Control Systems with Saturating Inputs: Analysis Tools and Advanced Design

Lecture Notes in Control and Information Sciences 424 Editors: M. Thoma, F. Allgöwer, M. Morari Maria Letizia Corradi...

Author: Maria Letizia Corradini | Andrea Cristofaro | Fabio Giannoni | Giuseppe Orlando

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Lecture Notes in Control and Information Sciences 424 Editors: M. Thoma, F. Allgöwer, M. Morari

Maria Letizia Corradini, Andrea Cristofaro, Fabio Giannoni, and Giuseppe Orlando

Control Systems with Saturating Inputs Analysis Tools and Advanced Design

ABC

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

Authors Prof. Maria Letizia Corradini Università di Camerino Scuola di Scienze e Tecnologie Via Madonna delle Carceri Camerino Italy

Prof. Fabio Giannoni Università di Camerino Scuola di Scienze e Tecnologie Via Madonna delle Carceri Camerino Italy

Dr. Andrea Cristofaro Università di Camerino Scuola di Scienze e Tecnologie Via Madonna delle Carceri Camerino Italy

Dr. Giuseppe Orlando Dipartimento di ingegneria dell’informazione Università Politecnica delle Marche Ancona Italy

ISSN 0170-8643 ISBN 978-1-4471-2505-1 DOI 10.1007/978-1-4471-2506-8 Springer London Heidelberg New York Dordrecht

e-ISSN 1610-7411 e-ISBN 978-1-4471-2506-8

Library of Congress Control Number: 2011945434 c Springer-Verlag London Limited 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied speciﬁcally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Saturation nonlinearities are ubiquitous in engineering systems: every physical actuator or sensor is subject to saturation owing to its maximum and minimum limits. Input saturation is an operating condition that is well known to the control community for its side effects, which cause conventional controllers to loose their closed loop performance as well as control authority in stabilization. Therefore, practical application of control theory cannot avoid taking into account saturation nonlinearities in actuators, explicitly dealing with constraints in control design. This book investigates the problem of actuator saturation from a practice-oriented alternative viewpoint. Analysis tools applicable to plants of arbitrary ﬁnite dimension are given providing an analytical estimate of the maximal null controllable region. Nonlinear control design techniques are presented with particular reference to robustness with respect to matched disturbances and/or uncertainties. Design approaches explicitly developed in the discrete-time framework are described in order to enhance the practical applicability of controllers. More speciﬁcally, the results that are to be presented in this book are outlined as follows. In Chapter 1, after a short introduction to linear systems with saturation nonlinearities in the actuator, we state and discuss some deﬁnitions and technical terms which will be useful in the following. Chapter 2 and 3 give explicit descriptions of the null controllable regions of a linear system driven by saturating actuators in the continuous-time and discrete-time framework respectively. The description of the null controllable region is obtained following an iterative procedure based on reversed-time evolution and convexiﬁcation. In both cases, Single Input planar case are addressed ﬁrst, then results are extended to n-dimensional Single Input plants and ﬁnally to Multi Input Systems. Chapter 4 addresses design issues in the continuous-time framework. First a design technique is proposed for linear plants subject to saturating actuators, such that the resulting linear controller has the property of having non-increasing norm along the closed-loop system trajectories. In particular the region of attraction associated to the saturating control is an unbounded strip and it can be straightforwardly characterized. Moreover it is shown how, once the saturation level is ﬁxed, it is possible to split the controller into a ﬁnite number of saturating components. The number

VI

Preface

of components can be a priori determined for any ﬁxed compact set of initial data. Next, the problem of controlling uncertain Multi-Input linear plant with saturating actuators is looked at from a different perspective. The objective is to construct time-varying feedback laws, derived imposing the achievement of a sliding motion onto a suitable time-varying sliding surface, able to ensure that saturation thresholds are never violated. It is proved there that a constructive procedure exists for designing the surface as to guarantee the asymptotical stabilization of the plant in the presence of bounded matched uncertainties, under the usual assumption of the saturation threshold being larger than the bound on uncertainties. The discrete-time counterparts of the approaches described in Chapter 4 are addressed in Chapter 5. It worths noticing that completely different technical solutions are required with respect to the continuous time case. The main features of the proposed sliding-mode based control law are: i) no restrictions are needed in the plant structure; ii) bounded matched disturbances are considered; iii) robust practical stabilization on the null controllability region can be achieved by means of a time-varying state feedback controller, derived imposing the achievement of a quasi-sliding motion onto a suitable time-varying quasi-sliding surface; iv) performing transient shaping is not subject to any condition and can be achieved simply by manipulating the dynamics imposed onto the quasi-sliding surface. It is proved that a constructive systematic procedure exists for designing the surface as to guarantee the ultimately boundedness of plant trajectories in the presence of bounded matched uncertainties. Wherever possible, the various topics are rounded out with results obtained through experiments with actual plants. Mathematical details which are outside the normal province of control engineers are presented in an appendix for the interested reader. The ideas formulated in this book could be of great practical help to professionals, researchers, practitioners and graduate students in control, electrical, and mechanical engineering, working directly with problems related to the control of plants with saturating actuators. Some ﬁrst–year graduate courses in linear systems and multivariable control or some background in nonlinear control systems would greatly facilitate the reading of this book. Camerino, Italy November, 2011

Maria Letizia Corradini Andrea Cristofaro Fabio Giannoni Giuseppe Orlando

Symbols and Notation

R the ﬁeld of real numbers || · || the Euclidean norm A a matrix deﬁned in the n × m-dimensional real vector space, n, m > 1 b a vector deﬁned in the r-dimensional real vector space x the state vector xi the i-th element of the state vector sat the saturation function for Single Input systems u the generic unconstrained control input v the generic constrained control input sat the saturation function for Multi Input systems

λ a generic eigenvalue BM the largest null controllable region for Single Input systems (continuous-time case) BM the largest null controllable region for Multi Input systems (continuous-time case) BM the largest null controllable region for Single Input systems (discrete-time case)

UM the set of admissible control inputs (continuous-time case) UM∗ the set of bang-bang control inputs (continuous-time case) WM the set of admissible control inputs (discrete-time case) WM∗ the set of bang-bang control inputs (discrete-time case) C0 the class of continuous functions

VIII

C0 the class of piecewise continuous functions ∞ the set of bounded sequences ∞ (Z) the set of bounded bilateral sequences j the imaginary unit

Symbols and Notation

Acronyms

ANCBC ARE LHP MI MIMO PWC REM RHP RRM SI SISO SM VSS

Asymptotically Null Controllable with Bounded Controls Algebric Riccati Equation Left Half Plane Multi Input Multi Input, Multi Output Piecewise Constant Maximum Region of Reachability Right Half Plane Maximum Region of Recoverability Single Input Single Input, Single Output Sliding Mode Variable Structure Systems

Contents

Symbols and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII Part I: Introduction and Analysis Tools 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Linear Plants with Actuator Saturation . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 5

2

Estimation of the Null Controllable Region: Continuous-Time Plants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Introductory Remarks and Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 The SISO Planar Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 The SISO n-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 The MIMO Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3

Estimation of the Null Controllable Region: Discrete-Time Plants . . . 3.1 The SISO Planar Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The SISO n-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The MIMO Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 34 38 44 48

Part II: Design Issues 4

Control Design Issues: Continuous-Time Plants . . . . . . . . . . . . . . . . . . . 55 4.1 Invariant Strips and Linear Feedback Laws . . . . . . . . . . . . . . . . . . . . . 56 4.2 Nonlinear Robust Controller Design via Sliding Modes for Continuous-Time Multi Input Plants . . . . . . . . . . . . . . . . . . . . . . . . 83

XII

5

Contents

Control Design Issues: Discrete-Time Plants . . . . . . . . . . . . . . . . . . . . . . . 95 5.1 Invariant Strips and Linear Feedback Laws . . . . . . . . . . . . . . . . . . . . . 96 5.2 Nonlinear Robust Controller Design via Quasi-sliding Modes . . . . . 98 5.3 Experimental Data: Stabilization of a Twin Rotor System . . . . . . . . . 107

Appendix A: Support Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A.1 Support Material for Chapters 2, 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A.2 Support Material for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Part I: Introduction and Analysis Tools

Chapter 1

Introduction

1.1 Linear Plants with Actuator Saturation The presence of actuator saturation in control systems, though frequently ignored, is due to inherent (and unavoidable) physical limitations of devices. The relevance of this issue from the practical viewpoint is more and more attracting the attention of control system researchers, as failure in accounting for actuator saturation may lead to severe deterioration of closed loop system performance, even to instability. As a matter of fact, classical control theory suffers from the limitation that, in reality, dynamical systems are frequently affected by a number of ignored factors which may invalidate, or at least severely limit its effective application. Among these factors, inherent saturation effects turn out to be almost ubiquitous, since the output of all physical devices are limited to some degree, and when a limiting value is reached saturation is said to occur. A classical example is the output torque of a servomotor, which is subject to a maximum value which can be supplied. In general, the performance of a closed-loop system where control design has been carried out ignoring saturation may seriously deteriorate. A well known example is the case of the PID controller applied in closed loop in the presence of saturation, producing the so called integrator wind-up. What happens is that, whenever the actuator saturates, the error variable is continuously integrated and this produces larger and larger values of the control input, exceeding the saturation threshold. The schemes known as anti-windup approaches are aimed at mitigating the adverse effects of saturation by usually adding extra feedback loops in order to constrain the control variable to vary within an assigned range. There are several possible models for saturating devices: a simple model given by the truncation operator is here considered: Tk (s) = max{−k, min{k, s}},

(1.1)

where k ∈ R is the saturation level. The function Tk (s) is symmetric, linear on a suitable interval containing the origin and constant outside this interval (see Fig. 1.1). Linear systems subject to asymmetric input saturation have also been M.L. Corradini et al.: Control Systems with Saturating Inputs, LNCIS 424, pp. 3–6. c Springer-Verlag London Limited 2012 springerlink.com

4

1 Introduction

k

Tk (s) 6

−k

k

s

−k

Fig. 1.1. Saturation function.

studied (see e.g. [1]). Denoting with A ∈ Rn×n the state matrix, with B ∈ Rn×1 the input matrix, with x ∈ Rn the state variable and with u ∈ R the control input, a SISO linear system subject to actuator saturation can be written as x˙ = Ax + Bu,

x(0) = x0 ,

(1.2)

with u = TM (v), where the constant M > 0 is the saturation level. Throughout this book, the saturation function for a scalar s ∈ R will be denoted as (1.3) satM (s) = TM (s) with a saturation level M, unless differently speciﬁed. With a slight abuse of notation, for a vector s = [s1 s2 . . . sm ]T ∈ Rm , the saturation function will be denoted as sat(s) = [sat(s1 ) sat(s2 ) . . . sat(sm )]T

(1.4)

1.2 Background Following [2], a system is said to be globally Asymptotically Null Controllable by Bounded Controls (ANCBC) if, for a given bound on the controls, every state can be driven to the origin either in a ﬁnite time or asymptotically by a bounded control. Since it has been proved [3] that a linear stabilizable system having all its poles in the closed left-half plane is asymptotically null controllable, such system is said ANCBC. Existing results pertaining to the problem of global and semi-global stabilization showed that, in general, global stabilization of a linear system subject to saturating actuators can be achieved by a nonlinear controller if and only if the linear system (without saturation) is asymptotically null controllable by bounded control, this condition being equivalent to classical stabilizability and to the requirement that the plant has stable open loop poles [4] [3] [5] [6] [7]. In this framework, a nested feedback technique for designing nonlinear globally asymptotically stabilizing controllers was ﬁrst proposed for a chain of integrators [8] and then fully generalized [9].

1.3 Outline of the Book

5

In the semi-global framework, early works proved that, under the appropriate conditions, stabilization of asymptotically null controllable with bounded control linear systems can be achieved using linear feedback laws [10] [11]. To this purpose, a number of design methods for low gain controllers have been proposed, as well as the so called low-and-high design technique [12]. An LQR/LQG theory for systems with saturating actuators has been recently presented [13]. An extensive bibliography on the subject is reported in [14]. If the assumption of asymptotically null controllability with bounded control is relaxed, however, results on stabilizability of uncertain linear systems are more fragmentary. Studying the null controllable region, [15] recently showed that, if a planar antistable plant is considered, for any given set in the null controllable region there exists a saturated linear feedback controller yielding a closed loop system having an asymptotically stable equilibrium whose domain of attraction includes this set. Linear plants with bounded controls have been widely studied in the ﬁeld of antiwindup approach [16] [17], where an additional feedback loop is introduced. In most cases, this technique requires both plant stability and/or complete knowledge even for linear time-invariant systems. Exponentially unstable systems have been also considered by [18] in the presence of reference and disturbance inputs. The disturbances, however, are restricted not to act on the exponentially unstable modes. Composite nonlinear feedback control has been recently used [19] for generic linear systems without uncertainties. In the wide literature addressing the problem of disturbance rejection for linear systems subject to actuator saturation, an interesting research line considers disturbances that are magnitude bounded. This line complements the research thrust addressing L p disturbances [20] [21]. In the former framework, [22] proved that semiglobal practical stabilization for a linear system subject to actuator saturation and input additive disturbances can be achieved as long as the open loop system is not exponentially unstable. For the same class of systems, Lin [7] constructed nonlinear feedback laws that achieve global practical stabilization. Very recently, it has been proved in [23] that a 2-dimensional linear systems subject to actuator saturation and bounded input additive disturbances can be globally practically stabilized by linear state feedback.

1.3 Outline of the Book This book investigates the problem of actuator saturation from a practice-oriented viewpoint. Analysis tools applicable to plants of arbitrary ﬁnite dimension are given for the determination of an analytical estimate of the maximal null controllable region. Nonlinear control design techniques are presented with particular reference to robustness with respect to matched disturbances and/or uncertainties. Design approaches explicitly developed in the discrete-time framework are described in order to enhance the practical applicability of controllers. More speciﬁcally, the results that will be reported in this book can be outlined as follows.

6

1 Introduction

After having presented, in the present Chapter, a short introduction to linear systems with saturation nonlinearities in the actuator, Chapter 2 and 3 provide the explicit description of the null controllable region of a linear system driven by saturating actuators in the continuous-time and discrete-time framework respectively. In both cases, Single Input planar systems are addressed ﬁrst, then results are extended to n-dimensional Single Input plants and ﬁnally to Multi Input systems. It is worth noting that characterizations of the null controllable regions have been already proposed in the literature in recent years (e.g. in terms of trajectories corresponding to bang-bang controls with a ﬁnite number of switchings [24] for a continuous-time system, in the framework of discrete-time systems [25], rational systems [26] and MPC controllers [27]). Since the problem requires a high computational burden, an alternative characterization is proposed in these Chapters, where the description of the null controllable region is made following an iterative procedure based on reversed-time evolution and convexiﬁcation. Chapter 4 addresses design issues in the continuous-time framework. First a design technique is proposed for linear plants subject to saturating actuators, such that the resulting linear controller has the property of having non-increasing norm along the closed-loop system trajectories. In particular the region of attraction associated to the saturating control is an unbounded strip and it can be straightforwardly characterized. Moreover it is shown how, once the saturation level is ﬁxed, it is possible to split the controller into a ﬁnite number of saturating components. The number of components can be a priori determined for any ﬁxed compact set of initial data. Next, the problem of controlling uncertain Multi-Input linear plant with saturating actuators is looked at from a different perspective. The objective is to construct timevarying feedback laws, derived imposing the achievement of a sliding motion onto a suitable time-varying sliding surface, able to ensure that saturation thresholds are never violated. It is proved there that a constructive procedure exists for designing the surface as to guarantee the asymptotical stabilization of the plant in the presence of bounded matched uncertainties, under the usual assumption of the saturation threshold being larger than the bound on uncertainties. Finally, the discrete-time counterparts of the approaches described in Chapter 4 are addressed in Chapter 5.

Chapter 2

Estimation of the Null Controllable Region: Continuous-Time Plants

2.1 Introductory Remarks and Deﬁnitions The description of the maximal stability region for a constrained system is a fundamental problem in control theory and a huge amount of literature pertaining the subject can be found (see for instance [5], [7], [28], [29], [30], [24]). A seminal work in this ﬁeld is represented by the paper [31], which introduces recoverable and reachable zones for linear systems subject to input constraints and exploits the main properties of such regions. A general constrained control problem can be formulated imposing that the input variable u is allowed to vary only in a prescribed, possibly bounded, subset U , the set of admissible controls. Let us consider the linear continuous-time plant ⎧ ⎨ x˙ = Ax + Bu (2.1) ⎩ x(0) = x0 and let us denote by φ (t, x0 , u) the solution of the system for t ∈ R. A saturating device is assumed to be acting on the system, i.e. the input u is subject to the constraint u = satM (v), M > 0

(2.2)

sath (s) = max{−h, min{h, s}}

(2.3)

with where the input u is available for direct manipulation. An element x0 in the state space Rn is a recoverable state in the time interval (t0 ,t f ) with respect to the ﬁnal state x f , if there exists an admissible control which will drive the system from x0 at time t0 to state x f at time t f . The maximum region of recoverability with respect to x f in (t0 ,t f ), denoted by RRM(x f ;t0 ,t f ), is the set of all recoverable states in (t0 ,t f ) with respect to x f . An element x f in the state space Rn is a reachable state in the time interval (t0 ,t f ) with respect to the initial state x0 , if there exists an admissible control which M.L. Corradini et al.: Control Systems with Saturating Inputs, LNCIS 424, pp. 7–32. c Springer-Verlag London Limited 2012 springerlink.com

8

2 Estimation of the Null Controllable Region: Continuous-Time Plants

will drive the system from x0 at time t0 to x f at time t f . The maximum region of reachability with respect to x0 in (t0 ,t f ), denoted by REM(x0 ;t0 ,t f ), is the set of all reachable states in (t0 ,t f ) with respect to x0 . The regions can be expressed as follows tf A(ξ −t0 ) A(t f −t0 ) n RRM(x f ;t0 ,t f ) = x ∈ R : x = e xf − e Bu(ξ )d ξ , u ∈ U t0

A(t0 −t f ) n REM(x0 ;t0 ,t f ) = x ∈ R : x = e x0 +

tf

e

A(ξ −t f )

t0

Bu(ξ )d ξ , u ∈ U

.

If t0 = 0 and x f = 0, or x0 = 0, we use the notation of RRM(t f ) and REM(t f ), respectively. Denoting by REM− the reachable set for the reverse time system, the following result holds. Proposition 2.1. The maximum recoverable set of the positive time system coincides with the maximum reachable set of the negative time system RRM+ (t f ) = REM− (t f ). Thanks to the Jordan canonical decomposition, the system can be always split into a stable and an unstable subsystem. Suppose that the state matrix A has m eigenvalues with positive real part and n − m eigenvalues with non-positive real part; then, up to a coordinates change, the system can be written in the equivalent form x˙ u Au 0 xu Bu = + u, (2.4) xs Bs x˙ s 0 As where xu , Bu ∈ Rm , xs , Bs ∈ Rn−m , Au ∈ Rm×m and As ∈ Rn−m×n−m . The following theorem on the maximum recoverable region has been proved by J. LeMay in [31]. Theorem 2.1. Given the system (2.4), the asymptotic maximum recoverable region (t f = +∞) is given by RRM(∞) = RRMu (∞) × Rn−m , where RRMu (∞) is the asymptotic maximum recoverable region of the system x˙ u = Au xu + Bu u. The theorem states that the recoverable region is completely determined by the unstable subsystem; in particular if the system has m unstable modes, the recoverable region contains a linear subspace of dimension n − m. In the presence of saturation constraints, the recoverable region is often denominated null controllable region, which is the largest set of initial data ensuring

2.2 The SISO Planar Case

9

asymptotic stability for a saturating system with an admissible control function. In order to deﬁne it rigorously we give some deﬁnitions. Deﬁnition 1. Given a saturation level M > 0, the set of admissible control inputs is deﬁned as UM = u ∈ C0 (R) : u∞ = sup |u(t)| ≤ M . t∈R

Deﬁnition 2. Given the system (2.1) we deﬁne saturated maximal region of attraction the set

BM = x ∈ Rn : ∃u ∈ UM with lim ϕ (t, x, u) = 0 . t→∞

Deﬁnition 3. The set UM∗ ⊂ UM is the set of admissible bang-bang control functions UM∗ = {u ∈ UM : u(t) ∈ {M, −M} ∀t ∈ R} . Deﬁning Bu as the region of attraction associated to the control variable u, the following identity holds Bu . BM = u∈UM

The problem of stabilization of linear systems with bounded controls has been extensively studied and completely solved for ANCBC systems (see [5], [6], [7]). In particular if all the eigenvalues of the state matrix have non positive real part, the saturated maximal region of attraction is the whole space; moreover semi-global stabilization for such systems can be achieved by a linear feedback. For systems having exponentially unstable modes, thanks to Theorem 2.1, it is enough to study the null controllable region of the antistable subsystem. In [24], the authors present a detailed analysis of the null controllable regions based on control inputs having a ﬁnite number of switching from maximum and minimum admissible values. The approach described in the present chapter is different (see [32]): starting from the knowledge of controlled invariant strips for the saturated system, thanks to an iterated convexiﬁcation technique, a recursive sequence of sets is deﬁned and the null controllable region is obtained as limit of such approximating sequence. The ﬁrst part of the chapter is dedicated to single input systems, while in the last section the MIMO case is presented.

2.2 The SISO Planar Case Consider a 2-dimensional controllable SISO linear system having, without loss of generality, the following structure ⎧ ⎨ x˙ = Ax + Bu (2.5) ⎩ x(0) = x0

10

2 Estimation of the Null Controllable Region: Continuous-Time Plants

where A=

0 1 a1 a2

0 B= , 1

,

x(t) = (x1 (t), x2 (t)) ∈ C0 (R, R2 ) is the state vector and v(t) ∈ C0 (R) is the input variable. Input saturation is assumed for the system; in particular the input v is subject to the constraint (2.2).

2.2.1 Real Eigenvalues For a planar system having real unstable eigenvalues it is possible to determine an invariant (controlled) parallelogram as external bound to the null controllable region. Moreover such bound is sharp, in the sense that the null controllable region and the parallelogram are tangent one to the other; setting the control u equal to the saturated values ±M and considering reversed-time evolution with the corner points of the parallelogram as initial positions, the boundary of BM is completely covered by the corresponding system trajectories. Consider the system (2.5) in the case when the state matrix A has real eigenvalues λ1 , λ2 ∈ R. Let us deﬁne the functions

ψ±M (t) = x1 (t) + mx2 (t) + q;

(2.6)

In order to determine constants m, q ∈ R such that, when the control variable u is taken identically equal to ±M, we have

ψ±M (t) = 0 ⇒ ψ˙ ±M (t) = 0. Since

ψ˙ ±M (t) = x˙1 (t) + mx˙2(t) = x2 (t) + m(a1x1 (t) + a2x2 (t) ± M);

(2.7)

imposing ψ±M (t) = 0 we get x1 (t) = −mx2 (t) − q. The equation ψ˙ ±M (t) = 0 can be written as

ψ˙ ±M (t) = x2 (t) + m(−a1mx2 (t) − a1q + a2x2 (t) ± M) = 0. and we obtain

⎧ M ⎪ ⎨q = ± a1 ⎪ ⎩ a1 m2 − a2 m − 1 = 0.

From the last equation we get m=

a2 ±

a22 + 4a1 2a1

;

(2.8)

2.2 The SISO Planar Case

11

the expression a22 + 4a1 is non negative if and only if A has real eigenvalues. In particular we have λ 1 + λ 2 ± |λ 1 − λ 2 | 1 1 m= ∈ − ,− . −2λ1λ2 λ1 λ2 1 , E 2 the (unbounded) strips We denote by EM M M 1 1 2 EM = x ∈ R : x1 − x2 < λ1 |λ 1 λ 2 | 1 M 2 2 EM = x ∈ R : x1 − x2 < λ2 |λ 1 λ 2 |

Let us denote by Γ±M the trajectory of ϕ (t, ±M/a1 , ±M) for t ∈ (−∞, 0]. Remark 2.1. Note that the two curves ΓM and Γ−M are symmetric with respect to the origin by construction. Theorem 2.2. Given the system (2.5) with λ1 , λ2 ∈ R, λ1 > 0 and λ2 > 0, the set ΓM ∪ Γ−M can be regarded as the boundary of a domain D. Moreover D coincides with the saturated maximal region of attraction BM and we have

∂ BM = ΓM ∪ Γ−M . Proof. We give the proof for λ1 = λ2 . Since a1 = −λ1 λ2 < 0 and λ1 , λ2 > 0, from (2.7) we obtain for j = 1, 2 λ

λ

λ

λ

j j ψ±M (t) < 0 ⇒ ψ˙ ±M (t) < 0 j j ψ±M (t) > 0 ⇒ ψ˙ ±M (t) > 0,

λ

j (t) is the function deﬁned by (2.6)-(2.7) and associated to the coefﬁcient where ψ±M m = 1/λ j , j = 1, 2; as a consequence we have

1 2 BM ⊆ EM ∩ EM .

Moreover, since for x2 < 0 we have x˙1 < 0 and for x2 > 0 we have x˙1 > 0, the set of points M x ∈ E1 ∩ E2 : |x1 | ≥ |λ 1 λ 2 | is not included in BM . By contradiction, let us suppose that (x1 , x2 ) = (−M/|a1|, s) ∈ BM with s < 0. In order to reach the origin, the derivative of the ﬁrst coordinate of the solution must change sign from negative to positive; this implies that the

12

2 Estimation of the Null Controllable Region: Continuous-Time Plants

trajectory must have an intersection with the x1 -axis in x = (x1 , 0), with / E1 ∩ E2 . We can conclude that x1 < −M/|a1 |, that is x ∈ BM ⊆ {x ∈ E1 ∩ E2 : |x1 | < M/|a1 |} . For u ≡ ±M the general structure of the system solution ϕ (t, x0 , ±M) is given by x1 (t) = C1 (x0 , M) exp (λ1t) + C2 (x0 , M) exp (λ2t) ∓

M a1

x2 (t) = λ1C1 (x0 , M) exp (λ1t) + λ2C2 (x0 , M) exp (λ2t), where C1 ,C2 are constants depending only on x0 and M; we see that, for any choice of the initial datum x0 , the solution veriﬁes M lim ϕ (t, x0 , ±M) = ∓ , 0 . t→−∞ a1 Let us ﬁx x0 = (M/a1 , 0) and consider the solution ϕ (t, x0 , M). Recall that, as a consequence of the unicity of the solution, there is no intersection between the integral curves. Let us denote by ΓM the trajectory of ϕ (t, x0 , M) for t ∈ [−∞, 0]. ΓM is a regular curve contained in the half-space {x2 ≤ 0} which separate into two disconnected components the region {x ∈ E1 ∩ E2 : x2 ≤ 0}. Let us choose z0 belonging to the lower part of {x ∈ E1 ∩ E2 : x2 ≤ 0}; the solution starting from z0 converges to (−M/a1 , 0) for t → −∞ by construction and since the trajectory must have no / E1 ∩ E2 . On the intersection with ΓM , there exists T > 0 such that ϕ (T, z0 , M) ∈ other hand, if z0 is taken in the region between ΓM and the x1 -axis, the trajectory of ϕ (t, z0 , M) intersects the x1 -axis in some x = (x1 , 0) with |x1 | < M/|a1 |. It is easy to see that, as θ varies in (0, M), the curves Γθ describe the whole region between ΓM and the x1 -axis. We can conclude that the boundary of BM is ΓM ∪ Γ−M , where Γ−M is the trajectory of the solution ϕ (t, x0 , −M) for t ∈ [−∞, 0] with x0 = (−M/a1 , 0). Remark 2.2. The curves ΓM and Γ−M constitute the boundary of the saturated maximal region of attraction also in the special case of coinciding real eigenvalues λ deﬁned λ1 = λ2 = λ > 0. Note that in this case we have only one invariant strip EM as 1 M λ = x ∈ R2 : x1 − x2 < 2 . EM λ λ Remark 2.3. For systems having real distinct eigenvalues, we can also derive equivalent bounds on the region of attraction starting from a diagonal state matrix setting. In particular it is very easy to obtain invariant strips for the equivalent system z˙ = diag(λ1 , λ2 )z + H−1Bu,

where H=

1 1 λ1 λ2

and z = H−1 x.

2.2 The SISO Planar Case

13

Remark 2.4. The asymptotes of the curves ΓM and Γ−M are the linear spaces satisfying the equations 1 M x2 ± . x1 = min (λ1 , λ2 ) a1

2.2.2 Complex Eigenvalues This section is devoted to the study of the saturated maximal region of attraction for a system having complex eigenvalues with strictly positive real part. Even if in this case no invariant parallelogram exists, the boundary of the null controllable region is still given by reversed-time system trajectories associated to the saturated control. Denoting by λ1 = α + jω and λ2 = α − jω the conjugate complex eigenvalues with α > 0, the matrix A can be written in terms of α and ω as follows 0 1 A= (2.9) −α 2 − ω 2 2α Without loss of generality we can assume ω > 0. We point out that the approach followed in the case of real eigenvalues fails for the complex case since there is no real solution to (2.8). Theorem 2.3. Consider the system (2.5) with state matrix given by (2.9) with α , ω > 0 and assume input saturation (1.3). Setting s± = ∓

M(1 + exp(−απ /ω )) , (1 − exp(−απ /ω ))(α 2 + ω 2 )

we denote by Ψ±M the trajectories of the solutions ϕ (t, x0 , ±M) for t ∈ (−π /ω , 0] and x0 = (s± , 0). The following identity holds

∂ BM = ΨM ∪ Ψ−M . Proof. For u ≡ ±M the solution ϕ (t, x0 , ±M) has the following structure x1 (t) = (C1 (x0 , M) cos ω t + C2 (x0 , M) sin ω t) exp (α t) ∓

M , a1

x2 (t) = (α (C1 (x0 , M) cos ω t + C2 (x0 , M) sin ω t)+ ω (−C1 (x0 , M) sin ω t + C2 (x0 , M) cos ω t)) exp (α t), where C1 ,C2 are real constants depending on x0 and M. We see that for any choice of x0 we have lim ϕ (t, x0 , ±M) = (∓

t→−∞

M , 0). a1

(2.10)

14

2 Estimation of the Null Controllable Region: Continuous-Time Plants

Let us ﬁx x0 = (s, 0) with s ∈ R. We obtain C1 (x0 , M) = s ± We have

M , a1

−α 2 − ω 2 x2 (t) = ω

α C2 (x0 , M) = − C1 . ω

M s± exp (α t) sin ω t a1

and we see that x2 (t) = 0 for t = kπ /ω , k ∈ Z. Setting t = −π /ω , we have M M x1 (t) = − s ± exp (−απ /ω ) ∓ . a1 a1 Imposing x1 (t) = −x0 = (−s, 0), we get M M s = s± exp (−απ /ω ) ± , a1 a1 that is s± = ∓

M(1 + exp(−απ /ω )) . (1 − exp(−απ /ω ))(α 2 + ω 2 )

The closed curve ΨM ∪ Ψ−M is the boundary of a regular domain D. We claim that D = BM . In particular the closed curves Ψθ ∪ Ψ−θ for θ ∈ [0, M] cover the whole domain D and constitute a family of invariant sets for the solution associated to the switching control u(t) = −θ S(x2(t)), where S : C0 (R) → { f (t) : f (t) ∈ {1, −1} ∀t ∈ R}, S(g(t)) = limτ →t − sign(g(τ )). We denote by Q2,4 the set Q2,4 = x ∈ R2 : x1 x2 < 0 . Let x0 be an arbitrary point in the inner part of D; by construction there exists θ1 ∈ (0, M) such that x0 lies on the set Ψθ1 ∪ Ψ−θ1 . Let us ﬁx δ > 0 such that θ1 + δ < M and apply the control feedback uδ ,θ1 (t) deﬁned by uδ ,θ1 (t) = −(θ1 + δ χ{t∈R:x(t)∈Q2,4 } )S(x2 (t)). Recall that for a given set E ∈ R, the characteristic function χE = χE (t) is deﬁned as follows ⎧ ⎨ 1 if t ∈ E, χE = ⎩ 0 if t ∈ / E.

