1 < t ≤ tf − hi

(8.225)

with vi (t) as defined in (8.151) which is different from the Krein space element vi (t). Then we have the following result. Lemma 8.4.3. Under the conditions of Theorem 8.4.2, Jtf of (8.177) can be rewritten as: Jtf = x (0)P (0)x(0) + (¯ v − v¯∗ ) Ry¯ (¯ v − v¯∗ ),

(8.226)

¯  and v¯∗ is where P (0) is the solution of RDE (8.184) at t = 0, Ry¯ = ¯ y, y ∗ obtained from v¯ with vi (t) replaced by vi (t) for i = 0, · · · , l and 0 ≤ t ≤ tf − hi .

8.4 H∞ Full-Information Control

197

Proof: Recall from (8.177) that Jtf = ξ  Pξ + (v − v ∗ ) Ry (v − v ∗ ).

(8.227)

Since for t < 0, wi (t) = 0 and ui (t) = 0, i.e. vi (t) = 0, it follows from (8.153) that ˆ (0), x(0) − x ˆ (0) = v˜(t) = 0, 0 ≤ t ≤ hl , which implies that v˜ = 0. Since x(0) − x P (0), we have ξ  Pξ = x (0)P (0)x(0). Further, it is straightforward to verify that (v − v ∗ ) Ry (v − v ∗ ) is equal to (¯ v − v¯∗ ) Ry¯ (¯ v − v¯∗ ). Hence, the result follows. ∇ ¯ Now we introduce the innovation sequence w(t) associated with the new obser¯ (t) as vation y ˆ ˆ ¯ (tf | tf ) = 0, ¯ (t | t), y ¯ ¯ (t) − y w(t) =y

(8.228)

ˆ ¯ (t | t) is the projection of y ¯ (t) onto the linear space L{¯ where y y(τ ), t < τ ≤ ¯ ), t < τ ≤ tf }. It is easy to observe that tf } = L{w(τ ¯ w (t)δ(t − τ ) = R ¯ v (t)δ(t − τ ), ¯ ¯ ) = R w(t), w(τ ¯ v (t) is as in (8.222). The observation y ¯ = {¯ where R y(t), 0 ≤ t ≤ tf } and the ¯ = {w(t), ¯ innovation w 0 ≤ t ≤ tf } have the following relationship, ¯ = Lw w, ¯ y

(8.229)

where Lw = I + kw is a causal operator, i.e. the kernel of kw (t, τ ) is zero for ¯ τ ≥ t. By using the innovation sequence w(t), we have Ry¯

¯ w L , ¯  = Lw R = ¯ y, y w

(8.230)

¯ w = w, ¯ w ¯ is a diagonal operator. where R Lemma 8.4.4. Under the conditions of Theorem 8.4.2, the linear quadratic form Jtf of (8.177) can be further rewritten as Jtf

= x (0)P (0)x(0) +

l 9  i=0

tf −hi

{vi (τ ) − vi∗ (τ )} R(i) {vi (τ ) − vi∗ (τ )} dτ,

0

(8.231) where vi∗ (τ ) is as in (8.213). Proof: From (8.226) and (8.230), it follows that Jtf

= =

ξ  Pξ + (¯ v − v¯∗ ) Ry¯ (¯ v − v¯∗ ) ¯ w (τ )L (¯ v − v¯∗ ) Lw R ¯∗ ). ξ  Pξ + (¯ w v−v

Similar to the discussion in [30], (8.231) follows immediately. Now we are in the position to give the main result of this section.

