Control Engineering Series Editor William S. Levine University of Maryland
Editorial Advisory Board Okko Bosgra Delft University, Netherlands
William Powers Ford Motor Company, USA
Graham Goodwin University of Newcastle, Australia
Mark Spong University of !/Iinois,
Petar Kokotovic University of California, Santa Barbara
lori Hashimoto Kyoto University, Japan
Manfred Morar; ETH, Zurich, Switzerland
Urbana~Champaign
Robust Kalman Filtering for Signals and Systems with Large Uncertainties Ian R. Petersen Andrey V. Savkin
Birkhauser Boston. Basel. Berlin
Ian R Petersen School of ElectrIcal Engmeermg AustralIan Defence Force Academy Umverslty of New South Wales Northcott Dnve Canberra, ACT 2600 AustralIa
Andrey V Savkm Department of Electncal and Electromc Engmeenng Umverslty of Western AustralIa Nedlands, WA 6907, AustralIa
Library of Congress Cataloging-in-Publication Data Petersen Ian R Robubt Kalman filtenng for signals and systems with large uncertamtles / Ian R Petersen, Andrey V Savkm p em - (Control engmeermg) Includes bibliographical referpn(,pq and mdex ISBN 0-8176-4089-4 (hardcover alk paper) 1 Feedback control systems 2 Kalman filtermg 3 Signal processmg I Savkm, Andrey V II Title III Senes Control engmeermg (Blrkhauser) TJ216 P462 1999 629 8'3-DC21
98-460l0 CIP
AMS Subject ClassificatIOns 49, 93 PrInted on aCid-free paper © 1999 Blrkhauser Boston
Birkhiiuser
$
®
All rIghts reserved ThiS work may not be tranqlated or copied m whole or m part without the wntten permission of the publisher (Blrkhauser Boston, c/o Sprmger-Verlag New York, Inc, 175 Fifth Avenue, New York, NY 10010, USA) except for bnef excerpts m connectIOn with reVieWS or scholarly analysIs Use III connectIOn with any form of mformatlOn ,torage and retneval, electromc adaptatIOn, computer software or by Similar or diSSimilar methodology now known or hereafter developed IS forbIdden The use of general descnptlve names, trade names, trademarks, etc, m thIS publication, even If the former are not espeCially IdentIfied IS not to be tdken as a sign that such names, as understood by the Trade Marks and MerchandIse Marks Act, may accordmgly be used freely by anyone ISBN 0-8176-4089-4 ISBN 3-7643-4089-4
SPIN 19901575
Typeset by the author usmg I;\1EX 2E Prmted and bound by Edwards Brothers, Inc, Ann Arbor MI Prmted m the Umted States of Amenca
9 8 7 6 5 4 3 2 1
Contents
Preface 1
2
3
Introduction 1 1 The Kalman Fllter 1 2 Robust State EstimatlOn 1 3 Guaranteed Cost State EstllnatlOn 14 Set-Valued State EstllllatlOn 141 Discrete-Contmuous Data 142 Structured U ncertamty 143 Nonlmear Systems 1 5 Model VahdatlOn for Uncertam Systems 1 6 Robust H:xJ Flltenng 1 7 ApphcatlOns of Robust Kalman Flltenng
i~
1 I
2 ~.
3 £;
.6 (]
()
7
8
Continuous-Time Quadratic Guaranteed Cost Filtering 2 1 IntroductlOn 22 A Guaranteed Co<;t State E<;hmatlOn Problem 23 Prehmmary Results 24 OptImal Guaranteed Cost Filter DesIgn 25 IllustratIve Example
11 11
Discrete-Time Quadratic Guaranteed Cost Filtering 3 1 IntroductlOn 32 Dlscrete-Tnne Quadratlc Guaranteed Cost Filtenng
35
1'2 17
2S 3'2 3,5
36
VI
Contents 33 34 35
4
5
PrelImInary Results Discrete-Tlme Optimal Guaranteed Cost Fllter Deslgn Illustrative Example
40 43 51
Continuous-Time Set-Valued State Estima60n and Model Validation 4 1 IntroductlOn 42 Model Validation and Set-Valued State Estimation Problem Statements 4 3 Deslgn of Set-Valued State Estimator 4 4 Tlme Invanant Set-Valued State EstImatlOn Discrete-Time Set-Valued State Estimation 5 1 IntroductlOn 5 2 Sum Quadratic ConstraInt UncertaInty DescnptlOn 53 Deslgn of DIscrete-TIme Set-Valued State Estimator 54 Dlscrete-Tlme Uncertam Systems with MISSIng Data 5 5 Deslgn of a Set-Valued State Estimator with MISSIng Data 5 6 A Robust Deconvolution Problem 5 7 A Robust DeconvolutlOn Problem with MISSIng Data
57 57
59 62 68
71 71 72
74 78 79 82 84
6
Robust State Estimation with Discrete and Continuous Measurements 89 6 1 IntroductlOn 89 6 2 UncertaIn Systems wlth Dlscrete and ContInUOUS Measurements 91 63 Deslgn of a Hybnd Set-Valued State EstImator 93 64 Uncertam Systems wIth Mlssmg Data 100 6 5 Model VahdatlOn and State EstlmatlOn wIth MISSIng DiscreteContInuous Data 102
7
Set-Valued State Estimation with Structured Uncertainty 7 1 IntroductlOn 72 Averaged Integral QuadratIC Constramts 73 S-Procedure for Averaged Integral Quadratic ConstraInts 7 4 DesIgn of a Set-Valued State EstImator 75 IIlustratlVe Example
8
107 107 109
113 118 125
Robust HOC Filtering with Structured Uncertainty 131 8 1 IntroductlOn 131 132 8 2 Robust H OO FIltenng 8 3 S-Procedure for NonlInear UncertaIn Systems on an Infimte TIme Interval 136
Contents 8.4 9
Design of Robust H OO Filters . . . .
Robust Fixed Order HOC> Filtering 9.1 Introduction . . . . . . . . . . . . . . 9.2 Fixed Order H OO Filters . . . . . . . 9.3 Nonlinear versus Linear Fixed Order Filtering.
VII
139
145 145 146 147
10 Set-Valued State Estimation for Nonlinear Uncertain Systems 153 10.1 Introduction. . . . . . 153 10.2 Problem Formulation. 154 10.3 The State Estimator . 156 10.4 A Robust Extended Kalman Filter 158 161 10.5 Local Results for the Robust Extended Kalman Filter 10.6 Illustrative Example . . . . . . . . . . . . . . . . . . . 170 11 Robust Filtering Applied to Induction Motor Control 11.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . 11.2 State Feedback Torque Control of Induction Motors 11.3 A Robust Kalman Filter for the Induction Motor 11.4 Simulation and Experimental Results. 11.5 Discussion . . . . . . . . . . . . . . . .
173 173 174 176 178 181
References
185
Index
199
Preface
A sigmficant shortcoming of the state space control theory that emerged in the 1960s was its lack of concern for the issue of robustness. However. in the design of feedback control systems, robustness is a critical issue. These facts led to great activity in the research area of robust control theory. One of the major developments of modern control theory was the Kalman Filter and hence the development of a robust version of the Kalman Filter has become an active area of research. Although the issue of robustness in filtering is not as critical as in feedback control (where there is always the issue of instability to worry about), research on robust filtering and state estimation has remained very active in recent years. However. although numerous books have appeared on the topic of Kalman filtering. this book is one of the first to appear on robust Kalman filtering. Most of the material presented in this book derives from a period of research collaboration between the authors from 1992 to 1994. However, its origins go back earlier than that. The first author (I.R.P.) became interested in problems of robust filtering through his research collaboration with Dr. Duncan McFarlane. At this time, Dr. McFarlane was employed at the Melbourne Research Laboratories of BHP Ltd .. a large Australian minerals, resources, and steel processing company. In discussing possible areas of joint research which would be of interest to BHP, it was suggested that Kalman Filtering might be a suitable area of practical importance. Since both Dr. McFarlane and the first author had a strong research interest in the area of robust control, it was decided that a good area of joint research would that of robust Kalman filtering. This led to a series of results on Robust Kalman Filtering. which are presented in Chapters 2 and 3 of this
x
Preface
book. Furthermore, it formed the launching point for much of the research carried out by the authors, which is presented in Chapters 4 to 11. Thus, it is useful to note that the underlying motivation for the research presented in this book was the fact that the Kalman Filter remains the most important result in modern systems and control theory from the point of view of practical applications. During the period from 1992 to 1994, the authors carried out collaborative research on various aspects of robust control and filtering theory. Much of this research resulted from a melding of ideas emerging from researchers in the old Soviet Union and ideas which had been developed in the West. In particular, it was the ideas of the absolute stability and integral quadratic constraints from the Leningrad School together with ideas from robust control and filtering theory developed in the West that led to the robust set-valued state estimation ideas, which form a large portion of this book. This book is primarily a research monograph which presents in a unified fashion, some recent research on robust Kalman filtering. The book is intended for researchers in robust control and filtering theory, advanced postgraduate students, and engineers with an interest in applying the latest techniques of robust Kalman filtering. In the production of this book, the authors wish to acknowledge the support they have received from the Australian Research Council. Also the authors wish to acknowledge the major contributions made by their colleagues Duncan McFarlane, Matthew James, Duco Pulle, and Reza Moheimani in the research that underlies much of the material presented in this book. Furthermore, the first author is grateful for the enormous support he has received from his wife Lynn and children Charlotte and Edward.
1 Introduction
1.1
The Kalman Filter
One of the most significant ideas to emerge in the area of systems and control theory is the Kalman Filter. Given a signal model that consists of a linear dynamical system driven by stochastic white noise processes, the Kalman Filter provides a method for constructing an optimal estimate of the system state. Thus. the Kalman Filter provides an optimal way of extracting a signal from noise by exploiting a state space signal model. A key feature of the Kalman Filter is that it involves finite dimensional recursive computations, which are straightforward to implement on a digital computer. The Kalman Filter can be applied to state estimation problems defined over a finite or infinite time interval. It is applicable for the case in which the signal model is a linear but possibly time-varying linear system. Also, an approximate version of the Kalman Filter referred to as the "Extended Kalman Filter" can be applied in the case of a nonlinear signal model; e.g .. see [2]. A precursor to the Kalman Filter was the Weiner Filter which was derived independently by Weiner [157] and Kolmogorov [70]. The Weiner Filter also gives a method of optimally extracting a signal from noise. However, the Weiner Filter is limited to time-invariant problems involving stationary noise processes. Also. the Weiner Filter is not as computationally straightforward as the Kalman Filter. The discrete-time Kalman Filter was originally described in the paper [64] and the continuous-time Kalman Filter was originally described
2
1. Introduction
in the paper [65]. Since then, the Kalman Filter has been the subject of thousands of papers. patents, and books. Some of the books that have been published on the Kalman Filter are as follows [2,8,17,21,23,25.29,31,43.47, 49,54,60.61,63,80,83,84,146]. Also. some of the applications of the Kalman Filter have included Inertial Navigation and Guidance [5,22,25,169]. Global Positioning Systems [36,66], Target Tracking [9.29,153], Communications and Signal Processing [24,78,106,146,152], Electrical Machines [46]. and Finance [156]. Furthermore, it has been applied to problems of stochastic optimal control, fault diagnosis, deconvolution filtering, channel equalization; e.g., see [2,28] and the references therein. An interesting brief description of the history of Kalman Filtering theory is contained in the book review [14]. Also, a number of articles concerning the history and applications of the Kalman Filter are contained in the book [4].
1.2
Robust State Estimation
In spite of the large number of papers published on the Kalman Filter in the last forty years, one issue that was not addressed to any significant degree until recently is robustness: see [67] for some early results on robust filtering. In particular, the standard approach to Kalman Filtering does not address the issue of robustness against large parameter uncertainty in the linear process model. For example, the performance of a Kalman Filter may be significantly degraded if the actual system model does not match the model on which the Kalman Filter was based: e.g., see [60]. The aim of this book is to present some of the recent research that has emerged in the area of robust Kalman filtering. In particular, this book looks at the various ways in which the standard Kalman Filter can be modified to make it robust against large parameter uncertainties. One of the critical assumptions required in the application of the Kalman Filter is the assumption that there exists an accurate signal model. If this assumption does not hold and there are errors in the process model, the Kalman Filter may lead to poor performance: e.g., see [100.154]. The robust state estimation problem arises out of the desire to estimate unmeasurable state variables when the only available model is uncertain. That is, the signal model is a system whose dynamics are unknown but bounded in some sense. Thus, the issue of robustness can be addressed by considering the optimal state estimation problem applied to a uncertain dynamical system rather than a known linear system.
1.4 Set-Valued State EstimlLtion
1.3
3
Guaranteed Cost State Estimation
The problem of robust state estimation can be regarded as an extension of the standard Kalman Filter to the case of uncertain systems. In recent years, the increased interest in robust and H OO control theory has led to the publication of a large number of papers addressing the robust state estimation problem; e.g., see [18-20,32.33,44,53,57,99,100,141,160]. In these papers, a number of approaches are considered. In particular, the approach considered in the papers [53,57,99,100] is concerned with constructing a state estimator that bounds the mean square estimation error. In references [53,57,99,100]' a fixed quadratic "Lyapunov function" is used to establish an upper bound on the state estimation error covariance. This motivates a notion of Quadratic Guaranteed Cost State Estimation defined in [99] and [100]. This definition is analogous to the concept of quadratic stabilization considered in robust control problems; e.g., see [68]. The main result of [99,100] is a Riccati equation approach to the construction of an "optimal" quadratic guaranteed cost state estimator. That is, the state estimator leads to a mean square error bound that is at least as good as the bound obtained from any other quadratic guaranteed cost state estimator. This is in contrast to most other results on robust state estimation that lead to state estimators which are sub-optimal in this sense. The main results of the papers [99,100] are presented in Chapter 2. The main results of the papers [101,102] extend the continuous time results of [99,100] to the case of uncertain discrete-time systems. The motivation for this extension is that from the point of view of applications, a discrete-time robust state estimator is much more useful than the corresponding continuous-time robust state estimator. Indeed, the use of digital hardware invariably requires that a state estimator must be implemented in discrete-time. These results are presented in Chapter 3.
1.4
Set-Valued State Estimation
Aside from the usual stochastic interpretation of the Kalman Filter, it is also possible to view the Kalman Filter from a deterministic point of view. In particular, it is useful to consider the deterministic interpretation of Kalman Filtering presented in [16]. In [16], the following problem is considered: Given output measurements from a time-varying linear system with noise inputs subject to an L2 norm bound, find the set of all states consistent with these measurements. The solution to this problem was found to be an ellipsoid in state space that is defined by the standard Kalman Filter equations. Thus, the results of [16] give an alternative deterministic interpretation of the standard Kalman Filter. This deterministic interpretation of the Kalman Filter has proved to be particularly useful in extending the
4
1 Introduction
standard Kalman Filter to obtain a robust Kalman Filter for uncertain systems. A useful reference on the deterministic set-valued approach to state estimation is the book [73]. In the papers [120.127,139]. the results of [16] are extended to obtain a robust Kalman Filter for a class of uncertain systems: see also [85,87] for an overview of some of the recent results obtained in the area of set-valued state estimation. A feature of the set-valued state estimation approach taken in [120-122,126.127,139] is that it leads to a different observer structure than that obtained in the HOC filtering approach such as presented in [89]. The class of time-varying uncertain systems considered in the papers [120,121,126,127] involves uncertainty. which is defined by a certain Integral Quadratic Constraint: e.g., see [117,119.136,165]. The integral quadratic constramt considered in [120,121. 126. 127] extends the definitions of [16] to include uncertainty of the type that originated in the work of Yakubovich. This class of uncertainties is a particularly rich uncertainty class allowing for nonlinear, time-varying. dynamic uncertainties. Furthermore, the integral quadratic constraint uncertainty description has recelved a significant amount of interest in the recent robust control literature: e.g .. see [115,117,119,136]. In Chapter 4. we present the main results contamed in the papers [120,121,126,127]. The class of uncertain systems considered in Chapter 4 allows for nonlinear dynamic uncertainties. Consequently, it is not straightforward to extend the stochastic approach to Kalman Filtering to such systems. In contrast, we show that the deterministic approach of [16] is straightforward to extend to this case. The main results of Chapter 4 show that the results of [16] can be modified directly to exactly construct the set of possible states for the class of uncertain systems under consideration. As in [16]. the solution is found to be an ellipsoid in the state space. This set is constructed via the solution of Kalman Filter like equations. In fact the modified Kalman Filter equations derived in Chapter 4 correspond directly to the state estimator equations arising in the output feedback H= control problem; e.g., see [77,97]. In the papers [122.139], discrete-time versions of the set-valued state estimation results of [120,121,126,127] are presented. In this case, the integral quadratic constraint uncertainty description is replaced by a corresponding sum quadratic constraint uncertainty description. These discrete-time set-valued state estimation results are presented in Chapter 5. Also considered in Chapter 5 is the set-valued state estimation problem for the case in which some of the measurement data are missing. This situation may arise in practical problems in which the sensors are connected to the state estimator via a communications network which may lose some data. The problem of state estimation with missing data has been considered in a number of cases for signal models without uncertainty; e.g., see [27,91]. The results presented in Chapter 5 are based on the robust state estimation results of [137] for the case of missing data.
1.4 Set-Valued State Estimation
1.4.1
5
Discrete-Contznuous Data
A useful extension to the set-valued state estimation results presented in Chapters 4 and 5 involves a set-valued state estimation problem with "hybrid measurements." In many cases. the signal model being considered is a continuous-time signal model but the measured output data are available only at discrete sampling instants. This leads to the problem of "hybrid'" set-valued state estimation. Furthermore, this problem can also be extended to allow for the case when some of the outputs can be measured continuously and other outputs can only be measured at discrete sampling instants. Such a situation may arise in complex hybrid processes involving both digital and analog blocks. Results on this problem of set-valued state estimation with discrete-continuous measurements originally appeared in the papers [125,138]. We will present the main results in this area of setvalued state estimation in Chapter 6. Since the results of Chapter 6 consider the case of both continuoustime and discrete-time outputs, these results contain as a special case, the sampled-data robust state estimation problem. This problem is related to the sampled-data H= filtering problem considered in [150]. Sampled data model validation problems have also been considered in [145] and [llO].
1.4.2
Structured Uncertainty
In any problem of robust filtering or state estimation, a critical issue concerns the uncertainty model used. If the uncertain system model used does not give an accurate representation of the true uncertainty in the problem but rather overbounds the true uncertainty. then this will lead to an overly conservative robust filter with a correspondingly poor performance. Thus, in order to obtain good results from robust filters. it is important to use robust filtering techniques that can be applied to uncertain system models which accurately reflect the uncertainty in the system. The uncertain system models considered in Chapters 2 - 6 of this book involve what is referred to as "unstructured uncertainty." That is, they allow for only a single uncertainty block. If there is only one significant source of uncertainty in the system model, this may be a reasonable assumption. However, if there are more than one independent source of uncertainty. this will lead to a situation with multiple independent uncertainty blocks. In this case, the uncertainty is referred to as "structured uncertainty": e.g., see [40]. Although it is possible to overbound structured uncertainty by unstructured uncertainty, this will lead to conservatism and poor performance. A better approach is use a robust filtering algorithm that exploits the structure in the uncertainty description. In the paper [132]' a set-valued state estimation problem is considered for a new class of uncertain systems satisfying a certain ""Averaged Integral Quadratic Constraint." This uncertainty description extends the standard integral quadratic constraint uncertainty
6
1. Introduction
description in such a way as to allow for structured uncertainty and yet still lead to a tractable set-valued state estimation problem: see also [124,133]. This approach to set-valued state estimation with structured uncertainty is presented in Chapter 7.
1.4. 3
Nonlinear Systems
Most of the robust filtering problems considered in this book involve uncertain system models for which the nominal system is linear (the uncertainties may however be nonlinear). However, in practice there arise many situations in which one wishes to construct an optimal state estimate exploiting knowledge about nonlinearities in the system. A useful approach to state estimation for nonlinear systems is the Extended Kalman Filter; e.g., see [10,55.88.140]. The results presented in Chapter 10 consider a robust set-valued state estimation problem for the case of nonlinear uncertain systems in which the uncertainty is described by an integral quadratic constraint. These results originally appeared in the papers [58.59]. Chapter 10 first considers the problem in a general nonlinear setting and derives an exact characterization of the set-valued state estimate using dynamic programming. Similar results on nonlinear set-valued state estimation but without integral quadratic constraint uncertainty have appeared in the paper [11]. Chapter 10 also considers a robust set-valued state estimation version of the Extended Kalman Filter. This result is obtained by approximating the set-valued state estimate by an ellipsoid and linearizing the nominal system dynamics about the current state estimate.
1.5
Model Validation for Uncertain Systems
The set-valued state estimation problem considered in Chapters 4 and 5 is closely related to a problem of model validation for uncertain systems. Many of the recent advances in the area of robust control system design assume that the system to be controlled is modeled as an uncertain system; e.g., see [38]. This motivates a problem of system identification in which the model sought is an uncertain system suitable for applying such robust control system design techniques. An integral part of any system identification process is model validation. Model validation is a process of determining if given input-output measurements are compatible with a given model. Thus, model validation can only determine if a model is invalid in that it is incompatible with a given data set. In practice, model validation can be used to choose between a collection of models and to determine bounds on uncertain parameters. The significant advances being made in the field of robust control the-
1.6 Robust HOC Filtering
7
ory have motivated a number of authors to study the model validation problem for uncertain systems; e.g., see [107,143,144,171]. The approach of [143,144] involves an uncertain system with structured uncertainty of the type that arises in the f1 framework. For this class of uncertain systems, [143,144] apply a frequency domain approach to convert the model validation problem into a f1 problem which can be solved via numerical optimization techniques. The approach of l107j involves a class of single-input single-output discrete-time uncertain systems with norm bounded uncertainty. For this class of uncertain systems, reference [107] applies a time domain approach to convert the model validation problem into a convex optimization problem. A similar approach is also considered in [171]. In Chapters 4 and 5 we present solutions to the model validation problem for uncertain systems in which the uncertainty is described by an integral quadratic constraint or a sum quadratic constraint. It should be noted that a number of new robust control system design methodologies have recently been developed for uncertain systems with integral quadratic constraints; e.g., see [15,117,119,136]. This underlines the importance of problems of system identification and model validation for uncertain systems described by integral quadratic constraints. The model validation results presented in Chapters 4 and 5 involve the solution of a Riccati differential or difference equation and the solution of a set of filtering state equations. This is in contrast to the other model validation results mentioned above which are based on convex optimization. In this sense, the results presented in Chapters 4 and 5 are closely related to results on observer based fault detection; e.g .. see [92]. However, the filter state equations arising in the model validation approach presented in Chapters 4 and 5 are not standard Kalman filter state equations. Instead they take the form of a robust state estimator. In addition to the computational advantages mentioned above, the approach to model validation presented in Chapters 4 and 5 has a number of other advantages over other methods. The first advantage is that this approach does not require zero initial conditions on the plant from which the data are measured. Also, in contrast to [143, 144], this approach requires input-output data defined only over a finite time interval. Also, in contrast to [143,144]' the integral quadratic constraint uncertainty description allows for nonlinear time-varying uncertainties.
1.6
Robust HOC! Filtering
With the advent of H oo control theory (e.g., see [13,41,95,97]) there has also arisen a great deal of interest in Hoo filtering: see [13,48,89]. This problem involves constructing a filter such that the Hoo norm from the disturbance inputs to the filter error output is minimized. The H= filter makes no
8
1. Introduction
assumptions about the spectral properties of the disturbance signals and the filter is designed to minimize the estimation error due to the worst case disturbance signal. Thus for applications in which the characteristics of disturbance signals are unknown, the HOG filter may be more suitable than the standard Kalman Filter. The standard Hoc filter such as considered in [13,48,89] still requires that an accurate system model is available. In the case in which there are uncertainties in the system model, this may lead to poor performance of the Hex:: filter. This fact motivates the consideration of a robust version of the Hoc filtering problem in which a filter is constructed to minimize the worst case Hac norm of the transfer function from the noise inputs to the estimation error output; e.g., see [32-34,44,99,160]. Chapter 8 presents results on a robust H XJ filtering problem in which the uncertain systeD1 under consideration contains structured uncertainty modeled via integral quadratic constraints. The results presented in Chapter 8 are derived froD1 the robust HOG control results presented in the paper [131]. As mentioned above, the consideration of an uncertain system model with structured uncertainty allows for less conservative filtering results to be obtained. The structured uncertainty model considered in Chapter 8 involves an integral quadratic constraint uncertainty description, which is somewhat simpler than the structured uncertainty model considered in Chapter 7. This simplified uncertainty model can be used in Chapter 8 because of differences between the set-v;,tlued state estimation problem considered in Chapter 7 and the H oo filtering problem considered in Chapter 8. Chapter 9 also considers a robust H OO filtering problem. However in this case, consideration is given to the problem of constructing a filter of fixed degree. Also, the uncertainty description considered in Chapter S) does not involve integral quadratic constraints or structured uncertainty but rather is closely related to the norm bounded uncertainty description considered in Chapter 2. The results presented in Chapter 9 originally appeared in the paper [134]. These results are primarily concerned with the question of whether a HcP filtering objective can be achieved with a nonlinear filter or a linear filter. It is shown that for the class of problems under consideration (involving fixed order filters), the existence of a nonlinear filter achieving the filtering objective will imply the existence of a linear filter achieving the same objective.
L 7 Applications of Robust Kalman Filtering As mentioned above, the standard Kalman Filter has been applied to a wide variety of problems. Many of these problems are also potential areaS of application for the robust Kalman filtering ideas developed in this book.
1.7 Applications of Robust Kalman Filtering
9
However in this book, we focus on one practical problem for which a robust Kalman Filter can be successfully applied. An area in which the Kalman Filter has been applied recently is in estimating the flux of an induction machine: e.g., see [46]. However, in practice the state space model of an induction machine is subject to considerable uncertainty due to the change in resistance parameters in the model as the machine heats up. Thus, the problem of flux estimation in an induction machine provides an interesting example to illustrate the application of robust Kalman filtering techniques. This application is presented in Chapter 11.
The application presented in Chapter 11 involves applying the robust set-valued state estimation results given in Chapter 4 to the problem of flux estimation in an induction machine. This application was originally considered in the paper [105]. The motivation for estimating the fluxes in an induction machine is that these variables form the state vector in the induction machine state space model. Once the state vector has been estimated, the problem of induction machine control is much simplified. Because of this, many proposed schemes for induction motor control involve first estimating the state vector. In particular, the results presented in Chapter 11 involve combining the robust state estimator with a certain state feedback control law which is a modification of the direct torque control method: see [108]. That is, the robust Kalman Filter is used as part of a feedback control scheme.
Continuous-Time Quadratic Guaranteed Cost Filtering
2.1
Introduction
Underlying each of the robust filtering algorithms presented in this book is a corresponding class of uncertain systems and filtering performance objective. A common class of uncertain systems in the robust control literature involves a linear time invariant system with a norm bounded uncertainty matrix. For such a class of uncertain systems. the robust control problem that is considered is the quadratic stabilizability problem. This problem involves finding a suitable state feedback controller so that the resulting closed loop system is stable with a single quadratic Lyapunov function. Further details on the quadratic stabilizability problem can be found in the references [12,26.68,75.96,98]. An important approach to the quadratic stabilization problem is the approach based on the algebraic Riccati equation. This approach was developed in [68.96,98]. Also, a connection has emerged between this Riccati equation approach to the quadratic stabilization problem and Hoc control theory: e.g., see [68]. This enables results on HX control such as in [41,94,97] to be applied to problems of robust stabilization. An extension to the quadratic stabilization problem involves designing a state feedback controller that not only guarantees quadratic stability but also guarantees a bound on the closed loop performance as measured by a quadratic cost function. This problem is referred to as the quadratic guaranteed cost control problem because a single quadratic Lyapunov function is also used in constructing the bound on the cost function. A Riccati
12
2. Continuous-Time Quadratic Guaranteed Cost Filtering
equation approach to the quadratic guaranteed cost control problem was presented in [100]. The approach of [100] yields a state feedback controller and corresponding bound on the closed loop cost function which is optimal in the sense that there exists no better bound based on a single quadratic Lyapunov function. Indeed, the controllers derived in [100] are referred to as optimal quadratic guaranteed cost controllers. Dual to the problem of optimal quadratic guaranteed cost control is the problem of optimal quadratic guaranteed cost filtering. This problem is also considered in the paper [100] and involves constructing a state estimator that leads to a certain bound on the estimation error covariance. This bound is constructed by using a single quadratic Lyapunov function. The result presented in [100] is a Riccati equation approach to the guaranteed cost filtering problem that leads to a state estimator which is optimal in the sense that no other linear state estimator leads to a better bound based on a single quadratic Lyapunov function. A preliminary version of this result was also presented in the conference paper [99]. This chapter explores problems of robust Kalman filtering by considering a steady state optimal quadratic guaranteed cost filtering problem using the approach of [100]. In particular, a continuous time uncertain system model is considered and the state estimation problem is solved over an infinite time interval. The main result presented in this chapter is a Riccati equation approach to the construction of an "optimal" quadratic guaranteed cost state estimator. This optimal state estimator leads to a mean square error bound that is at least as good as the bound obtained from any other quadratic guaranteed cost state estimator. From this point of view, the results described in this chapter can be regarded as an extension of the continuous time steady state Kalman Filter to systems with norm bounded uncertainty It should be noted that as well as the results of [100] which form the basis of this chapter, there exist other results in the literature on guaranteed cost filtering such as in [53]' [57], [161] and [162]. These results however lead to state estimators that are sub-optimal in a sense which will be defined in this chapter.
2.2
A Guaranteed Cost State Estimation Problem
This section describes a class of continuous time uncertain systems that underlies the corresponding optimal guaranteed cost state estimation problem. Note that the uncertain systems considered in this chapter are stochastic uncertain systems in that the uncertain systems are subject to stochastic white noise disturbances. In presenting these results, a heuristic "white noise" framework is used for stochastic systems following the style of [74]. However, it would be straightforward to reformulate these results using a
2.2 A Guaranteed Cost State Estimation Problem
13
rigorous "Wiener process" framework for stochastic systems such as in [6]. The class of uncertain systems under consideration is described by the state equations
[A + Bl6.(t)K]x(t) + W(t); X(tO) = XO: [C + D l 6.(t)K]x(t) + V(t):
x( t)
y(t)
(2.2.1)
where x(t) E Rn is the state, y(t) E Rf is the measured output, Xo is the initial condition which is assumed to be a zero mean Gaussian random vector, 6.(t) is a time varying matrix of uncertain parameters satisfying the bound 6.(t)' 6.(t) ::; I, and w(t) and v(t) are zero mean white Gaussian noise processes with joint covariance matrix
A feedback interpretation of this uncertainty structure is illustrated in the block diagram shown in Figure 2.2.1.
~(t)
v{t) Nominal System
y{t)
w(t)
FIGURE 2.2.1. Uncertain system block diagram.
A fundamental property of uncertain systems of the form (2.2.1) is the property of quadratic stability; e.g., see [68].
Definition 2.2.1 The uncertain system (2.2.1) is said to be quadratically stable if there exists a positive-definite matrix P such that
2x' P[A for all non-zero x
E
+ Bl6.K]x < 0
(2.2.2)
R n and all matrices 6. : 6.'6. ::; I.
Throughout the remainder of this chapter, it is assumed that the system (2.2.1) is quadratically stable. Note that this assumption is not overly restrictive because in practice, it is rare that a state estimator be applied "open loop" to an unstable system.
14
2. Continuous-Time Quadratic Guaranteed Cost Filtering
The state estimators under consideration are assumed to be of the lowing form: £(t) = Afx(t)
+ Bfy(t);
x(to) = xo;
fol~
(2.2.3)
where x(t) E Rn is the state estimate and Xo is the estimator initial condition which is assumed to be a zero mean Gaussian random vector. When such an estimator is applied to the uncertain system (2.2.1), the corresponding estimation error is defined by e(t) ~ x(t) - x(t). In terms of the state variables e and X, the state equations describing the augmented system formed from (2.2.1) and (2.2.3) are as follows: x(t)
= Ax(t) + Eh!J.(t)Kx(t) + Bow(t); x(to) = xo; (2.2.4)
where
x
6.
A
6.
K
6.
Since this chapter deals with the steady state filtering problem, it is assumed that the initial time approaches negative infinity: to --> -00. The following definition formally introduces the notion of a quadratic guaranteed cost state estimator which will be the main topic of this chapter. Definition 2.2.2 A state estimator (2.2.3) is said to be a quadratic guaranteed cost state estimator with associated cost matrix Q > a for the system (2.2.1) if the matrix Af is stable and there exists a matrix
Q=
[6~2 6~~]
>a
(2.2.6)
such that (A
+ B1!J.K) Q + Q (A + B1!J.K)' + BoBb < a
(2.2.7)
for all matrices !J. : !J.'!J. ::; I.
Throughout the remainder of this chapter, attention is restricted to state estimators in the class of quadratic guaranteed cost state estimators. The next result shows that if (2.2.3) is a quadratic guaranteed cost state estimator for the system (2.2.1) with cost matrix Q, then Q defines an upper bound on the steady state error covariance at time t, Q6.(t) ~
lim
to---+-CXJ
E {e(t)e(t)'}.
2.2 A Guaranteed Cost State Estimation Problem
15
Theorem 2.2.3 Suppose the system (2.2.1) zs quadratzcally stable and (2.2.3) zs a quadratzc guaranteed cost state estzmator wzth cost matrzx Q > 0 for thzs system. Then the correspondzng augmented system (2.2.4) wzll be quadratzcally stable and the steady state error covarzance at tzme t satzsjies the bound Qll(t) :::: Q for all admzsszbleuncertazntzes ,6.(t). Conversely, zf the system (2.2.1) zs quadratzcally stable, any state estzmator of the form (2.2.3) with A f stable wzll be a quadratzc guaranteed cost state estimator for this system wzth some cost matrzx Q > O.
Proof To establish the first part of the theorem, suppose (2.2.3) is a quadratic guaranteed cost state estimator with cost matrix Q > O. It follows that the matrix A f must be stable. Also. it has been assumed that the system (2.2.1) is quadratically stable. From this it is straightforward to verify that the augmented system (2.2.4) is quadratically stable. This can be shown using a Lyapunov matrix of the form
where PI defines a quadratic Lyapunov function for the uncertain system (2.2.1), P2 defines a quadratic Lyapunov function for the system (2.2.3), and a > 0 is a sufficiently large scaling constant. Now let ,6.(t) be any admissible uncertainty and observe that w(t) is a Gaussian white noise process with identity covariance. Also, assume the initial condition random vector for the system (2.2.4) has covariance matrix E {xox~} = Qo 2: o. It now follows from a standard result on linear systems driven by whIte noise (e.g., see Theorem 1.52 of [74]) that the corresponding state covariance of the augmented system (2.2.4) at time t is given by
Qll(t. to)
~ E {x(t)x(t)'} =
(t. to)Qo(t. to)' +
it
(t.T)BoBb(t, T)'dT
to
where <1>( t. T) is the state transition matrix associated with the system (2.2.4) (with the specified uncertainty ,6.(t)). Furthermore. using the fact that the system (2.2.4) is quadratically stable. it follows that lim
to~~::::x:J
(t,to) = 0
and hence. the steady state error covariance at time t is given by
16
2. Continuous-Time Quadratic Guaranteed Cost Filtering
Now let the time t be fixed and let
7]'Ot:..(t)7]
=
7]
be a given vector in R 2n. Then
Itex; 7]'if>(t. r)BoBbif>(t, r)'7]dT ItOG 7]'( r)BoBb7]( r)dr
where '1(T) ~ if>(t, T)'7] is the solution to the dual state equation r,(r) = -[A
+ B 1 l:!.(r)K]'7](r); 7](t) = 7].
-
t:..
(2.2.9)
-
Now with Q defined as in (2.2.6), let V(7]) = 7]'Q7]. It follows from (2.2.7) and (2.2.9) that - d
7](r)' BoBb7](r) ::: dT V(7](r)).
Therefore, using the fact that the system (2.2.4) is quadratically stable, the dual system (2.2.9) will be stable in reverse time and hence
Itoo 7]'(r)BoBb7](r)dr ::: V(7]) -
V(7]( -00)) = V(7]).
Thus, using (2.2.8)
7]' 0 t:.. (t)7] ::: 7]'07] for all 7] E R 2n. From this it follows that 0 t:..C t) ::: 0 for all admissible uncertainties l:!. (t). Now the state error covariance Q t:.. (t) is the (1,1) block of the matrix Ot:..(t) and cost matrix Q is the (1,1) block of the matrix O. Hence, (2.2.10) for all admissible uncertainties l:!.(t). This completes the proof of the first part of the theorem. Conversely, suppose the system (2.2.1) is quadratically stable and consider any state estimator of the form (2.2.3) with At stable. From this, it follows that the corresponding augmented system (2.2.4) is quadratically stable. Hence, there exists a matrix 0 > 0 such that
for all matrices l:!. : l:!.'l:!. ::: I. From this inequality, it is straightforward to verify that there exists a constant E > 0 such that
for all matrices l:!. : l:!.'l:!. ::: I. Thus, this state estimator is a quadratic guaranteed cost state estimator with cost matrix Q ~ 0/ E > O. 0
2 3 PreliIrunary Results
17
Remarks ThIS chapter IS concerned wIth constructmg a state estimator that mImmizes the nght hand sIde of the error covanance bound (2 2 10) However, (22 10) IS a matnx mequahty To obtam a scalar mmimizatlOn problem, conSIder the correspondmg bound on the steady state mean square error hm
to---+-OO
E{e(t)'e(t)} = tr{Q.6-(t)} ::; tr{Q}
(2211)
Thus, the results of thIS chapter are concerned wIth mmimIzmg tr{ Q} It should be noted that there may be situatlOns m whIch one IS only concerned wIth estImatmg a lImlted number of state vanables In thIS case, an output vanable z(t) = Hx(t) would be conSIdered and correspondmg .6-
output estImate would be z(t) = Hx(t) The correspondmg steady state mean square error bound IS then hm
E{(z(t) - z(t))'(z(t) - z(t))} = tr{H'Q.6-(t)H} ::; tr{H'QH}
to---+-CXJ
Thus m thIS case, It IS reqUIred to mmimize the quantIty tr{H'QH} However, the approach taken would remam vIrtually the same as If the quantIty tr{ Q} were to be mmimized Thus, throughout the sequel It WIll be assumed that the quantIty tr{Q} IS to be mimmized
2.3
PrelImmary Results
ThIS sectlOn contams a number of Important results relatmg to uncertam systems wIth norm bounded uncertamty, HOG control theory, and algebraIc Riccati equatlOns whIch WIll be useful when addressmg the problem of optImal guaranteed cost filtenng for the uncertam system (2 2 1) The followmg lemma IS a verSlOn of the small gam theorem whIch relates the robust stabIlIty of an uncertam system to an HOG norm bound conditlOn ThIS verSlOn of the small gam theorem was ongmally presented m [68]
Lemma 2.3.1 The unceriam system (22 1) zs quadratzcally stable zf and
only zf the followmg two condztzons are satzsjied (z) The matrzx A zs stable (that zs, all of zts ezgenvalue he m the open left half of the complex plane),
(zz) The HOG norm bound IIK(sI - A)-I BIII= < 1 zs satzsjied Proof See [68] 0 The followmg results relate to the algebraIC Rlccatl equatlOn and some of ItS properties
18
2. Continuous-Time Quadratic Guaranteed Cost Filtering
Notatzon
A symmetric matrix p+ is said to be a stabilizing solution to the Riccati equation A' P + P A - PM P + N = 0 If it satisfies the Riccati equation and the matrix A - AI P+ is stable. SImilarly a symmetric matrix P+ is said to be a strong solution to this Riccati equation if it satisfies the Riccati equation and the matrix A - M p+ has all of its eigenvalues in the closed left half of the complex plane. Note that any stabilizing solution to the Riccati equation WIll also be a strong solution. The following comparison result for stabilizing solutIOns to the algebraic Riccati equation was first presented in [109]. It is frequently used in the theory of quadratic stabilization and H= control. Lemma 2.3.2 Conszder the algebrazc Rzccatz equatzon
A' P + P A - P]\;1 P + f.r = 0 and (A, B) zs stabzl2zable. Suppose thzs
where M ;::: 0 a symmetrzc solutzon
P
(2.3.1)
Rzccati eqtI.atwn has
and define the matrzx
- [f.rA-
H=
A' _ ]. -M
Also conszder the Rzccatz equatzon A' P
+ PA
- PM P
+N
(2.3.2)
= 0
where M ;::: 0 and define the matrzx H
=
[~ ~~1]'
If H ;::: H, then the Rzccatz equatzon {2.3.2} will Iuwe ct Uf&ique strong solutzon P+ whzch satzsfies p+ ;::: P. Proof
See Theorems 2.1 and 2.2 of [109]. 0 The following corollary to this lemma is used in the sequel and in the proof of the Strict Bounded Real Lemma given below. Corollary 2.3.3 Suppose A zs stable and the Rzccatz equation (2.3.3)
has a symmetrzc solutzon Rzccatz equatzon A' P
P.