2.2 The SISO Planar Case

15

We denote by t1 the quantity t1 = min t ∈ (0, ∞) : ϕ (t, x0 , uδ ,θ1 ) ∈ {x2 = 0} ; by construction the point xt1 =(x1 (t1 ), 0) lies on the closed curve Ψθ2∪ Ψ−θ2 with θ2 < θ1 . Deﬁne t2 = t1 + min t ∈ (0, ∞) : ϕ (t, xt1 , uδ2 ,θ2 ) ∈ {x2 = 0} with δ2 = δ /2. In this way we can construct a sequence of parameters {θn } ⊂ [0, M] and a sequence of instants {tn } ⊂ (0, ∞), tn +∞, such that, if we deﬁne uδ (t) =

∞

∑ uδn ,θn (t)χ[tn−1 ,tn ) ,

t0 = 0, δn = δ /n,

n=1

we have lim ϕ (t, x0 , uδ ) = 0.

t→∞

We have shown that D ⊆ BM . / D. Without loss of generality we can assume x0 = Let us consider now x0 ∈ (x01 , x02 ) with x02 ≤ 0. We proceed by contradiction supposing the existence of u(t) ∈ UM and T > 0 such that ϕ (T, x0 , u) ∈ ΨM ∪ Ψ−M . If ϕ (T, x0 , u) ∈ ΨM , since ϕ˙ (0, x0 , u) = (x02 , a1 x01 +a2 x02 +u(0)), ϕ˙ (0, x0 , M) = (x02 , a1 x01 +a2 x02 +M), u(0) ≤ M and property (2.10) holds, there exists T0 ≤ T such that ϕ (T0 , x0 , M) ∈ ΨM and this contradicts the invariance of ΨM . If ϕ (T, x0 , u) ∈ Ψ−M , there exists T1 < T such / Ψ−M . We have that ϕ (T1 , x0 , u) = (x1 (T1 ), x2 (T1 )) = xT1 with x2 (T1 ) ≥ 0 and xT1 ∈ ϕ (T − T1 , xT1 , u) ∈ Ψ−M and ϕ (T2 , xT1 , −M) ∈ Ψ−M for some T2 ≤ T − T1 , that is impossible because of the invariance of Ψ−M . We can conclude that BM ⊆ D. The claim is proved.

2.2.3 Some Remarks Remark 2.5. The sets BM1 and BM2 for M1 = M2 are time-uniformly homothetic: there exists a continuous map Θ : R 2 → R 2 such that 1. Θ (x) is an homothety with

Θ (BM1 ) = BM2 =

M2 BM1 . M1

2. If the solution φ (t0 , x0 , ±M1 ) ∈ BM1 , for some t0 ∈ R and x0 ∈ BM1 , then we have Θ (φ (t0 , x0 , ±M1 )) = φ (t0 , M2 x0 /M1 , ±M2 ). Remark 2.6. Consider the system (2.5) and assume that the state matrix A has real eigenvalues λ1 = 0, λ2 > 0. For p, q ∈ R, we deﬁne the functions

ψ±M (t) = px1 (t) + x2 (t) + q.

16

2 Estimation of the Null Controllable Region: Continuous-Time Plants

Following the steps of Section 2.2.1, if we take p = 0 and q = ±M/λ2 , we obtain

ψ±M (t) > 0 ⇒ ψ˙ ±M (t) > 0 ψ±M (t) < 0 ⇒ ψ˙ ±M (t) < 0. We can conclude that the saturated maximal region of attraction for such systems is M BM = x ∈ R2 : |x2 | < . λ2 Remark 2.7. Consider the system (2.5) and assume the state matrix A has real positive eigenvalues λ1 , λ2 > 0. Let us ﬁx the saturation level M > 0 and the determinant of the state matrix −a1 > 0; if λ1 (or λ2 ) increases then the saturated maximal region of attraction contracts √ along the x2 -axis. In particular the widest region is obtained whenever λ1 = λ2 = −a1 . Remark 2.8. Consider the system (2.5) having complex conjugate eigenvalues α ,ω α ± jω . Let us denote with BM the corresponding saturated maximal region of attraction. If we take ω1 , ω2 ∈ R with ω1 > ω2 > 0 then we have α ,ω1 α ,ω2 BM ⊂ BM .

In particular the following identity holds

α ,ω α BM = BM ,

ω ∈(0,∞) α is the saturated maximal region of attraction of the system (2.5) with real where BM eigenvalues λ1 = λ2 = α > 0.

2.3 The SISO n-Dimensional Case Let us consider a general controllable single-input linear system described by x˙ = Ax + Bu x(0) = x0 with x ∈ Rn . In view of the controllability assumption, without loss of generality, let us suppose that the pair (A, B) is given in the Brunovsky canonical form. Denote by a1 , ..., an the characteristic coefﬁcients of the state matrix A. The description of the null controllable region has been obtained following several steps; the main ones are described below.

2.3 The SISO n-Dimensional Case

17

• Invariant strips for the saturated system are found with an algebraic method; the intersection of such strips gives an invariant closed hyperparallelogram, leading to an outer estimate for the null controllable region. • Any reachable state at time T > 0 for an admissible control u is proven to be arbitrarily close to a reachable state at time T > 0 for a control input switching a ﬁnite number of times between maximum and minimum admissible values. • A pair of symmetric limit cycles Γ±M having extrema in two vertices of the invariant hyperparallelogram is considered. The afﬁne hull of such curves is shown to be the whole state space, i.e. there is no afﬁne subspace containing entirely Γ±M . • The convex hull of ΓM ∪ Γ−M is a closed set containing the origin as interior point; this set can be enlarged taking for any boundary point the curve corresponding to an extremal trajectory for the reversed-time system. The null controllable region is obtained with an iterative procedure combining convexiﬁcation and reversedtime evolution. Let us denote by λ1 , ..., λn ∈ R the eigenvalues (not necessarily distinct) of the matrix A. Suppose the system is antistable, that is λi > 0 for any i ∈ {1, .., n}. Consider the family of surfaces n−1

ψ±M (t) = x1 (t) + ∑ mi xi+1 + q±M = 0,

mi , q±M ∈ R

i=1

and look for all the possible choices of coefﬁcients mi in order to have

ψ±M (t) = 0 ⇒ ψ˙ ±M (t) = 0 for u ≡ ±M.

(2.11)

The derivative of ψ±M (t) is given by n−2

n

ψ˙ ±M (t) = x2 + ∑ mi xi+2 + mn−1 ±M + ∑ ai xi . i=1

i=1

Using the equality ψ±M (t) = 0 and imposing ψ˙ ±M (t) = 0, the following set of algebraic equations is obtained ⎧ ⎨ q±M = ±M/a1 1 + mn−1(−m1 a1 + a2) = 0 ⎩ m j + mn−1 (−a1 m j+1 + a j+2) = 0, j = 1, ..., n − 2. Thanks to a recursion property, the system can be rewritten as ⎧ mn−2 = a1 m2n−1 − anmn−1 , ⎪ ⎪ ⎪ ⎪ m = a21 m3n−1 − a1an m2n−1 − an−1mn−1 , ⎪ ⎨ n−3 .. . ⎪ ⎪ n−3− j n−2 n−4 n−4− j ⎪ ⎪ an− j mn−1 , m = an−3 ⎪ 1 mn−1 − ∑ j=0 a1 ⎩ 2 m1 = (1 + mn−1a2 )/mn−1 a1 ,

18

2 Estimation of the Null Controllable Region: Continuous-Time Plants

where mn−1 satisﬁes the equation p∗A (mn−1 ) = 0, with n−2

n j+1 p∗A (λ ) = −an−1 = 0. 1 λ + 1 + ∑ a1 a j+2 λ j

(2.12)

j=0

Proposition 2.2. Let λ be a solution of the characteristic equation pA (λ ) = λ n − ∑nj=1 a j λ j−1 = 0. Then λ /a1 satisﬁes p∗A (λ /a1) = 0. Proof. The statement follows observing that n

j+1

n−2 λ λ j p∗A (λ /a1 ) = − + 1 + ∑ a1 a j+2 j+1 = a1 a1 j=0 n 1 n j−1 = 0. λ − ∑ a jλ =− a1 j=1

As a consequence, the solutions of (2.12) are given by

λj=

λj , n Πi=1 λi

j = 1, ..., n.

The above solutions lead to a set of n pairs of parallel hyperplanes; taking all the possible intersections, a closed hyperparallelogram containing the origin as interior point is found. Denote this set as FM . Thanks to the invariance property (2.11), if a trajectory starts from a point outside the set FM , there is no admissible control that can drive it inside, this meaning that BM ⊆ FM . The set ∂ FM intersects the x1 -axis in x± 0 = (±M/a1 , 0, ..., 0). It is immediate to verify that ∓ lim φ (t, x± 0 , ±M) = x0 t→−∞

The following technical lemma is needed. Lemma 2.1. Fix x0 ∈ Rn , u ∈ UM and T > 0; for any ε > 0 there exists a control function vε ∈ UM∗ such that 1. φ (T, x0 , u) − φ (T, x0, vε ) ≤ ε ; 2. vε has a ﬁnite number of switches in [0, T ]. Proof. The statement of the lemma is equivalent to the existence of a function vε ∈ UM∗ , with a ﬁnite number of switches in [0, T ], such that T T A(T −t) A(T −t) ≤ ε. J= e Bu(t)dt − e Bv (t)dt (2.13) ε 0

0

2.3 The SISO n-Dimensional Case

19

Consider a general partition Π = {0 = t1 < t2 < ... < tr = T } of the interval (0, T ) and a piecewise constant (PWC) matrix function Θ(t) associated to it; in particular Θ(t) =

r−1

∑ Hi χ(ti ,ti+1 ) (t),

i=1

where Hi are constant n × n matrices. The integral in (2.13) can be estimated as T A(T −t) e − Θ(t) B(u(t) − v (t))dt J≤ ε 0 T Θ(t)B(u(t) − vε (t))dt + = J1 + J2 . 0

The two terms will be treated separately. The ﬁrst one veriﬁes J1 ≤

T A(T −t) − Θ(t) B(u(t) − vε (t)) dt e 0 T

≤ 2MB

0

A(T −t) − Θ(t) dt e

for any vε ∈ UM . Since eA(T −t) is continuous, a partition Πε with corresponding PWC matrix function Θε can be chosen such that T A(T −t) − Θε (t) dt ≤ e 0

ε . 2MB

Let us consider the second term r−1 ti+1 r−1 ti+1 ε ε . J2 = ∑ Hi B(u(t) − vε (t))dt ≤ ∑ H B(u(t) − v (t))dt ε i i=1 ti i=1 ti Since Hεi B for any ﬁxed i is a constant vector, the function vε ∈ UM∗ can be deﬁned t t on any subinterval (ti ,ti+1 ) in order to have tii+1 Hεi Bu(t)dt = tii+1 Hεi Bvε (t)dt. It follows that J2 = 0. Note that the function vε can be designed having at most one switch in any subinterval (ti ,ti+1 ); this implies that the total number of switches in the whole interval [0, T ] is less or equal than 2r < ∞. n Proposition 2.3. Let x± 0 = (±M/a1 , 0, 0, ..., 0) ∈ R and consider the curve γ (t) = ± φ (t, x0 , ±M) for t ∈ (−∞, 0]. For s < n there is no s-dimensional afﬁne subspace containing γ .

Proof. A necessary and sufﬁcient condition for a curve to belong to an afﬁne jdimensional subspace is that its j-curvature is identically null. It will be shown that

20

2 Estimation of the Null Controllable Region: Continuous-Time Plants

the (n − 1)-curvature of γ (t) in x± 0 is different from zero. With a recursive procedure the derivatives of γ (t) can be computed. In particular it holds

γ (1) (t) = Aγ (t) ± BM and

γ ( j) (t) = Aγ ( j−1) (t) = A j−1 γ (1) (t)

j = 2, ..., n,

where j ∈ N denotes the derivative order. Let us denote by W(t) the square matrix having γ ( j) (t), j = 1, ..., n, as columns and by κ (n−1)(t) the (n − 1)-curvature of γ (t). The condition κ (n−1)(t) = 0 is equivalent to det W(t) = 0. Since in particular det W(0) = ±2M = 0, it has been proved that γ (t) is not contained in any (n − 1)-dimensional afﬁne subspace. Let us introduce the set-valued function Φ±M deﬁned on Rn as follows:

Φ±M (x) = {y ∈ Rn : ∃t ∈ (−∞, 0] : y = φ (t, x, ±M)} . ± In particular the identity γ (t) = Φ±M (x± 0 ) holds, where x0 = (±M/a1 , 0, ..., 0). + Corollary 2.1. Given the set Γ = Φ−M (x− 0 ) ∪ ΦM (x0 ) together with its convex hull S0 = Co(Γ ), there exists δ > 0 with Bδ (0) ⊂ S0 ; in particular |S0 | > 0.

Proof. The interior part of Co(Γ ) is nontrivial since the afﬁne hull of Γ (i.e. the smallest afﬁne space containing the set Γ ) is the whole space Rn by Proposition 2.3; this proves that |S0 | > 0. Let x be an interior point of Co(Γ ); by symmetry it follows that −x is still an interior point as well as any point xθ obtained as convex combination (see [33] page 45) xθ = θ x − (1 − θ )x, θ ∈ [0, 1]. 1 The result follows observing that the origin veriﬁes (2.14) with θ = . 2 The main result of the paper is stated in the next theorem.

(2.14)

Theorem 2.4. Deﬁne a monotone recursive sequence of sets {S j } j∈N S j+1 = Co(Φ−M (∂ S j ) ∪ ΦM (∂ S j )),

(2.15)

where S0 is given by S0 = Co(Γ ) and Γ is deﬁned in Corollary 2.1. The null controllable region of the system is given by BM = lim S j = j→∞

j∈N

S j =: S∞

2.3 The SISO n-Dimensional Case

21

Proof. First prove that BM ⊆ S∞ . Let x0 be an arbitrary point contained in BM and suppose that x0 ∈ / S0 ; by Corollary 3.6, there exists δ > 0 such that Bδ (0) ⊂ S0 ⊂ S∞ . Take 0 < δ1 < δ . Since x0 ∈ BM , there exist u ∈ UM and t > 0 such that φ (t, x0 , u) < δ1 for any t > t. Thanks to Lemma 2.1 with ε = δ − δ1 > 0, the existence of v ∈ UM∗ such that φ (t, x0 , v) < δ can be inferred. Moreover, since the trajectory is continuous, it must intersect the boundary ∂ S0 ; in particular φ (t 1 , x0 , v) = x˜ ∈ ∂ S0 for some 0 < t 1 < t. By construction (see Lemma 2.1) the control v ∈ UM∗ has a ﬁnite number of switches, say k ∈ N, as t varies in the bounded interval [0,t 1 ]. Consider the reversed-time evolution

φ (t, x˜ , v(t + t 1 )) for t ∈ [−t 1 , 0]; it veriﬁes φ (−t 1 , x˜ , v(0)) = x0 . Recalling the formula (2.15) for the sets S j , it follows immediately that x0 ∈ Sk , this proving that x0 ∈ S∞ . Let us show that the converse inclusion S∞ ⊆ BM holds too. Saying that x0 ∈ S∞ means there exists k such that x0 ∈ Sk . Moreover, by deﬁnition, there exists a control function in UM∗ having at most k switches such that the solution driven by it enters S0 in ﬁnite time. It remains only to prove that S0 ⊆ BM . Let x0 ∈ S0 . Since the solutions are continuously dependant on the system parameters, the interior part of S0 is completely covered by a family of homothetic closed surfaces ZM∗ which ∗ are generated starting from the points x± 0,M∗ = (±M /a1 , 0, ..., 0) and switching the ∗ ∗ control between the values ±M as M varies in [0, M]. Now, if x∗0 ∈ ZM∗ , by the Carath´eodory theorem (see [33], page 155), there exist θ j > 0, 1 < j < n + 1, with ∑n+1 j=1 θ j = 1 and t j ∈ (−∞, 0] such that x∗0 =

j∗

n+1

j=1

j= j +1

∑ θ j φ (t j , x+0,M∗ , M∗ ) + ∑∗

∗ θ j φ (t j , x− 0,M∗ , −M ),

(2.16)

with j∗ ∈ [0, n + 1]. Note that the elements of S∞ obtained as lim φ (t, x, ±M), for some x, belong to ∂ S∞ . It follows from (2.16) that ∗

x (t) =

j∗

n+1

j=1

j= j +1

∑ θ j φ (t + t j , x+0,M∗ , M∗ ) + ∑∗

t→−∞

∗ θ j φ (t + t j , x− 0,M∗ , −M )

satisﬁes the system equations for initial datum x∗0 and control input u∗ (t) = M ∗ j∗ ∗ (∑ j=1 θ j − ∑n+1 j= j ∗ +1 θ j ). This means that any point in ZM lies on a trajectory associated to a suitable admissible control input. Now, if x0 ∈ S0 , there exist M1∗ < M such that x0 lies on the invariant surface ZM1∗ under a suitable control input u∗ with |u∗ | ≤ M1∗ < M. In order to stabilize the system, it sufﬁces to make the trajectory jump to a lower level ZM2∗ with M2∗ < M1∗ increasing the control norm. This concludes the proof.

22

2 Estimation of the Null Controllable Region: Continuous-Time Plants

2.3.1 Remarks and Discussion It has been clear since earlier works (see [31] for example) that, in order to obtain a good description of null controllable region (maximal recoverable region), it is necessary to deal with time-reversed systems (maximal reachable region). Since the problem has a high computational complexity, the development of different techniques may be useful and interesting. The authors of [24], ﬁxing the saturation level to M = 1, describe the boundary of the null controllable region as the boundary of the reachable region for the reversed-time system 0 Aτ T Aτ n ∂R = z = e Bsign(c e B)d τ : c ∈ R \ {0} . (2.17) −∞

For systems having only real eigenvalues, the boundary can be described also in terms of trajectories corresponding to control input having at most n − 1 switches. The result is based on a technical lemma which states the maximum number of solutions for a class of exponential/polynomial algebraic equations. In this section a different characterization of the null controllable region is presented for antistable systems having real eigenvalues. First an outer estimate is found considering an invariant hyperparallelogram; the second step is to determine a pair of symmetric time-reversed trajectories Γ±M lying on the boundary of the null controllable region and having extrema in two opposite vertices of the hyperparallelogram. Finally, a bounded domain is obtained taking the convex hull Co(ΓM ∪ Γ−M ). The conclusion follows with an iterative procedure based on a simple algorithm: 1. enlarge the set considering time-reversed evolution starting from boundary points with control input u ≡ ±M; 2. take the convex hull of the new set. Regarding numerical implementation, an approximation for the null controllable region can be performed by means of Carath´eodory theorem with the desired accuracy. q Selecting a discrete set of points on ΓM ∪ Γ−M , say {γ j } j=1 q ∈ N, an approximated domain can be computed taking convex combinations of n + 1 elements : (1)

BM =

x∈R:x=

n+1

n+1

s=1

s=1

∑ θs γ js : ∑ θs = 1

.

The approximation can be further improved repeating the steps of the algorithm described above. In particular ﬁxing a discrete set of negative time instants Tq = {−∞ < t1 < ... < tq ≤ 0}, an increasing sequence of discrete sets can be derived: ( j)

BM =

x∈R:x=

n+1

n+1

s=1

s=1

∑ θs zs :

∑ θs = 1, zs ∈ Λ j−1

( j−1) . where Λ j−1 = z ∈ Rn : z = φ (t, x, ±M) : t ∈ Tq , x ∈ BM

,

2.4 The MIMO Case

23

The accuracy of the method is determined by several parameters: the number q ∈ N, the set Tq , the coefﬁcients θs for the convex combinations.

2.4 The MIMO Case This section is devoted to the presentation of null controllable regions for multi input continuous-time linear systems. The multi input null controllable region can be derived as combination of lower-order single input null controllable regions as well as by a direct computation. Both methods will be presented for sake of completeness. Let us consider the following multi input linear system ⎧ ⎨ x˙ = Ax + Bu (2.18) ⎩ x(0) = x0 with m < n, A ∈ Rn×n , B ∈ Rn×m , x ∈ Rn and u ∈ Rm ; the plant is assumed to be controllable, i.e. (2.19) rank[B AB A2 B · · · An−1 B] = n and antistable, that is ℜ(λi ) > 0 for any eigenvalue λi , i = 1, ..., n, of the matrix A. We will use the following notation: ⎡ ⎤ u1 ⎢ u2 ⎥ ⎢ ⎥ B = [B1 B2 · · · Bm ], u = ⎢ . ⎥ , ⎣ .. ⎦ um where Bi ∈ Rn and ui ∈ R for any i = 1, ..., m.

2.4.1 Method of Lower-Order Single Input Subsystems According to (2.18), we can deﬁne a family of single input systems Σi , i = 1, .., m, as follows ⎧ ⎨ x˙ = Ax + Bi ui Σi = ⎩ x(0) = x0 Let ni ∈ N be the dimension of the controllability subspace for the system Σi , ni := rank[Bi ABi A2 Bi · · · An−1 Bi ] ≤ n.

24

2 Estimation of the Null Controllable Region: Continuous-Time Plants

The system Σi can be transformed into a controllable/uncontrollable subsystems decomposition by a linear coordinates transformation; let Ri ∈ Rn×n the matrix associated to such transformation. We have ⎡ ∗⎤ Bi ! ⎢ 0 ⎥ (i) ni ×(n−ni ) A 0 ⎢ ⎥ c R−1 , R−1 i ARi = i Bi = ⎢ .. ⎥ (i) ⎣ . ⎦ 0(n−ni)×n Auc 0 (i)

(i)

with Ac ∈ Rni ×ni , Auc ∈ R(n−ni )×(n−ni ) , B∗i ∈ Rni and " # (i) 2 ∗ (i) ni −1 ∗ ∗ (i) ∗ rank Bi Ac Bi Ac Bi · · · Ac Bi = ni . (i)

As a consequence, the lower-order system (Ac , B∗i ) can be transformed in its controllability canonical form by a linear transformation Qi ∈ Rni ×ni , ⎡

0 ··· ⎢ 1 ··· ⎢ .. . . ⎢ (i) ⎢ ¯ . . Q−1 i Ac Qi = Ai := ⎢ .. ⎢ ⎣ 0 0 0 . (i) (i) (i) a1 a2 a3 · · · 0 0 .. .

1 0 .. .

0 0 .. .

⎤

⎥ ⎥ ⎥ ⎥, ⎥ ⎥ 1 ⎦ (i) a ni

⎡ ⎤ 0 ⎢0⎥ ⎢ ⎥ ⎢ .. ⎥ ∗ ¯ Q−1 ⎥. i Bi = Bi := ⎢ ⎢.⎥ ⎣0⎦ 1

Using the results of Section 2.3, we can compute the null controllable region for ¯ i , B¯ i ); let us denote this region by B (i) . The description of B (i) with the system (A M M respect to the original coordinates of the system Σi is given by the inverse transformation (i) (i) Ri Hi BM × {0n−ni } := DM , where the matrix Hi ∈ Rn×n is deﬁned as " # Qi 0ni ×(n−ni) Hi := (n−ni)×n . 0 I(n−ni ) (i)

By construction, the set DM is a convex set contained in a linear subspace Vi ⊂ Rn with dim(Vi ) = ni . Theorem 2.5. The null controllable region BM for the multi input system (2.18) is given by

m (i) (i) BM = ∑ DM = x ∈ Rn : x = x1 + x2 + · · · + xm , xi ∈ DM ∀ i = 1, ..., m . i=1

2.4 The MIMO Case

25 (i)

Proof. The set ∑m i=1 DM , as it is deﬁned as a ﬁnite sum of convex sets, is still con(i) (i) m vex (see for instance [33]). Let x ∈ ∑m i=1 DM , with x = ∑i=1 xi . Since xi ∈ DM , by construction, there exists ui ∈ UM such that, deﬁning φi (t, x0 , u) the dynamics of the system Σi driven by the control u, we have lim ||φi (t, xi , ui )|| = 0.

t→∞

Now, since

m

φ (t, x, u) = ∑ φi (t, xi , ui ), i=1

we can deduce the existence of u ∈ UM such that lim ||φi (t, x, u)|| = 0;

t→∞ (i)

in other words we have ∑m i=1 DM ⊆ BM . Let us show that the converse inclusion (i) BM ⊆ ∑m D holds too. Suppose that x ∈ BM ; by deﬁnition, there exists u = i=1 M (u1 , ..., um ) with |ui | ≤ M such that lim ||φi (t, x, u)|| = 0.

(2.20)

t→∞

Using the explicit formula for the solution we have

φi (t, x, u) = eAt x +

t 0

m

eA(t−s) ∑ Bi ui (s)ds. i=1

From (2.20) we obtain lim eAt

t→∞

x + e−At

t 0

m

eA(t−s) ∑ Bi ui (s)ds

=0

i=1

and necessarily one has lim e−At

t→∞

t 0

m

eA(t−s) ∑ Bi ui (s)ds = lim

t

t→∞ 0

i=1

m

e−As ∑ Bi ui (s)ds = −x. i=1

m Now, since || ∑m i=1 Bi ui (s)|| ≤ M ∑i=1 ||Bi || < ∞ and limits exist ∀i = 1, ..., m and they are ﬁnite:

t

lim

t→∞ 0

∞ 0

||e−As ds|| < ∞, the following

e−As Bi ui (s)ds := −xi .

Moreover, to ensure the asymptotic stability of the system, the input control u = (u1 , u2 , ..., um ) has to satisfy lim |u1 (s)| = lim |u2 (s)| = · · · = lim |um (s)| = 0

t→∞

t→∞

t→∞

26

2 Estimation of the Null Controllable Region: Continuous-Time Plants

and, as a consequence, the following property holds t At −As lim e xi + e Bi ui (s)ds = 0 ∀i = 1, ..., m. t→∞

0

(i)

(i)

m In conclusion, we have xi ∈ DM and x = ∑m i=1 xi , that is BM ⊆ ∑i=1 DM . The proof is completed.

2.4.2 Direct Method The direct method for the description of the multi input null controllable region follows the same steps presented in the single input case. In particular, the null controllable region can be computed with an iterative procedure combining reversed-time evolution with extremal controls and set convexiﬁcation. The following result, which constitutes the multi input version of Lemma 2.1, can be proved. Lemma 2.2. Fix x0 and u = (u1 , ..., um ) ∈ UM ; then for every T > 0 and for any ε > 0, there exists v(ε ) ∈ UM∗ such that ||φ (T, x0 , u) − φ (T, x0 , v(ε ) )|| ≤ ε . Moreover each component of v(ε ) has a ﬁnite number of switches in the interval [0, T ].

Proof. Using the explicit formula for the solution, the statement of the lemma is equivalent to the existence of a bang-bang control v(ε ) ∈ UM∗ with a ﬁnite number of switchings such that T ε A(T −s) (ε ) e B(u(s) − v (s))ds J := ≤ ε. 0

Consider now a general partition Πr = {0 = t1 < t2 < ... < tr = T } of the interval (0, T ) and deﬁne a piecewise constant matrix (PWC) function Θ (t) associated to it:

Θ (t) =

r−1

∑ Hk χ(tk ,tk+1 ) (t),

k=1

(2.21)

2.4 The MIMO Case

27

where Hi are constant n × m matrices to be ﬁxed and χ(tk ,tk+1 ) (t) are standard characteristic functions. Setting H (t) := eA(t−s) B − Θ (t), the integral J ε can be estimates as T ε (ε ) J ≤ H (s)(u(s) − v (s))ds 0

T (ε ) ε ε Θ (s)(u(s) − v (s))ds + =: J1 + J2 . 0 The two terms will be treated separately. The ﬁrst one veriﬁes J1ε ≤

T 0

H (s)(u(s) − v(ε ) (s)) ds ≤ 2M

T 0

H (s) ds

for any v(ε ) ∈ UM∗ . Since the matrix eA(t−s) B ∈ L1loc (R; Rn×m ) and the PWC functions deﬁned in (2.21) are dense with respect to L1 -norm (see [34]), suitable matrices Hεk and a partition Πrε can be found such that T 0

H (s) ds ≤

ε . 2M

(2.22)

Let us consider the second term rε −1 tk+1 ε ε (ε ) J2 = ∑ Hk (u(s) − v (s))ds k=1 tk ≤

rε −1 tk+1

∑

k=1

tk

Hεk (u(s) − v(ε ) (s))ds .

Since Hεk for any ﬁxed k is a constant matrix, the control vε can be deﬁned on any subinterval (tk ,tk+1 ) in order to have tk+1 tk

u(s)ds =

tk+1 tk

v(ε ) (s)ds;

this can be done following the simple rule ⎧ ⎨ M if tk ≤ t ≤ τki (ε ) vi (t)|(t ,t ) := k k+1 ⎩ −M if τki < t ≤ tk+1 ,

28

2 Estimation of the Null Controllable Region: Continuous-Time Plants tk+1

for i = 1, ..., m where

τki

=

tk

ui (s)ds + M(tk+1 + tk ) . It follows that, with the 2M

above choice for the components of the bang-bang control v(ε ) , the integral J2ε veriﬁes rε −1 ε tk+1 ε (ε ) J2 ≤ ∑ Hk (u(s) − v (s))ds = 0. t k=1

k

(ε )

Note that in any subinterval (tk ,tk+1 ) the functions vi have at most one switch by construction; this implies that the total number of switches in the whole interval [0, T ] is less or equal than 2(rε − 1). Let us denote by UM∗∗ the set of extremal controls; to this purpose we recall that a control input u = (u1 , ..., um ) ∈ UM∗ is extremal if ui ≡ ±M ∀i = 1, ..., m. Using the explicit formula for the solution φ (t, x, u) one can verify that ∀u ∈ UM∗∗ lim φ (t, x, u∗ ) = A−1 Bu∗ ∀x ∈ Rn .

t→−∞

Fixing u∗1 ≡ M, there are 2m−1 possible combinations for the other components of an extremal control u ∈ UM∗∗ ; in this way, we can deﬁne a set of 2m−1 points m

xM,u = A−1 B1 + ∑ ±Bi M i=2

such that

∗∗ lim φ (t, ∓xM,u , ±u) = ±xM,u ∀u ∈ UM,+

t→−∞

∗∗ := {u ∈ U ∗∗ : u = M}. For u ∈ U ∗∗ we deﬁne the where it has been set UM,+ 1 M M,+ family of curves γM,u (t) := φ (t, −xM,u , u), t ∈ (−∞, 0]

and the corresponding supports

ΓM,u := {x ∈ Rn : x = φ (t, −xM,u , u), t ∈ (−∞, 0]}.

(2.23)

∗∗ such that, for Proposition 2.1. There exists at least one extremal control u¯ ∈ UM,+ s < n there is no s-dimensional afﬁne subspace containing γM,u¯ (t). ∗∗ the multi input system (A, B) can be regarded as the Proof. For any ﬁxed u ∈ UM,+ single input system (A, Bu ) driven by u ≡ M with m

Bu = B1 + ∑ ±Bi ; i=1

(2.24)

2.5 Examples

29

the signs in the above formula are ﬁxed according to the signs of the components ui , i = 2, ...m. Since the plant (A, B) is assumed to be controllable, there exists ∗∗ such that the single input system (A, B ), with B deﬁned necessarily u¯ ∈ UM,+ u¯ u¯ by (2.24), is controllable as well. Now the curve γM,u¯ (t) can be regarded as the reversed-time evolution of the single input system (A, Bu¯ ) driven by the control u ≡ M and the conclusion follows applying the results given in Proposition 2.3 to such curve. ∗∗ , deﬁned on Rn as follows Introducing the set-valued function Φ±u , u ∈ UM,+

Φu (x) = {y ∈ Rn : ∃t ∈ (−∞, 0] : y = φ (t, x, ±u)} , one can design a monotone recursive sequence of sets {S j } j∈N ⎛ ⎞ S j+1 = Co ⎝

where S0 is given by S0 = Co

∗∗ uUM,+

(Φ−u (∂ S j ) ∪ Φu (∂ S j ))⎠ ,

(

∗∗ −ΓM,u ∪ ΓM,u uUM,+

(2.25)

with ΓM,u given by (2.23).