(8.232) ∇

198

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

Theorem 8.4.3. Consider the system (8.144)-(8.145) and the performance criterion (8.146). For a given γ > 0, suppose the RDE (8.184) has a bounded solution P (t), 0 ≤ t ≤ tf and for any τ > 0, the RDE (8.211) admits a bounded solution P τ (t), 0 ≤ t ≤ min{hl , tf − τ }, with initial value P τ (t) |t=hl = P (τ + hl ) for hl + τ ≤ tf or P τ (t) |t=tf −τ = P (tf ) = 0 for hl + τ > tf , where P (τ + hl ) is calculated by RDE (8.184). Then, there exists an H∞ controller that solves the H∞ full-information control problem if and only if Π −1 − γ −2 P (0) >

0,

(8.233)

where P (0) is the terminal value of P (τ ) of (8.184). In this case, a suitable H∞ controller (central controller) u∗i (τ ) is given by ui (τ ) = [Im , 0] × vi∗ (τ ) where vi∗ (τ ) is as in (8.213), i.e.  −1  vi∗ (τ ) = −R(i) Γ(i) × P τ (hi )[Ψ¯ τ (0, hi )] x(τ )+ 9 hi P τ (hi ) [Ψ¯ τ (s, hi )] {In − Gτ (s)P τ (s)} v˜τ ∗ (s)ds+ h0 , 9 hl τ τ τ τ τ ∗ [In − P (hi )G (hi )] Ψ¯ (hi , s)P (s)˜ v (s)ds , hi+1

(8.234) while v˜τ ∗ (·) is as in (8.207) with vi (·) replaced by vi∗ (·) for i = 0, · · · , l, i.e. ⎧ l ⎪ ⎨  Γ v ∗ (t + τ − h ), h ≤ t < h , i = 0, 1, · · · , l − 1 j i i+1  (j) j τ∗ v˜ (t) = j=i+1 ⎪ ⎩ 0, t ≥ hl (8.235) and Gτ (s) is as defined in (8.214), i.e., 9 s ! Ψ¯ τ (r, s) Γ (r)R−1 (r)Γ  (r)Ψ¯ τ (r, s)dr. Gτ (s) =

(8.236)

0

Proof: Substituting (8.231) into (8.148) yields Jt∞ f

=

x (0)Π −1 x(0) − γ −2 Jtf

x (0)[Π −1 − γ −2 P (0)]x(0) l 9 tf −hi   {vi (τ ) − vi∗ (τ )} R(i) {vi (τ ) − vi∗ (τ )} dτ, −γ −2 =

i=0

0

where vi∗ (τ ) is as in (8.234). Note  R(i) = diag{D(i) D(i) , − γ 2 Ir },

(8.237)

8.4 H∞ Full-Information Control



and vi (t) =

199

 ui (t) , wi (t)

can be further written as then Jt∞ f Jt∞ f

=

x (0)[Π −1 − γ −2 P (0)]x(0) l 9 tf −hi    −γ −2 {ui (τ ) − u∗i (τ )} D(i) D(i) {ui (τ ) − u∗i (τ )} dτ i=0

+

0

l 9 tf −hi  i=0



{wi (τ ) − wi∗ (τ )} {wi (τ ) − wi∗ (τ )} dτ.

(8.238)

0

Our aim is to find the controller ui (t) such that Jt∞ is positive for all x(0) f has minimum over and exogenous input wi (t), which is equivalent to that Jt∞ f {x(0); wi (t), 0 ≤ t ≤ tf − hi , i = 0, · · · , l}, and ui (t) can be chosen such that the minimum of Jt∞ is positive. Thus, the necessary and sufficient condition for f the existence of the H∞ controller is that Π −1 − γ −2 P (0) >

0.

(8.239)

In view of (8.238), a suitable controller (central controller) can be chosen such that ui (τ ) = u∗i (τ ) = [Im , 0] × vi∗ (τ ). Therefore, we have established the theorem.

(8.240) ∇

Remark 8.4.3. For delay-free systems, i.e. h1 = · · · = hl = 0, it is obvious that a suitable H∞ controller is given from (8.234) as u∗0 (τ )

= [Im , 0] × vi∗ (τ )   = −(D(0) D(0) )−1 B(0) P τ (0)[Ψ¯ τ (0, 0)] x(τ )   D(0) )−1 B(0) P (τ )x(τ ), = −(D(0)

(8.241)

where Ψ¯ τ (0, 0) = I has been used in the second equality. Note that P τ (0) = P (τ ) obeys the standard Riccati equation (8.184). Thus the obtained H∞ controller u∗0 (τ ) is the same as the well-known result for H∞ control [30]. Remark 8.4.4. Observe that a similar H∞ full-information control for systems with multiple input delays has been investigated in [44] using an operator Riccati equation approach. In the present work, we give an explicit solution based on a duality between the H∞ control and a smoothing problem. 8.4.5

Special Cases

In this subsection, we shall discuss some special cases of the H∞ control problem tackled earlier.