Furthermore, suppose
+ P A + P BB' P + Q =
Q ;::: Q
;::: O. Then the (2.3.4)
0
wzll have a unzque strong solutzon P and moreover, 0 :::; P :::;
P.
2 3 PrelImmary Results
19
Proof Let
X = -P
Hence, Rlccatl equatlOn (233) can be rewntten as A' X
+ X BB' X - Q = 0
Furthermore, smce A 1S stable the palI (A, B) must be stabilizable Hence, USIng Lemma 2 3 2, It follows that the Rlccatl equatlOn A' X
+XA
- X BB' X - Q = 0
WIll have a umque strong solutlOn X :::: X Now let P = -X It follows llnmedlately that P ::; F IS the umque strong solutlOn to (2 3 4) Moreover, USIng a standard result on Lyapunov equatlOns, It follows from (234) that P :::: 0 see Lemma 12 1 of [158] D The proof of the the maIn result of thIS chapter makes extensIve use of the follOWIng Stnct Bounded Real Lemma ThIS \erSlOn of thIS result was ongInally presented In [97] Lemma 2.3.4 (Stnet Bounded Real Lemma) The followmg statements are equzvalent (z) A zs stable and IIC(8I - A)
1
BII= <
1
(zz) There exzsts a matrzx F > 0 such that A' F+FA+FBB' P+C'C < 0 (zzz) The Rzccatz equatwn A'P+PA+PBB'P+C'C = 0
(2 3 5)
has a stabzlzzzng solutwn P :::: 0 Furthermore, zf these statements hold, then P < F Proof The eqUlvalence of the statements (z) - (m) IS estabhshed first (z) =* (zz) It follows from condltlOn (1) that there eXIsts an that C(JW - A)-l BB'( -JwI - A,)-lC' ::; (1 - E)I
€ ::::
0 such (236)
for all w ~ 0 Let JL ~ IIC(8I - A)-liloc Hence,
2:2C(JW - A)-l(-JwI - A,)-lC' ::; ~I
(237)
for all w ~ 0 Addmg equatlOns (2 3 6) and (2 3 7) It follows that glVen any w:::: 0, (238)
20
2. Continuous-Time Quadratic Guaranteed Cost Filtering
where
E is a non-singular
matrix defined by
EE' =
BB'
+ _10_ 1 . 2p,2
Furthermore, (2.3.8) implies E'( -jw - A')-IG'G(jwl - A)-IE:S; (1 - ~)I for
alI w ~ O.
6.
-
Now let 17 = II(s1 - A)-I Blloo. Hence (2.3.10)
Adding eque.tions (2.3.9) and (2.3.10), it follows that given any'U/I 2: 0,
E'(-jw - A')-IC'C(jw1 - A)-IE:S; (1- ~)1 (2.3.11)
where
C is a
non-singular matrix defined so that
C'C = G'G +
_10_
217 2
1.
Thus, IIC(s1 - A)-l Ell :s; 1. Furthermore, since E and C are non-singular, the triple (A, E. C) is minimal. Hence, using the standard Bounded Real Lemma given in [3], it follows that there exists a matrix P such that
A'p+ PA+ PEE'P+ C'C
=
o.
That is,
Hence, condition (ii) holds. (ii) =} (iii) : It follows from (ii) that there exist matrices it > 0 such that
A'p + PA+ PBB'P+ G'G+R = o.
P>
0 and
(2.3.12)
Hence, using a standard Lyapunov stability result, it follows that A is stable; see Lemma 12.2 of [158]. Furthermore, comparing equations (2.3.12) and (2.3.5), it follows from Corollary 2.3.3 that Riccati equation (2.3.5) will have a unique strong solution P :s; P. Moreover, using the fact that A is stable, it follows from a standard property of Lyapunov equations that P ;:::: 0: see Lemma 12.1 of [158]. It must now be shown that P is in fact the
23 PrelImmary Results ~
-
-
21
~
stablhzmg solutlOn to (235) FIrst let S = P - P ~ 0 and A = A + BB' P It follows from (2 3 5) and (23 12) that
A's + sA + SBB'S + H = 0
(2 3 13)
ThIs equatlOn wIll be used to show that A + BE' P has no eIgenvalues on the Imagmary aXIS Indeed, suppose A has an Imagmary aXIS eIgenvalue JW wIth correspondmg eIgenvalue x That IS, Ax = JWX It follows from (2 3 13) that -JWX* Sx
+ JWx* Sx + x* SBB'Sx + x* Hx =
0
and hence x* Hx = 0 However, thIS contradIcts the fact that H> 0 Thus, It follows that A + BB' P IS stable, and therefme, P ~ 0 IS the stabIllzmg solutlOn to (235) Hence condItion (m) holds
-
~
(m) =? (n) Suppose condltlOn (m) holds arId let C = B'P It follo,,"s that A + B6 = A + BB' P IS stable and hence the paIr (A 6) IS detectable Furthermore, It follows from (2 3 5) that A' P
+ P A + 6' 6
:s 0
Hence usmg a standard Lyapunov stablhty result, It follows that A IS stable see Lemma 12 2 of [158] In order to sho,," that IIC(sl _A)-l BII < 1 observe that equatlOn (2 35) Imphes B'(-Jw1 - A')-IC'C(Jw1 - A)-IB
= 1- [1-B'P(-Jw1-A)-lB]' [1-B'P(Jw1-A) IB] (2 3 14)
for all W ~ 0 It follows that I C(sl - A) - 1 B I :::: 1 Furthermore note that C(Jw1 - A)-l B ----> 0 as W ----> 00 Now suppose that there eXIsts an W ~ 0 such that IIC(Jw - A)-l BII = 1 It follows fronI (2314) that there eXIsts a non-zero vector z such that [I - B'P(Jw - A)-l]z = 0 Hence,
However, usmg a standard result on determmarJ.ts, It follows that det[Jw - A - BB' P] = det[Jw - A] det[1 - B' P(Jw - A)-I].
see for example SectlOn A 11 of [62] Thus, det[J~
- A - BB'P] = 0
However thIS contradIcts the fact that P IS the stabIllZmg solutlOn to (235) Hence, It follows that IIC(sl - A)-l BII < 1 Thus, condltlOn (~) has been estabhshed
22
2. Continuous-Time QuadratIc Guaranteed Cost Filtermg
In order to complete the proof of the lemma, now suppose that statements (2) - (m) hold. It must be shown that P < F. However, it has already been proyed that P <::; that
F.
Now suppose that there exists a non-zero vector z such
z'Fz = Z'pZ. That is,
Sz = 0 where S = F - P ~ 0 is as defined above. Applying this fact to equation (2.3.13), it follows that z'Hz = O. However, this contradicts the fact that H > O. Thus, P < F. This completes the proof of this lemma. D The proof of the main result of this chapter relies on the following result from H OO control theory which was presented in [97]. This result relates to a state feedback H= control problem in which the underlying system is described by the state equations :i;
Ax + Bl W
z
C1x
+ B 2 u.
+ D1Zu
(2.3.15)
where x is the state, w is the disturbance input, u is the control input. and z is the error output. For this system. define E ~ D~2D12' Lemma 2.3.5 Cons2der a system (2.3.15) such that thefollowzng assumptwns are sat2sfied:
2. The matrzx [
A - JwI
B2 ] has full column rank for all w ~ D12
C
1
o.
If there exzsts a state feedback control u = PI and a matrzx P > 0 such that
then the Rzccatz equatwn
(A - B 2 E- 1 D~2Cl)' X
+X
(BIB~ -
+ X (A - B 2 E- 1 D~2Cl) B 2 E- 1 B;) X + C~ (I - D 12 E- 1 D~2) C 1 = 0
has a stabthzzng solutwn X
~
0 such that X
< P.
Proof See Theorem 3.4 of [97].
D
2 4 Optimal Guaranteed Cost Filter Design
2.4
23
Optimal Guaranteed Cost Filter Design
This section presents the main result of this chapter. This involves constructing a guaranteed cost state estimator which minimizes the right hand side of the mean square error bound (2.2.11) Theorem 2.4.1 Suppose the uncertam system (2.2.1) zs quadratzcally stable. Then there exzsts a constant E* > 0 such that for all E E (0, E*) the Rzccatz equatzon
AS
1
+ SA' + ESK'KS + -BIB~ + W = 0
(2.4.1)
E
has a stabzlzzzng solutzon S+
> O. For any such
E,
the Rzccatz equatzon
(A - BID~ (EV + DID~)-I C) Q +Q (A - BID~ (EV + DID~)-I c)" +EQK'KQ - EQC' (EV 1 ( ( EV +~BI I -'DI
has a stabzlzzzng solutzon Q+ estzmator
iCt)
+ DIDD- I CQ
+ DIDI')-1) DI BI, + W = 0
(2.4.2)
> 0 such that Q+ ::; S+. Also, the state
+ EQ+ K' K) x(t) I + (EQ+C' + BIDD (EV + D 1 DD- (y(t) (A
- Cx(t)) (2.4.3)
has the followzng property: Gwen any t5 > 0, there exzsts a matnx Q > 0 such that Q+ ::; Q < Q+ + r5I and (2.4.3) zs a quadratzc guaranteed cost state estzmator for the system (2.2.1) wzth cost matnx Q. Conversely, gwen any quadratzc guaranteed cost state estzmator for the system (2. 2.1) wzth cost matnx Q, there exzsts a constant E > 0 such that Rzccatz equatzons (2.4.1) and (2.4.2) have stabzlzzzng solutzons S+ > 0 and Q+ > 0, respectwely, and Q+ < Q. Proof In order to establish the first part of the theorem, note that Lemma 2.3.1 and the quadratic ~tabIlity of the system (2.2.1) imply that A is stable and IIK(sI - A)-I BIiloo < 1. Hence, it follows that there exists a constant E* > 0 such that
Therefore, Lemma 2.3.4 implies that for all ditions hold:
E E
(0,
E*),
the following con-
24
2 Contmuous- TIme QuadratIC Guaranteed Cost FIlterIng
(1) There eXIsts a matnx S > 0 such that -
-
AS + SA'
-
-
1
+ ESK'KS + -BIB~ +W < 0 E
(244)
(u) Riccatl equatlOn (241) has a stablhzmg solutlOn S+ ~ 0 Furthermore, smce W IS posItlve-defimte, It follows ImmedIately from (241) that S+ IS posltlVe-defil1lte Now let E E (0, E*) be gIwn and consIder mequahty (244) It IS desIred to apply Lemma 2 3 5 to thIS mequahty In order to do thIS, consIder the correspondmg H oo control problem defined by the system
A'x + VEK'w
x z
[
+ VEC'U,
..LB' 1 [ D' 1 ~~ + i~V~ 1
X
(245)
U
It IS stratghtforward to venfy that thIS system lS a system of the form (2 3 15) whIch satlsfies assumptlOns (1) and (u) of Lemma 235 Suppose the state feedback control u := 0 lS applied to thlS system Usmg Lemma 2 3 5 It follows from mequahty (244) that Rlccatl equatlOn (242) has a stablhzmg solutlOn Q+ 2: 0 Furthermore, W > 0 lmplies Q+ > 0 - ~ 1 Now compare Rlccatl equatlOns (241) and (242) Lettmg ~ = (S+)- , It IS stratghtforward to venfy that f: > 0 IS the stabIlIzmg solutIOn to the Riccati equatlOn
-
-
A'~
-
-
~A
-
1-
-
-~BIB~~
-
-
-
~W~
- EK'K
=
0
E ~
1
(246) -
~
Also, let ~ = (Q+)- and mtroduce the notatlOn V = EV + DID~ It IS straightforward to venfy that ~ > 0 IS the stablhzmg solutlOn to the Rlccatl equatlOn
(247) Lemma 2 3 2 wIll be applIed to compare RIccati equatIOns (2 4 6) and (2 4 7) Thus, consIder the matnx
[-EK'K M= -A assocIated WIth Rlccatl equatIOn (2 4 6) and the matnx
- (A-BIDiV-IC)'
-~Bl (I - D~ V-I Dl) Bi - W
1
2.4 Optimal Guaranteed Cost Filter Design
25
associated with the Riccati equation (2.4.7). Hence, EC'V-lC [ BlD~ V-lC
M-M =
[t~:Di ]
C'V- l DlBi ] ~BlDi V-I DlBi
V-I [VEC
)£DlBi]
> O. Therefore, it follows from Lemma 2.3.2 that I: ;::: t and hence Q+ ::; S+. Now consider the augmented system (2.2.4) obtained when the state estimator (2.4.3) is applied to the uncertain system (2.2.1). In this case, the matrices A, Eo, 13 1 , and K are as follows: A - (EQ+C' + BlDi) V-lC [ (EQ+C'+BlDDv-1C
[:!
-EQ+ K' K ] A+EQ+K'K'
-(EQ+C'+BlDDV-lV! ] (EQ+C' + BlDD V-IV! '
Bl - (EQ+C' + B~DD V-I Dl ] [ (EQ+C' + BlDi) V-I Dl ' [K
K
J.
With these definitions, it is straightforward but tedious to verify that the matrix
Q ~ [Qo+
S+
~ Q+
] ;:::0
satisfies the Riccati equation -AQ
--
----
1--
--
+ QA' + EQK' KQ + -BIB~ + BoBb = o. E
(2.4.8)
Furthermore, the matrix
A+EQk'k A - (EQ+C' + BlDi) V-lC + EQ+ K' K [ (EQ+C' + BlDU V-lC + E(S+ - Q+)K' K
0 A
]
+ ES+ K' K
is stable since Q+ is the stabilizing solution to Riccati equation (2.4.2) and S+ is the stabilizing solution to Riccati equation (2.4.1). Thus, Q is the stabilizing solution to Riccati equation (2.4.8). Using Lemma 2.3.4, it now follows that A is stable and
Lemma 2.3.4 also implies that there exist matrices Ql > that
Q and if>
0 such
26
2. Continuous-Time Quadratic Guaranteed Cost Filtering
Thus given any p E (0 , 1), -AQI
--
----
1--
--
-
+ QlA' + fQIK' KQI + -BIB~ + BoB~ + pV < 0 f
and therefore, Lemma 2.3.4 implies that the Riccati equation
(2.4.9) has a stabilizing solution Qp > o. (The positive-definiteness of Qp follows from the positive-definiteness of v.) Furthermore, comparing Riccati equations (2.4.8) and (2.4.9), it follows from Corollary 2.3.3 that Qp ~ Q. Now since the stabilizing solution to Riccati equation (2.4.9) will be a continuous function of the parameter p, it follows that (2.4.10) Hence, (2 .4.9) and (2.4.10) imply that given any 6> 0 there exists a matrix Qp > 0 such that Q ::: Qp < Q + M and
(2.4.11) Also, letting Q > 0 be defined as the (1,1) block of the matrix Qp, then it follows that Q+ ::: Q < Q+ + M. Now let the matrix ~ be given such that ~'~ ::: I . Using a standard matrix inequality, Lemma 2.4.11 implies that
(A + fh~k) Qp + Qp (A + fh~k)' + f3of3~ --
--
----
1-
--
< AQp + QpA' + fQpK'KQp + -BID' + BoB~ f < O. Also, since the matrix A (which is the system matrix of the augmented system) is stable, then the matrix A f = A + fQ+ K' K (which is the system matrix of the state estimator) must also be stable. Thus, it follows that the state estimator (2.4.3) is a quadratic guaranteed cost state estimator for the system (2.2.1) with cost matrix Q. This completes the proof of the first part of the theorem. To establish the second part of the theorem, suppose (2 .2.3) is a given quadratic guaranteed cost state estimator with cost matrix Q > O. Hence, there exists a matrix
- = [QQ~2
Q
2.4 Optimal Guaranteed Cost Filter Design
27
such that
for all matrices ~ : ~'~ .:; 1. Here the matrices A, Bo, fh, and K define the corresponding augmented uncertain system as in (2.2.4) and (2.2.5). Using Lemma 3.1 and Observation 3.1 of [96]. this implies that
x' (AQ + QA' + BoBb) x + 21IB~xIIIIKQxll < 0 for all
x of. O.
Thus.
AQ + QA' + BoBb < 0 and
(x' [AQ + QA' + BoBb] x)2 > 4x' BIB~xx'QK' KQx for all x of. O. Hence, using Theorem 4.7 of [98], it follows that there exists an E > 0 such that
(2.4.12) Now pre-multiply this inequality by the matrix [1 1] and post-multiply by the matrix [1 I] '. It is straightforward to verify that this yields the inequality 1 AIT + ITA' + EITK' KIT + -BIB~ + W < 0 E
where IT ~ Q + QI2 + Q~2 + Q22 > O. Hence using Lemma 2.3.4, it follows immediately that Riccati equation (2.4.1) has a stabilizing solution S+ 2: O. Furthermore, W > 0 implies S+ > O. Now pre-multiply (2.4.12) by the matrix [I -QI2Q2l] and postmultiply by the matrix [1 -QI2Q221]'. It is straightforward to verify that this yields the inequality
(I + QI2Q2n BfC] ~ + ~ [A - (I + QI2Q2n BfC]' + (1 + Q12Q:;l) BfVBj (I + Q12Q221)' + W + E~K'K~
[A -
+ ~E
[BI -
(I + Q12Q2l) BfDd [BI - (I + QI2Q:;l) BfD 1 ]' < 0 (2.4.13)
where ~ ~ Q - Q12Q2lQ~2 > O. Hence, ~ .:; from (2.4.13) that ~ satisfies the inequality
Q.
Furthermore, it follows
[A
+ y'ELCj ~ + ~ [A + y'ELC], + E~K'K~ + ELVL' + W
+
[~E BI + LDI] [~E BI + LDI]' < 0
(2.4.14)
28
2. Continuous-Time Quadratic Guaranteed Cost Filtering
where L ~ - JE (I + Q12Q2l) B f · It is desired to apply Lemma 2.3.5 to this inequality. To do this, consider a state feedback HOC control problem defined by the system (2.4.5). In this case. suppose the state feedback control u = £Ix is apphed to this system. Using Lemma 2.3.5 it follows from inequality (2 4 14) that Riccati equation (2.4.2) has a stabilizing solution Q+ 2': 0 such that Q+ < ~ ::; Q. Furthermore since Q+ 2': 0 satisfies (2.4.2), W > 0 implies Q+ > O. This completes the proof of the theorem. 0 Observation 2.4.1 The above theorem states that the state est2mator defined by (2.4.2) and (2.4.3) has the followmg property: Gwen any 6 > 0, there eX2sts a matr2x Q > 0 such that Q+ < Q < Q+ + 6I and the state est2mator zs a quadratzc guaranteed cost state estImator wzth cost matnx Q. Hence Theorem 2.2.3 Implzes that the steady state error covanance matnx at tIme t sat2sfies the bound
for all adm2ssible uncertaint2es
for all adm2ssible uncertamt2es
~(t).
Th2s bound holds for all8 > O. Hence,
~(t).
There now follows a number of results which are useful in constructing the optimal value of the constant E in Riccati equation (2.4.2). Theorem 2.4.2 Suppose that for E = f' > 0, the R2ccati equatIOn (2.4.1) has a solutIOn S > 0 and the R2ccat2 equatIOn (2.4.2) has a solution Q > O. Then for any E EO (0. f'), the R2ccat2 equatIOn (2.4.1) wzll have a stab2l2zing solutIOn S+ > 0 and the R2ccat2 equatIOn (2.4.2) w211 have a stab21Izmg solutIOn Q+ > O.
Proof
> 0 and Riccati Let IT ~ tQ-l > O. It
Suppose the Riccati equation (2.4.1) has a solution S equation (2.4.2) has a solution Q > 0 with E = f'. follows that IT satisfies the Riccati equation
IT (A - B1D~ v-Ie) + (A - B1D~ V-Ie)' IT + K' K- e'v- 1e +ITBI A
(I - D~ V-I Dl) B~IT + f'ITWIT = 0
~
where V = f'V + DID~. Also, let Riccati equation
_
~
(2.4.15) _
= f'S > O. It follows that
~
satisfies the
(2.4.16)
2.4 Optimal Guaranteed Cost Filter Design
Now let
fr
E
29
E (0, E) be given and consider the Riccati equations
(A -
+frBl
BID~V-IC) + (A - BID~V-IC)' fr + K'K -
C'V- 1 C
(1 - D~ V-I Dl) B~ fr + EfrWfr = 0
(2.4.17)
and (2.4.18) ,
Ll.
where V = EV + DID~. Lemma 2.3.2 will be applied to compare Riccati equations (2.4.15) and (2.4.17). Thus, consider the matrix
- (A - B 1 D~ V-1 -BI (1 -
1
C) ,
D~V-IDI) B~ - EW
associated with the Riccati equation (2.4.15) and the matrix
- (A - B 1 D~ V -1 -Bl
1
C)'
(1 - D~V-1DI) B~ -
EW
associated with Riccati equation (2.4.17). It follows by a straightforward algebraic manipulation that H -
if = [
BC1D' 'I
] (V-I - V-I) [C
DIB~] + [00
(E-E 0) W ] . (2.4.19)
Using the fact that E < E, it now follows that H - if ;:::: O. Thus, using Lemma 2.3.2, it follows that Riccati equation (2.4.17) has a strong solution II+ ;:::: fr > O. To establish that II+ is in fact a stabilizing solution to (2.4.17), observe that Riccati equation (2.4.15) can be re-written in the form
- [1 fr] if [
A] = o.
Similarly, Riccati equation (2.4.17) can be re-writtea in the form
(2.4.20)
30
2. Continuous-Time Quadratic Guaranteed Cost Filtering
Thus using (2.4.19) and (2.4.20), it follows that ity
fI > 0 satisfies the inequal-
fIJH[AJ
-[1
- [1
fI
JH [
A] - [1 fI
J (H - H) [
~J
< O. That is, there exists a matrix N > 0 such that
fI (A - BID~V-le) + (A - BID~V-le)' fI +K'K - c'v- e +fIBl (1 - D~ V-I Dl) B~fI +EfIwfI +N = 0 (2.4.21) 1
Now subtracting Riccati equation (2.4.21) from (2.4.17), it follows that the matrix Z ~ II+ -
fI :2: 0 satisfies the equation AZ+ZA' +V
=0
where
and
N ~ Z{Bl(I - D~V-lDdB~ + EW}Z + N > o.
A standard Lyapunov argument now implies that the matrix A is stable; see Lemma 12.2 in [158]. Hence, II+ is a stabilizing solution to (2.4.17). Now let Q+ ~ ~(II+)-l > O. E
It is straightforward to verify using Riccati equation (2.4.17) that Q+ is a stabilizing solution to Riccati equation (2.4.2). In order to complete the proof of this theorem, Corollary 2.3.3 is now applied to compare Riccati equations (2.4.16) and (2.4.18). From this corollary, it follows that Riccati equation (2.4.18) has a strong solution ~+ such that 0 ::::; ~+ ::::; t. Also since W > 0, it follows from (2.4.18) that ~+ > O. Now, it follows from (2.4.16) that t satisfies the Riccati equation
(2.4.22) where N = (E - E)W > O. Subtracting (2.4.18) from (2.4.22), it follows that the matrix
2 4 Optimal Guaranteed Cost Filter DesIgn
31
satisfies the equatIOn where
A = A + 'L,+K'K
and
N=
(I: - 'L,+)K'K(I: - 'L,+) +N > 0
A standard Lyapunov argument now Imphes that the matnx A IS stable, see Lemma 122m [158] Hence, L,+ IS a stablhzmg solutIOn to (24 18) ThiS completes the proof of the theorem 0
Theorem 2.4.3 Suppose Rzccatz equatwn (24 2) has a posztzve-defimte stabzlmng solutwn Q+(t) for each t m the mterval (OJ) Then tr(Q+(t)) zs a convex functwn of t over (0, E)
Proof If Q+
that
> 0 IS a stablhzmg solutIOn to (242) It IS straightforward to venfy
n+ ~ (Q+) -I > 0
IS the stablhzmg solutIOn to the RlccatI equatIOn
- n (A - BID~ (tV + DID~)-l e) - (A - BID~ (tV + DID~)-l e)' n 1 ( 1- Dl, ( tV - ~nBl
- EK'K
+ DIDI,)-1) Dl Bl'II -
+ Ee' (EV + D1DD- 1 e =
IIwn
0
(2423)
f.
Now let n+ ~ n+, and n+ ~ ~ n+ DlfferentIatmg Rlccatl equatIOn (2423) tWice leads to the followmg equatIOn (after a some straightforward but tedIOUS algebraic mampulatlOns)
-An+ - n+A' + 2[(d + D~ V-I DD-l(D~ v-Ie - B~n+) + B~n+]' x(d + D~ V-I DJ}-l[(d + D~ V-I Dl)-l(D~ v-Ie - B~n+) + B~n+]
=0 where
However. smce the matnx A IS stable, It follows from a standard result on the Lyapunov equatIOn that n+ ::; 0 e g , see Lemma 12 1 of [158] c. -Q+ d c. -Q+ d2 Now let Q+ = and Q+ = Then de'
de 2
Q+ = -Q+n+Q+ and Q+ = 2Q+n+Q+n+Q+ - Q+n+Q+ Thus smce n+ functIOn of t
::; 0,
It follows that Q+
~
0 Hence, tr( Q+) Will be a convex
0
32
2. Continuous-Time Quadratic Guaranteed Cost Filtering
Remarks It follows from the above theorem that in minimizing tr(Q+) with respect to E, any local minimum will also be a global minimum of tr(Q+) and efficient numerical methods can be found to perform the minimization. Using the above results, it follows that the optimal guaranteed cost state estimator can be obtained by choosing E > 0 to minimize tr(Q+) where Q+ is the stabilizing solution to Riccati equation (2.4.2). [This convex minimization problem is also subject to the constraint that Riccati equation (2.4.1) has a stabilizing solution S+ > 0.] The required optimal estimator is then given by (2.4.3). Note however that there is no guarantee that a minimum will exist.
2.5
Illustrative Example
This section presents an example to illustrate the theory developed in this chapter. Indeed, consider the uncertain system described by the state equations
~7 5~(t)
x( t)
yet)
-1
] x(t)
+ wet):
1x(t) + vet);
where ~(t) is a scalar uncertain parameter subject to the bound 1~(t)1 :::; 1, and wet) and vet) are zero mean Gaussian white noise processes with joint covariance matrix
This system is of the form (2.2.1) where
A
[~1
K
[0
!l],C=[l 5
1' W
= [
-1 ] , Bl
1~0 1~0
= [
~
] , Dl = 0,
] , V = 1.
For this system, it is straightforward to verify that conditions (i) and (ii) of Lemma 2.3.1 are satisfied. Hence. this system is quadratically stable. Consider the problem of estimating Xl (t), the first component of the state of this system. In order to construct the required optimal quadratic guaranteed cost state estimator, it is required to find the value of the parameter E > 0 which minimizes the mean square error bound H1Q+ Hf where Hl = [1 0]. Here Q+ is the stabilizing solution to Riccati equation (2.4.2).
2.5 Illustrative Example
33
Riccati equations (2.4.1) and (2.4.2) were found to have positive-definite stabilizing solutions for E in the range (0,4 x 10- 4 ). A plot of H1 Q+ H{ versus E is shown in Figure 2.5.1. From this plot, the opt1mal value of f is found to be f1 = 2 88 X 10- 4 . 240,--,--,------,------,-----,------,------,------,-----0 I
I
H 1Q+H 1'
220
,
en
§
o
-
'. ,
""0
200
I
H2 Q+H2 '
I
tr(Q+) /
_.
.0 ~
g ~
JBO
(ij :J 0""
en
c
ro 160 Q)
~
140
120~----~----~------L-----~----~------~-----L----~
o
05
15
2
25
3
35
Parameter E
4 J(
FIGURE 2.5.1. Mean square error bounds versus
1.0-4
E.
Figure 2.5.1 also shows a plot of the mean square error bound H 2 Q+ H~ where H2 = [0.707 O. 707J. This bound would be used if the output variable z(t) = H2X(t) were to be estimated. In this case, the optimal value of f is found to be f2 = 1. 76 X 10- 4 . Thus, it can be seen that depending on the state variables to be estimated, a different optimum value of f will be obtained. A third case which is considered is the case in which the total state vector is to be estimated. In this case, the mean square error bound is given by trace( Q+). A plot of trace (Q+) versus f is also shown in Figure 2.5.1. The corresponding optimal value of f is found to be f3 = 2.31 X 10- 4 . With to = f3 = 2.31 X 10- 4 , the stabilizing solutions to Riccati equations (2.4.1) and (2.4.2) are
s+
= 103
x
[2.2551 0.0367
0.0367] 0.0606
>
0,
Q+ = [ 125.4738 59.7607
59.7607] O. 60.3444 >
The corresponding value of the mean square error bound is trace(Q+) = 185.8181. Also, equation (2.4.3) gives the optimal quadratic guaranteed
34
2. Continuous-Time Quadratic Guaranteed Cost Filtering
cost state estimator of the form (2.2.3) with
= [ -66.7130
A
0.5836
f
67.0582] B = [ 65.7130 ] -1.2351 ' f -0.5836'
To illustrate the robust filtering properties of the quadratic guaranteed cost state estimator, Figure 2.5.2 shows a plot of the mean square error as a function of the uncertain parameter 6.. From this plot. it can be seen that for all admissible values of the uncertain parameter, the corresponding mean square error is less than the calculated quadratic guaranteed cost bound. Note that in this plot, only constant values of the uncertain parameter were considered. However, the quadratic guaranteed cost bound also holds for time-varying uncertain parameters. If time varying uncertain parameters were allowed, one would expect the bound to be somewhat less conservative. For the sake of comparison, Figure 2.5.2 also shows a plot of the mean square error which would be obtained if a standard Kalman Filter (based on the system without uncertainty) were applied. From this plot, it can be seen that in the worst case, the robust filter is significantly better than the Kalman Filter. 500,----,-----,----,-----,-----,----,-----,-----r----,-----, 450
350
... o t: <)
/
- - Robust Filter - - Kalman Filter - - - Quadratic Cost Bound
400
/ /
/
/
,.,
300
~
'"C' 250 ;::l
I
/
'" r:::
~ 200 ~
------------------------7----------/
150
/
100,
50 OL-__ -1
~
_____ L_ _ _ __ L_ _ _ _
-08
-06
-04
~
-02
_ _ _ _L __ _
o
~
____
02
~
_ _ _ __ L_ _ _ _J __ _
04
06
~
08
Uncertain Parameter /';.
FIGURE 2.5.2. Mean square estimation error versus the uncertain parameter t1.
3 Discrete-Time Quadratic Guaranteed Cost Filtering
3.1
Introduction
In this chapter we build on the continuous-time robust filtering approach developed in Chapter 2 and consider a corresponding discrete-time optimal guaranteed cost filter. The motivation for considering problems of robust state estimation for discrete-time uncertain systems is that, in practical applications, a state estimator will usually be implemented digitally. Thus, it is useful to obtain a discrete-time version of the robust filter considered in Chapter 2. As in Chapter 2, we will consider a class of uncertain systems with norm bounded uncertainty and Gaussian white noise processes modeling the system noise and measurement noise. However, in this chapter. these uncertain systems will be discrete-time uncertain systems. Also, as in Chapter 2, we will consider a guaranteed cost approach to the steady state robust state estimation problem. In this approach, the robust state estimation problem is considered over an infinite time horizon and a quadratic Lyapunov function is used to establish an upper bound on the state estimation error. Furthermore, as in Chapter 2, we show how a discrete-time steady state optimal quadratic guaranteed cost state estimator can be constructed. Such a guaranteed cost state estimator is shown to be optimal in the sense that there exists no other estimator which achieves a better guaranteed estimation error bound using a quadratic Lyapunov function. From this point of view, the discrete-time optimal quadratic guaranteed cost state estimation results presented in this paper can be regarded as an extension of
36
3. Discrete-Time Quadratic Guaranteed Cost Filtering
the steady state discrete-time Kalman Filter to the case of discrete-time uncertain systems with norm bounded uncertainty. The discrete-time robust state estimation results presented in this chapter are based on results originally presented in the papers [101,102]. It should be noted that a number of other papers have considered the problem of robust state estimation for discrete-time uncertain systems: e.g., see [163]. However, the results of these papers do not lead to optimal guaranteed cost state estimators in the sense that is defined in the sequel. The approach taken in this chapter is closely related to the discrete-time optimal quadratic guaranteed cost control problem such as described in the papers [103,104]. Also, the algorithm used to construct the discrete-time optimal guaranteed state estimator involves the solution to an algebraic Riccati equation of the type which arises in the discrete-time H= control problem; see [13,147,148].
3.2
Discrete-Time Quadratic Guaranteed Cost Filtering
We now introduce the class of discrete-time uncertain systems to be considered in this chapter. As in Chapter 2, this class of uncertain systems will be described by a set of linear state equations which depend on a matrix of uncertain parameters. This matrix of uncertain parameters is subject to a bound on its matrix norm. The uncertain systems under consideration are described by state equations of the form
x(k
+ 1) y(k)
(A + Bl6.(k)K)x(k) Cx(k) + v(k)
+ w(k); (3.2.1)
where x(k) E R n is the state, y(k) E Rl is the measured output, 6.(k) E Rpxq is a time-varying matrix of uncertain parameters satisfying the bound 6.(k)'6.(k) ::; I, and w(k) and v(k) are zero mean Gaussian white noise processes with joint covariance matrix
E{[W(k)] v(k)
[ w(k)'
v(k)' ] }
=[~
~
] > O.
The initial condition x(k o) is assumed to be a zero mean Gaussian random variable independent of the white noise processes w(k) and v(k). The feedback interpretation of this discrete-time uncertain system is the same as that given for the continuous-time uncertain systems considered in Chapter 2; see Figure 2.2.1. However, note that in this chapter, we consider uncertainty only in the system matrix A with no uncertainty in the measurement matrix C. This corresponds to the assumptions made in [102].
3.2 Discrete-Time Quadratic Guaranteed Cost Filtering
37
The following definition introduces the discrete-time version of quadratic stability for uncertain systems of the form (3.2.1): see also [90,103,104]. This defimtion is the discrete-time analog of Definition 2.2.l. Definition 3.2.1 The uncertam system (3.2.1) zs smd to be quadratically stable zJ there exzsts a posztwe-dejimte matrzx P such that x/fA
+ Bl~K]'P[A + Bl~K]x -
Jar all non-zero x E Rn and all matrzces
~
x/Px
<0
(3.2.2)
: ~/ ~ :::; I.
As in Chapter 2. it will be assumed in this chapter that the uncertain system (3.2.1) is quadratically stable. We will restrict attention to linear state estimators of the following form: x(k
+ 1) = Afx(k) + Bfy(k)
(3.2.3)
where x(k) E Rn is the state estzmate and x(k o ) is the estzmator zmtwl condztwn which is assumed to be a zero mean Gaussian random vector. When such an estimator is applied to the uncertain system (3.2.1), the corresponding estzmatwn error vector is defined by e( k) ~ x( k) - x( k). In terms of the state variables e(k) and x(k), the state equations describing the augmented system formed from (3.2.1) and (3.2.3) are as follows: x(k
+ 1) =
Ax(k)
+ fh~(k)Kx(k) + BWk;
x(ko)
=
Xo
(3.2.4) where x(k)
~
A
~
K
~
(3.2.5)
Since we are dealing with a steady state filtering problem, it will be assumed in this chapter that the initial time approaches -00; i.e., ko -4 -00. The following definition introduces the concept of a discrete-time quadratic guaranteed cost state estimator. Definition 3.2.2 A state estzmator (3.2.3) zs sazd to be a quadratic guaranteed cost state estimator with assocwted cost matrix Q > 0 Jar the system (3.2.1) zJ the matrzx A f zs Schur stable (that zs, all oj lis ezgenvalues lze m the open umt dzsk) and there exzsts a matrix (3.2.6)
38
3. Discrete-Time Quadratic Guaranteed Cost Filtering
such that (3.2.7) for all matrices
~
: ~'~
:s: I.
Throughout this chapter, we restrict attention to state estimators in the class of quadratic guaranteed cost state estimators. Our next result shows that if (3.2.3) is a quadratic guaranteed cost state estimator for the system (3.2.1) with cost matrix Q. then Q defines an upper bound on the steady state error covariance matrix at time k:
Qc,.(k) ~
lim
ko~-oo
E {e(k)e(k)'}.
This result is the discrete-time version of Theorem 2.2.3. Theorem 3.2.3 Suppose the uncertam system (3.2.1) zs quadratzcally stable and (3.2.3) zs a quadratzc guaranteed cost state estimator wzth cost matrzx Q > 0 for thzs system. Then the correspondmg augmented system (3.2.4) wzll be quadratzcally stable and the steady state error covarzance matrzx at tzme k satzsfies the bound
Qc,.(k)
:s: Q
(3.2.8)
for all admzsszble uncertamtzes ~(k). Conversely, any state estzmator of the form (3.2.3) with AI Schur stable wzll be a quadratzc guaranteed cost state esizmator for the system (3.2.1) wzth some cost matrtx Q > O. Proof To establish the first part of the theorem, suppose (3.2.3) is a quadratic guaranteed cost state estimator with cost matrix Q > O. It follows that the matrix A I must be Schur stable. Also, it has been assumed that the uncertain system (3.2.1) is quadratically stable. From this it is straightforward to verify that the augmented system (3.2.4) will be quadratically stable. This fact can be established using the Lyapunov argument mentioned in the proof of Theorem 2.2.3 or by simply noting that the augmented system is formed by cascading a quadratically stable uncertain system with a stable linear time-invariant system. Now let ~(k) be any admissible uncertainty and observe that Wk is a Gaussian white noise process with identity covariance. Also, assume that the initial condition random vector for the system (3.2.4) has covariance matrix
E {x(ko)x(ko)'} = Qo ::::: O.
3.2 Discrete-Time Quadratic Guaranteed Cost Filtering
39
It now follows that at time k. the corresponding state covariance matrix for the augmented system (3.2.4) is given by
Qc,,(k, ko) ~ E {x(k)x(k)'} = if>(k, ko)Qoif>(k, ko)' +
k
L
if>(k,j)BB'if>(k,j)'
)=ko
where if>(k, J) is the state transition matrix associated with the system (3.2.4) [with the specified uncertainty realization ~(k)]; e.g., see Chapter 2 of [2]. Furthermore, using the fact that the system (3.2.4) is quadratically stable, it follows that lim if>(k.ko) = 0 ko---+-oo
and hence, the steady state error covariance matrix at time k is given by
Qc,,(k) ~
k
lim
ko---+-oo
Q6(k,ko) =
L
if>(k,j)BB'if>(k,j)'.
)=-00
Now let Hk ~ Q - (A
and
+ Bl8.(k)K)Q(A + B l 8.(k)K)' -
BB' > 0
Y(k, ko) ~ Q - Qc,,(k, ko).
It is straightforward to verify that Y (k, ko ) satisfies the Lyapunov difference equation
Now since the system (3.2.4) is quadratically stable, it follows that lim
ko---+-oo
Y(k. ko) = Q - QD.(k) ~ 0
independently of the initial condition Y(ko. ko). Hence Qc,,(k) :::; Q for all admissible uncertainties 8.(k). However, the state error covariance Qc,,(k) is the (1,1) block of the matrix Qc,,(k) and cost matrix Q is the (1,1) block of the matrix Q. Hence,
QD.(k):::; Q
(3.2.9)
for all admissible uncertainties ~(k). This completes the proof of the first part of the theorem. To establish the second part of the theorem, consider any state estimator of the form (3.2.3) with Af Schur stable. As above, since the system (3.2.1) is quadratically stable, it is straightforward to verify that the augmented system (3.2.4) will be quadratically stable. Hence, there exists a matrix
40
3. Discrete-Time Quadratic Guaranteed Cost Filtering
such that
(A + Ih6.K) Q (A + Ih6.K)' - Q < 0 for all matrices 6. : 6.'6. :::; 1. Therefore, there exists a constant that
f
> 0 such
(A + fh6.K) Q (A + fhf1K)' - Q + BE' < 0 E
E
for all matrices 6. : 6.'6. :::; I. Thus, this state estimator is a quadratic guaranteed cost state estimator with the cost matrix Qnl E 0
Remarks The main result of this chapter is concerned with constructing a state estimator that minimizes the right hand side of the error covariance bound (3.2.9). However, (3.2.9) is a matrix inequality. To obtain a scalar minimization problem, we therefore consider the corresponding bound on the steady state mean square error: lim
ko~-oo
E{e(k)'e(k)} = tr{Q,c,.(k)} :::; tr{Q}. (3.2.10)
Thus, the results of this chapter are concerned with constructing a quadratic guaranteed cost state estimator that minimizes the quantity tr{ Q}. As in Chapter 2, it should be noted that there may be situations in which not all of the state variables are required to be estimated. Instead, it may be required to estimate the value of an output variable z(k) = Hx(k). The solution to this problem will be an output estimate of the form i(k) ~ Hx(k). Furthermore. the corresponding steady state mean square error bound will be as follows: lim ko-'/>-cc
E{(z(k) - i(k))'(z(k) - i(k))} = tr{H'Q,c,.(k)H} :::; tr{H'QH}.
