Active Fault Tolerant Control Systems: Stochastic Analysis and Synthesis (Lecture Notes in Control and Information Sciences)

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

287

Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo

Mufeed M. Mahmoud, Jin Jiang, Youmin Zhang

Active Fault Tolerant Control Systems Stochastic Analysis and Synthesis With 45 Figures

13

Series Advisory Board A. Bensoussan · P. Fleming · M.J. Grimble · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · J.N. Tsitsiklis

Authors Prof. Mufeed M. Mahmoud University of Western Ontario Department of Electrical & Computer Engineering University of Massachusetts Lowell Lowell MA 01854 USA Prof. Jin Jiang Prof Youmin Zhang University of Western Ontario Department of Electrical & Computer Engineering Alexander Charles Spencer Engineering Science Building Richmond Str. 1155 London, Ontario N6A 5B9 Canada ISSN 0170-8643 ISBN 3-540-00318-5

Springer-Verlag Berlin Heidelberg New York

Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at . This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution act under German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2003 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Digital data supplied by author. Data-conversion by PTP-Berlin, Stefan Sossna e.K. Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper 62/3020Rw - 5 4 3 2 1 0

TO OUR WIVES, BELOVED CHILDREN AND DEAR PARENTS

Preface Modern technological systems rely on sophisticated control functions to meet increased performance requirements. For such systems, Fault Tolerant Control Systems (FTCSs) need to be developed. FTCSs can be broadly classied into passive and active. A Passive FTCS (PFTCS) can tolerate a predened set of faults while accomplishing its mission satisfactory without the need for control reconguration. Active FTCS (AFTCS), on the other hand, relies on a Fault Detection and Identication (FDI) process to monitor system performance, and to detect and isolate faults in the system. Accordingly, the control law is recongured on-line. The dynamic behavior of AFTCS can be modelled by Stochastic Dierential Equations (SDE), due to the fact that faults are random in nature, and the FDI decisions are non-deterministic. In general, SDE can be classied into two categories: SDE perturbed by white Gaussian noise (Ito dierential equations), and SDE whose coecients vary randomly with Markovian characteristics (hybrid systems). The dynamic behavior of an AFTCS belongs to the class of hybrid systems. It is hybrid because it combines both the Euclidean space for system dynamics and the discrete space for fault-induced changes. Stochastic stability of AFTCS is of prime importance. Substantial results for the stability of hybrid systems were obtained using the Lyapunov function approach and the supermartingale property. A major class of hybrid systems is Jump Linear Systems (JLS). In JLS, the random

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Preface

jumps in system parameters are represented by a nite state Markov chain representing the plant regime mode. The usual models of JLS assume the perfect knowledge of the plant regime, which implies that FDI algorithms detect faults instantaneously and always correctly. Unfortunately, this assumption is too idealistic in reality, since all FDI algorithms have non-zero detection delays and errors in detection. To relax this assumption, another class of hybrid systems has been de ned. This class of systems is known as Fault Tolerant Control Systems with Markovian Parameters. In FTCSMP, two random processes with dierent state spaces were de ned and used to characterize the behavior of AFTCS. The rst represents system component faults, and the second represents the FDI algorithm used for the control law recon guration. The main objective of this book is to study and to validate some important issues in real-time AFTCS through theoretical analysis and simulation. To achieve this objective, several FTCSMP models have been developed. These models take into consideration of practical aspects of the system to be controlled, performance deterioration in FDI algorithms, and limitations in recon gurable control laws. The rst topic considered in the book concerns with the practical issues of the system to be controlled. This involves: the inclusion of the environmental noises and multiple faults in dierent system components. These issues can be briey detailed as follows: For FTCSMP to be applicable in practical control systems, environmental noises must be taken into account. This vital issue was not studied adequately so far for FTCSMP. Hence, the eect of noise on the stability of FTCSMP is emphasized in this book. Both state-dependent and independent white Gaussian noises are considered. A method for modelling multiple component faults is presented. In practice, faults are random in size, time of occurrence, and locations (sensors, plant, or actuators). To relax the traditional assumption of a single component malfunction, the developed FTCSMP model allows the modelling of multiple faults at dierent locations

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IX

simultaneously. The
eld of FDI for dynamic systems has become an important topic of research. Many applications of qualitative and quantitative modeling and statistical signal processing techniques are being developed for complex engineering systems. Uniquely, in this book, the FDI algorithm is not assumed to operate perfectly. The impact of FDI imperfections, such as detection delays and errors in detection and identi
cation, on the stochastic stability of FTCSMP is studied in this book. Furthermore, the book addresses the issue of stabilizing FTCSMP with modelling constraints due to practical limitations. The
rst issue is actuator saturation. Valves are typical examples used in process industries. A control valve has a range of operation limited by being fully opened and fully closed. Unfortunately, the physical limitation of actuator saturation is unavoidable. If such a limit is not taken into account, a large overshoot in system response may be induced which could lead to an unstable closed-loop system. Stability conditions are derived and a procedure for stabilizing state-feedback control law is developed for FTCSMP driven by actuators with saturation. Another practical limitation is the inaccuracy of estimated system parameters. This inaccuracy arises due to modelling errors, environmental noises, statistical nature of the FDI algorithm, and the limited amount of time available for FDI algorithms to provide a reliable decision. As a result, FDI algorithms estimate system parameters with some uncertainties compared with the nominal parameters. This motivates the study and the design of a stabilizing controller for FTCSMP with parameter uncertainties. Finally, the book introduces several design approaches for fault tolerant control laws. The controller is
rst synthesized in noise-free environments, results are then extended to noisy environments. A Linear Quadratic Regulator (LQR) problem is formulated and is utilized as the main approach to obtain a fault tolerant control law. The LQR problem is solved

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Preface

using the matrix maximum principle for an equivalent deterministic cost function. Three scenarios have been considered. The
rst assumes that both the failure and FDI processes are accessible to the controller. In the second scenario, only the FDI process is accessible. The case where the FDI process itself is not able to provide any decision due to physical malfunction or excessive computational time is the third scenario. A computational algorithm is proposed to synthesize a fault tolerant control law. The design of a control law for FTCSMP in noisy environment is also considered in this book. Three types of noise are considered: state-dependent, control-dependent and purely additive white Gaussian noise. In particular, conditions for the existence of a control law for FTCSMP in the presence of noise are derived. The limiting behavior of the cost function, the Riccati-like and the covariance-like dierential equations is studied. The conditions that guarantee the
niteness of the cost function and the existence of steady-state solutions, for both Riccati-like and covariance-like dierential equations, are stated and veri
ed. A computational algorithm, to synthesize controller parameters, is constructed. The book adopts the Lyapunov second (direct) method to characterize the behavior of FTCSMP without explicit solution of the stochastic dierential equations. A prime focus of the book is the stochastic stability of FTCSMP. In particular, exponential stability in the mean square and almost sure asymptotic stability. The weak in
nitesimal operator and the supermartingale property of a stochastic Lyapunov function are employed to derive conditions for stochastic stability of FTCSMP. All theoretical results are validated by simulation examples. Researchers and industrial experts will appreciate the combination of practical issues and their mathematical treatment. Control engineers will bene
t from the results presented herein and may extend this work to dierent area of applications.

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Acknowledgments The rst author would like to thank the University of Western Ontario Faculty of Graduate Studies for providing Ontario Graduate Scholarship in Science and Technology (OGSST) and Special University Scholarship (SUS). He would also like to extend his thanks to the Ministry of Education-Ontario for providing Ontario Graduate Scholarship (OGS) throughout this research. The second author would like to express his appreciation towards the Natural Sciences and Engineering Research Council of Canada for funding this research.

Chapter Summary The book is organized as follows:

Chapter 1 presents a review of FTCS. The chapter de nes FTCS and classi es the existing design approaches into: PFTCS and AFTCS. The need for FTCS to increase reliability and automation level in modern engineering systems is also highlighted.

Chapter 2 provides the state-of-the-arts in AFTCS. A particular focus is on the major elements in AFTCS: Fault Detection and Identi cation (FDI) algorithms and Controller Recon guration Mechanisms (CRM).

Chapter 3 outlines the extension of deterministic stability concepts to stochastic systems briey. It de nes some stochastic terms and introduces the tools necessary to study the stability of sample solutions of stochastic systems. The chapter wraps up by highlighting some important stochastic de nitions and conditions.

Chapter 4 formulates the mathematical model of FTCSMP. This includes the dynamical system and both the failure and the FDI processes. The Markovian behavior of the failure and the FDI processes is also emphasized and justi ed. A method to calculate the conditional transition rates of the FDI process is also outlined.

Chapter 5 examines the stochastic stability of FTCSMP in noisy environment and in the case where the system to be controlled is subject to multiple failures in dierent

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components. Chapter 6 is devoted to imperfections of FDI algorithms. In particular, an analytical representation of detection delays and errors in detection is developed. The eect of these imperfections on the stability of FTCSMP is considered. Chapter 7 addresses some important practical limitations in modelling FTCSMP. It considers the case where the FTCSMP is driven by actuators with possible saturation. In addition, the chapter considers another important limitation in the performance of FDI algorithms which is uncertainties in the estimated system parameters. A quadratic upper bound is utilized and a stabilization algorithm is constructed. Chapter 8 presents a comprehensive design approach to fault tolerant control law. Here, an equivalent deterministic cost function is dened and minimized using matrix maximum principle. Both nite and innite time horizons are considered and a control law is designed in noise-free and noisy environments. Chapter 9 concludes the book by a few remarks and recommendations for future research.

Contents Dedication Preface Table of Contents List of Tables List of Figures Nomenclature

V VII XIII XVII XIX XXI

ÿ INTRODUCTION

ÿ.ÿ Fault Tolerant Control System ýFTCSü . . . ÿ.ÿ.ÿ Deûnition of FTCS . . . . . . . . . . ÿ.ÿ.ú Classiûcation of FTCS . . . . . . . . ÿ.ú Background and Motivations . . . . . . . . . ÿ.ø Advances in Fault Tolerant Control Systems ÿ.ù Scope of the Book . . . . . . . . . . . . . . . .

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þ ACTIVE FAULT TOLERANT CONTROL IN PROSPECTIVE ú.ÿ Introduction . . . . . . . . . . . . . . . . . ú.ú Faults in Dynamic Systems . . . . . . . . ú.ø Fault Detection and Identiûcation ýFDIü ú.ø.ÿ Approaches to FDI . . . . . . . . ú.ø.ú Diagnostic procedure of FDI . . . ú.ø.ø Performance measures in FDI . . ú.ù Control System Reconûguration . . . . . . ú.ó Applications of FTCS . . . . . . . . . . .

ý STOCHASTIC STABILITY

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ø.ÿ Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . úú ø.ú Deûnitions of Stochastic Stability . . . . . . . . . . . . . . . . . . . . . . úø ø.ø Conditions for Stochastic Stability . . . . . . . . . . . . . . . . . . . . . ú÷

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ÿ FTCS WITH MARKOVIAN PARAMETERS þFTCSMPý ÿ.ý Mathematical Models . . . . . . . . . . . . . . ÿ.ý.ý Dynamical system models . . . . . . . ÿ.ý.û Failure and FDI processes . . . . . . . ÿ.û Calculation of Transition Rates . . . . . . . . ÿ.ü The Uniqueness of FTCSMP Model . . . . . ÿ.ü.ý Errors in detection and identiøcation ÿ.ü.û Detection delays and false alarms . . ÿ.ÿ Stochastic Stability of FTCSMP . . . . . . . ÿ.ÿ.ý Supermartingales . . . . . . . . . . . ÿ.ÿ.û Stochastic Lyapunov functions . . . . ÿ.ÿ.ü The weak inønitesimal operator . . . ÿ.ÿ.ÿ Dynkinôs formula . . . . . . . . . . . ÿ.ÿ.ú Conditions for stochastic stability . .

ú STOCHASTIC STABILITY OF FTCSMP

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ú.ý Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ú.û Stochastic Stability of FTCSMP in the Presence of Noise . . . . . . . ú.û.ý Dynamical model of FTCSMP in the presence of noise . . . . ú.û.û Suòcient conditions for stochastic stability . . . . . . . . . . . ú.û.ü Necessary conditions for stochastic stability . . . . . . . . . . ú.û.ÿ A necessary and suòcient condition for exponential stability ú.û.ú A numerical example . . . . . . . . . . . . . . . . . . . . . . . . ú.ü Stability of FTCSMP with Multiple Failure Processes . . . . . . . . ú.ü.ý Dynamical model of FTCSMP with multiple failures . . . . . ú.ü.û Transition probabilities for failure and FDI processes . . . . . ú.ü.ü Stochastic stability . . . . . . . . . . . . . . . . . . . . . . . . . ú.ü.ÿ A necessary and suòcient condition for exponential stability ú.ü.ú Remarks and special cases . . . . . . . . . . . . . . . . . . . . . ú.ü.ó A numerical example . . . . . . . . . . . . . . . . . . . . . . . . ú.ÿ Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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üû

üý üý üü üú üù üù üù ü÷ üö üö ÿõ ÿý ÿü

ÿú

ÿú ÿó ÿù ÿö úü úó óý óù ó÷ ùõ ùõ ùû ùó ùö ÷û

ù PERFORMANCE AND STABILITY OF FTCSMP UNDER IMPERFECT FDI ÷ü

ó.ý Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ÷ü ó.û Eñects of Detection Delays on the Stability of FTCSMP . . . . . . . . ÷ÿ ó.û.ý Dynamical model of FTCSMP with detection delays . . . . . . ÷ú

Contents

ÿ.ý.ý Interpretation of detection delays in FDI process . . . . . . ÿ.ý.ú Exponentially distributed detection delays . . . . . . . . . . ÿ.ý.ù ÿ distributed detection delays . . . . . . . . . . . . . . . . . ÿ.ý.û A necessary and suøcient condition for stochastic stability ÿ.ý.ÿ A numerical example . . . . . . . . . . . . . . . . . . . . . . . ÿ.ú Eöects of Detection Errors on the Stability of FTCSMP . . . . . . ÿ.ú.õ A new FTCSMP model . . . . . . . . . . . . . . . . . . . . . ÿ.ú.ý The role of FDI in fault tolerant control law design . . . . . ÿ.ú.ú A necessary and suøcient condition for stochastic stability ÿ.ú.ù A numerical example . . . . . . . . . . . . . . . . . . . . . . . ÿ.ù Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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üû üÿ üÿ üü ü÷ ÷ý ÷ú ÷ù ÷ô ÷ô õóõ

ÿ FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES þýü ô.õ Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ô.ý Stability of FTCSMP Driven by Actuators with Saturation . . . . . ô.ý.õ Dynamical model of FTCSMP with actuator saturation limit ô.ý.ý Actuators with saturation limits . . . . . . . . . . . . . . . . . ô.ý.ú Exponential stability of FTCSSA . . . . . . . . . . . . . . . . ô.ý.ù A suøcient condition for exponential stability of FTCSSA . ô.ý.û Remarks and special cases . . . . . . . . . . . . . . . . . . . . . ô.ý.ÿ A numerical example . . . . . . . . . . . . . . . . . . . . . . . . ô.ú Stabilization of FTCSMP with Parameter Uncertainties . . . . . . . ô.ú.õ Dynamical model of FTCSMP with parameter uncertinities . ô.ú.ý The model of parameter uncertainties . . . . . . . . . . . . . ô.ú.ú Stabilization of uncertain FTCSMP . . . . . . . . . . . . . . . ô.ú.ù Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ô.ú.û A numerical example . . . . . . . . . . . . . . . . . . . . . . . . ô.ù Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

û SYNTHESIS OF FAULT TOLERANT CONTROL LAWS ü.õ Introduction . . . . . . . . . . . . . . . . . . . . . . . ü.ý Design Approaches to Fault Tolerant Control Laws ü.ý.õ The problem statement . . . . . . . . . . . . ü.ý.ý A control law on ònite time horizon . . . . . ü.ý.ú Control laws on inònite time horizons . . . . ü.ý.ù A numerical example . . . . . . . . . . . . . .

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õóú õóù õóû õóû õóÿ õó÷ õõú õõ÷ õýù õýÿ õýü õý÷ õúü õùõ õùÿ

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õùü õûó õûó õûú õûü õÿú

XVI

Contents

ÿ.ý A Fault Tolerant Control Law in Noisy Environment ÿ.ý.ü The problem statement . . . . . . . . . . . . . ÿ.ý.ù A control law on ønite time horizon . . . . . . ÿ.ý.ý A control law on inønite time horizon . . . . ÿ.ý.õ A numerical example . . . . . . . . . . . . . . . ÿ.õ Chapter Summary . . . . . . . . . . . . . . . . . . . . .

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üûú üûú ü÷ö ü÷ý üÿö üÿü

ÿ EPILOGUE

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BIBLIOGRAPHY INDEX

þýý ûúù

ú.ü Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . üÿõ

List of Tables 4.1 Conditional transition rates for FDI processes. . . . . . . . . . . . . . . 35 5.1 Stability parameters for dierent independent noise intensities. . . . . 64 5.2 Stability with increased intensities of dependent noise. . . . . . . . . . 66 7.1 7.2 7.3 7.4 7.5 7.6

Positive-de nite solutions of Riccati-like equations. . . . . . . . . . Constants in the sector a,1]. . . . . . . . . . . . . . . . . . . . . State feedback control laws in the sector 0.75,1]. . . . . . . . . . . . Solutions of Riccati-like matrix equations for uncertain FTCSMP. . Solutions of Riccati-like matrix equations for certain FTCSMP. . . State feedback control laws for uncertain and certain FTCSMP. . . ki

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121 122 124 143 143 145

8.1 Steady-state controller gains, in the three scenarios. . . . . . . . . . . . 168 8.2 Summary of the steady-state solutions of the Riccati-like, covariancelike, and control gains equations. . . . . . . . . . . . . . . . . . . . . . . 182

List of Figures 1.1 Classication of fault tolerant control systems. . . . . . . . . . . . . . 1.2 Schematic diagram for active fault tolerant control systems. . . . . . . 2.1 2.2 2.3 2.4 2.5

Location of potential faults in a control system. Dierent fault-induced changes. . . . . . . . . . . Diagnostic procedure in typical FDI process. . . General scheme of model-based fault detection. . Performance indices for FDI processes. . . . . . .

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3.1 Illustration of Lyapunov stability. . . . . . . . . . . . . . . . . . . . . . . 24 4.1 AFTCS subject to random faults in actuators . . . . . . . . . . . . . . . 33 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

AFTCS in noisy environment. . . . . . . . . . . . . . . . . . . . . . . . Bounded solutions for the state feedback gain = f
1 0]
3 5]g. . . . Unbounded solutions for the state feedback gain = f
1 0]
2 25]g. . Unbounded solutions for high intensity of state-dependent noise. . . Unbounded solutions for high intensity of control-dependent noise. . AFTCS subject to faults in plant components and actuators. . . . . Bounded solutions with 1 =
2], 2 =
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6.1 6.2 6.3 6.4 6.5 6.6 6.7

Bounded solutions, , with tolerable detection delays. . . . . . . . . Unbounded solutions, , with intolerable detection delays. . . . . . Solutions, , for dierent means of detection delays. . . . . . . . . . Steady-state gains, , FTCSMP with errors in detection. . . . . . . Bounded solutions, , for perfect fault detection and identication. Unbounded solutions, , under a false alarm. . . . . . . . . . . . . . Bounded solutions, , under a missed detection. . . . . . . . . . . .

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7.1 General schematic diagram for FTCSSA. . . . . . . . . . . . . . . . . . 106

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List of Figures

7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11

Characteristics of an actuator with saturation. . . . . . . . . . . . Characteristics of an actuator with saturation in sector a,1]. . . . Positive-de nite solutions in the sector 0.75,1]. . . . . . . . . . . . Positive-de nite solutions for FTCSMP without saturation. . . . The constants versus a. . . . . . . . . . . . . . . . . . . . . . . Schematic diagram for AFTCS with faults in plant components. . Plant with uncertainties in system matrix. . . . . . . . . . . . . . Positive-de nite solutions, P , for uncertain FTCSMP. . . . . . . Positive-de nite solutions, P , for certain FTCSMP. . . . . . . . Positive-de nite solutions, P , versus . . . . . . . . . . . . . . .

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8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10

The control gains, K , in scenario 1. . . . . . . . . . . . . . . . The solutions of Riccati-like equations, P , in scenario 1. . . . The solutions of covariance-like equations, X , in scenario 1. The control gains, K , in scenario 2. . . . . . . . . . . . . . . . The solutions of Riccati-like equations, P , in scenario 2. . . . The solutions of covariance-like equations, X , in scenario 2. . Steady-state control gain, K , scenario 3. . . . . . . . . . . . . . The solutions of Riccati-like equations, P , in scenario 3. . . . The solution of covariance-like equation, X , in scenario 3. . . The control gains, K . . . . . . . . . . . . . . . . . . . . . . . .

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Nomenclature Symbols Time-invariant fault-free system matrix. System matrix with possible faults in plant components. Admissible parameter uncertainties in system matrixý A t H t F t E t H t ÿ þnÿp and E t ÿ þqÿn are known constant matricesý and F t ÿ þpÿq is a Lipschitz measurable matrix function satisfyingü F t T F t ý Ipþ B Time-invariant fault-free system input matrix. Být System input matrix with possible actuator faults. D x t ü ý t ü t Wÿ t State-dependent white Gaussian noise. Dl The lth component of the state-dependent noise matrix. E u t ü ý t ü t Wþ t Control-dependent white Gaussian noise. El The lth component of the control-dependent noise matrix. f x t ü ý t ü u x t ü t ü t ü t Non-linear FTCSMP in noise-free environmentý f admits a unique solution x t for every initial x to xoü ý to ýo and to o. k fij t The holding time density functioný when the FDI transits from state i to state j while the failure process is in state k. A Aÿt A t ÿ ÿ þþ

ý

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ý

ÿüÿ þþ û

ÿüÿ þþ

ÿüÿ þþ

ÿüÿ þþ

ÿüÿ þþ

ÿüÿ þþ

ÿüÿ þþ

ÿüÿ þþ

ÿüÿ þþ

ÿ ÿ þþ ÿ

ÿ þ

ÿ þ

þ ú

ÿ þ

ÿ

ÿ þ

ÿ þ

þ ú

ÿ þ

ÿ

ÿ þ

ÿ þ

ÿ

ÿ þ üÿ þ

þ

þ

ÿ þ

û

ÿ þ

ÿ

þ û

üÿ

þ û ü

ÿ

þ

XXII

Nomenclature

The holding time distribution functionÿ when the FDI transits from state i to state j while the failure process is in state k. F ÿÿÿtþþWý ÿ ÿtþ Independent white Gaussian noise. gÿxÿtþþ ÿÿtþþ uÿxÿtþþ üÿtþþ tþþ tþ n ÿ m matrix-valued function. K ÿüÿtþþ Constant state-feedback controller gain reconügured based on FDI decision. K ÿÿÿtþþ üÿtþþ An analytical state-feedback controller gain reconügured based on the knowledge of both the actuator and the FDI states. Lÿxþ ÿþ üþ tþ A positive-deünite functionÿ continuous in x. M ÿxþ ÿþ üþ tþ A positive-deünite functionÿ continuous in xÿ and M ÿxþ ÿþ üþ tþ û ú only at x û ú. N ÿxþ ÿþ üþ tþ A positive-deünite functionÿ continuous in xÿ function of ÿÿtþ and üÿtþ. N ÿxþ ýþ ÿþ üþ tþ A positive-deünite functionÿ continuous in xÿ function of ý ÿtþÿ ÿÿtþ and üÿtþ. P A probability measure for the space ÿùþ þþ P þ. P ÿxþ ÿþ üþ tþ A positive-deünite symmetric matrixÿ ý ÿ û k ü S and ü û i ü R. q FDI conditional transition rate at which it will decide that the next state is j leaving the state iÿ while the actuator failure process is currently in state k. q Conditional transition rates of the FDI process given that ý û j and ÿ û kü Qÿxþ ÿþ üþ tþ A positive-deünite symmetric matrixÿ ý ÿ û k ü S and ü û i ü R. F ÿtþ k ij

k ij

jk

iv

Nomenclature

R f ÿ ÿ þþþÿ rg Rijk t ÿ

þ ý

ü û

S f ÿ ÿ þþþÿ sg Sat u x t ÿ t ÿ t u x t ÿ t ÿt ul t uH uL u x t ÿý t ÿ t ÿt V x t to ÿ xo ÿ w vk t v xÿ ýÿ ÿ t V xÿ ýÿ ÿ t ÿ

þ ý

ú

ü

ü

ü û ùü û

ü û ùü û

ûø

û

ü û

ü

ü û

ü û ùü û

ü ÷

ûû

ü

ü û

ü

ù

ü

û

ù

û

V xÿ üÿ ýÿ ÿ t ü

ù

û

Vt xÿ ýÿ ÿ t Vxx xÿ ýÿ ÿ t wik t wki t Wt W xÿ üÿ ýÿ ÿ t xt xl t ü

ù

ü

û

ù

û

ü û

ü û

ü û

ÿ

ù

ü

ü û

ù

The ÿnite state space of the FDI process. The reliability functioný when the FDI transits from state i to state j while the failure process is in state k. The ÿnite state space of the actuator failure process. System input with saturation characteristics. System input function of the FDI process. The lth component of the control input. An upper limit of system actuator with saturation. A lower limit of system actuator with saturation. A proposed analytical system input. A stochastic Lyapunov function. The probability distribution for the actuator failure process. A positive-deÿnite symmetric scalarý ÿ ý k þ Sÿ i þ R. A stochastic Lyapunov function in terms of the failure and the FDI processes. It is a quadratic function in this book. A stochastic Lyapunov function in terms of the plant component and the actuator failure processes and the FDI processes. It is a quadratic function in this book. The ÿrst partial derivative of V xÿ ýÿ ÿ t with respect to t. The second partial derivative of V xÿ ýÿ ÿ t with respect to x. The conditional probability distribution for the FDI process. Solutions of scalar diûerential equations. A Wiener process. A positive-deÿnite functioný continuous in x. State trajectory. The lth component of the state trajectory. ü

ü û

ü

û

û

XXIII

ù ÿ

û

ù

û

XXIV

Nomenclature

Xki ÿtþ zki ÿtþ Z ý fûþ úþ ýýýþ zg ükj ûjh úki ùt ù ÿtþ ùn ÿtþ ÿÿtþ øki ý üS ÿR øki ÷ öÿtþ oÿùtþ õ ÷ki ÿtþ õ þ üÿtþ ÿõþ ûþ P þ õ û

kxkpp E ÿxþ

Covariance-like matricesþ deýned asü EfxÿtþxÿtþT j ÿ ÿtþ ý kþ üÿtþ ý ig. Scalar diúerential equations used to force the additive white Gaussian noise to vanish at t ÿ þ. The ýnite state space of the plant component failure process. Actuator failure rates. Plant component failure rates. Positive real constants. Inýnitesimal transition time interval. Markovian failure process for plant components. m-dimensional white Gaussian noise. Markovian failure process for actuators. An indicator function used to denote that the failure process is in state k and the FDI process is in state i. Constants represent the means of random detection delays. A positive constant. A homogeneous Markov process. Inýnitesimal terms of the order higher than that of ùt. The FDI transition-free time. The transition matrix associated with Aöki. Markovian FDI process. A probability space. The space of elementary events. A ôúýeld which consists of all subsets of õ that are measurable. The púnorm of a vector x. The mathematical expectation of x.

Nomenclature

XXV

The pÿnorm of a vector x. E ÿxþ The mathematical expectation of x. ÿV ÿxÿtý to þ xo þ wþþ The weak inþnitesimal operator of a stochastic Lyapunov function. ÿV ÿxþ ýþ üþ tþ The weak inþnitesimal operator of the Lyapunov function V ÿxþ ýþ üþ tþ. ÿV ÿxþ üþ ýþ üþ tþ The weak inþnitesimal operator of the Lyapunov function V ÿxþ üþ ýþ üþ tþ. ÿu V ÿxþ üþ üþ The weak inþnitesimal operator of the uncertain FTCSMP. ÿc V ÿxþ üþ üþ The weak inþnitesimal operator of the certain FTCSMP. ÿupper V ÿxþ üþ üþ The upper bound of the weak inþnitesimal operator for uncertain FTCSMP. ûÿV ÿxþ üþ üþ The part of the weak inþnitesimal operator of the uncertain FTCSMP caused by parameter uncertainties. ÿûÿV þbound The upper bound of ûÿV ÿxþ üþ üþ with the maximum uncertainty. kxkpp

Abbreviations AFTCS CF CRM CUSUM DD FA FDI FTCS FTCSMP FTCSSA

Active Fault Tolerant Control Systems. Correct Fault detection rate. Controller Reconþguration Mechanisms. Cumulative SUM test. Detection-Delay rate. False Alarm rate. Fault Detection and Identiþcation. Fault Tolerant Control Systems. Fault Tolerant Control Systems with Markovian Parameters. FTCS with potentially Saturated Actuators.