Theorem 2.6. The null controllable region for the multi input system (2.18) is BM = lim S j = j→∞

S j =: S∞

j∈N

Proof. Using Lemma 2.2, the dynamics of the system driven by an admissible control can be approximated with the desired accuracy by the dynamics corresponding to extremal controls; due to this result, the arguments to prove the theorem are analogous to those employed to demonstrate the validity of Theorem 2.4 and for sake of brevity the detailed proof is omitted.

2.5 Examples Example 2.1. We consider a planar system having real eigenvalues λ1 = 1 and λ2 = 3, therefore the state matrix is 0 1 A= −3 4 Fixing the saturation level to M = 4, the equations of ΓM = −Γ−M are ⎧ 2 2 ⎪ ⎨ x1 (t) = (et − 1)2 (et + 2) + et (e2t − 3), 3 3 ΓM = ⎪ ⎩ x2 (t) = 4et (e2t − 1), t ∈ (−∞, 0]

30

2 Estimation of the Null Controllable Region: Continuous-Time Plants

Figure 2.1 shows a plot of the saturated maximal region of attraction.

x2 1.5

1.0

0.5

1.0

0.5

0.5

1.0

x1

0.5

1.0

1.5

Fig. 2.1. Example of saturated maximal region of attraction with λ1 = 1, λ2 = 3 and M = 4.

Example 2.2. Let us consider a planar systems with complex eigenvalues λ1 , λ2 ∈ C with ℜ(λ1 ) = ℜ(λ2 ) = 0.5 and ℑ(λ1 ) = −ℑ(λ2 ) = 1.2. The canonical form of the state matrix is 0 1 A= −1.69 1 Fixing the saturation level to M = 2, the equations of ΨM = −Ψ−M are ⎧ 5π + t (−12 cos 6t +5 sin 6t ) ⎪ 12 2 5 5 100 e ⎪ ⎪ x1 (t) = 6+ , ⎪ 5π ⎪ ⎪ 507 ⎨ e 12 − 1 ΨM = ⎪ 5π t ⎪ ⎪ 10e 12 + 2 sin 6t5 ⎪ ⎪ ⎪ x (t) = ⎩ 2 5π 3(e 12 − 1)

2.5 Examples

31

The null controllable region for the system is depicted in Fig.2.2. x2

2

1

2

1

1

2

x1

1

2

Fig. 2.2. Example of saturated maximal region of attraction with ℜ(λ1 ) = ℜ(λ2 ) = 1.5, ℑ(λ1 ) = −ℑ(λ2 ) = 1.2 and M = 2.

Example 2.3. Let us consider the antistable system (A, B) with ⎛ ⎞ ⎛ ⎞ 0 1 0 0 A = ⎝ 0 0 1 ⎠, B = ⎝ 0 ⎠. 0.1 −0.8 1.7 1 The open-loop system eigenvalues are

λ1 = 0.2,

λ2 = 0.5,

λ3 = 1.

Setting the saturation level M = 1, the starting points for the algorithm are given by x± 0 = ±(10, 0, 0).

32

2 Estimation of the Null Controllable Region: Continuous-Time Plants

Pictures of the null controllable region at different iteration steps are given in Fig.2.3. The simulation has been performed prescribing symmetric convex combinations, i.e. θ1 = θ2 = · · · = θN+1 = 1/(N + 1) and using the convhulln MatLab built-in function.

(a) - Curve Γ

(b) - Initial approximated set S0

(c) - Approximation after 10 iterations

(d) - Approximation after 30 iterations

Fig. 2.3. Approximation of the null controllable region

Chapter 3

Estimation of the Null Controllable Region: Discrete-Time Plants

This chapter is devoted to the study of the null controllable region for a saturating discrete-time plant. In the SISO case, analogously to the continuos-time case, the null controllable region is characterized making resort to a recursive formula deﬁning sets whose union completely describes it, wherever are located the complex eigenvalues of the matrix A with respect to the unit circle. The following notations will be used in the following. Denote by ∞ (Z) the set of bounded bilateral sequences. For any arbitrary control input u ∈ ∞ (Z and a ﬁxed initial condition x0 , let us denote by φ (k, x0 , u) the solution of the system x(k + 1) = Ax(k) + Bu(k) (3.1) x(0) = x0 with x = (x1 , ..., xn ) ∈ Rn . In view of the controllability assumption, without loss of generality, the pair (A, B) is given in canonical form. Given a real positive value M > 0, the system is said to be subject to saturation at level M if the control variable u(k) is forced to have norm less than M. By the notation WM will indicate the set of admissible control functions: ∞ WM = u ∈ (Z) : u∞ = sup |u(k)| ≤ M . k∈Z

is deﬁned as the maximal As in the continuous case the null controllable region BM region of attraction for the system under the input constraint, i.e. = x ∈ Rn : ∃u ∈ WM : lim φ (k, x, u) = 0 . BM k→+∞

The set WM∗ ⊂ WM is the set of admissible switching control functions WM∗ = {u ∈ WM : u(k) ∈ {M, −M} ∀k ∈ Z} .

M.L. Corradini et al.: Control Systems with Saturating Inputs, LNCIS 424, pp. 33–52. c Springer-Verlag London Limited 2012 springerlink.com

34

3 Estimation of the Null Controllable Region: Discrete-Time Plants

It is worth noting that, albeit an algorithm describing the null controllable region has been already given in [35], the approach described in the present chapter is different and follows the lines of the procedure used in the continuous case. As in the previous chapter, the SISO planar case is treated ﬁrst, then the null controllable region for multidimensional systems having positive real eigenvalues with norm larger than 1 is described. Finally the MIMO case is addressed in the last section.

3.1 The SISO Planar Case Let us consider the following planar single-input discrete-time linear system x(k + 1) = Ax(k) + Bu(k) (3.2) x(0) = x0 with x = (x1 , x2 ) ∈ R2 . In view of the controllability assumption, without loss of generality, the pair (A, B) is given in canonical form. The system is assumed to be subject to input saturation, i.e. u(k) = satM (v(k)). The elements a1 , a2 on the last row of the matrix A are the coefﬁcients of the charac teristic polynomial, i.e. pA (λ ) = −λ 2 + a2 λ + a1 . The null controllable region BM in this case is deﬁned as 2 BM = x ∈ R : ∃u ∈ WM : lim φ (k, x, u) = 0 k→+∞

Let us deﬁne the functions

ψ±M (t) = x1 (t) + mx2 (t) + q; our aim is to determine constants m, q ∈ R such that, when the control variable u is taken identically equal to ±M, we have

ψ±M (t) = 0 ⇒ ψ±M (t + 1) = 0. We have

ψ±M (t + 1) = x1 (t + 1) + mx2(t + 1) + q = x2 (t) + m(a1x1 (t) + a2x2 (t) ± M) + q;

(3.3)

imposing ψ±M (t) = 0 we get x1 (t) = −mx2 (t) − q. The equation ψ±M (t + 1) = 0 can be written as

ψ±M (t + 1) = x2 (t) + m(−a1mx2 (t) − a1q + a2x2 (t) ± M) + q = 0.

3.1 The SISO Planar Case

We obtain

35

⎧ mM ⎪ ⎨q = ± a1 m − 1 ⎪ ⎩ a1 m2 − a2 m − 1 = 0.

From the last equation we get m=

a2 ±

a22 + 4a1 2a1

;

(3.4)

the expression a22 + 4a1 is non negative if and only if A has real eigenvalues. In particular we have 1 1 λ 1 + λ 2 ± |λ 1 − λ 2 | ∈ − ,− m= . −2λ1λ2 λ1 λ2 1 , E 2 the (unbounded) strips We denote by EM M M 1 1 2 EM = x ∈ R : x1 − x2 < λ1 |λ 1 λ 2 − λ 1 | M 1 2 EM = x ∈ R2 : x1 − x2 < . λ2 |λ 1 λ 2 − λ 2 |

If it is assumed that |λ1 |, |λ2 | > 1, as a consequence of the previous analysis, the system cannot be driven inside the strips by a control input satisfying the constraint 1 or the set E 2 . In other |u(k)| < M if the initial condition is taken outside the set EM M 1 and R2 \ E 2 cannot words, the states belonging to the complementary sets R2 \ EM M be steered to the origin under any admissible control −M ≤ u(k) ≤ M, this leading to an outer estimate for the maximal region of attraction, as described in the next statement. Proposition 3.1. If the eigenvalues λ1 , λ2 of the state matrix A are real numbers placed outside the unit circle, i.e. |λ1 |, |λ2 | > 1, then the following set inclusion holds 1 2 ⊂ EM ∩ EM . (3.5) BM 1 and ∂ E 2 have intersection along the main bisector x = x The boundary sets ∂ EM 1 2 M and such intersection is given by a pair of symmetric points

x± 0 = ±(

M M , ). 1 − a1 − a2 1 − a1 − a2

(3.6)

Referring to the system (3.2), one can deﬁne the reversed-time dynamics for k ≤ 0 as follows: x∗ (k − 1) = A−1 x∗ (k) − A−1 Bu(k)

36

3 Estimation of the Null Controllable Region: Discrete-Time Plants

The solution of the above system, with the control input set to the saturation level u(k) ≡ ±M, is given by the formula |k|

x∗ (k) = A−|k| x0 ∓ ∑ A−i BM.

(3.7)

i=1

Using the above formula, it is easy to verify that, since A−1 is a Schur matrix, for any x0 ∈ Rn one has lim x∗ (k) = ∓[(I − A−1 )−1 − I]BM = ±(I − A)−1 BM.

k→−∞

In particular, for x± 0 given by (3.6), the following property holds: − lim φ (k, x+ 0 , −M) = x0

k→−∞

+ lim φ (k, x− 0 , M) = x0 .

k→−∞

Remark 3.1. It is worth to note that the above property is still valid for systems having a pair of complex unstable eigenvalues, even if no invariant parallelogram exists in this case. the discrete sets of points Let us denote by Γ±M − Γ±M = {x ∈ R2 : x = φ (k, x± 0 , ∓M), k ∈ Z }.

(3.8)

Theorem 3.1. Suppose that the eigenvalues of the state matrix A are complex numbers with |λ1 | > 1 and |λ2 | > 1. Then we have BM = Co(ΓM ∪ Γ−M ).

Proof. The proof well be given for the general case of n-dimensional systems, see Theorem 3.3.

3.1.1 Systems with Positive Real Eigenvalues The case of systems having two positive eigenvalues can be treated separately since the description of the null controllable region is very simple. In order to simplify the notation, we set

φ (−|k|, x± 0 , ∓M) := φ± (k); together with the sets (3.8), we consider their interpolations

Γ ±M :=

∞

{x ∈ R2 : x = θ φ± ( j − 1) + (1 − θ )φ±( j), θ ∈ [0, 1]}

j=1

3.1 The SISO Planar Case

37

Let us denote by φ±,1 (k), φ±,2 (k) the components of φ (−|k|, x± 0 , ∓M):

φ (−|k|, x± 0 , ∓M) = (φ±,1 (k), φ±,2 (k)). It can be easily veriﬁed that

φ±,2 (k) = φ±,1 (k − 1)

(3.9)

and

φ±,1 (k) = ±

h k−1−h −M λ1k λ2k + 2M ∑ki=0 λ1i λ2k−i − 2M ∑k−1 h=0 λ1 λ2 , k k (λ1 − 1)(λ2 − 1)λ1 λ2

(3.10)

where we have used the identities a1 = −λ1 λ2 , a2 = λ1 + λ2 . The next theorem gives a characterization of the null controllable region for planar systems with two real positive unstable eigenvalues. Theorem 3.2. Suppose that the eigenvalues of the state matrix A are real numbers with λ1 , λ2 > 1. Then we have ∂ BM = Γ M ∪ Γ −M .

Proof. It is easy to see that Γ M ∪ Γ −M is the boundary of a convex domain. This can be proved observing that the incremental ratio

β+ (k) :=

φ+,2 (k) − φ+,2 (k − 1) φ+,1 (k) − φ+,1 (k − 1)

is nonnegative and nondecreasing for any k ≥ 1. Using the formulas (3.9), (3.10) one can computes β (k) explicitly as follows

β+ (k) =

λ1k λ2 − λ2k λ1 ≥ 0; λ1k − λ2k

moreover, since λ1 , λ2 > 1 by assumption, we have d β+ (k) λ1k λ2k (λ1 − λ2)(log λ1 − log λ2 ) = ≥ 0. dk (λ1k − λ2k )2 By analogous computations, we obtain that the incremental ratio

β− (k) :=

φ−,2 (k) − φ−,2 (k − 1) φ−,1 (k) − φ−,1 (k − 1)

is nonpositive and nonincreasing. The above analysis shows that the domain enclosed by the polygonal curves Γ M , Γ −M is convex and let us denote this set by D, / D. Since the i.e. ∂ D = Γ M ∪ Γ −M . Let be x = (x1 , x2 ) ∈ R2 with x1 ≥ x2 and x ∈

38

3 Estimation of the Null Controllable Region: Discrete-Time Plants

lower part of the boundary of D, given by Γ M , corresponds to the trajectory of the system driven by the extremal control u ≡ M, we can conclude that

φ (k, x, u) ∈ / D ∀k ∈ N, ∀u ≤ M; in the same way, if x = (x1 , x2 ) ∈ / D with x1 ≤ x2 , we have

φ (k, x, u) ∈ / D ∀k ∈ N, ∀u ≥ −M, this meaning that

BM ⊆ D.

one can observe that the inner part of D in To prove the converse inclusion D ⊆ BM covered by a family of omothetic polygonal curves Γ M∗ , Γ −M∗ with M ∗ ∈ (0, M). Now if x ∈ D, there exists M1 < M such that x ∈ Γ M1 ∪ Γ −M1 ; ﬁxing u(0) = M ∗ with M1 < M ∗ < M one has

φ (1, x, M ∗ ) ∈ Γ M2 ∪ Γ −M2 , M2 < M1 . Iterating the procedure, we can deﬁne a decreasing sequence Mk with limk→∞ Mk = 0 and an admissible control u(k) ∈ WM such that

φ (k, x, u(k)) ∈ Γ Mk+1 ∪ Γ −Mk+1 and as a consequence lim ||φ (k, x, u(k))|| = 0.

k→∞

3.2 The SISO n-Dimensional Case Let us consider a general controllable single-input discrete-time linear system described by x(k + 1) = Ax(k) + Bu(k) (3.11) x(0) = x0 with x = (x1 , ..., xn ) ∈ Rn . In view of the controllability assumption, without loss of generality, the pair (A, B) is given in canonical form. The elements a1 , ..., an on the last row of the matrix A are the coefﬁcients of the characteristic polynomial, i.e. pA (λ ) = ∑ni=1 ai λ i−1 − λ n. For an arbitrary control input u ∈ ∞ (Z) and a ﬁxed initial condition x0 , the solution of the system will be denoted by φ (k, x0 , u). As in the planar case, the input v(k) is assumed to be preceded by a saturating device: u(k) = satM (v(k)),

M > 0.

3.2 The SISO n-Dimensional Case

39

Let us denote by λ1 , ..., λn ∈ R the eigenvalues (not necessarily distinct) of the matrix A. Suppose the system is antistable, that is |λi | > 1 for any i ∈ {1, .., n}. Consider the family of surfaces n−1

ψ±M (k) = x1 (k) + ∑ mi xi+1 (k) + q±M = 0, mi , q±M ∈ R i=1

and look for all the possible choices of coefﬁcients mi in order to have

ψ±M (k) = 0 ⇒ ψ±M (k + 1) = 0 for u ≡ ±M.

(3.12)

The forward step ψ±M (k + 1) is given by n−2

ψ±M (k + 1) = x2 (k) + ∑ mi xi+2 (k)+ i=1 n

+mn−1 ±M + ∑ ai xi (k) + q±M . i=1

Using the equality ψ±M (k) = 0 and imposing ψ±M (k + 1) = 0, the following set of algebraic equations is obtained ⎧ ±Mmn−1 ⎪ q±M = ; ⎪ ⎪ ⎪ a 1 mn−1 − 1 ⎨ 1 + mn−1(−m1 a1 + a2) = 0; ⎪ ⎪ ⎪ ⎪ ⎩ m j + mn−1 (−a1 m j+1 + a j+2) = 0,

j = 1, ..., n − 2.

Thanks to a recursion property, the system can be rewritten as ⎧ mn−2 = a1 m2n−1 − anmn−1 , ⎪ ⎪ ⎪m ⎪ = a21 m3n−1 − a1an m2n−1 − an−1mn−1 , ⎪ ⎨ n−3 .. . ⎪ ⎪ j n−2 n−4 n−4− j ⎪ ⎪ m2 = an−3 an− j mn−3− ⎪ 1 mn−1 − ∑ j=0 a1 n−1 , ⎩ m1 = (1 + mn−1a2 )/mn−1 a1 , where mn−1 satisﬁes the equation p∗A (mn−1 ) = 0, with n−2

n j+1 p∗A (λ ) = −an−1 = 0. 1 λ + 1 + ∑ a1 a j+2 λ j=0

j

(3.13)

40

3 Estimation of the Null Controllable Region: Discrete-Time Plants

Proposition 3.2. Let ν be a solution of the characteristic equation pA (λ ) = λ n − ∑nj=1 a j λ j−1 = 0. Then ν /a1 satisﬁes p∗A (ν /a1 ) = 0. Proof. The statement follows observing that n−2 νn ν j+1 + 1 + ∑ a1j a j+2 j+1 = a1 a1 j=0 n 1 ν n − ∑ a j ν j−1 = 0. =− a1 j=1

p∗A (ν /a1 ) = −

As a consequence, the number of real solutions of Equation (3.13) is exactly the number of real eigenvalues the state matrix A; assuming that λ j ∈ R for j = 1, ..., ν ≤ n, then the real solutions of (3.13) are given by

λˆ j = (−1)n+1

λj , n Πi=1 λi

j = 1, ..., ν .

The above solutions lead to a set of n pairs of parallel hyperplanes; note that for mn−1 = λˆ j with j = 1, ..., ν , the equation for q±M is well-posed. Taking all possible intersections, a convex polytopic set containing the origin as interior point is found. Denote this set as FM ; let us point out that, if ν < n, then FM is an unbounded set. Due to the invariance property (3.12), if a trajectory starts from a point outside the set FM , there is no admissible control that can drive it inside. Proposition 3.3. Suppose that the system eigenvalues verify λi ∈ R and |λi | > 1 for any i = 1, ..., n. Then ⊆ FM . BM The set ∂ FM intersects the main bisector γ (given by x1 = x2 = · · · = xn ) in a pair of symmetric points given by x± 0 =±

n M ei = ±M(I − A)−1 B, ∑ n 1 − ∑i=1 ai i=1

(3.14)

where the sum represents a linear combination of the euclidean basis vectors ei . The solution of the reversed-time system (k ≤ 0) x∗ (k − 1) = A−1 x∗ (k) − A−1 Bu(k) with the control input set to the saturation level u(k) ≡ ±M veriﬁes the identity |k|

x∗ (k) = A−|k| x0 ∓ ∑ A−i BM. i=1

(3.15)

3.2 The SISO n-Dimensional Case

41

Using the above formula, it is easy to verify that, since A−1 is a Schur matrix, for any x0 ∈ Rn one has lim x∗ (k) = ∓[(I − A−1 )−1 − I]BM = ±(I − A)−1 BM.

k→−∞

In particular, for x± 0 given by (3.14), the following property holds: − lim φ (k, x+ 0 , −M) = x0

k→−∞

+ lim φ (k, x− 0 , M) = x0 .

k→−∞

Remark 3.2. The above property is valid for any antistable discrete-time system; in particular the points x± 0 deﬁned in (3.14) turn out to be the end-points of any reversed-time trajectory corresponding to u ≡ ±M also for systems having only complex eigenvalues. The points x± 0 play a central role in the description of the null controllable region BM . A uniﬁed approach for systems having both real and complex eigenvalues is presented. ± Proposition 3.4. Let±x0 given by (3.14) and consider the countable set of points {γ (k)}k≤0 = φ (k, x0 , ±M) k≤0 . For s ∈ N with s < n there is no s-dimensional afﬁne subspace containing γ .

Proof. Let us deﬁne the vectors wi = γ (−i) − x± 0 and consider the matrix H(i1 , ..., in ) ∈ Rn×n , for in > · · · > i2 > i1 ≥ 1, given by H(i1 , ..., in ) = [wi1

wi2

···

win ] .

The statement of the proposition is equivalent to the existence of a multiindex (i∗1 , · · · , i∗n ) such that det H(i∗1 , · · · , i∗n ) = 0. Let us recall that the inverse matrix A−1 has the following structure ⎛ ⎜ A−1 = ⎝

−

a 1 ⎞ a1 a1 ⎟ ⎠

In−1

with a = [a2

a3

···

an ];

0

using the explicit expression of A−1 in the recursion formula (3.15), the quantities wi , i = 1, ..., n, can be easily obtained:

42

3 Estimation of the Null Controllable Region: Discrete-Time Plants

w1 = w2 =

+ + "

w3 =

2M a1

0 0

2M(a1 −a2 ) a21

0

···

2M a1

0

, 0 0

2M(a21 +a22 −a1 (a2 +a3 )) a31

,

···

0

2M(a1 −a2 ) a21

# 2M a1

···

0

0

.. . " wn−1 =

2Mqn−1 (a1 ,...,an−1 ) an−1 1

2Mqn−2 (a1 ,...,an−2 ) an−2 1

··· wn = [

2Mqn (a1 ,...,an ) an1

···

2M(a1 −a2 ) a21

2Mqn−1 (a1 ,...,an−1 ) an−1 1

2M(a21 +a22 −a1 (a2 +a3 )) a31

··· ,

2M a1

0

··· 2M(a1 −a2 ) a21

# 2M a1

where the function qi (·) is a suitable homogeneous polynomial of degree i − 1. The conclusion follows observing that 2M n detH(1, ..., n) = = 0. a1 The following result can be proved: Proposition 3.5. Let x0 ∈ Rn and x−1 = φ (−1, x0 , u0 ) for some u0 ∈ [−M, M]. There exists θ ∈ (0, 1) such that x−1 = θ φ (−1, x0 , M) + (1 − θ )φ (−1, x0, −M). Proof. The proof is trivial: the coefﬁcient θ is given by

θ=

u0 + M . 2M

Let us introduce the set-valued function Φ±M deﬁned on Rn as follows:

Φ±M (x) = {y ∈ Rn : ∃k ∈ (−∞, 0] : y = φ (k, x, ±M)} . + Proposition 3.6. Given the set Γ = Φ−M (x− 0 ) ∪ ΦM (x0 ) together with its convex hull S0 = Co(Γ ), there exists δ > 0 with Bδ (0) ⊂ S0 ; in particular |S0 | > 0.

3.2 The SISO n-Dimensional Case

43

Proof. The interior part of Co(Γ ) is nontrivial since the afﬁne hull of Γ (i.e. the smallest afﬁne space containing the set Γ ) is the whole space Rn by Proposition 3.4; this proves that |S0 | > 0. Let x be an interior point of Co(Γ ); by symmetry it follows that −x is still an interior point as well as any point xθ obtained as convex combination (see Appendix A.1) xθ = θ x − (1 − θ )x, θ ∈ [0, 1]. 1 The result follows observing that the origin veriﬁes (3.16) with θ = . 2 Theorem 3.3. Deﬁne a monotone recursive sequence of sets {S j } j∈N S j+1 = Co(Φ−M (∂ S j ) ∪ ΦM (∂ S j )),

(3.16)

(3.17)

where S0 is given by S0 = Co(Γ ). The null controllable region of the system is given by = lim S j = S j =: S∞ BM j→∞

j∈N

Proof. First prove that BM ⊆ S∞ . Suppose x0 ∈ BM and x0 ∈ / S0 . Since x0 ∈ BM , there exists u ∈ UM such that lim φ (k, x0 , u) = 0.

k→∞

/ S0 and xk := φ (k, x0 , u) ∈ S0 In particular there exists k such that xk := φ (k, x0 , u) ∈ for any k ≥ k + 1. Now xk = φ (−1, xk+1 , u). By Proposition 3.5, there exists 0 ≤ θ ≤ 1 such that xk = θ φ (−1, xk+1 , M) + (1 − θ )φ (−1, xk+1 , −M).

(3.18)

Since xk+1 ∈ S0 and by construction φ (−1, xk+1 , M) ∈ Co(Φ−M (∂ S0 ) ∪ ΦM (∂ S0 )), identity (3.18) implies that xk ∈ S1 . Iterating the above procedure it can be easily proved that x0 ∈ Sk ⊂ S∞ . holds too. Saying that x0 ∈ S∞ Let us show that the converse inclusion S∞ ⊆ BM means that exists k such that x0 ∈ Sk . Moreover, by deﬁnition, there exists a control function in WM∗ having at most k switches such that the solution driven by it enters . Let x0 ∈ S0 . Since the soluS0 in ﬁnite time. It remains only to prove that S0 ⊆ BM tions are continuously dependant on the system parameters, the interior part of S0 is completely covered by a family of homothetic closed surfaces which are generated

44

3 Estimation of the Null Controllable Region: Discrete-Time Plants

starting from the points x0,M∗ = ±(I − A)−1 BM ∗ and switching the control between the values M ∗ as M ∗ varies in [0, M]. Denote these sets as ZM∗ . In particular it holds ZM∗ = ∂ S0,M∗ , where

+ ∗ S0,M∗ = Co(Φ−M∗ (x− 0,M∗ ) ∪ ΦM (x0,M∗ ))

and S0 =

ZM∗ ,

ZM1 ∩ ZM2 = 0/

0≤M∗ ≤M

for M1 = M2 .

Let x∗0 ∈ ZM∗ ; by the Carath´eodory theorem (see Appendix A.1), there exist θ j > 0, 1 < j < N + 1, with ∑N+1 j=1 θ j = 1 and k j ∈ (−∞, 0] such that x∗0 +

j∗

=

∑ θ j φ (k j , x+0,M∗ , M∗ ) +

j=1 N+1

∑

j= j ∗ +1

(3.19) ∗ θ j φ (k j , x− 0,M∗ , −M ),

with j∗ ∈ [0, N + 1]. Note that the elements of S∞ obtained as limk→−∞ φ (k, x, M), for some x, belong to ∂ S∞ . It follows from (3.19) that x∗ (t) = +

j∗

∑ θ j φ (k + k j , x+0,M∗ , M∗ ) +

j=1 N+1

∑

j= j ∗ +1

∗ θ j φ (k + k j , x− 0,M∗ , −M )

satisﬁes the system equations for initial datum x∗0 and control input u∗ (k) = M ∗ ∗ ∗ (∑ jj=1 θ j − ∑N+1 j= j ∗ +1 θ j ). This means that any point in ZM lies on a trajectory associated to a suitable admissible control input. As a consequence, from the invariance + ∗ ∗ of the closed curve Φ−M∗ (x− 0,M∗ ) ∪ ΦM (x0,M∗ ), the invariance of the whole set ZM can be deduced. Now, if x0 ∈ S0 , there exist M1∗ < M such that x0 lies on the invariant surface ZM1∗ under a suitable control input u∗ with |u∗ | ≤ M1∗ < M. In order to stabilize the system, it sufﬁces to make the trajectory jump to a lower level ZM2∗ with M2∗ < M1∗ increasing the control norm. This concludes the proof.

3.3 The MIMO Case As for continuous-time systems, the description of the null controllable for discretetime MIMO systems can be obtained with a direct method as well as by a method based on lower-order single input subsystems. Since the technical results are

3.3 The MIMO Case

45

analogous to those presented in Section 2.4, some details have been omitted here. Let us consider the following multi input linear system ⎧ ⎨ x(k + 1) = Ax(k) + Bu(k) (3.20) ⎩ x(0) = x0 with m < n, A ∈ Rn×n , B ∈ Rn×m , x ∈ Rn and u ∈ Rm ; the plant is assumed to be controllable, i.e. (3.21) rank[B AB A2 B · · · An−1 B] = n and antistable, that is |λi | > 1 for any eigenvalue λi , i = 1, ..., n, of the matrix A. We will use the following notation: ⎡ ⎤ u1 ⎢ u2 ⎥ ⎢ ⎥ B = [B1 B2 · · · Bm ], u = ⎢ . ⎥ , ⎣ .. ⎦ um where Bi ∈ Rn and ui ∈ R for any i = 1, ..., m.

3.3.1 Method of Lower-Order Single Input Subsystems According to (3.20), we can deﬁne a family of single input systems Σi , i = 1, .., m, as follows ⎧ ⎨ x(k + 1) = Ax(k) + Bi ui (k) Σi = ⎩ x(0) = x0 Let ni ∈ N be the dimension of the controllability subspace for the system Σi , ni := rank[Bi ABi A2 Bi · · · An−1 Bi ] ≤ n. The system Σi can be transformed into a controllable/uncontrollable subsystems decomposition by a linear coordinates transformation and one can easily compute (i), the maximal stability region DM of the single input system Σi (see Section 2.3 (i), for details). By construction, the set DM is a convex region contained in a linear n subspace Vi ⊂ R with dim(Vi ) = ni . for the multi input system (3.20) is Theorem 3.4. The null controllable region BM given by m

(i),

= ∑ DM BM i=1

(i), = x ∈ Rn : x = x1 + x2 + · · · + xm , xi ∈ DM ∀ i = 1, ..., m .

46

3 Estimation of the Null Controllable Region: Discrete-Time Plants (i),

Proof. The set ∑m i=1 DM , as it is deﬁned as a ﬁnite sum of convex sets, is still (i), (i), m convex (see for instance [33]). Let x ∈ ∑m i=1 DM , with x = ∑i=1 xi . Since xi ∈ DM , by construction, there exists ui ∈ WM such that, deﬁning φi (k, x0 , u(k)) the dynamics of the system Σi driven by the control u, we have lim ||φi (k, xi , ui (k)|| = 0.

k→∞

Now, since

m

φ (k, x, u(k) = ∑ φi (k, xi , ui (k)), i=1

we can deduce the existence of u ∈ WM such that lim ||φi (k, x, u(k))|| = 0;

k→∞

(i),

in other words we have ∑m i=1 DM ⊆ BM . Let us show that the converse inclusion (i), BM ⊆ ∑m i=1 DM holds too. Suppose that x ∈ BM ; by deﬁnition, there exists u = (u1 , ..., um ) with |ui | ≤ M such that

lim ||φi (k, x, u(k))|| = 0.

(3.22)

k→∞

Using the explicit formula for the solution we have k−1

m

j=0

i=1

φi (k, x, u(k)) = Ak x + ∑ Ak−1− j ∑ Bi ui ( j). From (3.22) we obtain lim A

k−1

x+ ∑ A

k

k→∞

−1− j

j=0

m

∑ Bi ui ( j)

=0

i=1

and necessarily one has k−1

m

j=0

i=1

∑ A−1− j ∑ Bi ui ( j) = −x. k→∞ lim

m ∞ −1− j < ∞, the folNow, since || ∑m i=1 Bi ui ( j)|| ≤ M, ∑i=1 ||Bi || < ∞ and ∑ j=0 A lowing limits exist ∀i = 1, ..., m and they are ﬁnite: k−1

lim

∑ A−1− j Bi ui ( j) = −xi.

k→∞ j=0

3.3 The MIMO Case

47

As a consequence one has lim A

k−1

xi + ∑ A

k

k→∞

−1− j

Bi ui ( j)

= 0 ∀i = 1, ..., m,

j=0

(i),

(i),

m that is xi ∈ DM ; now, since by construction x = ∑m i=1 xi , we have BM ⊆ ∑i=1 DM and the statement is proved.