Thus a quadratic guaranteed cost state estimator would be constructed to minimize the quantity tr{H'QH}. Throughout the sequel, we will assume that the state estimator is to be constructed to minimize the quantity tr{ Q}. However, it can be seen that only a minor modification is required if instead a quantity such as tr{H'QH} is to be minimized.
3.3
Preliminary Results
This section contains a collection of results relating to discrete-time uncertain systems with norm bounded uncertainty and related results on discrete-time Hoo control and algebraic Riccati equations. Most of the results presented in this section are analogous to corresponding results presented in Section 2.3. The following result is a discrete-time version of the small gain theorem 2.3.1.
3.3 Preliminary Results
41
Lemma 3.3.1 The uncertam system (3.2.1) zs quadratzcally stable zf and only zf the following two condztwns are satzsfied:
(z) The matrzx A zs Schur stable. (zz) The dzscrete-tzme H OO norm bound I/K(zI -A)-lBllI oo < 1 is satisfied. Proof See [90,103,104]. 0 The following matrix inversion lemma is a useful matrix identity which is extensively used in problems involving discrete-time linear systems.
Lemma 3.3.2 (Matrzx Inverswn Lemma). If A E R nxn and C E Rmxm are nonsmgular matrzces, then for matrices Band D of approprzate dzmenswns:
Proof This identity can be verified by substitution using the definition of the matrix inverse. 0 Notation A symmetric matrix P+ ~ is said to be a strong solution to the discrete-time algebraic Riccati equation
°
APA' - P - APC'(M- 1
+ CPC')-lCPA' + N =
0
(3.3.1) if it satisfies the Riccati equation and the matrix (3.3.2) has all of its eigenvalues in the closed unit disk. The solution P+ is said to be a stabilizing solution if the matrix (3.3.2) is Schur stable Furthermore if P > 0, then it can be shown using the Matrix Inversion Lemma (Lemma 3.3.2) that Riccati equation (3.3.1) is equivalent to the Riccatl equation A(P- 1
+ C'MC)-l A' -
P
+ N = O.
(3.3.3)
In this case, a symmetric matrix P+ > 0 is the stabilizing solution to Riccati equation (3.3.3) if it satisfies this Riccati equation and the matrix (I + C' MCp+)-l A' is Schur stable. The following comparison result for stabilizing solutions to the discretetime algebraic Riccati equation is the discrete-time analog of Lemma 2.3.2.
Lemma 3.3.3 Conszder the dzscrete-tzme algebrazc Rzccatz equatzon A'PA - P - A'PB(M + B'PB)-lB'PA + IV =
° (3.3.4)
42
3 DIscrete-TIme Quadratic Guaranteed Cost FIltermg
where £if ::;> 0 and (A, B) zs stabzhzable. Suppose thzs Rzccati equation has a symmetrzc solutwn F such that
111 + B'FB > 0 and define the matrzx
Also conszder the Rzccatz equatwn A'PA - P- A'PB(M + B'PB)-lB'PA+N= 0 (3.3.5) where 111 ::;> 0 and define the matrzx
If H ::;> iI, then the Rzccatz equatwn (3.3.5) wzll have a umque strong solutwn P+ such that £if +B'P+B > 0 whzch satzsfies p+ ::;> F. Proof See Theorems 3.1 and 3.2 of [109] 0 The following result is a discrete-time version of the strict bounded real lemma (Lemma 2.3.4). Lemma 3.3.4 (Dzscrete-Tzme Strzct Bounded Real Lemma, see (35) for proof.) The followmg three condztwns are equwalent:
(z) The matrzx A zs Schur stable and [[G(zI - A)-lB[[oo < 1. (n) There exzsts a matrzx P > 0 such that p- 1 A'(P- 1
-
BB')-l A - P
-
BB' > 0 and
+ G'G <
0
(m) The algebrazc Rzccatz equatzon A'PA - P
+ A'PB(I -
B'PB)-l B'PA
+ G'G = 0
has a stabzlzzzng solutwn P+ ::;> 0 such that I - B' p+ B >
o.
Furthermore, zf these condztzons hold, then P+ zs umque and satzsfies p+ < P where P> 0 zs any matrzx satzsjymg condztzon (zz).
3.4 DIE>crete-Time Optimal Guaranteed Cost Filter Design
43
The proof of the main result of this chapter will use a result on the standard discrete-tirne state feedback H OO control problem. The underlying linear system for this problem is as follows: x(k
+ 1)
+ B1W(k) + B2U(k); C 1x(k) + D 12 U(k). Ax(k)
z(k)
(3.3.6)
This system is assurned to satisfy the following assumptions:
(i)
C~ D12
= 0:
(ii) DbD12 > 0; (iii)
q
C1
> o.
Lemma 3.3.5 Conszder a system of the form (3.3.6) satisfymg assumptzons (z) - (zzz) above. Furthermore, suppose there exzsts a state feedback control law u(k) = Fx(k) that solves the HOO control problem defined by the system (3.3.6). That zs, the followmg condztzons are satzsfied: 1. The matrzx A
+ B2F
zs Schur stable;
Then the Rzccatz equation
\,;).,;).7) has a stabzlzzmg solutwn p+
> 0 whzch satzsfies the conditwn (3.3.8)
Proof
See Theorem 3.7 of [13) or Theorem 3.1 of [35].
3.4
o
Discrete-Time Optimal Guaranteed Cost Filter Design
This section presents the main result of this chapter. This is a Riccati equation approach to constructing the quadratic guaranteed cost state estimator which minimizes the right hand side of the mean square error bound (3.2.10).
44
3. Discrete-TIme Quadratic Guaranteed Cost FIltering
Theorem 3.4.1 Suppose the uncertam system (3.21) zs quadratzcally stable. Then there exzsts a constant > 0 such that for all E (0, the Rzccatz equatwn
f*
f
f*),
(3.4.1 )
has a stabzlzzmg solutwn S+ > 0 satzsfymg the condztzon (S+)-1 - fK' K > O.
For any such
f,
A(Q-l
(3.4 2)
the Rzccatl equatwn
+ C'V- 1C
+ ~BIB~ + w;:::
- fK' K)-1 A' - Q
f
° (3.4.3)
has a stabzlzzmg solutwn Q+ > 0 whzch satzsfies the conddwn (3.4.4)
and furthermore, Q+ ::; S+. Also, the state estzmator x(k
+ 1)
A(I + f((Q+)-1 +A((Q+)-l
+ C'V- 1C
+ C'V-1C -
- fK' K)-1 K' K)x(k)
fK' K)-IC'V-I(y(k) - Cx(k)) (3.4.5)
has the followmg property: Gwen any b > 0, there exzsts a matrzx Q > 0 such that Q+ ::; Q < Q+ + bI and (3.4.5) zs a quadratzc guaranteed cost state estzmator for the system (3.21) wzth cost matrzx Q. Conversely, gwen any quadratzc guaranteed cost state estzmator for the system (3.2.1) wzth cost matrzx Q, there exzsts a constant f > 0 such that Rzccatz equatwns (3.4.1) and (3.4.3) have stabzlzzzng solutwns S+ > 0 and Q+ > 0 satzsfymg condztwns (3.4.2) and (3.4.4) and Q+ ::; Q. Proof In order to establish the first part of the theorem, note that Lemma 3.3.1 and the quadratic stability of the system (3.2.1) imphes that the matrix A is Schur stable and IIK(zI - A)-lB11I= < 1 Hence. it follows that there exists a constant > 0 such that
f* Ilvf* K(zI -
A)-1 [
v~. Bl W~] 11= < 1
Therefore, Lemma 3.3.4 implies that for all f E (0, f*), the following conditions hold: (i) There exists a matrix
A(5- 1 -
5 > 0 such that 5- 1 -
EK'K)-IA' -
fK' K > 0 and
5 + ~BIB~ + W < O. f
(3.4.6)
3.4 Discrete-Time Optimal Guaranteed Cost Filter Design
45
(ii) The Riccati equation
ASA' - S
+ EASK'(I -
EKSK')-l KSA'
+ ~EIE~ + W =
0 (3.4.7)
E
has a stabilizing solution S+ 2: 0 such that 1 - EK S+ K' > O. Furthermore, since W > 0, it follows from (3.4.7) that S+ > O. Hence, using (3.4.7) and the Matrix Inversion Lemma (Lemma 3.3.2). we can conclude that S+ is also the stabilizing solution to Riccati equation (3.4.1). Moreover, since 1 - EK S+ K' > 0, it follows immediately that S+ satisfies condition (3.4.2). Now let E E (0, E*) be given and consider inequality (3.4.6). Associated with this inequality is a state feedback HOC control problem defined by the system:
x(k
+ 1) z(k)
A'x(k)
[
+ JEK'w(k) + JEC'u(k);
t! 1 ...LE'1
x(k)
[
+ . 0tEV~
1
u(k).
(3.4.8)
Using inequality (3.4.6) and Lemma 3.3.4, it is straightforward to verify that the control law u(k) == 0 solves the H= control problem associated with this system. Hence, using Lemma 3.3.5, it follows that Riccati equation (3.4.3) has a stabilizing solution Q+ > 0 such that (Q+)-l-EK' K > O. Also, by comparing Riccati equations (3.4.1) and (3.4.3), it is straightforward to verify using the differential game interpretation of these Riccati equations that Q+ :S S+; see Theorem 3.7 of [13]. We now consider the augmented system (3.2.4) obtained when the state estimator (3.4.5) is applied to the uncertain system (3.2.1). In this case, the matrices ..4, B, BI , and K are given by:
A (I - MC'V-IC) [ AMC'V- 1 C
[ :~ [K where
-EAMK'K ] A (1 + EMK'K) ,
-AMC'V-! AMC'V-! ] : Bl = [ K
~1
] ;
1
M ~ (Q+)-l
+ C'V- 1 C -
EK'K)-I.
With these definitions, it is straightforward but tedious to verify that the matrix - ~ [ Q+ 0 ] Q= 0 S+ _ Q+ 2: 0
3. Discrete-Time Quadratic Guaranteed Cost Filtering
46
satisfies the Riccati equation
AQA' - Q + fAQR' (I - fRQR') -1 RQA' + ~Ed~~ + EE' =
o. (3.4.9)
c
Furthermore,
A + cQR' (I - cRQR') -1 R A (I =
+ Q+[C'V- 1 C -
EK'KJ)-l
[ A ((1 - cS+ K' K)-l - (I
+ Q+[C'V- 1C - eK'K]) -1)
~(1 -
fS+K'K)-l ].
However, this matrix is Schur stable since Q+ is the stabilizing solution to Riccati equation (3.4.3) and S+ is the stabilizing solution to Riccati equation (3.4.1). Thus, Q is the stabilizing solution to Riccati equation (3.4.9). Using Lemma 3.3.4, it now follows that A is Schur stable and
Lemma 3.3.4 also implies that there exist matrices Q1 > Q and that
V>
0 such
A (Ql 1 - fR'R)-l A' - Q1 + ~E1E~ + EE' + V = 0 c
and
Ql1 - cR'R > O. Thus given any
p E (0,1),
A (Ql 1 - fR'Rr A' - Q1 + ~Bd]~ + BB' + pV < o. 1
c
Therefore using Lemma 3.3.4, it follows that the Riccati equation
A (Q~l - cR'R)-l A' - Qp + ~B1B~ + EB' + pV c
= 0
(3.4.10)
has a stabilizing solution Qp > 0 such that Q;l - cR'R > O. Furthermore, using the Matrix Inversion Lemma (Lemma 3.3.2), this Riccati equation can be re-written as
AQpA' - Qp + cAQpR' (I - cRQpR') -1 RQpA' 1- c
+-B1B~
--
-
+ BB' + pV =
O.
(3.4.11)
Now, comparing Riccati equations (3.4.9) and (3.4.11), it follows from Lemma 3.3.3 that Qp ?:: Q. Also, since the stabilizing solution to Riccati equation (3.4.10) is a continuous function of the parameter p, it follows
3.4 Discrete-Time Optimal Guaranteed Cost Filter Design
47
that limp-->o Qp = Q. Hence, using (3.4.10) it follows that given any r5 > 0, there exists a matrix Qp > 0 such that
(3.4.12) (3.4.13) and Q ::; Qp < Q + r5I. Also, if we let Q > 0 be defined as the (1,1) block of the matrix Qp, we must have Q+ ::; Q < Q+ + r5I. Now let the matrix A be given such that A' A ::; 1. It is straightforward to verify that
(A + E1AK) Qp (A + E1AK)' - Qp + EE' < AQpA' + AQpK' A' E~ + E1AKQpA' + E1AKQpK' A' E~ 1-
-
--B1AA'B~ E
1- -
--
-
+ -B1B~ + BB' - Qp E
AQpA' + ~E1E~ + EAQpK' (1 - KQpK') -1 KQpA' E
+EE' - HI'HI - Qp where
Hence. using the Matrix Inversion Lemma (Lemma 3.3.2) and inequalit¥ (3.4.12), we obtain
(A + E1AK) Qp (A + E1AK)' - Qp + EE' < AQpA' + ~E1E~ + EAQpK' (I - KQpK') -1 KQp.J.' + EE' - Qp E
A (Q-1 - EK'K)-l A' - Qp + ~E1E~ + EE' p E < O. Also, since the matrix A (which is the system matrix for the augmented system) is Schur stable, then the matrix
must also be Schur stable. Thus, we can now conclude that the state estimator (3.4.5) is a quadratic guaranteed cost state estimator for the system (3.2.1) with cost matrix Q. This completes the proof of the first part of the theorem.
48
3. Discrete-Time Quadratic Guaranteed Cost Filtering
To establish the second part of the theorem, we suppose (3.2.3) is a given quadratic guaranteed cost state estimator with cost matrix Cd > O. Hence, there exists a matrix
such that
(A + fhAK) Q (A + fhAK)' - Q + BB' < 0 (3.4.14) for all matrices A : A' A ::; I. Here the matrices A, B, Eh, and K are defined as in (3.2.4) and (3.2.5). Using Theorem 2.2 of [103]' inequality (3.4.14) implies that there exists an E > 0 such that
(3.4.15) and
Q-1 _ EK'K > o.
(3.4.16)
Also, the Matrix Inversion Lemma (Lemma 3.3.2) implies
and hence inequality (3.4.15) becomes
AQA' + AQK'
(~ -
KQK') -1 KQA' - Q +
t
~B1B~ + BB' < O. t
(3.4.17)
Now pre-multiply this inequality by the matrix [I I] and post-multiply by the matrix [I 11'. It is straightforward to verify that this yields the inequality
,)-1, A -
1
A (II- - tK K where
t>
-
II = Q + Q12
1,
II + -B1B1 t
+W < 0
,
+ Q 12 + Q22 > O.
Also, it follows from (3.4.16) that I -EKQK' > 0 and hence I -EKIIK' > O. Therefore, II- 1 - EK' K > O. Thus, using Lemma 3.3.4, it follows immediately that the Riccati equation
ASA' - S
+ f.ASK'(1 - f.KSK,)-lKSA' + ~B1B~ + W = 0 f.
3.4 Discrete-Time Optimal Guaranteed Cost Filter Design
49
has a stabilizing solution S+ ::::: 0 such that 1- EKS+ K' > o. However, since W > 0, it follows that S+ > 0 and hence using the Matrix Inversion Lemma (Lemma 3.3.2), we can conclude that S+ is the stabilizing solution to Riccati equation (3.4.1) and condition (3.4.2) is satisfied. We now return to (3.4.17) and pre-multiply by the matrix
and post-multiply by the matrix
It is straightforward to verify that this yields the inequality
1 (A - BfC) (~-1 - EK'Kr (A - BfG)' - ~ 1 -(3.4.18) +-B1B~ + BfVBi + W < 0 E
where
Hence, E :::::
Q.
Also, we can write ~-1
(>-1
=
[
-
_Q-1Q' ~-1 22
~-lQ 12 Q-1 22
-1 Q 22
12
]
+ Q-1Q' ~ 22 12L.
1Q 12 Q-1 22
and therefore (3.4.16) implies ~-1 [
_
_~-lQ12Q221
cK' K
-Q221Q~2~-1
- EK' K
Q2l
-
EK' K
+ Q221Q~2~-lQ12Q2l- EK' K
]
> O.
Taking the (1,1) block of this inequality, we conclude ~-1 _
EK'K >
o.
Using this fact, is straightforward to verify that
(A - BfC) (~-1 1 +-B1B~ E
-
EK' K) -1 (A - BfC), - E --
+ BfVBi + W
A(~-l + C'V- 1C -
EK' K)-l A' -
~ + !B1B~ + W E
where
TV =
-Bi
+ [C(~-1
-
EK' K)-lC'
+ V] -1 C(~-1
-
EK' K)-l A'.
50
3. Discrete-Time Quadratic Guaranteed Cost Filtering
Hence, (3.4.18) implies A(L:- 1 + C'V-1C - fK'K)-lA' - L:
+ ~BIB~ + W:::: o. f
That is. there exists a Z
~
0 such that
A(L:- 1 + C'V-1C - EK'K)-lA' - L:
+ ~BIB~ + W + Z
= O.
f
We now compare this Riccati equation with Riccati equation (3.4.3) using the differential game interpretation of these Riccati equations. Indeed using the fact that L:- 1 - fK' K > 0, it follows from Theorem 3.7 of [13] that Riccati equation (3.4.3) has a solution Q > 0 such that condition (3.4.4) is satisfied and Q < L: :::: Q Furthermore, it follows from Theorem 3.8 of [13] that there exists a state feed back controller that solves the Hoo control problem defined by the system (3.4.8). This state feedback controller can also be regarded as a full information controller. Hence, using Theorem 9.2 of [148], it follows that Riccati equation (3.4.3) has a stabilizing solution Q+ satisfying the condition (3.4.19)
However, since Q satisfies condition (3.4.4). it must also satisfy condition (3.4.19). Hence using Lemmas 3.5 and 3.8 of [147], it follows that Q+ :::: Q. Therefore, Q :::: Q implies Q+ :::: Q. Furthermore. since Q satisfies condition (3.4.4), we can conclude that Q+ must also satisfy (3.4.4). This completes the proof of the theorem. D Observation 3.4.1 The above theorem states that the state estimator defined by equatwns (3.4.3) and (3.4.5) has the followmg property: Gwen any 5 > 0, there ensts a matrzx Q > 0 such that Q < Q < Q + 5I and the state estzmator is a quadratzc guaranteed cost state estzmator wzth cost matrzx Q. Hence Theorem 3.2.3 zmplzes that steady state error covarzance matrzx satzsfies the bound
Q6.(k) :::: Q < Q + 5I for all admzsszble uncertamtzes ll(k). However, thzs bound holds for all 5> O. Hence, we must have
Qdk):::: Q for all admzsszble uncertamtzes ll(k). Remarks Using the above results. we can now see that the optimal quadratic guaranteed cost state estimator can be obtained by choosing the parameter f > 0 to minimize tr( Q+) where Q+ is the stabilizing solution to Riccati equation (3.4.3). This optimization is to be carried out for f contained in the
3.5 Illustrative Example
51
interval (0, E*) over which Riccati equation (3.4.1) has a stabilizing solution satisfying condition (3.4.2). The required optimal estimator is then given by (3.4.5). Note however, that there is no guarantee that the minimum will actually be achieved. It is of interest to note that the equations defining the optimal quadratic guaranteed cost state estimator (3.4.5) can also be re-written in a '·predictorcorrector" form as is common for the standard Kalman filter; e.g., see [2]. This predictor-corrector form is as follows:
Predictor: x(k + 11k) = Ax(klk) Corrector: x(klk) = (I
+ EM K' K)x(klk -
where M = ((Q+)-l
3.5
+ C'V-IC -
1)
+ MC'V- 1 (1I(k) - Cx(klk -
1))
EK' K)-l.
Illustrative Example
In this section, we present an example that illustrates the approach to discrete-time optimal guaranteed cost state estimation which has been presented in this chapter. The example we consider is obtained by discretizing the continuous-time example considered in Section 2.5. Indeed, the continuous-time uncertain system considered in Section 2.5 can be represented as in the block diagram shown in Figure 3.5.1.
z(t)
~(t)
Continuous-time
y(t)
Nominal system
w(t) v(t) FIGURE 3.5.1. Continuous-time uncertain system.
In this uncertain system, the nominal system is described by the state equations X
Ax
+ BIt. + w,
z
Kx,
y
Cx +v,
52
3. Discrete- Time Quadratic Guaranteed Cost Filtering
where
~1
!l],C=[l
A
[
K
[05],W=[
100 0
1~0 ]
u' ]
}=[:
Here
[WI
E {[ : ]
=[~ ], = ,v =
-1 ] , Bl
Dl
0,
1.
~]>O
and ~ = ~z where II~II :::; 1. We now approximate this continuous-time uncertain system with a discrete-time uncertain system with sampling period h = 0.028 as shown in Figure 3.5.2.
f,(kh)
z(kh)
A
f,( t)
z(t)
ZOH
Nominal
w(t)
Sample
y(t)
System
y(kh)
Sample
I;
v(t) FIGURE 3.5.2. Approximate discrete-time uncertain system.
This approximate uncertain system will be described by state equations of the form
x(k
+ 1)
(.4. + ih~(k)K)x(k) + w(k);
y(k)
Cx(k)
+ v(k)
where (e.g., see [7]) Ah
e
C
= [0.9802
0
0.0196]. 0.9802'
[1 - 1]; K = [0 5]; V = 1.
3.5 Illustrative Example
53
Also,
where (e.g., see [7])
W
is the solution to the Lyapunov equation
AW + WA ' = eAhWeA'h
-
W.
In this example, we obtain
W=
[1.9608 0.0195
00195] 19605 .
Furthermore. lJ.(k) is a scalar uncertain parameter subject to the bound 1lJ.(k)1 :s: 1. To find the optimal quadratic guaranteed cost state estimator, we solve Riccati equations (3.4.1) and (3.4 3) for a series of values of E > 0 and plot tr(Q+) versus E This plot is shown in Figure 3.53 From this 20,-,--,,----,-----,-----,-----,-----,-----,-----, 18
16 14 "0
c:
:::l
o
12
.c
elii
10
Q)
Ca
8
:::l
g c: t1l
6
Q)
::!: 4
oL---~L---~----~~--~----~----~----~,---~.
o
05
15
2
25
3
Parameter E FIGURE 3.5.3. tr(Q+) versus
35
4 x 10-'
Eo
plot. we determine the optimal value of E to be E = 5.98 X 10- 6 . With this value of E, the required stabilizing solutions to Riccati equations are S+
= 10 3
x
[1. 7737 0.0441
0.0441] 0.0662
>
0
54
3. Discrete-Time Quadratic Guaranteed Cost Filtering
Q+
=
[
134.9989 65.2477
65.2477 65.9187
J
> O.
Then, using equation (3.4.5), we obtain the optimal quadratic guaranteed cost state estimator
-(k 1) = [ 0.3301 x + 0.0063
0.6796 0.9837
J -(k) x
+
[0.6501] (k) -0.0063 Y .
To illustrate the robust filtering properties of this quadratic guaranteed cost state estimator, Figure 3.5.4 shows a plot of the mean square error as a function of the uncertain parameter ~. From this plot, it can be seen that for all admissible values of the uncertain parameter, the corresponding mean square error is less than the calculated quadratic guaranteed cost bound. Note that in this plot, we have only considered constant values of the uncertain parameter ~. However, the quadratic guaranteed cost bound also holds for time-varying uncertain parameters. If uncertain parameter was allowed to be time-varying, one would expect the mean square error bound to be somewhat less conservative. For the sake of comparison, Figure 3.5.4 also shows a plot of the mean square error which would be obtained if a standard Kalman Filter (based on the nominal system without uncertainty) were applied. From this plot, we can see that in the worst case, the robust filter is significantly better than the standard Kalman Filter.
3.5 Illustrative Example
55
450r----,----~----_.----,_----~----r_---,----~-----.----,
I
400
/ I
/
- - - Robust Filter - - Kalman Filter - - - Quadratic Cost Bound
300
250
8
E200-
/
/ /
_
-
.L /
~
,
'"0-150
/ ;I
/
/ '"c: ~ 100~____________________________~~~----------------------~
50 ~1L-----o~~~--__-0.L6----_-OL4----_~OL2--~0L----Q~2~---Q~.L4----~~e----O~.~8----~
Uncertain Parameter ll.
FIGURE 3.5.4. Mean square estimation error versus the uncert8.in parameter fl..
4 Continuous-Time Set-Valued State Estimation and Model Validation
4.1
Introduction
In the previous chapters, the problem of robust state estimation was addressed by extending the standard steady state Kalman Filter to the case in which the underlying signal model is an uncertain system. The uncertain systems being considered were uncertain systems with norm bounded uncertainty and subject to stochastic white noise disturbances. In particular. the robust Kalman Filters of Chapters 2 and 3 are concerned with constructing a state estimator which bounds the mean square estimation error. However, these results may be conservative in that only an upper bound is obtained for the mean square estimation error. Also, these results do not extend in a straightforward way to the case of finite time horizon state estimation problems or robust state estimation problems with structured uncertainty. A somewhat different approach to the robust state estimation problem is the approach of [120]. Reference [120] builds on the deterministic interpretation of Kalman Filtering presented in [16]. This deterministic approach to Kalman Filtering also forms the launching point for the results of this chapter and the remaining chapters of the book. In [16], the following deterministic state estimation problem is considered: Given output measurements from a time-varying linear system with noise inputs subject to an L2 norm bound, find the set of all states consistent with these measurements. Such a problem is referred to as a set-valued state estzmatwn problem. The solution to this problem was found to be
58
4. Continuous-Time Set-Valued State Estimation
an ellipsoid in state space which is defined by the standard Kalman Filter equations. Thus, the results of [16] give an alternative interpretation of the standard Kalman Filter. In [57], an attempt was made to extend the results of [16] to the case of uncertain systems containing norm bounded uncertainties. The main result of [57] leads to the construction of a set which over bounds the true set of possible states. This set is also constructed via Kalman Filter like equations. However. because the set of states obtained using the methods of [57] is only an overbound of the true set of possible states, the results may be conservative (and no indication is given as to the degree of conservatism). The results presented in this chapter originally appeared in the papers [120,121,126,127]. In particular, we consider an approach to robust set-valued state estimation which builds on the results of [16]. However following [120], we consider a different class of uncertain systems to that considered in [57]. The class of time-varying uncertain systems considered in this chapter contain uncertainty which is defined by a certain Integral Quadratic Constraint (IQC); e.g., see [165-168]. This class of uncertain systems originated in the work of Yakubovich and is a particularly rich uncertainty class allowing for nonlinear, time-varying, dynamic uncertainties. Furthermore, a number of new robust control system design methodologies have recently been developed for uncertain systems with integral quadratic constraints; e.g., see [115,117,119,123,136]. The integral quadratic constraint approach to robust state estimation considered in this chapter extends the results of [16] to include uncertainty of this type. An interesting feature of this set-valued state estimation approach is that it also leads to the solution to a certain problem of model validation. This model validation problem is concerned with determining if an uncertain system model is consistent with a given set of input - output measurements. The problem of model validation for uncertain systems is motivated by the problem of system identification for uncertain systems in which an uncertain system model is to be constructed on the basis of a collection of measured input - output data. An integral part of any system identification process is model validation. Model validation is a process of determining if given input - output measurements are compatible with a given model. Model validation can actually only determine if a given model is invalid in that it is incompatible with a given data set. In practice, model validation can be used to choose between a collection of models and to determine bounds on uncertain parameters. The significant advances being made in the field of robust control theory have motivated a number of authors to study the model validation problem for uncertain systems: e.g., see [107,143,144,171]. The approach of [143,144] involves an uncertain system with structured uncertainty of the type which arises in the J-L framework. For this class of uncertain systems, [143,144] applies a frequency domain approach to convert the model validation problem into a J-L problem which can be solved via numerical op-
4.2 Model Validation and Set-Valued State Estimation Problem Statements
59
timization techniques. The approach of [107] involves a class of single input single output discrete-time uncertain systems with norm bounded uncertainty. For this class of uncertain systems. reference [107J applies a time domain approach to convert the model validation problem into a convex optimization problem. A similar approach is also considered in [171 J. The main results presented in this chapter relate to the problems of model validation and robust state estimation for continuous time uncertain systems in which the uncertainty is described by an integral quadratic constraint. These results were originally presented in the papers [118.120, 121,126.127, 135J. The results of this chapter involve the solution of a Riccati differential equation and the solution of a set of filtering state equations. This is in contrast to the existing model validation results mentioned above which are based on convex optimization. In this sense. the results of this chapter are more closely related to results on observer based fault detection: e.g .. see [92J. However, the filter state equations arising in this chapter are not standard Kalman Filter state equations. Instead they take the form of the related robust state estimator. This may lead to a considerable computational advantage as compared to existing techniques for model validation. In addition to the computational advantages mentioned above. our approach to model validation has a number of other advantages over existing methods. The first point to mention is that our approach does not require zero initial conditions on the plant from which the data is measured. Also. in contrast to [143, 144], our approach requires input - output data defined only over a finite time interval. Furthermore. the integral quadratic constraint uncertainty description considered in our approach allows for nonlinear time-varying uncertainties. The robust state estimation results presented in this chapter may be preferable over those of Chapter 2 in situations in which a finite horizon state estimation problem is to be solved. Also, the IQC uncertainty model considered in this chapter allows for a richer class of uncertainties than the norm bounded uncertainty models considered in Chapters 2 and 3.
4.2
Model Validation and Set-Valued State Estimation Problem Statements
Consider the time-varying uncertain system:
x( t)
z( t) y(t)
A(t)x(t) + Bl(t)W(t) + B2(t)U(t); K(t)x(t) + G(t)u(t): C(t)x(t) + v(t)
(4.2.1)
where x(t) E Rn is the state. w(t) E RP and v(t) E Rl are the uncertainty inputs. u(t) E Rh is a known control mput. z(t) E Rq is the uncertainty
60
4 Contmuous-Time Set-Valued State EstImatIOn
output and y(t) E Rl IS the measured output, A( ), BI ( ), B 2 ( and C( ) are bounded plece'Wlse contInUOUS matnx functIOns
),
K(·), G(·)
System Uncertamty
The uncertaInty In the above system IS descnbed by an equation of the form w(t) ] = ¢(t,x()) [ v(t) where the folloWIng Integral Quadratic ConstraInt IS satisfied Let N = N > 0 be a given matnx, Xo E Rn be a given vector, d > 0 be a given constant, Q( ) = Q( )' and R( ) = R( )' be given bounded piecewise contInUOUS matnx weightIng functIOns satisfYIng the followmg conditIOn there eXists a constant b > 0 such that Q(t) ~ M, R(t) ~ M for all t For a given fimte time Interval [0, s], we Will consider the uncertamty Inputs w( ) and v( ) and ImtIal conditions x(O) such that
1 8
(x(O) - xo)' N(x(O) - xo)
1Ilz(t)11
+
(w(t)'Q(t)w(t)
+ v(t)' R(t)v(t))dt
8
::::: d +
2
(422)
dt
Here II I denotes the standard Euchdean norm The uncertamty In the uncertaIn system (4 2 1), (4 2 2) can be regarded as a feed back mterconnectIOn between the nomInal lInear system (4 2 1) and an uncertamty block which takes the uncertaInty output z and produces the uncertaInty Inputs wand v ThiS IS Illustrated m Figure 4 2 1
Uncertamty
Nommal System
z
FIGURE 42 1 Block diagram representatIon of uncertam system
Note that the above uncertaInty descnptIOn allows for uncertaInties In 'Which the uncertaInty mputs w() and v() depend dynamically on the uncertaInty output z() In thiS case, the constant d may be Interpreted as
4.2 Model Validation and Set-Valued State Estimation Problem Statements
61
a measure of the SIze of the initial conditions on the nominal system and uncertainty dynamics. It is clear that the uncertain system (4.2.1), (4.2 2) allows for uncertainty satisfying a standard norm bound constraint. In thIS case. the uncertain system would be described by the state equations
x( t) y(t)
[A(t) + BI(t)6.I(t)K(t)]x(t) +[B2(t) + Bdt)6.I(t)G(t)]u(t); [C(t) + 6. 2(t)K(t)]x(t) + 6.2(t)G(t)u(t); 11[6. I (t)' 6. 2 (t)'] II :::; 1
(4.2.3)
where 6.1(t) and 6. 2 (t) are the uncertainty matrices and 11·11 denotes the standard induced matrix norm. Also. the initial conditions would be required to satisfy the inequality
(x(O) - xo)' N(x(O) - xo) :::; d. To verify that such uncertainty is admissible for the uncertain system (4.2.1), (422), let w(t) = 6. 1 (t)z(t). v(t) = 6. 2 (t)z(t) where
11[6. I (t)' 6. 2 (t)']11 :::; 1 for all t E [0, T]. Then condition (4.2.2) is satisfied with Q(.)
I and
R(·) == I. Notatzon Let u(t) = uo(t) be a fixed control input and y(t) = yo(t) be a fixed measured output of the uncertain system (4.2.1), and let the finite time interval [0. s] be given. Then,
denotes the set of all possible states x( s) at time s for the uncertain system (4.2.1) with uncertainty inputs and initial conditions satisfying the constraint (4.2.2)
Definition 4.2.1 The unceriam system (4.2.1), (4.2.2) zs sazd to be strictly verifiable on [0. T], if for any vector Xo ERn, any tzme s E (0. T], any constant d > 0, any fixed control mput u(t) = uo(t) and any fixed measured output y(t) = yo(t). the set Xs[xo, uo(-)I~. Yo(-)I~, d] zs bounded. Definition 4.2.2 Let Xo E R n and d > 0 be gwen. Also, let uo(-) and Yo(-) be gwen vector functzons defined over a gwen tzme mterval [0. s]. The input - output pazr [uo(-). Yo(-)] zs sazd to be realizable wzth parameters Xo and d if there exzst [xU, w(·),v(·)] satzsfying condztzons (4.2.1), (4.2.2) wzth u(t) = uo(t) and y(t) = yo(t).
62
4. Continuous-Time Set-Valued State Estimation
Model Validation Problem We consider the following problem: Given an input - output pair [uoe) , YoO], determine if this pair is realizable for the uncertain system (4.2.1), (4.2.2).
Remarks The restriction of strict verifiability as defined above is a reasonable one since if it does not hold, then it is unlikely that any reasonable conclusions can be obtained about the system model from measured data. In particular, note that within our integral quadratic constraint framework, if the strict verifiability condition is not satisfied, then the model may be such that any pair Uo (-), Yo (-) is possible.
Set- Valued State Estimation Problem The main results of this chapter also concern the following set-valued state estimation problem: Given an input - output pair [uaU. yoU]' for any S E [0, Tj, find the set
of all possible states xes) at time s for the system (4.2.1) with uncertainty inputs and initial conditions satisfying the constraint (4.2.2).
Remarks In the sequel. we will show that the solution to this set-valued state estimation problem leads to a set Xs[xo, uo(·)I~. Ya(')I~, d] which is an ellipsoid in Rn. If a point-valued state estimate is required rather than a set-valued state estimate, then the center of this ellipsoid can be used to provide a point-valued state estimate. The connection between the above set-valued state estimation problem and the model yalidation problem can be seen from he following simple observation. If for a given input-output pair [uo('),Yo(')]' the corresponding set of possible states Xs[xo,uo(-)I~,YoUI~,d] is empty, the given measurements must be incompatible with the uncertain system model. Hence, the uncertain system model has been invalidated.
4.3
Design of Set-Valued State Estimator
The solution to the above model validation and set-valued state estimation problems involve the following Riccati differential equation:
Pet)
B1(t)Q(t)-1 Bl(t)'
+ A(t)P(t) + P(t)A(t)'
+P(t)[K(t)' K(t) - G(t)' R(t)G(t)]P(t).
(4.3.1)
4.3 Design of Set-Valued State Estimator
63
Also, we consider the filter state equations 5:(t)
+ P(t)[K(t)' K(t) - G(t)' R(t)G(t)]] x(t) +P(t)G(t)' R(t)Yo(t) + [P(t)K(t)'G(t) + B2(t)]UO(t).
=
[A(t)
(4.3.2) The following theorem gives a solution to both the set-valued state estimation problem and the model validation problem. Theorem 4.3.1 Let N = N' > 0 be a gwen matnx, and QU = QU' and RU = R(o)' be gwen matnx functwns such that the condztwn Q(t) :::: 8I and R(t) :::: 8I holds on the tzme wterval [O,Tj. Cons2der the uncertaw system (4.2.1), (4.2.2). Then the followwg statements hold:
M The system (4.2.1),
(4.2.2) 2S strzctly venfiable on [0. T] 2f and only 2f the solutwn P( ) to the R2ccatz equatwn (4 3.1) wzth zmtzal condztwn P(O) = N- 1 2S defined and posdwe-defimte on the wterval [0. T].
(n) Suppose the system (4.2.1), (4.2.2) zs stnctly venfiable on [0, T]. Also, let 8 E (0, Tj be gwen and let Xo E R n be a gwen vector, d > 0 be a gwen constant, and uo(t) and yo(t) be gwen vector functwns defined on [0,8]. Then, the pmr [uo( ),Yo(')] 28 reahzable zf and only zf Ps[uoU. Yo(')] :::: -d where
61' [
Ps[uo( ), yoU]
=
0
[[CK(t)x(t) + G(t)uo(t))[[2 ] -(C(t)x(t) - yo(t))'R(t)(G(t)x(t) - yo(t)) dt
(4.3.3) and x( ) Xo·
2S
defined by the equatwn (4.3.2) wzth wztwl cond2twn X(O)
=
(iii) If the system (4.2.1), (4.2.2) zs stnctly venfiable, then
(4.3.4) for all s E [0, T]. Proof (z) Necesszty In this case. we must establish the existence of a positivedefinite solution to Riccati equation (4.3.1). This will be achieved by showing that the cost function in a corresponding linear quadratic optimal control problem is bounded from below.
64
4. Continuous-Time Set-Valued State Estimation
First let s E (0, T] be given and consider ~he uncertain system (4.2.1), (4.2.2) defined on [0, s]. Given a pair luoU. YoU]' we have by the definition of Xs[xo, uoe)I~, yo(-)I~. d]. that
Xs
E
Xs[xo, uoC) I~, YOn I~. d]
if and only if there exist vector functions x(·), w(·) and v(·) satisfying equation (4.2.1) and such that x(s) = x s , the comltraint (4.2.2) holds. and
yo(t)
= C(t)x(t) + 1!(t)
( 4.3.5)
for all t E [0, s]. Substitution of (4.3.5) into (J.2.2) implies that
Xs
E
Xs[xo, uo(·)I~. Yonl~, d]
if and only if there exists an uncertainty input we) E L 2 [0, s] such that
( 4.3.6)
J[xs, wC)] ::; d where J[xs,w(-)] is defined by
J[xs, wU]
~
(x(O) - xo)' N(x(O) - xo) + t ( w(t)'Q(t)w(t) - 11(j{(t)x(t) + G(t)uo(t))112 ) dt Jo +(Yo(t) - C(t)x(t))' R(t)(Yo(t) - C(t)x(t)) (4.3.7)
and xC) is the solution to (4.2.1) with uncertainty input we) and boundary condition x( s) = Xs' Now consider the functional (4.3.7) with Xo = 0, uo(-) == and Yo(-) == 0. In this case, J is a homogeneous quadratic functional with a terminal cost term. Also, consider the set Xs [0, 0. 0,1] corre@ponding to Xo = 0, uo(-) == 0, Yo e) == and d = 1. Since Xs [0,0,0. 1] is bounded. there exists a constant hs > such that all vectors Xs E R n with Ilxsll = hs do not belong to the set X.[O, 0. 0.1]. Hence,
°
°°
( 4.3.8) for all Xs E Rn such that Ilxs I = hs and for all we) E L2 [O, sJ. Since. J is a homogeneous quadratic functional, we have
J[axs, aw(.)] = a2J[xs' we)] and (4.3.8) implies that inf
w( )EL 2 [O,sJ
J[xs, w(·)] >
°
( 4.3.9)
for all s E [0, T] and all Xs "I- 0. Note that in the above infimum. the vector Xs "I- is fixed and using (4.3.8). this enables us to conclude the strict inequality.