XXVI

GLR IF JLS MF PFTCS PIM SPRT

Nomenclature

Generalized Likelihood Ratio test. Incorrect Fault detection rate. Jump Linear Systems. Missed Fault detection and identiþcation rate. Passive Fault Tolerant Control Systems. Pseudo-Inverse Method. Sequential Probability Ratio Test.

Terminology Active fault tolerant control systems Adaptive control

Control systems where faults are explicitly detected and accommodated through changing of the control laws. A systematic approach for automatic adjustment of the controller in real-timeü to achieve or maintain a desired level of performance of the control system when parameters of the plant dynamics are slowly changing with time. Analytical redundancy Use of more than oneü not necessarily identical ways to determine a variableü where one way uses a mathematical process model in an analytical form. Availability Probability that a system or equipment will operate satisfactory and eûectively at any point in time. Availability measureü Aü is MTBF A ÿ MTBF þ MTTR MTTRú mean time to repair. MTBF ú mean time between failures. Elementary outcome An individual point w in ý. Events The measurable subsets A in the collection ÿ.

Nomenclature

Failure

XXVII

A permanent interruption of a systemÿs ability to perform a required function under speciþed operating conditions. Failure state Particular way in which a failure may occur. Fault An unpermitted deviation of at least one characteristic property or parameter of the system from the acceptableü normalüstandard condition. Fault detection Determination of the presence of faults in a system. Fault identiþcation Determination of the size and time-variant behavior of a fault. Usually follows the fault detection. Fault-induced changes The consequences of a fault on the operationú functionú or status of an item. Fault tolerant control AFTCS with two Markovian processesù one represents systems with Markovian random faults in the systemú and the other represents parameters decision of the FDI process. Fault tolerant system A system where a fault can be accommodated withüwithout performance degradationú so that a single fault at subsystem level does not develop into a failure on a system level. Field A family of ÿ of subsets of ÿ which satisþesù Fÿþ ÿ þ ÿ. Fýþ If S þ ÿú then S c ÿ ÿ ý S þ ÿ. Füþ If Si þ ÿú þ ü i ü nú then Sniÿþ Si þ ÿ. Graceful degradation The ability of a control systemú in case of a faultú to automatically reduce its demand on the level of performance so that the high priority objectives can be maintained þrst. Hardware redundancy Multiple independent hardware channels used with a majority vote selection for healthy system channels.

XXVIII

Nomenclature

Hybrid system

A system which combines both the Euclidean space for system dynamics and the discrete space for event-induced changes. Jump linear systems Dynamic systems with randomly varying coeýcients. Jumps in the coeýcients can be represented by a Markov chain known as plant regime mode. Maintainability The need for a system to be repaired and the ease with which repairs can be made. Malfunction An intermittent irregularity in the fulülment of systemsûs designed functions. Measurable space The sample space ÿ ÿ provided that ÿ is a þþ üeld of subsets of . Monitoring A continuous real-time task of determining the conditions of a physical systemú by analyzing the informationú recognizing and indicating anomalies in its behavior. Passive fault tolerance A fault tolerant system where faults are not explicitly detected and accommodatedú but the controller is designed to be insensitive to a certain set of faults. Reconüguration Ability of a system to modify its structureùparameters to account for the detected faults in the system. Redundancy Ability of a system to overcome lost capabilities with the remaining resources. There are two forms of redundancyø physical and analytical. Reliability Ability of a system to perform a required function under stated conditionsú within a given period of time. Reliability is measured byø MTBF ý ý rate of failure. ÿþ

ý

þ

ü

ú

û

Nomenclature

Residual

XXIX

A fault indicatorÿ based on the deviation between measurements and calculated values based on system models. Robustness Ability of a system to maintain satisfactory performance in the presence of parameter variations. Safety Ability of a system not to cause any danger to human operatorsÿ equipment or the environment. Sample space The set of all possible outcomes of some experimentÿ denoted as ÿ. Separable The measurable space þÿÿ ÿý is separable if there exists a countable system of sets S which separates the points of the space ÿ and generates a corresponding þþ algebra ÿ. Survivability The likelihood of conducting an operation safely. Symptom A change in observable quantities from normal behaviors. þþýeld A ýeld with the condition üFÿû replaced with a stronger condition If S ý ÿÿ i ý Zÿ ÿ füýûÿ ýýýÿ gÿ then Sÿþý S ý ÿ. i

i

i

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Index actuator |, 3, 6, 8, 10, 12, 20, 21, 31, 33, 67{ 69, 72, 77, 78, 82, 103{105, 113{ 115, 117, 119, 123, 146, 186 dynamics, 78 failure process, 31, 32, 34, 47, 70, 85, 152, 155{157, 172 failure rates, 34, 61, 70, 89, 97, 119, 163, 180 no dynamics, 77 non-linear saturation, 103, 105, 146 transition probabilities, 27, 34, 67, 70, 85, 128, 138, 155 with saturation, 103{105, 107, 113, 117, 118, 123 algorithm computational, 149, 162, 163, 179, 183, 187 stabilization, 112, 137, 140, 145, 147, 186 component failure process, 70, 128 condition growth, 32, 48 Lipschitz, 27, 32, 48

cost function | , 84, 95, 96, 149, 151, 152, 156{160, 169, 170, 173, 175, 176, 178, 179, 182 equivalent deterministic, 148, 151, 153, 169, 182, 187 performance index, 94, 151 quadratic, 19, 95, 152, 173 cost-critical, 4, 5, 20 distribution exponential, 84 Gamma, 84, 86 dynamic programming principle, 8, 83, 93 Dynkin's formula, 41, 42, 52, 55, 133 eigenvalue, 19, 133 equations covariance-like dierential equations, 149, 154, 155, 157, 158, 161, 162, 169, 173, 178, 179, 182 Riccati-like dierential equations, 116, 155, 160, 178 Riccati-like matrix equations, 94, 98, 99, 103, 105, 109, 110, 112, 120,

INDEX

208

131, 132, 134, 135, 137{139, 142, 143, 146, 147, 154

exponentially distributed detection delays, 86, 88

actuator, 6, 12, 45, 67, 68

false alarm, 9, 38, 93, 98{100 Gamma distributed detection delays, 87, 88

additive , 13

hypothesis tests, 30, 92

fault-induced changes, 2, 3, 8, 13, 30, 92

memory-less tests, 83 missed detection, 37, 83, 93, 99, 101

multiplicative, 13

performance measures, 17

plant component, 45

process, 14, 17, 30{38, 46, 47, 49, 62, 67{71, 76, 79, 80, 82{86, 89, 93, 96{99, 101, 103, 105, 113, 127, 128, 138, 141, 148, 150{153, 155{ 157, 164{166, 169, 170, 180, 182, 185{187

faults

plant component , 12 random, 10, 31, 33, 67, 68, 77, 78, 82, 105, 127, 150, 153, 186 sensor, 7, 10, 12, 67 FDI algorithm, 4, 9, 10, 13, 18, 31, 83, 85, 88, 93, 124, 125, 127, 138, 148, 157, 186 approaches, 13

sequential tests, 31, 36, 83, 84 FTCS active, 2, 69 applications, 20

conditional transition probabilities, 34, 35, 67, 70, 79, 85, 128

classi cation, 2

conditional transition rates, 34, 35, 62, 85{89, 91, 98, 120, 141, 164, 180, 185

fault tolerant control systems, 1{3, 6, 9, 184

de nition, 1

diagnostic procedure, 16

FTCS driven by actuators with saturation, 105{107, 109, 110, 112, 117{121, 123 FTCS with Markovian Parameters, 38

errors in detection, 9, 34, 37, 67, 92, 93

FTCS with Markovian parameters, 9, 22, 30{34, 37{39, 43{59, 61, 67,

detection delays, 9, 34, 37, 38, 46, 83{ 92, 104, 186

INDEX 69{74, 82{85, 88{93, 95, 97{99, 101, 103{105, 114, 115, 117, 120, 123{131, 133, 134, 136{138, 140{ 151, 153, 159{164, 169, 170, 172, 173, 178, 181, 182, 185, 186 passive, 2 Hamiltonian, 154, 173, 177 horizon nite, 149, 153, 169, 170, 172, 182 in nite, 150, 159, 162, 173, 178, 181 hybrid systems, 8, 9, 37, 38, 46, 76, 83, 125, 173, 186 imprecise estimated parameters, 103 JLS Jump linear systems, 8, 9, 37, 38, 46, 82, 83, 105, 114, 125, 139, 147, 186 Lyapunov function approach, 8, 26, 27, 38, 41, 45, 50 Markov process, 27, 28, 30{34, 37, 40, 42, 43, 48, 68, 73, 106, 113, 128 matrix maximum principle, 8, 9, 148, 154, 169, 170, 173, 182, 187 multiple failure processes, 45, 67 noise control-dependent, 48, 49, 65{67, 173

209

environmental noises, 9, 45{47, 82, 104, 185 independent, 49, 63, 64, 82, 185 state-dependent, 48, 49, 64{66 white Gaussian noise, 46, 47, 57, 174 optimal control problem, 151, 170 Optimality Lemma, 96 parameter uncertainties |, 103, 104, 124{128, 130, 135, 140, 142, 145, 147, 148 norm bounded, 125, 126, 128, 135, 138, 142, 147, 186 positive-de nite solutions, 89, 110, 112, 115{117, 120{123, 132{140, 142, 144{146 property availability, 1, 4, 21, 93 maintainability, 2 reliability, 1, 2, 4, 7, 12, 20, 35, 36, 85{87 survivability, 2 recon guration |, 181 |, 13, 18{20, 83, 96, 124, 155, 185, 186 control recon guration, 83, 96, 155, 181, 185, 186 mechanisms, 7, 18

INDEX

210

redundancy |, 2{4, 6, 7, 10, 12, 20, 184 analytical, 4 hardware, 3, 7, 12 safety-critical systems, 4, 5, 20 stability deterministic, 23, 80 stability almost sure asymptotic, 26, 53, 56 almost sure asymptotic Lyapunov, 25 asymptotic in probability, 25, 28, 56 asymptotic in the mth moment, 25 asymptotic Lyapunov, 23, 25 exponential in the mean square, 50, 56, 82, 88, 97, 98, 126, 141, 186 exponential of the mth moment, 26 in probability, 24 in the mth moment, 25 Lyapunov, 23{25 deterministic, 26, 62, 63, 80, 185 stochastic Lyapunov function, 27, 38{42, 44, 50{52, 54, 56, 57, 61, 71, 73{ 75, 82, 107, 108, 129, 132 stochastic systems, 4, 8, 25{28, 38, 41 supermartingales positive, 39, 42, 50{52 property, 8, 27, 38, 42, 50 transition matrix, 33, 60, 161

transition probabilities, 68, 70, 78, 79, 113 volume-critical, 4, 5, 20 weak innitesimal operator, 28, 40, 41, 54, 55, 57{59, 61, 73{76, 82, 95, 107{ 111, 113, 118, 129, 130, 132, 135, 139

Chapter 1 INTRODUCTION Modern technological devices rely on sophisticated control systems to meet increased performance requirements. For such systems, the consequences of faults in system components can be catastrophic. Reliability of such systems can be increased by ensuring that faults will not occur, however, this objective is unrealistic and often unattainable because faults may arise not only due to components aging and wear, but also as human errors in connection with installation and maintenance. Therefore, it is necessary to design control systems that are able to tolerate possible faults in such systems to improve reliability and availability. This type of control systems is often known as Fault Tolerant Control Systems (FTCSs), which can be classied into two categories: active and passive. In this book, we are concerned with the design and analysis of active FTCSs.

1.1 Fault Tolerant Control System (FTCS) 1.1.1 Denition of FTCS

A FTCS is a control system that can accommodate system component faults and is able to maintain stability and acceptable degree of performance not only when the system is fault-free but also when there are component malfunctions. FTCS prevents faults in a subsystem from developing into failures at the system level. M.M. Mahmoud, J. Jiang, Y. Zhang: Active Fault Tolerant Control Systems, LNCIS 287, pp. 1−10, 2003.  Springer-Verlag Berlin Heidelberg 2003

2

CHAPTER 1. INTRODUCTION

FTCS may be called upon to improve system reliability, maintainability, and survivability 169]. The objectives of FTCS may be dierent for dierent applications. A FTCS is said to improve reliability if it allows normal completion of tasks even after component faults. FTCS could improve maintainability by increasing the time between maintenance actions and allowing the use of simpler repair procedures. 1.1.2 Classication of FTCS

The design techniques for FTCS can be classi ed into two approaches: PFTCS and AFTCS 151]. A particular approach to be employed depends on the ability to determine the faults that a system may undergo at the design phase, the behavior of fault-induced changes, and the type of redundancy being utilized in the system. Figure 1.1 shows classi cation of FTCS approaches. Fault Tolerant Control Systems

Passive (PFTCS)

Active (AFTCS)

On-line Controller Selection

On-line Controller Redesign

Figure 1.1: Classi cation of fault tolerant control systems. In this approach, a system may tolerate only a limited number of faults which are assumed to be known prior to the design of the controller. Once the controller is designed, it can compensate for the anticipated faults without any access of on-line fault information. PFTCS treats the faults as if they were sources of modelling uncertainty 151]. PFTCS:

3

1.1. FAULT TOLERANT CONTROL SYSTEM (FTCS)

PFTCS has a very limited fault tolerance capability. When running on-line, a passive controller is robust only to the presumed faults. Therefore, it is quite risky to rely on PFTCS alone 151]. In general, PFTCS has the following characteristics:

1 Robust for anticipated faults. 2 Utilizes hardware redundancy (multiple actuators and sensors, etc.). 3 More conservative. AFTCS: In most conventional control systems, controllers are designed for fault-free systems without considering the possibility of fault occurrence. In other cases, the system to be controlled may have a limited physical redundancy and it is not possible to increase or change the hardware conguration due to cost or physical restrictions. In these cases, an AFTCS could be designed using the available resources and employing both physical and analytic system redundancy to accommodate unanticipated faults. Figure 1.2 shows a general schematic diagram for AFTCS. Reconfiguration Mechanism

FDI

Input

Controller +

Actuators

Plant

Sensors

Outputs/states

_

Controller

Figure 1.2: Schematic diagram for active fault tolerant control systems. AFTCS compensates for the e ects of faults either by selecting a pre-computed control law, or by synthesizing a new control law on-line in real-time. Both approaches need a Fault Detection and Identication (FDI) algorithm to identify the fault-induced changes and to recongure the control law on-line.

4

CHAPTER 1. INTRODUCTION

AFTCS involves signi
cant amount of on-line fault detection, real-time decision making and controller recon
guration, it accepts a graceful degradation in overall system performance in the case of faults 88, 201]. Generally, AFTCS has the following characteristics: 1

Employs analytical redundancy.

2

Utilizes FDI algorithm and recon
gurable controller.

3

Accepts degraded performance in the presence of a fault.

4

Reduces conservationist.

AFTCS is a complex interdisciplinary
eld that covers a wide range of research areas, such as stochastic systems, applied statistics, risk analysis, reliability, signal processing, control and dynamical modelling. 1.2

Background and Motivations

FTCS is a strategy for highly reliable control system synthesis. It is the synonym for a collection of recent techniques that have been developed to increase plant reliability and availability and reduce the risk of safety hazards. FTCS is designed to accommodate faults in early stage of their development, such that minor faults in a sub-system do not develop into failures at the system level. FTCS may command either a safe shutdown, or a continued operation with graceful degradation in performance. The need for FTCS to increase reliability and automation level of modern control systems is becoming more apparent. FTCS is being utilized in various control applications such as: safety-critical systems (nuclear reactors, aircraft, missile guidance systems), cost-critical systems (large space structures, space vehicles, autonomous underwater vehicles), and volume-critical systems (assembly processes in auto-industry, mobile communication networks, automated highways).

1.2. BACKGROUND AND MOTIVATIONS

5

In nuclear power plants, as an example of safety-critical systems, the occurrence of a fault may lead to human/environmental disaster. A proper control action could be a safe shutdown for the plant in the presence of emergency. In cost-critical and volume-critical systems, the occurrence of a fault may cause a huge economical loss. Autonomous underwater and unmanned space vehicles are examples where the vehicle must be able to deal with unexpected events, such as faults, in a predictable manner. In the worst case scenario, the vehicle should be able to return safely (not completing the mission). In industry, there are several examples where minor faults have caused a shutdown or even damage to equipment and thereby an expensive production stop. Among the abundant examples of fatal fault situations, two cases are cited here: 1. On 26th of April 1986, there occurred near the Ukrainian town of Chornobyl a tremendous explosion at a huge nuclear power plant, followed by a gradual meltdown of the reactor No. 4. The main cause for this tragedy was the faulty outdated technology and the lack of a fault handling mechanism 183]. 2. The Ariane 5 rocket exploded on June 4th, 1996 thirty seven seconds after lift-o. The reason was a software exception in the Inertial Reference Unit (IRU) that provides attitude and trajectory information for the control system. The exception caused the normal attitude information to be replaced by some diagnostic information that the control system was not designed to understand 27]. The catastrophic consequences of these fault cases could have been avoided or at least mitigated if the control system is designed with certain degree of fault tolerant capabilities. In the rst case, a FTCS could have been designed to lead to a safe shutdown of the reactor. In the second case, the malfunction of the IRU could be

6

CHAPTER 1. INTRODUCTION

considered as a sensor fault and a voting mechanism could have been implemented to perform a consistency check of the attitude information against the expected attitude.

1.3 Advances in Fault Tolerant Control Systems As mentioned previously, the main task in FTCSs is the design of a controller to guarantee the system stability and acceptable performance, not only when the system is fault-free but also when there is a component malfunction (sensor, actuator, plant component, or any combination). This task was tackled by employing robust control techniques to design a controller that guarantee a fail-safe operation of the system. A system is said to be fail-safe if the closed-loop system remains stable in the event of component faults. These design methods exploit the high level of redundancy and diversity in sensors and actuators. MacFarlane, 120], used such design methodology to dene an important concept known as system integrity. A feedback controller for systems with integrity was designed in 172], whereas necessary and su cient conditions for the stability of systems possessing integrity were derived in 55, 67]. When FTCSs start to attract the attention of researchers as a vital modern control tool, the concept of system integrity was further extended to design PFTCSs. A design approach for PFTCS against actuator faults was developed using state feedback control based on the solution of Riccati-like equations in 165], and based on the solution of Lyapunov equation in 164]. However, these design approaches are only applicable to open-loop asymptotically stable systems. This limitation was relaxed under the assumption that the actuator fault modes are restricted to a specic set and using linear quadratic control in 92, 179], and with H1 norm bound performance measure in 180]. The problem of simultaneous stabilization where a single control law is used to stabilize multiple systems was dened and studied in 181]. While the reliable stabi-

1.3. ADVANCES IN FAULT TOLERANT CONTROL SYSTEMS

7

lization problem, the dual problem, concerns with the design of multiple controllers to stabilize a single system was studied in 182]. These methods can be extended employing systems hardware redundancy for the design of a PFTCS 93]. A graphical design scheme was proposed in 1], this approach can guarantee the closed-loop stability of the system in case of actuator/sensor faults by proper selection of controller structure and parameters. For situations where there is a limited hardware redundancy in the system and a limited knowledge (process history and faults statistics) of possible faults, AFTCS shall be used to improve system reliability. Generally speaking, there has been signicant research for AFTCS. Unfortunately, most of this research is scattered in the literature and focuses only on individual aspects of AFTCS. A comprehensive survey of the design methods for basic blocks of AFTCS is detailed in Chapter 2 with special focus on FDI algorithms and controller reconguration mechanisms. AFTCS is a topic of active and ongoing research. Signicant research completed to retain system stability against component faults in ight control systems, such as a self-repairing ight controller to accommodate surface malfunctions 81]. This controller distributes control authority to the remaining eectors. In 117], system performance in frequency domain is used to design an automatic re-structurable controller. To recover system performance subsequent to a fault, a pseudo-inverse based method was developed in 59]. A major drawback for this method is that the stability of the impaired system cannot be guaranteed. The method has been modied in 60], such that the dierence between the nominal and the estimated closed-loop system after the fault is minimized subject to stability constrains. To enhance performance recovery, an eigenstructure assignment based algorithm was proposed in 87] and investigated further in 202, 203, 204]. Promising results were obtained by employing the linear matrix inequality technique in 25, 131]. In practical control systems, faults are random in nature. They may occur at

8

CHAPTER 1. INTRODUCTION

random instants of time, have unknown size and can happen to any system component (sensors, plant, or actuators). Moreover, control systems in practice are subject to measurement noises and disturbances from the surroundings and the communication channels. In addition, the majority of FDI algorithms relies on statistical hypothesis tests. All these facts necessitate that AFTCSs be best modelled as stochastic systems. The dynamic behavior of an AFTCS can be modelled by stochastic dierential equation and can be viewed as a general hybrid system. It is hybrid because it combines both the Euclidean space for system dynamics and the discrete space for faultinduced changes. Stability of hybrid systems was rst studied using the Lyapunov function approach in 95], and the supermartingale property of Lyapunov functions in 17, 107]. A major class of hybrid systems is Jump Linear Systems (JLS). In JLS, the random jumps in system parameters are represented by a nite state Markov chain known as the plant regime mode. The research in JLS can be broadly classied into two categories: the rst focuses on deriving necessary and/or sucient conditions for the existence of optimal quadratic regulator. An optimal control law was designed using the dynamic programming principle in 190], and using the matrix maximum principle in 173]. Sucient conditions for the existence of a steady-state optimal JLS controller were derived in 191]. A control strategy for JLS in the presence of disturbances was developed in 13], and in the presence of noise in 174]. The design of H1 and H2 control law for JLS were considered in 32, 33, 35, 36, 50]. The linear matrix inequality approach was employed to design a control law for uncertain JLS in 58]. A constrained state feedback control law was synthesized in 34]. The second category deals with the properties of this class of systems such as stability 10, 14, 31, 48, 49, 68, 114], controllability and observability 30, 86], and the solution of resulting Riccati equations 41, 42]. It is important to mention that the models of JLS assume perfect knowledge of the regime.

1.4. SCOPE OF THE BOOK

9

Unfortunately, all FDI algorithms have non-zero detection delays and error probabilities associated with their decisions. Therefore, the JLS model which assumes perfect regime knowledge do not t the AFTCS exactly. As a result, sucient conditions for exponential stability in the mean square of JLS with detection delays were derived in 19, 133]. However, the results are based on the assumption of certainty of correct fault identication. To relax this assumption for better description of AFTCS, another class of hybrid systems was dened in 168]. This class of systems is known as Fault Tolerant Control Systems with Markovian Parameters (FTCSMP). In FTCSMP, two random processes with dierent state spaces were dened and used to study the behavior of AFTCS in 168]: the rst represents system component faults, and the second represents the FDI algorithm used for the control law reconguration. Very recently, the superiority of FTCSMP model to describe the dynamical behavior and to study AFTCS was demonstrated in several works. The vital issue of environmental noises and how they aect the overall performance of FTCSMP was studied in 128]. FTCSMP with multiple fault processes was dened to account for independent faults in dierent system components in 122, 129]. Imperfections in the performance of the FDI algorithms including detection delays, false alarm and errors in detection were also considered in 125]. A control law for FTCSMP using the matrix maximum principle and for dierent operating scenarios was designed for noise-free and noisy environments in 123] and 126], respectively. The eect of imprecise FDI estimation on the stability of FTCSMP was considered in 124], and the design of a stabilizing controller that can tolerate uncertainties in the estimated parameters in 89, 90, 127]. 1.4

Scope of the Book

FTCS is a control technique that belongs to the domain of complex systems. Various approaches are employed to design PFTCS and AFTCS. The methods used to de-

10

CHAPTER 1. INTRODUCTION

sign elements of AFTCS (FDI algorithm, recongurable controller, and decision logic unit) are diverse and range from simple to complex. Principles of their functionality are numerous (binary logic, probabilistic methods, articial neural networks, etc). In addition, the system to be controlled may be simple or large-scale, lumped or distributed, static or dynamic, linear or nonlinear. All these facts create diculties when trying to develop a general model for FTCS. Generally speaking, research on FTCS can be conducted in both deterministic and stochastic domains. A comprehensive overview of the status of FTCS up to 1997 was provided in 151]. The subject of this book falls within the area of AFTCS that are modelled by stochastic linear dierential equations with randomly varying parameters. Random parameters are due to random faults that may occur and the non-deterministic nature of the FDI algorithms. The random nature of the FDI process can be assumed to have a strong Markovian characteristics. A primary concern, in this book, is the faults in system actuators and plant components. Sensors integrity is assumed in all work completed. It will be claried in Chapter 2 that multiple redundancy with majority voting is sucient to accommodate sensor faults. Hence, the assumption of sensor integrity is not a limitation to the work conducted. Moreover, this book only considers the case where the system to be controlled is linear.

Chapter ACTIVE FAULT TOLERANT CONTROL IN PROSPECTIVE . Introduction There has been signi cant research for AFTCS design. Unfortunately most of research is scattered in the literature and considers only the individual elements of AFTCS. Some work has been completed to combine dierent blocks of AFTCS together and to study performance of the overall AFTCS. The issue of merging blocks of FTCS seems to be easy task in principle unfortunately this is not the case in reality    . The main diculty that arises when combining the blocks in AFTCS is that each individual block is assumed to operate perfectly and is readily available to provide decisions actions instantaneously to other blocks. This chapter reviews the modelling of faults in dynamic systems presents dierent design methods for FDI and controller recon guration and wraps-up by highlighting the need for fault tolerant control in modern dynamic systems.

.

Faults in Dynamic Systems

A fault is de ned as an unpermitted deviation of at least one characteristic property or parameter of the system from the acceptable behavior. The fault is a state that may lead to a malfunction or a failure in the system. Faults may take place in M.M. Mahmoud, J. Jiang, Y. Zhang: Active Fault Tolerant Control Systems, LNCIS 287, pp. 11-21, 2003.  Springer-Verlag Berlin Heidelberg 2003




CHAPTER . ACTIVE FAULT TOLERANT CONTROL IN PROSPECTIVE

any system component actuators sensors plant components or any combination. Faults can be classied based on several criteria such as the time characteristics of faults physical locations in the system and the eect of faults on system performance. When faults are classied according to their physical locations three main faults can be dened actuator faults sensor faults and plant component faults. Figure
. shows possible fault locations in the system to be controlled. Input

Controller +

Actuators

Plant

Sensors

Fault

Fault

Fault

Outputs/States

_

Controller

Figure
.  Location of potential faults in a control system. Faults in actuators range from loss of partial control eectiveness stuck at a xed value to a complete loss of control. Since an actuator is often considered as the entrance to the system actuator faults have severe consequences on the system performance. Moreover actuators are usually expensive large in size and need a large driving signal. Therefore it is generally very di cult to add extra hardware redundancy to increase reliability. Sensor faults include incorrect reading due to malfunction in sensor circuit elements or transducers. Three types of sensor faults can be identied dynamic changes in transducer gain reduction and unknown bias. Fortunately sensors are usually small in size and do not need large driving signals. As a result sensor reliability can be increased by utilizing parallel hardware redundancy followed by a majority voting scheme. Plant component faults cause changes in the dynamical relationship among the system variables. These faults are caused by physical parameter changes in the sys-




.. FAULT DETECTION AND IDENTIFICATION FDI

tem such as resistance inductance amplier gain etc. These changes will appear as coecient changes in the dynamical model of the system to be controlled. If faults are to be classied according to their induced eects on the system performance they can be classied into two types as shown in Figure . additive and multiplicative  . Additive faults result in changes only in the mean value of the system output signal. Whereas multiplicative faults result in changes in variance correlations of the system output signal as well as changes in the spectral characteristics and dynamics of the system.

X

+ +

fa

fm

+

X a = X + fa

Additive fault

X

+

a

Xm = ෈a + fm෉X

Multiplicative fault

Figure . Dierent fault-induced changes. .