200

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

H∞ control for single input delay systems. We consider the following time invariant system for the H∞ control problem. = Ax(t) + B1 w(t) + B2 u(t − h) % & Cx(t) z(t) = , D D = Im . Du(t − h)

x(t) ˙

(8.242) (8.243)

It is clear that (8.242)-(8.243) is a special case of the system (8.144)-(8.145) with l = 1, G(0) = B1 , G(1) = 0, B(0) = 0, B(1) = B2 , D(0) = 0, D(1) = D and Q = C  C. Associated with the system (8.242)-(8.243) and performance index (8.146), we introduce the following notations as in the last section, R(0) = −γ 2 Ir , R(1) = Im , v0 (t) = w(t), v1 (t) = u(t), Γ(0) = B1 , Γ(1) = B2 , and 

v τ (t) = 

v˜τ (t) = 

Γ (t) =

⎧ ⎨  v0 (t + τ ),  0 ≤ t < h, (t + τ ) v 0 ⎩ , t≥h v1 (t + τ − h) 1 B2 u(t + τ − h), 0 ≤ t < h, 0, t≥h 1 0≤t


R(t) =

[ Γ(0)

Γ(1) ] , t ≥ h

R(0) ,

0 ≤ t < h,

diag{R(0) , R(1) }, t ≥ h.

(8.244)

(8.245)

(8.246)

(8.247)

Following a similar discussion as in Theorem 8.4.3, we have Theorem 8.4.4. Consider the system (8.242)-(8.243) and performance index (8.146). Given a scalar γ > 0, assume that: 1) RDE −

dP (t) dt

=

A P (t) + P (t)A + C  C − P (t)Γ (t)R(t)−1 Γ (t) P (t), P (tf ) = 0, (8.248)

has a bounded solution P (t) for 0 ≤ t ≤ tf . 2) RDE −

dP τ (t) dt

=

A P τ (t) + P τ (t)A + C  C +

1 τ P (t)B1 B1 P τ (t), (8.249) γ2

has a bounded solution for 0 ≤ τ < tf , 0 ≤ t < min{h, tf − τ }, where for τ + h < tf the initial value P τ (t) |t=h = P (τ + h) which is calculated by (8.248), while for the case of τ + h ≥ tf the initial value is as P τ (t) |t=tf −τ =P (tf ) = 0.

8.4 H∞ Full-Information Control

201

Then, there exists an H∞ controller that solves the H∞ control problem if and only if Π −1 − γ −2 P (0) >

0,

(8.250)

where P (0) is the terminal value of P (t) of (8.248). In this situation, a suitable H∞ controller is given by u(τ ) =

−B2 P τ (h)[Ψ¯ τ (0, h)] x(τ ) 9 h  τ −B2 P (h) [Ψ¯ τ (s, h)] {[In − Gτ (s)P τ (s)} B2 u(s + τ − h)ds, 0

(8.251) where

9

s

[Ψ¯ τ (r, s)] Γ (r)R(r)−1 Γ (r) Ψ¯ τ (r, s)dr 0 9 s 1 = − 2 [Ψ¯ τ (r, s)] B1 B1 Ψ¯ τ (r, s)dr, (8.252) γ 0   ¯τr = − 12 P τ (r)B1 B1 + A . and Ψ¯ τ (r, .) is the transition matrix of −Φ γ Gτ (s) =

Remark 8.4.5. We note that when tf → ∞, (8.248) and (8.249) will be replaced respectively by ; < 1    B1 B1 − B2 B2 P + C  C = 0 (8.253) A P + PA + P γ2 and − P˙ τ (t)