°
4.3 Design of Set-Valued State Estimator
65
The optimization problem (4.3.9) with an unconstrained terminal condition xes) i and subject to the constraint defined by the system (4.2.1) is a linear quadratic optimal control problem in which time is reversed. In this linear quadratic optimal control problem, a sign indefinite quadratic cost function is being considered. We now use a known result from linear quadratic optimal control theory which states that if the infimum of the cost function is strictly positive for all terminal conditions, then there exists a solution to the corresponding Riccati equation. Furthermore, the terminal value of the Riccati solution is positive-definite; e.g., see page 23 of [30]. Thus, we conclude that condition (4.3.9) implies that there exists a solution X (.) to the Riccati equation
°
- Xes)
K(s)' K(s) - C(s)' R(s)C(s) +X(s)A(s) + A(s)' Xes) + X(s)B(s)Q(s)-l B(s)' Xes) (4.3.10)
with initial condition X(O) = N. Furthermore, since (4.3.9) holds for any terminal time s E [0, TJ, this solution is positive-definite on [0, T]. From this, it follows that the required solution to Riccati equation (4.3.1) is given by P (.) ~ XU -1. Note that in order to establish the existence of a suitable solution to Riccati equation (4.3.10), it was only necessary to consider the case Xo = 0, uoU == 0, yoU == and d = 1. This completes the proof of this part of the theorem. (i) Sufficiency For a given time interval [0, s], we have shown above that
°
if and only if there exists an uncertainty input wU E L 2 [0, s] such that condition (4.3.6) holds for the functional (4.3.7). Now consider the following minimization problem inf w(-)EL 2 [O,s]
J[x s , w( .)]
(4.3.11)
where the infimum is taken over all xC) and wC) connected by (4.2.1) with the boundary condition xes) = Xs' This problem is a linear quadratic optimal tracking problem in which the system operates in reverse time. \Ve wish to convert the above tracking problem into a tracking problem of the form considered in [76] and [16]. First define Xl(t) to be the solution to the state equations
(4.3.12) Now let
x(t) ~ x(t) - Xl(t).
66
4. Continuous-Time Set-Valued State Estimation
Then, it follows from (4.2.1) and (4.3.12) that x(t) satisfies the state equations
xCt) = ACt)x(t)
+ Bl(t)W(t)
(4.3.13)
where X(O) = x(O). Furthermore, the cost function (4.3.7) can be re-written as
i[xs, w(-)] (x(O) - xo)' N(x(O) - xo) t ( w(t)'Q(t)w(t) + v(t)' R(t)v(t) ) + Jo -11(K(t)[x(t) + Xl(t)] + G(t)uO(t))1I2 dt
J[xs, w(·)]
(4.3.14) where xes) = Xs = Xs - Xl(S) and
vet) ~ Yo(t) - C(t)[x(t)
+ Xl(t)].
(4.3.15)
Equations (4.3.13) and (4.3.14) now define a tracking problem of the form considered in [76] where Yo (.), uo (-) and Xl (.) are all treated as reference inputs. In fact, the only difference between this tracking problem and the tracking problem considered in the proof of the result of [16] is that in this theorem, we have a sign indefinite quadratic cost function. The solution to this tracking problem is well known (e.g., see [76]). Indeed, if the Riccati equation (4.3.1) has a positive-definite solution defined on [0, T] with initial condition P(O) = Xo\ then the matrix function
XC) = p(-)-l > 0 is the solution to Riccati equation (4.3.10) with initial condition X(O) From this, it follows that the infimum in inf
w(-)EL2[O,S]
= N.
i[xs, w(-)]
will be achieved for any Xo, uoC) and Yo(-). Furthermore as in [16]' we can write min
w(')E L 2[O,S]
i[xs, w(-)]
= (xs - :Tl(S))'X(s)(xs - :Tl(S)) - Ps[uo(')' Yo(·)]
(4.3.16)
where
.) (.)] ~ t ( - Jo
Ps [Uo ( ,Yo
II(K(t)[Xl(t) + :TI(t)] -v(t)' R(t)v(t)
+ G(t)uo(t)) 112
) dt
4 3 DeSIgn of Set-Valued State Estimator
67
and Xl (s) IS the solutIOn to state equatIOns
Xl(S)
[A(s) + P(s)[K(s)' K(s) - C(s)' R(s)C(s)]] Xl(S) +P(s)[K(s)' K(s) - C(s)' R(s)C(S)]Xl(S) +P(s)C(s)' R(s)yo(s) + P(s)K(s)'G(s)uo(s) (4317)
=
with initial condition Xl (0) = Xo Now let
Using the fact that Xs = Xs -Xl (s). It follows that (4.3.16) can be re-written as mf
w( )EL2[O s]
J[xs,w()] = (xs - x(s))'X(s)(xs - xes)) - Ps[uo(·),yo(-)J
where t>
t [ -(C(t)x(t) II(K(t)x(t) + G(t)uo(t))112 - yo(t))' R(t)(C(t)x(t) _ yo(t))
Ps[uo( ), Yo( )] = Jo
] dt
and x( s) IS the solutIOn to state equatIOns (4 3 2) WIth mltIal condItIOn X(O) = Xo From thIS we can conclude that the set Xs[xo, uo( )I~. Yo( )I~, d] IS gIven by
Xs[xo, uo( )I~ Yo( )I~, d] {xs ERn mm
w( )EL2[O s]
=
J[xs, w( )] ::; d}
Xs ERn (xs - x(s)),p(s)-I(x s - xes)) { ::; d + Ps [uo ( ), Yo ( )]
}
(4.318)
ThIS completes the proof of thIS part of the theorem (n) It IS clear that the pair [uo( ),Yo()] defined on an mterval [O,s] IS realIzable If and only If the set Xs[xo,uo( )I~,yo( )1~,d]IS not empty Smce the system (4 2 1), (4 2 2) IS stnctly venfiable, It now follows that the set
IS defined by (4318) Hence, the set Xs[xo, uo( )I~, Yo( )I~, d]ls not empty If and only If Ps[uo( ), Yo( )] 2 -d ThIS completes the proof of thIS part of the theorem (m) In the proof of the part (I) of the theorem, we have already proved that the set
Xs[xo,uo( )I~,yo( )I~,d] IS descnbed by (4 3 18) ThIS completes the proof of the theorem
0
Remarks Note that the only difference between the tracking problem considered in the proof of the above theorem and the tracking problem considered in the proof of the result of [16] is that in this chapter, we have a sign indefinite quadratic cost function.
4.4
Time Invariant Set-Valued State Estimation
We now consider a time-invariant version of the set-valued state estimation problem. In this problem, we consider the following time-invariant uncertain system defined on the infinite time interval [0,00):
i( t) z(t) y(t)
Ax(t) + BIW(t) Kx(t); Gx(t)
+ B 2 u(t);
+ v(t)
(4.4.1)
where x(t) E Rn is the state. w(t) E RP and v(t) E Rl are the uncertainty inputs, u(t) E Rh is the control input, z(t) E Rq is the uncertainty output and y(t) E Rl is the measured output. Let Q = Q' > 0 and R = R' > 0 be given matrices associated with the system (4.4.1). Then the uncertainty inputs and initial conditions for this uncertain system are required to satisfy the following integral quadratic constraint:
1 8
(x(O) - xo)' N(x(O) - xo) ::; d +
is
+
(w(t)'Qw(t)
+ v(t)' Rv(t))dt
Ilz(t)11 2 dt.
(4.4.2)
Theorem 4.4.1 Consider the uncertain system {4.4.1} with weighting matrices Q = Q' > 0 and R = R' > 0 and suppose that the pair (A, B) is stabilizable. If the algebraic Riccati equatwn
AP + PA'
+ P[K'K - G'RG]P + BIQ-IB~ = 0 (4.4.3)
has a solutwn P > 0 such that the matrix [A' - [G'RC - K'K]P] is stable. then for any matrix N = N' > 0 such that N- 1 ::; P, the set Xs[xo,uoUI~,yo(-)I~,d] of all possible states x(s) at time s corresponding to a input-output pair [uo(-, yoU] is described by n
Xs[xo, uO(')I~, YO(')I~, d] =
R : (xs - x(s))' P(s)-l(x s - x(s)) ::; d + Ps [uoO, yoU]
Xs E {
}
4.4 Time Invariant Set-Valued State Estimation
69
where P(·) zs the solutwn of the Rzccatz dzjjerentwl equatwn
p(t) = AP(t) + P(t)A' + P(t)[K' K - C'RClP(t) + B1Q-l B~ with zmtial condztwn P(O)
.i:( t)
[A
= N-l, x(·)
+ P(t)[K' K
+P(t)C'RYo(t) wzth zmtzal condztwn x(O)
=
zs the solutwn to the state equations
- C'RCll x(t)
+ [P(t)KG + B2l uo(t)
( 4.4.4)
Xo and
Ps [uoO· Yo 0]
~
t [ II(Kx(t) + Guo(t))112-
io
] dt
(Cx(t) - yo(t))'R(Cx(t) - yo(t))
.
Moreover, PC) zs defined on [0,00) and has the property P(s) s -+ 00.
-+
P
as
Proof
The first part of the theorem follows directly from Theorem 4.3.1. Hence, to complete the proof of this theorem, it remains only to show that P(·) is defined on [0. 00) and has the property P (s) -+ Pass ---> 00. Using the fact that P > 0 is a solution to Riccati equation (4.4.3), it follows that P also satisfies the Riccati equation AP + P A'
+ P K' K P + Q =
0
where A = A - PC'RC and Q = PC' RC P + BQ-l B' ~ O. Furthermore. since the matrix [A' - [C'RC - K'K]P] is stable. it follows that the matrix A + PK' K is stable. From this, we can use the Strict Bounded Real Lemma (Lemma 2.3.4) to conclude that the matrix A = A - PC'RC is stable. Hence, the pair (C, A) must be detectable. Now using the fact that the pair (A, B) is stabilizable and the pair (C, A) is detectable, it follows from Theorem 4.1 of [159] and the remarks following that theorem, that PC) is defined on [0,00) and has the property P( s) ---> P as s ---> 00. D Remark
The above theorem shows that for a time-invariant uncertain system defined over an infinite time interval, the state estimator (4.4.4) converges asymptotically to a time-invariant state estimator. Furthermore, this timeinvariant state estimator can be constructed via the solution to an algebraic Riccati equation.
5 Discrete-Time Set-Valued State Estimation
5.1
Introduction
In this chapter, we consider the problem of set-valued state estimation for uncertain discrete-time systems. As in Chapter 4. the starting point for our approach is the deterministic interpretation of the discrete-time Kalman Filter given in [16]. In [16]. the Kalman Filter is shown to give a state estimate in the form of an ellipsoidal set of all possible states consistent with the given process measurements and a deterministic description of the noise. We extend the approach of [16] to consider discrete-time uncertain process models which have a deterministic description of the noise and uncertainty. This uncertainty description is referred to as the Sum Quadratic Constraint (SQC) uncertainty description and is the discrete-time version of the IQC uncertainty description considered in Chapter 4. As for the IQC uncertainty description, the SQC uncertainty description allows for a large class of nonlinear, dynamic uncertainties. Furthermore, our main result shows that for this uncertainty description, our robust filter can be used to determine if the assumed model is consistent with the given output measurements. Such model validation results cannot be obtained with a stochastic description of noise and uncertainty. One of the main results of this chapter is a set-valued state estimator which gives a state estimate in the form of an ellipsoidal set of all possible states consistent with the given process measurements and our deterministic description of the noise and uncertainty. In particular, this result gives an exact characterization of this set. As in the standard Kalman Filter, our
72
5. Discrete-Time Set-Valued State Estimation
state estimate is constructed recursively from the output measurements. However, the form of our state estimator equations is somewhat different from the standard Kalman Filter equations. The result on discrete-time set-valued state estimation presented in this chapter was originally presented in the papers [122,139]. Some other results on set-valued robust discrete-time filtering can be found in [86]. As well as considering the standard set-valued state estimation problem for discrete-time uncertain systems, this chapter also considers a set-valued state estimation problem for uncertain discrete-time systems for the case in which some of the measured data points are missing. In many practical filtering problems, there is a possibility that some of the observation data may be missing. This problem of missing data may arise from temporary sensor failure or congestion of the communications network connecting the sensors to the processor. The standard Kalman Filtering problem with missing measurement data has been considered by a number of previous authors: e.g., see [27.91]. In this chapter, we also consider a problem of robust Kalman Filtering which allows for missing measurement data. This result on discrete-time set-valued state estimation with missing data which is presented in this chapter originally appeared in the papers [130,137].
5.2
Sum Quadratic Constraint Uncertainty Description
Let T E {1. 2. 3, ... }. Consider the time-varying uncertain discrete-time system defined for k = 0.1. .... T:
x(k
z(k)
A(k)x(k) + Bdk)w(k); K(k)x(k);
y(k)
C(k)x(k)
+ 1)
+ v(k)
(5.2.1)
where x(k) ERn is the state, w(k) E RP and v(k) E Rl are the uncertainty inputs, z(k) E Rq is the uncertainty output, y(k) E Rl is the measured output, and A(k). Bl (k), K(k) and C(k) are given matrices such that A(k) is non-singular for k = 0.1, .... T. System Uncertainty
The uncertainty in the above system is described by an equation of the form: w(k) ] _ - q;(k.x(·)) [ v(k) where the following Sum Quadratic Constraint is satisfied. Let
N=N'>O
5.2 Sum Quadratic Constraint Uncertainty Description
73
be a given matrix, Xo E Rn be a given vector, d > 0 be a given constant, and Q(k) and R(k) be given positive-definite symmetric matrices defined for k = 0, L ... , T. Then we will consider the ullcertainty inputs w(-) and v(·) and initial conditions x(O) such that the following Sum Quadratic Constraint is satisfied:
(x(O) - xoY N(x(O) - xo) T-l
+L
(w(k)'Q(k)w(k) + tI(k + l}'R(k + l)t1(k + 1))
k=O T-l
~d+
L
Ilz(k + 1)112.
(5.2.2)
k=O
As in Chapter 4, this uncertainty description can be represented by the block diagram shown in Figure 4.2.1. Also, as in Chapter 4, the uncertain system (5.2.1), (5.2.2) allows for uncertainty satisfying a standard norm bound constraint. For example, consider an uncertain system described by the state equations
x(k
+ 1) y(k)
+ Bll (k)6.(k)K(k)]x(k) + Bdk)nl (k); C(k)x(k) + n2(k); 116.(k)'Q(k)~ I ::; 1 [A(k)
(5.2.3)
where 6.(k) is the uncertainty matrix, nl(k) and n2(k) are noise sequences, B 1 (k) = [B 11 (k) B 12 (k)], and II . II denotes the standard induced matrix norm. Also suppose, initial conditions and noise sequences satisfy the inequality
(x(O) - xo)' N(x(O) - xo) T-l
+L
+ x(O)' K(O)'Q(O)K(O)x(O) T
nl (k)'Q(k)nl (k)
k=O
+L
n2(k)' R(k)n2(k) ::; d.
k=1
To verify that such uncertainty is admissible for the uncertain system (5.2.1), (5.2.2), let
w(k) = [
6.(k)~~Z?x(k)
]
for k = 0,1, ... ,T and v(k) = n2(k) for k = 1,2,. ,. ,Twhere
116.(k)'Q(k)~ II ::; 1 for all k. Then condition (5.2.2) is satisfied. It is clear that the Sum Quadratic Constraint (5.2.2) is a discrete--time version of the Integral Quadratic Constraint (4.2.2).
74
5. Discrete-Time Set-Valued State Estimation
Set- Valued State Est2matwn Problem The first result of this chapter concerns the following state estimation problem. Let y(k) = yo(k) be a fixed measured output of the uncertain system (5.2.1), (52.2) for k = 1,2, .. ,T. Then, find the corresponding set XT[xo, Yo(-)Ii, d] of all possible states x(T) at time T for the system (5.2.1) with uncertainty inputs and initial conditions satisfying the constraint (5.2.2).
Definition 5.2.1 The system (5.2.1), (5.2.2) 2S smd to be strictly veriT fiable 2j the set XT[xo, Yo(-)1 1 ,d] 2S bounded jor any Xo, any Yo(')' and any d. Let yo(k), k = 1,2, ... ,T be a gwen output sequence. The output Yo(') 2S smd to be realizable 2j there exzst sequences [xO, w(·), vC)] satzsjymg conditions (5.2.1), (5.2.2) wzth y(k) = yo(k). In addition to solving the state estimation problem mentioned above. the main result of this chapter also solves the following problem: Given an output sequence Yo('), determine if this output is realizable for the uncertain system (5.2.1), (5.2.2). If a given measured output sequence is not realizable for the given uncertain system model, we can say that this model is invalidated by the measured data. Thus, the results of this chapter are useful in the question of model validation.
5.3
Design of Discrete-Time Set-Valued State Estimator
Our solution to the above state estimation problem involves the following Riccati difference equation:
F(k
+ 1)
[B1(k)'S(k)B1(k) + Q(k)] # ih(k)'S(k)A.(k),
S(k
+ 1)
A.(k)'S(k) [A.(k) - iJr(k)F(k + +C(k + 1)' R(k + l)C(k
1)]
+ 1) - K(k + I)' K(k + 1),
S(O) = N
where
(5.3.1)
A.(k) ~ A(k)-l, ih(k) ~ A(k)-lB1(k)
and 0# denotes the Moore - Penrose pseudo-inverse; e.g., see [2]. Solutions to this Riccati equation will be required to satisfy the following condition:
Bl(k)'S(k)Bl(k) + Q(k) N(ih(k)'S(k)ih(k) + Q(k))
> 0& c N(A.(k)'S(k)ih(k))
(5.3.2)
for k = 1,2, ... ,T. Here N(·) denotes the operation of taking the null space of a matrix.
5.3 Design of Discrete-Time Set-Valued State Estimator
75
Also. we consider a set of state equations of the form
[A(k) - Bl(k)F(k +
T)(k+1)
+G(k Nxo,
v(O) g(k + 1)
g(k)
1)], T)(k)
+ I)'R(k + l)yo(k + 1),
+ yo(k + 1)'R(k + I)Yo(k + 1)
-7)(k)' ih(k) [B1(k)' S(k)Bl(k) -+- Q(k)] # Bl(k)'1](k),
g(O)
=
x~Nxo.
(5.3.3)
Note that if the matrix ih(k)'S(k)B1(k) + Q(k) is positive-definite, then condition (5.3.2) holds automatically and the pseudo-inverse in (5.3.1) and (5.3.3) can be replaced by a normal matrix inverse. This situation will hold in almost all cases for which a suitable solution exists to Riccati equation
(53.1). Theorem 5.3.1 Gons~der the uncertam system (5.2.1), (5.2.2). Then the
followmg statements hold: ~s strzctly verzfiable if and only ~f there ex~sts a solutwn to Riccah equatwn (5.3.1) sat~sfymg cond~twn (5.3.2) and S(T) > O.
(z) The uncertam system (5.2.1), (5.2.2)
(n) Let yo(k), k = 1,2, ... ,T be a gwen output sequence and suppose the system (5.2.1), (5.2.2) ~s strzctly verzfiable. Then, Yo (-) ~s realzzable zf and only ~f PT(YO(-)) ;::- -d where PT(YO(-)) ~ 7)(T)'S(T)-l7)(T) - g(T) and 7)(T) and g(T) are defined by the equatwns (5.3.3). (iii) If the uncertam system (5.2.1), (5.2.2) is stnctly venfiable, then
XT[XO, yo(-)IJ, d] =
XT ERn: } II(S(T)h T - S(T)-~7)(T))112 . { :::; p(Yo(·)) + d (5.3.4)
Proof
Given an output sequence Yo (-) , we have by the definition of XT[xo, yo(· )IJ, d], that XT E XT[xo, Yo(-)IJ, d] if and only if there exist sequences xU, w(·) and v(·) satisfying equation (5.2.1) and such that x(T) = XT, the constraint (5.2.2) holds, and
yo(k) = G(k)x(k)
+ v(k).
76
5. Discrete-Time Set-Valued State Estimation'
Substitution of this into (5.2.2) implies that
if and only if there exists an input sequence w(-) such that J[XT' w(·)J ::; d where J[XT' we)J is defined by
(X(O) - xo)' N(x(O) - xo)
J[XT' w(-)J
+
T-l ( w(k)'Q(k)w(k) ) -x(k + 1)' K(k + I)' K(k + l)x(k +1) k=O +v(k + 1)' R(k + l)v(k + 1)
L
(5.3.5)
where v(k
+ 1) ~
yo(k
+ 1) -
C(k
+ l)x(k + 1)
and x(·) is the solution to (5.2.1) with input we-) and boundary condition x(T) = XT. Now suppose the uncertain system (5.2.1), (5.2.2) is strictly verifiable and consider the functional (5.3.5) with Xo = and Yo(-) == 0. In this case, J is a homogeneous quadratic functional with a terminal cost term. Also, consider the set XT[O,O, 1] corresponding to Xo = 0, Yo(-) == and d = 1. Since XT[O, 0, 1J is bounded, there exists a constant hT > such that all vectors XT E Rn with IlxT11 = hT do not belong to the set XT[O,O, 1J. Hence, J[XT,W(·)J > 1 for all XT E Rn such that IlxrII = hT and for all wO. Since. J is a homogeneous quadratic functional, we have
°
°°
J[aXT, awOJ = a2 J[XT, wO] and the condition, J[XT' w(·)] > 1 for for all XT i 0 where
IlxT11
= hT, implies that
m(xT) > 0
m(XT) ~ inf J[XT'W(')]' w(·)
The optilnization problem inf J[XT' w(·)]
w(·)
subject to the constraint defined by the system (5.2.1) is a linear quadratic optimal control problem in which time is reversed. In this linear quadratic optimal control problem, a sign indefinite quadratic cost function is being considered. Using a known result from linear quadratic optimal control theory, we conclude that the condition m(xT) > implies that there exists a solution to Riccati equation (5.3.1) satisfying condition (5.3.2) and SeT) > 0; e.g., see [30,76].
°
5.3 Design of Discrete-Time Set-Valued State Estimator
77
We have shown above that an output sequence YoU is realizable if and only if there exists a vector XT E R n and an uncertainty input u{) such that the condition J[XT. w(·)] :::; d holds. Now With J defined as in (5.3.5), consider the optimization problem inf J[XT, wO]·
w(-)
This problem is a linear quadratic optimal tracking problem in which the system operates in reverse time. In fact, the O~ly difference between this tracking problem and the tracking problem cO~sidered in [76] is here. we have a sign indefinite quadratic cost function and time is reversed. The solution to this tracking problem is well know!), (e.g., see [76]). Indeed, if there exists a solution to Riccati equation (5.3.1) satisfying (5.3.2) and S(T) > 0, then the infimum inf J[XT'W(')]
m(/
will be achieved for any Xo and Yo('); e.g., se~ [30,76]. Furthermore, as in [76], we can write
m(XT) = min J[XT' w(·)] = x~S(T)XT .~ 2x~7)(T) w()
+ g(T) (5.3.6)
where [7)(-),g(')] is the solution to state equaticms (5.3.3). However, since S(T) > 0, it follows that the set
is bounded. Since, xo, Yo(-), and d were arbitrar.'y, it follows that the uncertain system must be strictly verifiable. Conver:,\ely, we have proved above that if the uncertain system (5.2.1), (5.2.2) is strictly verifiable, then there exists a solution to Riccati equation (5.3.1) satiSfying (5.3.2) and S(T) > O. Thus, we have established statement (i). Now suppose the system is strictly verifiable. As above, it follows that there will exist a solution to Riccati equation (5.3.1) satisfying (5.3.2) and S(T) > O. Also, it is clear that the output yoU is realizable if and only if there exists a vector XT E Rn such that m(xT) :s: d. This and (5.3.6) imply that the realizability of Yo (.) is equivalent to the condition
PT(YO(')) 2: -d. Thus we have established statement (ii). Also, we have shown above that XT[xo,Yo(-)li',d] is the set of all XT E R n SUth that m(xT) :::; d. Then from (5.3.6), condition (5.3.4) follows immedial;ely. Thus, we have proved statement (iii). 0
78
5.4
5. Discrete-Time Set-Valued State Estimation
Discrete-Time Uncertain Systems with Missing Data
We now consider a problem of set-valued state estimation for uncertain discrete time systems which have missing measurement data. Consider the time-varying uncertain discrete-time system (5.2.1), (5.2.2). We suppose that the measurement sequence y(-) is incomplete. That is, let
be a given vector for k = 1,2, .... T such that M'(k) E {O, 1} for any i = 1. ... ,l and any k = 1, ... ,T. Then, the ith component y'(k) of the output vector y(k) is known if M'(k) = 1 and y'(k) is unknown if M'(k) = O. The matrix
M ~ [M(l)
M(2)
M(T) ]
is referred to as the incompleteness matrix for the system (5.2.1). Associated with the incompleteness matrix M, we also define two sequences of matrices E(k) and E(k) as follows: For each k, the matrix E(k) is defined to be the diagonal matrix whose diagonal elements are given by the elements of the vector M (k). To define the matrix E(k ), let S be the set of standard unit vectors e, in Rl such that !vItek) = O. Then the columns of the matrix E(k) are the unit vectors in the set S in numerical order. If ]\I(k) is a vector of ones, then E(k) is the zero vector. It follows from this definition that the range space of the matrix E(k) corresponds to the unknown elements of the output vector y(k).
Set- Valued State Estimatwn with Missing Data The set-valued state estimation problem with missing data is defined as follows: Let !II be a given incompleteness matrix and let y(k) = yo(k) be the output of the uncertain system (5.2.1), (5.2.2) for k = 1,2, .... T. Also, define a corresponding known output sequence Yo (k) such that Yo (k) = yO( k) if M'(k) = 1 and y'6(k) = 0 if M'(k) = O. That is. the known output sequence is obtained by setting unknown elements in the output vector to zero. It follows from this definition that we can write yo(k) = E(k)yo(k) where the matrix E(k) is defined as above. We consider the problem of finding the corresponding set XT[M, xo, Yo(') d] of all possible states x(T) at time T for the system (5.2.1) with the incompleteness matrix M and uncertainty inputs and initial conditions satisfying the constraint (5.2.2).
Ii,
Remark The above robust state estimation problem with missing data includes the problem of robust prediction as a special case; see also [87]. This follows if we consider the incompleteness matrix to be such that M(k) = 0 for
5.5 Design of a Set-Valued State Estimator with Missing Data
79
k = To + LTo + 2, ... ,T. Then, the set XT[M,xo,yoOli.d] gives the predicted set of possible states at time k = T given measurements up to time k = To. Definition 5.4.1 The system (5.2.1). (5.2.2) is said to be strictly verifiable with the incompleteness matrix M if the set XT[M,xo,yoOli,d]
is bounded for any xo, any known output sequence YoO and any d.
5.5
Design of a Set-Valued State Estimator with Missing Data
Our solution to the above set-valued state estimation problem with missing data involves the foUowing Riccati difference equation:
F(k
+ 1)
[B 1(k)'5(k)B 1(k)
S(k
+ 1)
A(k)'S(k) [A(k) - B1(k)F(k +
+ Q(k))", Bl(k)'5(k)A(k),
1)]
+C(k + 1)' E(k + I)R(k + l)E(k -K(k + 1)' K(k + 1), 5(0) = N
+ I)C(k + 1) (5.5.1)
where
R(k+l)
~
R(k+l)-R(k+l)E(k+l) x (E(k
+ 1)'R(k + I)E(k + 1)) -1 E(k + 1)'R(k + 1)
> O. If E(k+ 1) = 0, we let R(k + 1) = R(k + 1). Note, (.)# denotes the MoorePenrose pseudo-inverse. Solutions to this Riccati equation will be required to satisfy the following condition:
+ Q(k) > + Q(k)) c
Bl(k)'5(k)B1(k) N(Bl(k)'5(k)B1(k) for k = 1. 2, . .. . T. Here space of a matrix.
N (.) denotes the
0&
N(4(k)'5(k)B1(k))
(5.5.2)
0P~ration of taking the null
80
5. Discrete-Time Set-Valued State Estimation
Also, we consider a set of state equations of the form
7](k+I)
=
[liCk) - Bl(k)F(k
+
1)]' 7](k)
+C(k + I)' E(k + I)R(k + 1)yo(k + 1),
v(O)
Nxo,
g(k + 1)
g(k) + yo(k + 1)'R(k + I)Yo(k + 1) -7](k)'Bl(k) [B1(k)'S(k)B1(k)
g(O)
=
+ Q(k)t Bl(k)'7](k),
x~Nxo.
(5.5.3)
Note that if the matrix Bl(k)'S(k)Bdk) + Q(k) is positive-definite, then condition (5.5.2) holds automatically and the pseudo-inverse in (5.5.1) and (5.5.3) can be replaced by a normal matrix inverse. This situation will hold in almost all cases for which a suitable solution exists to Riccati equation (5.5.1).
Theorem 5.5.1 Cons2der the uncertam system (5.2.1), (5.2.2) wdh m2SSmg data and let Iv! be a gwen mcompleteness matT'tX. ThpTl the followmg statements hold: (z) The uncertam system (5.2.1), (5.2.2) zs strzctly verzjiable wzth the mcompleteness matrzx M zf and only 2f there exzsts a solutzon to R2ccati dziJerence equatwn (5.5.1) satzsfymg condztzon (5.5.2) and SeT) > O. (n) If the unceriam system (5.2.1), (5.2.2) 2S strzctly verzjiable wzth the mcompleteness matrix M, then
where T)(T) and geT) are defined by the equatzons (5.5.3). Proof
Given a known output sequence yo(·), we have by the definition of the set
XT[M, Xo, yoOli, d], that XT E XT[M.xo,YoOli,dj if and only if there exist sequences YO('), xC,), w(-) and v(-) satisfying equation (5.2.1) and such that x(T) = XT, the constraint (5.2.2) and
Yo(k) = E(k )yo( k). Now since
yo(k) = E(k)[C(k)x(k)
+ v(k)],
5.5 Design of a Set-Valued State Estua.tor"with Missing Data
81
it follows that we can write
v(k)
= Ya(k) - E(k)C(k)x(k) + E(k)J-l(k)
where the vector J-l(k) is arbitrary. Substitution of this into (5.2.2) implies that XT E XT[M, Xa, Ya(')li, d] if and only if there exists an input sequence w(·) and a sequence J-lC) such that
(x(O) - xa)' N(x(O) - xa) T-l (
+L
k=a
w(k)'Q(k)w(k) -x(k + 1)' K(k + 1)' K(I v(k)' R(k + l)v(k)
+ l)x(k + 1)
) :::; d (5.5.5)
where
v(k) ~ Ya(k
+ 1) -
E(k
+ I)C(k + l)x(k + 1) + E(k + 1)J-l(k + 1).
Minimizing the left hand side of this inequality with respect to J-l(k + 1), it then follows that XT E XT[M, Xa, YaOli, d] if and only if there exists an input sequence w(·) such that J[XT, wO] :::; d where J[XT' wO] is defined by il.
J[XT' w(·)] = (x(O) - xa)' N(x(O) - xa) T-l (
+L
k=a
w(k)'Q(k)w(k) - x(k + 1)'K(k + 1)'K(k + l)x(k + 1) ) + [jJo(k + 1) - E(k + I)C(k + l)x(k + 1)]' xR(k + 1) [Yo(k + 1) - E(k + I)C(k + l)x(k + 1)]
and x(·) is the solution to (5.2.1) with input w(·) and boundary condition x(T) = XT. The remainder of the proof of this theorem is similar to the proof of Theorem 5.3.1 in Section 5.3. 0
Remark It should be noted that the Riccati difference equation (5.5.1) is such that to determine S(k), the matrices E(k) and E(k) are required. These matrices will depend on which data are missing at time k. Thus. in order to operate our robust state estimator in real time, the Riccati difference equation (5.5.1) together with the state equations (5.5.3) must be solved on-line. The coefficients in equations (5.5.1) and (5.5.3) would then be adjusted depending on which measurements were available at time step k.
82
5.6
5. Discrete-Time Set-Valued State Estimation
A Robust Deconvolution Problem
To illustrate the results of this chapter on set-valued state estimation for discrete time uncertain systems. we now consider a robust deconvolution problem similar to those considered in [28]. However, we use a different uncertainty description than was considered in [28]. A block diagram of the system under consideration is shown if Figure 5.6.1. In this robust deconvo-
Signal Model
Channel Model
u(k)
+
y(k)
0.4 z-0.2
0.707
+
State Estimator
0.0127
u(k) FIGURE 5.6.1. Robust deconvolution system.
lution problem, the uncertain parameter ~(k) represents the uncertainty in the signal model natural frequency. Combining the signal model and the channel model, we obtain the following uncertain system of the form (5.2.3) :
(5.6.1) where T
T-l
(10 + 0.0127 )llx(0)11 2
2
+L
nl(k)2
and the uncertain parameter
~(k)
+L
n2(k)2 ~ 1
(5.6.2)
k=l
k=O
satisfies
11~(k)11
:::; 1.
(5.6.3)
5.6 A Robust Detonvolution Problem
83
In this state space description, xI(k) and x2(k) are the state variables of the signal model. The required signal u(k) cotresponds to xI(k). Also, x3(k) is the state variable of the channel model. We consider this system over a finite time interval of T = 100 samples. To apply our results to this deconvolution problem, we consider a corresponding uncertain system of the form (5.2.1) in which the uncertainty satisfies the sum quadratic constraint. In this case. the matrices A, N, B I , XO, K. C, Q, and Rare given by
A
c
[ [
1.98 1 0.4
-1 0 0
0
N
= [ 010 10
0.2
0
0
~o ]
0707] ] o : Xo = [ 0 0
0.707 0 0
[a a
o o] :
o
I
]:
0 K
= [0.0127
a a 1:
Q = I and R = I. (5.6.4)
Also, the constant d is given by d = 1. To illustrate the performance of our state estimator, we consider the uncertainties and noise signals to be such that D,.(k) == 1, nI (0) = 0.5. nICk) = 0 for k = 1,2, ... ,100, and 1
.
n2(k) = 105 sm(k/10) for k = 1, 2.. .. 100. With the initial condition :teO) ward to verify that the uncertainty input sequences
= 0, it is straightfor-
w(k) = [x(k)'K'D,.' nl(k)']' and v(k) = n2(k) satisfy the sum quadratic constraint (5.2.2). We now apply our state estimator to the linear system corresponding to this uncertainty realization. Figure 5.6.2 shows the resulting estimate of the signal u(k). upper and lower bounds on u(k), and the true value of u(k) for k = 1, 2, .... 100. The estimated value of the st''1te vector corresponds to the center of the ellipsoid of possible states desctibed by equation (5.3.4). Indeed, referring to equation (5.3.4), the state estimate at time k is given by i(k) = S(k)-IT}(k). In this example, the required estimate of the signal u( k) corresponds to the first component of this estimated value of the state vector. Also, the upper and lower bounds on the signal u(k) are obtained by projecting the ellipsoidal set of possible values of the state vector onto its first component. In addition to the uncertainty realization described above we also considered another uncertainty realization in which th~~ value of D,. was replaced by D,. = -1. Apart from this change. which corre:sponds to a change in the
81
5. Discrete-Time Set-Valued State Estimation
-
true value of u(k)
- - estimated value of u(k) -
10
upper bound on u(k) lower bound on u(k)
,
,
\
I \
,
\
,
o
-5
o
10
20
30
40
50 time step
60
70
80
90
100
FIGURE: 5 .6.2. Estimated value of u(k) with D. = 1.
natural frequency of the signal model , the uncertainty realization is the same as above. Figure 5.6.3 shows the corresponding simulations for this case. Note that in both cases, a good estimate is obtained for the actual signal u(k) in spite of large uncertamty in the signal model.
5.7
A Robust Deconvolution Problem with Missing
Data To illustrate the results of this chapter on set-valued state estimation with missing data, we consider a robust deconvolution problem similar to that considered in Section 5.6. However, unlike Section 5.6, we will assume that over the time interv;:tl of interest, there exist periods of missing data. The block diagram of the system under consideration remains as shown in Figure 5.6.l. In this robust deconvolution problem, the uncertain parameter 6.(k) represents the uncertainty in the signal model natural frequency. Combining the signal model and the channel model. we obtain the uncertain system (5.6.1) , (5 .6.2) , (5.6.3). This is an uncertain system of the form (5.2.3). We consider this system over a finite time interval of T = 100 samples. Furthermore, we assume that the measured signal y( k) is unavailable to the state estimator during the time periods:
k E {5, 6, ... ,10,31,32, .. . ,35,76,77, ... ,SO}.
5.7 A Robust Deconvolution Problem with Missing
DMa
85
15'---~---'----'----'----'----'----'----r---''---,
-
true value of u(k)
- - estimated value of u(k)
10
- . upper bound on u(k) lower bound on u(k)
~
'5"
5
/
\
\
I
0;
,
\
C
0>
,
iii
-5
o
10
20
30
40
60
50
70
80
90
100
time step
FIGURE 5.6.3. Estimated value of u(k) with Do = -1.
To apply our results to this deconvolution problem, we consider a corresponding uncertain system of the form (5.2.1) in which the uncertainty satisfies the sum quadratic constraint. In this case. the matrices A, N, B I , Xc, K, C, Q, and R are given by (5.6.4). Also, the constant d is given by d = 1. The incompleteness matrix corresponding to this problem is a row vector M = [M(l) M(2) ... M(lOO)] where M(k) = 0 for
k
E
{5, 6, ... ,10,31. 32, ... ,35,76.77, ... ,SO}
and M(k) = 1 otherwise. Corresponding to this matrix M, the matrices E(k) and E(k) are as defined in Section 5.4. To illustrate the performance of our state estimator, we consider the uncertainties and noise signals to be such that 6;.(k) == 1, nI(O) = 0.5, nICk) = 0 for k = 1,2, ... ,100, and
1
.
n2(k) = 105 sm(k/10) for k = 1,2, ... ,100. With the initial condition x(O) = 0, it is straightforward to verify that the uncertainty input sequences
w(k) = [x(k)'K'6;.' nI(k)']'
86
5 Discrete-Time Set-Valued State Estimation
and v(k) = n2(k) satisfy the sum quadratic constraint (5.2.2). We now apply our state estimator to the linear system corresponding to this uncertainty realization. Figure 5.7.1 shows the resulting estimate of the signal u(k), upper and lower bounds onu(k), and the true value of u(k) for k = 1. 2 .... ,100. The estimated value of the state vector corresponds to the center of the ellipsoid of possible states described by equation (5.5.4). Indeed. referring to equation (5.5.4), the state estimate at time k is given by x(k) = S(k)-l7](k). In this example, the required estimate of the signal u(k) corresponds to the first component of this estimated value of the state vector Also, the upper and lower bounds on the signal u(k) are obtained by projecting the ellipsoidal set of possible values of the state vector onto its first component. In addition to the uncertainty realization described above. we also considered another uncertainty realization in which the value of ~ was replaced by ,6. = -1. Apart from this change, which corresponds to a change in the natural frequency of the signal model. the uncertainty realization is the same as above. Figure 5.7.2 shows the corresponding simulations for this case. Note that in both cases, a good estimate is obtained for the actual signal u( k) in spite of large uncertainty in the signal model. 30 -
~5
true value of u(k)
- - est, mated value of u(k) - - upper bound on u(k)
gO
lower bound on u(k) L -_ _ _ _ _ _ _ _
15
,
I
I,
10
I
2 :J Cii c:
,
I I
5
I,
~
,,
I' I
I I
'I
,
/
,
,
Ol
en
Q
-5 -10
-15 _20L--~-~--~--L--L--~-~--~--L-~ ~ ~ 70 80 90 100
o
ro
w
W
W
time step
FIGURE 57.1. Estimated value of u(k) with ll.
=
1.
5.7 A Robust Deconvolution Problem with Missing Data
30
25
-
true value of u(k)
- - estimated value of u(k)
.~
- - upper bound on u(k) lower bound on u(k)
1&
?
,I
10
I
'I
'5 Cii c en
, ,
,
5
in
r
0
,
I /
I
I,
I,
-.$ -lit;) -~I~
_20L-__- L_ _ _ _L -_ _- L_ _ _ _L -_ _- L_ _ _ _L -_ _- L_ _ _ _ o 10 20 30 40 50 60 70 80
~
_ _ ~_ _~
90
time step
FIGURE 5.7.2. EstImated value of u(k) with .6.
= -1.