Fault Detection and Identication FDI

The main task of AFTCS is on-line reconguration of the controller. For this to be possible detailed information about fault-induced changes is required. In this context an FDI algorithm plays crucial role in AFTCS. The FDI algorithm monitors system performance to detect the occurrence of faults and to determine their magnitudes.

.. Approaches to FDI Several FDI approaches have been developed. In general these approaches can be categorized into signal-based and model-based techniques

 . Signal-based methods detect faults by testing specic properties of measurement signals. Bandpass lters and spectral analysis are examples of the several techniques employed in signal-based methods.




CHAPTER . ACTIVE FAULT TOLERANT CONTROL IN PROSPECTIVE

Model-based methods have wider range of application and are normally performed in two steps residual generation and residual evaluation decision-making. Figure . sketches the procedure for typical FDI processes. Residual is generated by comparing the expected behavior of the system with the measured behavior where the expected behavior is obtained from a model of the system. System Measurements Analytical Model

+ _

Residual Generation

Decision Making

Fault

Information

Figure . Diagnostic procedure in typical FDI process. Two major approaches have been used in model-based residual generation Qualitative heuristic methods and quantitative analytic methods   . To design an AFTCS precise knowledge about the plant dynamics need to be known after the occurrence of a fault. Hence more emphasis has been placed upon quantitative model-based FDI approaches. Three main classes of model-based residual generators exist observer-based approaches parity space approaches and parameter estimation approaches. The principle of observer-based approaches is to estimate the system variables with Luenburger observer for the deterministic case or a Kalman lter for the stochastic case and to use the estimation errorsinnovations as residuals. Various methods for the design of a proper observer gain have been suggested eigenstructure assignment    unknown input observer 
 Kronecker canonical form 
fault sensitive lters

and frequency domain optimization apporach based on a factorization of the input-output transfer matrix  . Recent developments in the application of Kalman lters are found in   

  . A bank of observers or Kalman lters with distinct properties can be used in parallel to isolate faults  
 . The number and nature of faults to be detected

.. FAULT DETECTION AND IDENTIFICATION FDI




and isolated necessitate dierent observer structures 
. A survey of nonlinear observer methods is provided in    . In the parity space approaches residuals are computed as the dierence between the measured outputs and estimated outputs and their associated derivatives. The method reshapes the primary residual signals using a transformation matrix to make the residual insensitive to unknown disturbances and to increase fault identi cation ability. The parity space approach has been developed in frequency domain in 
 and in time domain in  . The parameter estimation methods for FDI are based on the concept that faults typically aect the physical coe cients of the process. By continuously estimating the parameters of a process model residuals are computed as the parameter estimation errors. To successfully isolate faults the mapping from the model coe cients to the process parameters must exist and be known. Dierent methods for parameter estimation in FDI have been studied least squares estimation instrumental variable approach output error methods    sliding mode estimation  neural network estimation  and extended Kalman lters   
. Recently several interesting approaches have been utilized to design and implement FDI algorithms such as a geometric approach for bilinear systems  a linear matrix inequality approach    and frequency domain approaches   . The role of residual evaluation decision-making is to detect when the residuals have changed su ciently to make a reliable fault detection. Several decision-making methods have been used such as binary decision and statistical decision. A binary decision is made from a comparison between the residual and a xed threshold. Adaptive thresholds can be used to increase the robustness to modelling uncertainties  . Surveys of adaptive threshold techniques are provided in 

  . A variety of detection techniques are available in the eld of statistical decision




CHAPTER . ACTIVE FAULT TOLERANT CONTROL IN PROSPECTIVE

theory such as generalized likelihood ratio test GLR sequential probability ratio test SPRT 2-test and cumulative sum test CUSUM. Surveys and signicant results on this topic are available in        . A class of model-free-based FDI approaches has also been developed. Various algorithms have been implemented employing fuzzy logic   
    

 and articial neural networks        

. There is a huge body of applications for the dierent FDI algorithms in practical control systems and industry. Examples include power generation   communication    process industry 
  automotive and transportation industry     energy   agriculture    large complex industries    and several other applications. The interested reader may nd plenty of applications in the literature. ..

Diagnostic procedure of FDI

The diagnostic procedure which is sketched in Figure . is based on the observed analytical and heuristic symptoms and the heuristic knowledge of the process. Fault

Fault

Fault Noise

Input + -

Actuators

Plant

Sensors

Process Model Model Based Fault Detection Residual Generation

Residuals Normal Behaviour

Fault Detection

Analytical Symptoms

Figure . General scheme of model-based fault detection.

Output

.. FAULT DETECTION AND IDENTIFICATION FDI




Analytic symptoms generation is a quantiable analytical information about the process obtained by data processing of the measurable signals. If the input and output signals to a process are measurable model-based methods can then be used. However if only output signals of a process can be measured signal-based methods should be applied. Heuristic symptoms are represented as linguistic variables e.g. small medium large or as vague numbers e.g. around a certain value. The symptoms can be produced using qualitative information from human operators through human observations in the form of acoustic noise vibrations or optical impressions such as color or smoke. The heuristic knowledge of the process includes process history and faults statistics. Process history includes the past information of running time load measures last maintenance or repair. If fault statistics exist they describe the frequency of certain faults for the same or similar process. Fault statistics can be used as analytical or heuristic symptoms.

.. Performance measures in FDI To evaluate the performance of a FDI process several performance indices have been dened and used   such as Correct Fault CF detection False Alarm FA rate Missed Fault MF detection Incorrect Fault IF detection and Detection Delays DD. The major emphasis in quantitative model-based FDI has been placed upon methods of detecting and isolating faults rapidly and accurately   . A good FDI process should possess certain degree of robustness in its decisions. The robustness in decision means that the FDI process should have high CF and low FA MF IF and DD.




CHAPTER . ACTIVE FAULT TOLERANT CONTROL IN PROSPECTIVE

Figure . Performance indices for FDI processes. Figure . shows the interpretation of the performance indices CF FA MF and IF for a system with a normal mode of operation and three possible faults.

. Control System Reconguration The purpose of controller reconguration is to compensate for the eects of the failed component. Reconguration mechanisms can be classied as on-line controller selection and on-line controller calculation methods. In the rst controllers associated with preassumed fault conditions are computed a priori in the design phase and initiated on-line based on the real-time information from the FDI algorithm. These methods are sometimes known as projection-based methods. In the second class of methods controllers are synthesized on-line and in real-time after the occurrence of faults. Control law re-scheduling multiple models and interacting multiple models approaches are examples of projection-based approach. In this approach pre-computed control laws are selected based on an estimate of the system impairment status. This approach has been considered in several works   
  . This approach is highly dependent on prompt and correct operation of the FDI algorithm. Any false missed or error in detection may lead to degraded performance or even a complete loss of the stability of the closed-loop system. Recently in an

.. CONTROL SYSTEM RECONFIGURATION




attempt to deal with these FDI robustness problems and to design a stability guarateed AFTCs methods employing Linear Matrix Inequality LMI theory have been proposed in   . One of the on-line controller design methods is the Pseudo-Inverse Method PIM. The principle of PIM is to recompute the controller gain matrix such that the recon gured system approximates the nominal system in some sense. A severe drawback of this method is that the stability of the recon gured system is not guaranteed 
. To overcome this stability problem a modi ed PIM method was proposed in which the di erence between the closed-loop matrices is minimized subject to the stability constraints  . An Eigenstructure Assignment EA based algorithm was proposed in . In this approach the post-fault eigenvalues and eigenvectors are assigned in an optimal way such that performance recovery of the original system is maximized. Extension to integrated FDI and recon gurable control design using EA algorithms has been developed in    An approach that is related to PIM is the control distribution concept reported in . In this method a control distribution matrix is computed based on the minimization of L2ᝇnorm of a quantity that provides a measure for the di erence between failed and healthy systems. Another on-line recon guration method is the model-following approach. In this approach controller gains are calculated on-line either by enforcing system trajectories to follow the desired trajectories explicit model following    or by minimizing a quadratic cost function of the actual and the modelled states implicit model following . Recently AFTCS design schemes with explicit consideration of graceful performance degradation using explicit model-following approach have been proposed in   .




CHAPTER . ACTIVE FAULT TOLERANT CONTROL IN PROSPECTIVE

Feedback linearization is an established on-line reconguration technique applied to nonlinear systems 
     . Here an adaptive base-line controller is modied on-line by the output of a parameter estimation algorithm. To overcome diculties in existing on-line methods and to integrate the FDI scheme and on-line recongurable control law in a coherent manner without any pre-assumption of the knowledge of the post-fault system several integrated design approaches have been proposed recently    



 . An on-line reconguration method that does not require the use of FDI algorithms is the hybrid adaptive linear quadratic control proposed in  . Even though this design method does not need explicit fault information it has an on-line fault accommodation capability. A hybrid approach which designs a fault tolerant control system utilizing system redundancy has been employed  .

. Applications of FTCS FTCS is designed to increase reliability and automation level of modern control systems. The need to achieve the objectives of FTCS maintaining stability and satisfactory performance is apparent in several applications such as hazardous chemical plants   control of nuclear reactors     spacecraft   aircrafthelicopter 
 
  
    computer systems 
 space structure 
    missile guidance systems   underwater vehicles   automated highway systems     industrial systems        . As pointed out earlier these applications can be grouped into safety-critical systems cost-critical systems and volume-critical systems. In aircraft as an example of safety-critical systems the maneuvering capabilities were increased at the cost of reduced static stability. Since the system is open-loop unstable a well designed stabilizing FTCS is required to accommodate any possible fault in aircraft actuators sensors or ight computers hardware.

.. APPLICATIONS OF FTCS




Dependability is one of the key elements for practical usage of autonomous vehicles which are used for space and ocean exploration. Space structures as an example contains hundreds of sensors and actuators mounted on their surfaces. The huge cost to reach these structures enforces a long in-orbit mission time to minimize economical losses due to failed components. Availability has become a key incentive for automated production processes and supply utilities. With increasing level of automation to achieve desired quality plants have become more assailable to faults in system components. For such systems FTCS oers a possible way of obtaining increased availability hence protability. The substantial body of research in the area of FDI and recongurable control has lead to signicant results for each mechanism individually. Unfortunately the combination of FDIcontrol reconguration functions is not well addressed in the literature. This issue is very complex and vital. Complexity arises due to the stochastic behavior of the combined AFTCS and vitality is the natural result of practical constraints arising from application studies. Modelling AFTCS to account for practical limitations and stochastic stability of the developed AFTCS models are the prime focus of this book. To better characterize the stochastic behavior of AFTCS dierent concepts and tools used to study stochastic stability for general stochastic systems are presented with some detail in next chapter.

Chapter 3 STOCHASTIC STABILITY 3.1

Introduction

Stability is a qualitative property of a system, which can be studied without a direct recourse to solve the di
erential equations that describe the system. Stability is often the most important property of a dynamical system. In control engineering, one needs to make sure that the closed-loop system is stable (or at least stabilizable) before we can consider the design of a regulatory or tracking control device. As mentioned earlier, FTCSMP can be described by stochastic di
erential equations. The stochastic modelling is needed to capture the uncertainty about the environment in which FTCSMP are operating (measurement noise, disturbances, etc.), the randomly varying structure of FTCSMP due to faults and FDI decisions, and modelling uncertainties of FTCSMP due to inaccurate estimated parameters of the physical process being controlled. Therefore, study of the stability properties of FTCSMP leads to the study of the stability of the solutions to stochastic di
erential equations. Signicant e
orts have been dedicated to study the stochastic stability in the past. Substantial results have been reported in the literature with applications to physical and engineering systems. The concepts of stochastic stability are extensions to the deterministic counterparts in each of the common mode of convergence of probability theory. Since there M.M. Mahmoud, J. Jiang, Y. Zhang: Active Fault Tolerant Control Systems, LNCIS 287, pp. 22−29, 2003.  Springer-Verlag Berlin Heidelberg 2003

23

3.2. DEFINITIONS OF STOCHASTIC STABILITY

are three modes of convergence: convergence in probability, convergence in the mean, and almost sure convergence, there are at lease three times as many denitions for the stochastic stability as there are for the deterministic stability 104]. 3.2

Denitions of Stochastic Stability

In this book, we are concerned with the Lyapunov concept of stability. In simple words, Lyapunov stability criterion is concerned with the deviation of the system state vector from a given solution when the initial conditions are close to the initial conditions of the given solution. To dene stochastic stability one needs to state the concept of Lyapunov stability for deterministic systems. Without loss of generality, the equilibrium solution, x = 0, is considered as the solution whose stability property is being tested. Let xo be the initial state at the initial time to, the solution with initial state xo at time t will be denoted as x(t xo to) 2 Rn. Moreover, unless specied otherwise, the norm is dened as: kxkm =

Xn j xi jm i=1

The following denitions were developed in 104]. De nition 3.1

(Lyapunov stability) The equilibrium solution is said to be stable

if, for a given " > 0, there exists (" to ) such that for kxo k < , it follows that sup kx(t xo to )k < "

t to

De nition 3.2

(Asymptotic Lyapunov stability) The equilibrium point is said

to be asymptotically stable, if it is stable and if there exists  such that for kxo k < , one has lim kx(t xo to )k = 0

t!1

The denitions above can be interpreted geometrically as follows. As illustrated in Figure 3.1, given the circle A2 with radius R2 , then the equilibrium point is stable if

24

CHAPTER 3. STOCHASTIC STABILITY

it is possible to nd another circle A1 with radius R1 such that the trajectory which starts in A1 will never leave A2 regardless how large t becomes. The equilibrium point is asymptotically stable if it is stable and in addition a circle A3 can be found such that all trajectories starting in A3 will eventually tend to the origin x = 0. In the later case, if all trajectories while moving towards x = 0 have exponential envelop then the solution is exponentially stable.

R1 R3

R2 Stability in probability Asymptotic stability Exponential stability

A1 A2 A3

x=0

Figure 3.1: Illustration of Lyapunov stability. To extend the above concepts to the stochastic domain, the above denitions are expressed in terms of the three forms of convergence. Moreover, the solution x(t xo to ) is rewritten as x(t xo to w) to emphasize the fact that it is a stochastic process with a random variable w: Based on this, we can state the following stochastic stability concepts: The equilibrium solution is said to be stable in probability, if for a given " > 0,  > 0 there exists ("  to ) such De nition 3.3

(Lyapunov stability in probability)

that for kxo k < , it implies Prf sup kx(t xo to w)k > g < " t to

3.2. DEFINITIONS OF STOCHASTIC STABILITY

De nition 3.4

25

(Lyapunov stability in the mth moment) The equilibrium point

is said to be stable in the mth moment if the mth moment of the solution vector exists and for a given " > 0, there exists (" to ) such that for kxo k < , it implies Ef sup kx(t xo  to  w)km g < "

t to

De nition 3.5

(Almost sure Lyapunov stability) The equilibrium point is said

to be almost surely stable, if Prf lim sup kx(t xo  to w)k = 0g = 1

kxok!0t
to

The almost sure Lyapunov stability means that the equilibrium solutions are stable almost for all sample systems. That is, almost sure stability is the one that is closest to the deterministic system stability. Similarly, the asymptotic (deterministic) Lyapunov stability can be extended to deal with stochastic systems by rewriting Denition 3.2 in the following three modes of convergence: De nition 3.6

(Asymptotic stability in probability) The equilibrium point is

said to be asymptotically stable in probability, if it is stable in probability and if there exists  such that for kxo k < , it implies lim Prfsupkx(t xo  to  w)k > g = 0

!1 t
De nition 3.7 (Asymptotic stability in the mth moment) The equilibrium point is said to be asymptotically stable in the mth moment, if it is stable in the mth moment and if there exists  such that for kxo k < , it implies lim Efsupkx(t xo  to  w)km g = 0

!1 t
De nition 3.8 (Almost sure asymptotic Lyapunov stability) The equilibrium point is said to be almost surely asymptotically stable, if it is almost surely stable and if there exists  such that for kxo k < , it implies lim Prfsupkx(t xo  to  w)k > g = 0

!1

t


26

CHAPTER 3. STOCHASTIC STABILITY

An important asymptotic stability concept that has been studied and used for stochastic systems in engineering and applied science is the exponential stability of the mean which is dened as follows 96]: De nition 3.9

(Exponential stability of the mth moment) The equilibrium

point is said to be exponentially stable of the mth moment, if there exists constants a b > 0 such that for kxo k < ~ and all t to it implies

~ > 0,

and

Efkx(t xo  to)km g
bkxokm expfa(t  to)g Despite large number of stochastic stability concepts and denitions, not all will be of interest in the context of FTCS, because they may be too weak to be of practical signicance 104]. The proper concepts of stochastic stability as well as use of descriptive terminology remains to be settled as the subject evolves 107]. For applications to practical systems, we prefer those stability properties which are as close to the deterministic stability as possible. Therefore, we are more interested in almost sure sample stability properties. In stochastic systems, exponential stability in the mean square implies almost sure asymptotic stability 118]. As a result, in this book, we will continue to seek for the conditions that guarantee almost sure asymptotic stability and exponential stability in the mean square. Similar to the deterministic case, the Lyapunov's second (direct) approach is employed to study stability properties of stochastic systems. The Lyapunov function approach has the physical intuition that if the energy of a physical system is always decreasing near an equilibrium point, then that equilibrium point will be stable. 3.3

Conditions for Stochastic Stability

The rst attempt to extend the deterministic Lyapunov function approach to stochastic systems was made by Betram and Sarachick 11] and Kats and Krasovskii 96].

27

3.3. CONDITIONS FOR STOCHASTIC STABILITY

These two works concerned with the moment stability. Bucy 17] was the rst to realize that stochastic Lyapunov function must have the supermartingale property while studying nonlinear stochastic stability. A systematic treatment was carried out by Kushner 107, 108] and by Khasminskii 99] to study stochastic systems with white noise. Other works include Wonham 188, 189], Khashminskii 98], Kats 95], and Rabotnikov 157]. The Lyapunov function approach for stochastic systems is brie y outlined as follows. (3.1)

_( ) = ( ( ) ( ) )

x t

f x t

y t

t

where ( ) 2 0 `V (x y) = k(x y)  0 for all x 2 Qm .

3.3. CONDITIONS FOR STOCHASTIC STABILITY

Theorem 3.3

29

(Exponential stability in probability)

The equilibrium point, =

0 is exponentially asymptotically stable if V

( ) 0 x y

for all 2 m, and for some x

Q

>

 >

`V

0

:

( )= x y

V

( )
0 x y

x

Chapter 4 FTCS WITH MARKOVIAN PARAMETERS (FTCSMP) In Chapter 2, it was mentioned that faults may occur randomly at any instant of time, in any form, with various degree of severity, and in any system component. To represent the stochastic behavior of faults and fault-induced changes, a stochastic process, (t), is dened. The decisions of the FDI process are based on statistical tests, and they are not deterministic. A stochastic process, (t), is dened to represent these decisions. Both (t) and (t) are assumed to have Markovian characteristics with nite state spaces S = f1
2
:::
sg and R = f1
2
:::
rg, respectively. The assumption of Markovian behavior for the failure process is practical for physical systems. Typically, a control system is designed assuming that system components are fault-free. However, the system components may undergo certain faults. Fortunately, the occurrence of faults is not frequent instead is rare. For rare events, if we dene a small time interval h, then there is a denite probability that an event will take place and almost zero probability for the occurrence of two or more events in this time interval. Naturally, the behavior of rare events is best described by a Poisson process, which is also a Markov process 175]. An FDI process can be interpreted as a stochastic hypothesis test 168]. This test can be implemented using single sample test, moving window, or sequential test. M.M. Mahmoud, J. Jiang, Y. Zhang: Active Fault Tolerant Control Systems, LNCIS 287, pp. 30−44, 2003.  Springer-Verlag Berlin Heidelberg 2003

31

4.1. MATHEMATICAL MODELS

For single sample tests, information is collected, processed, and discarded. If the information is corrupted with additive white noise, the FDI process will be memoryless. That is, the future decision of the FDI process is independent of the past decision for the xed state of (t) and (t). Under these conditions, a Markov process can be used to describe the transition behavior of the FDI process 168]. Also, an FDI process can be reasonably represented by a Markov process if the time required by the FDI process to make a decision is su ciently smaller than the time interval between consecutive faults. In some physical systems, the Markovian assumption seems to be strong, however, it may still be satised by a change in the state structure. Hence, no physical system can ever be classied absolutely as either Markovian or non-Markovian 78]. There are several work that have been conducted to study the issue of the convergence of non-Markovian to Markovian processes 79, 171].

4.1 Mathematical Models 4.1.1 Dynamical system models The fault-free system to be controlled is described by: x_ (t) = f (x(t)
u(x(t)
t)
t)

(4.1)

If the system (4.1) is subject to random faults in actuators, and an FDI algorithm is used to monitor system performance, a FTCSMP can then be described by 168] x_ (t) = f (x(t)


(t)
u(x(t)
(t)
t)
t)

(4.2)

where x(t) 2 t)

= Prf Prtf+ >ttg

g = Prft
t

Prf

g

t + t

g

>t

(4.14)

The properties of the distribution function imply k Fij (t + t

j

> t)

=

k Fij (t +  t )


F (t )

1
Fijk (t)

The conditional transition rate of the FDI process is k qij

(

(t) = lim t

!0

k Fij (t + t

t

j

> t)

k ij

)

(4.15)

(4.16)

Therefore, it follows that k qij (t )

=

k fij (t)

1
Fijk (t)]

(4.17)

In terms of the reliability function k qij (t)

=


(dRijk (t)=dt) k Rij (t)

(4.18)

The calculation of the holding distribution and the reliability function can be found in the literature. For semi-Markovian FDI processes, the transition rates are calculated from the probability mass functions for the decision time statistics associated with various sequential test 184].

4.3. THE UNIQUENESS OF FTCSMP MODEL

4.3

37

The Uniqueness of FTCSMP Model

The model of FTCSMP resembles practical FTCS by dening one Markov process to represent the random failure process and the other Markov process to represent the FDI process. Ideally, the FDI should operate perfectly. That is, at any instant of time the state of the FDI process is the same as the state of the failure process, therefore, both processes can be represented by a single Markov process (similar to the existing results in the literature). However, practical FDI process cannot be expected to provide perfect information on the failure process. Detection delays, false alarms, missed detection, or errors in identication often exist in all FDI processes. Therefore, the existing results for hybrid systems and JLS cannot be directly applied to FTCS.

4.3.1 Errors in detection and identication In reality, the FDI process may provide incorrect decision, which means that the state of the failure process is k 2 S , and the state of the FDI process is i 2 R, and i 6= k. If one tries to deal with this situation using the existing results by assuming the prior knowledge of all error combinations for the failure process and FDI process, then the state of the failure process is implicitly assumed to be directly measurable. This is unrealistic in practical systems. Under this strong hypothetical assumption a controller that stabilizes the system can be designed independent of the errors in detection and identication, which is generally not true. A study for this case was carried out in 125].

4.3.2 Detection delays and false alarms It is even more di cult to represent time-delays and false alarms in FDI by the existing results, these two imperfections are largely related to the characteristics of the residual generation, the method used for residual evaluation, the choice of a decision threshold, and the environmental noise and disturbance. These factors cause

38

CHAPTER 4. FTCS WITH MARKOVIAN PARAMETERS (FTCSMP)

random detection delays and false alarm rates with any possible size. That is, the state space to represent these imperfections can be so large that it is impossible to pre-dene all possible combinations to apply the existing results to deal with practical FTCS. Another factor that limits the usage of the existing results is that there is a conict between sensitivity of the FDI process (to noise) and its alertness (to fault), which create contradictory objectives that cannot be achieved by the existing results. The impact of the detection delays on the stability of FTCSMP was studied in 125]. To summarize, the existing results of hybrid systems and conventional JLS assume that the plant regime is known. If one views these systems as special types of the FTCSs, it implies instantaneous detection and perfect identication of faults, which is not practical from FTCS point of the view. In fact, detection delays, and errors in fault detection and identication have to be considered in FTCSs. In most of the conclusions presented in this book, the results of JLS will be derived as interesting special cases of FTCSs. In the next section, we will study stability properties of FTCSMP, particularly exponential stability in the mean square 104]. It was mentioned that, in stochastic systems, moment stability implies almost sure stability 118], therefore, we will also study almost sure asymptotic stability of FTCSMP.

4.4

Stochastic Stability of FTCSMP

The Markovian assumption has been justied for FTCSMP. Following the techniques used to study stochastic systems, we will use Lyapunov's second (direct) method to seek for conditions for the stability of FTCSMP (4.7). Stochastic Lyapunov function approach is used to derive necessary and sucient conditions for exponential stability in the mean square, while the supermartingale property is employed for almost sure asymptotic stability. Before stating the conditions of exponential stability in the

39

4.4. STOCHASTIC STABILITY OF FTCSMP

mean square and almost sure stability of FTCSMP, we will dene some tools and properties that are useful to complete the analysis.

4.4.1 Supermartingales Dene a probability space (
= ).
is the space of elementary events, = is a eld which consists of all subsets of
that are measurable, and is the probability measure. Let t
= be a family of algebras of events in
dened for each  0 such that s
t for . The following denitions and theorems are due to Doob 43]. P




P










t

s < t

De nition 4.1 Let ( ) be a stochastic process with nite expectation Ef ( )g  t w

1, such that (

 t w

((

 t w

)

t

 t w




t

Theorem 4.1 If ( ) is a positive Supermartingale, then tlim !1 ( ) = 1 almost surely exists and is nite. Moreover, tlim !1Ef ( )g = Ef 1 g Theorem 4.2 If f ( ) t  0g is a non-negative supermartingale, then for any  t

 t w

 t w

 t w

 >







:

t

0, we have

Prf sup (

 1

0 t

 t w

)  g  Ef (0 )g 



w



4.4.2 Stochastic Lyapunov functions The stochastic Lyapunov function method is a way of obtaining information on a family of solutions of stochastic di erential equations without explicitly solving the equations. The conditions that a stochastic function must meet to qualify as a stochastic Lyapunov function are:

40

CHAPTER 4. FTCS WITH MARKOVIAN PARAMETERS (FTCSMP)

of the joint Markov process f
g quali es as a stochastic Lyapunov function candidate if the following conditions hold

De nition 4.2 The random function V (x

for some xed




= Vt((t)
t) + h((t)
t) = `V ((t)
t)

exists pointwise, and satis es lim !oEf Vt((t +  )
t +  )g + h((t +  )
t +  ) = Vt((t)
t) + h((t)
t)

where Ef V ((t +  )
t +  ) j((t)
t) g is the conditional expectation of the stochastic Lyapunov function at time t +  , given its value at time t. h((t)
t) is the weak innitesimal operator of the function V ((t)
t) , V (x(t)
(t)
(t)
t) for xed t.

4.4.4 Dynkin's formula Dynkin's formula plays a key role to extend the concept of Lyapunov function approach from deterministic to stochastic systems. The major diculty, in early development, was to nd a connection between the Lyapunov function V (x(t)
(t)
(t)
t), the weak innitesimal operator `V (x(t)
(t)
(t)
t), and the martingale theorem. Dynkin's formula clearly indicated this connection and played the same role to prove the stochastic analogous of the following deterministic equation which is crucial for Lyapunov's second (direct) method ( )

V xo

( )=

Zt

( ) =

Zt

0

_( )

V xs ds

k xs ds

V xt

0

42

CHAPTER 4. FTCS WITH MARKOVIAN PARAMETERS (FTCSMP)

De nition 4.4 45] For a right continuous strong Markov process fx(t)
(t) (t)g if a bounded function V (x(t)
(t)

t t) is in the domain of `, then for every random

( )

t = minft  g with Eft g < 1 and
xed s < 

EfV =

x t)
(t ) (t) t ) j Ns g
V (x(s)
(s)

( (


t

EfRs fVt x

t

(



s) + h(x




s s)

( )

s)g ds j Ns g

(4.19)

EfRs `V x s
s s s ds j Nsg where Ns is the 
algebras generated by the process fx
g up to time s. Equation =

( ( )

( ) ( )

)



(4.19) is called Dynkin's formula 45]. It should be noted that not every stochastic Lyapunov function possesses the supermartingale property. Lemma 4.1 states the condition for which the supermartingale property is guaranteed for a given stochastic Lyapunov function.

Lemma 4.1 Let V (x
 t) be a stochastic Lyapunov function and suppose that `V (x




t)  0. Then the process V (x(t )
(t ) (t) t ) is also a supermartingale.