= A P τ (t) + P τ (t)A + C  C +

1 τ P (t)B1 B1 P τ (t), (8.254) γ2

where 0 ≤ t ≤ h and the terminal condition P τ (h) = P . In this case, it is known from [87], where the infinite horizon H∞ control of systems with single input delay is considered, that an H∞ controller exists if (8.253) admits a positive semi-definite stabilizing solution P and (8.254) admits a bounded non-negative definite solution P τ (t), 0 ≤ t ≤ h. Noting that P τ (h) = P , it follows from (8.251) that u(τ ) =

−B2 P [Ψ¯ τ (0, h)] x(τ ) 9 h   ¯ τ (s)P τ (s) B2 u(s + τ − h)ds,(8.255) −B2 P [Ψ¯ τ (s, h)] I + G 0

¯ τ (s) is where G ¯ τ (s) = G =

−Gτ (s) 9 s 1 [Ψ¯ τ (r, s)] B1 B1 Ψ¯ τ (r, s)dr. γ2 0

(8.256)

Note that Ψ¯ τ (r, s) is the transition matrix of −Φ¯τr = − γ12 (P τ (r)B1 B1 + A ), ! thus Ψ¯ τ (r, s) is the transition matrix of 12 (B1 B1 P τ (s) + A) [77]. Let h = 1, γ

202

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

then the controller (8.255) is the same as the controller in [87]. We note that there is a typing mistake in G(s) of [87]. The correct form should be (8.256). H∞ control with preview. H∞ control with preview is concerned with the following system x(t + 1) = z(t) =

Ax(t) + B1 w(t − h) + B2 u(t), & % Cx(t) , D D = Im . Du(t)

(8.257) (8.258)

It is clear that (8.257)-(8.258) is a special case of the system (8.144)-(8.145) with l = 1, Φ = A, G(0) = 0, G(1) = B1 , B(0) = B2 , B(1) = 0, D(0) = D, D(1) = 0, and Q = C  C. Denote R(0) = Im , R(1) = −γ 2 Ir , v0 (t) = u(t), v1 (t) = w(t), Γ(0) = B2 , Γ(1) = B1 , and 

v τ (t) = 

v˜τ (t) = 

Γ (t) =

⎧ ⎨  v0 (t + τ ),  0 ≤ t < h, (t + τ ) v 0 ⎩ , t≥h v1 (t + τ − h) 1 B1 w(t + τ − h), 0 ≤ t < h, 0, t≥h 1 0≤t


R(t) =

[ Γ(0)

Γ(1) ] , t ≥ h 0 ≤ t < h,

R(0) ,

diag{R(0) , R(1) }, t ≥ h.

(8.259)

(8.260)

(8.261)

(8.262)

Following a similar discussion as in Theorem 8.4.3, we have Theorem 8.4.5. Consider the system (8.242)-(8.243) and performance criterion (8.146). Given a scalar γ > 0, assume that the RDE −

dP (t) dt

= A P (t) + P (t)A + C  C − P (t)Γ (t)R(t)−1 Γ (t) P (t),(8.263)

with P (tf ) = 0 has a bounded solution P (t) for 0 ≤ t ≤ tf . Then, there exists a controller that solves the H∞ control problem with preview if and only if Π −1 − γ −2 P (0) >

0,

(8.264)

where P (0) is the terminal value of P (τ ) of (8.263). In this situation, a suitable H∞ preview controller is given by 9 h  τ τ   ¯ Ψ¯ τ (0, s)P τ (s)B1 w(s + τ − h)ds, u(τ ) = −B2 P (0)[Ψ (0, 0)] x(τ ) − B2 0

(8.265)

8.5 Conclusion

203

where Ψ¯ τ (s, ·) is the transition matrix of − Φ¯τs = K τ (s)Γ  (s) − Φ = P τ (s)B2 B2 − A