100
87
Robust State Estimation with Discrete and Continuous Measurements
6.1
Introduction
In many filtering and state estimation problems, the process being considered has a continuous-time model but the measured output data are available only at discrete sampling instants. Hence, we are motivated to consider "hybrid" problems of problems of state estimation and model validation in which the underlying process model is continuous but the available measured data are available at discrete sampling times. Furthermore, we extend this problem to allow for the case when some of the outputs can be measured continuously and other outputs can only be measured at discrete sampling instants. Such a situation may arise in complex hybrid processes involving both digital and analog blocks. Also, there may arise situations in which some of the sensors supply data at a very fast rate (which can be approximated as a continuous-time signal) whereas other sensors supply data at a slower sample rate. In this case, it is important that the signal processing algorithm be able to simultaneously handle both types of measured data. The main results of this chapter extend the setvalued state estimation and model validation results of Chapters 4 and 5 to allow for hybrid discrete-continuous data. The results of this chapter originally appeared in the papers [125,128,138]. Another issue which arises in many real time signal processing problems is the issue of missing data. As mentioned in Chapter 5, this issue may arise when the connection between the sensors collecting the data and the computer processing the data involves computer networks which are subject
90
6. State Estimation with Discrete and Continuous Measurements
to overloads. This issue may also arise when the connection between the sensors and the computer processing the data involves communications channels which are subject to frequent loss of data packets. In both cases, it is important that the signal processing algorithm being used have the capability to cope with frequent loss of measurement data. The issue of missing data was addressed in the purely discrete-time case in Chapter 5. In this chapter, we will again address this issue but for the case of discretecontinuous data. ThE' results of this chapter consider a number of signal processing problems in which the above practical problems are addressed; that is, missing measurement data, continuous and discrete measurements. and robustness against model uncertainties. The main results of this chapter extend the results of Chapters 4 and 5 to allow for a more general information structure. In particular. we allow both continuous and discrete data. Since the results of this chapter consider the case of both continuoustime and discrete-time outputs, our results contain as a special case, the sampled-data robust state estimation problem. This problem is related to the sampled-data H= filtering problem considered in [150]. Also. note that the results of this chapter enable us to consider a sampled-data robust state estimation problem with missing measurement data. In this chapter, we again consider the model validation problem for uncertain systems. The model validation problem for uncertain systems was previously considered in Chapters 4 and 5; see also the papers [107.143.144. 171]. However, we now consider the model validation problem in a hybrid situation with a continuous-time model and some discrete-time measurements. The uncertain systems considered in this chapter are uncertain systems in which the uncertainty is described by a sum-integral quadratic constraint. This uncertainty description is a generalization of the integral quadratic constraint uncertainty description given in Chapter 4 and the sum quadratic constraint uncertainty description given in Chapter 5. The robust state estimation and model validation results of this chapter involve the solution of a jump Riccati differential equation and the solution of a set of jump state equations. The jump Riccati equation being considered behaves like a standard Riccati differential equation between the sampling instants. However at the sample times, its solution exhibits finite jumps. Also. the jump state equation being considered exhibits similar jump behavior. The solution to these equations leads to a recursive solution to the state estimation and model validation problems under consideration. These recursive calculations are closely related to the recursive calculations in the standard Kalman Filter. It should be noted that the coefficients in the jump Riccati differential equation will depend on which measurements are available at any given time instant. However. the jump Riccati differential equation can be solved on-line while the data are being processed. Hence the results of this chapter
6.2 Uncertain Systems with Discrete and Continuous Measurements
91
can be used to give an on-line robust state estimate for the case of missing data and discrete-continuous measurements. Also, these results can be applied to the problem of model validation for uncertam system. This will be useful in verifying uncertain system models for later use in robust controller design.
6.2
Uncertain Systems with Discrete and Continuous Measurements
Let Nd be a gIven integer and let 0 < tl < t2 < ... < tNd < T be given sample times. Consider the following time-varying uncertain system defined on the finite time interval [0. T]:
x(t)
zc(t) Zd( t J ) Ye(t) Yd( t J )
A(t)x(t) + BI(t)W(t) + B2(t)U(t); Ke(t)x(t) + Ce(t)u(t). Kd(tJ)x(t J ) +Cd(tJ)u(t J ) \/J = 1,2, ... ,Nd; Ce(t)x(t) + vc(t); Cd(tJ)X(t J ) +Vd(t J ) \/J = 1,2, ... ,Nd
(6.2.1)
where x(t) E R n is the state, wet) E RP, ve(t) E Rk and Vd(t) E Rl are the uncertamty mputs, u(t) E Rh is the control mput, zc(t) E Rq is the contmuous uncertamty output, Ye(t) E Rk is the contmuous measured output, Zd(t J ) E RT is the dzscrete unceriamty output, Yd(t J ) E Rl is the dzscrete measured output. A(-). Bl(-)' B 2(-), K e(-), CeO. and C e(-) are bounded piecewise continuous matrix functions, Kd(')' Cd(')' and CdC) are matrix sequences
System Uncertamty The uncertainty in the above system is required to satisfy the following Sum-Integral Quadratic Constraint (SIQC). Let N = N' > 0 be a given matrix, Xo E Rn be a given vector. d > 0 be a given constant. RdO = Rd( )' 2': 0 be given matrix weighting sequence. QO = Q(-)' and ReO = R e(·)' 2': 0 be given bounded piecewise continuous matrix weighting functions satisfying the following condition: there exists a constant J > 0 such that Q(t) 2': JI for all t. For a given finite time interval [0. s]. we will consider the uncertainty inputs w(·), vJ) and Vd(-) and initial conditions
92
6. State Estimation with Discrete and Continuous Measurements
x(o) such that (x(O) - xo)'N(x(O) - xo)
+L
+ is (w(t)'Q(t)w(t) + vc(t)'Rc(t)Vc(t))dt
Vd(t J )' Rd(tJ)Vd(t J ) (6.2.2)
Notatwn Let u(t) = uO(t) be a fixed control input, yc(t) = y~(t) and Yd(t J ) = y~(tJ) be fixed measured outputs of the uncertain system (6.2.1). and let the finite time interval [0, s] be given. Then,
denotes the set of all possible states x( s) at time s for the uncertain system (6.2.1) with uncertainty inputs and initial conditions satisfying the constraint (6.2.2). The following definitions extend corresponding definitions in Chapters 4 and 5 to the case of discrete-continuous measurements. Definition 6.2.1 The d2screte-contmuous uncertam system (6.2.1), (6.2.2) 2S sa2d to be strictly verifiable on [0, TJ, 2f for any vector Xo E Rn, t2me s E (0, TJ, constant d > 0, control mput u(t) = uO(t), and measured outputs yc(t) = y~(t) and Yd(t J ) = y~(tJ)' the set Xs[xo, uO(')I~, y~OI~, y~OI~, d] is
bounded. State Esttmatwn Problem The set-valued state estImation problem considered in this chapter is the problem of constructing the set
corresponding to given input and output measurements u(t) = uO(t), Yc(t) y~(t) and Yd(t J ) = y~(tJ)' Definition 6.2.2 Let Xo E R n and d
=
> 0 be gwen. Also, let uOO and
y~(-) be gwen vector functwns defined over a gwen t2me mterval [0, s] and let y~(tJ) be a gwen vector sequence defined for t J ::; s. The mput - output tnple [uO(.),y~( ),y~(.)] 2S sa2d to be realizable w2th parameters Xo and
d 2f there eX2st [x(·), w(-), vcO, Vd(')] sat2sfymg condztwns (6.2.1), (6.2.2) wzth u(t) = UO(t),yc(t) = y~(t) and Yd(t J ) = y~(tJ)'
6.3 Design of a Hybrid Set-Valued State Estimator
93
Model Val2datwn Problem The uncertain system model validation problem considered in this chapter is as follows: Given an input - output triple
determine if this triple is realizable for the uncertain system (6.2.1), (6.2.2).
Notatwn Let get) be a given matrix or vector function which is right continuous and may be left discontinuous Then, g(t~) denotes the value of get) just before t J , i.e.,
assuming that the limit exists.
6.3
Design of a Hybrid Set-Valued State Estimator
Our solution to the above set-valued state estimation and model validation problems involve the following Riccati differential equation which contains jumps in its solution:
Pet)
= A(t)P(t) + P(t)A(t)' + B1(t)Q(t)-1 BI(t)' +P(t)[Ke(t)' Ke(t) - Ce(t)' Re(t)Ce(t)]P(t) jor t oJ t J :
pet))
=
[P(t~)-l + Cd(t))'Rd(t))Cd(t)) - Kd(t))' Kd(t)) for J = 1,2 ..... N d.
]-1 (6.3.1)
Such a Riccati equation is referred to as a Jump R2ccati equatwn Also, we consider a set of state equations:
£(t)
=
[A(t)
+ P(t)[Ke(t)'Ke(t) - Ce(t)'Re(t)Ce(t)] ] x(t)
+P(t)Ce(t)' Re(t)y~(t) +[P(t)Ke(t)'Ce(t) + B2(t)]UO(t) for t oJ t): x(t))
+ P(t~) [ Kd(t))'Kd(t)) - Cd(t))'Rd(t))Cd(t)) ] x(t~) + P( t~ )Cd(t))' Rd(t) )y~(t)) + P( t~ )Kd( t))' C d(t) )uO (t))
x(t~)
for J = 1,2, ... ,Nd' Such a state equation is referred to as a Jump state equatwn.
(6.3.2)
94
6 State Estimation with Discrete and Continuous Measurements
The followmg theorem gives a solutlOn to the problems of model validation and set-valued state estimation in the case of discrete-continuous data.
°
°°
Theorem 6.3.1 Let N = N' > be a gwen matllx, Rd(t J ) = Rd(t J )' :::;, be a gwen matllx sequence and Q(.) = QU' and Rc() = Rc( )' :::;, be gwen matllx functzons such that the condztzon Q(t) :::;, M holds on the tune mterval [0, T]. Also, conszder the unceTtam system (6.2.1), (6.2.2). Then the followmg statements hold:
M The
system (6.2.1). (6.2.2) zs stllctly vellfiable on [0, T] zf and only zf the solutzon P(·) to the Jump Rzccatz equatzon (6.3.1) wzth zmtzal conddzon P(O) = N- I exzsts and zs posztwe-defimte on the mterval [O,T].
(ii) Suppose the system (6.2.1), (6.2.2) zs stllctly vellfiable on [0, T] Also, let s E (0, T] be gwen and let Xo E Rn be a gwen vector, d > 0 be a gwen constant, y~(tJ) be a gwen vector sequence defined for t J :s: sand uO(t) and y~(t) be gwen vector junctzons defined on [O,s]. Then, the tllple [uO( ),y~( ),y~()] zs realzzable zf and only zf Ps[uOU,y~( ),y~(-)]:::;' -d where Ps[u°(-), y~(-), y~(-)] ~
r[
Jo
II(Kc(t)x(t) + G c (t)uO(t))112 -(Cc(t)x(t) - y~(t))' Rc(t)(Cc(t)x(t) -
y~(t))
] dt
+ '"' [ II(Kd(t J )x(tJ ) + Gd(tJ)U°(tJ)) 112 L
t]s,s
-(Cd(tJ)x(t J ) - y~(tJ))' Rd(tJ)(Cd(tJ)X(tJ) - y~(tJ))
]
(6.3 3)
and x(-) zs defined by the Jump state equatzon (6.3.2) wzth mztzal condztzon x(O) = Xo.
(iii) If the unceTtam system (6.2.1), (6.2.2) zs stllctly vellfiable, then
(6.3.4)
Proof (z) Necessity In this case, we must establish the existence of a positivedefinite solution to the jump Riccati equation (6.3.1). This wIll be achieved by showing that the cost function in a corresponding linear quadratic optimal control problem is bounded from below.
6.3 Design of a Hybrid Set-Valued State Estimator
95
First let s E (0, T] be given and consider the system (6.2.1), (6.2.2) defined on [0, s]. Given rune), y~(-), Y~(-)]' we have by definition that
Xs
E
Xs[xo, uO(')I~, Y~(-)I~, Y~(')I~, d]
if and only if there exist signals xC), w(·). v e (,), and Vd(') satisfying equation (6.2.1) and such that xes) = x" the constraint (6.2.2) holds, and (6.3.5) and y~(tJ)
= Cd(t J ) + Vd(t J ) 'ItJ :s; s.
(6.3.6)
Substitution of (6.3.5) and (6.3.6) into (6.2.2) implies that
Xs
E
Xs[xo, uO(')I~, Y~c)I~, Y~c)I~, d]
if and only if there exists an uncertainty input w(·) E L2 [0, s] such that (6.3.7) where J[xs' wO] is defined by
J[x s , w(·)]
~
(x(O) - xo)' N(x(O) - xu)
1 s
+
w'Qw -11(Kcx
(
+ '" ( L
tJ
:Ss
+ Gc uO)112 + (y~ - Ccx)'Rc(Y~ - Ccx) ) dt
(y~(tJ) - Cd(tJ)x(tJ»)'Rd(tJ)(y~(tJ) - Cd(tJ)x(t J)) ) -11(Kd(tJ)X(t J ) + Gd(tJ)uO(tJ»112 (6.3.8)
and x(·) is the solution to (6.2.1) with uncertainty input w(·) and boundary condition xes) = Xs· Now let E > 0 be given such that E < min{tl' t2 - tl,'" . tNd - tNd-d, and let g(t J ) be a given matrix or vector sequence. Then, ge(.) denotes a corresponding function defined so that
and
ge(t)
=0
for all other t E [0, s].
Using this notation, we introduce the following uncertain system with purely continuous outputs:
x( t)
ze(t) YE (t)
A(t)x(t) + Bl(t)W(t) + B2(t)U(t); KE(t)X(t) + GE(t)u(t); CE(t)x(t) + vE(t)
(6.3.9)
where
CE(t)
~
GE(t) Here the uncertainty inputs w(·) and VEe) and initial conditions x(o) satisfy the following integral quadratic constraint:
(x(o) - xo)' N(x(O) - xo)
+ las (w(t)'Q(t)w(t) + vE(t)' RE(t)vE(t))dt
<::: d + las IlzE(t)112dt.
(6.3.10)
Let [u°(-). y~(-), y~(-)] be an input - output triple for the system (6.2.1) and introduce the corresponding input - output pair [uOO, y?O] for the system (6.3.9). where
°( ) =
YE
Do [
•
y~ (.) y~E ( . )
]
(6.3.11)
.
Also. introduce the corresponding cost functional
JE[X w(·)] S '
~
(x(O) - xo)' N(x(O) - xo)
+ las (w'Qw -11(KEx + G uo)11 2 + (y~ - C£X)'RE(Y~ E
G€x)) dt (6.3.12)
where xC) is the solution to (6.3.9) with uncertainty input w(·) and boundary condition x(s) = Xs. Then, it follows immediately that for any Xs and wC), we obtain the following limit (6.3.13) uniformly in Xs. Now consider the functional (6.3.8) with Xo = 0, d = 1, u°(-) == 0, y~O == and y~(-) == 0. In this case. J is a homogeneous quadratic functional with a terminal cost term. Also. consider the corresponding set X s [O,O,O.O,l]. Since Xs [0.0,0,0.1] is bounded. there exists a constant hs > such that all vectors Xs E Rn with Ilxs II = hs do not belong to the set Xs [0. 0, 0, 0. 1]. Hence,
°
°
(6.3.14)
6.3 Design of a Hybrid Set-Valued State Estimator
97
for all Xs ERn such that Ilxsll = hs and for all wO E L 2 [0, s]. Since, J is a homogeneous quadratic functionaL we have J[axs, aw(·)] = a 2 J[xs, w(·)] and (6.3.14) implies that inf
w( )EL 2 [0,sj
J[xg, wO] >
°
(6.3.15)
for all s E [0, T] and all Xs oF 0 From (6.3.15) and (6.3.13), it follows that there exists a constant EO > 0 such that
JE[X S , w(·)] > 0
mf w( )EL 2
(6.3.16)
[0,sj
for all E < EO. S E [0, T] and all Xs oF O. The optimization problem (6.3.16) with an unconstrained terminal condition x(s) oF 0 and subject to the constraint defined by the system (6.3.9) is a linear quadratic optimal control problem in which time is reversed. In this linear quadratic optimal control problem, a sign indefinite quadratic cost function is being considered. We now use a known result from linear quadratic optimal control theory which states that if the infimum of the cost function is strictly positive for all terminal conditions, then there exists a solution to the corresponding Riccati equation. Furthermore, the terminal value of the Riccatl solution is positive-definite; e.g., see page 23 of [30]. Thus, we conclude that condition (6.3.15) implies that for all E < EO, there exists a solution X E ( . ) to the Riccati equation
- Xes)
X.(s )A(s) + A(s)' X.(s) + X.(s )B(s )Q(s)-l B(s)' X.(s) +KE(S)' KE(S) - GE(s)' RE(S)G,(s) (6.3.17)
with initial condition X,(O) = N. Furthermore. since (6.3.16) holds for any terminal time s E [0. TJ, this solution is positive-definite on [0, T]. Now the infimum in (6.3.16) is achieved and equal to x~X,(s)xs (see, e.g., [76]). From this and (6.3.13). it follows that the limit Xes) ~ limE ...... o X,es) exists and is positive-definite on [0, T]. Indeed, it follows from (6.3.13) that
as f -> O. Also, note that since X,es) > 0 for all E, it follows that the quantity lim x~X,(s)xs = inf J[x s . wO] ' ...... 0
w( )EL 2 [0,sj
is bounded from below and hence the limit exists. Furthermore. since the set
{Xs ERn : x~X(s)xs :S I} = {Xs ERn:
inf w( )EL 2 [0.sj
J[xs, w(.)] :S I}
98
6. State Estimation with Discrete and Continuous Measurements
is bounded. then we must have Xes) > O. Now observe that Xes) satisfies the following jump Riccati equation
- Xes)
X(tj)
X(s)A(s) + A(s)' Xes) +X(s)B(s)Q(S)-1 B(s)' Xes) -Cc(s)'Rc(s)Cc(s) s i= t J:
+ Kc(s)' Kc(s)
X(t;) - Kd(tJ)'Kd(t J ) + Cd(tJ)'Rd(tJ)Cd(t J ) J = 1,2, ... ,Nd (6.3.18)
with initial condition X(O)
= N. From this, it follows that the required
solution to jump Riccati equation (6.3.1) is given by P(·) ~ X(-)-I. This completes the proof of this part of the theorem. (z) Sufjiczency For a time interval [0, s], we have shown above that
E L 2 [0, s] such that condition (6.3.7) holds for the functional (6.3.8). Since there exists a solution P(·) to the jump Riccati equation (6.3.1) with initial condition P(O) = N-l, then X(·) = p(-)-I is a solution to (6.3.18) with initial condition X(O) = N. Using continuity, it follows that there exists a constant EO > 0 such that for any 0 < I' < EO. there exists the solution X E (·) to (6.3.17) with XE(O) = Nand
if and only if there exists an uncertainty input w(·)
lim XE(s)
E->O
= Xes) 'is
E
[0. T].
(6.3.19)
Now consider the uncertain system (6.3.9) with the integral quadratic constraint (6.3.lO). Also, consider the following minimization problem min w( )EL 2 [0.s]
JE[X S' w(·)]
(6.3.20)
where the functional JE[X S, wO] is defined by (6.3.12), (6.3.11) and the minimum is taken over all x(-) and wC) connected by (6.3.9) with the boundary condition x( s) = Xs. This problem is a linear quadratic optimal tracking problem in which the system operates in reverse time. We wish to convert the above tracking problem into a standard tracking problem such as considered in [76] and [16]. First define XI(t) to be the solution to the state equations (6.3.21) Now let x(t) ~ x(t) - XI(t). Then, it follows from (6.3.9) and (6.3.21) that x(t) satisfies the state equations
x(t) = A(t)x(t)
+ Bl(t)W(t)
(6.3.22)
6.3 Design of a Hybrid Set-Valued State Estimator
where X(O) written as
= x(O).
J,[X s ' w(·)J
99
Furthermore. the cost function (6.3.12) can be re-
=
J,[x s ' w(-)J
=
(x(O) - xo)' N(x(O) - xo)
t (
+ )0
w'Qw - II (K,[x + XIJ + G, uO)11 2 +(y~ - Gf[x + Xl])' R,(y~ - Gf[x
)
+ Xl])
d t
(6.3.23) where xes) = Xs = Xs - XI(S). Equations (6.3.22) and (6.3.23) now define a tracking problem of the form considered in [76]) where y~(-) , u°(-) and Xl (.) are all treated as reference inputs. In fact , the only difference between this tracking problem and the tracking problem considered in the proof of the result of [16J is that in our case. we have a s2gn indefimte quadratic cost function . The solution to this tracking problem is well known (e.g .. see [76)). Indeed , if the Riccati equation (6.3.17) has a positive-definite solution defined on [0. T] with initial condition X,(O) = X o , thel1 the minimum in
J,[X., w(-) j
min w ( )EL 2 [0,s]
will be achieved for any xo. uO(.) and y~(-) . Furthermore. as in [16], we can write
Jf[X S , w(-)J
min w( )EL 2 [0,s]
(xs - X1(S))'X,(s)(xs - X1(S)) - p~[u°(-) , y~(-)J
=
(6.3.24) where
p~[u°(-). y~( ' )J
t ( II(K, [XI + Xl] + G,uO)11
~
2
)
-(G, [xl+xd - y~)'R,(C, rXl+xd-y~)
-)0
dt
and X1(S) is the solution to state equations
=
(
A (s)
+ X , (s)-
1 [
K , (s)' K,(s) ] -G,(s)' R, (s)C,(s)
+ X , ( s) -lC, (s)' R,( s)y~ (s) +[X, (S)-l K,(s)'G,(s) + B 2 (s)JuO(s)
(6.3.25)
with initial condition Xl(O) = Xo . Now let xes) ~ XI(S) + 1:1(S) . Using the fact that Xs = Xs - Xl(S) , it follows that (6.3.24) can be re-written as
J,[x" w(·)J
min w( ) EL
2
[0,s]
= (xs - £(s))' X,(s)(xs - xes)) '- p~[u°(-). y~(.)]
100
6. State Estimation with Discrete and Continuous Measurements
where p~[uOO, y~O]
~
1 S
[II (K,x + G,UO) 112 -
(C,X -
y~)' R,(C,x - y~) ] dt
and x( s) is the solution to state equations (6.3.2) with initial condition x(O) = Xo. From this we can conclude that the corresponding set
of all possible states of the system (6.3.9), (6.3.10) is given by
X:[XO, u°(-)I~, y~OI~, d] {Xs E R n : min
Jdxs, wO] ::; d}
w(· )EL 2 [O,sl
Xs {
ERn:
(xs - x(s))'X,(s)(xs - x(s)) } ::; d + p~[uOO, y~(-)] .
(6.3.26)
Also, it is clear that
as E ---+ O. Hence by taking the limit as E ---+ 0, it follows from (6.3.26) and (6.3.19) that the set all possible states of the system (6.2.1), (6.2.2) is given by (6.3.4) and is bounded. This completes the proof of this part of the theorem. (ii) It is clear that the triple [uOO, y~O, y~Ol defined on an interval [O,s] is realizable if and only if the set Xs[xo,uOOI~,y~OI~'Y~(')I~,d] is not empty. Since the system (6.2.1), (6.2.2) is strictly verifiable, it now follows that the set Xs[xo, uO(')I~, y~OI~, Y~(')I~, d] is defined by (6.3.4). Hence, Xs[xo,uoOI~,y~OI~,Y~(-)I~,dl is not empty if and only if
(iii) We have proved above that if the system (6.2.1), (6.2.2) is strictly verifiable, then the set
is defined by (6.3.4). This completes the proof of the theorem.
6.4
D
Uncertain Systems with Missing Data
In this section, we will extend the results of the previous section to consider the uncertain system (6.2.1), (6.2.2) with missing data.
Incomplete Measurements We will consider the case when the measurements Yd(-) and ycC) are incomplete. That is, let Md(t J ) = [MJ(t J ) MJ(t J ) ... Md(tj)]' be a given vector for j = 1,2, ... ,Nd such that Md(t J ) E {0,1} for any i = 1, ... ,l and any j = 1,2, ... , N d . Then, the ith component y'd(t J ) of the output vector Yd(t J) is known if Md(t J ) = 1 and y'd(t J ) is unknown if Md(t J ) = O.
The matrix Md ~ [Md(tl) M d(t2) Md(tN,,) 1is referred to as the incompleteness matrix for the discrete output of the system (6.2.1). Also, let Mc(t) = [Ml(t) M;(t) ... M~(t)]' be a given vector for t E [0, T] such that M~(t) E {0,1} for any i = 1, ... ,k and any t E [O,T]. Then, the ith component y~(t) of the output vector Yc(t) is known if M~(t) = 1 and y~(t) is unknown if M~(t) = O. The matrix function Mc(-) is referred to as the incompleteness matrix function for the continuous output of the system (6.2.1). Associated with the incompleteness matrix Md and the incompleteness matrix function Mc(-), we also define matrices Ed(t J ) and Ec(t) as follows: For each j, the matrix Ed(t J ) is defined to be the diagonal matrix whose diagonal elements are given by the elements of the vector M d (t J) and for each t, the matrix Ec(t) is defined to be the diagonal matrix whose diagonal elements are given by the elements of the vector Mc(t).
Notation Let Md be a given incompleteness matrix, M c (') be a given incompleteness matrix function and let Yc(t) = y~(t) and Yd(t J ) = y~(tj) be the outputs of the uncertain system (6.2.1), (6.2.2). Also, dE'!fine corresponding known outputs y~ (t) and y~ (t J ) such that y~' (t) = y~t (t) ~f M~ (t) = 1 and y~t(t) = 0 if M~(t) = O. That is, the known output is obtained by setting unknown elements in the output vector to zero. It follow~ from this definition that we can write y~(t) = Ec(t)y~(t) where the matri~ Ec(t) is defined as above. Also, we define the known output y~(tJ) such that y~(tJ) = Md(tJ)y~(tj). Definition 6.4.1 Let Xo E Rn and d
>
0 be given. Also, let Uo(.) and
y~(-) be given vector functions defined over a gi'lJen time interval [0, s] and let y~(tJ) be a given vector sequence defined for t J ::; s. Also, let y~(-) and y~(-) be the corresponding known outputs. The input - output triple [u°(-), y~(.), y~(.)] is said to be realizable with parameters Xo and d if
there exist [x(-), w(·), vc(-), Vd(-)] satisfying conditions (6.2.1), (6.2.2) with u(t) = uO(t),Mc(t)Yc(t) = y~(t) and Md(tJ)Yd(t J ) = y~(tJ)' Model Validation with Missing Discrete-Continu,ous Data The model validation problem for an uncertain system with missing discretecontinuous data is defined as follows: Given a measured input - output triple [uO (-), y~ (.), y~(-)], determine if this tripl~ is realizable for the uncertain system (6.2.1), (6.2.2) with incompleten~ss matrix Md and incom-
102
6. State EstImation with Discrete aad. Colninuous Measurements
pleteness matrix function MeO.
Notatwn Let u(t) = uO(t) be a fixed control input, fj~(t) and fj~(tJ) be fixed known outputs for the uncertain system (6.2.1) with incompleteness matrix Md and incompleteness matrix function AfeU· Als!). let the finite time interval [0, s] be given. Then,
denotes the set of all possible states x( s) at time s for the uncertain system (6.2.1) with uncertainty inputs and initial conditions satisfying the constraint (6.2.2). Definition 6.4.2 The dzscrete-contmuous system (62.1), (6.2.2) zs sazd to be strictly verifiable wzth the mcomplete"ess matrzx Md and the meVfIT:p.icifcr;:css l'lfx?i!r'Cz /a:r;:cifcU'l';} Jf'c ( } U'l'i: [ro, 7J1, tf for «rtf[ flector X'o ERn, any tzme s E (0, T]. any constant d > 0, any fixto.d control mput u(t) = uO(t) and any fixed known outputs fj~(t) and fj~(tJ)' the set
X s [xo, UO (.) I~. fj~ (.) I~, fj~ (.) l~. d] zs bounded. Set- Valued State Estzmatwn wtth Mlssmg DtSCTete-Contmuous Data The set-valued state estimation problem for an uncertain system with missing discrete-continuous data is defined as follO\1;s: Given a measured inputoutput triple [uO(.), fj~(-), fj~()] for the uncertain system (62.1). (6.2.2) with incompleteness matrix Mq and incomplet~ness matrix function J\;fe (.), construct the set
Remark The above robust state estimation problem with missing discrete-continuous data, includes as a special case the problem of to bust prediction with missing discrete-continuous data. This follows if we consider the incompleteness matrix to be such that Md(t J ) = 0 for J = Nd,o +1, Nd,O+2, ... . Nd and the incompleteness matrix function to be such that Me(t) = 0 for t E [To, T].
6.5
Model Validation and State Estimation with Missing Discrete-Continuous Data
Our solutions to the above problems of set-valued state estimation and model validation with missing discrete-continuous data involve the follow-
65 Model Validation and State Estimation with Missing Discrete-Continuous D..- 103 ing Riccati differential equation which contains jumps in its solution:
F(t)
= A(t)P(t) + P(t)A(t)' +P(t) [Ke(t)' Ke(t) - Ce(t)' Ee(t)Rc(t)Ee(t)Ce(t)]P(t) +Bl(t)Q(t)-lBl(t)' Jar t =f t J ;
P(t~)-l + Cd(tJ)'Ed(tJ)Rd(tJ)Ed(tJ)Cd(tJ) [ -Kd(tJ)'Kd(t J ) Jar J = 1,2 .... ,Nd.
]-1
(6.5.1)
Such a Riccati equation will be referred to as a Jump Rzccatz equatwn. Also, we consider a set of state equations which contains jumps in its solution
5:(t)
= [A(t) + P(t)[Ke(t)' Ke(t) - Ce(t)' Ee(t)Rc(t)Ee(t)Ce(t)]] x(t) + P( t)Ce(t)' Ee(t)Re(t )Y~ (t) +[P(t)Ke(t)'Ge(t) + B2(t)]UO(t) Jar t =f t J .
~( )
x tj
'( -) x tJ
+
P( -) [ Kd(tJ),Kd(t J ) tJ -Cd(t j )' Ed(tj)Rd(tJ)Ed(tj)Cct(tJ)
]
A(t-)
X
j
+ P(t~ )Cd(t J )' E d(t J)Rd(tJ )Y~( t J ) + P(t~ )Kd( t J ), G d( t J )uo (t J ) Jar J = 1,2 ..... N d·
( 6.5.2)
Such a state equation is referred to as a Jump state equatzon. The following theorem, which is the main result of this section, provides a solution to the problems of set-valued state estimation and model validation for uncertain systems with missing discrete-continuous data. This theorem extends the results of Theorem 6 3.1 to allow for missing data.
= N' > 0 be a given matrzx. Rd(t J ) = Rd(t J )' ~ 0 be a gwen matrzx sequence and Q(.) = Q(.)' and ReU = ReU' ~ 0 be gwen matrzx Junctzons such that the conddzon Q(t) ~ i5! holds on the tune znterval [0. T]. Also, consider the uncertazn system (6.2.1)' (6.2.2) with a gwen zncompleteness matrzx llId and zncompleteness matrzx Junctzon Me (.). Then the Jollowzng statements hold:
Theorem 6.5.1 Let N
(z) The system (6.2.1), (6.2.2) zs strzctly verzjiable wzth the zncompleteness matrzx AId and the zncompleteness Junctzon MeU on [0, T] zJ and only zJ the solutzon P(·) to the Jump Rzccatz equatzon (6.5.1) wzth inztial condztzon P(O) = N- 1 exzsts and zs posztwe-dejinzte on the interval [0. T].
(ij) Suppose the system (6.2.1), (6.2.2) zs strzctly verzjiable on [0, T]. Also, let s E (0. T] be gwen and let Xo E Rn be a gwen vector, d > 0 be a gwen constant, y~(tJ) and Ye(t) be gwen known outputs and
104
6. State Estimation with Discrete and Continuous Measurements
UO(t) be a gwen vector functwn defined on [0, s]. Then, the tnple [U°(-) , y~(-), y~(-)] zs realzzable zf and only zf
where
and x(·) is defined by the Jump state equatwn (6.5.2) wzth mitial condItwn x(O) = xo.
(m) If the uncertam system (6.2.1), (6.2.2) IS stnctly venfiable, then
Xs[XO, u°(-)I~, Y~(')I~, Y~(')I~, d] _ {xs ERn: (xs - x(s))'p(s)-l(x s - xes)) }. (6.5.4) :::; d + Ps[uO(.), Y~(-), Y~U] Proof First let s E (0, T] be given and consider the uncertain system (6.2.1), (6.2.2) defined on [0, s]. Given a triple [uOU, Y~(')' Y~(')], we have by the definition of Xs[xo, u°(-)I~, Y~(-)I~, Y~(-)I~, d] that
Xs
E
Xs[xo, uO(')I~, Y~(-)I~, Y~(-)I~, d]
if and only if there exist x(·), w(·), ve(')' and Vd(') satisfying equation (6.2.1) and such that xes) = x s , the constraint (6.2.2) holds, and
= Me(t) [Ce(t)x(t) + ve(t)] 'it E [0, s] y~(t]) = Md(t]) [Cd(t]) + Vd(t])] 'it] :::; s.
y~(t)
(6.5.5)
Substitution of (6.5.5) into (6.2.2) implies that
Xs
E
Xs [xo, uO(.) I~, y~(-) I~, y~(. )I~, d]
if and only if there exists an uncertainty input w(-) E L2 [0, s] such that
J[xs, w(·)] :::; d
6.5 Model Validation and State Estimation with Missing Discrete-Continuous Data 105
where J[xs, wU] is defined by
and xC) is the solution to (6.2.1) with uncertainty input w(·) and boundary condition x(s) = Xs. Now the rest of the proof of this theorem is similar to the proof of Theorem 6.3.1. 0
7 Set-Val ued State Estimation with Structured Uncertainty
7.1
Introduction
In this chapter, we will present a new formulation of the robust set-valued state estimation problem which enables us to present set-vaIUf~d state estimation results for a class of uncertain systems with structured uncertainties. This class of uncertain systems is one in which the uncertainty is described by an "Averaged Integral Quadratic Constraint" (AIQC). This uncertainty description extends the standard integral quadratic constraint uncertainty description given in Chapter 4. The standard integral quadratic constraint defines a class of uncertainties which is extremely rich and allows for nonlinear time-varying dynamic uncertainties. Our new uncertainty description also allows for such a rich uncertainty class. Furthermore. it enables a tractable solution to be obtained for the set-valued state estimation problem in the case of structured uncertainty. Such problems have been found to be intractable using other representations of structured uncertainty. An important element of this chapter is the class of uncertain systems under consideration. The choice of an uncertain system structure to model a real system or process is like any modeling problem. The model structure must be chosen so that it can capture the essential features of the real system and the uncertainty in that system. Also, the model structure must be chosen so that it leads to a mathematically tractable state estimation problem. In introducing a new uncertainty class in this chapter, the essential features which we are trying to capture are as follows: Firstly, the
108
7. State Estimation with Structured Uncertainty
nominal system should be linear and possibly time-varying. Secondly, the uncertainty in the system should be structured (that is, the uncertainty can be broken up into a number of independent uncertainty blocks). Also, it is required that each uncertainty block be subject to an induced norm bound. Provided these conditions are satisfied, the uncertainty class is then chosen so that it leads to the most tractable solution to the state estimation problem. These considerations lead us to the "uncertainty averaging" uncertainty description considered in this chapter. Another direct motivation for our uncertainty averaging uncertainty description arises from the fact that in many instances, the model of the system and system uncertainty will be obtained by taking a series of data measurements and using some sort of averaging process. Given that the system model may be obtained via an averaging process, it is therefore quite natural to use an uncertainty description based on the use of averaging. It should be noted that the uncertainty averaging uncertainty description does capture the essential features we require in an uncertain system model. Namely, the nominal system is linear and time-varying and the uncertainty is structured with the individual uncertainty blocks satisfying an induced norm bound. Indeed, the principal difference between our new uncertainty description and some common existing uncertainty descriptions for structured uncertainty can be related to the initial conditions on the uncertainty dynamics. In the J1 type uncertainty description of [39,113]' the initial condition on the uncertain dynamics is implicitly assumed to be zero. In the standard integral quadratic constraint uncertainty description such as occurs in Chapter 4, the initial conditions on the uncertainty dynamics are implicitly assumed to be unknown but bounded. One interpretation of our new uncertainty description is that the initial condition on the uncertainty dynamics have an elementary probabilistic characterization. In most applications, the description of the initial conditions on the uncertainty dynamics is not an important consideration and thus it is most reasonable to use the formulation which is most tractable. This has been the main motivation behind our introduction of the uncertainty averaging uncertainty description. Also, it should be noted that in situations where the initial conditions on the uncertainty dynamics are important, the uncertainty averaging approach presented in this chapter may often be the most reasonable approach. For example, it may be the case that the bounds on the size of the initial conditions on the uncertainty dynamics are to be estimated via a series of measurements and the use of averaging. In this case, our probabilistic description would be a natural way to describe the resulting bound on the uncertainty initial conditions. For the class of uncertain systems with averaged integral quadratic constraints considered in this chapter, we solve a robust set-valued state estimation problem using a similar approach to that of Chapter 4. However, rather than obtaining the set of all possible values for the state as was done in Chapter 4, we obtain a set of all possible values of the averaged state.
7 2 Averaged Integral Quadratic Constramts
109
ThIS IS because of our use of the averaged mtegral quadratic constramt uncertamty descnptlOn One advantage of thIS approach IS that It mduces convexIty mto a problem whIch would be non-convex usmg normal uncertamty descnptlOns ThIS enables us to handle prevIously mtractable robust state estlmatlOn problems ansmg from tlme-varymg nommal systems contammg structured uncertamtIes The results presented m thIS chapter ongmally appeared m the papers [116 132J
7.2
Averaged Integral Quadratic Constraints
We conSIder the followmg uncertam tlme-varymg system defined on the fimte tIme mterval [0, TJ
k
+ L B 8(t)f;8(t),
x(t)
A(t)x(t)
yet) zo(t) Zl (t)
C(t)x(t) + f;o(t), Ko(t)x(t) Kl(t)X(t),
Zk(t)
Kk(t)X(t)
8=1
(72 1)
where x(t) E Rn IS the state, yet) E Rl IS the measured output, zo(t) E Rho, Zl(t) E Rh" ,Zk(t) E Rhk are the uncertaznty outputs, t;o(t) E Rl, 6 (t) ERr, ,t;k(t) E Rrk are the uncertaznty znputs, and A( ), B 1 ( ), ,Bd ), C(), K o(), K 1 (), ,Kk () are bounded pIeceWIse contmuous matnx functlOns defined on [0, TJ The structured nature of the uncertamty IS Illustrated m FIgure 7 2 1 In thIS case, the uncertamty mputs are the outputs of the uncertamty blocks and the uncertamty outputs are the mputs to the uncertamty blocks The bounds on these uncertamty blocks are descnbed below
System Uncertaznty The uncertamty m the above system IS descnbed by a set of equatlOns of the form
t;o(t) 6(t)
cPo(t,x( )), cPl(t,X( )),
(722) Alternatively, the uncertamty mputs and outputs may be collected together L).
mto two vectors That IS, we define t;(t) = [t;o(t)' 6(t)'
t;k(t)'
l'
and
110
7. State Estimation with Structured Uncertainty .6-
z(t) = [zo(t)' zdt)' ... zdt)'
1'. Then (7.2.2) can be re-written in the more
compact form ~(t) =
(t. x(·».
(7.2.3)
The uncertainty described by the above equations is required to satisfy a certain "Averaged Integral Quadratic Constraint". That is. we consider finite sequences of uncertainty functions of the form (7.2.3) such that the following constraint is satisfied: A veraged Integral Quadratzc Constramt
°
Let do > 0.d 1 > 0, ... ,d k > be given positive constants and Vo(')' V1 0, .... Vk (.) be given piecewise continuous bounded symmetric matrix weighting functions defined on [0, T], associated with the system (7.2.1). We will consider sequences of uncertainty functions S = {<1>1 ( .), <1>2 ( .), ... <1>q (.) }. The number of elements q in any such sequence is arbitrary. A sequence of uncertainty functions of the form S = {I(.). 2(.), ... q(.)} is an admzsszble uncertamty sequence for the system (7.2.1) if the following conditions hold: Given any "(-) E S and any corresponding solution {x"(-).C(·)} to equations (7.2.1), (7.2.3) defined on [0, T], then ~"O E L 2 [0, T], and
-1 L iT (~b(t)'Vl(t)~b(t) q
q "=1
Il zb(t)112) dt < do:
0
-1q L iT (~W)'V2(t)~W) -
Ilzl(t)112) dt < d 1 ;
-1q L iT (~k(t)'Vk(t)~k(t) -
Ilzk(t)ln dt < dk·
q
"=1
0
q
,=1
(7.2.4)
0
The class of all such admissible uncertainty sequences is denoted ::::. A block diagram representing this uncertainty structure is shown in Figure 7.2.1. Remarks The above definition extends the definition of the integral quadratic constraint given in Chapter 4: see also [114,115,117,119,165,167,168]. In the standard IQC uncertainty description such as given in Chapter 4, only individual uncertainty functions are considered rather than sequences of uncertainty functions. One interpretation of the uncertainty class described above is a probabilistic one. That is. given any admissible uncertainty sequence S E ::::. each uncertainty function ' (.) E S is assigned an equal probability. Then condition (7.2.4) amounts to a bound on the expected
7.%, Averag6cl Integral QfladratidJtmstraints
111
lJncertainty Blocks
¢k(-)
¢1 (.)