Proof: Applying Dynkin's formula

EfV =

x  t )
( t )

( (

R
t

Efs `V

t) t ) jNs g
V (x(s)
(s) (s) s)

(

x s
(s)

( ( )

s s)ds jNs g  0:

(4.20)

( )

The left hand side yields

EfV

x t )
(t )

( (

t ) t)jNs g  V (x(s)
(s)

(

s s ):

( )

(4.21)

Since V (x(t)
(t ) (t ) t) is a stochastic Lyapunov function, it is positive-denite for all t  0: By Denition 4.1, V (x(t )
(t ) (t) t) is a positive Supermartingale. If the condition in Lemma 4.1 is satised for the chosen stochastic Lyapunov function, we can therefore apply the supermartingale theorems to derive conditions

43

4.4. STOCHASTIC STABILITY OF FTCSMP

for almost sure asymptotic stability and exponential stability in the mean square for the FTCSMP (4.7). 4.4.5 Conditions for stochastic stability

It was shown that f ( ) ( ) ( )g is a joint Markov process. However, only stability properties of ( ) are of prime interest. Therefore, we have to consider this fact when dening stability concepts for FTCSMP. In the following, we will dene almost sure asymptotic stability and exponential stability in the mean square for the FTCSMP (4.7). Conditions for both types of stability were developed in 168]. These theorems are stated here without proof to avoid repetition. Readers may refer to the mentioned citation for details. Without loss of generality, let's assume that the equilibrium point, = 0, is the solution at which the stability properties are examined. x t


t

t

x t

x

De nition 4.5 The equilibrium point,

almost surely stable for (

   to

t

0 If for any :


o

) 0 such that for any jj o = ( >

x
o

x

= 0 of the FTCSMP (4.7) is said to be

x

2

o o )jj t

Prf sup k (  0 t

1

x t xo to

De nition 4.6 The equilibrium point,

almost surely asymptotically stable for

x

)k

x t xo to

x

0 and some positive constants

o o )k  , the following inequality holds 8

a >



x t xo to


o

0 and

(

Efjj ( 

 >

0 there exists

0 we have



= 0 of the FTCSMP (4.7) is said to be

o )

t

g



t

0 and

) = 0g = 1

(

x
o

and

< 

exponentially stable in the mean square, if, for any >

 >

= 0 of the FTCSMP (4.7) is said to be

Prftlim ( !1


o

R

0, if it is almost surely stable and

t

De nition 4.7 The equilibrium point,

o 2

S

t

2 and o 2 S

b >

to

b t

there exist

0, such that when k

)jj2 g  jj o jj2 expf (  o )g a x

R

t

xo

=

CHAPTER 4. FTCS WITH MARKOVIAN PARAMETERS (FTCSMP)

44

Theorem 4.3 The equilibrium point, = 0 of the FTCSMP (4.7) is almost surely x

asymptotically stable if there exists a stochastic Lyapunov function,

V

(x




 t),

such

that `V

where

(

k x


 t) > 0

(x




 t) =

(

k x


is continuous in and x

Theorem 4.4 The equilibrium point,

(

 t)
0

k x


 t) = 0

only at

x

= 0:

stable in the mean square 8 

0

of the FTCSMP (4.7) is exponentially if there exists a stochastic Lyapunov function

for certain

0,

such that

t

V

(x




 t)

k1 k2 k3 >

(a )

k1

k

(b )

`V

(x

x

=0

( ) k2
V (x

x t




 t)





k3

k

 t)
k2 ( ) k2

x t

k

( ) k2

x t

Chapter STOCHASTIC STABILITY OF FTCSMP .

Introduction

Practical issues that arise in practical control systems add extra complexities when analyzing the behavior of FTCSMP. In this chapter
we will study the stochastic stability of FTCSMP subject to constraints due to the environment and the system to be controlled. Namely
we are concerned with the environmental noises and the existence of multiple faults in the system to be controlled. In Section .
the problem of the inclusion of environmental noises in the dynamical model of the FTCSMP is addressed. Three types of noises are considered state-dependent
control-dependent and purely additive Gaussian white noise. Section . deals with FTCSMP with multiple failure processes. In particular
two independent failure processes with Markovian characteristics are considered one for plant component faults and the other for actuator faults. The analysis is carried out using the Lyapunov function approach. A dynamical FTCSMP model
in each case
will be developed rst. Conditions for the stochastic stability of the FTCSMP will be stated and derived subsequently. The theoretical results will further be illustrated by numerical examples. M.M. Mahmoud, J. Jiang, Y. Zhang: Active Fault Tolerant Control Systems, LNCIS 287, pp. 45−82, 2003.  Springer-Verlag Berlin Heidelberg 2003


.

CHAPTER . STOCHASTIC STABILITY OF FTCSMP

Stochastic Stability of FTCSMP in the Presence of Noise

The FTCS can be modelled by stochastic dierential equations and can be viewed as general hybrid systems. The stochastic stability of hybrid systems was studied in    and for JLS in   
  . FTCSMP being another class of hybrid systems was de ned as one type of FTCSs. Sucient conditions for the stability of FTCSMP with detection delays were derived in . These results are unfortunately based on the assumption of certainty of correct fault identi cation. To overcome this limitation two random processes with dierent state spaces were de ned and used to analyze the stochastic stability of FTCSMP in 
. Signi cant results were obtained in 
 without consideration of noise. The FTCSMP model was proposed to better describe practical FTCS where modeling the failure process and the FDI process separately makes the consideration of delays and errors in detecting and isolating the faults much simpler. However practical FTCS are also subject to environmental noises and disturbances which could also aect the behavior of such systems considerably. In this section we will examine the stochastic stability of FTCSMP in the presence of white Gaussian noise. In particular necessary and sucient conditions for the stochastic exponential stability in the mean square are derived. The signi cance of these results is that they are applicable to FTCS without the restrictive assumptions of instantaneous fault detection certainty of correct identi cation or noise-free environment.




.. STOCHASTIC STABILITY OF FTCSMP IN THE PRESENCE OF NOISE

.. Dynamical model of FTCSMP in the presence of noise Consider general AFTCS as shown in Figure .. The noise-free system was given in  . which is rewritten here for convenience

xt

_෈ ෉ =

fxt
t uxt ෈

෈ ෉

෈ ෉

෈

t t t

෈ ෉ ඵ෈ ෉

෉

.

෉

where
t is the actuator failure process and t is the FDI process. The two processes were dened in Section ... To account for environmental noises and unknown disturbances a di usion term is added. The system can then be represented by

෈ ෉

dx t

෈ ෉ =

ඵ෈ ෉

fxt
t uxt ෈

෈ ෉

෈ ෉

෈

t t t dt g x t
t u x t

෈ ෉ ඵ෈ ෉

෉

෉

+

෈

෈ ෉

෈ ෉

෈

.

t t t n t dt

෈ ෉ ඵ෈ ෉

෉

෉

෈ ෉

where g x t
t u x t t t t is an n ථ m matrix-valued function and n t represents m-dimensional white Gaussian noise. ෈

෈ ෉

෈ ෉

෈

෈ ෉ ඵ෈ ෉

෉

෉

෈ ෉

Reconfiguration Mechanism

FDI

Plant

Actuators

Input +

Outputs/States

Sensors

_

Faults

Noise

Controller

Figure . AFTCS in noisy environment. Recall that the white Gaussian noise is dened as n t

෈ ෉ =

dW t dt

. 

෈ ෉

where W t is a Wiener process. The system can be written as

෈ ෉

dx t

෈ ෉ =

fxt
t uxt ෈

෈ ෉

෈ ෉

෈

t t t dt g x t
t u x t

෈ ෉ ඵ෈ ෉

෉

෉

+

෈

෈ ෉

෈ ෉

෈

t t t dW t

෈ ෉ ඵ෈ ෉

෉

෉

෈ ෉

. 




CHAPTER . STOCHASTIC STABILITY OF FTCSMP

In order for the system .  to have a solution at x෈t෉ = 0 it is necessary that f ෈0
෈t෉ u෈0 ඵ෈t෉ t෉ t෉ = 0

.

g෈0
෈t෉ u෈0 ඵ෈t෉ t෉ t෉ = 0

.

and

Furthermore it is assumed that the growth condition

k f ෈x෈t෉
෈t෉ u෈x෈t෉ ඵ෈t෉ t෉ t෉ k+k g෈x෈t෉
෈t෉ u෈x෈t෉ ඵ෈t෉ t෉ t෉ k k෈1+ k x k෉ . and the uniform Lipschitz condition

k f ෈~x
u෈~x ඵ t෉ t෉
f ෈x
u෈x ඵ t෉ t෉k +

.


kg෈~x
u෈~x ඵ t෉ t෉
g෈x
u෈x ඵ t෉ t෉k kkx~
xk are both satis ed for some constant k. Under these two conditions the solution of the system equation .  determines a family of continuous stochastic processes x෈t෉ one for each choice of the random variable x෈to ෉. Moreover the joint process fx
ඵg = fx෈t෉
෈t෉ ඵ෈t෉ t0 t tf g is a Markov process. A linear form of the FTCSMP .  with the assumption of possible actuator fault can be represented as

x_ ෈t෉

=

A෈t෉x෈t෉ + B ෈
෈t෉෉u෈t෉ + D෈x෈t෉
෈t෉ t෉ W_ 1 ෈t෉ +

.

E ෈u෈t෉
෈t෉ t෉ W_ 2 ෈t෉ + F ෈
෈t෉෉ W_ 3 ෈t෉

where Wi෈t෉ are independent Wiener processes in n i = 1 2 3. D෈x෈t෉
෈t෉ t෉ E ෈u෈t෉
෈t෉ t෉ and F ෈
෈t෉෉ are functional matrices of dimension nථn . Three types of noise are considered in equation . state-dependent noise D෈x෈t෉
෈t෉ t෉W_ 1 ෈t෉ control-dependent noise E ෈u෈t෉
෈t෉ t෉W_ 2 ෈t෉ and state and control independent noise i

i

.. STOCHASTIC STABILITY OF FTCSMP IN THE PRESENCE OF NOISE




F ෈ ෈t෉෉ W_ 3 ෈t෉ which will herein be referred to as independent noise. D෈x෈t෉
෈t෉
t෉ and E ෈u෈t෉
෈t෉
t෉ are assumed to be linear in x෈t෉ and u෈t෉ respectively as follows D෈x෈t෉
෈t෉
t෉ = P Dl ෈ ෈t෉෉ xl ෈t෉ n

E ෈u෈t෉
෈t෉
t෉ =

l=1 m

P El

l=1

. 

t ul ෈t෉

෈ ෈ ෉෉

where xl ෈t෉ ul ෈t෉ Dl and El are the lth component of the state trajectory the control input the state-dependent noise and the control-dependent noise distribution matrices respectively. In FTCSMP the control law is recon gured according to the decision of the FDI process ඵ෈t෉. Furthermore under the assumption that perfect state information is available the representation of linear FTCSMP becomes x_ ෈t෉ = A෈t෉x෈t෉ + B ෈ ෈t෉෉u෈x෈t෉
ඵ෈t෉
t෉ D෈x෈t෉
෈t෉
t෉W_ 1 ෈t෉ + E ෈u෈t෉
෈t෉
t෉W_ 2 ෈t෉ + F ෈ ෈t෉෉W_ 3 ෈t෉

+

. 

u෈x෈t෉
ඵ෈t෉
t෉ = K ෈ඵ෈t෉෉x෈t෉

In the sequel the following notations will be used B ෈ ෈t෉෉ = Bk  F ෈ ෈t෉෉ = Fk  D෈ ෈t෉෉ = Dk and E ෈ ෈t෉෉ = Ek when ෈t෉ = k
S and u෈x෈t෉
ඵ෈t෉
t෉ = ui when ඵ෈t෉ = i
R. Also denote x෈t෉ = x
෈t෉ =
ඵ෈t෉ = ඵ and the initial conditions x෈to෉ = xo
෈to ෉ = o
ඵ෈to ෉ = ඵo. The inner product of two vectors will be denoted by h
i. .. Sucient conditions for stochastic stability

It is important to emphasize that the asymptotic stability cannot be achieved in the presence of the independent noise F ෈ ෈t෉෉ W_ 3  In this case only bounded-input bounded-output stability can be established for example obtaining bounded variance for the state process. Therefore in this section the conditions for the asymptotic stochastic stability of FTCSMP are analyzed without considering this noise term. The




CHAPTER . STOCHASTIC STABILITY OF FTCSMP

eect of this term is considered later by suitable choice of the stochastic Lyapunov function as discussed in Section ... Almost sure asymptotic stochastic stability and the stochastic exponential stability in the mean square are dened for the FTCSMP  . . Su cient conditions are derived employing the stochastic Lyapunov function approach and the positive Supermartingale property. Recall that not every stochastic Lyapunov function possesses the supermartingale property. Lemma . stated the condition for which the supermartingale property is guaranteed for a given stochastic Lyapunov function. If this condition is satised we can apply the supermartingale theorems to derive conditions for the almost sure asymptotic stability and the exponential stability in the mean square for the FTCSMP  . . De nition
. The equilibrium point

be almost surely stable for exists 

   to

t


of the system equation
. is said to

x


If for any



such that for any jj 

o



xo

x o

Prf sup k   0යtය1

x t xo to

De nition
. The equilibrium point

t

Prftlim  !1

o o jj

k

t

g



 

 

k

xo



x o



t



x

x t xo to

t

 


there


we have


of the system equation
. is said a 

o


and

the following inequality holds 

Efjj  

and


and


g 


and some positive constants

o o k 

 



to be exponentially stable in the mean square if for any exist  o o 

R


if it is almost surely stable and

x t xo to

De nition
. The equilibrium point

o


S


of the system equation
. is said to

x

be almost surely asymptotically stable for




t


and o
S

b 

to

jj2 g  jj o jj2 expf   o g a x

b t

t

R

there


such that when




.. STOCHASTIC STABILITY OF FTCSMP IN THE PRESENCE OF NOISE

Theorem . The equilibrium point


of the FTCSMP
. is almost surely

x

asymptotically stable if there exists a stochastic Lyapunov function

V



x


  such t

that 

V

Proof  By Lemma .
  V

. for any

 

x


 



t

 


N x


t



  is a positive Supermartingale. Using Theorem

x


t


it follows that

Prf sup

to යtයtf

V



 
g  Ef 

x


t

V



o
o

x

 o o g

 .


t



Since    is a stochastic Lyapunov function it satis es the conditions of De nition . i.e.    
as 
. De ne V

x


t

V

x


t

x

V





o o 

o
o

x

 .


t



For the positive-de nite function    and using C eby sev inequality  as f   for some 1
 equation  .
 becomes V

t



x


t



Prf sup k   k
1g  to යtය1

x t



 .




According to De nition .
 the system is almost surely stable. Theorem .
implies that almost surely lim Ef        g

t!1

Therefore 

V

x t


t

1  

 .


         

 .


t

t

V

 and



s  t

t

Ef        j sg V

x t


t

t

t

N

V

x s


s

s

s

De ne an open set s f      

g  f     time s f s     sg  s   o  For  it is true that O



t

x t

O

Ef sg  Ef



Z1



to

x t

t

N x


t



t




s

s

x

V

x


t

 

g and the



       

N x s

 

s ds

j sg   N

V

o
o

x

o o   t



 .





CHAPTER . STOCHASTIC STABILITY OF FTCSMP

Therefore s
and s
as to
with probability one This is equivalent to say that any state trajectory x
t of the system  . beginning in the region 
k x k
m will almost surely reach the boundary of this domain in nite time for any suciently small m  and   . This implies that N
x   t has an innitesimal upper limit hence tlim !1N
x   t   Since the function N
x   t is by assumption continuous positive denite then tlim !1 k x
t k  The proof is completed.

Theorem . The equilibrium point x   of the FTCSMP
. is exponen-

tially stable in the mean square  t  if there exist a stochastic Lyapunov function V
x   t and some k1  k2  k3 

such that


a k1 k x
t k2  V
x   t  k2 k x
t k2
b V
x   t  k3 k x
t k2

 . 

Proof  By Lemma .  V
x
t  
t  
t t  is a positive Supermartingale. Theorem . states that lim !1EfV
x
t 
t 
t tg  V1


t

 . 

almost surely exists with a nite expectation. Applying Dynkins formula to this bounded function it follows that

EfV
x
t 
t 
t t j
xo o og  V
xo o o  Ef0 V
x
s 
s 
s s ds j
xo o og Rt

 .


Dierentiating  .
 with respect to t we have d dt EfV
x
t 
t 
t t g  EfV
x
t 
t 
t t g

 .


Taking the expectation and using the assumptions a and b in Theorem .
 we have




.. STOCHASTIC STABILITY OF FTCSMP IN THE PRESENCE OF NOISE

i
1 Efk k2 g Ef k

x

ii
Ef

V

x





t

V




x


t

g

k2

Efk k2 g x

g
3 Efk k2 g k

x

Substituting ii
for the right-hand side of  . d dt

Ef

V

x t
t


t

Ef




t

g
3 Efk k2 g k

 .

x

By Inequality i
 V

x


t

k2

g Efk k2 g

 .


x

That is d dt

Ef

V

x t
t

Integrating with respect to for t

Ef

V

x t
t


t

t

g

to

V

g
3 Ef

V

x






t

t



it can be shown that

xo
o

k


o

to 

t

g

 . 

k2

f
3 g k

exp

k2

 

t

 . 

t

According to De nition .
 the system is exponentially stable in the mean square. The proof is completed. .. Necessary conditions for stochastic stability

In this section necessary conditions for exponential stability in the mean square of the FTCSMP  .  are derived. The result is then used to prove the necessary condition for the exponential stability in the mean square and to provide the sucient condition for almost sure asymptotic stability in probability of the FTCSMP  ..

Theorem . If the equilibrium point

of the FTCSMP
. is exponentially stable in the mean square and the functions
  and
  have continuous bounded derivatives with respect to up to the second order then x



f x
u x

t

t

g x
u x

t

t

x

there exists a function in Theorem ..

V

x





t

  and
 satisfying conditions


S

R

a

and

b




CHAPTER . STOCHASTIC STABILITY OF FTCSMP

Select a stochastic Lyapunov function as follows

Proof

tZ+T

V




x


t



Efk

x 

k2 g

 .

d

t

We will prove that this function satises the conditions of Theorem . for some . For exponentially stable system in the mean square and for some  and  T 

a 

b 

tZ+T

V

x





t

k

a

k2 expf


x t

b 


g t



d

k

a

x t

tZ+T

k2

exp

t

or some

T 

f



g

b 

t

t

d

 . 





ᝇbT
 k  k2  2 k  k2  .
 At the equilibrium point   both
 and
 are equal to zero as indicated in  .  and  .. They also satisfy the growth and the Lipschitz conditions and have bounded partial derivatives with respect to that is k
k 1 k k V

x





a

t

e

b

x t

x

f x
u

k

t

x t

g x
u

t

x

f x
u

k

g x
u

t




t

k

C

x t

C

1 kx tk

 . 

For every realization of FTCSMP  .
 the weak innitesimal operator associated with the function
 Efk  k2 g is V

V

x





t



h

x


t

f x
u




x t

t

V

x




x


t

i   h

 x
u




t

V

xx

where
 
 T

 x xx rst and the second partial derivatives with respect to . Therefore  x
u

t

g x
u

t g

x
u

t  V

x


t




x


V

x





t

i  .

t

are the

x

k

V

x





kh

f x
u

k

f x
u

t







k t

t

x

V

kk

x


V

x




x


t

i  12 h




t

k  21 k




xx

t

V

 x
u




 x
u

t

x


kk




V

xx

t

 .

ik

x





t

k

.. STOCHASTIC STABILITY OF FTCSMP IN THE PRESENCE OF NOISE

Since derivatives are bounded up to the second order
we have

k

x ෈x

V




ඵ

t෉

k

f

k

k

k k

x෈t෉

V

xx ෈x

ඵ

t෉

k2 =

C2




k

 .

g

k

That is

k Efk 

x෈t෉

k2 g k

k

f C1

k

k2 + 21

x෈t෉

Applying Dynkins formula to the function k

Efk


x෈t

C2

j ෉ k2 g
k

+T

t+ RT t

x෈t෉

t

Efk

x෈ ෉

k2 g

d

=

k

2

g 1

k C

x෈t෉

k2 =




C2 V

x෈t෉

x෈t෉

k2

 .

k2 we have

t+ RT

Ef k 

t ෈x

k




ඵ

x෈ ෉

k2 g  d

 .

t෉

Since the FTCSMP  . is exponentially stable in the mean square
there exists some 0 such that  T 

Efk

x෈t

+T

j ෉ k2 g t



k

x෈t෉

k2

 ෈0 1෉



 . 

Applying this inequality in  .
we have V

෈x

ඵ




t෉

 1k k

x෈t෉

k2

 . 

This proves the inequality a in Theorem .. To verify the required smoothness of the function ෈ ඵ ෉ in inequality b
we use the following corollary  . V

x


t

Corollary . The weak in nitesimal operator of the function tZ+T V

෈x




ඵ

t෉

=

Efk

x෈ ෉

k2 g

d

t

is V

෈x




ඵ

t෉

=

Efk

x෈t

+T

j ෉ k2 g
k t

x෈t෉

k2

From this corollary and  . 
we obtain V

෈x




ඵ

t෉

෈


1෉ k

x෈t෉

k2 =
3 k k

x෈t෉

k2

This completes the proof of Theorem .. An important result for linear FTCSMP  .

 is stated in Theorem ..

 .




CHAPTER . STOCHASTIC STABILITY OF FTCSMP

Theorem . If the FTCSMP
. is exponentially stable in the mean square then for any given positive-denite function

L෈x


ඵ

t෉

which is bounded and continuous

and ඵ  there exists a positive denite function ෈ same order which satises the conditions in Theorem
. such that t




L෈x

0









ඵ

S

t෉

R

V

x


V

ඵ ෈x

t෉




of the ඵ

t෉

=

 0

Proof The proof can be completed using the similar arguments in the proof of Theorem . and by selecting the stochastic Lyapunov function as tZ+T V

෈x




ඵ

t෉

=

EfW ෈x෈ ෉


෈ ෉

ඵ෈ ෉

g

 ෉ d

 .

t

The positive-de nite function ෈ ඵ ෉ in fact satis es the conditions in both Theorem . and Theorem .. This result con rms the one reported in

  which states that exponential stability in the mean square of a stochastic system implies its almost sure asymptotic stability in probability. In other words a sucient condition for the almost sure asymptotic stability in probability for the equilibrium point of the FTCSMP  .

 can be established by establishing the exponential stability in the mean square. Therefore only one set of conditions needs to be satis ed to guarantee both types of stochastic stability. L x


t

.. A necessary and sucient condition for exponential stability Theorems . - . were obtained for general FTCSMP  .. However the results are dicult to test and verify. In this section we will derive a testable necessary and sucient condition for the exponential stability in the mean square for the linear FTCSMP in  .

.

.. STOCHASTIC STABILITY OF FTCSMP IN THE PRESENCE OF NOISE




The weak innitesimal operator of the FTCSMP  . with the stochastic Lyapunov function V x

t at the point fx
 k
 i tg is V x





t



Vt x k i t  h Ax  Bk ui  Vx x k i ti

P kj V x j i t 

j 2S j



V x k i t

=k

1

P qk

j 2R j

6

2



=i

ij V x k j t V x k i t

 . 

6

xT  Vxx k i t x  21 uTi  Vxx k i t ui  12 tracefFkT Vxx x k i tFk g

where





Vxx k i t  tracefDklT Vxx x k i tDkc g
l c     n

 . 

Vxx k i t  tracefEklT Vxx x k i tEkc g
l c     m

Dkj and Ekj are dened in  .  for
t  k. For the FTCSMP  . a quadratic form with an added scalar term is selected as follows V x





t  xT P





t x  v





t

 .

where P

t and v

t are a positive symmetric matrix and a positive scalar respectively

 k  S and
 i  R The scalar term is introduced to account for the independent white Gaussian noise term F
t W 3 t. To simplify the notation the time dependence of P

t and v

t is dropped. For this stochastic Lyapunov function the weak innitesimal operator under the state feedback control




CHAPTER . STOCHASTIC STABILITY OF FTCSMP

law ui

Ki x is

V
x


t xT Pki x  xT Pki Ax  xT AT Pki x xT Pki Bk Ki x xT KiT BkT Pki x

xT tracefDklT Pki Dkcgx  xT KiT tracefEklT Pki EkcgKi x xT f P kj Pji Pki 


j 2S j

j




j 2S j

j 2R

=k 6

vki P kj vji vki  =k

=i

qijk Pkj Pki gx

6




P

j 2R j

6




P

=i

qijk vkj vki  tracefFkT Pki Fk g

6

 . Dene Aki A Bk Ki

1 2

I

P

j 2S j

=k

kj

1 2

I

P

j 2R j

6

=i

qijk

6

ki
Pki
k
i tracefDklT Pki Dkc g
l
c
  
n

 . 

ki 
Pki
k
i tracefEklT Pki Ekc g
l
c
  
m The weak innitesimal operator can then be rewritten as V
x


t xT fPki  ATki Pki  Pki Aki 

P

j 2S j




vki  P kj vji vki  j 2S j

=k 6

=k 6

P

j 2R j

=i

kj Pji 

P

j 2R j

=i

qijk Pkj  ki  KiT kiKi gx

 .

6




qijk vkj vki  tracefFkT Pki Fk g

6

Theorem . A necessary and su cient condition for the exponential stability in the mean square of the FTCSMP
. under the control law ui

Ki x
is that there

exist bounded steady-state solutions Pki  to the following coupled matrix dierential




.. STOCHASTIC STABILITY OF FTCSMP IN THE PRESENCE OF NOISE

k
S i
R

equations

ATki Pki t Pki t Aki

Pki t _

t  

as

෈ ෉ + ~

෈ ෉ ~

෈ ෉+

X +

j 2S j

X


kj Pji t

෈ ෉+

=k

j 2R j

6

qijk Pkj t

෈ ෉+ද

T ki + Ki ᝇki Ki + Qki = 0

=i 6

 .  Qki  Aki

for

~

0

X

vki t _

෈ ෉ +

j 2S j

ද

ki

ki

and ᝇ


kj vji t

෈ ෉

are de
ned in .. Moreover

 vki ෈t෉




X +

j 2R

=k

j

6

qijk vkj t

෈ ෉

 vki ෈t෉




tracefFkT Pki t Fk g

+

෈ ෉

= 0

=i 6

 . Pki

with

and

෈0෉ = 0

vki

෈0෉ = 0

k
S i
R

Assume that the FTCSMP  . is exponentially stable in the mean square under the control law ui Kix i
R. Let L x  t xT Q  t x  . Then by Theorem . there exists a function that satises both the boundedness and the smoothness conditions of Theorem . such that Proof of Necessity

=

෈

ඵ

෉

෈

ඵ

෉

=

0

V x  ෈

ඵ

t

෉ =

L෈x



ඵ

t

෉

 k
S

0

=

ඵ =

i
R

 .

For the quadratic function  . the weak innitesimal operator is given in  .. From  . we have xT fPki t _

ATki Pki t

෈ ෉ + ~

෈ ෉+

Pki t Aki ෈ ෉ ~

+

P

j 2S j

=k

P

j 2S j

+

P

j 2R j

=i 6

qijk vkj t

෈ ෉

 vki ෈t෉




+

෈ ෉ +

=k

P

j 2R j

6

ki + KiT ᝇki Ki gx + v_ ki ෈t෉ + v_ ki ෈t෉ +

+ද


kj Pji t


kj vji t

෈ ෉

=i

qijk Pkj t

෈ ෉

6

 vki ෈t෉




6

tracefFkT Pki t Fk g ෈ ෉

+

xT Qki x

= 0

 . 




CHAPTER . STOCHASTIC STABILITY OF FTCSMP

The left-hand side must sum to zero for any fx k S i xT fP
ki t  ATki Pki t  Pki tAki 

tg that is

P
kj Pjit

j 2S j

=k 6

 P qk Pkj t  ki  K T ki Ki  Qkigx  j 2R j

=i

ij

.

i

6

and v
ki t 

X

j 2S j





kj vji t
vkit 

=k

X

j 2R j

6

qijk vkj t
vkit  tracefFkT Pki tFk g 


=i 6

.
For the quadratic term .  de ne the transition matrix associated with Aki as

ki t    expAki t
   k S i R

.