3.3.2 Direct Method The direct method for the description of the multi input null controllable region follows the same steps presented in the single input case. In particular, the null controllable region can be computed with an iterative procedure combining reversed-time evolution with extremal controls and set convexiﬁcation. Let us denote by WM∗∗ the set of extremal controls; to this purpose we recall that a control input u = (u1 , ..., um ) ∈ WM is extremal if ui ≡ ±M ∀i = 1, ..., m. Using the explicit formula for the solution φ (k, x, u(k)) one can verify that ∀u ∈ WM∗∗ lim φ (k, x, u∗ ) = (I − A)−1 Bu∗ ∀x ∈ Rn .

k→−∞

Fixing u∗1 ≡ M, there are 2m−1 possible combinations for the other components of an extremal control u ∈ WM∗∗ ; in this way, we can deﬁne a set of 2m−1 points m

xM,u = (I − A)−1 B1 + ∑ ±Bi M i=2

such that

∗∗ lim φ (k, ∓xM,u , ±u) = ±xM,u ∀u ∈ WM,+

k→−∞

∗∗ := {u ∈ W ∗∗ : u = M}. For u ∈ W ∗∗ we deﬁne the where it has been set WM,+ 1 M M,+ family of polygonal curves having vertices

γM,u (k) := φ (k, −xM,u , u), k ∈ Z \ N and the corresponding supports

ΓM,u := {x ∈ Rn : x = φ (k, −xM,u , u), k ∈ Z \ N}.

(3.23)

∗∗ such that, for Proposition 3.1. There exists at least one extremal control u¯ ∈ WM,+ s < n there is no s-dimensional afﬁne subspace containing γM,u¯ (k). ∗∗ the multi input system (A, B) can be regarded as the Proof. For any ﬁxed u ∈ WM,+ single input system (A, Bu ) driven by u ≡ M with

48

3 Estimation of the Null Controllable Region: Discrete-Time Plants m

Bu = B1 + ∑ ±Bi ;

(3.24)

i=1

the signs in the above formula are ﬁxed according to the signs of the components ui , i = 2, ...m. Since the plant (A, B) is assumed to be controllable, there exists ∗∗ such that the single input system (A, B ), with B deﬁned necessarily u¯ ∈ UM,+ u¯ u¯ by (3.24), is controllable as well. Now the curve γM,u¯ (t) can be regarded as the reversed-time evolution of the single input system (A, Bu¯ ) driven by the control u ≡ M and the conclusion follows applying the results given in Proposition 3.4 to such polygonal curve. ∗∗ , deﬁned on Rn as follows Introducing the set-valued function Φ±u , u ∈ WM,+

Φu (x) = {y ∈ Rn : ∃k ∈ Z \ N : y = φ (k, x, u)} , one can design a monotone recursive sequence of sets {S j } j∈N ⎛ ⎞

S j+1 = Co ⎝

∗∗ uWM,+

where S0 is given by S0 = Co

(Φ−u (∂ S j ) ∪ Φu (∂ S j ))⎠ ,

(

∗∗ −ΓM,u ∪ ΓM,u uWM,+

(3.25)

with ΓM,u given by (3.23).

Theorem 3.5. The null controllable region for the multi input system (3.20) is BM = lim S j = j→∞

S j =: S∞

j∈N

The proof is omitted. It can be derived using Proposition 3.1 and applying the same arguments used to prove Theorem 3.3.

3.4 Examples Example 3.1. Let us consider the antistable system described by the matrices ⎛ A=⎝

0

1

−3.15 3.6

⎞ ⎠,

⎛ ⎞ 0 B = ⎝ ⎠. 1

with M = 1. The open-loop eigenvalues are

λ1 = 1.5

λ2 = 2.1

−1 and the initial points x± 0 = ±M(I − A) B are given by

x± 0 = ±(1.81, 1.81).

3.4 Examples

49

The next Fig.3.1 shows the null controllable region for the above system.

1.5

1.0

0.5

1.5

1.0

0.5

0.5

1.0

1.5

0.5

1.0

1.5

Fig. 3.1. Boundary of the null controllable region for real positive eigenvalues λ1 = 1.5, λ2 = 2.1 and saturation level M = 1.

Example 3.2. Let us consider the system expressed by the following parameters ⎛ A=⎝

0

1

⎞ ⎠,

6.12 1.9

⎛ ⎞ 0 B = ⎝ ⎠. 1

The system eigenvalues are

λ1 = −1.7

λ2 = 3.6

−1 and the initial points x± 0 = ±M(I − A) B are given by

x± 0 = ∓(1.42, 1.42). A picture of the null controllable region for the system is shown in the Fig. 3.2 below.

50

3 Estimation of the Null Controllable Region: Discrete-Time Plants

0.15

0.10

0.05

0.15

0.10

0.05

0.05

0.10

0.15

0.05

0.10

0.15

Fig. 3.2. Boundary of the null controllable region for real eigenvalues λ1 = −1.7, λ2 = 3.6 and saturation level M = 1. Dashed lines represent extremal trajectories corresponding to control inputs u ≡ ±M.

Example 3.3. Let us consider the following system matrices ⎛ A=⎝

0

1

−2.02 1.8

⎞

⎛ ⎞ 0 B = ⎝ ⎠. 1

⎠,

The open-loop eigenvalues are a pair of complex conjugate numbers:

λ1 = 0.9 + 1.1j

λ2 = 0.9 − 1.1j;

−1 the initial points x± 0 = ±M(I − A) B are given by

x± 0 = ±(0.82, 0.82). A picture of the null controllable region for the system is showed in Fig. 3.3. Example 3.4. In the last example the system matrices are set as ⎛ A=⎝

0

1

−1.94 −2.6

⎞ ⎠,

⎛ ⎞ 0 B = ⎝ ⎠. 1

The eigenvalues of A are a pair of conjugate complex numbers given by

λ1 = −1.3 + 0.5j

λ2 = −1.3 − 0.5j

3.4 Examples

51

1.0

0.5

1.0

0.5

0.5

1.0

0.5

1.0

Fig. 3.3. Boundary of the null controllable region for complex eigenvalues λ1 = 0.9 + 1.1j, λ2 = 0.9 − 1.1j and saturation level M = 1. Dashed lines represent extremal trajectories corresponding to control inputs u ≡ ±M.

−1 and the initial points x± 0 = ±M(I − A) B are given by

x± 0 = ±(0.18, 0.18). Figure 3.4 shows a picture of the null controllable region for the system. Remark 3.3. In the ﬁrst example (systems with real positive eigenvalues) the null controllable turns out to be equal to the region bounded by ΓM ∪ Γ−M ; on the other hand, such region is in general not convex for systems with real negative or complex eigenvalues and BM is equal to the convex hull Co(ΓM ∪ Γ−M ) as showed in the other examples.

52

3 Estimation of the Null Controllable Region: Discrete-Time Plants

0.5

0.5

0.5

0.5

Fig. 3.4. Boundary of the null controllable region for complex eigenvalues λ1 = −1.3 + 0.5j, λ2 = −1.3 − 0.5j and saturation level M = 1. Dashed lines represent extremal trajectories corresponding to control inputs u ≡ ±M.

Part II: Design Issues

Chapter 4

Control Design Issues: Continuous-Time Plants

Earlier in the book, attention has been focussed on the analysis of the basin of attraction of a linear system subject to actuator saturation, and mostly on its geometrical determination and numerical computation. It is now time to drive our attention to the stabilization of a linear plant on its null controllable region with saturating actuators. As a ﬁrst look to the problem, a planar unstable plant without uncertainties is considered (the extension to linear plants of arbitrary ﬁnite dimensions is discussed next). A design technique is proposed for ﬁnding a linear state feedback controller with the property of having nonincreasing norm along the closed-loop system trajectories [36]. Interestingly, the set of initial conditions satisfying the saturation constraint turns out to be invariant for the closed-loop system evolution (results on invariant sets in control theory can be found in [37] and [38]). In particular, the region of attraction associated to such controller is an unbounded strip and it can be straightforwardly characterized. Furthermore, it is shown show how, once the saturation level is ﬁxed to M > 0, it is possible to split the controller into a ﬁnite number of saturating components. The number of components can be a priori determined for any ﬁxed compact set of initial data. By a practical point of view, this scheme can be regarded as the model of a plant equipped with a controller followed by a ﬁnite number of saturating devices with the same structure. In the presence of disturbances affecting the plant, we are primarily concerned on boundedness of trajectories in face of the perturbing term, and mostly on achieving disturbance rejection, i.e. on designing feedback laws having the ability to completely reject at least a given class of disturbances. In the wide literature addressing the problem of disturbance rejection for linear systems subject to actuator saturation, an interesting research line considers disturbances that are magnitude bounded. This line complements the research thrust addressing L p disturbances [20] [21]. In the former framework, [22] proved that semiglobal practical stabilization for a linear system subject to actuator saturation and input additive disturbances can be achieved as long as the open loop system is not exponentially unstable. For the same class of systems, Lin [7] constructed nonlinear feedback laws that achieve global practical stabilization. Very recently, it has been proved in [23] that a 2-dimensional linear systems subject to actuator M.L. Corradini et al.: Control Systems with Saturating Inputs, LNCIS 424, pp. 55–94. c Springer-Verlag London Limited 2012 springerlink.com

56

4 Control Design Issues: Continuous-Time Plants

saturation and bounded input additive disturbances can be globally practically stabilized by linear state feedback. Variable Structure Control techniques have been rarely used to address the problem of robustly controlling plants with saturating actuators. It is worth mentioning that higher order sliding mode techniques have been effectively employed to achieve plant stabilization in the presence of unmodeled actuator dynamics [39]. In this chapter, a design technique will be described ensuring the robust asymptotical stabilization of a Multi-Input linear plant [40] [41]. The time-varying state feedback control law is derived imposing the achievement of a sliding motion onto a suitable time-varying sliding surface [42] [43]. It can be proved that a constructive procedure exists for designing the surface as to guarantee the asymptotical stabilization of the plant in the presence of bounded matched uncertainties, under the usual assumption of the saturation threshold being larger than the bound on uncertainties.

4.1 Invariant Strips and Linear Feedback Laws 4.1.1 Planar Systems Consider a 2-dimensional controllable SISO linear system having, without loss of generality, the following structure ⎧ ⎨ x˙ = Ax + bu (4.1) ⎩ x(0) = x0

where A=

0 1 a1 a2

,

0 b= , 1

x(t) = (x1 (t), x2 (t)) ∈ C0 ([0, +∞), R2 ) is the state vector and u(t) ∈ C0 ([0, +∞), R) is the input variable. We will suppose a1 > 0; this implies that the eigenvalues of A are real numbers with opposite signs. Input saturation is assumed to be present in the system; in particular any input u is subject to the constraint u = satM (v), M > 0,

(4.2)

where satM (s) is given by (1.3) and u(t) ∈ C0 ([0, +∞), R). Lemma 4.1. Given the system (4.1) with a1 > 0 there exists a linear feedback u = K, x such that the closed-loop system driven by A + bKT is asymptotically stable in the origin and moreover the following condition holds: • if K, x ≥ 0 then K, x˙ ≤ 0, • if K, x ≤ 0 then K, x˙ ≥ 0.

4.1 Invariant Strips and Linear Feedback Laws

57

Proof. Given a saturation level M > 0, we consider the level set K, x = M; our aim is to construct K such that K, Ax + bu ≤ 0 over the previous set, that is K, Ax + Mb ≤ 0 ∀x such that K, x = M.

(4.3)

Let us denote by k1 , k2 the components of K. In order to achieve condition (4.3), the two afﬁne subspaces {x ∈ R2 : K, x = M} and {x ∈ R2 : K, Ax + Mb = 0} must have no intersection, in particular they must be parallel, and this can hold if and only if K is an eigenvector of the matrix AT . Moreover the corresponding eigenvalue λ must be real with λ ≤ −k2 . Parallelism between the spaces {x ∈ R2 : K, x = −M} and {x ∈ R2 : K, Ax − Mb = 0} gives the same condition on K. The eigenvalue λ is characterized by λ k1 = a 1 k2 (4.4)

λ k2 = k1 + a 2 k2 .

(4.5)

From the ﬁrst equation we get

λ=

a 1 k2 k1

and we can transform the last one in a second degree equation for the ratio k2 /k1 a1

k22 k2 − a2 − 1 = 0. k1 k12

(4.6)

Since a1 > 0, in order to obtain stabilization of the system we must require k1 < 0 (recall that a1 + k1 represents the opposite of the product between the eigenvalues of the closed-loop system). Moreover we will assume k2 < 0 in both cases a2 > 0 and a2 < 0. The positive solution of (4.6) is given by + a22 + 4a1 a 2 k2 = . (4.7) k1 2a1 4.1.1.1 CASE tr(A) > 0 If a2 > 0 we search for K given by k1 = −a1 − η ,

k2 = −a2 − ε

with η , ε > 0.

Using these expressions of the coefﬁcients of K in (4.7), we obtain a1 a2 − a1 a22 + 4a1 + 2a1ε η= a2 + a22 + 4a1

(4.8)

(4.9)

58

4 Control Design Issues: Continuous-Time Plants

which is positive if we take

ε>

a22 + 4a1 − a2 2

(> 0).

(4.10)

4.1.1.2 CASE tr(A) < 0 In the complementary case a2 < 0 we will choose k1 as before and k2 = −ε < 0. a22 + 4a1 + 2a1ε a2 + a22 + 4a1

We ﬁnd

η=

(4.11)

−a1 a2 − a1

(4.12)

and then ε must satisfy

ε>

a22 + 4a1 + a2 2

(> 0).

(4.13)

This concludes the proof of lemma.

Remark 4.1. A vector K with the desired properties can be found also in the border case a1 > 0, a2 = 0; it is enough to choose K = (−a1 − η , −ε ) with

ε , η > 0 and

ε2 1 = . 2 (−a1 − η ) a1

Theorem 4.1. Given the system (4.1) with a1 > 0 there exists a vector K ∈ R2 such that for any M > 0 1. ΩM = x ∈ R2 : |K, x| ≤ M is an invariant set for the solution φ (t, x0 , K); 2. if x0 ∈ ΩM we have limt→∞ φ (t, x0 , K) = 0 under the saturation constraint (2.2). The expression of the coefﬁcients of K is given by (4.8)-(4.9)-(4.10) and (4.11)(4.12)-(4.13) for a2 > 0 and a2 < 0, respectively. See Remark 4.1 for the case a2 = 0. Proof. We give the proof for a2 > 0. We set K = K(ε ) = (−a1 − η (ε ), −a2 − ε ) with ε satysfying (4.10). datum x0 ∈ ΩM , since K, x˙ ≤ 0 on the Taking the initial boundary component x ∈ R2 : K, x = M , the control cannot exceed the level M; similarly K, x˙ ≥ 0 on x ∈ R2 : K, x = −M and so ΩM is an invariant set. T

Moreover the closed-loop system A + bK has eigenvalues with negative real parts by construction, so that asymptotic stability is proved.

4.1 Invariant Strips and Linear Feedback Laws

59

4.1.1.3 Brief Discussion on Stable Systems A few words about the application of the latter method to stable systems can be here useful. If a1 , a2 < 0, following the procedure used for the unstable case one ﬁnds a a22 + 4a1 2 k2 = . (4.14) k1 2a1 In order to have wellposedness of the previous relation one should require a22 + 4a1 > 0; note that this condition forces the state matrix A to have real eigenvalues. Choose k1 = −η , k2 = −ε , that is

η= Set K = K(ε ) = −ε ( √2a21 a2

a2 +4a1

2ε a 1 > 0. a2 a22 + 4a1

, 1). Application of the linear feedback u = K(ε ), x

to (4.1) with a1, a2 < 0 improves systemstability. Moreover, if we take the initial condition x0 ∈ x ∈ R2 : |K(ε ), x| < M , the control variable satisﬁes |u| < M for any t > 0. If a1 = 0 and a2 < 0 the same result holds with K given by k1 = a2 ε , k2 = −ε < 0. In particular K turns out to belong to ker(A). It should be emphasized that in the stable case one has K( ε go to zero, one can make ε ) = O(ε ) therefore, letting the width of the strip x ∈ R2 : |K(ε ), x| < M arbitrarily large. In other words semiglobal asymptotic stabilization by bounded control is achieved.

4.1.2 A Simulation Example Consider the continuous-time system reported in [44], [45] setting the time-delay constant τ = 0. Up to this assumption the system matrices A and b are given by 1 1.5 0 −1 10 A= + B= (4.15) 0.3 −2 0 0 1 The pair (A, b) is controllable and its canonical form is 0 1 0 A= B= 2.15 −1 1

(4.16)

√ √ The eigenvalues of the system are λ1 = −0.5 − 2 0.6 and λ2 = −0.5 + 2 0.6 with λ2 > 0 and λ1 + λ2 < 0. According to Lemma 4.1, taking

60

4 Control Design Issues: Continuous-Time Plants

ε =2> we get

a22 + 4a1 − a2 2

≈ 1.05

√ K = (−1 − 4 0.6, −2).

Having ﬁxed the saturation level to M = 1, we have √simulate the closed-loop system evolution with initial state x0 = (5, −2.99 − 2 15) and control input u(t) = K, x(t). Note that x0 ≈ 11.8 and K, x0 ≈ 0.98, so that u(0) is very close to the saturation level M. A plotting of the evolution of the system variables during the time interval (0, 5) is shown in Fig. 4.1-4.3. The region of attraction associated to u(t) in the presence of saturation on the actuator is the unbounded strip

√ E = (x1 , x2 ) ∈ R2 : |x1 + 4 0.6x1 + 2x2| < 1 . The closed-loop state matrix, corresponding to the linear feedback without saturation, is given by 0 √ 1 T A + BK = . 1.15 − 4 0.6 −3

u 1.0

0.8

0.6

0.4

0.2

1

2

3

Fig. 4.1. Control variable u(t).

4

5

t

4.1 Invariant Strips and Linear Feedback Laws

61

x1

1.5

1.0

0.5

1

2

3

4

5

4

5

t

Fig. 4.2. State variable x1 (t).

x2 1

2

3

1

2

3

4

Fig. 4.3. State variable x2 (t).

t

62

4 Control Design Issues: Continuous-Time Plants

4.1.3 Multidimensional Systems Consider a n-dimensional (n ≥ 2) controllable SISO linear system having, without loss of generality, the following structure ⎧ ⎨ x˙ = Ax + Bu (4.17) ⎩ x(0) = x0 where ⎛ ⎞ ⎞ ⎛ 0 0 1 ··· 0 ⎜ .. ⎟ ⎜ .. .. .. .. ⎟ ⎟ ⎜ ⎟ ⎜ B = ⎜ . ⎟, A = ⎜ . . . . ⎟, ⎝0⎠ ⎝ 0 ··· 0 1 ⎠ a1 a2 · · · an 1 x(t) = (x1 (t), x2 (t), ..., xn (t)) ∈ C0 ([0, +∞), Rn ) is the state vector and v(t) ∈ C0 ([0, +∞), R) is the input variable. Let us denote with {λ1 , λ2 , ..., λn } the eigenvalues of A. We will suppose Re(λ j ) < 0 for all j ≤ n − 1, λn ∈ [0, +∞).

(4.18)

As a consequence, we have a1 ≥ 0 for any n ∈ N. Input saturation is assumed in the system; in particular the input u is subject to the constraint u = satM (v), M > 0

(4.19)

where satM (s) is given by (1.4) and u(t) ∈ C0 ([0, ∞), R). Lemma 4.2. Let be K( j) an eigenvector associate to the real eigenvalue λ j of the ( j) matrix AT , with A given by (4.17). For σ ∈ R we deﬁne the matrix Aσ = A + ( j) σ b(K( j) )T ; the eigenvalues of Aσ are ( j)

λ1 , ..., λ j−1 , λ j + σ kn , ..., λn , ( j)

where kn is the nth component of K( j) . Proof. The eigenvector K( j) satisﬁes the following system of equations ( j)

( j)

a 1 kn = λ j k1 , ( j)

( j)

( j)

ks−1 + askn = λ j ks

(4.20)

∀ 1 < s ≤ n;

(4.21)

( j)

we have denoted by ks the sth component of K( j) , 1 ≤ s ≤ n. With a recursive procedure from equations (4.21) we can derive an equivalent system given by ( j)

( j)

ks = kn

n−1

μ n−s − ∑ μ r−s j j ar+1 r=s

∀ 1 ≤ s < n.

(4.22)

4.1 Invariant Strips and Linear Feedback Laws

63

Consider the polynomial q(λ ) = ∏(λ − λr ) = λ n−1 + r = j

n−2

∑ br+1λ r . Recalling the

r=0

explicit expressions of the coefﬁcients ar and br in terms of the eigenvalues { μr }nr=1 , we see that n−1

μ n−s − ∑ μ r−s j j ar+1 = bs

∀ 1 ≤ s < n.

(4.23)

r=s

( j)

The characteristic polynomial of the matrix At

n−1

n−1 ( j) ( j) pσ (λ ) = λ n − ∑ ar+1 + σ kr+1 λ r = r=0

= λ − ∑ ar+1 λ n

is given by

r

( j) − σ kn

λ

r=0

n−1

n−2

+∑

μ n−s − j

s=1

n−1

∑

μ r−s j ar+1

λ

s−1

,

r=s

where we have used the equations (4.22); by formula (4.23) we get n ( j) ( j) ( j) pσ (λ ) = ∏ (λ − λr ) − σ kn q(λ ) = (λ − μ j ) − σ kn q(λ ). r=1

( j)

We obtain that equation pσ (λ ) = 0 is satisﬁed for λ = μr , with r = j, and for ( j) λ = μ j + σ kn . Lemma 4.3. Given the system (4.17) with the linear feedback u = K, x, the control norm |u| is non-increasing along the closed-loop system trajectories if and only if K is an eigenvector of AT with λ < −kn , where λ ∈ R is the corresponding eigenvalue and kn is the nth component of K. Proof. Non-increasing property of |u| can be written as • if K, x ≥ 0 then K, x˙ ≤ 0, • if K, x ≤ 0 then K, x˙ ≥ 0. Given M > 0, we consider the level set K, x = M; our aim is to construct K such that K, Ax + Bu ≤ 0 over the previous set, that is K, Ax + MB ≤ 0.

(4.24)

The two afﬁne subspaces K, x = M and K, Ax + MB = 0 must be parallel and this can hold if and only if K is an eigenvector of the matrix AT . Denoting with λ ∈ R the eigenvalue associate to K, for K, x = M we have K, Ax = M λ ,

K, MB = Mkn

We see that condition (4.24) holds if the eigenvalue λ satisﬁes λ ≤ −kn . Parallelism between the spaces K, x = −M and K, Ax − MB = 0 gives the same condition on K. This concludes the proof.

64

4 Control Design Issues: Continuous-Time Plants

Theorem 4.2. Given the system (4.17) satisfying condition (4.18), there exists a vector K ∈ Rn such that for any M > 0 1. ΩM = x ∈ Rn : |K, x| ≤ M is an invariant set for the solution φ (t, x0 , K); 2. if x0 ∈ ΩM we have limt→∞ φ (t, x0 , K) = 0 under the saturation constraint (4.19). Proof. Let K(n) be an eigenvector of AT , associated to the positive eigenvalue μn (n) (n) and suppose without loss of generality kn > 0. We set K = σ K n , with σ ∈ R to be determined. Accordingly to Lemma 4.3, the set ΩM = x ∈ R : |K, x| ≤ M is invariant for the solution if we take

σ ≤−

μn (n)

kn

.

Moreover, by Lemma 4.2, the eigenvalues of the closed-loop system { μ j } are given by μ j = μ j ∀ 1 ≤ j < n, (n) μ n = μn + σ kn . Recall that, by assumption, Re(μ j ) < 0 for any 1 ≤ j < n. Choosing

σ <−

μn (n)

kn

,

we obtain asymptotic stability under the feedback u = K, x = σ K(n) , x. The Theorem is proved.

4.1.4 Semiglobal Stabilization by a Finite Number of Actuators The aim of this section is to introduce a battery of singularly saturating actuators, with predeterminable cardinality, such that taking their sum we reconstruct the linear feedback K, x ensuring semiglobal asymptotic stability of the closed-loop system. To this purpose we modify the saturation constraint as follows u = ∑ u j where u j = satM (v j ), with M > 0 and

(4.25)

j

v j (t) ∈ C0 ([0, +∞), R), j ∈ N. Denoting by DM the domain of attraction associated to K, x with saturation level ﬁxed to M > 0, if we assume j = 1, ..., r in (4.25) then the domain of attraction associated to u is DrM . In Section 4.1.3 we have shown how to design a stabilizing linear feedback decreasing (in norm) along the closed-loop system trajectory. Thanks to the invariance of the level sets of such feedback, we can state the following result.

4.1 Invariant Strips and Linear Feedback Laws

65

Theorem 4.3. Assume the saturation constraint (4.25) on system (4.17) and suppose condition (4.18) holding; consider the linear feedback K, x, where the vector K is given by Theorem 4.1. For any ﬁxed M > 0, δ > 0 arbitrarely small and x0 ∈ Rn there exists r = r(x0 , K, M, δ ) ∈ N such that we can ﬁnd a set of r functions { f j (x)}rj=1 with the following properties: 1. supx∈Rn | f j (x)| < M − δ ∀ j = 1, .., r; 2. Setting u(x) = ∑rj=1 f j (x) = ∑rj=1 satM ( f j (x)), we have u(φ (t, x0 )) = K, φ (t, x0 ). " # |K, x0 | In particular r = + 1, where [s] stands for the integer part of the M−δ number s. Proof. In combination with the truncation operator (1.1) we consider the remainder Gh (s) = s − sath (s) Given the system (4.17) under assumption (4.18) we take K as designed in Lemma 4.2 and Lemma 4.3. Let us ﬁx a real number δ > 0 arbitrarily small and deﬁne the following partition of Rn : Rn =

∞

Ω δj

j=1

with

Ω δj = {x ∈ Rn : ( j − 1)(M − δ ) ≤ |K, x| < j(M − δ )} ,

(4.26)

where M is the saturation level of the actuators. Note that the level sets of the linear function K, x are hyperplanes; this implies that any Ω εj consists of a pair of unbounded strips. Let us deﬁne the sequence of functions f j , j ≥ 1 f j (x) = satM−δ (G j(M−δ ) (K, x)). It is easy to verify that f j (x) ≤ M − δ for any j. Fix the initial condition x0 ∈ Rn ; we have x0 ∈ Ωrδ for some r ∈ N. In particular, recalling that for any x ∈ Ω δj we have j−1 ≤ r is given by

K, x < j, M−δ

"

# |K, x| r= + 1. M−δ

Setting r

u(x) = satM−δ (K, x) + ∑ f j (x), j=1

(4.27)

66

4 Control Design Issues: Continuous-Time Plants

by construction it veriﬁes u(x) = K, x for any x ∈

r

Ω δj .

j=1

Recalling that under the feedback K, x the sets ΩC , C > 0, are invariant for the solution of the closed-loop system in the presence of saturation constraint (2.2) and that K has been designed in order to guarantee the asymptotic stability of the system, Theorem 4.3 is proved. Remark 4.2. Note that if the solution φ (t, x0 , u) lies in Ω δj0 for some j0 then f j (φ (t, x0 , u)) ≡ 0 for any j > j0 . In particular any f j (φ (t, x0 , u)) with j > 1 is different from zero at most as t varies in a bounded interval. Remark 4.3. Since the level set K, x = M lies in the semispace K, Ax + Mb < 0, if the initial condition x0 belongs to the strip |k1 | δ , r(M − δ ) ≤ K, x < r(M − δ ) ⊂ Ωr+1 a1 we have asymptotic stabilization under the feedback r

u = satM−δ (K, x) + ∑ f j (x). j=1

Note that (a1 k2 /k1 ) < −k2 is the positive eigenvalue of the state matrix A. Remark 4.4. The real parameter δ > 0 has been introduced to saturate each actuator on a level which is less than the maximum one. This has been done in order to prevent an eventual fault of the plant due to overload of the input devices.

4.1.5 A Planar Example Let us set the saturation level to M = 2 and deﬁne the state matrix A choosing the parameters a1 = 3, a2 = 2, that is 0 1 A= . 3 2 With this choice the eigenvalues of A are μ1 = −1, μ2 = 3. An eigenvector of AT associated to μ2 is given by K = (σ , σ ), with σ ∈ R \ {0}. In order to have asymptotic stability of the system driven by the feedback u = K, x, we need to impose σ < −μ2 = −3; for sake of simplicity we take σ = −4. The eigenvalues of the closed-loop matrix A = A + bKT are μ 1 = μ 2 = −1.

4.1 Invariant Strips and Linear Feedback Laws

67

The initial condition x0 = (3/2, 3/4) is considered, and the saturation constraint (4.25) on the control variable is assumed; it is easy to verify that K, x0 = −9 therefore, according to (4.26), one has x0 ∈ Ω5δ for δ ∈ R sufﬁciently small. Figure 4.4 shows the evolution of the system in the (x1 , x2 )-plane for the time interval [0, 7]; the straight lines are the boundary components of the sets Ω δj with δ = 0. Figure 4.5 represents the control u(t) as t varies in the interval [0, 7]. Since the norm of the control v = ∑5j=1 satM−δ (G j(M−δ ) (K, x)) is decreasing by construction, the number of employed actuators reduces from ﬁve to one during the system evolution; in particular any intersection of the state x(t) with the boundary lines of the strips corresponds to the deactivation of one actuator (see Figure 4.4). Note that for t = 7 both x(t) and |v(t)| are close to the origin: we have √ 9 3 929 ≈ 0.0208453 and |v(7)| = 7 ≈ 0.00820694. x(7) = 7 4e e

x2 1.0

0.5

0.5

0.5

1.0

1.5

2.0

x1

0.5

1.0

1.5

Fig. 4.4. Evolution of (x1 (t), x2 (t)) for t ∈ [0, 7].

4.1.6 Extension to Multi-Input Systems Let us consider a n-dimensional multi-input (MI) linear system having the following structure ⎧ ⎨ x˙ = Ax + Bu (4.28) ⎩ x(0) = x0

68

4 Control Design Issues: Continuous-Time Plants

u 1

2

3

4

5

6

7

t

1

2

3

4

5

6

Fig. 4.5. Control variable u(t) for t ∈ [0, 7].

where A ∈ M n×n , B ∈ M n×m , x(t) ∈ C0 (R, Rn ) is the state vector and u(t) ∈ ˜ ∈ M n×n , the specL∞ (R, Rm ) is the input variable. For an arbitrary square matrix A ˜ ˜ trum of A, i.e. the set of eigenvalues, is indicated with σ (A) ⊂ C; in addition, set Rm,+ = {x ∈ Rm : xi > 0, i = 1, ..., m} and C− = {ω ∈ C : Re(ω ) < 0}. In this section it is discussed the problem of extending to multi input systems the results on the existence of controlled invariant strips. To this purpose the following general deﬁnition can be useful. Deﬁnition 4. The control feedback u = u(x(t)) = (u1 (x(t)), ..., um (x(t))) is said to satisfy the Sublevel Sets Invariance and Stabilization property (SSIS) if the following conditions hold: 1. System (4.28) driven by u(x(t)) is asymptotically stable. 2. For any M = (M1 , ..., Mm ) ∈ Rm,+ , the set EM = {x ∈ Rn : |ui (x)| < Mi , i = 1, ..., m}

(4.29)

is a controlled invariant set for the solution of System (3.11) driven by u(x(t)). Remark 4.5. The SSIS condition means asymptotic stability of the closed-loop system together with the invariance of the control norm sublevel sets; it is a generalization of the invariance condition presented in Section 4.1.3 for single input systems.

4.1 Invariant Strips and Linear Feedback Laws

69

There is a strict connection between the structure of the saturated maximal region of attraction for a system (or null controllable region) and the existence of SSIS controls. It follows from the deﬁnition that if u(t) is a SSIS control, then the sublevel set EM (deﬁned in (4.29)) veriﬁes EM ⊆ BM

∀ M ∈ Rm,+ .

(4.30)

The existence of SISS controls will be proved for special classes of systems ﬁrst and then some general conditions of existence will be presented.