¢o( .)
';0
Zo
6
Zl
Nominal System ';k
Zk
,'Jj
FIGURE 7.2.1. Block diagram of an uncertain sy&tem with structured uncertainty.
112
7. State Estimation with Structured Uncertainty
value of the "measure of mismatch" between given uncertainty inputs and the following L 2 [O. T] induced norm bound condition:
~s(-)
e.g., see also Chapter 4, [115]' [119] and [117]. It should be noted that the averaged integral quadratic constraint on the uncertainty allows for nonlinear time-varying dynamic uncertainty. Indeed, the average measure of mismatch bound ds can be regarded as a bound on the average size of the initial condition for the uncertainty dynamics.
Initial Conditions The initial conditions on the system (7.2.1) are described as follows: Let Po > 0 be a given symmetric positive-definite matrix, Xo E Rn a given nominal initial condition and d x > 0 a given constant. Vie will consider sequences of solutions of the system (7.2.1) with initial conditions satisfying the following property: Given any S E ::::, let q be the number of elements in this uncertainty sequence and let {x' (O)} 1 be a corresponding sequence of initial conditions, then we require
;=
~
q
t
(x'(O) - xo)' Po (x'(O) - xo) ::::; dx .
(7.2.5)
,=1
The class of all such initial condition sequences is denoted Woo
Remark The above averaged uncertainty description lends itself to a procedure for constructing the uncertainty bounds based on a series of measurements on a physical system. That is, if the signals ~(-) and z(·) are available for measurement, then the uncertainty description would be would be determined so that condition (7.2.4) is satisfied for the available measurements.
Systems with Non-Dynamlc Norm Bounded Structured Uncertainty It is straightforward to verify that the above uncertainty description allows for structured non-dynamic uncertainty satisfying a standard norm bound condition. In this case, the uncertain system would be described by the state equations k
x(t)
[A(t)
+L
B8(t)~8(t)Ks(t)]x(t);
8=1
y(t) (x(O) - xo)' Po(x(O) - xo)
[C(t)
< dx :
+ ~o(t)Ko(t)]x(t) (7.2.6)
7.3 S-Procedure for Averaged Integral Quadratic Constraints
113
where lls(t) are uncertainty matrices satisfying Illls(t)11 ::::; 1 for s = 0,1, .... k. To verify that such uncertainty is admissible for the uncertain system (7.2.1), (7.2.2), we consider a sequence of uncertainty functions S of length one. This "sequence" is defined so that the uncertainty inputs are given by ~s(t) = lls(t)Ks(t)x(t) where Illls(t)11 ::::; 1 for all t ~ O. Then the uncertainty inputs ~s(-) satisfy condition (7.2.4) with any any d s > O. Similarly, the corresponding initial condition sequence is a sequence of length one {x(O)} defined so that (x(O) - xo)' Po(x(O) - xo) ::::; dx . From this, it immediately follows that condition (7.2.5) is satisfied. Robust State Estlmatwn Problem
Suppose that we have a given measured output y(t) = yo(t) for the uncertain system (7.2.1), (7.2.3). We will consider the following problem' Describe the set X T of all possible values of the averaged state vector
for the system (7.2.1), (7.2.3). Here {x'(-)};=l is a sequence of solutions corresponding to an admissible uncertainty sequence S E :=: and initial condition sequence {xt(O)};=l E 'lTo and such that yt(t) := yo(t) for all t E [0, T] and for l = 1, .. , ,q. The closure of the set XT is denoted cl(XT)' One interpretation of the set X T is a probabilistic one which arises when we consider the probabilistic interpretation of the averaged integral quadratic constraint. In this case. the set X T is interpreted as the set of all possible expected values of the state of the uncertain system (7.2.1), (7.2.3).
7.3
S-Procedure for Averaged Integral Quadratic Constraints
In order to prove the main result of this chapter, we will use an S-procedure result. The term "S-procedure" originated in the Russian literature in the book [1]. More recently, such results have been used extensively in the western literature and the name S-procedure has remained; e.g., see [81, 119,123,142]. At an intuitive level, such results enable problems involving structured uncertainty to be converted into parameter dependent problems involving unstructured uncertainty. In particular, note that the S-procedure result of [81] could also be applied to uncertain systems with structured uncertainties. However, the result of [81J cannot be applied to problems involving a time-varying nominal system or finite-time problems. Thus, one of the main motivations for the new class of uncertain systems introduced in this chapter is that it allows for the following S-procedure result to
114
7. State Estimation with Structured Uncertainty
be used. Furthermore, this S-procedure result applies to a much larger class of problems than the S-procedure result of [81] and has been applied to a number of robust control problems in addition to the robust state estimation problem considered in this chapter. see e.g. [124,133]. Theorem 7.3.1 Cons2der a set P = {v = [VO VI '" Vp]'} C RP+l w2th the following property: Gwen any fimte sequence {vI, v 2 , ... ,vq } c P such that 2:;=1 vt :::: 0, ... , 2:;=1 v~ :::: 0, then 2:;=lvO :::: O. Then there eX2st constants TO :::: 0, T1 :::: 0, ... ,Tp :::: 0 such that 2:~=0 Ts > 0 and
(7.3.1) for all V
=
[VO VI .. , Vp]' E P.
In order to prove this theorem, we will use the following preliminary Convex Analysis result. However. we first introduce some notation; e.g., see [111]. Notatwn
Given sets S
c
Rn and TeRn and a constant A E R. then S+T ~ {x+y : ~
~
x E S,y E T},).S = {Ax: XES}, and cone(S) = {ax:
X E S,a:::: O} (the conic hull of the set S). Also, cl(S) denotes the closure of the set S.
Lemma 7.3.2 Cons2der a set M C RP+1 wzth the property that a + b E cl(M) for all a, b E M. If Xo :::: 0 for all vectors [XO Xl Xp E M such that Xl > 0 .... xp > 0, then there exist constants TO :::: 0, T1 :::: 0, ... ,Tp :::: 0 such that 2:~=0 Ts > 0 and
J'
for all [XO
Xl
Xp
J' E M.
Proof
In order to establish this result. we first establish the following claims: Clazm 1
Given any two sets S C RP+l and T C RP+l and any scalars a E Rand (3 E R, then acl(S) + ,Bcl(T) c cl(aS + ,BT). To establish this claim, let ax +,By E acl(S) + ,Bcl(T) be given. That is, there exist sequences {X'}~l C S and {Y'}~l C T such that x, -> x and y, -> y. Hence. ax, + ,By, -> ax + (3y. However, ax, + (3y, E as + (3T for all z. Thus, we must have ax + (3y E cl( as + (3T). This completes the proof of the claim.
7.3 S-Procedure for A~ Integral Qua
U5
Claim 2 The set M has the property that cl(M)
+ cl(M) c
cl(M).
To establish this claim, first note that it follows from Claim 1 that cl(M)
+ cl(M) c
cl(M
+ M).
However, since a+b E cl(M) for all a, bE M, we must haveM+M C cl(M) and hence cl(M + M) c cl(M). Combining these two facts, it fQUows that cl(M) + cl(M) c cl(M). Claim 3 Given any set S C RP+l, then cone(cl(S))
C
cl(cone(S)).
To establish this claim, let x E cone( cl( S)) be given. We can write x = ay where 0 ?: and y E cl(S). Hence, there exists a sequence {Y,}~l C S such that y, --> y. Therefore ny, --> QY = x. However, oy, E cone(S) for all i. Thus, x E cl( cone( S)). This completes the proof of the claim.
°
Claim 4 The set d( cone( M)) is convex. To establish this claim, let A E [0, 1J be given and consider two points Xl, X2 E cone(M). We can write Xl = 0lZl and X2 = Q2Z2 where Ql ?: 0, Q2 ?: 0, Zl E M, and Z2 E M. Hence Zl E cl(M) and Z2 E cl(M). It follows from Claim 2 that for all positive integers nand m. Now consider a sequence of rational numbers {;;",
}:l such that
Since, n, and m, are positive integers for all i, it follows that
for all i. Therefore, (1 -
A)Q2
n, Zl ( -m,
+ Z2 )
=
(1-
A)Q2
m,
(n,zl
+ miz2)
E cone(cl(M»
116
7. State Estimation with Structured Uncertainty
for all z. Hence, using Claim 3, it follows that (1 - >.)a2 (n'.. Zl
. mi
+ Z2)
E
rl(cone(M»
for all i. However, (1 - >.)a2 ( ( Aal) Zl 1- A a2 AalZl + (1 - A)a2z2 AXl + (1 - A)X2.
+ Z2 )
Thus, since cl(cone(M» is a closed set, it follows that AXl + (1 - A)X2 E cl(cone(M». Moreover, Xl E cone(M) and X2 E cone(M) were chosen arbitrarily and hence, it follows that
Acone(M)
+ (1
- A)cone(M)
C
cl(cone(M).
Therefore,
cl(Acone(M)
+ (1
- A)cone(M» C cl(cone(M».
Now using Claim 1, we conclude Acl(cone(M»
+ (1
- A)cl(cone(M» C cl(cone(M».
Thus, cl(cone(M» is a convex set. Using the above claim. we are now in a position to complete the proof of the lemma. Indeed suppose Xo 2': 0 for all vectors Xp
J' E M
such that Xl > 0, ... ,xp > O. Also, let the convex cone Q C RP+l be defined by Q ~ {r
=
[ro,rl, ... ,rp]': ro
< O,rl >
0,r2
> 0 ....
,rp
> O}.
It follows that the intersection of the sets Q and M is empty. Also, since Q is a cone, it is straightforward to verify that Q n cone(M) = 0. Furthermore, the fact that Q is an open set implies Q n cl( cone(M» = 0. Thus, using the Separating Hyperplane Theorem (Theorem 11.3 of [111]), it follows that there exists a hyperplane separating the convex cones Q and cl(cone(M» properly. Moreover using Theorem 11.7 of [111], it follows that the hyperplane can be chosen to pass through the origin. Thus there exists a non-zero vector b E RP+l such that
x'b ::; 0
(7.3.2)
7.3 S-Procedure for
for all x
E
cl( cone(
M»
A.~
latflgal Quadratic ~ts
and
;'6 ~ 0 for all x
E
117
(7.3.3)
Q. Now write
°
It follows from (7.3.3) that T, ~ for all i. Also, since b '# 0, this implies' that 2:~=o Ts > 0. Moreover, it follows from (7.3.2) that TOXO ~ T1Xl
for all
[xo
Xp
Xl
+ T2X2 + . , . + TpXp
J' E M. This completes the proof of the lemma.
o
Proof of Theorem 7.3.1. Let the set M set P. That is
C RP+1
be formed by taking finite sums of elements in the
q
M ~ {v
=
LV' : {V'};=1 is any finite sequence contained in P }. ,=1
The set M has the following property: Given any b E Jlvl and c E M, then b + c EM. Indeed. if bE M and c E M, then we can write h
r
b=
LU\ c= LY' ,=1 ,=1
where {u'}f=l C P, {Y'}~=1 C P. Therefore. r+h
b+c=
LZ' ,=1
where z' = v' for 1 :::: z :::: rand
for r + 1 :::: i :::: r + h. Thus, we must have b + c E M. Also, it follows from the assumed property of the set P that the set M will have the property that Vo ~ for all vectors [vo VI ... v p ]' E M such that Vo ~ 0, VI ~ 0, ... , Vp ~ O. Hence, using Lemma 7.3.2, there exist constants TO ~ 0, Tl ~ 0, ... ,Tp ~ 0 such that 2:~=o Ts > and the inequality (7.3.1) holds for all V = [vo VI '" v p ]' E M. However. any point V E P, will also be contained in the set M. Hence, the inequality (7.3.1) holds for all V E P. This completes the proof. 0
°
°
Observation 7.3.1 If assumption of the above theorem is satisfied and there exists an element v = [VO VI '" V p ]' E P such that VI > 0, ... ,vp > 0, then the constant TO zn (7.3.1) may be taken as TO = 1.
118
7. State Estimation with Structured Uncertainty
7.4
Design of a Set-Valued State Estimator
In this section, we present the main result of this chapter which characterizes the set of all possible values of the averaged state for the uncertain time-varying system (7.2.1), (7.2.4). (7.2.5). This characterization is given in terms of the solutions to a parameter dependent Riccati differential equation. Our main result requires that the uncertain system and associated weighting matrix function functions Vo(-), . .. , Vk (·) satisfy the following additional assumptions.
Assumptwns 4.1 The matrix functions Vs (') are strictly positive: i.e., there exists a constant E > such that Vs(t) :;:.. E1 for all t E [0, T] and for s = 0,1, ... ,k.
°
4.2 For any j = 1, ... ,k, there exists a vector x~ E Rn and a constant E > such that the following condition holds: For any
°
x~ E {x~ E R n
:
Ilx~
-
x~11 :S E}
there exists a time t* E [0. T) and an input the corresponding solution to the system
~; (-) E L2 [t*,
with boundary condition x(T) = x~, satisfies x(t*) =
T] such that
°
and
Assumption 4.2 is a technical assumption required in the proof of the main result of this chapter. It can always be satisfied by changing either C(t) or Ko(t) on an infinitesimal time interval [t*, T]. This indicates that this assumption is quite weak. The Riccati equation under consideration is defined as follows: Let TO :;:.. 0, T1 > 0, ... ,Tk > 0, Tk+1 :;:.. be given constants and consider the Riccati equation
°
- ?(t)
A(t)' P(t)
+ P(t)A(t) + P(t)B(t)B(t)' P(t)
+K(t)' K(t) - C(t)'Vo(t)C(t), P(o)
Tk+lPO
(7.4.1)
7.4 Desiga of a Set-Valued State Estimator
119
where
a
K(t)
=
~~~~~~ 1, Vo(t) a ToVo(t), ~
[ vfTkKk(t)
=
[ }r,-B1(t)V1(t)-~
B(t)
Also, our state estimator will be defined by state equations of the form
+ B(t)B(t)' pet)]' vet) + C(t)'Vo(t)yo(t),
vet)
- [A(t)
v(O)
Tk+1POXO:
get)
yo(t)'Vo(t)yo(t) - v(t)' B(t)B(t)'v(t),
g(O)
Tk+1X~POXO'
(7.4.3)
Notatwn The set
r
rc
R k+2 is defined as follows:
> 0, }_ 0 & there exists a solution to Riccati equation (7.4.1) on [0. T] which satisfies peT) ::::: 0
T
~
=
T2
{
Also, for any
T
=
[To,T1, ... ,Tk+d: TO::::: 0,T1
> 0, ....
Tk
> O. Tk+1
[TO, .. . , Tk+1] E
r
:::::
we introduce the set Xf defined as
(7.4.4)
where veT) and geT) are defined by the equations (7.4.3). The following theorem is the main result of the chapter. The first part of the theorem relates to the question as to whether the closure set of all possible averaged states for the uncertain system (7.2.1), (7.2.4), (7.2.5) (consistent with the measured data) is equal to the entire state space. If this set is equal to the entire state space, then no useful state estimate can be obtained since any point in the state space would be consistent with the measured data and an admissible uncertainty. If the set is not equal to the entire state space, then the theorem states that the set r defined above is non-empty. The second part of the theorem shows that. in this case, the closure of the set of all possible averaged states for the uncertain system can be constructed by taking the intersection of a family of ellipsoids. These ellipsoids are constructed from the solutions to equations (7.4.1) and (7.4.3). This then gives a formula for constructing a set-valued state estimate for the uncertain system.
120
7. State Estimation with Structured Uncertainty
Theorem 7.4.1 Consider the uncertain system (7.2.1), (7.2.4), (7.2.5) with given output y(t) = Yo(t) for t E [0, T] and suppose that Assumptions 4.1-4.2 are satisfied. Then the following statements hold:
(i) The closure of the set of all possible values of the averaged state, cl(XT) is not equal to R n if and only if the set r is non-empty.
(ii) If cl(XT ) is not equal to Rn, then cl(XT ) = nTErXf. Proof (i) : Necessity. In order to prove this result, we first establish the following claim. Claim
The set cl(XT) is convex. Let the points xT' x~ E X T be given. It follows that there exist sequences {x~(-)) ~~(')};:1 and {Xl,(-) , ~!;(')};:1 of solutions to the system (7.2.1) such that y~(-) == Yo(-) and y!;(.) == Yo(') for all i, conditions (7.2.4) and (7.2.5)
t
q: 2:;:1
hold, and x~(T) = x T and 2:;~1 xl,(T) = x~. Let q ~ 2qaqb and consider a new sequence {x'(-)'~'(')};=l where every pair {x~(-)'~~(')};:1 is taken qb times and every pair {xl,('))~b(')};:1 is taken qa times. For such a sequence, conditions (7.2.4) and (7.2.5) obviously hold and
From this it follows that the set cl(XT ) is convex. This completes the proof of the claim. Now consider a vector x~ which does not belong to cl(XT)' Since cl(XT ) is convex, it follows from the Separating Hyperplane Theorem that there exists a vector r ERn and a constant c E R such that r' x~ + c > 0 and r'xT + C < 0 for all XT E cl(XT): e.g.) see [111]. Let Sxo ~ {XT E R n : r'XT
+ c 2': O}.
For the system (7.2.1), a corresponding set follows:
n ~ { A(·) =
[x(·) ~(-)]:
n c
L2(O,T) is defined 'as
{x(·), ~(-)} are related by (7.2.1), } L 2 [O, T] and x(T) E Sxo .
~(-) E
7.4 Design of a Set-Valued State Estimator
121
Also, we consider the following set of functionals mapping from D into R: Ll.
fo (xU, ~U)
=
II (x(·), ~(-))
~
!k (x(·), ~(.))
~
fk+1 (x(·), ~(.))
Ll.
iT ( lT o
[C(t)x(t) - yo(t)]' Vo(t) ) . x [C(t)x(t) - yo(t)] - Il zo(t)112 dt - do,
(1Iz1(t)112 - 6(t)'V1(t)6(t)) dt
lT
(IIZk(t)112 -
+ d1;
~k(t)'Vk(t)~k(t)) dt + dk;
- (x(O) - xo)' Po (x(O) - Xo)
+ dx.
(7.4.5)
Now consider a sequence of vector functions {x'U,C(-)};=l cD such that q
L II (x'O, CO)
> 0,
fk+1 (x'(·), ~'O)
> o.
,=1
q
L
,=1
Since %2::;=1 x'(T) E Sxo and the intersection of cl(XT ) and Sxo is empty, using the substitution ~o(t) = yo(t) - C(t)x(t) and constraints (7.2.4) and (7.2.5), we must have q
L
,=1
fo (x'(·), CO) ~ O.
We now apply Theorem 7.3.1 to the set P
C
(7.4.6)
Rk+2 defined as follows:
P ~ {x = [fo(x(-)'~(·)), ... ,fk+1(X(·), ~U)], : [xU, ~U] ED}. From this definition and condition (7.4.6), it follows that the condition required by this theorem is satisfied. Using this result, it now follows that there exist constants TO ~ 0, T1 ~ 0, ... ,Tk+1 ~ 0 such that 2::~~~ Ts > 0 and
(7.4.7) for all x = [xo Xl
Xk+1]'
E
P. Hence,
k+1 Tofo (x(·), ~(.)) ~ LTsfs (x(·),~(·)) s=l
('7.4'.8)
122
7. State Estimation with Structured Uncertainty
for all [x(·),~(-)] E n. Using the definitions of the functionals (7.4.5), inequality (7.4.8) can be rewritten as FT(XT·~U)
~
Tk+l (x(O) ~ xo)' Po (x(O) ~ xo)
r
T
Jo
[
+
L~=l Ts (~s(t)'Vs(t)~s(t) ~ Il zs(t)112) +
TO([YO(t) ~ C(t)x(t)]' Vo(t) [Yo(t) ~ C(t)x(t)] ~ Il zo(t)112)
]
dt
k
> LTsds + Tk+ldx 8=0
> O.
(7.4.9)
\Ve now prove that Ts > 0 for s = 1. 2, ... ,k. This fact will be established by contradiction. Suppose T J = 0 for some j and consider an input function ~(-) E L 2 [O.T] defined so that ~J(-) f. 0 and ~s(-) == 0 for s f. j. Now recall Assumption 4.2 and let x T be vector such that r' x T f. 0 and the conditions of Assumption 4.2 are satisfied. Also, let ~; U be the corresponding input defined as in Assumption 4.2. Then there exist constants ao > 0 and a E {~1,1} such that aaxT E Sxo for all a ~ ao. Now let ~J(t) == ~;(t) for t E [t*, T] and ~J (t) == 0 for t ::; t* where t* is as defined in Assumption 4.2. We define a functional
If the function ~ ( .) is as defined above. then it follows from Assumption 4.2 and the quadratic nature of the functional Fa (~(.). that Fa(~(-)) -> ~OO
as a
-> +00.
This contradicts (7.4.9). Hence. we can conclude that Ts > 0 for all s = 1.. .. ,k. In order to establish the existence of a suitable solution to Riccati equation (7.4.1), we consider the system
yet)
+ B(t)w(t): C(t)x(t) + ~o(t):
i(t)
K(t)x(t)
x(t)
A(t)x(t)
(7.4.10)
where
i(- ) w(-)
[Fozo(-)'. VTlZl(-)'.· .. ,y0izk(')']" ~
[VTlVl(-)~6(')"'" ,JTkVd')~~k(')']"
and the matrices BU and K(-) are defined as in (7.4.2). Since we have established above that Ts > 0 for s = 1, ... ,k, this system is well defined.
7.4 Design of a Set-Valued State Estimator
123
It is straightforward to establish that with this notation, the state equations (7.4.10) are equivalent to the state equations (7.2.1). Furthermore, condition (7.4.9) may be rewritten as k
F [XT, w(-)] :::::
L Tsds + Tk+l dx
(7.4.11)
s=O where F[XT' w(-)] is defined by
FlxT, w(·)]
~
Tk+l (x(O) - XO)' Po (x(O) - xo) + T ) ( Ilw(t)112 - x(t)' K(t)' K(t)x(t)+ io (Yo(t) - C(t)x(t))'Vo(t)(yo(t) - C(t)x(t)) dt
r
(7.4.12) and x(·) is the solution to (7.4.10) with input w(·) and boundary condition x(T) = XT E Sxo, Now consider the functional Ja[XT'W(')] = F[XT'W(')] with Xo = 0 and Yo(-) == O. Then, Ja[XT, w(-)] is a homogeneous quadratic functional with a terminal cost term. Also, condition (7.4.11) implies that Jr;(XT.·) ::::: 0 for all XT E Sxo, However, since Ja[XT. w(-)] is a homogeneous quadratic functional, we have
for all a E R. Hence, inf
we· )EL
2
[O,sJ
Ja[XT, w(·)] ::::: 0
(7.4.13)
for all XT: " XT of o. Furthermore, using the continuous dependence of Ja[XT, w(-)] on XT, it is straightforward to verify that condition (7.4.13) holds for all XT. The optimization problem (7.4.13) subject to the constraint defined by the system (7.4.10) is a linear quadratic optimal control problem in which time is reversed. In this linear quadratic optimal control problem, a sign indefinite quadratic cost function is being considered. Using a known result from linear quadratic optimal control theory, we conclude that condition (7.4.13) implies that there exists a solution P(-) to the Riccati equation (7.4.1) with initial condition P(O) = Tk+lPO such that P(T) ::::: 0; e.g., see page 23 of [30]. This completes the proof of this part of the theorem. (ii) Sufficiency Let T E r be given and consider the corresponding functional (7.4.12). Now consider the following optimization problem inf
w(·)ELdo.TJ
F[XT' w(·)]
(7.4.14)
where F[XT, w(·)] is defined as in (7.4.12) and the infimum is taken over all x(·) and w(-) connected by (7.4.10) with the boundary condition x(T) =
124
7. State Estimation with Structured Uncertainty
XT. This problem is a linear quadratic optimal tracking problem in which the system operates in reverse time. The solution to this tracking problem is well known (e.g., see [76]). Indeed, if the matrix function PC) is the solution to the Riccati equation (7.4.1) with initial condition P(O) = Tk+lPO, then the infimum in (7.4.14) will be achieved for any Xo and any YoU. Furthermore as in [76], we can write min
w(- )EL 2 [O.T]
F[XT, w(·)] = x~P(T)xT - 2x~v(T)
+ geT)
where veT) and geT) are defined as in (7.4.3). From this we can conclude that the set Xf defined in equation (7.4.4) satisfies k
Xf = {XT
E
Rn
:
min
w(- )EL 2 [o,T]
F[XT, w(·)] ::;
Ld
8
+ Tk+ldk+I}.
8=0
Therefore, since peT) ~ 0, the set Xf is not equal to R n. On the other hand, the constraints (7.2.4) and (7.2.5) with the substitution ';o(t) = Yo(t) - C(t)x(t) and non-negativeness of the constants T8 implies that cl(XT) C Xf· Hence, the set cl(XT ) cannot be equal to Rn. This completes the proof of this part of the theorem. (ii) We have proved above that cl(XT ) c Xf for any T E r. Furthermore, we have established that if XT does not belong to cl(XT) then there exists aTE r such that XT does not belong to Xf· Hence, cl(XT ) = TEr Xf. This completes the proof of the theorem. 0
n
Remark For the case in which Riccati equation (7.4.1) has a positive-definite solution on [0, T], it is straightforward to verify that the state estimator equations (7.4.3), (7.4.4) can be rewritten in the following Kalman Filter form:
where
£(t) = [A(t)
+ P(t)-l[K(t)' K(t)
- C(t)'Vo(t)C(t)]] x(t)
+p(t)-lC(t)'VO(t)Yo(t); x(o) = Xo and
PT = loT [IIK(t)x(t)112 -
(C(t)x(t) - yo(t))' Vo(t) (C(t)x(t) - yo(t))] dt.
In order to apply the main result of this chapter, it is required to calculate the set Xf for an infinite number of values of the parameter T in order to
7.5 Illustrative Example
125
obtain the state estimation set cl(XT)' Obviously this is not possible in practice. However, one can obtain an upper bound for the set cl(XT) by considering only a finite set of values for the parameter vector T and then taking the intersection of the resulting sets X T. As the number of values taken for T is increased, the resulting estimate for cl(XT ) becomes more accurate. The calculations required for this numerical scheme would require the parallel implementation of multiple versions of the equations (7.4.1) and (7.4.3). Such computations would be amenable to parallel processing.
7.5
Illustrative Example
In this section, we present an example to illustrate the main results of this chapter. The motivation for this example is the fact that Kalman Filtering is often applied to signals which have some underlying oscillatory behavior. Thus, it is natural to use a signal model which consists of a simple oscillator. However, if the oscillation frequency varies or is not precisely known, then the standard Kalman Filtering approach breaks down. In this example, we consider a signal model which consists of an oscillator with uncertainty in the frequency of oscillation and uncertainty in the measurement equation. Probably the most natural way to model such an uncertain system is as a system with non-dynamic norm bounded uncertainty of the form (7.2.6). In particular, we consider the following uncertain system
x(t)
[ -2
y(t)
[1
/~l(t) ~] x(t);
+ O.I~o(t)
- 1 + O.l~o(t)]x(t)
where II~o(t)11 :::: 1 and II~l(t)11 :::: 1. Also, the initial condition system is assumed to satisfy the bound
on
this
In this uncertain signal model, the uncertain parameter ~1 (t) represents the uncertainty in the oscillation frequency, the uncertain parameter ~o(t) represents the uncertainty in the measurement and the uncertainty in the initial condition represents uncertainty in the magnitude and phase of the sinusoid. We will consider a robust filtering problem corresponding to this uncertain signal model over the finite time interval [0,10]. To apply the results of this chapter to this problem, we first construct the corresponding uncertain system of the form (7.2.1), (7.2.2). This system is
126
7. State Estimation with Structured Uncertainty
defined as follows:
x(t)
[_~ ~] x(t) + [ ~
y(t)
[1 - l]x(t) + ~o(t);
zo(t)
[0.1 O.l]x(t);
Zl(t)
[10]x(t).
] 6(t);
The uncertainty in this system is required to satisfy the averaged integral quadratic constraint (7.2.4) with Vo = 1, do = 0.01, VI = 1 and d 1 = 1. Also, the initial condition on this system satisfies condition (7.2.5) with 10
Po = [ 0
0 ] 0.04 '
Xo = 0 and dx = 1. We consider this uncertain system over the finite time interval [0, 10]. Verification of Assumptions 4·1 and 4·2 It is straightforward to verify that this uncertain system satisfies Assumptions 4.1 and 4.2. Indeed, to verify that Assumption 4.2 is satisfied, we first observe that for this example, the pair (A, B 1 ) is controllable. Hence. we can use a standard formula to find an input ~i (.) which steers this system from an initial state of x(O) = 0 to a state of x(l) = [II]'. We then define GO according to the feedback law ~i(t) = [3 O]x(t) for t E [1,10]. Now let x~ be the state of the corresponding closed loop system at time t = 10 and let t* = O. With xr = x~ and ~i (-) defined as above, we calculate
1 1
°(IIC(t)X(t) 112 - IIKo(t)x(t)112)dt
11 1
(1IC(t)x(t)112 -IIKo(t)x(t)112)dt 10
+
(1IC(t)x(t)112 - IIKo(t)x(t)112)dt
0.1560 - 1.3132 x 106 < O. Moreover, if we perturb xr in a neighborhood of x~, it is straightforward to verify that the condition will remain satisfied if we modify ~i (.) on the interval [0,1] so that the system state is steered into the point
instead of the point [1 1]'.
75 IllustratIve Example
127
Szmulatwn Results To Illustrate the performance of our state estimator apphed to the above uncertam system we consIder the uncertamtIes to be such that ,6.dt) == -1 ThIs corresponds to changmg the frequency of the oscIllator from the nommal value of 2 rad/s to a value of 1 rad/s Also, we let ,6.o(t) == 1 and x(O) = [0 5]' Correspondmg to these values for the uncertam parameters we obtam the followmg "true" system
x(t)
[_~ ~] x(t).
yet)
[09 - 1 l]x(t)
We now apply our state estImator to the measured sIgnal Yo( ) obtained from thIs "true" system In order to apply our state estImator the Riccati equatIOn (741) and the state equatIOns (743) are solved over the time mterval [0,10] for vanous values of the vector T = [TO TI T2J' > 0 For each value of the vector T, thIs defines an ellIpsOId Xf accordmg to equatIOn (744) It follows from Theorem 74 1 that thIs ellIpsOId gIves an upper bound on the set of all possIble values of the averaged state at tIme T Also. the center of thIS ellIpsOId can be used as an estimate for the averaged value of the state vanable at time T FIgures 7 5 1 - 7 5 3 each show the resultmg estImate of the state vanable X2, upper and lower bounds on the value of X2 and the true value of X2 as a functIOn of tIme over the mterval [0,10] In FIgure 7 5 4, we show three of the ellIpSOIdal boundmg sets of the form (744) boundmg the true state of the system at tIme t = 10 Each of these elhpsOIdal boundmg sets was obtamed for a dIfferent value of the vector T = [TO TI T2]' > 0 From thIS figure, we can see that by mtersectmg the boundmg sets obtamed for dIfferent values of the vector T, a sIgnIficantly smaller boundmg set can be obtamed
128
7. State Estimation with Structured Uncertainty
10r----r----,----,----.----,--~T_~~----,_--_,----,
\ /
\
/ I
5
,,
/
\
/
\
,
\
/
\
/
,
\
/
\
\ \
\
0
\
\
\ \
\
,
\
C\I
\
><
\
,,
-5
\
- - true value of )(2 - - estimated value of x2
-10
- - upper bound on x2 lower bound on x2 _15L---~--~~
o
__
~
__
~----~---L----~---L--~
2345678 Time (Seconds)
_ _~
10
9
FIGuR£ 7.5.1. Estimated value of X2 as a function of time with
T
=
[10 1 1J.
20r---,----.----r---,----,----.----r---,~--._--_,
-
15
true value of )(2
- - estimated value of x2 - - upper bound on x2
10
lower bound on x2 /""
/
5' - ,
,
, \
/
\
/
\
\
0
- ..........
/
\
""x
-
/
\
\
\ \ \
\
I
,
,,
/
/
,
,
/
/
-5
,,
, "- -..-
-10
-15
0
2
3
4 5 6 Time (Seconds)
FIGURE 7.5.2. Estimated value of
X2
7
B
9
as a function of time with
10
T =
[2 1 1].
7.5 Illustrative Example
129
10 /
I
,
~
,
I
\
5
\
~
,
\
\
I \
\
\
/
I
,
I
I
\
\
I
\
I
0
I
\ \
\ \
N
-5
x
\ /
"
/
-10 -
true value of x2
- - estimated value of x2
-15
- - upper bound on x2 lower bound on x2
-20
0
2
3
4 5 6 Time (Seconds)
FIGURE 7.53. Estimated value of
7
8
9
as a function of time with
X2
10
T =
2
--
0 /"
/"
I
'-
/
\' \ \ \ \
-2
I
\
I
'" )(
-4
I
I
X
I
I
I I
T=[IO III
-
I
- -
-6
/"
,,-,-
,\
T= [211] - T= [511]
/"
X
true state value
-8
_10L-----L-----L-----L-----~----~----~----~----~
-10
-8
-6
-4
-2
o
2
4
xl
FIGURE 7.5.4. Ellipsoidal bounds for x(lO).
6
[5 1 1J.
8 Robust HOC) Filtering with Structured Uncertainty
8.1
Introduction
In this chapter. we present another approach to the problem of robust filtering for uncertain systems with structured uncertainty. In particular. we take a HOC filtering approach rather than the set-valued state estimation approach taken in Chapters 4 ~ 7. The problem of Hoc filtering has received considerable interest in recent years: e.g .• see [48.89]. This problem involves constructing a filter such that the Hoc norm from the disturbance inputs to the filter error output is minimized. Thus, no assumptions are made about the spectral properties of the disturbance signals. This is as opposed to the Kalman Filtering approach in which the spectral properties of the disturbance signals are assumed known. Thus in many applications. the Hoc filtering approach will be more suitable than the standard Kalman Filtering approach. Although the Hoc filtering approach does not require the spectral characteristics of the disturbance signals to be known. it still requires that an accurate system model is available. However in practice. there may be uncertainties in the system model. This fact has motivated a number of authors to consider a robust version of the Hcf'J filtering problem: e.g., see [34,44]. In this chapter, we consider a robu,;t Hoc problem in which the uncertain system under consideration contairlS structured uncertainty modeled via integral quadratic constraints. We consider linear time-invariant uncertain systems with structured uncertainty satisfying a collection of integral quadr<ic constraints. This un-
132
8. Robust H= Filtering with Structured Uncertainty
certainty description extends the uncertainty description given in Chapter 4 to allow for multiple integral quadratic constraints. This allows us to handle the problem of structured uncertainty. In contrast to Chapters 4 7, in this chapter, we consider a robust filtering problem over an infinite time interval. As in Chapter 7, a key technical result used in this chapter is a version of the "S-procedure." In particular, in this chapter, we use an S-procedure result which is an extension of the result of [81]. This result enables us to consider the problem of robust filtering with a specified level of disturbance attenuation for the case of uncertain systems with structured uncertainty but without the need to use the notion of uncertainty averaging as in Chapter 7. Furthermore, this result also allows us to consider the possible use of nonlinear filters. The advantage of this approach is that it allows for non-conservative results to be obtained for the case of uncertain systems with structured uncertainty. Furthermore, using this framework, it is shown that the use of a nonlinear filter does not give any advantage over the use of a linear filter. The results of this chapter can be regarded as an extension of the standard results on Hoc; filtering (see, e.g., [13,89]) to the realm of uncertain systems with structured uncertainty. In fact, the main result of this chapter is an immediate consequence of a result on Hoc control with structured uncertainty originally published in [131]. The solution to the robust HOC filtering problem presented in this chapter is obtained by solving a pair of parameter dependent algebraic Riccati equations of the game type.
8.2
Robust Hoo Filtering
We consider a robust HOC filtering problem for an uncertain syStem of the following form: k
x(t)
Ax(t)
+ Bow(t) + L
B"e.(t);
8=1
((t)
Lx(t);
Zl(t)
K1X(t);
KkX(t); Cx(t)
+ Dw(t)
(8.2.1 )
where x(t) E Rn is the state, wet) E RP is the disturbance input, ((t) E Rq is the error output, Zl (t) E Rhl, ... ,Zk(t) E Rhk are the uncertainty outputs, 6 (t) E RT1, ... ,~k (t) E RTk are the uncertainty inputs and y( t) E Rl is the measured output.
System Uncertainty The uncertainty in this system is dE!3Cribed by a set of equations of the form
6(t) 6(t)
¢l(t,X(-» ¢2(t,XC))
~k (t)
¢k(t, xC)
(8.2.2)
where the following Integral Quadratic Constraints are satisfied: Given any locally square integrable disturbance input we), and any corresponding solution to the system (8.2.1), (8.2.2), let (0, t*) be the interval on which this solution exists. Then there exist constants d 1 ~ 0, ... ,dk ~ 0 and a sequence {tt} ~ 1 such that tt --> t*, tt ~ 0 and
t, (1I zs(t)112 -11~s(t)112)dt ~ -ds Vi 'Is
Jo
= 1, ... ,k.
(8.2.3) Note that tt and t* may be equal to infinity. The class of all such admissible uncertainties is denoted 3. A block diagram representing this uncertainty structure is shown in Figure 8.2.l. In references [166] and [71]. a number of examples are given of physical systems in which the uncertainty naturally fits into the above framework. Also, note that the above uncertainty description allows for uncertainties in which the uncertainty input ~s depends dynamically on the uncertainty output ZS. In this case, the constant d s may be interpreted as a measure of the size of the initial condition on the uncertainty dynamics. Also, it is clear that the uncertain system (8.2.1), (8.2.3) allows for uncertainty satisfying a norm bound condition. In this case, the uncertain system would be described by the state equations k
+ L Bs!1 s (t)Kslx(t) + Bow(t);
x(t)
[A
yet)
Cx(t)
s=l
+ Dw(t); II!1 s (t)11 ::::: 1 'It
(8.2.4)
where !1 s (t) are the uncertainty matrices. Indeed, let ~s(t) = !1 s (t)K s x(t). Then the uncertainties ~s(·) satisfy conditions (8.2.3) with d s = 0 and with any tt. For the uncertain system (8.2.1), (8.2.3), we consider a problem of robust filtering with a specified level of disturbance attenuation. The class of filters
134
8. Robust H oo Filtering with Structured Uncertainty
¢k(-)
¢2(· )
¢IC)
21
~
6 ~k
w
22
N aminal System
2k
y
FIGURE 8.2.1. Block diagram of an uncertain system with structured uncertainty.
8.2 Robust IP"
~
135
considered are nonlinear dynamic filters of the form A(Xj(t), y(t)); ).,(Xj(t), y(t))
(8.2.5)
where A(x j, y) and ).,(x j, y) are continuous vector functions. Note that the dimension of the filter state vector Xj(t) in (8.2.5) may be arbitrary. Notatwn
In this chapter, is,
Ilq(-) 112
denotes the L2[0, 00) norm of a function q(.). That
We now introduce a new definition of robust observability corresponding to the robust Hoo filtering problem; c f Definition 5.2.1.
°
Definition 8.2.1 Let "Y > be a gwen constant. The uncertam system (8.2.1), (8.2.3) ~s sazd to be robustly observable with disturbance attenuation"Y (vza a nonlmear Jilter) ~f there ex~sts a Jilter of the form (8.2.5) and constants C1 > 0 and C2 > such that the followmg cond~tions hold:
°
(z) For any ~mtial cond~twn [x(O).Xj(O)], any admzssible uncertamty mputs';(·) and any d~sturbance mput w(·) E L2 [0.00), the correspondmg solution
to the system (8.2.1). (8.2.5) belongs to L 2[0, 00) (hence, t* = 00) and k
Ilx(-)ll~ + Ilxj(-)ll~ + II((-)II~ +
L
lI';s(-)ll~
s=l
k
< cIlllx(0)11 2 + Ilxj(0)11 2 + Ilw(-)ll~ + Ldsl.