Then the solution of the coupled di erential equations Pki t under the boundary condition Pki    is Pki t 

Z0

ᝇ1

X

X

Tkit   
kj Pji    j 2S j

=k 6

j 2R j

qijk Pkj    ki  KiT kiKi  Qki kit   d

=i 6

. Similar type of coupled ordinary di erential equations were studied in . The conclusion is that for a non-singular transition matrices ki and a positive-de nite matrices Qki the solutions are unique continuous for t 
  and Pki t   k S i R. Once Pki are known equation .
can be solved backwards in time with the boundary conditions vki   .

Proof of Su ciency


If the steady-state solutions fPki t  k S i Rg for the coupled matrix di erential equations under the boundary conditions Pki   exist then the function V x  t  xT P  tx v t is a stochastic Lyapunov




.. STOCHASTIC STABILITY OF FTCSMP IN THE PRESENCE OF NOISE

function satisfying the condition a in Theorem .. Under the control law u the weak innitesimal operator
V
x   t is given by ..

i

ᝇK x i

The steady-state solutions for P
t imply that they satisfy the coupled matrix di erential equations in Theorem .. Adding . and .  with simple manipulation we obtain . . That is ki


V
x   t  L
x   t  8  2 S  2 R

Since L
x   t   then
V
x   t FTCSMP .

 under the control law u square 8 t ර  Hence the proof.

i

. 

ᝇL
x   t   by Theorem . the ᝇK x is exponentially stable in the mean i

By virtue of Theorem . the existence of positive steady-state solutions P
t for a given state feedback control law satises the necessary and sucient condition for the exponential stability in the mean square of the FTCSMP .

. Theorem . guarantees that these bounded solutions are also sucient for the almost sure asymptotic stability in probability. ki

..

A numerical example

To illustrate the theoretical results presented in the previous sections lets consider a rst order system with a possible fault in the actuator. The system and the related parameters are given as



 B1   Normal A  B

 B2   Faulty The actuator failure rates are chosen to be 12  21  




CHAPTER . STOCHASTIC STABILITY OF FTCSMP

The conditional transition rates of the FDI process are 1

q ij

2 66

 64

 


3 7 7
7 5

 


q

2

ij

2 6  
 
6 6 4




3 7 7 7 5




The noise distribution matrices are assumed to be D1






D2 F




 


1

E1
F2




 





E2








The positive-denite matrices are given by 
  . Note that the open-loop system is deterministically unstable. The objective is to investigate the stochastic stability of the closed-loop system by testing the existence of the steady-state solutions f
  g under some control law
 . Furthermore the eect of noise on the stochastic stability is examined. Qki

Pki 

i


k

S
i

k

S
i

R

R

ui

Ki x

R

Case
stochastic stability with dierent feedback control gains

In this case the stochastic stability of the system is examined for two sets of gains f
  g and 2 f
  g. The solutions
 are shown graphically 1 in Figure .
and Figure .  respectively. For the rst set of gains steady-state solutions exist. However the solutions are unbounded as   for the second set of gains. According to Theorem . the system is exponentially and almost surely asymptotically stable for 1  but not for 2 . The scalars associated with 1 are  

12    21   22    they are obtained by solving 11 equation . backwards in time with the boundary conditions

. It is worthwhile to mention that the deterministic stability does not imply the stochastic stability. It is easy to verify that the deterministic stability for closed-loop system is guaranteed as long as 1 
 and 2 
Even thought the selected gains in the above two sets satisfy the deterministic closed-loop stability however the stochastic stability is only ensured for 1 . Similar results which state that the K

K

Pki 

t

K

v




v




K

v




K

v

vki

K i 

K i 

K




.. STOCHASTIC STABILITY OF FTCSMP IN THE PRESENCE OF NOISE

deterministic stability is neither sucient nor necessary for the stochastic stability were obtained in several other works   

  .

0.6

0.35 P11

Solution of differential equations

Solution of differential equations

0.7

0.5 0.4 0.3 0.2 0.1 0

-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time (sec)

0.2

0.1 0.05 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time (sec)

0

5 P21

Solution of differential equations

Solution of differential equations

P12

0.15

0

0

7 6

0.3 0.25

5 4 3 2 1 0-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time (sec)

4.5 4

P22

3.5 3 2.5 2 1.5 1 0.5 0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time (sec)

0

Figure .  Bounded solutions for the state feedback gain

K

f
 


0


 g.

Case
stochastic stability with dierent independent noise intensities

In this case the intensity of the independent noise is increased as follows F1


 




F2


 

Table . illustrates that the steady-state solutions  exist and their magnitudes are the same regardless of the amount of increase in the noise intensity. However the magnitudes of the scalars increased. This means that the additive noise will not alter the stochastic stability of the system but it will enlarge the variance of the state trajectory. Pki 

vki




CHAPTER . STOCHASTIC STABILITY OF FTCSMP 40

Solutions of differential equations

Solutions of differential equations

45 P11

35 30 25 20 15 10 5 0

P12

16 14 12 10 8 6 4 2 0

-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 Time (sec)

14000

-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 Time (sec)

10000 P21

12000

Solutions of differential equations

Solutions of differential equations

20 18

10000 8000 6000 4000 2000 0

9000 8000 6000 5000 4000 3000 2000 1000 0

-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 Time (sec)

P22

7000

-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 Time (sec)

Figure . Unbounded solutions for the state feedback gain

f
 


K


 g.

Table . Stability parameters for dierent independent noise intensities.




F1 F2



P




v

F1 F2



   



P

v

    


    

  


    

       

   

   

    

Case
stochastic stability with dierent state-dependent noise intensities

In this case if the intensity of the state-dependent noise is increased to D1


 




D2


 

.. STOCHASTIC STABILITY OF FTCSMP IN THE PRESENCE OF NOISE




The steady-state solutions cannot be reached and become unbounded. This means that increasing the state-dependent noise will alter the stochastic stability of the closed-loop system. If the intensity is too high the stochastic stability could be lost. This case is shown in Figure
.. Pki

250

Solutions of differential equations

Solutions of differential equations

350 P11

300 250 200 150 100 50 0 -100

150 100

-90 -80 -70 -60 -50 -40 -30 -20 -10 0 Time (sec)

Solutions of differential equations

Solutions of differential equations

8

P21

7 6 5 4 3 2 1 0

50 0 -100

-90 -80 -70 -60 -50 -40 -30 -20 -10 0 Time (sec)

4 9 x 10

10 x 10 9

P12

200

-100-90 -80 -70 -60 -50 -40 -30 -20 -10 0 Time (sec)

8 7

P22

6 5 4 3 2 1 0 -100-90 -80 -70 -60 -50 -40 -30 -20 -10 0 Time (sec)

Figure
. Unbounded solutions for high intensity of state-dependent noise.

Case
stochastic stability with dierent control-dependent noise intensities

In this case only the control-dependent noise is increased as E1


 




E2


 

The solutions become unbounded therefore the closed-loop system is stochastically unstable. Figure
.
illustrates the results. Pki

9 7 x 10

9 10 x 10 9 P11 8 7

Solutions of differential equations

Solutions of differential equations

CHAPTER . STOCHASTIC STABILITY OF FTCSMP

6 5 4 3 2 1 0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0

5 4 3 2 1 0 -100-90 -80 -70 -60 -50 -40 -30 -20 -10 0 Time (sec)

2.5

P21

2 1.5 1 0.5 0 -100

12 x 10

12

Solutions of differential equations

Solutions of differential equations

Time (sec) 3 x 10

P12

6

3

P22

2.5 2 1.5 1 0.5

0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 Time (sec)

-90 -80 -70 -60 -50 -40 -30 -20 -10 0 Time (sec)

Figure
.
 Unbounded solutions for high intensity of control-dependent noise. Table
. Stability with increased intensities of dependent noise. state-dependent noise

f

control-dependent noise

g f
 
 g f

D1

D2

Pki

 2 




vki




E1

E2

g f
 
 g


 3 

Pki

 10 

vki




 11 

P11

 

v11

 

P11

 

v11

 

P12

 

v12

 

P12





v12

 

P21

 

v21

  

P21

  

v21

 

P22

 

v22

 

P22

 

v22

 

















































Table
. summarizes the solutions and after  seconds when noise intensities have been increased by a factor of . Note that the solutions increase Pki

vki

Pki

.. STABILITY OF FTCSMP WITH MULTIPLE FAILURE PROCESSES




signicantly with the increased state-dependent or control-dependent noise.

.

Stability of FTCSMP with Multiple Failure Processes

The main objective of FTCS is to maintain the system stability and acceptable degree of performance in the event of malfunctions in sensors actuators plant components or any combination. It has been emphasized that faults are random in nature they may occur at any instant of time with various sizes and in any system component. To represent the random nature of the actuator faults a stochastic process ෈t෉ was dened. ෈t෉ is assumed to have Markovian characteristics with nite state space S = f1
2

sg and transition probabilities as given in . . For the FTCSMP model to better approximate practical control systems faults in other system components have to be considered. However it was mentioned in Chapter that sensor faults can be more eectively dealt with by other means henceforth they are not treated in this book. As a result we are left with possible random faults in plant components which have to be taken into account. Another scenario arises when a FTCSMP is driven by actuators with non-negligible internal dynamics. Unfortunately the model proposed in  considered faults only in actuators and actuators are assumed to have no internal dynamics. In the following analysis two failure processes are considered one for plant components and the other for actuators. The main reason for using two independent failure processes is that it allows the modeling of faults at dierent locations with independent failure characteristics. Furthermore it permits the construction of conditional transition probabilities of the FDI process when there are delays or errors in detection with respect to each failure process individually. Under some special conditions the two failure processes may




CHAPTER . STOCHASTIC STABILITY OF FTCSMP

share a common state space. In this case they may be replaced by an equivalent failure process as will be shown. In the following analysis a stochastic FDI process with Markovian characteristics is dened. The transition probabilities of the FDI process are conditioned on the current states of the two failure processes. In particular a necessary and sucient condition for stochastic exponential stability in the mean square is derived. These results are obtained without the restrictive assumptions of instantaneous fault detection certainty of correct identication limitation to fault in actuators with no internal dynamics or actuator faults only.

.. Dynamical model of FTCSMP with multiple failures A FTCS subject to faults in both plant components and actuators is shown in Figure . . The system under normal operation can be described by xt

_෈ ෉ =

Atxt ෈ ෉

෈ ෉ +

.

Btut ෈ ෉

෈ ෉

where A
nථn B
nථm x t
n and u t
m. It is important to emphasize that the location of faults and the nature of the faulty components are important when determining the appropriate dynamical model to describe the system. Here the random faults in plant components are represented by a homogeneous Markov process
t with the nite state space Z f  zg. On the other hand random changes in the actuators are represented by the Markov process  t  and the nondeterministic FDI decisions are represented by the process t . The failure processes
t   t  and the FDI process t are dened in Section .. . ෈ ෉

෈ ෉

෈ ෉

=

1 2

෈ ෉

ඵ෈ ෉

෈ ෉

෈ ෉

ඵ෈ ෉

If the plant dynamics undergo a change due to a component fault the system matrix A will change as well. If the actuators are also subject to random variations due to malfunctions then system input matrix B will also change. The FDI process has to consider the combinations of changes in A and B . For example a plant




.. STABILITY OF FTCSMP WITH MULTIPLE FAILURE PROCESSES Reconfiguration Mechanism

FDI

Plant

Actuators

Input +

Outputs/states

Sensors

_

Faults

Faults

Controller

Figure .  AFTCS subject to faults in plant components and actuators. component may undergo one type of fault resulting in two system matrices A1 for normal operation and A2 for fault condition. The corresponding failure process will also have two states Z = f1 2g. Similarly if the faults in the actuators result in three forms for the input matrix B1 normal B2 and B3 with faults the corresponding failure process will have three states S = f1 2 3g. In this case the FDI process will have six states corresponding to dierent faults encountered in the system i.e. R = f1 2 3 4 5 6g. It is apparent that the control law for active FTCS is only a function of the measurable FDI process ඵ෈t෉ as shown in Figure . . Therefore the FTCSMP can be modeled as x_ ෈t෉ = A෈
෈t෉෉x෈t෉ + B ෈෈t෉෉ u෈x෈t෉ ඵ෈t෉ t෉

.

u෈x෈t෉ ඵ෈t෉ t෉ = K ෈ඵ෈t෉෉x෈t෉

In the sequel A෈
෈t෉෉ = A when
෈t෉ = j
Z B ෈෈t෉෉ = B when ෈t෉ = k
S and u෈x෈t෉ ඵ෈t෉ t෉ = u when ඵ෈t෉ = i
R. Also denote x෈t෉ = x
෈t෉ =
෈t෉ =  ඵ෈t෉ = ඵ and the initial conditions are x෈t ෉ = x
෈t ෉ =
෈t ෉ =  ඵ෈t ෉ = ඵ  j

k

i

o

o

o

o

o

o

o

o




CHAPTER . STOCHASTIC STABILITY OF FTCSMP

.. Transition probabilities for failure and FDI processes

Recall that the random processes ෈t෉
෈t෉ and ඵ෈t෉ are assumed to be homogeneous Markov processes with nite state spaces Z
S
and R respectively. The transition probability for the plant component failure process ෈t෉
is dened as pjh ෈තt෉ = jh තt + o෈තt෉

pjj ෈තt෉ = 1

ᝇ P jh j =h

තt + o෈තt෉

6

෈j = h෉

.

෈j = h෉

6

while the transition probability for the actuator failure process ෈t෉
is pkl ෈තt෉ = kl තt + o෈තt෉

ᝇ P kl

pkk ෈තt෉ = 1

k =l

තt + o෈තt෉

6

෈k = l ෉

. 

෈k = l ෉

6

where kl represents the actuator failure rate and jh is the plant component failure rate. Given that = j and  = k
the conditional transition probability of the FDI process ඵ෈t෉
is jk

jk

piv ෈තt෉ = qiv තt + o෈තt෉

jk pii ෈තt෉ = 1

ᝇ P qjk i=v

iv තt + o෈තt෉

6

෈i = v ෉

. 

෈i = v ෉

6

Note that the rates qivjk are dependent on the FDI process used. These rates are vital in deciding the stochastic stability of the closed-loop system     . .. Stochastic stability

As explained in Chapter  in FTCS it is important to consider the almost surely asymptotic stability and the stochastic exponential stability in the mean square. This section provides the necessary denitions and theorems which guarantee both forms of stochastic stability for the FTCSMP in ..




.. STABILITY OF FTCSMP WITH MULTIPLE FAILURE PROCESSES

De nition
. The equilibrium point

  of FTCSMP
. is said to be

x t


almost surely asymptotically stable if for any

Z
o

o

there exists    to
  such that for any jjxo  x P r f sup 0යtය1

k

o
o 

x t xo  to
k g

S



o

o

R     



 to
jj  




and P rf

t

lim sup k x t x !1 t

රo

o

t

De nition
.
The equilibrium point

x t


 to


k g  

  of FTCSMP
. is said to be

exponentially stable in the mean square if for any Z
S  R and some    there exist two constants a   and b   such that when kx  x
   t
k o

o

o

o


 the following inequality holds t  t Efjjx t x

o

 to
jj2 g

o

o

o

o

o


bjjx jj2 expfa t  t
g o

o

For a nite state Markov FDI process the following theorems of stochastic stability are applicable to the FTCSMP .. These theorems were originally derived and proved in   for the stochastic Lyapunov function V x t
 t
 t
where t
is a Markov jump process. An extension to the proof of these theorems was carried out for the proposed Lyapunov function V x t
 r t

t
 t
for noise-free FTCSMP in 
 and for FTCSMP with noise in 
 . Using similar arguments we can prove the theorems for the stochastic Lyapunov function V x t
 t

t
  t
 t
. However they are not shown here to avoid repetition. The readers may refer to the above mentioned references for details.

Theorem
. Assume that V

x t


t

t
  t
 t


is a stochastic Lyapunov func-

tion let V x t
 t

t
  t
 t

N x t
t

t
 t
 t
  in an open set O for FTCSMP
. when t
Z
t
S and  t
R where N x t
t

t
 t
 t



is continuous in x t
 t   and

N x t


t

t
  t
 t


  only if

x t


  then




CHAPTER . STOCHASTIC STABILITY OF FTCSMP

the equilibrium point stable.

x෈t෉

=0

of the FTCSMP
.

 is almost surely asymptotically

Theorem . The equilibrium point

of FTCSMP
.

 is exponentially stable in the mean square if and only if there exists a Lyapunov function ෈ ෈ ෉ ෈ ෉ ෈ ෉ x෈t෉

= 0

V

ඵ෈t෉

that satises for some constants

t෉

a 1 k b

k2

c

x෈t෉

V

෈x෈t෉

V

෈x෈t෉


෈t෉  ෈t෉


෈ t ෉  ෈t ෉

ඵ෈t෉

ඵ෈t෉


3k

t෉

c

t෉

0

 c1  c2

c2

x෈t෉

k2

k

x෈t෉

x t


t

 t

0

c3 

k2



The following theorem is used to dene the necessary condition for exponential stability in the mean square of the FTCSMP ..

Theorem . If FTCSMP
.

 is exponentially stable in the mean square then for any given positive-denite function continuous   o ෈ ෉  function ෈ ෈ ෉ ෈ ෉ ෈ ෉ ඵ෈ ෉ Theorem
. and such that V

t

t

x t


t


t

 t

Z

t

t෉

V

ඵ෈t෉

t෉

which is bounded and

and ඵ෈ ෉   there exists a positive-denite of the same order which satises the conditions of ෈ ෉ ෈ ෉ ෈ ෉ ඵ෈ ෉ ෉ =
෈ ෈ ෉ ෈ ෉ ෈ ෉ ඵ෈ ෉ ෉.

 ෈t ෉

෈x



W ෈x෈t෉
෈t෉  ෈t෉

t

S


t

t

 t

R

t

t

W x t


t

 t

t

t

Furthermore the positive-denite function ෈ ෈ ෉ ෈ ෉ ෈ ෉ ඵ෈ ෉ ෉ satises both conditions in Theorem . and Theorem . . Therefore the exponential stability in the mean square of FTCSMP . provides a su cient condition for almost sure asymptotic stability. V

x t


t

 t

t

t

.. A necessary and sucient condition for exponential stability In this section a necessary and su cient condition for the exponential stability of FTCSMP . under the state feedback =
  is derived. The stability should be maintained not only under the normal operation but also when there are faults in plant components actuators or any combination thereof. ui

Ki x

i

R




.. STABILITY OF FTCSMP WITH MULTIPLE FAILURE PROCESSES

Let V x t

t
 t
 t
t
be a stochastic Lyapunov function of the joint Markov process fx t

t
 t
 t
g From Denition .
 the weak innitesimal operator for the system . at the point fx  x
 j   k   i tg is given by V x




t




V t



hf x j k i t
V x i 

X X= q



l

2S

X

jh

2Z



V x h k i t
V x j k i t


h=j

h

6

kl



V x j l i t


V x j k i t


l k 6



2R

jk iv



V x j k v t


. 

V x j k i t


v =i

v

6

In the following we will state and derive a testable necessary and su cient condition for the exponential stability in the mean square for the FTCSMP employing the weak innitesimal operator dened in . .

Theorem . A necessary and su cient condition for exponential stability in the K x i
R is that there exist steady-state solutions P t
  j
Z k
S and i
R as t   to the following coupled matrix dierential equations

mean square of FTCSMP
. under the control law u

i



i

jki

P

t
 A P

t
 P

T

jki

jki

jki

jki

t
A

jki

X



2Z

jh

P

hki

l

jki

and Q   with A


 

jki

A

jki



A

j

k

i



I

given by

X jki

BK

X

jli

2Z



jh

h =j

h

6



I

2S

l =k 6

jkv

t
 Q

jki



6

X l

2R

jk

iv

v =i

v

6

6

where P

2S

kl

l =k

h=j

h

X

t
  P t
 q P

kl



I

Xq 2R

jk

iv

v =i

v

6

Proof of necessity Assume that the dynamic system . is exponentially stable in the mean square under the control law u  K x  i
R. By Theorem . there exists a quadratic function V x t

t
 t
 t
t
  such that i

i

V x t

t
 t
 t
t
 W x t

t
 t
 t
t





CHAPTER . STOCHASTIC STABILITY OF FTCSMP

Consider the following quadratic stochastic Lyapunov function for the FTCSMP . .

V x t

t
 t
 t
t
 x t
P
t
 t
 t

x t
T

The weak in nitesimal operator in .  can be written as V x


t
 x P T



x A P



T

T

T

t
x  u B P

jki

j

t
x  x P

jki

T

T

i

k

jki

t
A x  x P

P

t
x  x f T

kl

2S

jli

t
P

t
gx

t
P

t
 gx

hki

jk

T

jki

2R

jkv

iv

v =i

v

jki

.

jki

6

6

where

i

k

6

l =k

l

t
B u

jki

P

t
gx  x f P q P

x f P  P t
P T



jh

2Z

h=j

h



T

j

jki

P x t

t
 t
 t
t
 x P x

t
 j  Z  t
 k  S  t
 i  R T

jki

With the state feedback control lawu comes V x
 x A P



T

T j



T

T

T

T

T

i

k

x f  P t
P



T

kl

2S

jli

T

P

2R

v =i

v

2Z

jh

h=j



t
B K x k

i

t
P

jki

jki

P

hki

h

Pq

T

T

j

t
x  x f

jki

t
gx

.

6



P

I

X

jk iv

t
P

jkv

jki

t
gx

6

6

De ne

t
A x x P

jki

t
gx  x f

jki

l =k

l

i

t
x  x P

jki

t
x x K B P

jki

P

t
 x P

K x the weak in nitesimal operator be-



i

A

jki



A

j

BK

 i

k



I

X 2Z



jh



h =j

h

2S



kl



l =k

l

6

I

P 2Z

h =j

h

6

jh

P



hki

t
 x fP T

t


P 2S

l =k

l

6

kl

jki

t
 A P

P t
 jli

T

jki

Pq 2R

v=i

v

6

jki

jk

iv

P

.

jk

iv

2R

v =i

v

6

6

where I is the identity matrix. Rearranging terms we have V x


Xq

t
 P

jki

jkv

t
gx

t
A

jki



.





.. STABILITY OF FTCSMP WITH MULTIPLE FAILURE PROCESSES

Let 
.


W x t

t
 t
 t
t
 x t
Q
t
 t
 t

x t
  T

Setting V x t

t
 t
 t
t





.

W x t

t
 t
 t
t


We have x fP T

P 2Z

h =j

h

6

jki

jh

P

t
 A P T

jki

jki

hki

t


P 2S

l =k

l

kl

t
 P

t
A

P t


Pq

jki

jli

jki

2R

v =i

v

jk iv



P

jkv

t
 Q gx  


. 

jki

6

6

where Q x t

t
 t
 t
t
 x Q x

t
 j  Z  t
 k  S  t
 i  R T

jki

Similar coupled ordinary dierential equations have been studied in   and Section
. . For non-singular matrices  t 
 exp A t 

and positive-denite matrices Q the solutions are unique continuous on t    and P t
 
j  Z k  S and i  R. These solutions are monotonically increasing on    as t  . They are bounded and will converge to steady-state solutions. jki

jki

jki

jki

Proof of su ciency
Assume that steady-state solutions fP

t
  j  Z k  S i  Rg for the coupled matrix dierential equations under the boundary conditions P 
  exist then V x t

t
 t
 t
t
 x t
P
t
 t
 t

x t
is a stochastic Lyapunov function and satises conditions a-c in Denition . and condition a in Theorem
. . That is V x t

t
 t
 t
t
is positive-denite bounded continuous and in the domain of the weak innitesimal operator. Furthermore the steady-state solutions of P t
imply that fP t
  j  Z k  S i  Rg satisfy the coupled matrix dierential equations in Theorem
.  jki

T

jki

jki

jki




CHAPTER . STOCHASTIC STABILITY OF FTCSMP

that is P

jki

P

2Z

t

jh

h=j

h

A P

෈ ෉ + ~T jki

P

jki

t

hki ෈ ෉ +

x fP T

P

2Z

h =j

P
2S

kl

l =k

l

6

or

t

෈ ෉+

P

jki

tA

෈ ෉ ~jki +

Pq

P ෈t෉ + jli

jh

P

t

A P

෈ ෉ + ~T jki

hki

t

෈ ෉+

h

P
2S

l =k

l

6

jki

iv

P

t

jkv ෈ ෉ +

.


Q

jki

=0

6

6

jki

2R

v =i

v

jk

t

෈ ෉+

kl

P

jki

P ෈t ෉ + jli

tA

෈ ෉ ~jki +

Pq 2R

v=i

v

jk

iv

P

jkv

t gx = x Q x T

෈ ෉

.


jki

6

6

For u = K x
i  R the weak in nitesimal operator V ෈x෈t෉  ෈t෉ ෈t෉ ඵ෈t෉ t෉ is given by .
 with A~ de ned in .
. Therefore it follows that i

i

jki

V ෈x෈t෉  ෈t෉ ෈t෉ ඵ෈t෉ t෉

=

x ෈t෉Q෈ ෈t෉ ෈t෉ ඵ෈t෉෉x෈t෉  0 T

. 

By Theorem . the system under the control law u = K x
i  R is exponentially stable in the mean square
t  0 Hence the proof is complete. For a given control law and using Theorem . one can verify the existence of the steady-state solutions of fP ෈t෉  0 j  Z k  S i  Rg. If the bounded solutions exist the system . is exponentially stable in the mean square and also almost surely asymptotically stable. i

i

jki

..

Remarks and special cases

Under certain assumptions several special cases can be derived from the above general result. Some of these cases were considered by other researchers in studying the stochastic stability of hybrid systems. Others are new and not available in the literature. These special cases deal with various aspects of FTCS the nature of faulty components the occurrence of faults at dierent locations in the system and the nature of the FDI process.

.. STABILITY OF FTCSMP WITH MULTIPLE FAILURE PROCESSES

I. Plant components faults In this case
only system matrix A can be subject to change due to random faults in one or more plant components. The system model becomes xt

Axt


 


 

j

Bu t

. 

j2Z




The coupled matrix dierential equations in Theorem become P

t

ji
 

A P t T

ji

ji
 

P tA ji




ji 

P l

2S

l

8P


t 2 ᝇ1


ji
  




jl

=j

P t

li
 

v

2R

v

6

j

iv

=i

P t

vj
 

6

Q

ji  

. 



where A is dened as 

ji

A

Pq



ji 

A ᝇ BK ᝇ I 

j

i



X l

2S

l

Xq

ᝇ I 

jl

=j



v

2R

v

6

j iv

. 

=i 6

The necessary and su cient condition for the stochastic exponential stability of the system under such conditions is the existence of steady-state solutions to equation . .

II. Faults in actuators with no dynamics In this case
integrity of the plant components is assumed
and the actuators have no internal dynamics. Therefore
only the input matrix B may change as a result of random faults in the actuators. The system can then be described by xt


 

Ax t


 

B ut

with A dened as 

ki

A



ki



AᝇB K ᝇ I 

k

i



X l

2S

l

=k 6

. 

k2S




k

kl

ᝇ I

Xq v

2R

v

=i 6

k

iv

. 




CHAPTER . STOCHASTIC STABILITY OF FTCSMP

The bounded solutions of the following coupled matrix dierential equation P

ki

t
 A P t
 P t
A T

ki

ki

ki

ki

P



l

2S

l

8 Pki


 


t2

kl

P t
 li

=k

Pq

v

2R

v

6

k iv

=i

P t
 Q vk

ki



6

. 

ᝇ1


provide the necessary and sucient condition for the stochastic exponential stability of the system . . This leads to a similar result obtained earlier 
.

III. Faults in actuators with dynamics In this case random faults can occur in actuators and the actuators have internal dynamics. The system can be modeled by x t
 A x t
 B u t
k

. 

k2ZS

k

Note that both the system matrix A and the input matrix B have the same failure index. This means that one fault induces simultaneous changes in both matrices. In other words the two failure processes are replaced with one failure process in the system .. The transition probability for the failure process becomes p p

kl

kk

t






kl

t




t o t


k 6 l


 

ᝇ

P

k

= 6

.


t o t
k  l


kl  

l

The necessary and sucient condition for the stochastic stability of this system is the existence of bounded solutions to the following coupled matrix dierential equation P

ki

t
 A P t
 P t
A T

ki

ki

ki

ki



P

l

2S

l

8 Pki


t2


 

kl

=k

P t
 li

Pq

v

2R

v

6

k

iv

=i

P t
 Q vk

6

ki



. 