4.1.6.1 Systems with a Single Unstable Mode This section deals with the problem of stabilizing a controllable MI system having a single unstable real mode. In particular, without loss of generality, it is assumed that Re(ν j ) < 0 ∀ 1 ≤ j ≤ n − 1,

ν := νn ≥ 0, where {ν j }nj=1

= σ (A). The main aim is to design a linear feedback able to stabilize semiglobally the system on the saturated maximal region of attraction. Note that, since there is only one unstable direction, in this case the saturated maximal region of attraction is a region bounded by two parallel (n − 1) dimensional hyperplanes. Let P ∈ M n×n the matrix associated to the following change of coordinates (Jordan canonical decomposition): A = P−1 AP, B = P−1 B with

⎛ A=⎝

A11 0

⎞

⎛

⎠

B=⎝

0 ν

B1 · · · Bm

⎞ ⎠

b1 · · · bm

The matrix A11 ∈ M (n−1)×(n−1) has stable eigenvalues by deﬁnition. The new state variables are z = (z1 , zn ) := (z1 , ..., zn−1 , zn ) with z = P−1 x. Due to plant controllability assumption, at least one of the coefﬁcients b j , j = 1, ..., m, is different from zero. Without loss of generality assume that b j = 0 ∀ 1 ≤ j ≤ s with s ≤ m. Lemma 4.4. For any ε > 0 and for 1 ≤ j ≤ s, the linear feedback j−1

( j) uε (z) =

m− j

- ./ 0 1 - ./ 0 − (0, ..., 0, (ν + ε )zn , 0, .., 0) bj

70

4 Control Design Issues: Continuous-Time Plants

ensures asymptotic stabilization of the saturating plant (A, B, M) on the region M j |b j | ( j) n Eε = (z1 , zn ) ∈ R : |zn | ≤ . ν +ε ( j)

Proof. It is straightforward to verify that the control uε (z) satisﬁes the saturation ( j) ( j) m×n the matrix whose constraints any state z ∈ Eε . Let us denote by F ∈ M

for ( j) entries fik satisfy ( j)

fik = δi j δkn , where δi j are the standard Kronecker symbols. The asymptotic stability of the plant follows observing that the closed-loop system is given by z˙ = (A − (ν + ε )BF( j) )z, whose eigenvalues are

σ (A − (ν + ε )BF( j) ) = σ (A11 ) ∪ {−ε } ⊂ C− .

The lemma shows that the system can be stabilized using separately s different SSIS controllers, where s is the number of non-null coefﬁcients b j ; next result illustrates how to obtain semiglobal asymptotic stabilization over the saturated maximal region of attraction. Let us deﬁne the set 1 s n Emax = z ∈ R : |zn | < ∑ Mi |bi | . ν i=1 Theorem 4.4. The set Emax is the saturated maximal region of attraction for the system (A, B, M); moreover for any z0 ∈ Emax there exists a linear feedback uz0 ensuring asymptotic stabilization of the plant. / Emax ; without loss of generality assume Proof. Suppose that z0 = (z1,0 , ..., zn,0 ) ∈ that 1 s zn,0 ≥ ∑ Mi |bi |. (4.31) ν i=1 Now, for an arbitrary control input u m

z˙n = ν zn + ∑ bi ui (t). i=1

For plant stabilization, since zn (0) > 0 by (4.31), it is required z˙n (0) < 0,

4.1 Invariant Strips and Linear Feedback Laws

71

that is equivalent to m

s

i=1

i=1

ν zn,0 < − ∑ bi ui (t) ≤ ∑ Mi |bi |. This condition is incompatible with assumption (4.31). It has been proved proved that, if the initial data are choosen outside the set Emax , there exists no admissible control input that can ensure plant stability. Let us deﬁne the control vector uε = uε (z0 ) = (uε ,1 , ..., uε ,m ) as follows. −(ν + ε )zn b1 b1 −(ν + ε )zn uε ,2 (z) = − TM1 (uε ,1 (z)) b2 b1 uε ,1 (z) =

.. . uε ,s (z) =

bs−1 bs

−(ν + ε )zn − TMs−1 (uε ,s−1 (z)) bs−1

uε , j (z) ≡ 0 ∀ s < j ≤ m. It is easy to verify that the eigenvalues of the closed-loop system are still σ (A11 ) ∪ {−ε }. Moreover, the saturation constraints are fulﬁlled over the set s 1 Eε = z ∈ Rn : |zn | < ∑ Mi |bi | . ν + ε i=1 The conclusion follows observing that lim Eε = Emax .

ε →0+

4.1.6.2 Eigenstructured Systems Let us discuss now the case of exactly m ≤ n real unstable modes. Without loss of generality it can be assumed that Re(ν j ) < 0 for any j = 1, ..., n − m and ν j ≥ 0 for j = n − m + 1, ..., n. For sake of semplicity, set

λ j = νn−m+ j ,

j = 1, ..., m.

The system can be easily splitted into a stable/unstable subsystems partition as follows: ⎞ ⎛ ⎛ ⎞ As 0 Bs ⎠ A=⎝ B = ⎝ ⎠, Bu 0 Au

72

4 Control Design Issues: Continuous-Time Plants

where As ∈ M (n−m)×(n−m), Au ∈ M m×m , Bs ∈ M (n−m)×m , Bu ∈ M m×m and A = R−1 AR, B = R−1 B for a suitable invertible matrix R ∈ M n×n . Let us focus the analysis on the unstable subsystem (Au , Bu ). Proposition 4.1. If Au Bu = Bu Au and rank(Bu ) = m then the system (A, B) admits a family of SSIS linear feedbacks. Proof. Since Au and Bu commute, they are simultaneously diagonalizable, i.e. there exists S ∈ M m×m such that S−1 Au S = diag(λ1 , ..., λm ) =: ΔA S−1 Bu S = diag(b1 , ..., bm ) =: ΔB . Denote by z = (z, z1 , ..., zm ) the new coordinates. From the condition rank(Bu ) = m follows b1 · b2 · · · bm = 0. It is straighforward to verify that for any ε > 0 the control u(z(t)) = (u1 (z(t)), ..., um (z(t)))

u j (z) = −

(λ j + ε ) zj bj

satisﬁes the SSIS property with respect to system (ΔA , ΔB ). Set λ1 + ε λm + ε Δε = −diag , ..., b1 bm Consider now the system (S−1 Au S, S−1 Bu ). Deﬁne the control u(x(t)) = SΔε x(t). The validity of the SSIS condition follows observing that the system (S−1 Au S, S−1 Bu ) driven by the control u(x(t)) is equivalent to system (ΔA , ΔB ) driven by the control Δε x(t). Finally, to obtain a SSIS control for the system (A, B) one can proceed as follows. Deﬁne the matrices H ∈ M n×n and Kε ∈ M m×n as ⎛ ⎞ I(n−m)×(n−m) 0 ⎠ H=⎝ 0 S Kε = (0

SΔε ) .

4.1 Invariant Strips and Linear Feedback Laws

73

The linear feedback u(x(t)) = Kε x(t) is a SSIS control for the system (R−1 H−1 ASR, R−1 H−1 B)

and the statement is proved.

From the Proposition 4.1, it follows that the existence of SSIS controls for a general system (A, B) with m unstable modes is ensured if one can ﬁnd a linear transformation R ∈ M n×n with both R−1 Au R and R−1 Bu R diagonal. Let us denote by wλ1 , ..., wλm the eigenvectors of the matrix A associated to the non-negative eigenvalues λ1 , ..., λm . Set Λ ⊥ = {span(wλ1 , ..., wλm )}⊥ . For j = 1, ..., m let us deﬁne the subspaces

Λ j = Λ ⊥ + span(wλ j ); Ξ j = {span(wλ j )}⊥ . Proposition 4.2. Let us denote by B j , j = 1, ..., m the column vectors of the matrix B. If, up to an order rearrangement, for any j = 1, ..., m the following conditions hold ⎧ ⎨ Bj ∈ Λj (4.32) ⎩ Bj ∈ / Ξ j, then there exists a family of SSIS linear feedbacks for system (A, B). Proof. The vectors {wλ j }, j = 1, ..., m are independent and constitute a basis for the subspace Λ ; take in addiction n − m independent vectors yi ∈ Λ ⊥ . Since Rn = Λ ⊕ Λ ⊥ , we have Rn = span(y1 , ..., yn−m , w1 , ..., wm ). Deﬁne the matrix H ∈ M n×n as H = (y1 y2 · · · w1 · · · wm ). Due to condition (4.32), the transformed matrices A = HAH−1 , following special form ⎞ ⎛ ⎛ ⎞ As 0 Bs ⎠ A=⎝ B = ⎝ ⎠, ΔB 0 ΔA with ΔA = diag(λ1 , ..., λm ) and ΔB = diag(b1 , ...., bm ), where b j = B j , wλ j = 0 ∀ j = 1, ..., m.

B = HB have the

74

4 Control Design Issues: Continuous-Time Plants

Since the unstable subsystem (ΔA , ΔB ) turns out to be diagonal, the SSIS control can be derived as in the proof of Proposition 4.1. Remark 4.6. Condition (4.32) can be achieved only if for any j = 1, ..., m, there exists s j ≤ (n − m + 2) such that rank[B j AB j · · · As j −1 B j ] < s j .

4.1.6.3 Examples Example 4.1. Consider the following system (A, B) with n = 5, m = 4 : ⎛ 29 12 ⎞ 32 2 5

⎜ ⎜ ⎜ − 22 5 ⎜ ⎜ √ ⎜ A = ⎜ − 35 + 23 ⎜ ⎜ ⎜ 9 √3 ⎜−5 − 2 ⎝ − 12 5

and

5

5

2

− 21 5

− 15

−1

√ 7−5 3 10

1 5

√ √ − 3 −2 + 3

√ −19+5 3 8 10 5

√ √ + 3 −1 − 3

− 65

⎛

− 15

−1

0 − 16 3

1

−

5

⎟ ⎟ ⎟ ⎟ √ ⎟ 3 + 3⎟ ⎟ 10 ⎟ √ ⎟ ⎟ 9 10 − 3 ⎟ ⎠ − 45

− 15

⎞

⎟ ⎜ ⎜ 0 1 1 −3 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ −1 0 1 1 B=⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ 0 −2 − 12 0 ⎟ ⎠ ⎝ 3

0

1 3

1

√ The eigenvalues of the state matrix are ν1,2 = −12i, ν3,4 = − 3 and ν5 = 3. By a coordinates change, the system can be rewritten as ⎞ ⎛ −1 −2 0

0

0

⎟ ⎜ ⎜ 2 −1 0 0 0⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 0 0 1 0 A=⎜ ⎟ ⎜ ⎟ ⎜ √ ⎟ ⎜ ⎜ 0 0 −3 −2 3 0 ⎟ ⎠ ⎝ 0

0

0

0

3

4.1 Invariant Strips and Linear Feedback Laws

⎛

and

75

1

11 6

1 3

−7 1

0

0

4

⎞

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 3 −1 0 10 3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ B = ⎜ 0 1 − 43 −1 ⎟ . ⎟ ⎜ ⎟ ⎜ ⎜ 9 −5 − 13 14 ⎟ ⎜ 6 3 ⎟ ⎠ ⎝ Set

3+ε x5 , −7 3+ε 3+ε x5 − TM1 − x5 u2 (x) = −7 − −7 −7 u1 (x) = −

and u3 = u4 = u5 ≡ 0. Moreover let us ﬁx the saturation constraints M1 = 2,

M2 = 3

and choose the initial datum x0 = (5, −4, 11, 8, 4). The system evolution is shown in Figure 4.6 and Figure 4.7.

xt

15

12.5

10

7.5

5

2.5

t 2

4

6

Fig. 4.6. Norm of the solution x(t) in the time interval (0,8)

8

76

4 Control Design Issues: Continuous-Time Plants

u1t,u2t 2

1

t 0.2

0.4

0.6

0.8

1

-1

-2

Fig. 4.7. Controls u1 (dotted line) and u2 (continuous line) in the time interval (0,1)

Example 4.2. Consider the following system (A, B) with n = 5, m = 3 : ⎞ ⎛ −3.25 −3.416 −0.375 0 −1.25

⎜ ⎜ −7.5 ⎜ ⎜ ⎜ A=⎜ ⎜ 7.5 ⎜ ⎜ ⎜ 1.5 ⎝

⎟

−1

0

0 −2.5 ⎟ ⎟

3

2

0 2.5 ⎟ ⎟

0.8

0

3 0.5 ⎟

⎟ ⎟ ⎟ ⎟ ⎠

6.75 −0.25 1.125 0 2.75

and

⎛

0.083 0.5 −0.625

⎞

⎟ ⎜ ⎜ 4 0 −2.5 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 2.5 −3.6 B=⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ −0.8 −0.4 0.5 ⎟ ⎠ ⎝ −2.25 −0.3 2.875

The eigenvalues of A are ν1,2 = −15i, ν3 = 3, ν4 = 2 and ν5 = 0.5, so that the system has three unstable modes. Changing the coordinates the new expressions are

4.1 Invariant Strips and Linear Feedback Laws

⎛

77

−1 5 0 0 0

⎞

⎜ ⎟ ⎜ −5 −1 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ 0 0 3 0 0 A=⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 02 0 ⎟ ⎝ ⎠ 0

and

⎛

0 0 0 0.5

−2 1.3 1

⎞

⎟ ⎜ ⎜ 8 0 −5 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟. 0 0 0.3 B=⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ 0 −2 0 ⎟ ⎠ ⎝ 0

Set the control variables

0 −5

u1 (x) = −3(2 + ε )x3, u2 (x) =

3+ε x4 , 2

0.5 + ε x5 . 5 and the saturation constraints M1 = 2, M2 = 4, M3 = 1.8. Figure 4.8 and Figure 4.9 illustrate the system evolution starting from x0 = (10, −6.5, −0.2, 1.8, −5.7). u3 (x) =

4.1.6.4 General Conditions for Set Invariance In the previous sections conditions for the existence of SSIS linear feedbacks have been given for special classes of systems, namely systems with a single unstable mode and systems with the number of unstable real modes equal to the number of inputs. Consider now an arbitrary MI system. The SSIS condition is in general not easy to be met by the system matrices. It is well known (see for instance [46], [47]) that the invariance of the control input sublevel-sets is equivalent to the existence of a solution F to the matricial equation AK + KBK = FK; moreover, in order to ensure positive invariance, the coefﬁcients of F are required to satisfy some positivity constraints.

78

4 Control Design Issues: Continuous-Time Plants

xt

12

10

8

6

4

2

t 1

2

3

5

4

6

Fig. 4.8. Norm of the solution x(t) in the time interval (0,6)

u1t,u2t,u3t

3

2

1

t 1

2

3

4

-1

Fig. 4.9. Controls u1 (continuous line), u2 (dashed line) and u3 (dotted line) in the time interval (0,4)

4.1 Invariant Strips and Linear Feedback Laws

79

The main result presented in this section illustrates a different approach toward investigation on the existence of SSIS linear controls. For a multi input linear system having distinct real unstable modes, it is shown that, if the columns of the input matrix are sufﬁciently close to a set of eigenvectors corresponding to the positive eigenvalues of the state matrix A, a SSIS linear feedback can be found. Assume the existence of m vectors K j , j = 1, ..., m with K j , x(t)K j , x˙ (t) ≤ 0.

(4.33)

The condition above is equivalent to the invariance of the set {x ∈ Rn : |K j , x| < M j ∀ j = 1, ..., m}, which guarantees the validity of SSIS property for the control u(t) = (KT1 x(t), ..., KTm x(t)). Suppose that for any j = 1, ..., m we have K j , x(t) = M j ; from (4.33), it follows that (4.34) K j , x˙ (t) ≤ 0 ∀ j = 1, ...m. 1

Now

K j, A + 1 =

A +

x(t)

=

x(t) + K j , B j KTj x(t) = 2

∑ Ki BTi

K j , x(t) + M j K j , B j

i = j

From (4.34) follows the inequality 1 T

2

∑ Bi KTi i = j

AT +

m

K j , A + ∑ B j KTj j=1 2

K j , x˙ (t) = 1 =

∑

i = j

2

Ki BTi

K j , x(t)

≤ −M j KTj B j

whenever K j , x(t) = M j . These conditions can be together fulﬁlled only if the spaces 1 2 AT +

∑ KiBTi

K j , x(t)

i = j

and K j , x(t) = 0

=0

(4.35)

80

4 Control Design Issues: Continuous-Time Plants

are parallel; it turns out that K j must be an eigenvector for the matrix AT + ∑ i = j Ki BTi . In particular for j = 1, ..., m the identity AT +

∑ KiBTi i = j

K j = μ jK j

(4.36)

holds. From (4.35), it follows that the eigenvalues μ j must satisfy

μ j ≤ −KTj B j .

(4.37)

The following statement summarizes the previous development. Proposition 4.3. Suppose that the system (4.28) admits an SSIS linear feedback controller, say u = (u1 , ..., um ), where u j = K j , x with K j ∈ Rn for j = 1, ..., m. Then the vectors K j satisfy (4.36) and (4.37). Remark 4.7. If the column vectors B j are a family of eigenvectors for the state matrix A, it is very easy to verify that condition (4.36) is satisﬁed. As shown in Proposition 4.2, the SSIS feedback is constituted by a set of dual eigenvectors K j , i.e. AT K j = λ j K j . Now, since Bi ∈ ker(A − λi I), K j ∈ Im(AT − λi I) and ker(A)⊥Im(AT ), it follows that for any j = 1, ..., m Bi , K j = BTi K j = 0

∀i = j.

The existence of SSIS linear controls is ruled by nonlinear algebraic equations (4.36)-(4.37); nevertheless, the following invariance result holds. Proposition 4.4. The SSIS property is invariant under orthogonal coordinates transformations. Proof. Let H be a orthogonal matrix, i.e. HHT = HT H = I, and let A, {Bi }, {Ki } and {μi } satisfy the SSIS conditions; by the change of coordinates associated to H, the relations (4.36) and (4.37) can be rewritten as 3 T T 4 H A H33 + ∑ i = j HT Ki BTi H 4 HT K j − μ j4HT K j = = HT AT + ∑ i = j Ki BTi K j − μ j K j = 0 and

μ j ≤ −KTj HHT B j = −KTj B j .

Corollary 4.1. If A, {Bi }, {Ki } and { μi } satisfy the SSIS conditions and an arbitrary change of coordinates is applied to the system by the transformation associated to the matrix H with det(H) = 0, then {H−1Ki } are SSIS linear feedbacks for the system (H−1 AH, {HT H(H−1 Bi )}). The next theorem states the local existence of SSIS linear feedbacks for an arbitrary system having distinct unstable real eigenvalues.

4.1 Invariant Strips and Linear Feedback Laws

81

Theorem 4.5. Suppose that A has m unstable real modes; due to Proposition 4.4 and Corollary 4.1, it can be assumed without loss of generality that A is given in Jordan canonical form. Let us denote with Bi , i = 1, ..., m, a family of independent eigenvectors corresponding to the positive distinct real eigenvalues ν1 ,..., νm . There exist ρi > 0 such that for Bi ∈ Cρi (Bi ) := V ∈ Rn : ||V − Bi || < ρi , the system (4.28) admits SSIS linear feedbacks.

Proof. The existence of SSIS controls will be proved applying the general implicit (i) function theorem. The following notations are employed: Bi = {b j } j=1,...,n , Ki = (i)

(i)

{k j } j=1,...,n, ki = χi and A = diag(Au , As ) with Au = diag(ν1 , ..., νm ) and As = {ars }r,s=m+1,...,n. The SSIS conditions (4.36) can be rewritten as the following set of relations for i = 1, ..., m : ⎧ ⎪ (i) ( j) ( j) (i) ⎪ ⎪ Φr = νr − μi + ∑ kr br kr ⎪ ⎪ ⎪ ⎪ j =i ⎪ ⎪ ⎪ n ⎪ ( j) ( j) (i) ⎪ ⎪ + ∑ r = 1, ..., m, ⎪ ∑ kr b s ks = 0 ⎪ ⎪ ⎨ s=1, s =r j =i ⎪ ⎪ ⎪ ⎪ (i) ( j) ( j) (i) ( j) ( j) (i) ⎪ ks + arr − μi + ∑ kr br kr + ⎪ Φr = ∑ ∑ kr bs ⎪ ⎪ ⎪ 1≤s
r<s≤n

Let us consider the n · m dimensional vector (1)

(1)

(2)

(2)

( j)

(m)

j , ..., kn , μm ) y = (k2 , k3 , ..., μ1 , k1 , k3 , ..., k j−1 , k j+1 ( j)

given by μ j and all entries of K j except k j . Let us consider in addiction the mapping F : Rn·m × Rn·m → Rn·m given by

(1)

(1)

(m)

(m)

F (B1 , ..., Bm , y) = (Φ1 , Φ2 , ...., Φn−1 , Φn ) The SSIS constraint (4.36) is equivalent to the equation F (B1 , ..., Bm , y) = 0. Referring to Remark 4.7, it can be seen that, setting Bi = λi ei , μ i = νi and Ki = hi ei , it follows F (B1 , ..., Bm , y) = 0. In order to apply the implicit function theorem, the following non-singularity condition has to be satisﬁed det

∂F ∂y |

(B1 ,...,Bm ,y)

= 0

82

4 Control Design Issues: Continuous-Time Plants

The following quantities have to be taken into account: ⎧ (s) (s) ⎪ δ jr νr − μi + ∑s =i kr br ⎪ ⎪ ⎪ ⎪ (s) (s) ⎪ +(1 − δ jr ) ∑s =i kr b j ⎪ ⎪ ⎪ ⎪ ⎪ (i) ⎨ ∂ Φr (s) (s) = δ − μ + k b a ∑ r r jr rr i s = i (i) ⎪ ⎪ ∂kj ⎪ (s) (s) ⎪ ⎪ +(1 − δ jr ) ∑s =i kr b j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (s) (s) ar j + ∑s =i kr b j (i)

∂ Φr

(s) ∂kj (i)

∂ Φr ∂ μi

(s) (i)

=

δr j ∑ bh kh

=

−kr

=

0

h =r

r≤m

r > m,

j≤r

r > m,

j>r

s = i

(i)

(i)

∂ Φr ∂ μs

s = i

Computing the above derivatives and performing the evaluation in (B1 , ..., Bm , y) a block diagonal matrix is obtained: ∂F = Diag(D1 , ..., Dm ); ∂y | (B1 ,...,Bm ,y)

each block D j is a n × n matrix given by ⎛ j−1 0 j−1×m− j 0 j−1×n−m diag(νs − ν j + hsλs )s=1 ⎜ 01× j−1 01×m− j 01×n−m ⎜ Dj = ⎝ 0m− j× j−1 diag(νs − ν j + hs λs )m s= j+1 0m− j×n−m 0n−m×m− j As − ν j In−m 0n−m× j−1

⎞ 0 −h j ⎟ ⎟ 0 ⎠ 0

Moreover, the block D j is itself a block-diagonal matrix up to a row ordering rearrangement. Now m ∂F = ∏ detD j . det ∂y | j=1 (B1 ,...,Bm ,y)

Let us ﬁx ε j > 0 and choose h j such that condition (4.37) holds in the quantitative form (4.38) −K j B j − μ j = ε j > 0, that is h j = −(ν j + ε j )/λ j . Now νs − ν j + hs λs = εs − ν j = εs + |ν j | > 0. It follows that det D j = (−1)n− j+1h j · det(As − ν j In−m )

m

∏

s = j,s=1

(εs + |ν j |) = 0.

4.2 Nonlinear Robust Controller Design via Sliding Modes

83

By the implicit function theorem there exists a neighborood J of (B1 , ..., Bm ) such that (4.36) is veriﬁed with y = y(B1 , ..., Bm ). Moreover, as a consequence of sign permanency, there exist J1 ⊆ J such that (4.37) is satisﬁed too. This concludes the proof. The global existence of SSIS controls can be investigated following a criterion introduced ∞ in [48]. Suppose that a real continuous positive function z : [0, ∞) → [0, ∞) 1 ds = ∞ can be found such that with 1 z(s) ||F (B1 , ..., Bm , y)|| · ||(F (B1 , ..., Bm , y))−1 || < z(||y||) for Bi ∈ Ji and y ∈ Jy , where Ji ⊂ Rn and Jy ⊂ Rn·m . Then there exists an implicit function y = y(B1 , ..., Bm ) satisfying F (B1 , ..., Bm , y(B1 , ..., Bm )) = 0 for any (B1 , ..., Bm ) ∈ J1 × J2 · · · × Jm . Remark 4.8. As in the SISO case (see [49]), the widest region of attraction associated to SSIS controls is the open set EM = {x ∈ Rn : |Ki , x| < Mi , i = 1, ..., m} where Ki veriﬁes (4.36) and where condition (4.37) is satisﬁed with equality for any i = 1, ..., m.

4.2 Nonlinear Robust Controller Design via Sliding Modes for Continuous-Time Multi Input Plants This section is devoted to the description of some results about the stabilization of multi-input plants by sliding modes [50], [51]. As well known, the main advantage of the sliding mode control technique is robustness, i.e. invariance with respect to bounded matched disturbances. It will be proved that robust stabilization of linear plants can be achieved by means of a time-varying state feedback controller, derived imposing the achievement of a sliding motion onto a suitable time-varying sliding surface [42] [43].The presented results can be found in [52] [53] [41]. Consider the following continuous-time, time invariant, controllable, uncertain def ˆ ˆ B}: Multi-Input plant S = {A, ˆ x + B(I ˆ + Δ B)( ˆ uˆ + d(t)) ˆ x˙ˆ = Aˆ

(4.39)

where: xˆ = [xˆ1 · · · xˆn ]T ∈ Rn is the state vector (assumed available for measurement), ˆ ∈ Rn×n , B ˆ ∈ Rn×m are the state uˆ ∈ Rm is the control input, with n > m, and A ˆ ∈ Rm matrix and the input matrix respectively. The matched uncertain terms d(t) m×m and Δ Bˆ ∈ R represents plant uncertainties and/or external disturbances affecting the system, and I is the m dimensional identity matrix. Under the controllability

84

4 Control Design Issues: Continuous-Time Plants

hypothesis, there exist two smooth changes of coordinates: x = Tx xˆ , u = Tu uˆ such that state and the input matrices of plant (4.39) have a canonical structure depending on the controllability indices of the system ρ j , j = 1 . . . m [54] [55]. Rearranging rows and columns [55], plant (4.39) can be further transformed into: x˙ = Ax + B(I + Δ B)(u + d(t))

(4.40) "

# A1,1 A1,2 , with A1,1 ∈ A2,1 A2,2 R(n−m)×(n−m)5, A1,2 ∈ R(n−m)×m , A2,1 ∈Rm×(n−m) , A2,2 ∈ Rm×m , where the block 6 matrix A1 = A1,1 A1,2 has elements ξ⎡ i, j , i = 1, . . . , n − m, ⎤ j = 1, . . . , n which a . . . a a 1,1 1,2 1,n 5 6 ⎦. ... are either 1 or 0, and A2 = A2,1 A2,2 = ⎣ am,1 am,2 . . . am,n The plant is preceded by a saturating device u = f (v) with known threshold M > 0, i.e. for the i-th component: 5 ˆ and B = 0(n−m)×m with d(t) = Tu d(t)

ui = satM (vi )

I

6T

, and: A =

i = 1, . . . , m.

(4.41)

Assumption 4.1. The uncertain terms d(t) and Δ B are bounded by known constants, i.e. ||d(t)|| ≤ ρ¯ , ||Δ B|| ≤ δ . Exploiting the presence of the saturation, system equations can be rearranged as follows x˙ = Ax + B(u + d1(t)) (4.42) with d1 (t) = d(t) + Δ B(u + d(t)), ||d1 (t)|| ≤ ρ¯ + δ (M + ρ¯ ). Consider a matrix C ∈ Rm×n of the form ⎤ ⎡ c1,1 . . . c1,n−m ε1 0 . . . 5 6 ... ... ⎦ C = C1 C2 = ⎣ (4.43) cm,1 . . . cm,n−m 0 0 εm with C1 ∈ R(n−m)×m , C2 ∈ Rm×m , ci, j , εi ∈ R, i = 1, . . . , m, j = 1, . . . , n − m. Coefﬁcients appearing in the C1 matrix can be designed such that, when a sliding motion [50] is achieved on the following sliding surface: sˆ(x) = Cx = 0

(4.44)

the corresponding reduced order system has assigned stable eigenvalues, and, as a consequence, system (4.42) is stable. Indeed, it has been shown [56] [57] that the design problem of C is equivalent to an eigenvalue assignment problem which, under the controllability hypothesis, can always be solved. In the absence of the saturating device (4.41), i.e. if the control input u could be directly manipulated, the following control law, obtained by imposing the inequality sˆT (x)s˙ˆ(x) < 0, uc = −(CB)−1 CAx − ρ¯ sign(BT CT sˆ(x))

(4.45)

4.2 Nonlinear Robust Controller Design via Sliding Modes

85

would ensure the achievement of a sliding motion on (4.44), hence plant stabilization1 . Since the input v only can be manipulated, the control problem addressed in this paper consists in ﬁnding a feedback controller v guaranteeing the robust stabilization of the system (4.39) in the presence of a saturating nonlinearity in the actuator.

4.2.1 A Time Varying Sliding Surface Deﬁne the matrix D ∈ Rm×n as

⎡

⎤ d1,1 . . . d1,n−m 0 0 ... ... ⎦ D = D1 0 = ⎣ dm,1 . . . dm,n−m 0 0 5

def

6

(4.46)

¯

¯ and C(t) = (C + De(−λ t) ) = [c¯i, j ]. Consider the following time-varying sliding surface, with λ¯ > 0: , + ¯ ¯ (4.47) s(x(t), x(0),t) = C(t) x(t) − x(0)e(−λ t) = 0 ⎡ ⎤ c¯1,1 . . . c¯1,n−m + , ⎢ ⎥ .. (−λ¯ t) =⎣ x(t) − x(0)e ⎦ . diag{ε1 , . . . , εm } c¯m,1 . . . c¯m,n−m ¯

where c¯i, j = (ci, j + di, j e(−λ t) ). It can be proved (see Appendix) that, for any choice of di, j 0, i = 1, m, j = 1, . . . , n − m, constraining the system to the surface s(x(t), x(0),t) = 0 implies plant asymptotical stabilization. Moreover, since s(x(0), x(0), 0) = 0, the surface (4.47) is such that no reaching phase exists [42] [43], as the plant state is on the hyperplane from the very beginning. What motivates ¯ the introduction of the vanishing term De(−λ t) x with respect to standard surfaces is the need of modulating the control input in order to cope with the saturation limitation. Roughly speaking, we are aiming at constraining the system on a sliding surface which, besides being asymptotically stabilizing, has a tunable part such that the control input is able to constrain the plant state on the sliding hyperplane without violating the saturation bounds. The following section is therefore devoted to show that the coefﬁcients of the D matrix can always be found as to satisfy the saturation limits, still preserving the persistence of the sliding motion. For the surface (4.47), the control input ensuring the achievement of a ﬁnite-time sliding motion is, similarly to (4.45): 1

The sign(·) symbol above denotes a vector whose i-th entry contains the sign of the i-th component of the vector BT CT sˆ(x).

86

4 Control Design Issues: Continuous-Time Plants

+ , ¯ −1 ¯ ¯ (C(t)A − λ¯ De(−λ t) )x + ϕ (x(0),t) v = − (C(t)B) ¯ T s(x(t), x(0),t)) − ρ¯ sign(BT C(t)

(4.48)

¯ ¯ where: ϕ (x(0),t) = λ¯ (C + 2De(−λ t) )x(0)e(−λ t) . With some abuse of notation, the sign function used in (4.48) denotes a vector containing signs of the component of argument entries. As already discussed, for the asymptotically stabilizing surface (4.47) no reaching phase exists. Hence, the plant is in sliding motion from t = 0, and the dynamics of the state variables are governed by sliding mode. It follows that state trajectories are bounded.