(8.266)

and P τ (t) for t ≤ min{h, tf − τ } is a solution to the RDE: −

dP τ (t) dt

= A P τ (t) + P τ (t)A + C  C − P τ (t)B2 B2 P τ (t), (8.267)

for any 0 ≤ τ < tf and t ≤ min{h, tf − τ }, where for τ + h < tf the initial value P τ (t) |t=h = P (τ + h) which is calculated by (8.263), while for the case of τ + h ≥ tf the initial value is as P τ (t) |t=tf −τ =P (tf ) = 0. Remark 8.4.6. We note from [26] that if the RDE (8.263) has a bounded solution P (t), then P (t) ≥ 0, which is in fact a necessary and sufficiency solution for the solvability of the standard H∞ control (h = 0). Also, since (8.267) is a LQ-type of RDE, its existence of a bounded solution is guaranteed. Remark 8.4.7. When tf → ∞, (8.263) and (8.267) will be replaced respectively by ; < 1    A P + P A + P B B − B B (8.268) 1 2 1 2 P +C C = 0 γ2 and − P˙ τ (t) =

A P τ (t) + P τ (t)A + C  C − P τ (t)B2 B2 P τ (t),

(8.269)

where 0 ≤ t ≤ h and the terminal condition P τ (h) = P . We note that the H∞ preview control problem in the infinite horizon case has been studied in [45, 88]. In [45], a sufficient solvability condition of the H∞ preview control is related to the existence of a stabilizing solution to (8.268) and the non-singularity of a matrix. The latter is shown to be equivalent to the existence of a stabilizing solution of an operator Riccati equation. An interesting necessary and sufficient condition for the infinite horizon H∞ control with preview has been given in [88].

8.5 Conclusion In this chapter we have studied the LQR problem for systems with multiple input delays. Firstly, an analytical solution to the LQR problem for input delayed systems has been derived by establishing a duality between the LQR and a smoothing estimation and applying standard projections. Next, the LQG control problem has been approached by applying a separation principle and the Kalman filter for systems with measurement delays discussed in Chapter 7. Finally, the H∞ full information control problem for multiple input delay systems has been solved by extending the result for an indefinite LQR problem.

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Index

ARMA, 46 ABR, 46 ATM, 46 augmented system, 67 augmentation, 8 augmentation approach, 26 auto-regressive(AR), 48, 111

differential equation, 129, 138 discrete-time systems, 17, 36, 44, 53 disturbance inputs, 100 downstream, 186 duality, 27, 29, 33, 34, 36, 163 duality principle, 27 dynamic programming, 39, 51, 194

backward delay, 48 backward iteration, 22, 121, 130 backward RDE, 38 backward stochastic state-space model, 30, 90, 188 boundary conditions, 145, 154 buffer, 47 buffer length, 48

energy gain, 53 estimation error, 53 Euclidean, 55 exogenous input, 88, 96 exogenous noise, 87

causal, 185 CBR(constant bit rate), 46 central estimator, 154 central smoother, 63 congested switch, 47 congestion control, 47, 51 continuous-time systems, 115, 130, 132, 141 cost function, 42, 109 covariance matrix, 12, 13, 19, 20, 48, 57, 59, 83 cross-covariance matrices, 128 cross-Gramian, 3

fictitious observation, 78, 124 finite dimensional approach, 145 finite-horizon, 108, 134 finite element method, 157 fixed-interval smoothing, 145 gain matrix, 34 game theoretic theory, 162 Gaussian, 42, 48, 131 Gaussian sequence, 47 Gramian, 2 matrix, 2, 3 operator, 189 Hamlitonian matrix, 163 H∞ :

delay free system, 53, 179 delayed inputs, 43 delayed measurements, 7, 18, 26, 43 difference equation, 60

control, 1 estimation, 15 filter, 80 filtering, 53, 77

212

Index

fixed-lag smoothing, 15, 26, 53, 63, 115, 132, 140 full-information control, 87, 104 multiple step-ahead prediction, 4, 53 performance, 115 prediction, 143, 161 H2 fixed-lag smoothing, 67 H2 estimation, 83, 141 Hilbert spaces,1, 2, 15, 70, 134 indefinite quadratic form, 87, 185 indefinite quadratic optimization, 143 indefinite space, 85 inertia, 64, 72, 80 infinite-dimensional systems, 143, 163 infinite horizon, 203 initial condition, 44, 75 initial estimate, 54 initial state, 8, 30, 39 innovation, 73, 81 innovation covariance, 61, 63, 147 innovation sequence, 14, 19 input delays, 27 input noises, 53, 69 integral operators, 166 i/o, 51 isotropic vector, 4 J-factorization,