ᝇ1


where A

ki



A

k

ᝇ Bk Ki ᝇ



I

X l

2S

l

=k 6

kl

ᝇ



I

Xq v

2R

v

=i 6

k iv

.





.. STABILITY OF FTCSMP WITH MULTIPLE FAILURE PROCESSES

IV. Perfect FDI performance In this case the FDI process is assumed to be able to instantaneously detect and always correctly identify faults. Therefore the two failure processes and the FDI process will share the same state space. This situation is similar to the one considered by JLS. The transition probability for the common failure process is p l
t  il
t  o
t

i  l

pii
t  


i  l

i

P

i=l

il
t  o
t

.

6

The conditional transition probability of the FDI process for this case will become qivk



  v  k 
v
k 

  v  k

. 

and Aii  Ai
BiKi
 I

X l 2R l

il

. 

=i 6

This result is similar to the one reported in 
 .

.. A numerical example Consider a scalar system with one possible fault in the actuator i.e. S  f g and one possible fault in the plant components i.e. Z  f g. Both failure processes are assumed to have Markovian transition characteristics. The FDI process is also Markovian with four states R  f   g. The following numerical values are used A1    A2     B1    B2   12    21   12    21  




CHAPTER . STOCHASTIC STABILITY OF FTCSMP

11

q iv

3 2 66



 7 7 66 7 7 66    

 77 12 66 7 q iv 7 66 7 7  
  
 66 7 7 4 5

2 3 6    

 77 6 6 7 6 7 6



 77 6 6 7 6 7 6 7 6 7
    
 6 7 6 7 4 5

3 2 66  
  
 7 7 66 7 7 66
    
 77 22 66 7 q 7 iv 66 7 7



 66 7 7 4 5

2 3 6  

   77 6 6 7 6 7 6 7
  
   6 7 6 7 6 7 6 7 6 7

     6 7 7 6 5 4



q

21

iv









































Note that the open-loop system is deterministically unstable. The objective is to test the existence of the steady-state solutions f



g under a certain pre-computed control law 
. As per Theorem . the existence of the steady-state solutions guarantees the exponential stability in the mean and the almost sure asymptotic stochastic stability. Since the FDI process has four states there are four associated controller gains. The rst set is 1 
 2 
 3 
 and 4 
 The second set is 1 
 2 
 3 
 and 4 
. The solutions of
under the boundary conditions

for the two sets of gains are shown in Figure . and Figure .  respectively. For the rst set of controller gains the steady-state solutions exist. However the solutions are unbounded as   for the second set of controller gains. According to Theorem . the system is exponentially stable in the mean square and almost sure asymptotically stable for 1 
 2 
 3 
 4 
 but is not for 
 2 
 3 
 4 
. 1 Pjki


ui

Ki x

i



j

Z

k

S



K

K

K

R

R

K

 K

i

 K

 K

 K

Pjki




Pjki

t

K

K

 K

 K

 K

 K

 K

 K

Note that deterministic stability does not imply stochastic stability. It is easy to check that the deterministic stability is guaranteed as long as 1    2 
 K




K







.. STABILITY OF FTCSMP WITH MULTIPLE FAILURE PROCESSES

K3
and K4

The selected controller gains in both sets guarantee the deterministic closed-loop stability.


 

P111 P113 P114 P112

0.8 0.7 0.6 0.5 0.4 0.3 0.2

0.1 0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time (sec)

0.4

0.3 0.25 0.2 0.15 0.1

1

Solutions of differentail equations

0.4

0.3 0.25 0.2 0.15 0.1

P113 P114 P112

1.5 1 0.5

 K2

2500 2000 1500 1000 500



0





.



P123 P124 P122

4 3.5 3 2.5 2 1.5 1

0

P221 P223 P224 P222

0.5 0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time (sec)

0

Figure .  Unbounded solutions with K1

 and K4

 

P121

4.5

0.05

 K3



0

0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time (sec)

0

P211 P213 P214 P212

-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time (sec)

0

P221 P223 P224 P222

2

 

P111

0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time (sec)

0.35

2

Solutions of differentail equations

2

4

Solutions of differentail equations

Solutions of differentail equations

3

6

0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time (sec)

6

4

8

0

Figure . Bounded solutions with K1

5

10

2.5

0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time (sec)

7

P123 P124 P122

12

0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time (sec)

0.05

8

14

0

P211 P213 P214 P212

0.35

P121

16

Solutions of differential equations

Solutions of differential equations

Solutions of differential equations

Solutions of differential equations

0.9

 K2





 K3



 and K4

 



0

.






CHAPTER . STOCHASTIC STABILITY OF FTCSMP

. Chapter Summary In this chapter the stochastic stability of FTCSMP has been analyzed by taking into consideration the eect of environmental noises and the possible multiple failures which may occur in dierent system components. The analysis has been carried out employing the weak innitesimal operator for a quadratic stochastic Lyapunov function. The conventional FTCSMP model with separate Markovian processes to represent random faults and the decisions of the FDI process is used. In Section .
 three types of noise have been considered statedependent control-dependent and state and control independent. The almost sure asymptotic stability and the exponential stability in the mean square have been considered. Specically a testable necessary and su cient condition for the exponential stability in the mean square has been derived. It has been concluded that the closedloop system stability of a FTCSMP may be lost if state or control dependent noises were neglected. Whereas the independent noise do not alter the stability of the FTCSMP even though it eects the variance signicantly. Section . considered the case where the system is subject to random faults in both plant components and actuators. A dynamical model for FTCSMP in such case was developed. It has been shown that the exponential stability in the mean square is su cient for almost sure asymptotic stability. In addition a necessary and su cient condition for exponential stability in the mean square for FTCSMP has been derived. Some existing results were shown to be special cases of this general result. Some of these results included JLS models and systems with dynamic actuators. The above theoretical developments were illustrated by two numerical examples.

Chapter PERFORMANCE AND STABILITY OF FTCSMP UNDER IMPERFECT FDI .

Introduction

In Chapter
it was outlined that FTCS can be classied into PFTCS and AFTCS. AFTCS is mainly composed of two parts an FDI algorithm which monitors the performance of the system
detect the occurrence of faults
and identify the faulty components
and a control reconguration mechanism which recongures the controller on-line after the occurrence of the fault. AFTCS can be viewed as a general hybrid system. Two types of hybrid systems were considered The JLS  





and the FTCSMP  

. In FTCS
the state feedback control law is recongured upon the decision of the FDI process. However
the FDI process cannot be expected to provide perfect information on the fault process. Detection delays
false alarms
or missed detection often exist in a FDI process. In this chapter
the eect of detection delays and errors on the stochastic stability of FTCSMP is studied. A mathematical representation of the detection delays in a FDI process is developed. FDI processes employing memory-less statistical and sequential test are considered. The dynamic programming principle is used to minimize M.M. Mahmoud, J. Jiang, Y. Zhang: Active Fault Tolerant Control Systems, LNCIS 287, pp. 83−102, 2003.  Springer-Verlag Berlin Heidelberg 2003




CHAPTER . PERFORMANCE AND STABILITY OF FTCSMP UNDER IMPERFECT FDI

a stochastic cost function and to synthesize a fault tolerant control law. To study the eects of errors in fault detection both the failure and the FDI processes are assumed to be available to the controller. It will be shown that the stability of FTCSMP could be lost in the event of long fault detection delays or large detection errors. Numerical examples are presented to demonstrate the theoretical developments.

.

Eects of Detection Delays on the Stability of FTCSMP

The eect of fault detection delays on the stochastic stability of FTCSMP is considered rst in this chapter. A work was completed in  to study the stability of FTCS with possible detection delays. Unfortunately the study was made under the assumption that the fault identication is always correct. In the current work the FTCSMP model relaxes such restriction so that one can obtain an important conclusion of the eect of detection delays. In this context a mathematical representation for detection delays associated with the FDI process is constructed. In practice detection delays are random variables which can be modelled by a probability distribution function. In this book two distributions are adopted namely exponential and Gamma distributions. The exponential distribution is chosen because of its unique memory-less property therefore it can be used to represent those memory-less FDI processes such as algorithms employing single sample tests  . On the other hand the Gamma distribution is selected for those FDI processes employing sequential test i.e. tests with data memory. The Gamma distribution is selected because of its ability to approximate several other discrete and continuous distributions. Moreover the Gamma distribution can be considered as a generalization of the exponential distribution . In particular the sum of independent exponential random delays leads to a Gamma distribution .




.. EFFECTS OF DETECTION DELAYS ON THE STABILITY OF FTCSMP

.. Dynamical model of FTCSMP with detection delays Consider the linear FTCSMP represented in . which is rewritten here for convenience xt

_෈ ෉ =

Ax t

෈ ෉ +

B t u x t
t
t ෈ ෈ ෉෉

u x t
t
t ෈

෈ ෉ ඵ෈ ෉

K

෉ =

෈

෈ ෉ ඵ෈ ෉

෉

 . 

t xt

෈ඵ෈ ෉෉

෈ ෉

where x t
n  u x t
t
t
m t is the actuator failure process and t is the FDI process. Both processes are assumed to have Markovian transition characteristics with nite state spaces S f


sg and R f


rg respectively. The transition probability for t is given in .  and the conditional transition probabilities for t is given in .

. ෈ ෉

෈

෈ ෉ ඵ෈ ෉

=

෉

෈ ෉

ඵ෈ ෉

1 2

=

1 2

෈ ෉

ඵ෈ ෉

.. Interpretation of detection delays in FDI process Detection delay is dened as the time needed after the occurrence of a fault for the FDI algorithm to provide a decision. In this section a mathematical formulation of the FDI conditional transition rates to describe detection delays is developed. The conditional transition rates of the FDI process as given in .  are qijk t

fijk t Fijk t

 .

෈ ෉

෈ ෉ = 1

෈ ෉

In terms of the reliability function as represented in .  qijk t

෈ ෉ =

෈

dRijk t dt Rijk t ෈ ෉

෈ ෉

෉

 .

The calculation of the distribution and the reliability functions are detailed in  .




CHAPTER . PERFORMANCE AND STABILITY OF FTCSMP UNDER IMPERFECT FDI

.. Exponentially distributed detection delays

If detection delays of the FDI process are exponentially distributed random variables with




mean


.

k ij

The probability density function can be represented as fijk



 expf t 

  k ij

k ij

tg t



.

t


the probability distribution function will be F

k ij




expf t

 

k ij

tg t



.


t


and the reliability function will be k Rij t

expf

k ij

tg t


From 
. the conditional transition rates become expf tg q t  expf tg k ij

k ij

k ij

k ij


.


mean

k ij


. 

It is clear that the conditional transition rates are constant which re ects the memory-less property. ..

distributed detection delays

If detection delays are assumed to have Gamma distribution their density functions will be



 t  t 
 r 
 k ij

f

k ij

k ij

k ij

k

rij

ᝇ1 expf

k ij

tg t
 t



. 




.. EFFECTS OF DETECTION DELAYS ON THE STABILITY OF FTCSMP

For an integer r


the probability distribution function is


r Pᝇ1 
 
k tm expf kij tg t
m=0 m ij Fijk
t 

 t k ij

.

and the reliability function becomes rijkPᝇ1

.

 k m
 t expf kij tg t
m=0 m ij

Rijk
t 

The time derivative of the reliability function can be calculated as dRijk
t dt



k rX ij ᝇ1

m=0

 k m+1 m m
ij 
t expf

kij tg 

m
k m
tmᝇ1 expf k tg ij ij m m=1 

r ᝇ1  r ᝇ1 k k X X   ij   ij
k tmᝇ1
k tm expf kij tg 
m  ij m  ij m=0 m=1 k ij

k ij



k rX ij ᝇ1

expf



k kij tg k ij
kij tr ᝇ1
rij 

.

k ij

Therefore the conditional transition rates can be represented as

   k  kij ij r ᝇ1 r ᝇ1 k k k  expf ij tg
rk 
ij t  
rk 
ij t   ij    ij     r ᝇ1 r Pᝇ1 P 
k tm   k m 
ij t expf kij tg m=0 m  m=0 m ij  
k r
tr ᝇ1   ij  .    r Pᝇ1  k m
 t
r k  k ij

qijk
t



k ij

k ij



ij

k ij

k ij

k ij

k ij

m=0 m ij

The conditional transition rates of the Gamma distributed detection delays are time-varying. However we can approximate the behavior of the time-varying conditional transition rates by considering their steady-state values. The steady-state

CHAPTER . PERFORMANCE AND STABILITY OF FTCSMP UNDER IMPERFECT FDI

conditional transition rates are

  



kij r
tr ᝇ1
kij r
tr ᝇ1 


  kij
 lim qijk
t  tlim 
ᝇ 1  !1
 t !1 r Pᝇ1    
k tm 
kij t
rijk ᝇ  k
rijk ᝇ 
r ᝇ  ij m=0 m ij k ij

k ij

k ij

k ij

rk ij

k ij


. Equation
. approximates the behavior of the transition rates of the FDI algorithm after a short transient time interval. It can be seen that the behavior of FDI algorithms with Gamma distributed detection delays approaches FDI algorithms with exponentially distributed detection delays. This approximation reects the additional time needed by sequential FDI algorithms to arrive at a decision.

.. A necessary and sucient condition for stochastic stability A necessary and sucient condition for the exponential stability in the mean square of the FTCSMP
. was derived in    . This condition can be modied such that the conditional transition rates are replaced by the means of the exponentially distributed
Gamma distributed detection delays. For bounded kij  i.e. non-zero detection delays a necessary and sucient condition for the exponential stability in the mean square can be given in Theorem ..

Theorem . A necessary and su cient condition for the exponential stability in the mean square of the FTCSMP
. under the control law ui  ᝇKi x
8 i 2 R
is that there exist bounded steady-state solutions Pki   8 k 2 S
i 2 R as t ! ᝇ1 to the following coupled matrix dierential equations Pki
t  A Tki Pki
t  Pki
tA ki 

X

j 2S j= 6 k

kj Pji
t 

with Pki
  
Qki   and A ki is de ned as A ki  A ᝇ Bk Ki ᝇ  I

X

j 2S j= 6 k

X k ij Pkj
t  Qki  

j 2R j= 6 i

kj ᝇ  I

X k

j 2R j= 6 i

ij




.. EFFECTS OF DETECTION DELAYS ON THE STABILITY OF FTCSMP Proof

see   
.

..

A numerical example

Consider the following system A
 
B1
 
B2
 

The actuator failure rates are assumed to be 12  
21  

The means of the detection delays of the FDI process are



 
k
i
j 


k ij

The conditional transition rates are q

 
k
i
j 


k ij

The weighting matrices Q and R are ki

ki

fQ11
Q12
Q21
Q22 g

R

ki

 


f 


 
 
 g


i
j
f
g

The e ect of detection delays on the stability of FTCSMP will be studied. The stochastic stability is examined by solving for positive-denite solutions P . The existence of bounded solutions is a necessary and su cient condition for the exponential stability in the mean square and the su cient condition for almost sure asymptotic stability    . Note that errors in fault detection and identication are not emphasized in this particular example but will be detailed in the next section. ki




CHAPTER . PERFORMANCE AND STABILITY OF FTCSMP UNDER IMPERFECT FDI

Case
Stability of FTCSMP with tolerable FDI detection delays

For the given system parameters stability of the FTCSMP is examined with the following feedback gain matrices

K1


 





 

K2

In this case positive-denite steady-state solutions exist as shown in Figure .. From Theorem . one can conclude that the FTCSMP is exponentially and almost surely asymptotically stable. In other words the FTCSMP can tolerate the given detection delays.

0.12

Solutions of matrix differential eqns

0.6

P11

0.5 0.4 0.3 0.2 0.1

Solutions of matrix differential eqns

0 -100

-90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

0.1

0.06 0.04 0.02 0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

0

4.5

0

1.5

4 3.5

P12

0.08

Solutions of matrix differential eqns

Solutions of matrix differential eqns

0.7

P21

3 2.5 2 1.5 1 0.5 0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

Figure . Bounded solutions

0

Pki

P22 1

0.5

0 -100

-90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

0

 with tolerable detection delays.




.. EFFECTS OF DETECTION DELAYS ON THE STABILITY OF FTCSMP

Case
Stability of FTCSMP with intolerable FDI detection delays

In this case the same state feedback gains 1
  and However the means of detection delays are increased to K








k ij



k i j

K2


  are used.

 

and the conditional transition rates become



Solutions of matrix differential eqns

3.5

P11

3 2.5 2 1.5 1 0.5 0

16

0

x 10

14 12

P21

10 8 6 4 2 0 -100

-90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)



2.5 x 10 2

5 P12

1.5 1 0.5 0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

Pki

0

250 200

P22

150 100 50 0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

0

Figure . Unbounded solutions



k i j

7 4 x 10

-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

Solutions of matrix differential eqns



Solutions of matrix differential eqns

Solutions of matrix differential eqns

k q ij

0

 with intolerable detection delays.

Figure . illustrates that there are no bounded solutions and the FTCSMP has lost its stochastic stability.




CHAPTER . PERFORMANCE AND STABILITY OF FTCSMP UNDER IMPERFECT FDI

Case
stochastic stability versus FDI detection delays

The matrix dierential equations in Theorem . are solved for a range of FDI detection delays having means between .





k  ij

2





k
i
j  


Figure . conrms that stochastic stability of the FTCSMP may be lost for large means. In this particular example the FTCSMP will lose stability if the average delay is longer than . seconds. Solutions of matrix differential eqns

Solutions of matrix differential eqns

9 x 10

4.5

P11

4 3.5 3 2.5 2 1.5 1 0.5 0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Detection-delay (sec)

12 10 8 6 4 2 0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Detection-delay (sec)

Figure . Solutions .

Pki

10 8 6 4 2 0 0

Solutions of matrix differential eqns

Solutions of matrix differential eqns

P21

14

P12

12

1

11 18 x 10 16

5 14 x 10

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Detection-delay (sec)

1200 1000

P22

800 600 400 200 0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Detection-delay (sec)

for dierent means of detection delays.

Eects of Detection Errors on the Stability of FTCSMP

AFTCS rely on FDI schemes to identify the fault-induced changes and to recongure the control law accordingly. The fact that an important part of the FDI algorithm is a hypothesis test necessitates the existence of error probabilities associated with its




.. EFFECTS OF DETECTION ERRORS ON THE STABILITY OF FTCSMP

decisions  . Clearly FDI algorithm may provide incorrect decisions about the actual faults in the system. To study the e ect of errors in detection and identication on the stability of FTCSMP a control law is designed using the dynamic programming principle. The controller is assumed to have access to the states of both the FDI algorithm and the failure process. In this section of the book an analytical approach is used to analyze the behavior of FTCSMP in the presence of incorrect FDI decisions.

.. A new FTCSMP model To study the e ect of errors in fault detection on the stochastic stability of the linear FTCSMP .  a new design methodology is proposed. In this methodology it is assumed that both the actual states of the failure process and the decisions of the FDI process are available. That is the control gain is calculated based on the knowledge of the both processes. However it is important to emphasize that the control law in FTCSMP is recongured only upon the decisions of the FDI process and the assumption of fault process availability is only to study analytically the e ect of errors in detection. For this specic purpose the FTCSMP . has been modied as xt

_෈ ෉ =

Ax t

෈ ෉ +

B t u x t
t
t ෈ ෈ ෉෉

u x t
t
t
t ෈

෈ ෉

෈ ෉ ඵ෈ ෉

෉ =

෈

෈ ෉ ඵ෈ ෉

෉

ᝇK t
t x t ෈ ෈ ෉ ඵ෈ ෉෉

.

෈ ෉

In .  if the certainty of the correct FDI decision is assumed then t t. Otherwise t 6 t . The later case can be interpreted as false alarm missed detection errors in detection etc. ඵ෈ ෉ =

ඵ෈ ෉ =

෈ ෉

෈ ෉




CHAPTER . PERFORMANCE AND STABILITY OF FTCSMP UNDER IMPERFECT FDI

The objective herein is to synthesize a fault tolerant control law for the system . that minimizes the following performance index J

Zt

f

Ef L
x
t

t 
t t dtg to tf

Z

Ef xT
tQ


t 
tx
t  uT
tR


t 
tu
t dtg to

.

where Q


t 
t and R


t 
t are semi-positive and positive-de nite matrices

respectively. Without loss of generality lets assume that the nal time is tf  and the initial time is to. .. The role of FDI in fault tolerant control law design

A fault tolerant controller for the FTCSMP . which minimizes the performance index . is given in the following Theorem.

Theorem . The control law that minimizes
.
 is uදki
t

Rkiᝇ1BkT Pki
tx
t

Kki
tx
t

. 

where the Pki
t are the solutions to the following Riccati-like matrix equations

Pki
t  A Tki Pki
t  Pki
tA ki 

P kj Pji
t  P qk Pkj
t  K T RkiKki  Qki j 2R

j 2S

=

=

ij

ki



j i

j k

6

6


Pki
tf   .  with

X X A ki A Bk Kki  I kj  I qijk j 2S j 2R =

j k 6

=

j i 6

. 




.. EFFECTS OF DETECTION ERRORS ON THE STABILITY OF FTCSMP Proof

Dene a cost function .

V x t

t
 t
t
 min u J

The principle of dynamic programing  states that V x
 t
satises min u V x t

t
 t
t
 L x t

t
 t
t
  f

g

. 

The weak innitesimal operator of the FTCSMP .
 for the stochastic cost function V x
 t
at the point x
 k   i t is f

g

V x
 t
 Vt x k i t
 Ax  Bk ui
Vx x k i t
h



P  V x j i t
V x k i t
 kj

j 2S j






=k

P qk V x k j t
V x k i t
 ij

For a quadratic cost function

.




j 2R j

6

i

=i 6

V x t

t
 t
t
 x t
T P
t
 t
t
x t


. 

The weak innitesimal operator V x t

t
 t
t
can be written as V x
 t
 xT Pki t
x  xT AT Pki t
x  xT Pki t
Ax  uTki BkT Pki t
x  xT Pki t
Bk uki xT

f

.

P  P t
P t
 x  xT P qk P t
P t
 x kj ji ki ki ij kj


j 2S j

g

f

=k

From .  we have




j 2R j

6

g

=i 6


T Pki t
x  xT AT Pki t
x  xT Pki t
Ax  uT B T Pki t
x

x


ki k






P
   xT Pki t
Bk ui  xT kj Pji t
Pki t
 x j 2S

  u
=




P


T k T T x qij Pkj t
Pki t
 x  x Qki x  ui Rkiui


j 2R  = f




j

f




j

6

i

6

k

g

g

.





CHAPTER . PERFORMANCE AND STABILITY OF FTCSMP UNDER IMPERFECT FDI

Therefore the optimal solution is uki t
 Rkiᝇ1BkT Pki t
x t
 Kki t
x t



.


where the Pki t
are obtained from the Optimality lemma in    which necessitates that the following condition must hold for the controller to be optimal. V x t

 t

 t

t




L x t

 t

 t

t
 


. 

This leads to xT Pki t
x  xT AT Pki t
x  xT Pki t
Ax  uTki BkT Pki t
x  xT Pki t
Bk uki 

xT f P kj Pji t
Pki t
gx  xT f P qijk Pkj t
Pki t
gx j 2S j



j 2R

=k

j

6

=i


.

6

xT Qki x  uTkiRki uki  

Under the control law uki t
 Kki t
x t
and with A ki de ned in 
.  we have Pki t
 A Tki Pki t
 Pki t
A ki 

X

j 2S j

=k

kj Pji t


X qk P

j 2R

6

j

T ij kj t
 Kki Rki Kki  Qki  

=i 6


.  The boundary condition is obtained from the de nition of the cost function at t  tf  V x tf

 tf

 tf

tf
 x tf
T P  tf

 tf

tf
x tf
 

or P  t f

 t f

t f
 

The proof is complete. For on-line control recon guration the controller needs to have access to the FDI process. As a result two scenarios arise if the FDI process is assumed to be perfect then  t
  t
 and 
.  reduces to uk t
 Rkᝇ1BkT Pk t
x t




Kk t
x t



.

.. EFFECTS OF DETECTION ERRORS ON THE STABILITY OF FTCSMP




This is the ideal case which represents the certainty of correct fault detection and identication. On the other hand if
t 6
t it means that the FDI process has incorrectly identied the current fault in the system. Hence the resulting controller may lead to a degraded system performance or even to a complete loss of the stability as will be illustrated by a numerical example.

.. A necessary and sucient condition for stochastic stability A necessary and sucient condition for the exponential stability in the mean square of FTCSMP . is stated in Theorem . without proof. Readers who are interested in proofs may refer to      for details.

Theorem . A necessary and su cient condition for the exponential stability in the mean square of the FTCSMP
. under the control law uki
t  ᝇKki x
t is that there exist bounded steady-state solutions Pki
 8 k 2 S i 2 R as t ! ᝇ1 to the following coupled matrix di erential equations

Pki
t  ATki Pki
t  Pki
tAki 

P kj Pji
t  P qk Pkj
t  Qki  

j 2S j

=k 6

j 2R j

=i

ij

6

8Pki
   with Qki
 and Aki is de ned in
..

.. A numerical example To illustrate the theoretical results lets consider a system with one possible fault in the actuator as follows A    B1    B2   

The actuator failure rates are assumed to be 12    21  




CHAPTER . PERFORMANCE AND STABILITY OF FTCSMP UNDER IMPERFECT FDI

The conditional transition rates of the FDI process are 1

q ij

2 66

 64




The weighting matrices Qki

Qki

and




Rki

3 7 7
7 5

q

2

ij

are

2 66 



64

  



3 7 7
7 5

Rki

2 6 
 
 6 6 4







2 6 



6 6 4







3 7 7 7 5

3 7 7 7 5

The objective is to design a fault tolerant control law for the given FTCSMP. The eect of detection errors will be studied. The existence of positive-denite bounded solutions
 for the dierential equations in Theorem . is a necessary and su cient condition for the exponential stability in the mean square and su cient for the almost sure asymptotic stability. Pki 

Case
Design of a fault tolerant control law

The Riccati-like matrix equations in Theorem . are solved iteratively as shown in Figure .. The steady-state solutions are K11

K12

K21

K22

   
    The gain matrices
f g are calculated under the assumption that both the failure and the FDI processes are accessible. They may be interpreted as follows 11  22 represent the correct fault detection by the FDI process when the system is operating under normal and faulty conditions respectively. 12 can be thought to refer to the case where the system is operating normally while the FDI process incorrectly indicates that there is a fault. This case is known as false alarm FA. Similarly 21 corresponds to the case where the fault has occurred in the system while the FDI process is not able to detect it. This case is often known as missed detection Kki
k
i

K




K

K

K

.. EFFECTS OF DETECTION ERRORS ON THE STABILITY OF FTCSMP


MD. In the following we will study the stochastic stability of the FTCSMP when there exist a FA or a MD. 1.2

1.4

K11

1.0 0.8 0.6 0.4 0.2 0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

0.6 0.4 0.2

4

K21

3 2.5 2 1.5 1 0.5 0

0.8

0

4.5

Fault tolernat control law

Fault tolernat control law

4

1

0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

0

4.5

3.5

K12

1.2

Fault tolernat control law

Fault tolernat control law

1.4

K22

3.5 3 2.5 2 1.5 1 0.5

-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

0

0

Figure . Steady-state gains ki  FTCSMP with errors in detection. K

Case
Perfect fault detection and identication This is the ideal case. The FDI process correctly detect and identify the fault. The controller gains are normal

K

K1


 




fault

K

K2


 

using this state feedback gains the Riccati-like matrix equations are solved. Positivede nite steady-state solutions are shown in Figure . .

Case 
Errors in fault detectionᝇfalse alarm If the system operates normally however the FDI process has incorrectly indicated that there is a fault in the system. The controller gains used are K1

K11


 




K2

K12


 





CHAPTER . PERFORMANCE AND STABILITY OF FTCSMP UNDER IMPERFECT FDI 0.6

0.12

P11

Solutions of Riccati-like equations

Solutions of Riccati-like equations

0.7

0.5 0.4 0.3 0.2 0.1 0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

0.02 0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

1.5

3 2.5 2 1.5 1 0.5 0

0.04

0

P21

4 3.5

0.06

Solutions of Riccati-like equations

Solutions of Riccati-like equations

4.5

P12

0.1 0.08

-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

P22

1

0.5

0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

0

0

0

Figure . Bounded solutions P  for perfect fault detection and identication. ki

Figure . shows that the solutions are not bounded. Therefore the system is stochastically unstable.