Remark 4.9. For any initial condition x(0), suitable functions bounding state trajectories can be determined from x(0) itself and the sliding mode dynamics imposed by the control law (4.48). It follows that, for any x(0) there exist a function (max) ΔF ∈ R+ such that: (max) ||x(t)|| ≤ ΔF , ∀t. (4.49) The dependence of such function on the parameters used for tuning the control algorithm will be addressed later, after a deep description of the control law. Assumption 4.2. It is assumed that the saturation threshold M satisﬁes M − ρ¯ − def δ (M + ρ¯ ) = M¯ > 0

4.2.2 The Control Law The constraint induced by saturation (4.41) requires: |vi | ≤ M,

i = 1, . . . , m

(4.50)

−1 = ¯ ¯ Due to the formof the matrices B, C(t) (4.43), (4.46), one has (C(t)B) 1 1 diag ,..., . From expression (4.48), it follows: ε1 εm

n 1 ¯ n ¯ ¯ λ ∑ ci, j + 2di, j e(−λ t) x j (0)e(−λ t) + εi ∑ ai, j x j εi i=1 i=1 ! n−m n n−m ¯ + c¯i, j ξ j,k xk − λ¯ e(−λ t) di, j x j − ρ¯ sign(s¯i (x)) ≤ M

vi = −

∑

j=1

∑

k=1

∑

(4.51)

j=1

¯ T s(x(t), x(0),t) as s¯i (x) with a slight abuse denoting the i−th component of BT C(t) of notation. Taking the worst case, the condition (4.50) for the i−th component can be rewritten as

4.2 Nonlinear Robust Controller Design via Sliding Modes

87

n n ¯ ¯ ¯ λ ∑ ci, j + 2di, j e(−λ t) x j (0)e(−λ t) + εi ∑ ai, j x j i=1 i=1 n−m n n−m ¯ ¯ εi | + ∑ c¯i, j ∑ ξ j,k xk − λ¯ e(−λ t) ∑ di, j x j ≤ M| j=1 j=1 k=1

(4.52)

Taking again the worst case, one has, for i = 1, . . . , m: n

∑

λ¯ (|ci, j | + 2|di, j |)

i=1

+

n−m 5

∑

|x j (0)| (max)

ΔF

! + |εi ||ai, j |

6 n (|ci, j | + |di, j |) + λ¯ |di, j | ≤

j=1

M¯ (max)

mΔ F

|εi |

(4.53)

Theorem 4.6. It is given the uncertain system (4.39) preceded by the saturating device (4.41), under Assumptions 4.1, 4.2. For initial conditions belonging to a suitable neighborhood of the origin, coefﬁcients di, j , εi , i = 1, . . . , m, j = 1, . . . , n − m and a λ¯ > 0 can be chosen such that the feedback controller (4.48) guarantees the robust asymptotical stabilization of the plant. Proof. The proof is constructive. Condition (4.53) corresponds to: ! n ¯λ (|ci, j | + 2|di, j |) |x j (0)| + |εi ||ai, j | ∑ (max) ΔF i=1 +

n−m 5

∑

j=1 def

with M1 =

M¯ (max)

mΔ F

n−m+1 6 n (|ci, j | + |di, j |) + λ¯ |di, j | ≤ M1 |εi | ∑ m−1 j

(4.54)

j=1

, and where m j , j = 1, . . . , n − m + 1 are such that n−m+1

∑

m−1 j ≤ 1.

(4.55)

j=1

In the following, the imposition of condition (4.54) will be performed taking suitable n − m + 1 ”portions” of (4.54) itself, and designing control coefﬁcients di, j , mi , εi , λ¯ involved in each derived inequality in order to ensure the simultaneous fulﬁllment of all of them. Step 1. To start, consider a portion of condition (4.54) made of some terms corresponding to j = 1 in summations: |x1 (0)| λ¯ (|ci,1 | + 2|di,1|) (max) + |εi ||ai,1 | + λ¯ |di,1 | ≤ M1 |εi |m−1 1 ΔF

88

4 Control Design Issues: Continuous-Time Plants

Note that the terms n (|ci,1 | + |di,1|) + |εi ||ai,1 |θ2 have not been taken, since they will be considered in the next step. Deﬁne 0 ≤ θ1 ≤ 1 satisfying θ1 |ai,1 |m1 < M1 , and choose di,1 such that: M1 |x1 (0)| − θ1 |ai,1 | − λ¯ |ci,1 | (max) |εi | m1 ΔF λ¯ |di,1 | < (4.56) |x1 (0)| (1 + 2 (max) ) ΔF λ¯ |ci,1 |

|x1 (0)| (max)

ΔF def The previous inequality requires, in turn, that one sets |εi | > M = Qi,1 . 1 − θ1 |ai,1 | m1 (0)| |x def def i After these choices, and deﬁning θ2 = (1 − θ1 ) and 1 − (max) = gi (x(0)) > 0, ΔF i = 1, . . . , m, what remains of inequality (4.54) is: n|εi | M1 θ2 |εi ||ai,1 | + n|ci,1|g1 (x(0)) + ¯ − θ1 |ai,1 | + |εi ||ai,2 | m1 λ |x (0)| 2 + λ¯ (|c1,2 | + 2|d1,2|) (max) + λ¯ |di,2 | + n|ci,2| + n|di,2| ΔF ! n n−m (0)| |x j + ∑ λ¯ (|ci, j | + 2|di, j |) (max) + |εi ||ai, j | + ∑ [n (|ci, j | ΔF j=3 j=3 n−m+1 6 +|di, j |) + λ¯ |di, j | ≤ M1 |εi | ∑ m−1 j . j=2

Step 2. Following the lines of the previous step, consider j = 2 ( j even), deﬁne def |xi (0)| ξi (x(0)) = (max) and choose di,2 such that: ΔF |εi | (M1 μ2 − ν2 ) − n|ci,1|g1 (x(0)) − λ¯ |ci,2 |ξ2 (x(0)) λ¯ |di,2 | < 1 + 2ξ2(x(0))

(4.57)

n def def with μ2 = 1/m2 − n/(λ¯ m1 ) and ν2 = |ai,2 | + θ2 |ai,1 | − ¯ k θ1 |ai,1 |. As in the previλ ous step, consistency of condition (4.57) is always ensured if one sets: |εi | >

λ¯ |ci,2 |ξ2 (x(0)) + n|ci,1|g1 (x(0)) def = Qi,2 (M1 μ2 − ν2 )

4.2 Nonlinear Robust Controller Design via Sliding Modes

89

provided that M1 μ2 − ν2 > 0 (the numerator of Qi,2 is always positive). A possible strategy to ensure this latter condition is to select μ2 > 0 and ν2 < −λ¯ , corresponding to: nm2 def ¯ λ¯ > = λ1 ; m1

λ¯ < λ¯ 2 .

(4.58)

where λ¯ 2 is the positive solution of the equation: λ¯ 2 + (|ai,2 | + θ2 |ai,1 |)λ¯ − nθ1 |ai,1 | = 0. Again, what remains of (4.54) to be fulﬁlled in the previous steps is: |ci,1 | n|εi | n|ci,2 |g2 (x(0)) − n2 ¯ g1 (x(0)) + ¯ (M1 μ2 − ν2 ) λ λ (0)| |x 3 + λ¯ (|ci,3 | + 2|di,3|) (max) + |εi ||ai,3 | + λ¯ |di,3 | + n|ci,3| ΔF ! n (0)| |x j + n|di,3| + ∑ λ¯ (|ci, j | + 2|di, j |) (max) + |εi ||ai, j | ΔF j=4 +

n−m 5

∑

j=4

n−m+1 6 n (|ci, j | + n|di, j |) + λ¯ |di, j | ≤ M1 |εi | ∑ m−1 j

(4.59)

j=3

Step 3. The case j = 3 ( j odd) can be treated similarly, though some different choices are needed. As before, deﬁne def

μ3 =

1 n − μ2 ; m3 λ¯ k

nν2 def ν3 = − ¯ λ

(4.60)

and choose di,3 such that:

λ¯ |di,3 | < +

1 {|εi | (M1 μ3 − ν3 ) − λ¯ |ci,3 |ξ3 (x(0)) (1 + 2ξ3(x(0)))

n2 |ci,1 | g1 (x(0)) −n|ci,2 |g2 (x(0))} . λ¯

(4.61)

The previous inequality requires to set: |ci,1 | −n2 ¯ g1 (x(0)) + n|ci,2|g2 (x(0)) + λ¯ |ci,3 |ξi (x(0)) λ |εi | > (M1 μ3 − ν3 ) def

= Qi,3

and M1 μ3 − ν3 > 0, or equivalently suitable λ¯ 3 > 0.

(4.62) M1 n M1 θ1 |ai,1 | > ¯ k( + ¯ ) i.e. λ¯ > λ¯ 3 for a m3 λ m2 λ

90

4 Control Design Issues: Continuous-Time Plants

Step j. The above procedure can be generalized for any j = r ≤ n − m, but the cases j even and j odd need to be differentiated. Deﬁne:

μ1 =

1 ; m1

def

μr =

1 n − μr−1 ; mr λ¯ k

r = 2 . . . n − m;

n def def ν1 = θ1 |ai,1 |; ν2 = |ai,2 | + θ2 |ai,1 | − ¯ k ν1 λ ⎧ nν r−1 ⎪ ⎪ ⎨− λ¯ , 2 < r < n − m, r odd; def νr = 2 < r < n − m, n2 νr−2 r ⎪ ⎪ ⎩∑=r−1 |ai, | + ¯ 2 , r even. λ ⎧ n nνr−1 λ¯ |xn−m+i (0)| ⎪ ⎪ − ¯ + ∑ |ai, j | + , ⎪ ⎪ (max) ⎪ λ ⎪ Δ j=n−m ⎪ F ⎨ i f n − m odd; def νn−m = n ⎪ n2 νr−2 λ¯ |xn−m+i (0)| ⎪ ⎪ + |a | + , i, j ∑ ⎪ (max) ⎪ λ¯ 2 ⎪ Δ j=n−m ⎪ F ⎩ i f n − m even.

(4.63)

(4.64)

and choose di,r such that (k = r − − 1): 1 · |εi | [M1 μr − νr ] − λ¯ |ci,r |ξr (x(0)) 1 + 2ξr (x(0)) k r−1 1 k+1 + ∑ |ci, | (−n) g (x(0)) ¯ k λ =1

λ¯ |di,r | <

(4.65)

Again, consistency requires that: r−1

|εi | >

− ∑ |ci, | (−n) =1

k+1

k 1 g (x(0)) ¯ + λ¯ |ci,r |ξr (x(0)) λ (M1 μr − νr )

def

= Qi,r

and: M1 μr − νr > 0

(4.66)

This last inequality can be shown to imply (see Lemma A.1 in the Appendix) the following conditions: 0 < λ¯ < λ¯ r ,

f or r even λ¯ > λ¯ r > 0,

f or r odd

(4.67)

In Lemma A.1, reported in the Appendix, it will be shown that a suitable λ¯ satisfying (4.66) can always be found. Taking into account condition (4.65), the general inequality (4.54) reads:

4.2 Nonlinear Robust Controller Design via Sliding Modes r

+

∑

k+1

|εi | + ¯ (M1 μr − νr ) λ 6 5 λ¯ (|ci, j | + 2|di, j |) ξr (x(0)) + |εi ||ai, j |

− ∑ |ci, | (−n)k+2 g (x(0)) =1 n

91

1 λ¯ k

j=r+1

+

n−m

∑

j=r+1

5

n−m+1 6 n (|ci, j | + |di, j |) + λ¯ |di, j | ≤ M1 |εi | ∑ m−1 j j=r+1

|ε1 ||ai,r | i f r even − 0 i f r odd

(4.68)

Step n − m. To investigate the last step, consider r = n − m in the previous inequality, and recall that |di, j | = 0 for j > n − m. Moreover, |ci, j | = |εi | for j = n − m + i and |ci, j | = 0 otherwise. Considering (4.65), (4.63) (4.64), what remain of (4.54) |εi | is: n|ci,n−m | + n|di,n−m | < M1 |εi |m−1 − |ci,n−m |. n−m+1 , i.e. |di,n−m | < M1 n · mn−m+1 Note that this latter is the second condition imposed on |di,n−m |, and needs to be satisﬁed together with (4.65) for r = n − m. The last condition requires |εi | > n|ci,n−m |mn−m+1 def = Qi,n−m+1 . M1 Summing up, all components of the control input vi can be computed choosing parameter λ¯ satisfying conditions (4.66), (4.67), parameters |di, j |, j = 1, . . . , n − m, fulﬁlling conditions (4.56), (4.65), and parameter εi such that: def

|εi | > max {Qi,1 , Qi,2 , . . . , Qi,n−m+1 } = Qi

i = 1, . . . , m.

(4.69)

4.2.3 Some Remarks Theorem 4.6 has a semiglobal validity if the plant is assumed having all the eigenvalues with strictly negative real part, otherwise it holds for a compact set of initial states, known as null controllable region. Explicit characterization and description of such regions are available in the literature [15] for completely known plants, and one can expect that the recoverable region in the presence of uncertainties is a subset of the previous one. In the case of the proposed control technique, the shape of the null controllable region is also tied to the choice of the parameters εi , as discussed in the following. Some heuristic comments, obtained by simulation, have been added in the Results section addressing the recoverable region which can be dealt with the proposed time-varying controller. (max) on the tuning parameters of the Analyzing the dependence of the bound ΔF (max) control algorithm, it can be easily proved that the bound ΔF depends both on the ¯ parameter λ and on the parameters εi . Such dependence can be qualitatively studied

92

4 Control Design Issues: Continuous-Time Plants

considering the dynamics of the reduced order system (A.2) (see Appendix), and evaluating the behavior of the solutions of the differential equation. As far as the dependence on λ¯ is concerned, integrating the differential equation and taking the (max) λ¯ t) ), this worst case, what turns out is that ΔF (λ¯ ) depends on λ¯ as exp( 1−exp(− λ¯ ¯ showing that it tends to ∞ for λ tending to zero for any ﬁnite t, as one could expect. Following an analogous approach, it can be qualitatively found that the dependence (max) (max) of ΔF on ε is of the form ε1 . The described dependence ΔF ( ε1 ) implies that (max) (max) ∀ε > ε¯ it holds ΔF ( ε1 ) < ΔF ( ε1¯ ). It follows that, ﬁxed an a priori value ε¯ and (max) 1 ( ε¯ ),

accordingly ΔF

any ε produced by the algorithm is admissible due to the (max) 1 (ε )

inequality (4.69), since the corresponding ΔF

(max) 1 ( ε¯ ).

is smaller than ΔF

4.2.4 A Procedure for Determining the Coefﬁcients of the Sliding Surface The tuning of coefﬁcients di, j , j = 1, . . . , n − m, i = 1, . . . , m in (4.47) should be done following the lines of the proof of Theorem 4.6. Operatively, one need to compute ﬁrst the upper limit λ¯ M for λ¯ using plant coefﬁcients. Next, once the parameter γ has been ﬁxed, parameters m j , j = 1, . . . , n − m + 1 can be determined, and the lower bound λ¯ m can be computed. Finally, choosing an admissible λ¯ one can determine the parameters εi , i = 1, . . . , m and, accordingly, proper bounds for coefﬁcients of the D matrix can be obtained. Note that the parameter γ (see Lemma A.1) ﬁxed a priori could be inconsistent with the steps to follow, and could be necessary to increase it and repeat the procedure. This strategy can be expressed as a step-by-step procedure as follows: 1. Fix i = 1, and compute λ¯ M in (A.4) using λ¯ j with j even (only plant parameters are needed). 2. Set an arbitrary γ > 1, and ﬁx the largest odd mk , k < m − n + 1, and all even m j ’s according to mk = m j = n. 3. Starting from j = k, ﬁnd the bound mk−2,m on mk−2 ensuring λ¯ k < λ¯ M . 4. Set mk−2 > max{n, mk−2,m } 5. Proceeding in reverse order from the largest to the smallest odd mi , repeat the previous two steps. 6. Once all the mi ’s have been computed, determine λ¯ m . 7. Check if (A.7) is fulﬁlled. If this is not the case, increase γ and go back to step 3. The procedure is guaranteed to converge, since the increase of γ produces a corresponding linear increase of m1,m , m3,m , . . . , while any λ¯ j , j odd = k goes to √ ∞ as ( γ ). 8. Once a suitable γ has been found, choose λ¯ within the interval [λ¯ m λ¯ M ]. 9. Choose εi according to (4.69) 10. Finally, select di, j according to the inequalities (4.56), (4.57), etc. and iterate for i = 2, . . . , m.

4.2 Nonlinear Robust Controller Design via Sliding Modes

93

4.2.5 Practical Issues To ease the application of the suggested method, some hints about the practical application of the procedure will be given in the following: • A ﬁrst important choice consists in selecting εi large enough, in view of the presence of a lower bound only. With this choice, the feasibility range of all di, j can be arbitrarily widened (in other words, di, j can be almost arbitrarily ﬁxed as initial guesses). • In fact, since (CB)−1 tends to the zero matrix when εi tends to inﬁnity, the control activity can be partially reduced by increasing εi . • A further important issue is to choose initial guesses of the odd mi ’s (excluding the largest) not too larger than n. This makes easier the fulﬁllment of (A.7) and the ﬁnding of a bound for γ close to 1. • Since the parameter λ¯ M can be computed exactly using the plant parameters, n2 is one should choose initial guesses of the odd mi ’s such that the quantity m i signiﬁcantly lower than λ¯ M , in order to enlarge the feasibility interval for λ¯ .

4.2.6 Simulation Results The proposed control approach, based on sliding surface (4.47) and control law (4.48), has been applied by simulation to an unstable 2-inputs plant the form (4.40) with: n = 4, m = 2, a1,1 = 1; a1,2 = −0.2; a1,3 = 0.3; a1,4 = 0; a2,1 = a2,4 = 1; a2,2 = a2,3 = −1. The control input u feeding the plant is the output of a saturation device, with threshold M = 3. A disturbance term of the form d(t) = A sin(ω t) has been supposed to perturb the system, with |A| ≤ 0.1 and ω = 1. Following the procedure described in Section 4.2.4, having set ρ1 = ρ2 = 2, we found the following parameters: γ = 1.1, m1 = m#2 = m3 = 4. The matrix C has " 3000 20000 155340 0 been selected as C = while the entries of the ma1 1 0 600 trix D" have been designed # according to Theorem 4.6. Found maximal values are 0.0795 0.5084 0 0 D= 104 . The value of D used in simulations were obtained 5.1647 0.8541 0 0 scaling each entry by a factor −1/80. In order to avoid chattering, a boundary layer has been introduced [58]. Simulations have been performed with λ¯ = 3 and initial condition x(0) = [0.5 0.5 0 0]T . Note that no constraints are required between the (max) and the threshold M, as it happened in [40]. We could set bounding function ΔF (max) ΔF = 2.8 as needed, regardless the value of M. Results have been reported in Fig. 4.10, showing the ﬁrst two state variables, and in Fig. 4.11, which displays the available control inputs v. Performances of the proposed controller have been compared by simulation with a standard linear state feedback controller (assigning eigenvalues −0.1, −0.2, −0.3, −0.4). It was found that, in the presence of the disturbance, the set of initial

94

4 Control Design Issues: Continuous-Time Plants 2.5

2

1.5

1

0.5

0

−0.5

−1 0

50

100

150

time [s]

Fig. 4.10. State variable x1 (solid line) and x2 (dashed line) 4 3 2 1 0 −1 −2 −3 −4 0

20

40

60

80 time [s]

100

120

140

Fig. 4.11. Unavailable Control inputs u1 (solid line) and u2 (dashed line) with the threshold ±M

conditions from which the plant could be steered to the origin was larger for the time-varying controller. Just as an example, the initial condition x(0) = [1 0.65 0 0]T could be effectively managed by the proposed control technique (maintaining the settings described) but caused instability using the standard controller.

Chapter 5

Control Design Issues: Discrete-Time Plants

This chapter is devoted to the extension of the results reported in the previous Chapter 4 to the discrete-time framework. Needless to say, completely different proofs are required with respect to the continuous time case. As in the previous Chapter, a discrete-time linear n-dimensional plant without uncertainties is considered ﬁrst, and a design technique is proposed for ﬁnding a linear state feedback controller with the property of having non-increasing norm along the closed-loop system trajectories. In the presence of disturbances affecting the plant, we are again concerned on achieving ultimate boundedness of trajectories and ensuring disturbance rejection. In view of the widely recognized features of robustness, transient shaping and ease of application offered by the discrete-time counterpart of sliding modes [59] [56], often referred to as quasi-sliding modes, one should expect their use to be widespread also for ensuring transient performance requirements and robust stabilization in the case of discrete-time plants with saturating actuators, too. Nevertheless, very few results are available. In the vast literature addressing the stabilization problem for discrete-time linear systems subject to actuator saturation, two lines of research have been mostly pursued, as discussed in the present book. The ﬁrst line focusses on the the estimation, less conservative as possible, of the null controllable region, i.e. the set of state which can be driven towards the origin of the state space using saturating actuators [60], [61], [62], [63], [64]. This research thrust has been widely addressed in Part I. The other line of research focuses on the semi-global stabilization on the null controllable region using saturating actuators. In this latter framework, the problem has been completely studied for ANCBC plants, for which the null controllable region is the whole state space [10], [11], [65]. In particular, some results are available for general discrete-time systems about feedback laws achieving semi-global stabilization on the null controllable region [66], [67]. However, [68] showed that in general linear feedback can not achieve global stabilization for discrete-time unstable systems. The problem of disturbance rejection for linear systems subject to actuator saturation has been investigated also in the discrete time framework, mostly considering M.L. Corradini et al.: Control Systems with Saturating Inputs, LNCIS 424, pp. 95–124. c Springer-Verlag London Limited 2012 springerlink.com

96

5 Control Design Issues: Discrete-Time Plants

disturbances that are bounded in magnitude. Also, the problem of the robust semiglobal [15] and global [23] stabilization has been solved for planar continuous-time systems with two unstable open-loop poles, in the case when the constraints are posed on the inputs. A more general setting including input and state constraints has been studied in [69], where solvability conditions have been provided for semiglobal stabilization in the admissible set of systems subject to right-invertible and non-minimum phase constraints. Furthermore, it has been proved [70] that for a discrete-time linear system with saturating actuators, neither Lq semiglobal L p stabilization nor global or semiglobal external stabilization is possible whenever there is one controllable open-loop pole located strictly outside the unit circle. Finally, it has been shown [71] that for both continuous-time and discrete-time critically unstable linear systems with saturating actuators, one cannot simultaneously achieve the global internal stabilization (in the absence of disturbance) and the global ﬁnite-gain L p performance in the presence of disturbance whenever the external input (disturbance) is not input-additive. However, one can achieve the global internal stabilization and global L p stabilization (without ﬁnite-gain) for any p ∈ [1, ∞). Estimation methods of the domain of attraction in the uncertain case using a saturation-dependent Lyapunov function has been proposed in [64] [72]. Also, the enlargement of the domain of attraction by antiwindup compensation has been proposed [73]. In addition to simple stabilization, the issue of guaranteeing some performance requirements has been also addressed mostly using LMI’s for completely known discrete-time plants [74] [60], [67] and for uncertain plants [75], but with reference to particular classes of disturbances. These approaches basically convert constraints to the form of LMI’s, so that the eventual solution to the control problem is conditioned to the feasibility of a convex problem. The extension of the design technique presented in Chapter 4 to the discretetime framework presents some relevant features: i) no restrictions are needed in the plant structure; ii) bounded matched disturbances are considered; iii) robust practical semi-global stabilization on the null controllability region (as deﬁned in [10] [23]) can be achieved by means of a time-varying state feedback controller, derived imposing the achievement of a quasi-sliding motion onto a suitable timevarying sliding surface; iv) performing transient shaping is not subject to any condition and can be achieved simply by manipulating the dynamics imposed onto the quasi-sliding surface. It will be proved that a constructive systematic procedure exists for designing the surface as to guarantee the ultimately boundedness of plant trajectories in the presence of bounded matched uncertainties. Finally, results from experiments in a twin rotor system will be reported and discussed.

5.1 Invariant Strips and Linear Feedback Laws In this section we explain how to extend the previous results to discrete linear systems.

5.1 Invariant Strips and Linear Feedback Laws

97

We consider a n-dimensional controllable SISO linear discrete system having, without loss of generality, the following structure ⎧ ⎨ x(k + 1) = Ax(k) + bu(k) (5.1) ⎩ x(0) = x0 where

⎛

0 ⎜ .. ⎜ A=⎜ . ⎝0 a1

⎞ 0 .. ⎟ . ⎟ ⎟, ··· 0 1 ⎠ a2 · · · an 1 .. .

··· .. .

⎛ ⎞ 0 ⎜ .. ⎟ ⎜ ⎟ b = ⎜ . ⎟, ⎝0⎠ 1

{x(k)}k∈N ⊂ Rn is the state vector and {u(k)}k∈N ⊂ R is the input variable, which is subject to the saturation constraint u(k) = satM (v(k)), M > 0. Denote by {λ j } the eigenvalues of the matrix A. It will be supposed that |λ j | < 1

∀ 1 ≤ j < n, λn ∈ R with |λn | ≥ 1.

(5.2)

In the next result it will be discussed how to design a linear feedback such that the set of values satisfying the saturation constraint is invariant for the closed-loop system evolution. Lemma 5.1. Given the system (5.1) controlled by the linear feedback u = l, x with l ∈ Rn , the following propositions are equivalent: 1. |l, x(k)| ≤ M ⇒ |l, x(k + 1)| ≤ M, ∀k ∈ N, ∀M > 0; 2. l is an eigenvector of the matrix AT and the corresponding eigenvalue λ satisﬁes |λ + ln | ≤ 1, where ln is the nth component of l. Proof. Given M > 0, we consider the set ΩM = {x ∈ Rn : |l, x| ≤ M}; our aim is to construct l such that x(k) ∈ ΩM implies x(k + 1) ∈ ΩM , that is |l, Ax(k) + blT x(k)| ≤ M.

(5.3)

Rewriting the previous condition as |AT l, x(k) + ln l, x(k)| ≤ M. we see that the invariance of ΩM can be achieved if and only if l is an eigenvector of the matrix AT with a corresponding real eigenvalue μ such that |μ l, x(k) + ln l, x(k)| ≤ M. We can conclude that condition (5.3) holds if and only if we have |μ + ln | ≤ 1.

(5.4) ♦

98

5 Control Design Issues: Discrete-Time Plants

We can state the following theorem, which can be regarded as a discrete version of Theorem 4.2. Theorem 5.1. Given the system (5.1) satisfying condition (5.2), there exists a vector l ∈ Rn such that for any M > 0 1. ΩM = x ∈ Rn : |l, x| ≤ M is an invariant set for the solution {φ (k, x0 , l)}k∈N ; 2. if x0 ∈ ΩM we have limk→∞ φ (k, x0 , l) = 0 under the saturation constraint (2.2). Proof. Let us deﬁne l = σ l(n) , where l(n) is an eigenvector of the matrix AT associated to the eigenvalue λn and σ is a real parameter to be determined. Recalling result of Lemma 4.2 and since assumption (5.2) holds, we can obtain asymptotical stabilization of the system choosing σ such that (n)

|λn + σ ln | < 1.

(5.5)

Moreover, by Lemma 5.1 condition (5.5) is sufﬁcient also for ensuring invariance of the sub-level sets of the control norm |u|. ♦

5.2 Nonlinear Robust Controller Design via Quasi-sliding Modes 5.2.1 Problem Statement Consider the following time invariant, uncertain, discrete time, controllable multiinput plant given by: x(k + 1) = Ax(k) + B [sat(u(k)) + δ (k)]

(5.6)

where: x = [x1 (k) · · · xn (k)]T ∈ Rn is the state vector (assumed available for measurement), u(k) ∈ Rm is the control input, A ∈ Rn×n , B ∈ Rn×m are the state matrix and the input matrix respectively. The uncertain term δ (k) ∈ Rm represent matched disturbances affecting the system. Given the standard saturation function sat(·) with threshold M > 0 known, the i-th component of the vectorial function sat(·) : Rm → Rm is given by sat(ui ). Without loss of generality, the plant is given in a form obtained rearranging rows and columns of the controllable canonical form, which on the controllability indices of the system n j , j = 1 . . . m [54] [55]: " depends # 6T 5 A1 A= , B = 0(n−m)×m I , with A1 ∈ R(n−m)×n , A2 ∈ Rm×n , where the block A2 matrix A1 has elements ξi, j , i = 1, . . . , n − m, j = 1, . . . , n which are either 1 or 0, and ⎤ ⎡ a1,1 a1,2 . . . a1,n ⎦ ... A2 = ⎣ am,1 am,2 . . . am,n

5.2 Nonlinear Robust Controller Design via Quasi-sliding Modes

99

Assumption 5.1. The uncertain term δ (k) is bounded by a known constant, i.e. ||δ (k)|| ≤ ρ . Deﬁnition 5. Denote solutions of a general system x(k + 1) = f (x(k), k) as φ (k, k0 , x(0)) with initial condition x(0). Following [76], such solutions are deﬁned uniformly ultimately bounded (with bound B) if there exists a B > 0 and if corresponding to any α > 0 and for every k0 ∈ N, there exists a T = T (α ) > 0 (independent of k0 ) such that ||x(0)|| < α implies that ||φ (k, k0 , x(0))|| < B for all k ≥ k0 + T (α ). 6 6 5 5 Deﬁne the matrices C = Γ E ∈ Rm×n and D = Δ 0 ∈ Rm×n as ⎡ ⎤ ⎤ ⎡ d11 d12 . . . d1,n−m 0 0 c11 c12 . . . c1,n−m ε1 0 . . . 0 ⎦ ... ... ⎦ C = ⎣ ... ... D=⎣ dm,1 dm,2 . . . dm,n−m 0 0 cm,1 cm,2 . . . cm,n−m 0 . . . 0 εm (5.7) def k ¯ ¯ and C(k) = (C+ Dλ ) = [c¯i, j ]. Consider the following time-varying sliding surface, with |λ¯ | < 1: , + ¯ (5.8) s(x(k), x(0), k) =C(k) x(k) − x(0)λ¯ k = 0 It will be proved in the following that, for any choice of di, j 0, i = 1, m, j = 1, . . . , n − m, a quasi sliding motion onto the surface s(x(k), x(0), k) = 0 implies ultimate boundedness of state trajectories. What motivates the introduction of the vanishing term Dλ¯ k x with respect to standard surfaces is the need of modulating the control input in order to cope with the saturation limitation. The matrix C, as usual, is chosen as to assign the eigenvalues of the reduced order system (see proof of Theorem 5.3 in Section 5.2.4). Recall that the robust stabilization of the reduced order system can be achieved in the presence of matched disturbances [50].

5.2.2 The Control Law For the surface (5.8), the control input ensuring the achievement of a quasi sliding motion is [56]: , + ¯ + 1)B]−1 CAx(k) + Dλ¯ k+1Ax(k) − C(k ¯ + 1)x(0)λ¯ k+1 + u(n) (k) u(k) = −[C(k

+ , ¯ + 1)x(0)λ¯ k+1 + u(n) (k) (5.9) Γ + Δ λ¯ k+1 A1 + EA2 x(k) − C(k = −E−1 with (n) ui (k)

=

−θi (|si (k)| − ρ ) i f |si (k)| ≥ ρ 0 i f |si (k)| < ρ

|θi | ≤ 1 i = 1, . . . , m

(5.10)

100

5 Control Design Issues: Discrete-Time Plants

where, with some abuse of notation, the i-th component of the variable s(x(k), x(0), k) has been denoted by si (k). Control law (5.9)-(5.10) is obtained solv(n) ing s(k + 1) = 0 in nominal conditions, and then choosing ui (k) in order to impose ||s(k + 1)|| < ||s(k)|| [56]. The following Proposition is straightforward, in view of the absence of the reaching phase. Proposition 5.1. It is given the uncertain system (5.6) driven by the feedback controller (5.9) under Assumption 5.1 and the assumption that M > ρ . For any bounded set S of initial conditions S = {x(0) : ||x(0)|| ≤ ζ ; ζ > 0} belonging to the null (max) ∈ R+ , depending on the chocontrollability region, there exists a constant ΔF sen set S and on the parameter 0 < |λ | < 1, such that supx(0)∈S ,0<|λ¯ |<1 ||x(k)|| ≤ (max)

ΔF

, ∀k ≥ 0.