(8.2.6)
s=l
(n) The followmg H OO reqwrement ~s satzsJied: If x(O) = then
°
and Xj(O) = 0,
136
8 Robust H"" Filtering with Structured Uncertainty
Remark Condltion (1) of DefimtlOn 821 reqUlres that the uncertam system (82 1), (8 2 3) IS absolutely stable (e g , see [115,117]) and the filter (82 5) IS stable as well Observation 8.2.1 It follows from the above defimtIOn that If the uncer-
tam system (8 2 1), (8 23) IS robustly observable wIth dIsturbance attenuatIOn ,,(, then zt must have the property that x(t) --> as t --> 00 Indeed, smce [xC ), ~( )] E L 2 [O, 00), we can conclude from (821) that x( ) E L 2[O, 00) However, usmg the fact that x( ) E L 2 [O,00) and x( ) E L 2 [O,00), zt now follows that x( t) --> as t --> 00
°
°
8.3
S-Procedure for Nonlinear Uncertain Systems on an Infinite Time Interval
In thlS sectlOn, we present a result whlch extends the "S-Procedure" of [81] to the realm of nonlmear systems see also SectlOn 7 3 for another Sprocedure result The result of [81]lS related to Fmsler's Theorem concernmg palrs of quadratic forms, e g , see [155] and also [164] The mam result of [81] apphes to coliectlOn of mtegral quadratlc forms defined over the space of solutlOns to a stable lmear tlme-mvanant system In th18 sectlOn, we extend thlS result to a more general set of mtegral functlOnals defined over the space of solutlOns to a stable nonlmear tlme-mvanant system A shghtly weaker verSlOn of thIS result was pubhshed m [119,123] The mam result of thIS sectlOn apphes to a nonlmear, tlme-mvanant system of the form
h(t) = II(h(t), 1jJ(t»
(83 1)
where h(t) E RN IS the state and 1jJ(t) E RM IS the mput Associated wIth the system (8 3 1) IS the followmg set of functlOnals
fo (h( ),1jJ(»
h
(h( ),l/J(»
1: 1:
vo(h(t), 1jJ(t»dt, vl(h(t),1jJ(t»dt,
!k (h( ),l/J(» AssumptIOns The system (8.3.1) and assoClated set of functlOnals satlsfy the following assumptIOns
83 S-Procedure for Nonlmear Uncertam Systems on an Infimte Time Interval
3.1 The functIons IT( , ), lIo (
, ),
lIk ( , )
137
are contInuous
3.2 For all1/J( ) E L;dO, 00) and all InItIal condItIOns h(O) ERN the correspondIng solutIOn h( ) belongs to L z [0,00) and the correspondIng , !k (h( ), 1/J( )) are fimte quantItIes fa (h( ), 1/J( )), fr (h( ), 1/J( )), 3.3 GIven any c > 0 there eXIsts a constant b > 0 such that the follOWIng conditIOn holds For any Input functIon 1/Jo() E {1/Jo() E L 2[0, 00), II1/Jo( )II~ ::; b} and any ho E {h o E RN Ilholl::; b}, let ho(t) denotes the correspondmg solutIOn to (831) wIth mitial con,k dItIOn ho(O) = ho Then Ifs (h o( ) 1/Jo( )) I < c for s = 0,1,
Remark AssumptIOn 33 IS a stabIhty type assumptIOn on the system (831)
Notatwn For the system (831) satIsfymg the above assumptIOns, we define
nc
Lz[O, 00) as follows n IS the set of {he ), 1/J( )} such that 1/J( ) E L 2[0, 00) and h( ) IS the correspondIng solutIOn to (831) wIth mitial condItIon h(O) = 0 Theorem 8.3.1 Conszder the system (831) and assocwted functwnals and suppose Assumptwns :1 1 - :1:1 are satzsfied If fa (h( ), 1/J( » ~ a for all {he ), 1/J()} E n such that fr (h( ) 1/J()) ~ 0, ,!k (h( ), 1/J()) ~ 0, then there exzst constants TO 2 0, TI ~ 0, ,Tk ~ 0 such that 2::=0 Ts > 0 and (832)
for all {he ), 1/J( )} E
n
Proof In the order to prove thIS theorem we estabhsh the followmg claIm
Clazm GIven any fa > 0 and any mput 1/Jo() E Lz[O,oo), then there exists a constant bo > 0 such that the followmg condItIon holds For any
ho E {h o E W\
Ilholl::; bo},
let hI (t) denotes the correspondIng solutIOn to (8 3 1) wIth ImtIal condItIon ht(O) = 0 and let h2(t) denotes the correspondmg solutIOn to (831) wIth InItIal condItIOn h2 (0) = ho Then
Ifs (h I (), 1/Jo()) - fs (h z (), 1/Jo(») I < fa for s = 0,1,
,k
138
8. Robust Hoc Filtering with Structured Uncertainty
°
Indeed, let EO > be some constant and let 6 be the constant from Assumption 3.3 corresponding to c = -';f. According to Assumption 3.2, [hI (.), 1/Jo (.) 1 E L2 [0,(0), therefore, there exists T > such that I hI (j) II :::; ~ and
°
£= 11~0(t)112dt
:::; 6.
Assumption 3.1 implies that there exists a constant 80 > 0 such that for all Ilholl < 60, the solution h2(-) of the system (8.3.1) with input 1/Jo(-) and initial condition h2 (0) = ho satisfies condition (8.3.3)
for all s and (8.3.4)
g,
Since IIhl(T)1I :::; we have from (8.3.4) that Ilh2(T)1I 5 5. Furthermore, Assumption 3.3 and the time invariance of the system (8.3.1) imply that
l~T II/s (h 1 (t), 1/Jo (t)) -
: ; l~T
(h2 (t), 1/Jo (t) ) Idt
(1I/s(hl(t), 1/Jo(t»1
EO
EO
4
4
< - +-
1/s
+ Il/s(h2(t), 1/Jo(t»l)dt
EO
=-.
2
From this and inequality (8.3.3). the claim follows immediately. Now suppose fo (h(·), 1/J(.)) 2: for all {h(-), 1/Je)} En such that
°
h (h(-), 1/J(.)) 2:
0, ...
,h (h(·), 1/J(-)) 2:
0
and let
M:= {[ fo(h(·),1/J(·)
h (hC), 1/J(.))
J' E Rk+l : {h(·), 1/Je)} En} .
°
It follows from the assumption on the set n that Xo 2: for all vectors [xo ... Xk]' E M such that Xl 2: 0, ... ,Xk 2: 0. Now let
be given. Since haC) E LdO, 00), then there exists a sequence {Tt}~1 such that Tt > 0 for all z and Tt -+ 00 and ha(Tt) -+ 0 as i -+ 00. Now consider the corresponding sequence {h t (·), 1/Jt(-)}~l en. where
8.4 Design of Itobust n-"Filtert
t39
We will establish that
as 2 --> 00 for s = 0,1, ... ,k. Indeed, let s E {0,1. ... ,k} be given and fix i. Now suppose h/,(.) is the solution to (8.3.1) with input 'IjJ(.) = 'ljJbC) and initial condition h/,(O) = ha(T,). It follows from the time invariance of the system (8.3.1) that h,(t) == h/,(t - T,). Hence,
Using the fact that ha (T,)
as i
-+ 00.
-->
0, the above claim implies
Also,
Hence, It follows from the above, that the set M has the property that a + b E cl(M) for all a, b E M. Hence, Lemma 7.3.2 implies that there exist
°
°
constants TO 2- 0, ... ,T1 2- such that 2:::=0 Ts > and TOXO 2- T1X1 + .. .+ for all [xo . . . Xk]' E M. That is, condition (8.3.2) is satisfied.
TkXk
o
8.4
Design of Robust Hoo Filters
In this section, we present the main result of this chapter which establishes a necessary and sufficient condition for the uncertain system (8.2.1), (8.2.3) to be robustly observable with a specified level of disturbance attenuation. This condition is given in terms of the existence of solutions to a pair of parameter dependent algebraic Riccati equations. The Riccati equations under consideration are defined as follows: Let T1 > 0, ... , Tk > be given constants and consider the algebraic Riccati equations
°
(8.4.1)
8 Robust H= Filtering with Structured Uncertainty
140
wltere
v'fkl Bk
]. (8.4.3)
Assumptwns The uncertain system (8.2.1), (8.2.3) will be required to satisfy the following additional assumptions: 4.1 The pair (A, L) is observable. 4.2 E
> O.
4.3 The pair (A, Bo) is controllable. 4.3 The pair (A, C) is observable.
Theorem 8.4.1 Consider the uncertam system (8.2.1), (8.2.3) and suppose that Assumptwns 4.1-4.4 are sat~sfied. Then the followmg statements are equwalent: (~)
The unceriam system (8.2.1}'(8.2.3) is robustly observable w~th turbance attenuatwn I vwa nonlmear filter of the form (8.2.5).
d~s
(n) There ex~st constants 71 > 0, ... . 7k > 0 such that the R~ccat~ equatwns (8.4.1) and (8.4.2) have solutwns X > 0 and Y > 0 and such that the spectral mdzus of the~r product sat~sfies p(XY) < l. If conddwn (22) holds, then the uncertam system (8.2.1), (8.2.3) ~s robustly observable with d~sturbance attenuatwn I vw a lmear tzme-mvarwnt filter of the form Afxf(t)
+ Bfy(t),
Lx f(t)
+ (ih - BfiJ21)B~X (I - Y X)-l(yC' + B1iJ~1)E-l.
(8.4.4)
A - BfC
(8.4.5)
8.4 Design of Robust Hoc FIlters
141
Proof
(2):=} (n) ConsIder the seUl ofvectorfunctions A( ) = [xC ),xf( ),~( ),w(·)] m L 2 [0, 00) connected by (821), (825) and the Imtial condition [x(O),Xf(O)] = 0 CondItion (82 7) Implies that there eXIsts a constant 61 J < ,2 - 261 Let
> 0 such that
where Cl and C2 are the constants from DefimtIOn 8 2 1 ConSIder the functIOnals fo,it, ,ik from to R where
n
_(11((( ) - (( ))II~ - ,21Iw( )II~ + 6211A( )11 2), 2 Il z l( )II~ - 116( )II~ + 6211A( )11 ,
fO(A( )) it(A( ))
(84.6)
Here
2 IIA( )11 = Ilx( )II~
+ IIXf( )II~ + II~( )II~ + Ilw( )II~ for all A() E n such that fs(A())
We prove that fO(A()) ~ 0 s = 1,. ,k Indeed, If fs(A( )) the constramts (823) wIth ds DefimtIOn 8 2 1 Implies that
~
=
~ 0 for 0 then thIS vector functIOn A(') satisfies 6211>'( )11 2 and t, = 00 CondItIOn (1) of
2 IIA( )11 :::; 2(CI
+ l)llw(
)II~
(8.4.7)
CondItion (ll) of DefimtIOn 8 2 1 Implies that
n
for all >.() E such that fs(>'()) ~ 0 for s = 1, ,k Smce 62 :::; 6t(2c2k(Cl + 1))-1 and 62 :::; 6I(2(CI + 1))-1, the mequalIty (847) Implies that 6111w( )II~ ~ C2k6211A( )11 2 and 6Illw( )II~ ~ 6211>'( )11 2 Therefore, fo(>'( )) ~ 0 Now the augmented system (8 21), (8 2 5) can be rewntten as the system (8 31) where h() = [xC ),xf()] and 1/;() = [6(), ,~d ),w()] Then, accordmg to Theorem 831 (AssumptIOns 31-33 ObvIOusly follow from CondItIOn (1) of DefimtIOn 82 1 and the contmUlty of the coefficIents of the filter (825)). there eXIst constants TO ~ O. Tl ~ 0, ,Tk ~ 0 such that 2:::;=0 Ts > 0 and the mequalIty (8.32) IS satIsfied for all A(') En.
142
8. Robust H oo Filtering with Structured Uncertainty
Now we prove that Ts > 0 for all s the functionals (8.4.6) implies that
=
0.1, ... , k. Condition (8.3.2) on k
To(II(((-) - (.))II~ _,21Iw(-)II~) + LT8(1lz8(-)11~ - 11~8(-)11~) 8=1
k
::; -60(L 11~8(-)11~
+ Ilw(')II~)
(8.4.8)
8=1 ~k
.
where 60 = 62 L..-8=0 Ts > O. If Tj = 0 for some J = 1, ... ,k, then we can take w(·) =::: 0, ~s(-) =::: 0 for all s 1= j and ~j(-) 1= O. Then, the inequality (8.4.8) is not satisfied, because the left side of (8.4.8) is non-negative and the right side of (8.4.8) is negative. [Analogously, if TO = 0, we can take w(·) 1= 0 and ~(.) =::: 0.] Therefore, Ts > 0 for s = 0,1, ... ,k. In this case, we can take in (8.4.8) TO = 1. Consider the following linear control system with the disturbance input
+ Fhw(t) + Fhu(t); Kx(t) + D 12 U(t); Cx(t) + D21w(t)
x( t)
Ax(t)
z(t)
yet)
(8.4.9)
where u(t)
((t),
VTl6(-),···
,VTk~k(-)],
w(-)
[,w(-),
, z(·)
[((-), yTl Z1(-) .... , ylTk"zk(-)],
the matrix coefficients Fh, C\, D12 and D21 are defined by (8.4.3) and O. Then, the inequality (8.4.8) with TO = 1 may be rewritten as
Ilz(')II~
-
Ilw(')II~
::;
-601Iw(·)II~·
B2 =
(8.4.10)
This implies
(8.4.11) Therefore, the controller u(t)
= (t)
(8.4.12)
where ((-) is the output of the filter (8.2.5) with the initial condition x/CO) = 0 solves the standard output feedback H= control problem for the system (8.4.9). Condition (ii) of the theorem follows from this directly using Theorems 5.5 and 5.6 of [13]. This completes the proof of this part of the theorem.
8.4 Design of Robust Hoc Filters
143
(ii) => (i) It is a standard result from H X control theory, e.g .. see [13 ,97]. that if condition (ii) holds then the linear controller (8.4.12). (8.4.4) solves the standa rd output feedback Hoo control problem (8.4.11) for the system (8.4.9). Condition (8.4.11) implies that there exists a constant 80 > 0 such that the inequality (8.4.10) is satisfied for all the solutions of the augmented system defined by (8 .4.9) and (8.4.4) with w(·) E L2[0, 00) and the initial condition [x(O), xf(O)] = O. The a ugmented uncertain system defined by (8.2.1) and (8.4.4) may be rewritten as
h(t) where
h= [
= Ph(t) + Qw(t),
(8.4.13)
:f ].P = [B~C ;f] 'Q = [ %:2 ] .
Since the controller (8.4.12) , (8.4.4) solves the H OO control problem , the matrix P is stable. Condition (8.2.3) implies that any disturbance input we) E L2 [0,00) and admissible uncertainty inputs ';1 (.) .... . ';k (-) satisfy the following integral quadratic constraint: There exists a constant d :::: 0 and a sequence {tt}~1 : t, ........ t. such that (8.4. 14) where k
d = "?llw( ')II ~ + LTsds; T =
[k D12 L]
.
s=1
z
Also, since = Th. condition (8.4.10) and stability of the matrix Pimply that there exists a constant 8 > 0 such that
1=
(1ITh(t)112 - IIw(t)1I2)dt
: ; -81
00
(1Ih(t)1I2
+ IIw(t) 112)dt
(8.4.15<)
for all rhO, w( ·)] E L 2 [0,00) connected by (8.4.13) with h(O) = O. Using Theorem 1 of [167] , condition (8.4.15) and the stability of the matrix P imply absolute stability of the uncertain system (8.4.13), (8.4.14). Thus, there exists a constant Co > 0 such that for any initial condition [x(O) , xc(O)] and any uncertainty w(·) described by (8.4.14), then [xO, x c (-), w(-)] E L 2 [0,00) and
1=
Cllx(t)1I2+ Ilx c (t) 1I2+ IIw(t)1I2)dt ::; eolllx(0)1I2
+ II xc(0)11 2 + d].
(8.4.16)
144
8. Robust Hoc Filtering with Structured Uncertainty
Since u(t)
= ((t) = LXf(t) in controller (8.4.12), (8.4.4).
k
d
= 'lllw(')II~ + LTsds. s=1
condition (i) of Definition 8.2.1 follows from the inequality (8.4.16). We now establish Condition (ii) of Definition 8.2.l. First note that Condition (8.4.10) is satisfied for all solutions of the system (8.2.1), (8.4.4) with the zero initial condition. This may be rewntten as condItIOn (8.4.8) with TO = 1. Furthermore, all solutions to the system (8.2.1), (8.4.4) in LdO, (0) satisfy condition (8.2.3) with d s = max[O. -(llzs( )II~ - II~s(')II~)J and tt = 00. Therefore, condition (8.2.7) with C2 = maxh, ... ,TkJ follows 0 from (8.4.8). This completes the proof of the theorem. The following corollary is an immediate consequence of the above theorem. Corollary 8.4.2 If the uncertam system (8.2.1)'(8.2.3) zs robustly observable wzth dzsturbance attenuatwn -y vza a nonlmear filter of the form (8.2.5), then zt zs robustly observable wzth dzsturbance attenuatwn -y vza a lmear tzme-mvarzant filter.
The following corollary is an immediate consequence of the above theorem and the remarks following the definition of the uncertain system (8.2.1), (8.2.3). Corollary 8.4.3 The uncertam system wzth norm bounded uncertamtzes, (8.2.4) wzll be robustly observable wzth dzsturbance attenuatwn -y vza the lmear filter (8.4.4),(8.4.5) zf condztwn (n) of Theorem 8.4.1 zs satzsfied.
9 Robust Fixed Order Hoo Filtering
9.1
Introduction
In this chapter, we consider a robust H oo filtering problem with a constraint on the order of the filter. Unlike the integral quadratic constraint uncertainty description used in the robust HOO filtering problem considered in Chapter 8. in this chapter we consider uncertain systems with norm bounded uncertainty. This uncertainty description is similar to that considered in Chapter 2. However, in this chapter we consider a robust H OO filtering problem rather than a guaranteed cost filtering problem such as considered in Chapter 2. The standard approaches to the robust Hoo filtering problem lead to a filter whose order is the same as that of the underlying system model. However, in many practical problems, it is desired to use a filter of a given fixed order. In this chapter, we investigate whether in this problem of fixed order robust HOC! filtering, there is any advantage to be obtained by using a nonlinear filter as opposed to using a linear filter. Within the area of robust controL the problem of non-linear versus linear control has been widely studied; e.g., see [69,93,112,119,129.149]. However, this issue has not been widely studied in problems of filtering. The main result of this chapter shows that for the fixed order robust Hoo filtering problems under consideration. there is no advantage to be obtained by using a nonlinear filter as opposed to a linear filter. The main results of this chapter were originally published in [134]. An analogous result was given in Chapter 8 for the problem of robust Hoo filtering with structured uncertainty.
146
9 Fixed Order Filtenng
9.2
Fixed Order Hoo Filters
Consider the linear uncertain system
x(t) ((t)
Ax(t) + Bow(t) Lx(t),
z(t)
Kx(t),
yet)
Cx(t)
+ Bl~(t);
+ Dow(t) + Dl~(t)
(9.2.1)
where x(t) ERn IS the state, ~(t) E RP is the uncertamty mput, wet) E Rm is the dzsturbance mput, ((t) E RT is the error output, z(t) E Rq is the uncertamty output and yet) E Rl is the measured output. Let k E {O, 1,2, ... }. We now consider a problem of robust filtering for the uncertain system (9.2.1) via a nonlinear time-varying filter of order k of the form
xJ(t) = A(t, Xj(t), y(t»), ((t) = A(t, xJ(t), yet»~ where x J(t) E Rk, A(-, 0, 0) == 0 and A( ,0,0) filter of the same order of the form
(9.2.2)
== 0, or a linear time-invariant
= AjxJ(t) + Bjy(t), ((t) = CJxJ(t) + DJy(t)
Xj(t)
(9 2 3)
We now introduce a corresponding notion of observability for the uncertain system (9.2.1); c.f. DefInition 8.2.1.
Definition 9.2.1 The uncertam system (9 2.1) zs sazd to be quadratically observable with unity disturbance attenuation via a nonlinear time-varying filter of order k zJ there exzsts a Jilter oj the Jorm (9.2.2), a matnx H = H' > 0 and a constant e > 0 such that the dzmenswn oj H zs (n + k) x (n + k) and the Jollowmg condztwn holds: Let
h(t)
~
[X(t) ], V(h) x J(t)
and let V[h(t)] be the denvatzve oj V(h(t» mented system (9.2.1). (9.2.2). Then, V[h(t)]
+ 11((t) -
~ hi Hh along tra]ectones oj the aug-
((t)112 + Ilz(t)112 - 1I~(t)112 - IIw(t)1I2 ::: -e(llh(t)112 + IIE(t)11 2 + Ilw(t)112)
(9.2.4)
for all solutwns to the augmented system (9.2.1), (9.2.2). The uncertam system (9.2.1) 1S sazd to be quadratically observable with unity disturbance attenuation via a linear time-invariant filter of order k zf there extsts a filter of the form (9.2.3), a matrzx H = H' > 0 and a constant e > 0 such that condztwn (9.2.4) zs satzsJied wzth the Jilter (9.2.2) replaced by the {mear filter (9.2.3).
9.3 Nonlinear versus Linear Fixed Order Filterinc
147
Remarks Consider the following uncertain system with uncertainty satisfying a. norm bound condition:
+ Bl~(t)Klx(t) + Bow(t); [C + Dl~(t)Klx(t) + Dow(t);
i(t)
[A
yet)
11~(t)11 ~ 1
(9.2.5)
where 6(t) is the uncertainty matrix. Let ~(t) = 6(t)Kx(t). Then the system (9.2.5) may be rewritten in the form (9.2.1) with 11~(t)112 ~ Ilz(t)112. Hence, if condition (92.4) holds with some filter and some quadratic function, then this filter satisfies the HXJ requirement
for the uncertain system (9.2.5).
Nonlinear versus Linear Fixed Order Filtering
9.3
In this section, we present the main result of this chapter.
AssumptIOn We make the following assumption:
(Do
Dl
1 [~~ ~~] = l 0
I].
Assumption 3.1 is a standard HXJ type assumption (see, e.g., [56]).
Theorem 9.3.1 Consider the uncertazn system (9.2.1) and suppose that AssumptIOn 3.1 holds. Let k E {O, 1, 2 .... }. Then the followzng statements are eqmvalent: (~)
The uncertazn system (9.2.1) ~s quadrat~cally observable vw a nonlznear t~me-varymg filter of order k.
(n) The uncertazn system (9.2.1) ~s t~me-znvarwnt filter of order k.
(m) There
quadrat~cally
observable vw a lznear
matrzces X = X' > 0 and Y = Y' > 0 such that the of X and Y ~s n x n and the followzng cond~tIOns hold:
ex~st
d~mensIOn
+ XA + X(BoB; + BIB~)X + K' K < 0: (9.3.1) YA+A'Y + Y(BoB; +BIB~)Y +L'L+K'K - dc < 0, A'X
(9.3.2)
X ~ Y; rank(Y - X) ~ k.
(9.3.3)
148
9 Fixed Order Filtering
In order to prove this theorem. we will use the following lemma which can be found in the papers [45.50,56]. Lemma 9.3.2 Conszder the control system
+ BI w(t) + B2 u(t)j Kx(t) + D12U(t): Cx(t) + D21W(t)
x(t)
Ax(t)
i( t) y(t) where
w(t)
(9.3 4)
is the dzsturbance znput, u( t) zs the control znput. Let k E
{0.1, 2, ... } and suppose that E
L:l. ' = D12D12 > 0,
D21 [B~
D~l] =
[0 I]
°
°
and there exzst matncfS X = X' > and Y = Y' > such that the dzmenswn of X and Y zs n x n, condztwn (9.3.3) holds and '"
1'"
+ X(A
(A - B 2 E- Dl2K)' X
-1
AI'
'"
- B 2 E- Dl2K) -1
'
....
+ K (I - Dl2E DdK < 0: (9.3.5) 1 1 Ay- + y- A' + y-l (K' K - d C)y-l + BIB~ < 0. (9.3.6) "AI
"
+X(BIBI - B2E
"'I
B 2 )X
A'
Then, there exzsts a lmear tzme-lnvanant controller of order k oj the form
Xc (t)
Acxc(t)
+ Bcy(t),
u(t)
Ccxc(t)
+ Dcy(t)
(9.3.7)
whzch solves for the system (9.3.4) the followmg H= control problem: A
L:l.
J =
sup x(O)=O. w( )EL 2 [0.oo)
Proof of Theorem 9.3.1 (i)
=}
fo= \\i(t)\\2dt = < l. fo \\w(t)\\2dt
(iii)
(9.3.8)
°
Suppose that a filter of the form (9.2.2), a matrix H = H' > and a constant E > satisfy condition (9.2.4). Introduce matrices y, Z and W as follows.
°
H =
[~, ~]
where the dimension of Y is n x n, the dimension of Z is n x k and the dimension of W is k x k. Then. since H > 0, we have Y > O. Also, W > if k > 0. Let X ~ Y - Zw- 1 Z' if k > 0 and X ~ Y if k = 0. Then, conditions (9.3.3) obviously hold. Also. X > 0, because
°
and H > 0 Now we prove that the matrix X satisfies condition (9.3.1). Let Vx(x) ~ x'Xx and Vx[x(t)] be the derivative of VxC) along trajectories
9.3 Nonlinear versus Linear Fixed Order Filtering
149
of the system (9.2.1). Consider the Lyapunov derivative V[h(t)] for the system (9.2.1), (9.2.2) at a point h(t) = [x(t)' xJ(tY], such that xJ(t) = _W- 1 Z' x(t). Then
V[h(t)]
-ZW- 1
2x(t) [ I
] [
~,
Z ] [ x( t) ] xJ(t)
W
o ] [ x( t) ]
2x(t) [ X
x J(t)
Vx[x(t)]. However, since (9.2.4) must hold at all points h(t) E Rn+k, we have
+ 11«((t) - (t))112 + IIz(t)112 - 11~(t)112 - Ilw(t)112 :::: -E(llx(t)112 + 11~(t!112 + Ilw(t)112) (9.3.9)
Vx[x(t)]
for all solutions to the system (9.2.1). Furthermore, (9.3.9) implies that
x(T)' Xx(T) - x(O)' Xx(O)
+ <
i
T
(E 11 X(t)11 2 + IIKx(t)112
-E iT (l1~(t)112
_11~(t)112 -llw(t)112)dt
+ Ilw(t)112)dt
for any T > 0 and for all solutions to the system (9.2.1). Hence, the solution XT(-) to the Riccati differential equation
- XT(t) XT(T)
=
=
A'XT(t) X
+ XT(t)A + XT(t)(BoB~ + BIB~)XT(t) + f( k, (9.3.10)
is defined and positive-definite on [0, T] and XT(O) :::: X (see e.g. [76]). Here
k ~ [K' VEl]'.
Furthermore,
XT(O)
= XT(T) - iT XT(t)dt X -
10
T XT(t)dt.
Hence,
10 for any T
> O.
T
XT(t)dt 2: 0
Now it is clear that
T
lim T---+O
~ r T 10
XT(t)dt = X(O)
150
9. Fixed Order Filtering
where XC) is the solution to the equation (9.3.10) with initial condition X(O) = X. Hence, X(O) 2: O. Therefore.
A'X + XA + X(BoB~ + BIB~)X + f(k:s:
o.
Since k' k > K' K, it follows that the inequality (9.3.1) holds. Now we prove that the matrix Y satisfies condition (9.3.2). Condition (9.2.4) implies that
h(T)' Hh(T) - h(O)' Hh(O)
1
_11~(t)112 -lIw(t)112)dt
T
+
(II((t) - «(t)112 + Ilz(t)112
1 T
< -E
(llh(t)11 2 +
11~(t)112 + Ilw(t)112)dt
(9.3.11)
for any T > 0 and for all solutions to the augmented system (9.2.1), (9.2.2). Let D~l be a matrix such that
Such a D~l exists due to Assumption 3.1. Define for any vC) E L2 [0, T], a vector function
~(t)']' ~ - [Do D 1
[w(t)'
Cx(t)
]'
+ D~; v(t).
Then, for the system (9.2.1), (9.2.2) with these inputs ~(.) and w(·) and initial condition xf(O) = 0, we have y(.) == o. «(.) == 0 and condition (9.3.11) may be re-written as
x(T)'Y x(T) - x(O)'Y x(O)
( Ellx(t)112 + IILx(t)112 + IIKx(t)112 ) dt -IICx(t)112 -llv(t)112
(
+ io <
-E
I
T
(IICX(t)11
2
+ Ilv(t)112)dt
for any T > 0 and for all solutions to the system
x(t) = Ax(t) + [Bo
B1
]
DH v(t).
Hence, the solution P(·) to the Riccati equation
-Pet)
= P(t)A + A' pet) + P(t)(BoB~ + BIB~)P(t)
+L'L+k'k-dc. peO) =Y
9.3 Nonlinear versus Linear Fb(ed Order Filtering
151
is defined on [0,00) and peT) ?': Y for all T > O. Here
k ~ [K' JEl
J' .
From this we have that p(O) ?': O. Hence,
+ A'y + Y(BoB~ + BIB~)Y + L'L + k' k - dc:s: o.
YA
Since f( k > K' K, it follows that condition (9.3.2) holds. (iii) => (ii) Consider the system (9.3.4) with
fh D12
[Bo
Bl]'
-l,D21
fh =
= [Do
0,
k =
[L'
K']"
D 1 ].
'"r~~"'" "",,,,, ~",, ,,~"" 1;:.,~ ,,~_'w;.;j"J'pJV1S. r.JllJnitjD..DR (,
and (9.3.6) for the system (9.3.4). According to Lemma 9.3.2, there exists a controller of the form (9.3.7) which solves the problem (9.3.8). Furthermore, condition (9.3.8) may be re-written as the condition
~
J -
fo=(II(((t) - ((t))112 + Ilz(t)112)dt 1 < (9.3.12) [€(),w()]EL 2 [0,=) fo= (11~(t)112 + 111P (t ) 112) dt sup
for the system (9.2.1) with x(O) = O. ((.) == uO and [w(·), ';(')'], == w(·). It is well known. that if condition (9.3.12) holds for the linear timeinvariant system (9.2.1), (9.2.3), then there exist il matrix H = H' > 0 and a constant f > 0 such that condition (9,2.4) holds for the system (9.2.1), (9.2.3).
(ii) => (i) This statement is obvious from the definitions.
o
Remarks In the proof of the above theorem, we show thilt if there exist matrices X > 0 and Y > 0 satisfying conditions (9.3.1), (9,3.2), (9.3.3), then there exists a linear filter solving the fixed order robust g= filtering problem. The problem of finding the required matrices X and Y is a non-convex problem. However, there do exist algorithms which claim jiO solve this problem in a large percentage of cases; e.g., see [50]. Furthermore, once the matrices X and Y have been found, then based on the resultS of [50], it is straightforward to construct the required filter. Also, it should be noted that for the case in which k = 0, that is a constant filter is required, then the problem of constructing X and Y becomes a convex problem for which standard convex programming methods can be applied.
10 Set-Valued State Estimation for Nonlinear Uncertain Systems
10.1
Introduction
In previous chapters, we have concentrated on the problem of robust state estimation for the case in which the nominal model is linear. However, there are many applications of Kalman Filtering in which the underlying signal model is nonlinear. The approach which we will follow in this chapter is the set-valued state estimation approach such as presented in Chapter 4. We will model the uncertainty and disturbances in the system deterministically via a nonlinear integral constraint. This nonlinear integral constraint is an extension of the Integral Quadratic Constraint (IQC) considered in Chapter 4; see also [123J. However unlike Chapter 4, we will deal with the case in which the state equations defining the uncertain system are nonlinear. The results presented in this chapter are based on those in the papers [58,59]. It should be noted that the paper [11 J also considers a related nonlinear setvalued state estimation problem. However in [11 J, the disturbance model is a disturbance model of the type given in [16J rather than a nonlinear integral constraint uncertainty model which allows for uncertain dynamics. As in Chapter 4, the underlying technical idea used in this chapter is to view the set-valued state estimation problem as an optimal tracking problem. This tracking problem is then solved using methods of dynamic programming. In particular, the value function corresponding to this optimal tracking problem is the solution to a Hamilton Jacobi Bellman partial differential equation. The level sets of this value function then define the set-valued state estimates. The solution to the Hamilton Jacobi Bellman
154
10. State Estimation for Nonlinear Uncertain Systems
partial differential equation serves as an "information state" for the robust state estimation problem being considered. An important practical approach to dealing with nonlinear filtering problems is the Extended Kalman Filter (EKF). The standard Extended Kalman Filter involves linearizing the nonlinear model dynamics at each time step in the filter iteration. The linearization is carried out around the current state estimate. One of the main results presented in this chapter is a robust, set-valued state estimation version of the Extended Kalman Filter. This is achieved by approximating the value function mentioned above by a quadratic form. Also, the dynamics of the nonlinear model are linearized about the robust state estimate. This is the state achieving the minimum value of the value function.
10.2
Problem Formulation
We consider nonlinear uncertain systems of the form
K(x.u) C(x)
(10.2.1)
+v
defined on a finite time interval [0, s]. Here, x(t) E Rn denotes the state of the system, y(t) E RI is the measured output and z(t) E Rq is the uncertainty output. The uncertainty inputs are w(t) E RP, v(t) E RI. Also, u(t) E Rm is the known control input. We assume that all of the functions appearing in (10.2.1) are of class C 1 with bounded first derivatives. Additionally. we assume that K(x, u) is bounded. This assumption has been made to simplify the technical development in this chapter. In practice this assumption could be removed. However. this would lead to many technical problems in the proofs to be presented. The matrix B1 is assumed to be independent of x. The uncertainty in this system is defined by the following nonlinear integral constraint:
(10.2.2) where d ~ 0 is a positive real number: c.f. the IQC (4.2.2). Here. CP, L1 and L2 are bounded non-negative functions of class C 1 satisfying growth conditions of the type I¢(x) - ¢(x')1 ::; fJ(l
+ Ixl + Ix'l)lx -
x'i
(10.2.3)
10.2 Problem Formulation
155
where f3 > 0 , and ¢ = if>. L 1 , L 2 · Uncertainty inputs w(-), v(-) satisfying this condition are called admzsszble uncertazntzes. The uncertainty in the uncertain system (10.2.1), (10.2 .2) can be regarded as a feedback interconnection between the nominal nonlinear system (10.2.1) and an uncertainty block which takes the uncertainty output z and produces the uncertainty inputs wand v. This is illustrated in Figure 10.2.1.
Uncertainty
w
Nominal System
v
z y
FIGURE 10.2.1. Block diagram representation of uncertain nonlinear system. We consider the problem of characterizing the set of all possible states Xs of the system (10.2.1) at time s 2' 0 which are consistent with a given control input u°(-) and a given output path yo( .); i.e., x E Xs if and only if there exist admissible uncertainties such that if uO(t) is the control input and xC) and y(-) are resulting trajectories, then xes) = x and yet) = yO(t). for all 0 :::; t :::; s. Remark The linear version of this problem was presented in Chapter 4. In the case K = 0, this problem was considered in [16] for linear systems, and in [11] for nonlinear systems. Also, it should be noted that as in previous chapters, the nonlinear integral constraint uncertainty description includes as a special case, the standard norm bounded uncertainty description. In this case, this would lead to an uncertain nonlinear system of the form
y
A(x, u) + BW'':',.l (t)K(x, u) C(x) + 6.2(t)K(x, u) + Vo
+ B12WO;
where
is a matrix of uncertain parameters satisfying the norm bound
(10.2.4)
156
10. State Estimation for Nonlinear Uncertain Systems
c.f. (4.2.3). Also. wo(t) and vo(t) are noise signals satisfying the bound
+
los [wO(t)'Q2WO(t) + vo(t)' R2VO(t))]dt :S d.
It is straightforward to see that this uncertain system leads to the satisfaction of conditions (10.2.1). (10.2.2) when we let
A(x.u)+BIW L1(w,v)
10.3
A(x. u) w'
+ [Bll
B 12 ]W;
[~l ~2]
W
+ v'
w(t)
t:..l(t)K(x, u) ] . [ wo(t) .
v(t)
t:..2(t)K(x, u) ] [ vo(t) .
[~l ~2] v;
The State Estimator
In this section we show that the state estimation set Xs is characterized in terms of level sets of the solution V(x, s) of the PDE
a a,V
{
+
:axwm- _
{ V' x V . (A(x. un) L,(w,
yO _
+ BIW) + L,(K(x, U O)) } = 0
C(x))
(10.3.1 )
V( .0) -
The PDE (10.3.1) can be viewed as a filter, taking observations uO(t). yO(t), and producing the set Xs as an output. The state of this filter is the function V (" s): thus V is an mformation state for the state estimation problem. In shorthand notation, equation (10.3.1) can be written
o :S t :S s,
.
° °
V = F(Vu ,y ).
(10.3.2)
where F(V, uO. yO) is the nonlinear partial differential operator
This is illustrated in Figure 10.3.1. In considering the solutions to the PDE (10.3.1), it is convenient to allow for non-smooth solutions. In particular, we consider solutions in a viscosity sense. A precise definition of viscosity solutions to PDEs is beyond the scope of this book. However, the interested reader can refer to [79] for more details.
10.3 The State Estimator
157
v = F(V, u O, yO) Xs = {x : V(x, s) ::; d}
FfGURE 10.3.1 Uncertam set filter structure. Theorem 10.3.1 Assume that the uncertam system (10.2.1), (10.2.2) sattsfies the assumptwns gwen above. Then the correspondmg set of posstble states zs gwen by
Xs = {x
E
Rn
:
vex, s) ::; d},
(10.3.3)
where V(x,t) tS the umque VtSCOStty solutwn of (10.3.1) m C(Rn x [O,s]). Proof
Using the results of [79], it follows from the assumptions made on the uncertain system (10.2.1) (10.2.2) that if the PI)E (10.3.1) has a solution in C(Rn x [0, s]), then it will be unique in the viscosity sense. Define the function
Vex, t) =
mm _ lex, t, lV),
(10.3.4)
wEL 2 [O,t]
where x(·) solves (102.1) with given control input uOU and xCi) = x. Here the cost functional lex. t, w) il:> given by
lex. t,w) "" (x(O))
+ foi[Ll(W(t),V(t)) - L 2 (z(t))J dt
where v(-) = y0(-) - C(x(·)). Then by standard methods (see. e.g. [42]), V belongs to C(Rn x [0, s]) for any s 2:: 0 and is a viscosity solution of (103.1). (In particular. our assumptions ensure that Vex. t) is finite.) From this, it follows that the PDE (10.3.1) has a viscosity solution V(·). Furthermore, this solution is unique and given by the formula (103.4). Now let x E Xs be given It follows that there exists an uncertainty input w*(-) and an initial condition x(O) such that the nonlinear integral constraint condition (10.2.2) is satisfied with v(-) = yOU - C(x(·)) and x( s) = x. From this it follows that
lex, s, w*) ::; d. Therefore vex, s) ::; d. Conversely, suppose x E Rn is such that Vex, s) ::; d. It follows that there exists a function w* (-) such that lex, s, w*) ::; d. From this it follows that xC) the corresponding solution to (10.2.1) is such that the nonlinear integral constraint (10.2.2) is satisfied. Hence x E Xs' (j]
158
10. State Estimation for Nonlinear Uncertain Systems
10.4
A Robust Extended Kalman Filter
In the previous section. we considered a formula for the set Xs defined in terms of the PDE (10.3.1). In this section, we consider an approximation to this formula which leads to a Kalman Filter like characterization of the set Xs. In the linear case, this approximation is exact and the characterization reduces to the results of [120]. The approximation considered in the nonlinear case amounts to an Extended Kalman Filter version of the results of Chapter 4. The derivation of this robust EKF is similar to the derivations of the standard EKF given in [10.55.88]. In this section, we consider uncertain systems described by (10.2.1) and an integral quadratic constraint of the form
(x(O) - xo)' Xo(x(O) - Xo) 1
liS
+-
2
t
°
[w(t)'Qw(t)
+ v(t)' Rv(t))]dt (10.4.1)
:S:d+'2}o z(t)'z(t)dt
where N > O. Q > 0 and R > O. That is, we consider the special case in which the nonlinear integral constraint (10.2.2) is an IQC of the form (4.2.2). For the system (10.2.1). (10.4.1), the PDE (10.3.1) can be written
as gtV+\7xV.A(x.uO)+~\7xVBIQ-IB~\7xV' -~(yo
{
- C(x))' R(yO - C(x))
+ ~K(x, uO)' K(x, uO) = 0
V(x,O) = (x - xoY N(x - xo).
(10.4.2)
We now consider a function x(t) defined so that
x(t) ~ arg min V(x, t). x
From these definitions, it follows that
\7 xV(x(t).t) = 0
(10.4.3)
and 2
.
\7 xV(x(t), t)x(t)
a
+ at \7 x V(x(t), t)'
= O.
(10.4.4)
We now take the gradient with respect to x of equation (10.4.2). This yields
:t
\7 ",v' + \7 xA(x. uo) . \7 ",v' + \7;V . A(x, uO) + V';V B 1 Q-l B~ \7 x V'
+\7 x C(x)'R(yO - C(x)) \7 x V(x, 0) = (x - xo)' N.