1.6

2.5

P11

Solutions of Riccati-like eqautions

Solutions of Riccati-like eqautions

12 2 x 10 1.8 1.4 1.2 1 0.8 0.6 0.4

0.2 0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

0

P21

Solutions of Riccati-like eqautions

Solutions of Riccati-like eqautions

7

2

P12

1.5 1 0.5 0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

0

14 10 x 10

14 9 x 10 8

12 x 10

6 5 4 3 2 1 0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

0

P22

9 8 7 6 5 4 3 2 1

0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time(sec)

Figure . Unbounded solutions P  under a false alarm. ki

0




.. CHAPTER SUMMARY

Case
Errors in fault detectionᝇmissed detection This case arises when the system is in fact undergoing a fault state but the FDI process is not able to detect it. The state feedback gains are K1

K22


 




K2

K21


 

Positive-denite steady-state solutions of the Riccati-like matrix equations are shown in Figure .. Therefore the system is stochastically stable. For this particular example the missed detection did not aect the stochastic stability. However this is not a general conclusion and missed detection may very well lead to an unstable closed-loop FTCSMP in other cases. 0.6

0.12

Solutions of Riccati-like equations

Solutions of Riccati-like equations

0.7

P11

0.5 0.4 0.3 0.2 0.1 0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time (sec)

0

4 3.5

P21

3 2.5 2 1.5 1 0.5 0

P12

0.06 0.04 0.02 0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time (sec)

0

1.5

Solutions of Riccati-like equations

Solutions of Riccati-like equations

4.5

0.1 0.08

-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time (sec)

0.5

0 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 Time (sec)

0

Figure . Bounded solutions

P22 1

Pki

0

 under a missed detection.

. Chapter Summary In this chapter the eect of imperfect FDI algorithm on the stochastic stability of the closed-loop FTCSMP has been studied. A mathematical representation for detection delays in FDI algorithms has been developed and a fault tolerant control law has been




CHAPTER . PERFORMANCE AND STABILITY OF FTCSMP UNDER IMPERFECT FDI

designed. The designed controller allows the study of detection delays and errors. The results revealed that large delays and errors in fault detection may lead to a complete loss of the stochastic stability of FTCSMP. The ndings conrm that the integration of FDI algorithms and controller reconguration mechanisms should take into account of the real-time and stochastic aspects of these two parts. This is to say that the design of an integrated AFTCS is a problem which needs further investigation. Theoretical results were veried by several numerical examples.

Chapter FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES .

Introduction

In this chapter the behavior of FTCSMP subject to some physical limitations is studied. The rst problem to be addressed is the case where the system to be controlled is driven by actuators with saturation. The second issue is associated with imprecisely estimated parameters in FDI process. In Section . actuator with non-linear characteristics are considered they operate linearly within their certain limits and saturate to xed levels if the limits are exceeded. In practice estimation techniques employed by FDI algorithms can only provide system parameters with some inaccuracies. As a result the model of the system to be controlled will include parameter uncertainties. Section . is dedicated to study exponential stabilization of FTCSMP with time-varying unknown-but-bounded parameter uncertainties. In particular an algorithm that provides a necessary and sucient condition for exponential stabilization in the mean square is derived. The stabilizing controller is calculated based on the solutions of Riccati-like matrix equations. Several interesting special cases are derived from the general result. M.M. Mahmoud, J. Jiang, Y. Zhang: Active Fault Tolerant Control Systems, LNCIS 287, pp. 103−147, 2003.  Springer-Verlag Berlin Heidelberg 2003




CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

Numerical examples are presented to demonstrate the theoretical developments.

. Stability of FTCSMP Driven by Actuators with Saturation Recently some investigations have been conducted to characterize the behavior of practical FTCSMP. The stochastic stability of FTCSMP was studied in   the e ect of detection delays and false alarms was considered in 

. Stochastic stability with multiple faults was reported in   conditions for stability of FTCSMP subject to environmental noises was developed in   and with parameter uncertainties in  . A vital problem which often arises in practical control systems is actuator saturation. A control valve is a good example of actuator with saturation. A valve has a range of operation limited by being fully open and fully closed. Unfortunately actuator saturation is usually unavoidable. If such behavior is not taken into account in the control system design an integral wind-up may be induced which could lead to a large overshoot in system response a limit cycle or an unstable closed-loop system . The consequences of actuator saturation are more severe when the system encounters sudden changes such as the occurrence of a fault. Special e ort has been devoted to solve the problem of feedback control with nonlinear saturating actuators for deterministic systems. The stability analysis of a continuous system with saturating actuators has been carried out for SISO systems employing the Popov s criteria 
. The tracking problem under similar conditions was considered in 
. The problem was treated in time domain in   . The combined problem of actuator saturation with state-delays was reported in   and with parameter uncertainties in  . Despite of its practical importance to the best of our knowledge the problem of nonlinear saturation for FTCSMP has not been addressed. A FTCSMP model




.. STABILITY OF FTCSMP DRIVEN BY ACTUATORS WITH SATURATION

subject to random faults and saturation in actuators will be developed. In particular this work denes and derives sucient conditions for the exponential stability in the mean square of FTCSMP with potentially saturated actuators. The derivations are completed employing Lyapunovs second method. Sucient conditions involve the solution of Riccati-like matrix equations. Stochastic stability of JLS driven by potentially saturated actuators can be established as an interesting special case of this general result. The results are compared with those of FTCSMP with linear non saturating actuators.

.. Dynamical model of FTCSMP with actuator saturation limit A general Fault Tolerant Control System with potentially Saturated Actuators FTCSSA is shown in Figure . . Such a system subject to random faults in actuators can be described by xt


 

Atxt



 

Sat u x t
t
t 





 


B t Sat u x t
t
t



 



Sat K 





 




.

t xt








where x t
n is the system state u x t
t
t
m is the control input t represents the failure process and t denotes the FDI process. t and t are similarly dened as in Section . . . A t and B t are properly dimensioned matrices.






 




















In the sequel we will use the following notations B t and Sat u x t
t
t Sat ui when t i
R.



 







 


 









Bk when t


 

k
S


 

.. Actuators with saturation limits Figure . shows the characteristic diagram of a typical actuator with saturation. The actuator described by a static non-linear function which saturates at uH and




CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES Reconfiguration Mechanism

FDI

Reference input +-

Actuators

Plant

Output/State

Sensors

Faults

Controller

Figure .  General schematic diagram for FTCSSA. L  is dened as

u

Sat u
x
t 
t t 











H

u

K

tx
t

L

u

u
x
t 
t t
u

L


u

H

u
x
t 
t t

.


H u

L

u
x
t 
t t  u

In which the operation of the is linear for all  L H . L and H are the lower and the upper limits of the actuator respectively. In view of Figure . and the axioms of the norm function we have S at u
x
t 
t t

u

u



u

u
x
t 
t t

u

k

Sat u
x
t 
t t

 

u
x
t 
t t

k
k 



u
x
t 
t t

k

.

The system . when operated in the linear region  L H is assumed to satisfy both the growth and the uniform Lipschitz conditions. Under these conditions the joint process f g is a Markov process. It is assumed that the state variable is available for feedback. u
x
t 
t t

u

u



x
t 
t 
t

.. Exponential stability of FTCSSA As mentioned earlier we assume that the equilibrium point  is the operating point whose stability properties are being tested. At this equilibrium state the x  




.. STABILITY OF FTCSMP DRIVEN BY ACTUATORS WITH SATURATION

H

u

Sat u


u

L

u

Figure . Characteristics of an actuator with saturation. exponential stability in the mean square of FTCSSA .  is dened as

De nition
. The equilibrium point

of the FTCSSA
. is said to be exponentially stable in the mean square if for any o and o there exist  and  such that when k o  o o
 and some positive constants o o o
k
the following inequality holds   o x

 









x


a 

t



S

R

b 

t

x

t

Efjjx t xo to
jj2 g
ajjxo jj2 expfb t  to
g

Su cient conditions for the exponential stability in the mean square are stated in the following Theorem.

Theorem
. The equilibrium point stable in the mean square   
for some constants 1 t

V

x


a b

k1

V

k

k x k2
V x





x







 

x

 

of the FTCSSA
. is exponentially

if there exists a stochastic Lyapunov function  and 3  such that 2

k



k




k2 k x k2


k3 k x k2

The proof of this theorem employs the supermartingale properties of a stochastic Lyapunov function 
. Similar theorem has been derived in  therefore proof will not be detailed to avoid repetition. In Theorem . 
is the weak innitesimal operator of the FTCSSA . . V

x


V

x





CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

For a quadratic stochastic Lyapunov function V x



 

xT P


.

x




where P

 is positive symmetric matrices
 k
S and
 i
R the weak in nitesimal operator is V x





xT AT Pki x  xT Pki Ax  xT Pki Bk Sat ui   SatT ui BkT Pki x



xT f



X

j 2S j

kj Pji  Pki gx  xT f


X

j 2R

=k

j

6

The weak in nitesimal operator can be written as V x


qijk Pkj  Pki gx


. 

=i 6

xT AT Pki x  xT Pki Ax


 

xT Pki Bk Sat ui   12 ui  21 ui  Sat ui   12 ui  12 ui T BkT Pki x


.




xT f P kj Pji  Pki gx  xT f P qijk Pkj  Pki gx



j 2S j

or V x





j 2R

=k

j

6



=i 6

xT AT Pki x  xT Pki Ax   xT Pki Bk ui   ui T BkT Pki x  

xT Pki Bk Sat ui    ui   Sat ui    uiT BkT Pki x xT f

X

j 2S j

kj Pji  Pki gx  xT f


=k

X






.

=i 6

under the state feedback ui  Ki x it follows that V x





j 2R j

6

qijk Pkj  Pki gx

xT Pki Ax   xT Pki Bk Ki x  xT AT Pki x   xT KiT BkT Pki x xT Pki Bk Sat ui   ui   Sat ui    uiT BkT Pki x

 

xT f

X

j 2S j

De ne

kj Pji  Pki gx  xT f


=k

X

j 2S j

=k 6

qijk Pkj  Pki gx


j 2R j

6

A ki  A   Bk Ki   I

X

.

=i 6

kj   I

X

j 2R j

=i 6

qijk

. 




.. STABILITY OF FTCSMP DRIVEN BY ACTUATORS WITH SATURATION

The weak innitesimal operator can then be rewritten as V x






xT fATki Pki  Pki Aki 

X

j 2S j

xT Pki Bk Sat ui 

=k

X qk P

j 2R j

6



ij kj gx

=i 6

ui  Sat ui 





kj Pji 

 

ui T BkT Pki x

.


For the selected Lyapunov function . we have min P x


 k x k
V x



max P x


 k x k

. 

Note that .  satises the boundeness condition in Theorem in  . As per Theorem . the FTCSSA is exponentially stable in the mean square if the weak innitesimal operator .
 satises the smoothness condition in Theorem . Therefore the exponential stability of FTCSSA can be dened as follows De nition
. The FTCSSA
. is said to be exponentially stable in the mean

square under the linear feedback control law ui t  Ki x t if there exist positivedenite symmetric matrices Pki such that the following matrix inequality holds

xT fATki Pki  Pki Aki 

P kj Pji P qk Pkj gx

j 2S j

xT Pki Bk Sat ui 



=k 6


ui  Sat ui 

j 2R j

=i

ij



6


ui

T B T Pki x
 k x k
k

Aki is dened in
. and some constant  

.. A sucient condition for exponential stability of FTCSSA In this section a testable sucient condition will be derived. This condition involves the solution of a set of Riccati-like matrix equations. In the literature several algorithms are available to solve Riccati matrix equations. These algorithms can be easily extended to solve the Riccati-like matrix equations. The following theorem states a sucient condition for the exponential stability of the FTCSSA in terms of Riccati-like matrix equations.




CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

Theorem . If the Riccati-like matrix equations ATki Pki
Pki Aki


X

j 2S j

kj Pji


=k

X

j 2R j

6

qijk Pkj Pki Bk Rkiᝇ BkT Pki
Qki  

. 

=i 6

have positive-de
nite solutions Pki
for given positive-de
nite weighting matrices Qki and Rki Then the FTCSSA . will be exponentially stable in the mean square under the control law

uki t  Rkiᝇ BkT Pki xt

. 

max Pki 
minRki 

k Bk k
 

. 

 X k I qij

. 

if

minQki  where

Aki  A

 X I

j 2S

kj

=k

j

j 2R =i

j

6

6

Proof The weak in nitesimal operator of the FTCSSA .  is V x





xT AT Pki x
xT Pki Ax
 xT Pki Bk uki
 uTki BkT Pki x
xT Pki Bk Satuki 
xT f

X

j 2S j

 u 
Satuki  ki

Pki gx
xT f


kj Pji

=k

X

j 2R j

6

 u T BkT Pki x ki

qijk Pkj Pki gx


. 

=i 6

If we de ne Aki  A

 X I

j 2S j

=k 6

kj

 X k I qij

j 2R j

=i 6

. 

.. STABILITY OF FTCSMP DRIVEN BY ACTUATORS WITH SATURATION

under the control law
.  the weak innitesimal operator becomes V x


  xT ATkiPki x  xT Pki Aki x 1 2

h

1 2

iT

xT Pki Bk Rkiᝇ1BkT Pki x

Rkiᝇ1BkT Pki x BkT Pki x  xT Pki Bk Sat uki 

Sat uki 



1 2

ukiT BkT Pki x  xT f

P kj Pjigx

j 2S j

or V x








=k

1 2

uki 

xT f P qijk Pkj gx j 2R j

6


. 

=i 6

xT ATki Pki x  xT Pki Akix xT Pki Bk Rkiᝇ1BkT Pki x  T uki   Sat uki   uki T BkT Pki x x Pki Bk Sat uki  xT f



X

j 2S

kj Pji gx  xT f

j

=k

X qk P

j 2R j

6

ij kj gx


. 

=i 6

Since uki   Sat uki 



uki T BkT Pki x
R1

uki   Sat uki 

1 2

uki T BkT Pkix 

xT Pki Bk Sat uki 



xT Pki Bk Sat uki 

1 2






. 

Then

xT Pki Bk Sat uki 

1 2

uki  k xT Pki Bk Sat uki 

1 2


. 

uki  k

Using the axioms of norm and
. yield

k xT Pki Bk Sat uki 



uki kk xT Pki Bk kk uki kk xT Pki Bk kk Rkiᝇ1BkT Pki x k


.



Then V x


  xT ATkiPki x  xT Pki Aki x xT Pki Bk Rkiᝇ1BkT Pki x xT f



P kj Pjigx

j 2S j

=k 6



xT f

P qk Pkj gx

j 2R j

=i 6

ij



k xT Pki Bk kk Rkiᝇ1BkT Pki x k


. 




CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

From .
 we have


xT Qki x k xT Pki Bk kk Rkiᝇ BkT Pki x k

.



xT Qkix k Pki k
k Rkiᝇ kk Bk kk BkT kk x k


.


V x




or V x




The de nition of the induced Euclidean norm gives V x







"

min Qki 




ස


max PRki  k Bk k
k x k
min ki

.


De ne real constants ki ki  min Qki 

If condition .  is satis ed i.e.





max PRki  k Bk k


ki  

V x




min ki

.


 then


ki k x k
 

.


The conditions of Theorem . are satis ed therefore the FTCSSA is exponentially stable in the mean square. The proof is completed.

Stabilization algorithm The exponential stability in the mean square of the FTCSSA .  can be tested as follows  Select positive-de nite matrices Qoki and Rkio .
 Solve the Riccati-like matrix equations in Theorem .
. If positive-de nite solutions exist then FTCSSA is exponentially stable in the mean square. Otherwise proceed to Step .  Increase Qki by some factor. Say Qki  Qoki . Go back to Step
and iterate.  If the algorithm did not succeed to provide positive-de nite solutions then stop. Declare that the exponential stability of FTCSSA cannot be determined.




.. STABILITY OF FTCSMP DRIVEN BY ACTUATORS WITH SATURATION

.. Remarks and special cases Remark
Under the assumption of perfect FDI performance i.e. instantaneous fault detection and perfect fault identication both the failure process and the FDI process will have identical state spaces. That is the two random processes t
and  t
can be replaced by a single process denoted as r t

Similar to t
and  t
 the process r t
represents a continuous time discrete state Markov process with values in a nite set   f 


 N g with transition probability rate matrix    =1
ij i j

N

In this case the transition probability for the jump process r t
 can be dened as P

with P N

j

=1

j

ij






ii

 i

kj

t

.

t o t
k  j



 kj 

.

=i

With this assumption the system when driven by actuators with saturation can be modeled as 6

x t
 A t
x t
B r t

u x t
 r t
 t
s



u u x t
 r t
 t
 u

 u x t
 r t
 t


K r t

x t
u  u x t
 r t
 t
 u


u u x t
 r t
 t
 u H

.


H

s

L

H

L

L

and the weak innitesimal operator becomes V x 




x A P x x P A x
x P B Rᝇ1B P x T

T

i

T

T

i

i

i

T

i

i

i

i

i



x f T

X N

j

=1

j

ij

P gx j

=i 6

x P B Sat u

 u Sat u

 u B P x

.


A
 I

.






T

i

i

i



i

i

T

i

T

i

i

with A

i



i




CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

From . we have V x




PN xT ATi Pi x  xT Pi Ai x
xT Pi Bi Riᝇ BiT Pix  xT f ij Pj gx k xT Pi Bi kk Riᝇ BiT Pi x k j
j

=i 6

. Following the similar arguments Corollary . states a sucient condition for the exponential stability of the system . .

Corollary . The system with saturating actuators
. is exponentially stable in the mean square if there exist positive-denite symmetric solutions Pi
to the following Riccati-like matrix equation

ATi Pi  Pi Ai 

N X j
j

ij Pj
Pi Bi Riᝇ BiT Pi  Qi  



.


=i 6

and 

min Qi 
max PRi   k Bi k   min i

. 

 Qi   Ri  . The linear control law is given as ui 
Riᝇ BiT Pix t

.

The model of the system .  is similar to those of JLS. Therefore Corollary . can be used to examine the exponential stability of JLS driven by actuators with potential saturation.

Remark  If we consider FTCSMP without saturation non-linearities we have x t  A tx t  B  tu x t

t
t u x t

t
t 
K

t x t

.




Following similar analysis used to prove Theorem .  sucient condition for the exponential stability of the system . is




.. STABILITY OF FTCSMP DRIVEN BY ACTUATORS WITH SATURATION

Theorem . The control law uki that stabilizes the FTCSMP
.
 is uki t
 ᝇRkiᝇ1BkT Skix t


.

where Ski are the solutions of the following Riccati-like matrix equation

ATki Ski  Ski Aki 

X

kj Sji 

j 2S j

=k

X qk S

j 2R j

6

ᝇ1 T ij kj ᝇ Ski Bk Rki Bk Ski  Qki  

.

=i 6

with Aki given in
.


Now we will illustrate that the solutions of the Riccati-like matrix equation for FTCSMP with saturating actuators .  are bounded below by the corresponding solutions of those for FTCSMP without saturating actuators .. To demonstrate this we rst establish the following Lemma. To reduce notations a positive-de nite matrix Y will be denoted as Y
.

Lemma . Let Pki1 be positive-denite solutions of the Riccati-like matrix equation ATki Pki  Pki Aki 

X

j 2S

kj Pji 

=k

j

X qk P

j 2R

. 

=i

j

6

1 ij kj ᝇ Pki Wki Pki  Qki  

6

where Wki
 Then for   Wki  Wki  the following Riccati-like matrix equation 1

ATki Pki  Pki Aki 

X

j 2S j

2

1

kj Pji 

=k

X qk P

j 2R j

6

2 ij kj ᝇ Pki Wki Pki  Qki  

. 

=i 6

have positive-denite solutions Pki2
Pki1


Proof Let the positive-de nite solutions Pki2  Pki1  ki  Substituting in .  we

have

ATki Pki1  ki  Pki1  ki Aki 

P

j 2S j

P qk

j 2R j

=i 6

=k

kj Pki1  ki 

6

ij Pki  ki ᝇ Pki  ki Wki Pki  ki  Qki   1

1

2

1

. 




CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

P

ATki Pki1 Pki1 Aki

j 2S j

P qk
ki ᝇ P

j 2R j

=i

ij

=k

P qk P

kj Pki1

j 2R =i

j

6

1 T ij ki Aki
ki


ki Aki

P

j 2S j

6

=k

kj
ki

6

.

1 2 1 1 2 2 1 2 ki Wki Pki ᝇ Pki Wki
ki ᝇ
ki Wki Pki ᝇ
ki Wki
ki Qki  

6

but Pki1 are solutions to equation . then

P

ATki
ki
ki Aki

j 2S j

=k

kj
ki

P qk
ki

j 2R j

6

=i

Pki1 Wki1 Pki1 ᝇ

ij

.

6

Pki1 Wki2 Pki1 ᝇ Pki1 Wki2
ki ᝇ
ki Wki2 Pki1 ᝇ
ki Wki2
ki  

with some arrangements

ATki ᝇ Pki1 Wki2 
ki
kiAki ᝇ Wki2 Pki1 

P

j 2S j

P qk
ki ij

j 2R j

=i

=k

kj
ki

. 

6

Pki1 Wki1 ᝇ Wki2 Pki1 ᝇ
kiWki2
ki  

6

Dene Aki  Aki ᝇ Wki2 Pki1 

Then Aki
ki
kiAki

X j 2S j

=k 6

kj
ki

X qk


j 2R j

ij ki

.


Pki1 Wki1 ᝇ Wki2 Pki1 ᝇ
ki Wki2
ki  

=i 6

. For Wki1
Wki2
 these coupled Riccati-like di erential equations have positivedenite solutions
ki
 That is Pki2  Pki1
ki
Pki1

The proof is complete.

.

.. STABILITY OF FTCSMP DRIVEN BY ACTUATORS WITH SATURATION




Using the assignment

Wki1
Bk Rkiᝇ1BkT  Wki2

Bk Rᝇ1B T ki

k

1 2

Wki1
Wki1


.

and based on Lemma
.  the positive-de nite solutions for FTCSMP with actuators saturation over-bound the positive de nite solutions for FTCS without actuators saturation. The result is stated in Lemma
. .

Lemma . If Pki t are positive-de
nite solutions for . then  Skit Pki t


. 

where Ski t are positive-de
nite solutions of . .

Remark  If only part of actuator non-linearity is to be considered during the actual system operation then less conservative results can be obtained. In this case the non-linearity will be restricted to the sector a  with  a  instead of the original sector   as shown in Figure
. . The axioms of norm function gives the following inequality

k Satui

  aui k 

a k ui k


. 

Following similar arguments used to derive sucient conditions for the stability of FTCSSA 
.  with non-linearities in the sector   we obtain the following lemma

Lemma . The FTCSSA . with saturation non-linearities in the sector a  is exponentially stable in the mean square under the control law

uki t
Rkiᝇ1BkT Pki xt


. 




CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

uH


a 

1

a

u

Sat u
uL

Figure . Characteristics of an actuator with saturation in sector a . where Pki are bounded positive-de
nite symmetric solutions to the following Riccatilike matrix equation

ATki Pki  Pki Aki 

X

kj Pji 

j 2S j

=k

X

j 2R j

6

qijk Pkj ᝇ

a Pki Bk Rkiᝇ BkT Pki  Qki   . 




=i 6

Qki
 Rki
 and Aki is given in . if  min Qki


a
max PRki

k Bk k
 min ki

The weak in nitesimal operator of FTCSSA .  is

Proof

V x 




xT AT Pki x  xT Pki Ax  



xT Pki Bk Sat ui



xT f



a xT Pki Bk ui  




a ui T BkT Pki x




a ui   Sat ui






a ui T BkT Pki x




X

kj Pji
Pki gx  xT f qijk Pkj
Pki gx j 2R j 2S X

j

=k

j

6

.



=i 6

From .  we have xT Pki Bk Sat ui



Then

xT Pki Bk Sat ui

 

a ui   Sat ui






a ui T BkT Pki x 




a ui k xT Pki Bk Sat ui






V x 

min Qki



a ui  k




max Pki
 k B k 
a
k min Rki


.





k x k

. 

.. STABILITY OF FTCSMP DRIVEN BY ACTUATORS WITH SATURATION

Dene real constants




a ki

a ki

Hence there exist some


min
Qki  ᝇ
 ᝇ a 

max

PRki  k Bk k   min

a ki

ki

.

  such that V
x  


a ki

kxk 

. 

The conditions of Theorem . are satised therefore the FTCSSA with non-linearity dened in the sector a  is exponentially stable in the mean square. This result can be considered as a general form to examine exponential stability of FTCSSA under various conditions

 If a  all actuator non-linearity is considered we obtain the results in Theorem . .  If a  actuators are assumed to be linear without saturation we obtain the results in Theorem . . Remarks and can be easily extended to deal with FTCS driven by actuators with potential saturation in the sector a .

.. A numerical example Consider a system with one possible actuator fault. The system and other design parameters are given as 3 2 66   7 7  B
A 64 7 5




2 3 6  77 6  B 6 5 4 7

The actuator failure rates are assumed to be



3 2 7 6  

7 6 7 6 5 4






CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

2 6 6 ᝇ
 
 jk
 6 4 
 ᝇ


3 7 7 7 5

The FDI conditional transition rates are 2 6 ᝇ
 
 qij1
 6 64 
 ᝇ


2 3 6 ᝇ
 
 7 7  qij2
 6 6 7 4 5 
 ᝇ


3 7 7 7 5

The initial weighting matrices used are 2 6 Q11  6 64









3 2 77 6 75  Q12  664









3 2 7 6 7  Q21  6 7 6 5 4









3 2 7 6 7  Q22  6 7 6 5 4









3 7 7  7 5

R11  
 R12  
 R21  
 R22  


The stochastic stability of the FTCSSA with actuator fault is to be investigated. To illustrate the theoretical results developed in this chapter this example considers several dierent scenarios.

CASE
Actuator saturation and stability of FTCSMP If the conditions in Theorem .
are satised the FTCSSA is exponentially stable in the mean square otherwise the stability cannot be determined. Three scenarios are considered FTCSSA with actuator saturation in the sector  FTCSSA with actuator saturation in the sector a and FTCSMP without actuator saturation. Positive-denite solutions of Riccati-like matrix equations and associated constants  for the three scenarios are summarized in Table . and .
 respectively.  





ki

Sucient conditions for the stability of FTCSSA necessitate that the Riccati-like




.. STABILITY OF FTCSMP DRIVEN BY ACTUATORS WITH SATURATION

Table .  Positive-denite solutions of Riccati-like equations.

P11

a

2 66 64





2 66 64

2 66 64

P12


  
 
 

 



  

  



  


3 77 75

2 66 64

 
  

3 77 75

2 66 64


  

3 77 75

2 66 64

  


 

 

       

P21

P22

3 77 75

2 66 64

 


3 77 75

2 66 64

 

3 77 75

2 66 64


  






  


        

3 77 75

2 66 64



3 77 75

2 66 64

 
  

3 77 75

2 66 64


  


 


 

 

 

 
   

 

3 77 75

3 77 75

3 77 75

matrix equations have positive-denite solutions and the constants are positive for all 2 2 . Even though Table . shows that symmetric positive-solutions exist for the three scenarios we still have to check the positiveness of Table .
lists these constants.
ki

k

S i

R


ki

Therefore the FTCSSA is not stable for actuator saturation in the sector . On the other hand both FTCSSA with the actuator saturation in the sector   







CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

Table .
 Constants Sector a

 a  

ki

11

ᝇ



 

in the sector a .

12

ᝇ

21

ᝇ



 



22



 

a  

  






 



a 


 



 




 



ᝇ















 

P12 

P11 

3

Riccatti-like matrix equation

Riccatti-like matrix equation

and without any saturation are stable. Positive-de nite solutions for these stable scenarios are shown in Figures . and . respectively.