The fact that the initial conditions are assumed to belong to the null controllability region in Proposition 5.1 ensures that the state can be driven to the origin using bounded control inputs, this ensuring boundedness of the state of the saturated plant. (max)

Remark 5.1. The bound ΔF , which always exists for initial conditions belonging to the null controllability region, can be estimated exploiting available boundedness results for closed loop trajectories (see Section IV, eq. (5.40)). It should be noticed (max) that it is not necessary to achieve an accurate estimate of ΔF , since even a rough overestimated upper bound of trajectories can be enough for the determination of the coefﬁcients of the time varying sliding surface. The constraint induced by saturation requires: |ui (k)| ≤ M

∀k ≥ 0,

i = 1, . . . , m

(5.11)

Taking the worst case and considering (5.9) it follows that the condition (5.11) can be rewritten as: n−m n |εi ui (k)| = λ¯ k+1 ∑ ci, j + di, j λ¯ k+1 x j (0) + εi xn−m+i (0)λ¯ k+1 − εi ∑ ai, j x j (k)+ p=1 j=1 (5.12)

n−m n ¯ εi | − ∑ c¯i, j (k) ∑ ξ j, x (k) − θi |si (k)| ≤ M| p=1 =1

(5.13)

def where c¯i, j (k) = ci, j + di, j λ¯ k , j = 1, . . . , n − m, M¯ = M − |θi |ρ . Considering the expression of |si (k)|, and taking again the worst case, one has: ! n−m n n−m n−m ¯ εi |ai, j | + λ + 2θi + γi |ci, j | + 2λ¯ |di, j | + 2 |di, j | ≤ M1 εi (5.14)

∑

j=1

∑

p=1

∑

p=1

∑

p=1

5.2 Nonlinear Robust Controller Design via Quasi-sliding Modes

with M1 =

M¯ (max) ΔF

101

, σ1 = λ¯ + 1, γi = σ1 + 2θi . The following Theorem provides a

robust stabilizing controller designed as to fulﬁll the constraint |ui (k)| ≤ M associated to saturation. It will be given under the standard assumption that the plant is ANCBC (i.e., the eigenvalues of A are on or inside the unit circle), which is necessary for semiglobal stabilization. Without such an assumption on the open-loop eigenvalues, only local results are possible. Theorem 5.2. It is given the uncertain system (5.6), assumed ANCBC, preceded by the saturating device (4.41), under the assumption that M¯ > 0 and Assumption 5.1. For any given x(0) belonging to the null controllability region, proper coefﬁcients di, j , i = 1, . . . , m, j = 1, . . . , n − m and a suitable 0 < |λ¯ | < 1 can always be found such that the feedback controller (5.9) guarantees that plant trajectories are semiglobally uniformly ultimately bounded. Proof. The proof is constructive, and consists in n − m steps, providing n − m chained inequalities, for each i-th component of u(k). As it will be clear soon, the construction is systematic, since for the i-th component the same development is applied to all the terms with subscript j at the step j-th. Moreover, odd and even steps have to be treated differently. Deﬁne m j > 0, j = 1, . . . , m such that n−m

∑ m−1 j ≤ 1.

(5.15)

p=1

In the following, the imposition of condition (5.14) will be performed taking suitable n − m ”portions” of (5.14) itself, and designing control coefﬁcients di, j , m j , εi , λ¯ involved in each derived inequality in order to ensure the simultaneous fulﬁllment of all of them. • Consider ﬁrst j = 1. A stronger inequality than (5.14) is the following: n−m 5 6 1 1 εi |ai,1 | + λ¯ + 2θi + γi |ci,1 | + 2λ¯ |di,1 | + 2|di,1| + Gi2 ≤ M1 εi +∑ m1 p=2 mp (5.16) with Gi, = εi ∑np= |ai, j | + ∑n−m p= (γi |ci, j | + 2σ1 |di, j |), = 2, . . . , n. Choose di,1 such that: 2|di,1 | < εi (M1 μ1 − ν1 ) − γi |ci,1 | + K1 (5.17) 1 , ν1 = |ai,1 | + λ¯ + 2θi , and a same constant K1 , to be determined, m1 has been added and subtracted in (5.16). For the r.h.s. term of condition (5.17) to M1 def , K1 > γi |ci,1 | = K1∗ and be positive, one can require: m1 > ν1 where μ1 =

|εi | <

K1 − γi |ci,1 | def = Qi,1 . ν1 − M1 μ1

(5.18)

102

5 Control Design Issues: Discrete-Time Plants

• Consider j = 2. Replacing (5.17) in (5.16) one gets: n−m 3 4 1 K1 σ1 + γi |ci,2 |− λ¯ |ci,1 | +2λ¯ |di,2 |+2|di,2 |+Gi3 ≤ εi (M1 μ2 − ν2 )+M1 εi ∑ p=3 m p

(5.19) being: def

μj =

1 − λ¯ μ j−1 ; mj

def ν j = |ai, j | − λ¯ ν j−1

(5.20)

Choose di,2 such that: 2|di,2 | < εi (M1 μ2 − ν2 ) + γi αi,2 + K2 − σ1 K1

(5.21)

with αi, = ∑j=1 (−1) j++( mod 2) · λ¯ j−1 |ci,(+1− j) |. Condition (5.21) requires:

εi <

K2 − σ1 K1 + γi αi,2 def = Qi,2 ν2 − M1 μ2

(5.22)

def

provided that K2 is large enough: K2 > σ1 K1 − γi αi,2 = K2∗ , and provided that M1 μ2 − ν2 < 0. To this purpose, one can impose μ2 < 0 and ν2 > 0, i.e.:

λ¯ < λ¯ 2 =

|ai,2 | ; |ai,1 | + 2θi + 1

m1 m2 > ¯ . λ

(5.23)

• Consider j = 3. Replacing (5.21) in (5.19), one gets:

β2 σ1 − γi αi,3 + 2λ¯ |di,3 | + 2|di,3| + Gi,4 ≤ εi (M1 μ3 − ν3 ) + M1 εi

n−m

1 (5.24) m p=4 p

∑

with β = ∑j=1 (−1)1− j++( mod 2) · λ¯ j−1 K(+1− j). Choose d3 such that: 2|di,3 | < εi (M1 μ3 − ν3 ) + γi αi,3 + K3 − σ1 β2 .

(5.25)

Differently from the previous case, where μ2 < 0 and ν2 > 0, the condition 1 M1 μ3 − ν3 < 0 has now to be imposed explicitly, i.e. λ¯ (ν2 − M1 μ2 ) < |ai,3 | − M m3 M1 and recalling the step 2, both members are positive. where, setting m3 > |a3 | Substituting the expressions (5.20) with j = 2, one gets ¯λ λ¯ ( M1 − |ai,1 | − λ¯ − 2θi ) + |ai,2| − M1 < |ai,3 | − M1 (5.26) m1 m2 m3 M1 M1 1 , a strongest condition is λ¯ (|ai,2 | − M m2 ) < |ai,3 | − m3 which ν1 M1 , provides, for m2 > |ai,2 |

and, since m1 >

5.2 Nonlinear Robust Controller Design via Quasi-sliding Modes

λ¯ <

1 |ai,3 | − M m3

|ai,2 | −

M1 m2

103

def

= λ¯ 3 > 0

(5.27)

Condition (5.25) requires:

εi <

K3 − σ1 β2 + γi αi,3 def = Q3 (ν3 − M1 μ3 )

(5.28)

def

provided that K3 is large enough: K3 > σ1 β2 − γi αi,3 = K3∗ . • The above procedure can be generalized for any j = r ≤ n − m. Choose di,r such that: 2|di,r | < εi (M1 μr − νr ) + γi αi,r + Kr − σ1 βr−1 .

(5.29)

Condition (5.29) requires:

εi <

Kr − σ1 βr−1 + γi αi,r def = Qi,r (νr − M1 μr )

(5.30)

and: M1 μr − νr < 0, which implies the following conditions: ⎧ |ar | def ¯ mr−1 ⎪ ¯ ⎪ r even ⎪ ⎪λ < νr−1 = λr ; mr > λ¯ ⎪ ⎪ ⎪ M ⎨ |ar | − m1r def ¯ λ¯ < M1 = λr ; r odd ⎪ |a | − ⎪ r−1 m ⎪ r−1 ⎪ ⎪ M1 ⎪ ⎪ ∀r ⎩mr > |ar | Kr >σ1 βr−1 − γi αi,r = Kr∗ def

(5.31)

r = 2, . . . , n − m − 1

(5.32)

To allow the correct ending of the procedure, the constant Kn−m is chosen equal to 0, and for r = n − m condition (5.32) becomes: σ1 βn−m−1 < γi αi,n−m . This condition can be easily imposed by using one of the redundant coefﬁcients of the matrix C, i.e.: 5 6 γi λ¯ |ci,n−m−1 | > σ1 βn−m−1 + γi |ci,n−m | − λ¯ 2αi,n−m−2 (5.33) • Finally, taking into account (5.29) for r = n − m and (5.32), the last condition to be fulﬁlled in order to guarantee (5.14) is the following:

λ¯ [εi (M1 μn−m − νn−m ) + γi αi,n−m − σ1 βn−m−1 ] +

n

∑

=n−m+1

|ai, | < 0

(5.34)

104

5 Control Design Issues: Discrete-Time Plants

i.e.

λ¯ σ1 Kn−m−1 >

n

∑

=n−m+1

|ai, | + λ¯ εi (M1 μn−m − νn−m ) + λ¯ γi αi,n−m + λ¯ 2 σ1 βn−m−2 (5.35)

which provides a further constraint on Kn−m−1 . Note that (5.35) is the second condition imposed on Kn−m−1 , and needs to be satisﬁed together with (5.29) for r = n − m − 1. ♦

5.2.3 A Systematic Procedure According to the proof of Theorem 5.2, the following systematic operative procedure can be given for the determination of the coefﬁcients di, j , i = 1, . . . , m, j = 1, . . . , n − m, of the matrix D in (5.8). 1. Select the matrix C as to assign the eigenvalues of the reduced order system (see proof of Theorem 5.3 in Section 5.2.4) 2. Set i = 1. 3. Set λ¯ < min{1, mini λ¯ i , i = 2, . . . , n − m}; 4. Fix m1 = n − m, and compute all the further m j , j = 2, . . . , n − m, according to (5.31). 5. Compute ν j and μ j , j = 1, . . . , n − m according to (5.20). (max) 6. Determine numerically an (even rough) estimate of the bounding constant ΔF , based on the set S of initial conditions and the assigned eigenvalues (see Theorem 5.3 in Section 5.2.4). 7. Compute all K j ’s, j = 1, . . . , n − m, according to the general expression: K j > max K ∗j = max σ1 β j−1 − γi αi, j , , i = 1, . . . , n − m − 1 (5.36) j

j

8. Compute all Qi, j ’s, j = 1, . . . , n − m, and select εi according to the general expression: K j − σ1 β j−1 + γi αi, j |εi | < min Qi, j = min (5.37) j j (ν j − M1 μ j ) 9. Finally, select di, j , j = 1, . . . , n − m, according to the general expression (5.29): 2|di, j | < εi (M1 μ j − ν j ) + γi αi, j + K j − σ1 β j−1 . 10. Repeat the whole procedure for i = 2, . . . , m.

(5.38)

5.2 Nonlinear Robust Controller Design via Quasi-sliding Modes

105

5.2.4 Stability Analysis and Transient Shaping In the following, the ultimate boundedness of the closed loop plant trajectories is proved. Theorem 5.3. Consider the plant (5.6) subject to the saturation constraint (4.41). The control input (5.9) ensures that closed loop state trajectories are ultimately bounded. Proof. The dynamics of the plant (5.6) driven by (5.9) are: ˜ x(k + 1) =Ax(k) + ϕ¯ (x(k), x(0), k)

(5.39)

4 ˜ = A − B E−1Γ A1 + A2 , ϕ¯ (x(k), x(0), k) = −BE−1 Δ λ¯ k+1 A1 x(k) + with A ¯ + 1)x(0)λ¯ k+1 + B u(n) (k) + δ (k) . Note that |eig(A)| ˜ < 1 due to a proper E−1 C(k k k ˜ || ≤ N λ , being λM the largest eigenvalue of A, ˜ and choice of Γ . Therefore ||A M where N > 0 is a suitable constant. ||Δ ||λ¯ Since ||ϕ¯ (x(k), x(0), k)|| ≤ δ1 ||x(k)|| + δ2 , with δ1 = max{ εi } , following the development described in [56], p. 114, one gets: 3

||x(k)|| ≤ N(λM + N δ1 )k ||x(0)|| + δ2N

1 − (λM + N δ1 )k 1 − (λM + N δ1 )

(5.40)

The arbitrarily assigned eigenvalue λM can be always selected such that λM + ||Δ ||N λ¯ ¯ max{εi } < 1, where λ and εi are chosen compatibly with the procedure described in the previous section and such that are ultimately bounded according to

||Δ ||N λ¯ max{εi }

lim ||x(k)|| ≤

k→∞

< 1. It follows that the state dynamics

δ2 N 1 − (λM + N δ1 )

(5.41)

♦ The proposed control algorithm features the possibility of improving the time domain performance of the closed loop system. Indeed, the extra parameters contained in the time-varying terms of the sliding surface can enhance the transient shaping capabilities inherent to sliding mode based controllers. The following result proves that the discussed controller, beside ensuring robustness to bounded matched uncertainties, allows also performance requirements to be satisﬁed. It also gives clues on the correct choice of the tuning parameters. Corollary 5.1. Consider the plant (5.6) subject to the saturation constraint (4.41) and fed by the control input (5.9). The following shaping requirement on the transient response xt (k): ||xt (k)|| ≤ Ct η k can be always fulﬁlled for arbitrary Ct , η ∈ R suitably choosing the parameters of the surface (5.8).

106

5 Control Design Issues: Discrete-Time Plants

Proof. From the previous Theorem one has " # N δ2 N δ2 ||x(k)|| ≤ N||x(0)|| − λk + 1 − λ2 2 1 − λ2

(5.42)

with λ2 = λM + N δ1 (with reference to quantities deﬁned in the proof of ˜ the" previous theorem). # Setting λ2 = η and assigning A such that N satisﬁes: δ2 N ||x(0)|| − ♦ = Ct , the statement follows. 1 − λ2 Remark 5.2. Differently from static state feedback control laws, the time-varying controller presented allows transient characteristics to be assigned by means of the time-varying part of the control law. This is clearly visible in Corollary 5.1, where it is explicitly given the bound of the state and is shown that it depends on λ¯ . Moreover, the proof of Corollary 5.1 shows the dependence of this bound from initial conditions. Remark 5.3. The proofs of Theorem 5.3 and Corollary 5.1 provide an operative ||Δ ||λ¯ way of choosing ”good” controller parameters. In fact, deﬁning δ1 = , λM max{εi } ˜ and choosing λ¯ such that the bound (5.31) and the as the largest eigenvalue of A, following inequality: λM + N δ1 < 1 are satisﬁed, it can be shown that: " # N δ2 N δ2 ||x(k)|| ≤ N||x(0)|| − λk + 1 − λ2 2 1 − λ2 being λ2 = λM + N δ1 and δ2 a positive constant (see proof of Theorem 5.2). Therefore, transient shaping can " # be obtained, suitably assigning the values of N δ2 λ2 and of N||x(0)|| − by means of matrix C. Note that the quantity 1 − λ2 # " N δ2 depends on the initial state, as expected. N||x(0)|| − 1 − λ2

5.2.5 A Benchmark Test Consider the example (without uncertainties) treated in [11] and improved in [60], described by: ⎡ ⎤ 0 1 0 0 ⎢ 0 0 1 0 ⎥ 5 6T ⎥ (5.43) A=⎢ ⎣ 0 0 0 1 ⎦ B= 0 0 0 1 . √ √ −1 2 2 −4 2 2

5.3 Experimental Data: Stabilization of a Twin Rotor System

107

x1 40 30 20 10 0 −10 −20 −30 −40 −50 −60 0

20

40

60

80

100

samples

Fig. 5.1. State variable x1 (k): plant (5.43) driven by controller (5.9)

with saturation threshold M = 4 initial conditions x(0) = 5[−10 10 10 10]T . Set6 ting the as follows: λ¯ = 0.05, C = −0.08 0.66 −1.5 1 , 5 proposed algorithm 6 D = −20 100 −5 0 , the results reported in Figs. 5.1-5.2 have been obtained, showing the time response of the state variable x1 (k) and the control signal v(k) respectively. Simulations prove that a remarkable improvement of the transient response with respect to [11], [60] is achieved, both for response amplitude (peak of nearly 700 in [60] vs. 50 in our case) and transient duration (nearly 150 samples in [60] vs. 40 samples in our case). This is due to the transient shaping capability offered by the time-varying sliding surface.

5.3 Experimental Data: Stabilization of a Twin Rotor System In this section, a further discrete-time control law is proposed along with experimental tests on a twin-rotor system. The controller is based on a time-varying sliding surface, different from that one proposed in the previous section, and it can be shown to be able to provide ﬁnite time plant stabilization for completely known systems. An extension to the case when matched bounded uncertainties affect the plant will be also considered, and a discrete-time sliding mode controller will be next proposed ensuring ultimate boundedness of state trajectories. Finally, experimental tests will be proposed relative to a twin-rotor system, in order to provide an experimental validation of the controller.

108

5 Control Design Issues: Discrete-Time Plants

Control variable v 4 3 2 1 0 −1 −2 −3 −4 0

20

40

60

80

100

samples

Fig. 5.2. Control input v(k): plant (5.43) driven by controller (5.9)

5.3.1 Problem Statement def ˆ ˆ Consider the following discrete-time, time invariant SISO plant S = {A, B} described by: ˆ x(k) + Bu(k) ˆ xˆ (k + 1) = Aˆ (5.44)

where: xˆ (k) = [xˆ1 (k) · · · xˆn (k)]T ∈ Rn is the state vector (assumed available for meaˆ ∈ Rn×n , B ˆ ∈ Rn are the state and surement), u(k) ∈ R is the control input, and A input distribution matrices, respectively. Assumption 5.2. The plant is controllable. The plant is supposed to be preceded by a saturating device u(k) = satM (v(k)) (2.2) with threshold M known. Under the controllability hypothesis, there exists a smooth change of coordinates: x(k) = T2 T1 xˆ (k) such that by T1 the plant is transformed in the controllability form, and T2 is such that system (5.44) becomes: x(k + 1) = Ax(k) + Bu(k)

(5.45)

5.3 Experimental Data: Stabilization of a Twin Rotor System

with:

" # A11 A12 −1 −1 ˆ A = T2 T1 AT1 T2 = = ⎤ A21 A22 ⎡ 0 α1 0 . . . 0 ⎥ ⎢ 0 0 α2 . . . 0 ⎢ ⎥ = ⎢. ⎥, ⎣ .. αn−1 ⎦ ... a1 a2 a3 . . . an " # 0 B = T2 T1 Bˆ = |α1 | < 1; . . . |αn−1 | < 1 b

109

(5.46)

(5.47)

Since |α1 | < 1; . . . |αn−1 | < 1, it is straightforward that, when all the eigenvalues of A are equal to zero, i.e. when a1 = a2 = · · · = an = 0, matrix A has norm less than 1.

5.3.2 A Finite Time Stabilizing Controller with Saturating Inputs The basic idea pursued in this section is to design a time-varying sliding surface such that the achievement of a quasi sliding motion on it can be ensured with saturating input. The associated sliding mode based controller is used to drive the plant state toward a suitable neighborhood of the origin, where a standard state feedback controller can be used to achieve ﬁnite time stability. First of all, a set of initial states can be easily found, starting from which the state vector can be directly steered to the origin using a standard linear state feedback controller. To this purpose, with reference to the transformed plant (5.45), consider the following control law: u(k) = Kx(k) (5.48) def

where K is such that the matrix A + BK = N is nilpotent. In the following, the symbol || · || will denote || · ||2 . Lemma 5.2. It is given the discrete-time system (5.45) preceded by the saturating device (2.2) under Assumption 5.2. The deadbeat controller (5.48) guarantees ﬁnite time stabilization with saturating inputs for any initial condition belonging to the set: M def ¯ =M (5.49) I = x(0) : ||x(0)|| ≤ ||K|| Such set is an invariant set. Proof. Consider x(0) as the initial condition, and apply the deadbeat controller u(k) = Kx(k). The saturation constraint provides: |Kx(k)| ≤ M

(5.50)

110

5 Control Design Issues: Discrete-Time Plants

Moreover, the following chain of inequalities is straightforward: |Kx(k)| ≤ ||K|| · ||x(k)|| ≤ ||K|| · ||Nk || · ||x(0)|| ≤ ||K|| · ||N||k · ||x(0)|| ≤ ||K|| · ||x(0)||

(5.51)

since ||N|| < 1. In fact, N = A + BK is nilpotent and therefore all the elements of its last row are equal to zero, due to transformation matrix T1 : as a consequence ||N|| < 1, due to transformation matrix T2 , i.e. to the choice of coefﬁcients α1 < 1 . . . αn < 1. Expression (5.51) implies that: ||x(0)|| ≤

M M ⇒ ||x(k)|| ≤ ||K|| ||K||

i.e. the set I is invariant. In other words, if the initial state belongs to the set I and fulﬁlls the saturation constraints, the entire dynamics satisfy the same constraint. Moreover, the deadbeat controller ensures stabilization in ﬁnite time. ♦ As already mentioned, a time varying sliding surface will be introduced. As well known [77] [78], a vector C = [C1 C2 ] ∈ Rn can be chosen such that, when a sliding motion is achieved on the following sliding surface: s(k) ˆ = Cx(k) = C1 x1 (k) + C2 x2 (k) = 0

(5.52)

the corresponding reduced order system has assigned stable eigenvalues, and, as a consequence, system (5.45) is stable, too. It will be assumed here to choose C1 and C2 such that the matrix C1 N1 = A11 − A12 C2 has stable eigenvalues, and that, without loss of generality, C2 > 0. Starting from the classical sliding surface (5.52), always with reference to the transformed plant (5.45), the following time varying sliding surface can be introduced: s(k) = Cx(k) − λ k CAx(k − 1) = 0

(5.53)

where 0 < λ < 1 is a design parameter. It is straightforward that when s(k) = 0, the system is asymptotically stable, since for k → ∞ surface (5.53) tends to surface (5.52). The equivalent control ensuring the achievement of a sliding motion on (5.53) can be obtained imposing the condition s(k + 1) = 0, i.e.: ueq (k) = −(CB)−1 CAx(k)(1 − λ k+1)

(5.54)

After the application of the controller (5.54), the closed loop system for k ≥ 1 is described by: + , x(k + 1) = A − (1 − λ k+1)B(CB)−1 CA x(k) = F(λ , k + 1)x(k)

(5.55)

5.3 Experimental Data: Stabilization of a Twin Rotor System

111

Due to the choice of C and λ in (5.53), the closed loop system (5.55) is asymptotically stable. The following result can be proved: Theorem 5.4. It is given the discrete-time system (5.45) preceded by the saturating device (2.2) under Assumption 5.2. The controller (5.54) guarantees that any initial condition belonging to the set: J =

x(0) ∈

k−1

∏ Q(λ , k − j)

I

(5.56)

j=0

being Q(λ , k − j) = F(λ , k − j)−1 , is driven to the set I in k steps without violating the saturation constraints. Therefore, ﬁnite time stabilization with saturating inputs is guaranteed for any initial condition belonging to the set J by coupling (5.48) and (5.54). Proof. The proof consists in showing that a procedure exists for selecting the parameter λ such that the control law (5.54) drives the state into the invariant set I in a ﬁnite number of steps k. Let’s consider x ∈ ∂ I , hence ||x|| = M. • Step 1 Consider the initial state xˆ 1 xˆ 1 := Q(λ , 1)x := [A − (1 − λ )B(CB)−1CA]−1 x. Imposing that |ueq (0)| ≤ M, one gets ||(1 − λ )Q(λ , 1)|| ≤

M ||K|| = ||(CB)−1 CA|| M||(CB)−1 CA||

(5.57)

Denoting by

λ1 = inf{λ ∈ (0, 1) : (5.57) is fulﬁlled}, one immediately gets that initial conditions belonging to the set Q(λ , 1)I can be driven to the set I simply setting 1 > λ > λ1 in (5.54), in fact x(1) = Aˆx1 + Bueq (0) = F(λ , 1)ˆx1 = = F(λ , 1)Q(λ , 1)x = x. • Step 2 Deﬁne xˆ 2 as

(5.58)

xˆ 2 := Q(λ , 1)Q(λ , 2)x

The saturation constraints is fulﬁlled if the parameter λ in (5.54) is chosen as 1 > λ > λ2 , with λ2 solution of 1. ||(1 − λ )Q(λ , 1)Q(λ , 2)|| < μ , 2. ||(1 − λ 2)Q(λ , 2)|| < μ , ||K|| having deﬁned μ = ||(CB) −1 CA|| . It follows that all initial conditions belonging to Q(λ , 1)Q(λ , 2)I can be driven to the set I in 2 steps.

112

5 Control Design Issues: Discrete-Time Plants

• Step k As before deﬁne

k−1

xˆ k :=

∏ Q(λ , k − j)

x

j=0

and imposing that k−r (1 − λ r ) ∏ Q(λ , k − j) < μ j=0

∀r ≤ k

(5.59)

one gets that, denoting

λk := inf{λ ∈ (0, 1) : (5.59) is fulﬁlled}, and choosing 1 > λ > λk in (5.54), any initial condition belonging to the set J can be driven to the set I in k steps. Therefore the statement follows. ♦

5.3.3 Presence of Bounded Uncertainties The case when matched bounded disturbances or uncertainties affect the plant will be now considered. As well known, such class of disturbances is traditionally dealt with by sliding mode control, though may be restrictive for some plants. Reference is made here to the plant: x(k"+ 1) = Ax(k) # + B"[u(k) # + d(k)] = A11 A12 0 = x(k) + [u(k) + d(k)] A21 A22 b

(5.60)

under the following assumption: Assumption 5.3. The uncertain term d(k) is such that: |d(k)| ≤ ρ , being ρ a known constant. Moreover, ρ is such that

ρ<

M . n||B|| · ||K||

It is straightforward to verify that Lemma 5.2 can be extended to cope with the uncertain plant (5.60) as follows: Lemma 5.3. It is given the discrete-time system (5.60) preceded by the saturating device (2.2) under Assumptions 5.2, 5.3. The deadbeat controller (5.48) guarantees that for any initial condition belonging to the set:

5.3 Experimental Data: Stabilization of a Twin Rotor System

113

M def ∗ −L = M Iρ = x(0) : ||x(0)|| ≤ ||K||

(5.61)

with L = nρ ||B||, ultimate boundedness of the state trajectories is ensured according to (5.62) lim ||x(k)|| ≤ L k→∞

Following the lines of the previous section, the following result can be stated. Theorem 5.5. It is given the discrete-time system (5.60) preceded by the saturating device (2.2) under Assumptions 5.2, 5.3. For perturbing terms d(k) bounded by a constant satisfying: M M , ρ < min . (5.63) 2n||K|| · ||B|| nρ ||A|| the control law u(k) = ueq (k) + v(k), with ueq (k) given by (5.54) and v(k) of the form v(k) = ⎧ ⎨0 ⎩

(5.64) if k = 0

¯ λ , k + 1)(x(k) − ∏kj=1 F(λ , j)x(0)) if k ≥ 1 −DF(

¯ = [0 0 · · · 0 1 ], guarantees that any initial condition belonging to the set: with D b Jρ =

x(0) ∈

k−1

∏ Q(λ , k − j)

Iρ

(5.65)

j=0

is driven to the set Iρ in k steps without violating the saturation constraints. Therefore, ultimate boundedness of state trajectories is guaranteed for any initial condition belonging to the set Jρ by coupling (5.48), (5.54) and (5.64). Proof. The proof consists in showing that a procedure exists for selecting the parameter λ such that the control laws (5.54) and (5.64) drive the state into the invariant set Iρ in a ﬁnite number of steps k. Noticing that ¯ ¯ (I − BD)F( λ , 1) = (I − BD)A = N,

114

5 Control Design Issues: Discrete-Time Plants

since Nn = 0, the following expressions can be easily derived: x(1) = F(λ , 1)x(0) + Bd(0) x(2) = F(λ , 2) (F(λ , 1)x(0)) + Bd(0)) + Bv(1) + Bd(1) = F(λ , 2) (F(λ , 1)x(0) + Bd(0)) ¯ λ , 2)Bd(0) + Bd(1) − BDF( = F(λ , 2)F(λ , 1)x(0) + NBd(0) + Bd(1) x(3) = F(λ , 3)F(λ , 2)F(λ , 1)x(0) + F(λ , 3)(Bd(1) + NBd(0)) + Bd(2) + Bv(2) = F(λ , 3)F(λ , 2)F(λ )x(0) + Bd(2) ¯ λ , 3)(Bd(1) + NBd(0)) + (I − BD)F( = F(λ , 3)F(λ , 2)F(λ )x(0) + Bd(2) + NBd(1) + N2Bd(0) .. . k

n−1

j=1

i=0

x(k) = ∏ F(λ , j)x(0) + ∑ Ni Bd(k − i) Moreover, the control input (5.64) fulﬁlls the following inequality n−1 ¯ |v(k)| = DF(λ , k + 1) ∑ Ni Bd(k − i − 1) i=0 n−1

¯ · sup ||F(λ , 1)|| · ∑ ||Ni B|| ≤ ρ ||D|| λ ∈(0,1)

i=0

¯ · ||B|| · ||A|| = nρ ||A||. ≤ nρ ||D|| and accounting for the saturation constraint requires that

ρ<

M n||A||

therefore the condition (5.63) is found coupling the previous inequality and Assumption 5.3. Following the same approach of the proof of Theorem 5.4, it is now enough to impose ∀r ≤ k k−r M − nρ ||A|| r (5.66) (1 − λ ) ∏ Q(λ , k − j) < −1 CA||(M ∗ − nρ ||B||) ||(CB) j=0

5.3 Experimental Data: Stabilization of a Twin Rotor System

115

Denoting by

λk = inf{λ ∈ (0, 1) : condition (5.66) is fulﬁlled} one can conclude that choosing 1 > λ > λk in (5.54) any initial condition belonging to the set Jρ can be driven to the set Iρ in k steps using a control law satisfying |ueq (k)| ≤ M − nρ ||A|| < M. ¯ it In fact, if x¯ ∈ Iρ , then for the initial condition x(0) := ∏k−1 j=0 Q(λ , k − j) x holds k

n−1

x(k) = ∏ F(λ , j)x(0) + ∑ Ni Bd(k − i) j=1

k

= ∏ F(λ , j) j=1

i=0

k−1

∏ Q(λ , k − j) j=0

n−1

x¯ + ∑ Ni Bd(k − i) i=0

n−1

= x¯ + ∑ Ni Bd(k − i) i=0

hence

||x(k)|| ≤ ||¯x|| + nρ ||B|| < M ∗ − nρ ||B|| + nρ ||B|| = M ∗ .

♦

5.3.4 Experimental Results Previous theoretical results have been experimentally validated on the twin rotor shown in Fig. 5.3. The plant has been built in our laboratory for educational purposes, and is constituted of two metal arms: the ﬁrst is locked to the ground, while the second is linked to the ﬁrst one, and can move with two degrees of freedom. The c ), movements are generated by two brushless D.C. motors (produced by AIRPAX placed on the two ends of the free arm. Moreover, two potentiometers are in charge of measuring the angular displacements of the free arm. The overall control system is depicted in Fig.5.4. The controller code is writc , running on a Personal Computer (PC). The PC ten in MATLAB/SIMULINK is equipped with a Plug-and-Play general purpose board, namely NI-PCI6024e, c , which is connected to MATLAB/ produced by NATIONAL INSTRUMENTS c SIMULINK by means of Real Time Workshop and Real Time Windows Target c MATLAB packages. The NI-PCI6024e allows data exchange between PC and the plant, but it is not directly connected to the potentiometers and to the motors. An interface board, made in our Lab, is in charge to ﬁlter and to adapt the following signals: • signals coming from the sensors, before passing them to the NI-PCI6024e board, and therefore to PC; • signals coming from the controller, i.e. from the PC through the NI-PCI6024e board, before passing them to the power board.