+ \7xK(x.uo)'K(x,uO) = 0; (10.4.5)
10.4 A Robust Extended Kalman Filter
159
If we evaluate this equation at x = x(t) and use (10.4.3) and (10.4.4), we obtain
V';V(x(t), t)i:(t) x(O)
=
V';V· A(x(t).uo) + V'xC(x(t))'R(yO - C(x(t))) +V' xK(x(t), uO)' K(x(t). uo) (10.4.6) Xo.
Furthermore, if the matrix V'~V(:i:(t), t) is nonsingular for all t, we can re-write this equation as
('() °
~ x(t)
=
A x t ,u ) + V' x V x(t . t)
x(O)
=
Xo.
[2 (' )
]-1 { V'xC(:i:(t))'R(yO - C(:i:(t))) } +V' xK(x(t). uO)' K(x(t), u O) (10.4.7)
Also, if we take the gradient with respect to x of (10.4.5), we obtain
:t
+ V'xA(x,u°)'. V';V + V';A(x,uo)'. V'xV + V';V. V'xA(x.uO) +V';V . A(x. uo) + V';V B 1 Q-1 B~ V' xV' + V';V B 1 Q-1 B~V';V +V';C(x)'R(yO - C(x)) - V'xC(x)'RV'xC(x) + V';K(x,uo)'K(x,uo)
V';V
+V' xK(x, uO)'V' xK(x, un) = 0; V';V(-,O) = N.
(10.4.8)
Equations (10.4.7) and (10.4.8) define a nonlinear version of the robust Kalman Filter presented in Chapter 4. However, these equations do not simplify the problem of solving the PDE (10.4.2). To obtain an approximate solution to (10.4.2), we now approximate V(x. t) by a function the form -
1
V(x. t) = 2(x - :i:(t))' X(t)(x - .i(t)) + ¢(t)
(10.4.9)
where :i:(t) is defined as the solution to the state equation
-'-() x t
_ -
A - ) 0) -1 [V' xC(:i:(t))' R(yO - C(:i:(t))) ]. (x(t ,u +X (t) +V'xK(x(t), uO)'K(x(t), UO) .
x(O)
=
Xo.
(10.4.10)
X(t) is defined as the solution to the Riccati Differential Equation (RDE)
V' xA(x. uO)' X + XV' xA(x, uo) + XB 1 Q-1 B~X - V' xC(x)' RV' xC(x) + V' xK(x, uO)'V' xK(i. uo) = 0; X(O) = N (10.4.11)
X+
and
160
10. State Estimation for Nonlinear Uncertain Systems
The above expressions for the quantities, x(t), X(t) and
\7;V(x(t), t)
= X(t).
Similarly, the RDE for X(t) was obtained by replacing \7~V(x(t), t) by X(t) in (10.4.8), then evaluating at x = x(t) and using (10.4.3) and (10.4.4). Neglecting the higher order gradient terms in A(x, u), C(x) and K(x, u), this yields
x + \7 xA(x, uD)'X + X\7 xA(x, u D) + XB 1Q-1 B~X -\7 xC(x)'R\7 xC(x) X(O) = N.
+ \7xK(x,uD)'\7xK(x,uD)
= 0;
If we replace x by x in this equation, (10.4.11) then follows. Furthermore, the corresponding differential equation for P( t) = X (t) -1 is
P =
P(O)
P\7 xA(x, u D)' + \7 xA(x, uD)P + B 1Q-1 B~ +P[\7 xK(x, u D)'\7 xK(x, u D) - \7 xC(x)' R\7 xC(x)]P: N- 1 . (10.4.13)
To derive the formula (10.4.12) for
a-v at -(x - x)' X(t)i
+ ~(x - x)' X(t)(x - x) + ¢(t)
_( --)'X(t) (A(- D) x x X,u
+ X-
[ \7 xA(x, uD)' X
1
--(x-x)' 2
1(t)[\7x C (X)'R(yD_C(X»]) "K(-x,u D)'K(-x,u D) +vx
+ X\7 xA(x, uD)
+XB1Q-1B~X-\7xC(x)'R\7xC(i:)
+\7x K (x.u D)'\7x K (x,u D)
+¢(t).
1(x-x)
(10.4.14)
Also,
\7 xV = (x-x)'X(t). We now replace
gt V
by
gt V
(10.4.15)
and \7 x V by \7 x V in (10.4.2). Using the
10.5 Local Results for the Robust Extended Kalman Filter
161
formulas (10.4.14) and (10.4.15), this yields
¢(t)
= (x - i)' X(t) [A(x, uo) - A(i, uo) - V' xA(i, uo)(x - i)] +(x - i)'V'xC(i)'R(yO - C(i)) 1
-2(x - i) 'V' xC(i)' RV' xC(i)(x - i) 1
+2(YO - C(x))'R(yO - C(x)) + (x - i)'V'xK{i,uo)'K(i,uo) 1
+2(x - i) 'V' xK(i. uO)'V' xK(i, uO)(x - i)' 1 -2K(x. u°),K(x, u °).
(10.4.16)
We now use the following first order approximations in this equation for
¢( t): ~
A(x, uO) C(x) K(x, uO)
~
~
A(i, uO) + V' xA(i. uO)(x - i), C(i) + V' xC(i)(x - i). K(i, ?L o ) + V' xK(.i:, uO)(x - i).
Substituting these approximations into (10.4.16) leads to the equation
Integrating this equation from 0 to 8 leads immediately to the formula (10.4.12). Equations (10.4.10), (10.4.11) and (10.4.12) define our approximate solution to the PDE (10.4.2) according to the relation -
V(x, t)
=
1
2(x - i(t))' X(t)(x - i(t»
+ ¢(t).
Hence. it follows from Theorem 10.3.1 that an approximate formula for the set Xs is given by Xs
= {x
E R
1 n : 2(x - i(8))' X(8)(X - i(8» ::; d - ¢(8)}.
This formula amounts to an Extended Kalman Filter generalization of the results of Chapter 4.
10.5
Local Results for the Robust Extended Kalman Filter
The robust Extended Kalman Filter derived in the previous section was based on the linearization of the nonlinear uncertain system (10.2.1), (10.4.1).
162
10. State Estimation for Nonlinear Uncertain Systems
This process of linearization will introduce some errors in the corresponding state estimation set Xs as compared to the true state estimation set Xs derived in Section 10.3. However. one might expect that in the limit as the signals in the uncertain system (10.2.1). (10.4.1) approach zero. the state estimation set Xs will approach the true state estimation set Xs. This question is the subject of this section. We now consider the behavior of the uncertain system (10.2.1), (10.4.1) in the limit as all signals approach zero. To this end, we assume that the integral quadratic constraint (10.4.1) is replaced by the integral quadratic constraint
(x(O) - axo)' N(x(O) - axo) +
11°
~
r [w(t)'Qw(t) + v(t)' Rv(t))]dt
2Jo
8
< ex 2 d+-
z(t)'z(t)dt
(10.5.1)
2
where ex > 0 is a scaling parameter. We will consider the limit as ex ---7 O. Furthermore, we will assume that the control input is scaled according to the equation (10.5.2)
where u°(-) E L 2 [0, s] is given. This scaling of the control input and integral quadratic constraint involves scaling down the nominal initial condition and also scaling down the "additive noise" component of the uncertainty inputs. We will show that in the limit as ex ---7 O. this scaling leads to a "scaling down" of all signals in the corresponding uncertain system (10.2.1). (10.5.1) . In addition to the assumptions made in Section 10.2. we assume that the uncertain system (10.2.1), (10.5.1) satisfies the following additional assumption: • The functions A(·.·) and K(-,·) in (10.2.1) are continuously differentiable at the origin. We now define the following system which is obtained by linearizing the state equations (10.2.1) around the origin:
= V' xA(O. O)x + V' uA(O. O)u + Bl w, V'xK(O.O)x
+ V'uK(O.O)u, (10.5.3)
V' xG(O)x
+ v.
Theorem 10.5.1 Let [x O(.), w°(-), v O(.). zO(-), yO(.)] be a solution to the
uncertain linear system (10.5.3). (10.4.1) with control input u O(.). Also, given any a > 0, consider the state equations (10.2.1) with control input
10.5 Local Results for the Robust Extended Kalman Filter
>
u(t) = auO(t). Then given any a corresponding solutzon
163
0 sufficiently small, there e:ci8ts a
to (10.2.1) satisfying an integral quadratic constraint of the form
+
(x(O) - axo)'N(x(O) - axo)
s:
(a
2
r
+ b(a))d + "21 J
r [w(t)'Qw(t) +
~
2Jo
v(t)' Ru(t))ldt
(10.5.4)
z(t)' z(t) dt
o
where b(a) ~ 0 and
.
hm
a~O
Ilea) = O. a
(10.5.5)
-2-
Furthermore, thzs solution is such that 1 lim [-x a (·). a = [xo(.),
a~O
1 1 1 1 -w a (·). -v a ('), -za(-). -Ycr(-)J a a a a wO(.), vO(.). zoO, yO(.)J (10.5.6)
where the oonvergence is uniform on [0. sJ. Proof
Given any a > 0, let xa(O) ~ axO(O), waU ~ aw°C), vaC) ~ avo(.) and unO ~ au°C).
Also. let Xa (t), Za (t) and Ya (t) be the corresponding solutions to the state equations (10.2.1) with these initial conditions and inputs. (The existence of such a solution for a > 0 sufficiently small will be justified below.) By taking a first order Taylor series expansion about the origin of the functions. A(·,·) and K(-, .). we can re-write (10.2.1) in the following form:
v xK(O, O)x + V uK(O, O)u + r2(x, u), (10.5.7) V xC(O)x
+ r3(x) + v
where
. 11m
h
. 11m
!r2(x, u)!
(x, u)!
I[x' u'JI~o ![x'
u'J!
I[x' u'l! lim hex)! Ixl~O Ix!
I[x' u'JI~o
0, 0,
O.
164
10. State Estimation for Nonlinear Uncertain Systems
We now define functions
If we divide through by a in (10.5.7), we obtain 1 . -x a
= \7 T A(O ' O)~ a.
+ \7
\7 x K(O ' O)~ 0: 1L a
\7 x C(O)~ Q:
U
A(O ' OJ:!!. a:
+
r,(x,u) Q
+ B 10:' :!!l.
+ \7 K(O OJ:!!. + T"2(X,U) a' U
'Q
+ T"3(:r) + -"'ex-. Q:
Comparing this with (10.5.3), it follows using a standard property of differential equations that
where the convergence is uniform on [0, s]. From this, (10.5.6) follows. Furthermore, this argument also guarantees the existence of the solution Xa (.) to (10.2.1) for a > 0 sufficiently small. We now consider the integral quadratic constraint (10.5.4) where
8(a) " max { 0,
(10.5.9) From this definition, it follows immediately that [xa (t), Wa (t), Va (t), Za (t)] satisfies (10.5.4). Furthermore, recall that [X O(.), woO, v°(-), ZO(.)] satisfies (10.4.1). Hence, by comparing (10.4.1) and (10.5.9) and using (10.5.8), it must follow that (10.5.5) holds. 0 Remarks
It follows from the above theorem that in the limit as a ---> 0, the integral quadratic constraint (10.5.4) approaches the integral quadratic constraint (lO.5.1). Furthermore, there exists a solution to the uncertain system (lO.2.1), (10.5.4) such that Ya(-) ---> ay°(-). Thus, when considering the robust Extended Kalman Filter corresponding to the uncertain system (10.2.1), (10.5.1) in the limit as a ---> 0, we are motivated to assume that the measured signal is of the form Ya(') = ay°(-).
10.5 Local Results for the Robust Extended Kalman Filter
165
For each value of the scaling parameter a > 0, we now consider the Extended Kalman Filter state estimation set Xso: corresponding to the uncertain system (10.2.1). (10.5.1). In this state estimator. it is assumed that the control input is of the form auoU and the measured output is of the form ay°(-). Here uO(.), and yOU are given functions defined on [0. s]. As in the previous section. the set-valued estimate Xso: is given by
where
A(io:(t), auo)
+
X-let) [ VxC(ia(t))'R[ayO - C(ia(t))] ]. 0: +V xK(io:(t), auo)' K(ia(t), auO) .
axo,
(10 ..5.10)
Xo:(t) is the solution to the RDE Xo: + V xA(i", auo)' X" + X" V xA(ia. auo) +XaB1Q-l B~XQ - V xC(inY RV xC(in,) +V xK(ia, auo),V xK(ia. auo) = 0: X,,(O) = N
(10.5.11 )
and 6
¢a(t) =
~
rt [(ayO _ C(i a ))' R(ayo - C(ia)) -
J
2 o
K(ia, auo)' K(ia, auo)] dT. (10.5.12)
As noted above. the set Xsa is only an approximation to the true state estimation set Xsa defined in Section 10.3. We wish to consider the limiting behavior of the set Xsa in the limit as a -> O. In order to consider this limit, we define the state estimation set X so corresponding to the linearized uncertain system (10.5.3), (10.5.1). As in Chapter 4, this state estimation set is defined in terms of the equations
xO
V xA(O, O)io -1
+Xo (t)
+ V uA(O, O)uO
[V xC(O)' R(yO - VxC(O)iO) +VxK(O, O)'[VxK(O, O)iO
+ VuK(O,O)uO]
IDeO) = xo
] ;
(10.5.13)
and X o + V xA(O, 0)' X o + X oV xA(O, 0) + XOB l Q-l B~ X o
- V xC(O)' RV xC(O) + V xK(O. O)'V xK(O, 0) = 0: Xo(O) = N.
(10.5.14)
166
10. State Estimation for Nonlinear Uncertain Systems
Then X so
=
{x E R
~(x -
n
xO(s»)' Xo(s)(x - xo(s»
~ a. 2 d -
where
\Ve now present the main convergence result of this section. Theorem 10.5.2 Gwen any ex > 0, suppose QY O ( ) lS the measured output generated by the uncertam system (10.2.1), (10.5.1) wtth control input exuO(.) where UO(.) E L 2 [0. s] and yOU E L 2 [0. s] Also, suppose that gwen any ex > O. there exzst bounded solutwns to the dzfferentwl equatwns (10.5.10) and (l0.5.11) defined on [O,s]. Furthermore, suppose Xo(s) > 0, Xo:(s) > D jor all a> D and and .1'so be defined as above. Then
Jet.to:
lim d 0:->0
where
('!'Xso:. X ex
SO )
=0
(10.5.15)
d(A,B) ~ max{p(A,B),p(B.A)}
denotes the Hausdorff dzstance between two set.s A and
B. That is,
6
= sup d(x, B)
p(A. B)
xEA
for two bounded sets A. BeRn and d(x. A)
6
= infx'EA Ix -
xii.
In order to establish this result. we will need the following lemmas. Lemma 10.5.3 Let f E C(Rn), ft E C(Rn) be a sequence of functwns such that (10.5.16) for some (3
for each R St
>0
> 0, =
and
lim
sup
t->XJ
xEB(O,R)
where B(O, R)
=
Ift(x) - f(x)1 =
Ixl :S R}.
{x E Rn :
{x E R n : ft(x):S O}.
S
=
°
(10.5.17) Write
{x E R n : f(x) ~ O},
and assume these sets are non-empty. Then
lim p(St. S) t->CXl
=
o.
(10.5.18)
10.5 Local Results for the Robust Ext~nded ~
161
FUtel
Proof We note first that the sets S, and S are compact, Indeed, by d,tinition tkey are closed, and from the coercivity assumption (10.5.16)
XES, implies
cllxl iJ
-
C2 :::;
f,(x) :::;
o.
and hence
Ixi :::;
(C2/ C l)1/iJ.
This also holds for XES. Thus S" S c B(O, R) for R 2: (cdcd1/iJ. The proof is by contradiction. If the assert~\)n is false, there exists a subsequence i J and a number Q > 0 such that lim
sup
)->CXJ
xEB(O.R)
(10.5.19)
If'J (x) - f(x)1 = 0
and
(10.5.20) Now (10.5.20) implies that for each J there eX\sts "1 eB'J with
d(x J • S) 2: ~Q. X*
(10.5.21)
Since x J E B(O, R), we can assume without l~ of generality that as j - 7 00, with X* E B(O, R). We have
f(x J )
:::;
X1
-7
f,,(x J ) + If'J(x J ) - f(xJ)1
:::; sUPxEB(O R) If"J (x) -- f(x)l. Sending j -+ 00 we conclude that
f(x*) S This implies x*
E
o.
S. However, (10.5.21) impli~
d(x*, S) 2: ~Q, a contradiction.
o
Lemma 10.5.4 Suppose that for all 0 < Q < 1 there eX2sts a bounded solution [i,,(t), X,,(t)] to the equatwns (l0.5.10), (10.5.11) defined on [0, s]. Also, let [iO(t), Xo(t)] be the solutions to equations (10.5.13), (10.5.14).
Then and where the convergence is umform on [O,s].
168
10. State Estimation for Nonlinear Uncertain Systems
Proof
The proof of this theorem follows from standard results on the theory of differential equations (continuity with respect to pa,rameters: e.g., see [82]).
o
Lemma 10.5.5 Let f E C(Rn) be a strzctly convex functwn. Also, let f, E C(Rn) be a sequence of functwns such that
for some
f3 > 0 and lim
sup
,~oo
xEB(O,R)
If,(x) - f(x)1 = 0
(10.5.22)
for each R > O. Write S,
= {x
E R
n
:
f,(x)::; O},
S
= {x
f(x) ~ O},
E RR
and assume these sets are non-empty. Then
lim peS, S,) =
o.
(10.5.23)
,~oo
Proof
It follows from the assumptions of this lemma that the sets Sand S, are compact. We now establish (10.5.23) by contradiction. Indeed, if (10.5.23) does not hold then there exists a subsequence i J and a number ex > 0 such that lim
sup
J--+OO
xEB(O,R)
If'J (x) - f(x)1
=0
and
peS, S,J 2: ex > O. Now (10.5.24) implies that for each j there exists d(xJ' j
S,J 2:
(10.5.24) Xj
E
S with
~ex.
Since x J E S, we can assume without loss of generality that x J with x* E S. Furthermore, it follows from (10.5.25) that
(10.5.25) ---+
x* as
---+ 00,
d(x*, S,J 2:
~ex
( 10.5.26)
for all j sufficiently large. This implies that (10.5.27)
10.5 Local Results for the'RObust Extended Kalman Filter
169
for all x such that Ix - x* I ::; ~a. Now recall that for each J, the set S" is non-empty. Hence, for each j, there exists a point x) E S'J: i.e., (10.5 28) Furthermore, it follows from (10.5.26) that 1 x_ J - x * I > -a. - 4
1
Hence, for each J, x) is contained in the compact set {x E B(O, R) : Ix x* I : : : ;ta}. Now for each J, we define a point xJ on the line segment joining the points x* and x) such that Ix) - x* I = ~a. It follows from (10.5.27) (10.5.29) Now without loss of generality, we can assume that
x)
->
x E {x
x]
->
x
E
E
B(O,R):
Ix - :1;*1::::: ~a} 1
{xRn . Ix - x*1 = Sa}.
The point x will be on the line segment joining the points x* and X. Moreover, using the continuity assumption (10.5.22), it follows from (10.5.28) and (10.5.29) that f(x) ::; 0 and f(x) ::::: O. Furthermore, x* E S implies fex*) ::; o. Thus, we have three points x*, x, and x which lie on a line segment and such that Ix - x*1 = a , Ix - x*j : : : ta, f(x*) ::; 0, f(x) ::::: 0 and f(x) ::; O. This contradicts the strict convexity of the function f. Thus, the lemma has been established. D
i
Proof of Theorem 10.5.2. First define the continuous function
Then
Also, the set X so can be written as
where
Wo(x) =
~(x 2
xO(s))'Xo(s)(x - xO(s)) - d +
170
10. State Estimation for Nonlinear Uncertain Systems
Now observe that Lemma 10.5.4 implies that -bw(a. ax) --+ wo(x) as a ---> the convergence is uniform on compact sets in R n. Also, since Xa(s) > 0 and Xo(s) > 0, it follows that the coercivity and convexity conditions required by Lemma 10.5.5 are satisfied by the functions w( a, ax) and wo(x). Hence, using this lemma and Lemma 10.5.3, (10.5.15) follows.
o where
o
10.6
Illustrative Example
Robust Extended Kalman Filter applied to FM Demodulation We consider an FM demodulation problem to which the Extended Kalman Filter is commonly applied; e.g., see page 200 of [2]. A modified version of the problem will be considered to allow for uncertainty in the signal model. In particular, we consider the following uncertain nonlinear system of the form (10.2.4):
~(t) ]
[ 8(t)
yet) where
1~(t)1
[ -1 +1
~(t) ~] [ ~g?
h
+ 8(t)) + vet)
sin(t
]
+ [
~
] wo(t),
(10.6.1)
::; 1 and
r
iOO
Jo
[50wO(t)2
+ 100v(t)2]dt ::;
10.
(10.6.2)
The initial conditions on the system (10.6.1) are assumed to be known to be >"(0) = 0 and 8(0) = O. In this signal model, the variable A(t) is the signal to be estimated and y( t) is the measured FM signal. To illustrate the application of our robust Extended Kalman Filter when applied to this problem, the system (10.6.1) was simulated with white noise signals wo(t) and vet) satisfying condition (10.6.2) and ~ = l. The magnitude of each noise signal wo(t) and vet) was chosen so that each contributes an equal amount to the integral in (10.6.2). The resulting signal yet) was then processed by the robust Extended Kalman Filter defined by equations (10.4.10), (10.4.13). Note that since the initial conditions on the system (10.6.1) are assumed known as >"(0) = 0 and 8(0) = 0, the equations (10.4.10) and (10.4.13) are solved with zero initial condition. The results of these simulations is shown in Figure 10.6.1 which shows a plot of the true value of the signal >..(t) together with the estimated value obtained from the robust Extended Kalman Filter. Note that although the estimated signal is somewhat noisy, it still manages to track the true signal in spite of the large uncertainty in the signal model. For the sake of comparison, a standard Extended Kalman Filter was also applied to the above problem. In this case, the measured signal is generated
10.6 Illustrative Example
171
02,---,----,----,----,----,----,----r----,----,---,
A (I)
"
-
."
True Signal Estimated Signal
10
20
30
40
50
60
70
80
90
100
Time Seconds
FIGURE 106 .1. FM demodulation simulation results Kalman FIlter.
USlllg
robust Extended
as above with .6. = 1 but the Extended Kalman Filter was constructed based on the nominal signal model corresponding to .6. = O. The results of this simulation are plotted in Figure 10.6.2. It can be seen that in this case, the estimated signal does not track the true signal but rather exhibits Extended Kalman Filter divergence.
172
10. State Estimation for Nonlinear Uncertain Systems
018 016
-
True Signal Estimated Signal
014 012
A (t)
01 QW
ode 004 002
10
20
30
40 50 60 Time Seconds
70
80
90
100
FIGURE 10.62. FM demodulation simulation results using standard Extended Kalman Filter
11 Robust Filtering Applied to Induction Motor Control
11.1
Introduction
In this chapter, we present a practical application of robust filtering. This application involves the construction of a robust state estimate required in an induction motor control problem. This application of robust filtering to induction motor control was first presented in the paper [105]. The induction motor control problem investigated in [105] and in a related paper [108] involves the construction of a robust version of Direct Torque Control (DTC) for induction motors. The objective of the DTC induction motor control scheme is to ensure that the torque produced by the motor tracks a given torque reference signal and the magnitude of the stator flux vector tracks a given stator flux reference signal. The standard DTC approach to induction motor control has attracted a considerable amount of interest in the field of electrical machines; e.g., see [37,51,52,151,170]. However, the standard DTC induction motor control scheme is known to have problems with robustness against variations in the motor parameters. This motivated the authors of [105, 108] to investigate the possibility of a robust version of DTC induction motor control. The basic idea used in these papers is to replace the observer which forms part of the standard DTC induction motor control scheme by a robust Kalman Filter of the type presented in Chapter 4. In this chapter we will use this robust induction motor control problem as a case study illustrating the application of robust filtering. In particular, we will apply the results of Chapter 4 on continuous time set-valued state
174
11. Robust Filtering Applied to Induction Motor Control
estimation to construct a robust flux estimator. This robust flux estimator will form a central part of a robust induction motor control scheme. That is, this robust induction motor control scheme uses the robust Kalman Filter to estimate the induction motor flux vectors and then combines these flux estimates with a state feedback induction motor control law derived in the paper [108]. The robustness of this control scheme will be illustrated with simulations and experimental results for the case in which the control scheme is applied to a 2.2 kw 4 pole induction motor.
11.2
State Feedback Torque Control of Induction Motors
An important characteristic of induction motors is the fact that the rotor flux vector is not normally available for measurement. That is. not all of the induction motor state variables are available for measurement. If all of the state variables were available for measurement, the problem of torque control for an induction motor would be greatly simplified. The induction motor torque control problem which is considered in this chapter involves first constructing a state feedback control law which assumes that all of the induction motor state variables (fluxes) are available for measurement. This state feedback control law is then combined with a robust Kalman filter which provides estimates for the induction motor state variables. We begin by considering general state space models for induction motors. One such model is a stator coordinate based induction motor model in complex state space form as follows: (11.2.1a) (11.2.1 b) where [ W WsS ].•
;£
-R -Rs/La RR/La
A
B2
=
[ [ ~ ].
Rs/La -RR/ LMa + jW m
C = [ l/L a
and LMa = L
LMLa L
M+
-1/ La
]
], (11.2.2)
a
Here, the parameters RR, R s , Ll\f. La are electrical parameters which define the characteristics of the motor.
11.2 State Feedback Torque Control of Induction Motors
175
Note that in this state space model, the induction motor state variables are the complex flux vectors IlF sand IlF R. The use of such complex state space models is standard practice for induction motors and leads to a considerable reduction in the complexity of the resulting state equations. \Ve will see in this case study that it also leads to a considerable simplification in the robust Kalman Filter Riccati differential equation and filter state equations. In the above complex state space model and throughout this chapter, we follow the convention that a complex quantity is underlined. This notational convention is standard in the literature on induction motors. In the paper, [108]' the model (11.2.1), (11.2.2) was used to derive a state feedback control law of the form: f )• f( :.f, W m , Tref 'T,re e ,'±' S
1!.s --
(11.2.3)
where Wm is the motor speed, T:;e f is the reference torque signal and llF:e f is the reference stator flux signal. This control law was derived so that in the case of full state feedback control, the resulting closed loop system is such that the torque tracks a reference torque signal T:;e f and the stator flux magnitude tracks a reference stator flux signal llF:ef. Indeed in [108], the state feedback control law (11.2.3) is defined by the following rather complicated equations which were derived using the "backstepping" approach of [72]: 1!.s
=
1 -llFe T
.
+ Rsls +
where 1
)2 _ (2L~T;Ef) ( IlFref s 3pWR
2
2L a Tref e
.
3pllF2
'
R
and
q,R
RR - - L IlFR Afa
Wm
+
RR
+ -L IlFsd; a
RRllFsq LaIlFR'
Here, IlF R is the rotor flux magnitude and eR is the rotor flux angle. These quantities are related to the rotor flux vector by the relation
176
11. Robust Filtering Applied to Induction Motor Control
Furthermore, the quantities 'l'sd and 'l'sq represent the stator flux vector in 'dq' coordinates. These quantities are defined by the equations 'l'sd
+ j'l' sq
'l'dq
-s
'l'dq eJ& R
.'!'.s·
-s
The quantities .'!'.e and If, s in the formula (11.2.4) relate to the the stator flux vector and the reference stator flux as follows:
where - dq J & 'l' s e H.,
+ jlf, sq; 2L a rrej e .
If, sd If, sq
11.3
3p'l'R
'
A Robust Kalman Filter for the Induction Motor
In order to apply the above control law in a practical induction motor control system. the vector of rotor and stator fluxes JZ must be estimated. An important issue in this flux estimation process is the issue of robustness. In an induction motor, the rotor and stator resistances, RR and Rs are subject to large variations as the motor heats up. This leads to uncertainties in the system model. It is proposed to use the robust Kalman filter presented in Chapter 4 to provide a robust flux estimator. Indeed using the theory presented in Chapter 4, we can construct a flux observer which is robust with respect to variations in the rotor and stator resistances. In order to account for variations in the rotor and stator resistances. we let (11.3.1) where Rs and RR are the nominal resistance values. The variations are subject to the bounds Iqs(t)1 ::; p. IqR(t)1 ::; J.1. The model (11.2.1) then yields the following uncertain system model for the induction motor:
Ax + Bl6.(th + B21l.s + 1Q;
(1l.3.2a)
Cx+1i.
(1l.3.2b)
11.3 A..lWbust Kalman
F'Uter f(.r the Induction Motor
171
where
Furthermore, Y2 and 1L represent process and measurement noise vectors which are assumed to be energy bounded according to the following integral quadratic constraint:
where Q and E are weighting matrices which determine the relative size of the noises Y2 and 1L. The time T refers to the current time at which the state estimate is to be obtained. In practice. the matrices Q and R can be treated as design parameters and taken to be of the form:(11.3.3) where the parameter p is chosen to achieve a specified steady state filter bandwidth. Formally. the robust Kalman filtering theory developed in Chapter 4 leads to a set-valued state estimate X T which consists of the set of all possible values of the systems state !f.(T) which is consistent with the uncertain system model and given output measurements of is(t) on the time interval [0, T]. In Chapter 4. it was shown that the set XT is an ellipsoidal set of the form XT
={
!f.:
(!f. -
i(T))'P-1(t)(!f. - i:.(T)) }
::; 1 + PT (is)
.
(11.3.4)
In this state estimation set, the quantities PT(i,). i:.(T), and P are defined as follows (see (4.3.1)):
p
AP + PA'
+ p[J.l2 1- G'RG]P + BIB~ + Q-\
P(O) = 0;
(l1.3.5a)
PT(i,)
J:( t)
2 T [ J.l i:.(t)'i:.(t)] o (Gx(t)x - is(t))' R(Gx(t)x - is(t)) dt:
i
(11.3.5b)
[11 + J.l2 P(t)].i:.(t) + B21Ls(t) + K(t)[is(t) - Gx(t)]; i:.(0) =
Xo
(l1.3.5c)
178
11. Robust Filtering Applied to Induction Motor Control
where K(t) = P(t)C'E. is the complex Kalman gain matrix. Note that equation (l1.3.5a) is a complex Riccati differential equation with Hermitian solution matrix
P(t) = [Pll(t) p* (t) -12
-
At this point, the reader might recall that the robust Kalman filter formulated in Chapter 4 deals with real state space models. However, the state space model for an induction motor given above is a complex state space model. In fact, the complex Riccati equation (l1.3.5a) was obtained by simply extending the approach of Chapter 4 to the complex case. An alternative approach would have been to convert the complex state space induction motor model (11.2.1) into a real state space model by expanding out the states into their real and imaginary components. The advantage of the complex state space approach is that it leads to a considerable saving in computational time. For example, in the complex state space approach, the complex Riccati equation (11.3.5a) which must be solved on-line, is equivalent to solving a fourth order differential equation . In contrast, if the real state space approach were taken, a Riccati equation equivalent to a tenth order differential equation would have to be solved. Even though the robust Kalman filtering theory of Chapter 4 leads to a set-valued state estimate X T , we require a point valued state estimate for :r.(T) which will be used by the state feedback controller (11.2 .3). A suitable point valued state estimate can be obtained by simply choosing the center of the ellipsoid ;r(T) as the estimate for the flux vector :r.(T). Then the robust induction motor controller consists of the state feedback control law (11.2.3) with :r.(t) replaced by;r(t) which is calculated according to equations (11.3.5).
11.4
Simulation and Experimental Results
~Te
now consider the robust induction motor controller defined above when applied to a 2.2kW delta connected induction motor connected via an inverter to an ideal 586V DC bus. This motor has a state space model of the form (11.2 .1), (11.2.2) where the electrical parameters of the motor are as shown in Table 11.4.1:
RR Rs LlIf La
n
13.3554 8.94 0.752 H 0.0983 H
n
TABLE 11.4.1. 2.2 kW Induction Motor Electrical Parameters
11.4 Simulation and Experimental Results
179
This drive system was simulated with the SIMULINK/MATLAB package using the simulink model illustrated in Figure 11.4.1. Also, Figure 11.4.2 shows more detail of the robust Kalman filter part of the induction motor controller.
rom T ~
4
omega~m
...... ~
wm
~'S ;
-
Tload
v - s
...
PhRs
vs
is
is
-
'Ps -
•k•
A
'P
-
r----
Induction Motor Model
- State Feedback Controller
s
Te
Phis
PSI_S
-" PSI_R
-
L
TI--+
Load Model
vs
A
"""
wm xhat
IS
vs"--
R Robust_Kalman_Filter
FIGURE 11.4.1. Simulink drive model.
Simulations were ci:trried out with resistance variations of qs = qR = ±0.25 in equation (11.3.1). For compatibility with the experimental results to be described in the sequel, the resistances used in the controller and Kalman filter design were varied and the resistances in the plant model were held constant. This is because in the experimental implementation of the induction motor controller, it is easy to change the resistance parameters used in the controller design but difficult to change the resistance parameters in the actual motor. In the design of the robust Kalman Filter, the measurement noise weighting parameter pin (11.3.3) was chosen to be p = 0.1. This corresponds to a steady state Kalman Filter bandwidth of 125 Hz. The simulations presented correspond to a fully defluxed motor at startup. The torque reference is initially set to zero and held for O.ls so that the rotor flux can reach a specified value. The stator flux reference is set to the rated value of 1.8 Wb at startup (t = -O.ls) and held at this value. The torque reference value is switched from zero to the rated motor value of 14N m at t = Os. The simulations were carried out for three different resis-
180
11. Robust Filtering Applied to Induction Motor Control
wm ROE
P(tlh
Ptimesh u"21-C'RC
3l-----------1~K
vs
t------~+
B2
FIGURE 11.4.2. Simulink robust Kalman Filter model.
tance combinations in the controller and Kalman filter design as follows: Case Rl: Rs = 8.9400, RR = 13.35D; i.e., the resistances used in the controller design were equal to the resistance parameters in the motor model. Case R2: R s = 11 .17, RR = 16.69D: i.e., the resistances used in the controller design were 25% above the resistance parameters in the motor model. Case R3: Rs = 6.70, RR = 10.OlD; i.e., the resistances used in the controller design were 25% below the resistance parameters in the motor model. The simulations were first carried out for the full state feedback case in which it is assumed that all state variables are measurable. The results for this case are shown in Figure 11.4.4. The second set of simulations correspond to the use of the robust Kalman filter described above to estimate the flux vectors. The results for this case are shown in Figure 11.4.5. From
11.5 Discussion
181
these results, it can be seen that the results for this case are not significantly worse than for the full state feedback case. The performance of the robust induction motor controller has also been verified experimentally. This was achieved using the dSPACE DSP system which enabled the simulink block diagram shown in Figure 11.4.1 to be implemented in a real time control system. The continuous time controller block diagram shown in Figure 1l.4.3 was discretized using the Euler method and downloaded to the dSPACE system. The inputs to this control system are the measured line currents, bus voltage and shaft speed. The outputs of the control system are three PWM signals which are used to drive a modified commercial inverter. A sampling time of a.5ms was used in the control system.
,u Robust Kalman Flltet
State Feedback Controller
wmS Signal coodillomng
PWM3
FIGURE 11.4.3. Simulink block diagram of controller implemented using DSPACE system.
In Figure 11.4.6, the corresponding experimental results are shown. These results show that the controller exhibits similar robustness to that which was predicted via the simulations.
11.5
Discussion
It is well known that the parameters of an induction motor are subject to large variations. Thus, it was required to design a robust controller which can tolerate significant changes in the motor resistances. The simulations presented in this chapter have confirmed the potential of robust Kalman Filtering to yield a robust induction motor control scheme which will handle large resistance variations. It would be straightforward to extend this approach to give robustness against other parameter variations. One point which should be noted about the robust induction motor control scheme which was investigated in this chapter is that it requires the use of a speed sensor. This is required so that we can treat the induction
182
11. Robust Filtering Applied to Induction i\Iotor Control
20,-----,-----~----_,----_,------,_----,_----_.----_,
"8 16 14
r
-
-
-
-
________________
_
1 r - - - - - - -________________________________________--1
If 12
8 - - Rs= 8 9400 n, R A : 13 35n - -
Rs=1117n, R A =1669n
- - - Rs = 6 70 n, RR = 1001 n
2
OL-____
o
~
005
_____ L_ _ _ __ L_ _ _ _ 01
015
~
_ _ _ _ _ _L __ _ _ _
02
025
~
03
_ _ _ __ L_ _ _ _
035
~
04
Tlme(seconds)
FIGURE 11.4.4. Simulated torque response of state feedback induction motor control system.
motor model (11.2.1). (11.2.2) as a (time-varying) uncertain linear system and hence apply the linear robust Kalman Filtering theory of Chapter 4. If the speed can not be measured then the induction motor must be treated as a nonlinear uncertam system. To apply a robust filtering approach in this case. it would therefore require the use of a corresponding theory of robust Kalman Filtering for nonlinear systems such as presented in Chapter 10.
11.5 Discussion
......
---- .....
"'--......
'--
183
......-- .....
- - Rs; 0 505 n, RR; 0 165n - -
Rs; 0 404 Q, R,,; 0 132Q
- - - Rs; 0 606 Q, RR; 0 198Q
01
015
02
025
03
035
04
Tlme(seconds)
FIG URE 11.4.5 Simulated torque response with measurement feedback induction motor control using th~ robust Kalman FIlter. 20,-----r----,~__- ,_____,----_,----_,----_,----,
/~
>.
,1
',olt
I
"
I,
,--
12
E
~10 1-"
B ~
tl
- .II-
~
Rs; 8 9400 n, RR; 13 35Q Rs; 11 17 n, RR; 16 69Q
- - Rs; 6 70 Q, RA
0
10 OW
~
0
0
0.05
01
015
0.2
025
03
035
04
Tlme(seconds)
FIGURE 11.4.6. Experimental torque response with measurement feedback induction motor control Usmg the robust Kalman Filter.
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Index
absolute stability, 136 algebraic Riccati equation, 17, 41, 68 augmented system, 14 averaged state vector, 113 averaged integral quadratic constraint, rIa
backsteppmg, 175 channel model, 82, 84 coercivity assumption, 166 constant filter, 151 cost matrix, 14 discrete-continuous Il1easurements, 91 disturbance attenuation, 133 dq coordinates, 176 EKF divergence, 170 error covariance, 14 estimation error, 14, 37 Finsler's Theorem, 136 fixed order filter. 14'(-51 flux vectors, 175 flux estimator. 176 flux observer. See flux estimator
FM demodulation, 170 guaranteed cost estimatOl', 14, 9T H= controL 22, 43 H"" filtering, 132-6, 146-7 Hausdorff distance, 166
incompleteness matrix, 78, 101 incompleteness matrix function, 101 induction motor. 173-82 induction motor model, 174 information state, 156 integral quadratic constraint, 60, 13:\, 158 jump Riccati equation, 93 jump state equation, 93 linearized uncertain system, 162 Matrix Inversion Lemma, 41 mean square error, 17, 40 missing data, 78-81. 100 model validation, 62, 74, 93, 101 nonlinear integral constraint, 154 nonlinear uncertain system, 154
200
Index
norm bounded uncertamty, 13, 36, 61, 73, 112 133 147 155 partial dIfferentIal equatIOn, 156 PDE See partIal dIfferentIal equatIOn predIctor-corrector equatIOns 51 quadratIc observablhty, 146 quadratIc stabIlity, 13, 37 reahzable mput-output paIr, 61 realizable mput-output tnple, 92 101 reahzable output, 74 RIccatl dIfference equatIOn, 74 79 Rlccati dIfferentIal equatIOn, 62 robust observablhty, 135 robust deconvolutIOn, 82-6 robust extended Kalman filter, 15869 robust predIction, 78, 102 rotor reSIstance, 176 S-procedure, 113 7, 136 9 set-valued state estImate, 61, 62, 74, 78 92 102 113 155, 177
SIgnal model 82 84 Small Gam Theorem, 17,40-1 stablhzmg solutIOn, 18 41 state estimator 14 37 stator coordmates, 174 stator reSIstance, 176 Stnct Bounded Real Lemma, 18-22 42-3 strict venfiablhty, 61, 74-5, 79 92, 102 strong solutIOn, 18, 41 structured uncertamty, 110, 133 sum quadratIC constramt, 72 sum-mtegral quadratic constramt, 91 torque control, 174-6 trackmg problem, 65, 77, 98 uncertam frequency, 125 uncertam parameters, 13, 36 uncertam system, 13,59, 72,91,109, 132, 146, 154 uncertamty structure 13 VISCOSIty solutIOn, 156
Errata Robust Kalman Filtering for Signals and Systems with Large Uflcertainties \'b.~ R. ~~\~""'~~ Andrey V. Savkin
© 1999 Birkhauser BostOn
ISBN 0-8 I 76-4089-4
Note-Following are the corrected versions of ~everal diagrams that were inadvertently incomplete.
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