54, 163

Kalman: filter, 4, 14, 42, 44 filtering, 6, 7, 10, 15, 13-17, 20, 22, 27, 34, 43, 51 Kernels, 167, 189 Krein spaces, 1, 2, 5, 15, 53, 55, 71, 134, 145 LDU factorization, 103 linear combination, 146 linear least mean square error, 18 linear estimation theory, 33 linear optimal estimation problem, 18 linear quadratic form, 100 linear quadratic Gaussian (LQG), 28, 43, 44, 47, 51, 163 linear quadratic regulation(LQR), 28, 29, 34, 36, 41, 42

linear space, 35, 74 link capacity, 48 measurement delay, 77, 141 measurement feedback control, 178 measurement noises, 42, 53 measurement output, 30, 42, 54, 69 mean square error, 7 minimax, 54, 95, 185 minimizing solution, 33 minimum, 72, 125 Minkowski, 1 multiple input delays, 87, 105, 163, 164 multiple measurement delays, 17, 84 multiple state delays, 143 multi-step prediction, 26 negative definite, 146 networked congestion control, networked control, 27 non-negative definite, 28, 42 null space, 4

27, 87

operator Riccati equation, 115 observation, 10 observation sequence, 10 one-step ahead prediction, 4, 13, 59 operator approach, 162 operator Riccati equations, 143 operator Riccati differential equations, 165 optimal control, 33, 36, 38 optimal controller, 37, 39, 41, 45, 50 optimal estimate, 9 optimal estimator, 12, 24 optimal estimation, 19 optimal filter, 13, 16, 22 optimal filtering, 26, 34 optimal filtering problem, 22 optimal smoothing estimate, 35 optimal state estimator, 18 optimization, 29 orthogonality, 150 partial differential equations, 115 performance criterion, 54, 63 performance index, 28 predictor, 162 preview, 88, 108, 202 priority sources, 46

Index projection, 9, 14, 18, 23 projections, 1, 3 projection formula, 13, 14, 21, 22, 35 quadratic function, 5, 6 quadratic performance index, quality of service(QoS), 46 queue buffer, 111 queue length, 46, 47, 49

28

re-organized innovation, 1, 11, 20, 22 23, 26, 66, 85 re-organized innovation analysis, 7, 9 15, 53 re-organized innovation sequence, 137 re-organized measurements, 136 Riccati difference equation (RDE), 7, 12, 27, 28, 44, 136 Riccati difference recursions, 15 Riccati equation, 15, 20, 24, 60 Riccati Partial Differential Equation, 115 RM cells, 47 separation principle, 28, 164 single input delay, 88, 106 single measurement delay, 84 smoothing estimation, 27 smoothing gain matrix, 169 smoothing problem, 33 state augmentation, 53

213

state estimation error, 20 state feedback, 27, 87 state prediction error, 119 static output feedback control, 27, 87 source rates, 47, 111 source sharing, 48 stationary point, 5 stochastic backward state-space model, 35 stochastic optimization, 145 stochastic sequence, 19 stochastic system, 30, 55, 70 system augmentation, 15 target queue length, 111 time delay, 15, 17 time-horizon, 54, 133 time-varying, 130 traffic, 46 transmission capacity, 46 transition matrix, 121, 169, 171 uncertainty, 77 unilateral delay, 186 unitary matrices, 102 upstream, 186 VBR(variable bit rate), white noises, wind-tunnel,

46

10, 12, 17, 30, 48, 66, 70 186

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