116

5 Control Design Issues: Discrete-Time Plants

Fig. 5.3. The twin rotor

The power board, made in our Lab, too, mounts a PWM modulator and drives the motors with suitable voltages, corresponding to the control actions produced by c . The maximum range that the power the control law implemented in SIMULINK board can supply is ±12V . Nevertheless, the safer voltage saturation limit of ±7V was chosen, because of the problems we encountered during the testing phase of the overall control system. Indeed sudden changes in the control variables caused damages to the boards (microchips and capacitors, for example), and, more seldom, a risk occurred to burn the motors. Finally, note that the angular velocities, required by the state feedback control law, were obtained by ﬁltering and differentiating the signals coming from the potentiometer. The chosen sampling time was Tc = 0.05 s. The mathematical model of the plant can be derived using well known theoretical physics results. Making reference to Fig.5.5, and introducing the state vec5 6T 5 6T tor x = x1 x2 x3 x4 = θ φ θ˙ φ˙ , where θ and φ are the pitch and yaw angles, respectively, the twin rotor is described by the following nonlinear equation x˙ = f(x) + h(x)u, with:

5.3 Experimental Data: Stabilization of a Twin Rotor System

117

Fig. 5.4. The twin rotor control scheme

⎡

x3 ⎢ x 4 ⎢ ⎢ −mglc cosx1 − JL x2 cos x1 sin x1 − αθ x3 4 f(x) = ⎢ ⎢ JL ⎢ ⎣ 2JL x3 x4 cos x1 sin x1 − αφ x4 (cos2 x1 JL + JA) ⎤ 0 0 ⎥ ⎢ 0 0 ⎥ ⎢ p2 l1 p1 ⎥ ⎢ h(x) = ⎢ ⎥ ⎥ ⎢ JL JL ⎦ ⎣ p3 cos(x1 ) l2 p4 cosx1 (cos2 x1 JL + JA ) (cos2 x1 JL + JA)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(5.67)

⎡

where

(5.68)

6T 5 • u = u1 u2 is the input vector, i.e. the voltages of the two motors driving the twin-rotor system; • JL and JA are the inertia moments of the free arm and of locked one, respectively; • lc is the centre of gravity of the free arm; • l1 and l2 are the distances between the ends and the centre of the free arm; • g is the gravity acceleration; • αφ and αθ are the damper coefﬁcients for angles φ and θ , respectively;

118

5 Control Design Issues: Discrete-Time Plants

Fig. 5.5. The twin rotor model

• p1 , p2 , p3 , p4 are suitable coefﬁcients, correlating the voltages and the moments supplied by the motors. The physical parameters of the plant, described in Fig.5.5, are reported in Tab.5.1. Note that all the numerical values have been found experimentally. Table 5.1. Physical parameters of the twin rotor Parameter JL JA m lc l1 l2 g αφ αθ p1 p2 p3 p4

Value 0.1087 0 1.115 0.0033 0.375 0.375 9.81 0.05 0.05 0.08 0.005 0.005 0.08

Unit [kg · m2 ] [kg · m2 ] [Kg] [m] [m] [m] [m/s2 ] [N · m · s] [N · m · s] [kg · m · rad/(V · s2 )] [kg · m2 · rad/(V · s2 )] [kg · m2 · rad/(V · s2 )] [kg · m · rad/(V · s2 )]

5.3 Experimental Data: Stabilization of a Twin Rotor System

119

The non-linear model has been linearized with respect to the equilibrium point 6T 6T 5 xTe uTe = 0T4x1 1.24 −0.21 , obtaining the following continuous time linear time invariant state space representation:

5

d (Δ x) = AΔ Δ x + BΔ Δ u dt being Δ u = u − ue , Δ x = x − xe, and with: ⎡ ⎤ 001 0 ⎢0 0 0 ⎥ 1 ⎥ AΔ = ⎢ ⎣ 0 0 −0.46 0 ⎦ 000 −0.46

⎡

⎤ 0 0 ⎢0 ⎥ 0 ⎥ BΔ = ⎢ ⎣ 0.276 0.046 ⎦ 0.046 0.276

6T 5 Considering θ and φ as the system output, i.e. y = φ θ , the corresponding inputoutput transfer matrix is given by: ⎡ ⎤ 0.046 0.276 ⎢ 2 + 0.46s s2 + 0.46s ⎥ F(s) = ⎣ s 0.046 0.276 ⎦ 2 2 s + 0.46s s + 0.46s Since F(s) is a diagonally dominant matrix, the coupling terms between θ and φ dynamics have been neglected in the linearised plant. The twin rotor has been considered as made of two independent SISO plants, characterized by the same transfer 0.276 . In order to apply the proposed control law, function F11 (s) = F22 (s) = 2 s + 0.46s the above transfer functions have been discretized with a sampling time Tc = 0.05 s, obtaining two equal subsystems of the form (5.44), with: " # " # 1 0.0494 0.0003 ˆ ˆ A= , B= . 0 0.9773 0.0136 ˆ and Bˆ have been transformed as in (5.45), (5.46), (5.47), obtaining: Successively, A " # " # 0 0.8 0 A= , B= . −1.22 1.98 1.25 Finally, two controllers 5 have been6built according 5 6 to the approach described in Theorem 5.4, with K = 0.98 −1.58 , C = 1 1.2 and λ = 0.999988. The saturation threshold of actuators is M = 7 V , as explained at the beginning of this section. With reference to the theoretical development presented in Sections 5.3.2, the sets of initial conditions from which the state can be steered to the set I is reported in Fig.5.6. Accordingly, the initial conditions have been chosen as −31 deg for the pitch angle θ and 24 deg for the yaw angle φ , with null initial velocities, in order to drive the state to the set I in just one sampling time.

120

5 Control Design Issues: Discrete-Time Plants

3

2

1

3

2

1

1

2

3

1

2

3

Fig. 5.6. Sets of initials conditions from which the state can be steered to the set I (the circle) in 1 (black), 2 (red), 3 (purple), 4 (green) steps

Two experiments have been performed. In the ﬁrst one, the control law based on (5.53), and given by (5.54), (5.48), has been implemented for each subsystem. In the second one, a standard equivalent control law based on (5.52) has been implemented for each subsystem, i.e. ueq (k) = −(CB)−1 CAx(k)

(5.69)

Results of the ﬁrst experiment have been reported in Figs.5.7-5.8, showing the experimental yaw and pitch angles, and Figs.5.9-5.10 displaying the control inputs u1 and u2 . The corresponding variables for the second experiment are reported in Figs.5.11-5.14. It can be noticed that when using control law (5.54), (5.48), the initial value of the control variables is always 0, regardless of the initial state, while using control law (5.69) the initial control effort depends on the initial state (the farther the initial state is from the origin, the larger will be the initial control effort). This fact can be seen comparing Figs.5.9-5.10 with Figs.5.13-5.14. However, after few time instants, control variables produced by (5.69) assume values comparable with signal produced by control law (5.54), (5.48). Anyway, the smaller initial control activity produced by (5.54), (5.48) is paid by the presence of some overshoots in the case of the behavior of the pitch angle, arising when controlled by the same controller (5.54), (5.48).

5.3 Experimental Data: Stabilization of a Twin Rotor System

Fig. 5.7. Pitch angle θ (control law (5.54), (5.48))

Fig. 5.8. Yaw angle φ (control law (5.54), (5.48))

121

122

5 Control Design Issues: Discrete-Time Plants

Fig. 5.9. Control input u1 (control law (5.54), (5.48))

Fig. 5.10. Control input u2 (control law (5.54), (5.48))

5.3 Experimental Data: Stabilization of a Twin Rotor System

Fig. 5.11. Pitch angle θ (control law (5.69))

Fig. 5.12. Yaw angle φ (control law (5.69))

123

124

5 Control Design Issues: Discrete-Time Plants

Fig. 5.13. Control input u1 (control law (5.69))

Fig. 5.14. Control input u2 (control law (5.69))

Appendix A

Support Material

A.1 Support Material for Chapters 2, 3 This section contains some results from convex analysis and measure theory which have been employed for the characterization of null controllable regions in Chapter 2 and Chapter 3. The interested readers may refer to the textbooks [33], [34] and [79] for the proofs of the presented results and for further details.

A.1.1 Tools from Convex Analysis We recall that, given k elements x1 , x2 , ..., xk ∈ Rn , the vector sum

λ1 x1 + λ2 x2 + · · · + λk xk is called a convex combination if the coefﬁcients λi are non-negative ∀i and satisfy k

∑ λi = 1.

i=1

The following simple proposition states a fundamental characterization of convex sets. Proposition A.1. A subset of Rn is convex if and only if it contains all the convex combinations of its elements. The convexity property is invariant under linear operations, namely • if C is convex, so it is every translated set C + γ , γ ∈ R; • if C is convex, the for any α ∈ R the rescaled set α C = {α x : x ∈ C} is convex too; • if C1 ,C2 are convex sets, then so is their sum C1 + C2 where

126

A Support Material

C1 + C2 = {z = x + y : x ∈ C1 , y ∈ C2 } . Moreover the following proposition can be easily proved. Proposition A.2. Let C1 and C2 be convex sets in Rm and R p , respectively. Then C1 × C2 = {z = (x, y) : x ∈ C1 , y ∈ C2 } is a convex set in Rm+p . Given a set S ∈ Rn there exists a unique smallest afﬁne set A containing S; this is called the afﬁne hull of the set S and it is denoted by A = aff(S). The dimension of a convex set is deﬁned as the dimension of its afﬁne hull. Let us denote by D the n-dimensional closed unit ball D := {z ∈ Rn : ||z|| ≤ 1} . Given an arbitrary point x ∈ Rn and a non empty set A ⊂ Rn , for η > 0 we deﬁne the sets x + η D := {z ∈ Rn : ||z − x|| ≤ η } ; A + η D := {z ∈ Rn : ∃ a ∈ A : ||z − a|| ≤ η } =

[a + η D].

a∈A

The closure C and the interior part int(C) of a set C ⊂ Rn can be expressed in terms of the sets deﬁned above: C=

7

[C + η D]

η >0

int(C) = {x ∈ C : ∃ η > 0 : [x + η D] ⊂ C} The relative interior part of a convex set C, ri(C), is deﬁned as the interior part of the set C when it is regarded as a subset of its afﬁne hull aff(C). Denoting by C the closure of C, the set of inclusions ri(C) ⊂ C ⊂ C is straightforward to verify. Remark A.1. If C ⊂ Rn is a n-dimensional convex set, i.e. if aff(C) = Rn , then the relative interior part of C coincides with the standard interior part int(C). Theorem A.1. Let C be a convex set in Rn ; let x ∈ ri(C) and y ∈ C. Then

θ x + (1 − θ )y ∈ ri(C) for any 0 < θ ≤ 1.

A.1 Support Material for Chapters 2, 3

127

Proof. The proof is given for the case of a n-dimensional convex set; the general case can be deduced by the continuity of the projection operator. We must prove that θ x + (1 − θ )y + η D ⊂ C for some η > 0. Now, since y ∈ C, for any η > 0 we have

θ x + (1 − θ )y + η D ⊂ θ x + (1 − θ )[C + η D] + η D = θ [x + θ −1(2 − θ )η D] + (1 − θ )C. Since x ∈ int(C), we have [x + θ −1 (2 − θ )η D] ⊂ C for η sufﬁciently small and hence θ x + (1 − θ )y + η D ⊂ θ C + (1 − θ )C = C. ♦ Given an arbitrary set S ⊂ Rn , the smallest convex set C containing S is called the convex hull of S and it is denoted by C = Co(S). Proposition A.3. Given a set S ⊂ Rn , its convex hull C = Co(S) consists of all convex combinations of elements of S. The following theorem, which constitutes a fundamental result in convex analysis, states that there exists an upper bound for the number of elements involved in the convex combinations generating the convex hull of a given set. Theorem A.2 (Carath´eodory Theorem). Let S be any set of points in Rn , and consider the convex hull C = Co(S). Then x ∈ C if and only if x can be expressed as a convex combination of at most n + 1 points in S. Corollary A.1. Let {Ci }ki=1 be a collection of convex sets in Rn and let C be the convex hull of the union of elements in such collection, i.e. C = Co

k

Ci .

i=1

Then any point in C can be expressed as a convex combinations of at most s = min{k, n + 1} points, each belonging to a different set Ci . Proof. By the Carath´eodory Theorem, any point x ∈ C can be expressed as convex ( combination of at most n + 1 elements in ki=1 Ci : x = θ1 y1 + θ2 y2 + · · · + θn+1yn+1 . Suppose now that two points, say y1 , y2 for simplicity, belong to the same set C1 ; then, since C1 is a convex set, the term θ1 y1 + θ2 y2 can be expressed as θ z, with

θ = θ1 + θ2 ,

z=

θ1 θ2 x1 + x2 ∈ C1 . θ θ

128

A Support Material

The above procedure can be easily generalized to the other groups of points, this showing that the point x can be expressed as a convex combination involving only ♦ elements belonging to different sets Ci , i = 1, ..., k. Given a convex set C, a subset S ⊂ C is said to be an internal representation of C if C = Co(S). An fundamental internal representation of a convex set C is the one given by its extreme points. A face of a convex set C is a convex subset C ⊂ C such that every closed segment in C with a relative interior point in C has both endpoints in C . The zerodimensional faces are called extreme points. In particular an element x ∈ C is an extreme point if and only if the only way to express x as a convex combination x = θ y + (1 − θ )z with θ ∈ (0, 1) and y, z ∈ C is by taking y = z = x. Theorem A.3 (Representation Theorem). Let C be a closed bounded convex set and let S be the set of its extreme points. Then C ≡ Co(S).

A.1.2 Tools from Measure Theory and Functional Analysis A function s = s(x) on a metric space X whose range consists of ﬁnitely many points in (−∞, ∞) is called simple function. The general expression of a simple function is given by s(x) =

r

∑ αi χAi ,

j=1

where αi , i = 1, .., r are real constant coefﬁcients and Ai are disjoint measurable subsets satisfying the identity r

Ai = X.

j=1

Theorem A.4. Let f : X → R be a measurable positive function. There exists a sequence of simple functions {sh }∞ h=1 such that (a) 0 ≤ s1 ≤ s2 ≤ · · · ≤ f (b)

limh→∞ sh (x) = f (x) for every x ∈ X.

Proof. For h = 1, 2, ..., and for 1 ≤ i ≤ h2h , deﬁne the sets " i−1 i −1 Eh,i = f and Fh = f −1 ([h, ∞)) , 2h 2h

A.1 Support Material for Chapters 2, 3

and put

129

h2h

i−1 χEh,i + nχFh . h i=1 2

sn = ∑

(A.1)

It is easy to verify that the functions sh verify condition (a). On the other hand, for any ﬁxed x ∈ X, we have sh (x) ≥ f (x) − 2−h , ♦

provided that h is large enough; this proves (b).

Corollary A.2. Let Ω ⊂ Rn be a bounded closed set and f : Ω → R be a continuous function; then the simple functions sh deﬁned in (A.1) converge uniformly to f as h tends to inﬁnity. In particular one has

lim

h→∞ Ω

| f (x) − sh (x)| = 0.

The notion of simple function can be extended to vector-valued functions in a natural way; a vector-valued simple function is a function S : X → Rm assuming a ﬁnite number of values and its general expression is given below S(x) =

r

∑ a j χA j ,

j=1

(r

where j=1 A j = X, Ai ∩ A j = 0/ for i = j, and a j ∈ Rm . To conclude this overview of classical results from analysis, one may recall the general version of the implicit function theorem. Let Ω ⊂ Rn+m be an open set and let F = F(x1 , ..., xn , y1 , ..., ym ) : Ω → Rm be a differentiable application. We deﬁne the Jacobian matrices ⎡

∂ F1 ∂ x1 ∂ F2 ∂ x1

⎢ ∂F ⎢ =⎢ . ∂x ⎢ ⎣ ..

∂ F1 ∂ x2 ∂ F2 ∂ x2

∂ Fm ∂ x1

···

∂ F1 ∂ xn ∂ F2 ∂ xn

⎤

⎡

∂ F1 ∂ y1 ∂ F2 ∂ y1

⎢ ∂F ⎢ =⎢ . ∂y ⎢ ⎣ ..

⎥ ··· ⎥ ⎥ .. . . .. ⎥ , . . . ⎦ ∂ Fm ∂ Fm ∂ x · · · ∂ xn

∂ Fm ∂ y1

2

∂ F1 ∂ y2 ∂ F2 ∂ y2

···

∂ F1 ∂ ym ∂ F2 ∂ ym

⎤

⎥ ··· ⎥ ⎥ .. . . .. ⎥ . . . ⎦ . ∂ Fm ∂ Fm ∂ y · · · ∂ ym 2

Theorem A.5. If (x0 , y0 ) ∈ Ω is such that F(x0 , y0 ) = 0,

det

∂ F(x0 , y0 ) = 0 ∂y

then one can determine a neighboorod U of x0 in Rn and a neighborood V of y0 in Rm such that, for any x ∈ U there exists a unique y = f (x) ∈ V with F(x, y) = 0; moreover the function f : U → Rm is differentiable and veriﬁes

∂ f (x) = ∂x

∂ F(x, f (x)) ∂y

−1

·

∂ F(x, f (x)) . ∂x

130

A Support Material

A.2 Support Material for Chapter 4 This section contains some results needed in the proof of the main Theorem of Chapter 4. Proposition A.4. For any choice of di, j 0, i = 1, m, j = 1, . . . , n − m, constraining the plant (5.6) onto the surface s(x(k), x(0), k) = 0 implies plant asymptotical stabilization. Proof. The statement can be proved making use of a standard results (reported e.g. in [80], [76]). With reference to plant (5.6), partition the state variable x = [xT1 xT2 ]T according to the partitions of the matrices A,C,D. On the surface s(x(k), x(0), k) = 0 given by (5.8) one has + , ¯ x2 = C−1 −C1 (t)x1 + (C1 (t)x1 (0) + C2x2 (0))e(−λ t) 2 def

¯

where C1 (t) = (C1 + D1 e(−λ t) ). The dynamics of the plant restricted on the sliding surface are: ¯

¯

¯

x˙ 1 = Λ 1 x1 + Λ 2 e(−λ t) x1 + G1e(−λ t) + G2 e(−2λ t) = Λ (t)x1 + g(t)

(A.2)

where Λ 1 = A1,1 − A1,2 C−1 2 C1 has stable assigned eigenvalues by a suitdef

−1 able selection of the matrix C, Λ 2 = −A1,2 C−1 2 D1 , G1 = A1,2 (C2 C1 x1 (0) + def

def

x2 (0)), G2 = A1,2 C−1 2 D1 x1 (0). It can be easily proved (see e.g. [80], [76], [81]) that the linear homogeneous system x˙ 1 = Λ (t)x1 is globally exponentially stable. It follows that the state transition matrix Φ (t, τ ) of (A.2) [80], for any initial time τ ∈ [0, ∞) and for any initial condition ξ ∈ Rm , is such that ||Φ (t, τ )ξ || ≤ K1 e−γ1 (t−τ ) ∀ξ , ∀t ≥ τ for suitable γ1 , K1 > 0. Recalling [80] that the unique solution ϕ (t, τ , ξ ) of (A.2) satisfying ϕ (τ , τ , ξ ) = ξ ¯ ¯ is given by ϕ (t, τ , ξ ) = Φ (t, τ )ξ + τt Φ (t, η )(G1 e(−λ η ) + G2 e(−2λ η ) )d η one has t ||ϕ (t, τ , ξ )|| ≤ ||Φ (t, τ )ξ ||+(||G1 ||+||G2 ||) τ ||Φ (t, η )||d η ≤ K2 e−γ2 (t−τ ) for suitable K2 , γ2 > 0. This shows that the plant (5.6) restricted onto (5.8) is globally exponentially stable. ♦ Lemma A.1. A positive constant λ¯ can always be found such that (5.15) and (4.66) are simultaneously satisﬁed. def

Proof. According to the proof of the Theorem 5.2, the fulﬁllment of (4.66) will be imposed as follows: i) for j > 1 even, it will be imposed simultaneously ν j ≤ −λ¯ and μ j > 0; ii) for j > 1 odd, the condition (4.66) will be imposed explicitly. A suitable constant λ¯ > 0 can be always found according to the procedure given below. • It is straightforward to prove that the condition μ j > 0 for j > 1 is fulﬁlled if: 1 n − μr−1 > 0 mr λ¯

⇒

λ¯ = nγ j m j μ j−1 ,

γj > 1

(A.3)

A.2 Support Material for Chapter 4

131

• For j = 2, consider to impose the condition ν2 ≤ −λ¯ and μ2 > 0. The former inequality is satisﬁed for 0 ≤ λ¯ ≤ λ¯ 2 , with λ¯ 2 deﬁned in (5.23). The latter corresponds to (A.3) for j = 2, and requires that λ¯ > λ¯ 1 as deﬁned in (5.23), i.e. 1 n n m2 = γ2 λ¯ m ; ⇒ μ2 = (γ2 − 1) λ¯ m for a suitable γ2 > 1. 1

1

• For j = 3, the condition M1 μ3 − ν3 > 0 gives M1 nλ¯ (mγ2 −1) − |ai,2 | − θ2 |ai,1 | + 1 θ1 |ai,1 |n M1 λ¯ ¯ 2 > m3 n M1 nγ2 + θ1 |ai,1 | def ¯ 2 > 0. < , i.e. λ = λ 3 m3 M1 m1 λ¯ • For j = 4, the condition ν4 < −λ¯ , considering that ν2 < −λ¯ , gives |ai,3 | + |ai,4 | + 2

ν2 n2 λ¯ 2

2 = |ai,3 | + |ai,4 | − nλ¯ < −λ¯ ⇒ λ¯ < λ¯ 4 , while μ4 > 0 produces λ¯ > nμ3 m4 , 1 n m4 1 1 n 1 which is satisﬁed if λ¯ > n , i.e. > , i.e. = γ4 , γ4 > 1. m3 m4 m3 λ¯ m4 m3 λ¯ • For j = 5, the condition M1 μ5 − ν5 > 0, proceeding similarly to i = 3, is imposed n2 M1 (γ4 −1) 1 − νλ¯4 < M if one requires that m5 . Recalling that ν4 = |ai,3 | + |ai,4 | + λ¯ 2 m 3

n2 ν2 θ |a |n n2 M1 (γ4 −1) 1 + 1 λ¯ i,1 < M 4 m5 λ¯ 2 m3 ¯λ 2 , a stronger condition than the previous one is 1 and, using the inequality θ1 |a1,1 |n < λ¯ 2 M m3 coming from the step j = 3, one has def ¯ 2 2 5 λ¯ 2 > m m3 (γ4 n + 1) = λ5 > 0. All further odd j’s can be treated similarly. • For j = 6, the conditions μ6 > 0 and ν6 < 0, imposed similarly to j = 4, give m6 λ¯ > n m and λ¯ < λ¯ 6 , for a suitable λ¯ 6 > 0. All further even j’s can be treated 5 similarly. Summing up, λ¯ has to satisfy: def λ¯ < min λ¯ 2 , λ¯ 4 , . . . , = λ¯ M ; def λ¯ > max λ¯ 1 , λ¯ 3 , λ¯ 5 , . . . , = λ¯ m 1 n n 1 = γ2 ¯ ; = γ4 ¯ ...; m2 λ m1 m4 λ m3

(A.4)

γ = max{γ2 , γ4 , . . . } > 1

(A.5)

The value λ¯ M in (A.4), can be computed exactly, as depends only on plant parameters. To ensure that an admissible interval of values for λ¯ exists, one has to impose that λ1 < λ¯ M ; λ¯ 3 < λ¯ M ; λ¯ 5 < λ¯ M , . . . . Since λ¯ 1 , λ¯ 3 depend inversely on m1 , λ¯ 5 depends inversely on m3 and so on, the above conditions produce a lower bound on odd coefﬁcients m j , i.e. m1 > m1,m ; m3 > m3,m ; . . . m j > m j,m ; j odd, j = 1, . . . , n − m Note that for the largest index k = n − m + 1, the only requirement for the corresponding mn−m+1 is simply (5.15). It remains now to show that the constraint (5.15) is satisﬁed. To this purpose, choosing m j > n, j = 1, . . . , n − m + 1, then (5.15) is fulﬁlled (the converse is not true). Fix m j , for j odd and j ≤ n − m according to m1 > max{n, m1,m };

m3 > max{n, m3,m }; . . . ;

m j > max{n, m j,m };

j odd, j ≤ n − m

(A.6)

132

A Support Material

Recalling (A.5), the constraint (5.15) gives n λγ¯ m11 + m13 + . . . ≤ 1 − m11 − m13 − . . . γ m11 + m13 + . . . n . To ensure that λ¯ belongs to the feasibility interval i.e. λ¯ ≥ 1 − m11 − m13 − . . . λ¯ > λ¯ m , it is enough to impose λ¯ m 1 1 1 1 1 1 + + ···+ > − − ···− 1− m1 m3 mk nγ m1 m3 mk

i.e. nγ m11 + m13 + . . . > λ¯ m 1 − m11 − m13 . . . . It is enough to set: ⎫ ⎧ ⎨ λ¯ m 1 − m1 − m1 − . . . ⎬ 1 3 γ ≥ max 1, 1 1 ⎩ ⎭ n m1 + m3 + . . .

(A.7) ♦

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57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81.

Index

actuator saturation 3, 4, 5, 55, 95 admissible control 7, 9, 17, 18, 21, 29, 33, 35, 38, 40, 44, 71 ANCBC 4, 9, 95, 100 antistable 5, 9, 17, 22, 23, 31, 39, 41, 45, 48 anti-windup 3 asymptotically null controllable 4, 5 asymptotically stable 5, 56, 68, 110, 111 bang-bang control 6, 9, 26, 28 chattering 93 convex hull 17, 20, 22, 42, 51, 127 convex set 25, 46, 125, 126, 127, 128 disturbance rejection 5, 55, 95 domain of attraction 5, 64, 96 equilibrium 5, 119 exponentially unstable 5, 9, 55 extremal control 26, 28, 29, 38, 47 global stabilization 4, 95 integrator wind-up 3 invariant set 14, 55, 58, 64, 68, 98, 109, 111, 113 Jordan canonical form 8, 69, 81 level set 57, 63, 64, 65, 66, 68, 69, 77, 98 limit cycle 17 LMI 96 maximal region of attraction 9 examples 29, 48 null controllability 96, 100 null controllable region, continuous-time 20, 22 SISO 13, 17 MIMO 23, 25, 27, 29 example 29

null controllable region, discrete-time 20, 22 SISO 34, 36, 43 MIMO 45, 48 example 48 performance requirements 96, 105 practical stabilization 5 semi-global, problem 55 theorem 92, 101 example 94, 106 saturated control 13 saturation of actuator 4 saturation function 4 scalar 4 vector 4 semi-global practical stabilization problem 55 theorem 92, 101 example 94, 106 stability antistable 5, 9, 17, 22, 23, 31, 39, 41, 45, 48 asymptotically stable 5, 56, 68, 110, 111 state transition map 130 stabilization practical, problem 55 practical, theorem 92, 101 practical, example 94, 106 stabilizing controller, with saturating actuators finite-time 109 robust 87, 101 state-feedback 64 sliding surface 56, 85, 92, 99, 105 transient shaping 105 time reversed system 22 variable structure control 56

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Vol. 412: Fridman, L.; Moreno, J.; Iriarte, R.: Sliding Modes after the ﬁrst Decade of the 21st Century XXX p. 2011 [978-3-642-22163-7]

Vol. 401: Chesi, G.; Hashimoto, K. (Eds.): Visual Servoing via Advanced Numerical Methods 393 p. 2010 [978-1-84996-088-5]

Vol. 400: Tomás-Rodríguez, M.; Banks, S.P.: Linear, Time-varying Approximations to Nonlinear Dynamical Systems 298 p. 2010 [978-1-84996-100-4] Vol. 399: Edwards, C.; Lombaerts, T.; Smaili, H. (Eds.): Fault Tolerant Flight Control appro. 350 p. 2010 [978-3-642-11689-6] Vol. 398: Hara, S.; Ohta, Y.; Willems, J.C.; Hisaya, F. (Eds.): Perspectives in Mathematical System Theory, Control, and Signal Processing appro. 370 p. 2010 [978-3-540-93917-7] Vol. 397: Yang, H.; Jiang, B.; Cocquempot, V.: Fault Tolerant Control Design for Hybrid Systems 191 p. 2010 [978-3-642-10680-4] Vol. 396: Kozlowski, K. (Ed.): Robot Motion and Control 2009 475 p. 2009 [978-1-84882-984-8] Vol. 395: Talebi, H.A.; Abdollahi, F.; Patel, R.V.; Khorasani, K.: Neural Network-Based State Estimation of Nonlinear Systems appro. 175 p. 2010 [978-1-4419-1437-8] Vol. 394: Pipeleers, G.; Demeulenaere, B.; Swevers, J.: Optimal Linear Controller Design for Periodic Inputs 177 p. 2009 [978-1-84882-974-9] Vol. 393: Ghosh, B.K.; Martin, C.F.; Zhou, Y.: Emergent Problems in Nonlinear Systems and Control 285 p. 2009 [978-3-642-03626-2] Vol. 392: Bandyopadhyay, B.; Deepak, F.; Kim, K.-S.: Sliding Mode Control Using Novel Sliding Surfaces 137 p. 2009 [978-3-642-03447-3] Vol. 391: Khaki-Sedigh, A.; Moaveni, B.: Control Conﬁguration Selection for Multivariable Plants 232 p. 2009 [978-3-642-03192-2] Vol. 390: Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A.: Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems 197 p. 2009 [978-1-84882-780-6] Vol. 389: Bru, R.; Romero-Vivó, S. (Eds.): Positive Systems 398 p. 2009 [978-3-642-02893-9]

Vol. 388: Jacques Loiseau, J.; Michiels, W.; Niculescu, S-I.; Sipahi, R. (Eds.): Topics in Time Delay Systems 418 p. 2009 [978-3-642-02896-0] Vol. 387: Xia, Y.; Fu, M.; Shi, P.: Analysis and Synthesis of Dynamical Systems with Time-Delays 283 p. 2009 [978-3-642-02695-9] Vol. 386: Huang, D.; Nguang, S.K.: Robust Control for Uncertain Networked Control Systems with Random Delays 159 p. 2009 [978-1-84882-677-9] Vol. 385: Jungers, R.: The Joint Spectral Radius 144 p. 2009 [978-3-540-95979-3] Vol. 384: Magni, L.; Raimondo, D.M.; Allgöwer, F. (Eds.): Nonlinear Model Predictive Control 572 p. 2009 [978-3-642-01093-4] Vol. 383: Sobhani-Tehrani E.; Khorasani K.; Fault Diagnosis of Nonlinear Systems Using a Hybrid Approach 360 p. 2009 [978-0-387-92906-4] Vol. 382: Bartoszewicz A.; Nowacka-Leverton A.; Time-Varying Sliding Modes for Second and Third Order Systems 192 p. 2009 [978-3-540-92216-2] Vol. 381: Hirsch M.J.; Commander C.W.; Pardalos P.M.; Murphey R. (Eds.) Optimization and Cooperative Control Strategies: Proceedings of the 8th International Conference on Cooperative Control and Optimization 459 p. 2009 [978-3-540-88062-2] Vol. 380: Basin M. New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems 206 p. 2008 [978-3-540-70802-5] Vol. 379: Mellodge P.; Kachroo P.; Model Abstraction in Dynamical Systems: Application to Mobile Robot Control 116 p. 2008 [978-3-540-70792-9] Vol. 378: Femat R.; Solis-Perales G.; Robust Synchronization of Chaotic Systems Via Feedback 199 p. 2008 [978-3-540-69306-2] Vol. 377: Patan K. Artiﬁcial Neural Networks for the Modelling and Fault Diagnosis of Technical Processes 206 p. 2008 [978-3-540-79871-2]