2.5 2 1.5 1 0.5 0

10

20 Iterations

30

3.5 3 2.5 2 1.5 1 0.5

0

10

8 6 4 2 0

0

10

20 Iterations

30

P22 

10

Riccatti-like matrix equation

Riccatti-like matrix equation

P21 

20 Iterations

30

14 12 10 8 6 4 2

0

10

20 Iterations

30

Figure .  Positive-de nite solutions in the sector  . . Notice that the positive-de nite solutions given in Table . are consistent with the results of Remark
. That is solutions of Riccati-like matrix equations for FTC-




.. STABILITY OF FTCSMP DRIVEN BY ACTUATORS WITH SATURATION P11


Riccatti-like matrix equation

Riccatti-like matrix equation

2.5

2

1.5

1

0.5 10

Iterations

20

3 2.5 2 1.5 1 0.5 0

30

P21


10 8 6 4 2 0 0

10

10

20

30

20

30

Iterations

Riccatti-like matrix equation

Riccatti-like matrix equation

0

P12


3.5

20

P22


12 10 8 6 4 2 0

30

10

Iterations

Iterations

Figure . Positive-denite solutions for FTCSMP without saturation. SSA over bound those for FTCS without saturation. P

FTCS ki

P

FTCSSA ki




k i

f g  

The state feedback control laws which stabilize the FTCSMP driven by actuators with saturation are shown in Table ..

CASE
Sector size of nonlinear saturation and stability of FTCSSA In the following the behavior of FTCS for smaller sector size will be studied. The weighting matrices ki and ki are maintained the same while the lower bound of the sector is being varied. The positiveness of the constants ki is the criteria for the stability of the FTCSSA in the sector . Q

R

a




a 


As can be seen in Figure . the FTCSSA is exponentially stable in the mean square if . a






CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

Table . State feedback control laws in the sector . . Sector

K

2 66 64





5 0

T

K

11

T

 

 


3 77 75

2 6 6 6 4





T

K

21

2 6 6 6 4

3 7 7 7 5

 






5

11

5

5

10

10

2 6 6 6 4

3 7 7 7 5

20

 

15 20

25

25 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

a

0

0 10

21

10

20

30

30

40

40

a

0.6

0.8

1

0.6

0.8

1

22

ki

20



ki 




3 7 7 7 5

ki 

15

50 60 70

T

22

12

0

ki 

K

12

50 60

0

0.2

0.4

0.6

0.8

1

70

0

0.2

0.4

a

Figure .  The constants

a

ki



versus . a

. Stabilization of FTCSMP with Parameter Uncertainties To achieve a satisfactory performance in fault tolerant control an FDI algorithm should be able to provide the recon guration mechanism with the most accurate

.. STABILIZATION OF FTCSMP WITH PARAMETER UNCERTAINTIES




and up-to-date information after the occurrence of a fault. Since the FDI algorithm usually relies on statistical tests to detect and identify faults it takes time to make a reliable decision. In practice an FDI algorithm is not able to provide exact post-fault system information instantaneously 
. Such an inaccuracy can be interpreted as uncertainties in system parameters. As a result the controller designed may lead to system performance degradation or even loss of closed-loop stability. This issue provides enough motivation to develop an FTCSMP with certain degree of robustness to the model parameter uncertainties in the system to be controlled. The robust control problem has attracted a considerable amount of interest in the last twenty years. The matching condition which provides su cient conditions for the stabilization of uncertain systems was employed by several authors  

. A quadratic upper bound for the Lyapunov derivative was de ned and used to stabilize uncertain systems    . The results have been extended to H1 control in    time-varying systems in   and time-delay systems with normbounded uncertainties in  . For hybrid systems with uncertainties substantial results were developed recently in several work. In 

 a robust state feedback controller for JLS in the presence of structured uncertainties was designed. Discretetime uncertain JLS was studied in  . Robust stability of continuous-time JLS with structured and unstructured uncertainties has been considered in  . On the other hand the stability of FTCSMP with parameter uncertainties has been considered in 
. However this current study concerns with the derivation of stability conditions for a given state feedback control law. Three forms of uncertainty were considered norm bounded linear combination and value bounded. The results revealed that the stochastic stability of FTCSMP may be lost in the presence of large uncertainty. Unfortunately the investigation of the stabilization of FTCSs with given bounded parameter uncertainties has not been carried out in the literature. In the following the phrase admissible parameter is used to denote parameter




CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

uncertainties which satisfy the boundness conditions in Section ..
. The phrase uncertain FTCSMP  will be used to denote the FTCSMP with admissible parameter uncertainties in the system to be controlled and certain FTCSMP denotes the FTCSMP without such parameter uncertainties. In this chapter a FTCSMP designed to tolerate faults in plant components is considered. Because of the inaccuracies in estimated parameters the system to be controlled contains parameter uncertainties which are assumed to be time-varying unknown-but-bounded. The norm bounded uncertainty form is used and the Lyapunov s second method is employed to study the exponential stability in the mean square of such systems. Consequently an algorithm to test exponential stabilization of the uncertain FTCSMP is proposed. The test involves the solution of coupled Riccati-like algebraic equations. The main result is that the convergence of the proposed algorithm provides a necessary and sucient condition for exponential stabilization in the mean square of uncertain FTCSMP. When the algorithm converges a stabilizing controller can be obtained. Several interesting results are shown to be the special cases of the general result presented.

.. Dynamical model of FTCSMP with parameter uncertinities Let s assume that the system to be controlled can be described by the following linear time-invariant dynamical model xt

_෈ ෉ =

Ax t

෈ ෉ +

.

Bu t

෈ ෉

If the system . is subject to faults in plant components a FTCSMP developed to maintain its stability can be represented as xt

_෈ ෉ =

A t xt ෈ ෈ ෉෉

u x t
t
t ෈

෈ ෉ ඵ෈ ෉

෈ ෉ +

෉ =

Bu x t
t
t

ᝇK

෈

෈ ෉ ඵ෈ ෉

t xt

෈ඵ෈ ෉෉

෈ ෉

෉

.




.. STABILIZATION OF FTCSMP WITH PARAMETER UNCERTAINTIES

where x 2 n is the system state u
m is the control input t
is a Markovian process representing random faults and  t
is a Markovian process representing the decisions of the FDI process. t
and  t
are homogeneous Markov processes with nite state spaces Z  f


zg and R  f


rg respectively 

  . A schematic diagram of such a system is shown in Figure .. Reconfiguration Mechanism

FDI

Actuators

Input +

Plant

Sensors

Outputs/states

_

Faults Controller

Figure . Schematic diagram for AFTCS with faults in plant components. Once a fault occurs the FDI algorithm will estimate post-fault system parameters possibly with certain degree of inaccuracies. Let these inaccuracies be represented by time-varying unknown-but-bounded matrices A  t

. Therefore the FTCSMP with parameter uncertainties can be described as x t
 A t

A  t

x t
Bu x t

 t

t
u x t

 t

t
 K

t x t


. 





Note that the uncertain FTCSMP .  is modelled such that the system matrix A is subject to changes due to random faults in plant components. These changes may take place according to the transition rates of the failure process t
. At the same time  uncertainties in parameters are a function of the FDI process  t
 which has its own transition rates. This model emphasizes the practical aspect of AFTCS where the failure and FDI processes have separable characteristics.




CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

It is assumed that the uncertain FTCSMP .  satises both the growth and the uniform Lipschitz conditions. Under these conditions the joint process fx෈t෉
෈t෉ ඵ෈t෉g is a Markov process. It is also assumed that the system state is available for feedback. The transition probability for the plant component failure process
෈t෉ is dened as p

kj ෈ත

t෉ = 

.


t + o෈තt෉ ෈k = j ෉

kj ත

where  represents the plant component failure rate. Given that
෈t෉ = k
Z the conditional transition probability of the FDI process ඵ෈t෉ is dened as kj

p

t෉ = q

k ෈ත ij

. 

t + o෈තt෉ ෈i = j ෉

k ත ij

where q is the rate at which the FDI process will decide that the system post-fault mode is j while the system pre-fault mode is i given that the fault k has occurred. In the sequel we will use the following notations A = A when
෈t෉ = k
Z and u෈x෈t෉ ඵ෈t෉ t෉ = u තA෈ඵ෈t෉෉ = තA when ඵ෈t෉ = i
R. k ij

k

i

i

.. The model of parameter uncertainties To study analytically the stabilization of uncertain FTCSs we will rst assume that the admissible parameter uncertainties have a norm bounded Uncertainty NBU form. This is the most adopted form in robust stability analysis 
 . A block diagram representation for a plant with such uncertainties in the system matrix is shown in Figure .. In this form the admissible parameter uncertainty is modelled as A

t

ත ෈ඵ෈ ෉෉ =

.

H ෈ඵ෈t෉෉F ෈ඵ෈t෉෉E ෈ඵ෈t෉෉

where H ෈ඵ෈t෉෉
 ථ F ෈ඵ෈t෉෉
 ථ and E ෈ඵ෈t෉෉
 ථ . H ෈ඵ෈t෉෉ E ෈ඵ෈t෉෉ are known constant matrices and F ෈ඵ෈t෉෉ is a Lipschitz measurable matrix function satn

p

p

q

q

n




.. STABILIZATION OF FTCSMP WITH PARAMETER UNCERTAINTIES

u

R

B

x

A + තA

Figure . Plant with uncertainties in system matrix. isfying the condition F ෈ඵ෈t෉෉T F ෈ඵ෈t෉෉ Ip
t  0 ඵ෈t෉  R

.

To simplify notations denote H ෈ඵ෈t෉෉ = Hi F ෈ඵ෈t෉෉ = Fi  E ෈ඵ෈t෉෉ = Ei when ඵ෈t෉ = i  R. .. Stabilization of uncertain FTCSMP

In this section exponential stabilization in the mean square of uncertain FTCSs is analyzed. Exponential stability is de ned and conditions are derived in
. Interested readers may refer to
 for more detail. In this section new necessary and sucient conditions for exponential stabilization of the uncertain FTCSMP .

are derived. A test algorithm is constructed. Without loss of generality we assume that the equilibrium point x = 0 is the solution whose stability properties are being examined. For a quadratic stochastic Lyapunov function V ෈x
ඵ෉ = xT P ෈
ඵ෉x

.

where P ෈
ඵ෉ is a positive symmetric matrix

= k  Z and ඵ = i  R the weak in nitesimal operator is i h V ෈x
ඵ෉ = xT ATk Pki + Pki Ak x + 2xT Pki තAix + 2xT Pki Bui

+xT f

X

j 2Z j

=k 6

kj Pji  Pki gx + xT f


X

j 2R j

=i 6

qijk Pkj  Pki gx


.




CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

under the control law ui = ᝇKi x the weak innitesimal operator becomes h i V ෈x

ඵ෉ = xT ATk Pki + Pki Ak x + 2xT Pki තAix ᝇ 2xT PkiBKi x

+xT f

X

j 2Z j

kj Pji Pki gx + xT f


X

j 2R

=k

j

6

qijk Pkj Pki gx


.

=i 6

Dene X X A^ki = Ak 21 I kj 21 I qijk j 2Z j 2R j

=k

j

6

.

=i 6

rearranging the terms in equation . leads to V ෈x

ඵ෉ = xT fA^Tki Pki + Pki A^ki +

X

j 2Z j

=k 6

kj Pji +

X

j 2R j

qijk Pkj 2Pki BKi + 2Pki තAigx

=i 6

.

Note that due to the presence of the parameter uncertainties the weak innitesimal operator of the uncertain FTCSMP contains an additional term in comparison with that of the certain FTCSMP that is u V ෈x

ඵ෉ = c V ෈x

ඵ෉ + ත V ෈x

ඵ෉

.

where cV ෈x

ඵ෉ is the weak innitesimal operator of the certain FTCSMP and ත V ෈x

ඵ
t෉ is the part of the weak innitesimal operator associated with the parameter uncertainties. This part has an upper bound ෈ත V ෉bound  that can be dened for dierent uncertainty forms. Then u V ෈x

ඵ෉
upper V ෈x

ඵ෉ = c V ෈x

ඵ෉ + ෈ත V ෉bound

.

where upperV ෈x

ඵ෉ is the upper bound of the weak innitesimal operator with the maximum uncertainty values. From equation .  the term due to parameter uncertainties is

ත V ෈x
k
i෉ = 2xT Pki තAi x
 = k  Z
ඵ = i  R To determine ෈ත V ෉bound  the following lemma is in order

.





.. STABILIZATION OF FTCSMP WITH PARAMETER UNCERTAINTIES

Lemma .   Let L M N be real matrices of appropriate dimensions. Then
for any
 and for all the functional matrices satisfying M T
tM
t I we have xT PLM
tNx


xT PLLT Px 

xT N T Nx

For the norm-bounded uncertainty form with the condition FiT
tFi
t Ip

 . 

the upper bound of V
x k i is xT Pki Ai x  xT Pki Hi Fi Ei x

xT 
iPki HiHiT Pki 

i EiT Ei x

 . 

De ne Di  HiHiT

 . 

Gi  EiT Ei

 .

and

Therefore
V
x

k ibound  xT f
i Pki DiPki 

i Gi g x

 . 

The following theorem states a sucient condition for the exponential stability of the uncertain FTCSMP  . .

Theorem . Suppose that there exist some constants
i 


i  R such that

Riccati-like matrix equations

X

X

j 2Z

j 2R

A Tki Pki Pki A ki  kj Pji  qijk Pkj Pki BRkiᝇ1B T Pki 
iPki Di Pki 
 Gi  Qki  j

=k 6

j

=i

i

6

have bounded positive-denite symmetric solutions
Pki
for symmetric positive-denite Qki Rki  Di and Gi are dened in . and .
respectively Then
the uncertain FTCSMP . is exponentially stabilizable in the mean square under the control law

uki
t  Rkiᝇ1B T Pki x
t




CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

Suppose that the Riccati-like matrix equations have positive-denite solutions Pki 0. Then the function

Proof

.

V ෈x

ඵ෉ = xT P ෈
ඵ෉x

is a stochastic Lyapunov function and satises the conditions of Theorem  in   . This means that V ෈x

ඵ෉ is positive-denite bounded with innitesimal upper limit continuous and in the domain of the weak innitesimal operator. Under the control law uki = ᝇRkiᝇ1BPkix෈t෉ and from equation .  the weak innitesimal operator is upper V ෈x
k
i෉ = A^Tki Pki + Pki A^ki +

P kj Pji + P qk Pkj ᝇ 2Pki BRᝇ BT Pki + iPkiDiPki +

j 2Z j

j 2R

=k

j

6

=i

1 ki

ij

1

G

i i

6

.

Moreover the existence of the steady-state solutions Pki  implies that they satisfy the coupled matrix equations in Theorem .  xT fA^Tki Pki + Pki A^ki +

P kj Pji + P qk Pkj

j 2Z j

+i Pki Di Pki + 1

G

+ i i

=k 6

j 2R j

=i

ij

ᝇ1 B T P 2Pki BRki ki

6

.

Qki gx = 0

therefore upper V ෈x

ඵ෉

+ xT Q෈
ඵ෉x = 0

.

From equation .  we obtain uV ෈x

ඵ෉
xT Q෈
ඵ෉x

For the quadratic Lyapunov function V ෈x

ඵ෉ = xT P ෈
ඵ෉x we have

.






.. STABILIZATION OF FTCSMP WITH PARAMETER UNCERTAINTIES u


  ᝇ x x


 x

V x



V

T

Q 


T

P 



x  
x

.

Taking the maximum we obtain u


 x




V x



V

"

ස

max
x Q 

x 
min x P 

x 2 2 T

T

k

Z i

ය

R

max P

ර


 
   .  



min Q 


where min Q 

 is the eigenvalue of the matrix Q 

 with the smallest real part and max P 

 is the eigenvalue of the matrix P 

 with the largest real part for all   k  Z

 i  R.  is positive constant and represents the rate of convergence. Equation .  can be written as u

V x





V

x



. 




From Dynkins formula and the Gronwell-Bellman lemma 
 we have

EfV




t
tg exp
 t
t V

x t
 t


o

xo
o


.



t o

or

Efjjx t x
t jj g o

o

jj jj expf
 t
t g

c xo

.

o

As dened in  the uncertain FTCSMP .  is exponentially stable in the mean square. The proof is complete. It is important to mention that Theorem . provides a sucient condition for the exponential stability of the uncertain FTCSMP .  for a chosen set of  . This means that if symmetric positive-denite solutions exist for a particular selection of   then the system is exponentially stable. If they do not exist nothing can be said about the stability of the uncertain FTCSMP. In the following a necessary condition will be derived and a testable algorithm will be constructed. In doing so the necessary and sucient condition for exponential stabilization of the uncertain FTCSMP .  in the mean square can be established. It will also be shown later that if the uncertain FTCSMP .  is exponentially stable in the mean square there should exist positive constants   such that the i

i

i




CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

Riccati-like matrix equations have positive-denite symmetric solutions. Theorem . formally states this necessary condition.

Theorem . The uncertain FTCSMP
. is exponentially stable in the mean square for any given symmetric positive-denite matrices Qki and Rki if there exist positive constants iද
 8 i 2 R such that the Riccati-like matrix equations

X

X

j 2Z

j 2R

A
Tk Pki Pki A
k  kj Pji  qijk Pkj ᝇ Pki BRkiᝇ1B T Pki i Pki Di Pki   Gi  Qki  i j

=k 6

j

=i 6

.  have bounded positive-denite solutions Pki  for all i 2   iද .

To complete the proof of the theorem we need the following Lemmas.

Lemma .  Let X Y and Z be given k ථ k symmetric matrices such that X and Z are positive semi-denite and Y is negative denite. Furthermore assume that T Y 2 ᝇ T XT Z


for all non-zero  2 Rk . Then there exists a constant 
such that the matrix 2 X  Y  Z is negative denite.

Lemma . Finslers Theorem  Let X and Y be given k ථ k symmetric matrices such that X is positive semi-denite T Y 
for all non-zero  2 Rk and T X   Then there exists a constant 
such that the matrix Y  X is positive-denite.

Lemma .  Given any x 2 Rn  n

o maxxT PDFEx2 F T F ය I  xT PDDT PxxT E T Ex

Proof of Theorem . Assume that the uncertain FTCSMP .  is exponentially




.. STABILIZATION OF FTCSMP WITH PARAMETER UNCERTAINTIES

stable in the mean square. Then there exist positive-denite solutions Pki that satisfy the Riccati-like matrix equations . . For the admissible parameter uncertainties and under the control law uki ᝇKki x the weak innitesimal operator is given in equation . . Since the system is exponentially stable in the mean square we have xT fATki Pki  Pki Aki 

u V
x






Xkj Pji  X qijk Pkj  PkiAi

j 2Z j

ki k x k


j 2R

=k

j

6

Pki BKki gx

=i 6

. 

2

For the norm bounded uncertainty the weak innitesimal operator is xT fATki Pki PkiAki 

u V
x



Xkj Pji Xqijk Pkj PkiHiFiEi

j 2Z




j

ki k x k


j 2R

=k

j

6

Pki BKki gx

=i 6

2

. 

for all x  p where p is a linear space dened as follows De nition
. Given any n  n positive-de
nite symmetric matrices Pki  let

p fx  Rn B T Pki x g

. 

Moreover de
ne to be a matrix whose columns form a set of basis vectors for the linear space p 

Therefore xT fATki Pki  Pki Aki 

P kj Pji  P qk Pkj gx

j 2Z j

j 2R

=k

j

6

=i

ij

6

.


 xT Pki HiFiEi x   maxfxT Pki HiFi Eix FiT Fi
I g


From Lemma . it follows

2 C BB T T X X B
x fAki Pki  Pki Aki  kj Pji  qijk Pkj gxCCA  xT Pki HiHiT Pki xxT EiT Eix j 2Z j 2R j

=k 6

j

=i 6

. 




CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

Dene the following symmetric matrices Xki = ථT Pki HiHiT Pki ථ Yki = ථT fA^Tki Pki + Pki A^ki +

P

j 2Z j

+

kj Pji

=k

P qk Pkj gථ

j 2R j

6

=i

ij

.

6

Zki = ථT EiT Ei ථ

From Lemma . there exist constants
i  0 such that
2i xT Pki HiHiT Pki x +
i xT fA^Tki Pki + Pki A^ki +

X

j 2Z j

kj Pji

=k

X

+ qijk Pkj gx + xT EiT Eix  0 j 2R j

6

=i 6

. For the linear space p we have xT Pki BB T Pki x = 0 for x = 0. Then from Lemma . and inequality . there exist constants i  0 such that Q~ ki = 2iPki BB T Pki

2i Pki HiHiT Pki



ifA^Tki Pki + Pki A^ki +

P

j 2Z j

or ATkiPki + Pki Aki +

=k

P

=k 6

P qk Pkj g
ET Ei  0

j 2R j

6

j 2Z j

kj Pji +

kj Pji

+

=i

ij

.

i

6

P qk Pkj

j 2R j

=i

ij

.

6


2෈
෉Pki BB T Pki +
iPki Di Pki +
1 iGi + Qki = 0 i

i

In equation . the weighting matrix Rki =
Iki  k  Z i  R. Theorem . guarantees that the existence of positive-denite solutions is not a ected by a particular choice of the weighting matrices Qki and Rki . That is if the uncertain FTCSMP . is exponentially stable for a given Qki Rki and i then the stability is maintained for any other set of positive-denite symmetric weighting matrices Qki

Rki and a proper selection of i. i

i




.. STABILIZATION OF FTCSMP WITH PARAMETER UNCERTAINTIES

Theorem . Let the Riccati-like matrix equations

X kj Pji Xqijk Pkjᝇ PkiBRkiᝇ BT Pki

ATki Pki
Pki Aki


j 2Z j




j 2R

=k

1



i Pki Di Pki


=i

j

6






iGi
Qki  

6

. have positive-de
nite symmetric solutions Pki  for given positive-de
nite symmetric weighting matrices Qki  Rki and positive constants
i   8 i 2 R. Then for any given positive-de
nite symmetric matrices Q ki and Rki there exist constants
iද   8 i 2 R such that for any given  
i ය
iද the Riccati-like matrix equations

X kj Pji Xqijk Pkj

ATki Pki
Pki Aki


j 2Z




j 2R

=k

j

=i

j

6

ᝇ1 T ᝇ Pki B R ki B Pki

iPki Di Pki





i Gi
Qki   

6

.  have positive-de
nite symmetric solutions Pki.

Proof The proof can be carried out using similar arguments as in Remark  in Section .. . This theorem implies that there exist positive constants
iද   such that for any
i 2 
iද  the Riccati-like matrix equations

X kj Pji Xqijk Pkj

ATki Pki
Pki Aki


j 2Z j




=k 6

j 2R j

ᝇ1 T ᝇ Pki BRki B Pki

iPkiDi Pki


=i




iGi
Qki  

6

.   have positive-de nite solutions Pki . The proof is completed.

Stabilization algorithm Based on the results of Theorems . and . an algorithm to examine the stabilizability of the uncertain FTCSMP can be constructed as follows Select symmetric positive-de nite matrices Qki  Rki and initial positive constants
io Compute Di and Gi de ned in equations . and . respectively.




CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

 Solve the Riccati-like matrix equations in Theorem .. If positive-denite solutions exist then the uncertain FTCSMP is exponentially stable in the mean square. The control gain is calculated as per Theorem .. Otherwise proceed to Step
.
Reduce

o i

by some factor. Say

1

i

=

o i


2

Return to Step .

 If it is not possible to nd that provides positive-denite solutions then the uncertain FTCSMP with norm bounded uncertainty is not stabilizable. i

If the algorithm succeeds for some positive constants the uncertain FTCSMP is exponentially stabilizable in the mean square. If the algorithm fails to nd positivedenite solutions the uncertain FTCSMP is not exponentially stabilizable. This result is formally stated in Corollary .. i

Corollary . A necessary and su cient condition for the uncertain FTCSMP
.

to be exponentially stabilizable in the mean square is that the algorithm converges to positive-denite solutions for the Riccati-like matrix equations for a su ciently small . i

.. Special cases

If we assume that the FDI algorithm can detect the occurrence of faults instantaneously and can always identify faults correctly the failure process  ෈t෉ and the FDI process ඵ෈t෉ will have identical state spaces. A single process ෈t෉ can be used to describe their Markovian behavior. ෈t෉ has a nite state space S = f1 2  sg with transition probability as follows

Special case 

pij ෈තt෉ = ij තt + o෈තt෉

෈i = j ෉

. 

where  represents the rate at which ෈t෉ will change from mode i to mode j . ij




.. STABILIZATION OF FTCSMP WITH PARAMETER UNCERTAINTIES

The corresponding uncertain system can be described as xt

A t


  



  

u x t
t
t








 

A t xt




 

Bu x t
t
t










. 


ᝇK t x t






The weak innitesimal operator becomes V x



 

xT fATi Pi Pi Ai 



X P

j 2S

ij j

Pi Ai

 





Pi BiKi gx

.  

The necessary and su cient conditions for the exponential stabilizability of the uncertain model . 
 are given in Corollaries .
and .  respectively. 

i  S
is exponentially stable in the mean square under the control law ui Riᝇ1BiT Pi x t  where Pi are bounded positive-de nite symmetric solutions to the following Riccati-like matrix equations

Corollary . The uncertain system
. for given constants i





ATi Pi Pi Ai 



X P

j 2S

ij j



Pi BiRiᝇ1BiT Pi i PiDi Pi  Gi Qi i 










.  



for symmetric positive-de nite Qi and Ri. Corollary . If the uncertain system
. is exponentially stable in the mean

square then for any given positive-de nite matrices Qi and Ri
there exist positive constants iද  equations



such that for all i 
iද

i  S  the Riccati-like matrix

ATi Pi Pi Ai 






X P

j 2Z

ij j





Pi BiRiᝇ1BiT Pi i Pi DiPi  Gi Qi i 









have positive-de nite solutions Pi .

Since a JLS has a similar model structure the above results with the constructed algorithm provide a necessary and su cient condition for exponential stabilization of an uncertain JLS.




CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

Given any admissible parameter uncertainties the success of the stabilization algorithm implies the exponential stability of the uncertain FTCSMP . . If there is no parameter uncertainty i.e. තA෈t෉ = 0 the system becomes the certain FTCSMP which is represented by

Special case


x_ ෈t෉ = A෈ ෈t෉෉x෈t෉ + Bu෈x෈t෉
ඵ෈t෉
t෉ u෈x෈t෉
ඵ෈t෉
t෉ = ᝇK ෈ඵ෈t෉෉x෈t෉

. 

To derive the condition for the stability of the certain FTCSMP .  as a special case of the general result we set Di = 0 and Gi = 0 In this case i is not used and the search for suitable i is not needed. The existence of positive-denite solutions needs to be tested only once without any iteration. Based on this the conditions of Theorems .
and . can be combined to provide a necessary and su cient condition for exponential stability in the mean square for the certain FTCSMP .  as follows Corollary . The certain FTCSMP
. under the control law ui ෈t෉ = ᝇKi x෈t෉


is exponentially stable in the mean square if and only if the following coupled matrix equations

ATk Pki + Pki Ak +

Xkj Pji + X qijk Pkj ᝇ 2KiT RkiKi + Qki = 0

j 2Z

j 2R

have positive-de nite symmetric solutions Pki  0

If we select Rki = I
and dene

A~ki = Ak ᝇ BKi ᝇ 12 I

X qijk ᝇ 1 I Xkj 2

j 2R j

=i

j 2Z j

6

. 

=k 6

then the condition in Corollary . can be written as A~Tki Pki + Pki A~ki +

Xkj Pji + Xqijk Pkj + Qki = 0

j 2Z j

=k 6

j 2R j

=i 6

. 




.. STABILIZATION OF FTCSMP WITH PARAMETER UNCERTAINTIES

Similar results were obtained in  . As a result if the certain FTCSMP is not stabilizable there is no point in investigating the stabilizability for the uncertain FTCSMP. Clearly the reverse is not true. This result is formally stated as follows Corollary . The exponential stability in the mean square of the certain FTCSMP

is necessary for the exponential stability in the mean square of the uncertain FTCSMP.

.. A numerical example Consider a system with a potential fault in its plant dynamics. The system and other design parameters are given as A1

2 3 66

7 7 64 7 5

A2

 ᝇ

3
77 7 5

2 6  6 6 4

B

 ᝇ



Plant parameter failure rates are assumed to be

2 6 ᝇ
 
 jk  664



ᝇ



2 3 6
77 6 6 5 4 7



3 7 7 7 5

and conditional transition rates for the FDI process are

q 1  ij

2 66 ᝇ
 
 64




ᝇ


3 7 7 7 5

q 2  ij

2 6 ᝇ

 

 6 6 4



 ᝇ



Assume that the system matrix uncertainty is represented by

 A1

2 6 1 sin 41 t 16 2 36 4 1 1 4

sin 4 t

3 cos t 77 7 5 1 1

1 4

1 4

2

cos 4 t

A2

2

Initial constants  and weighting matrices are i



o i



 Q

ki

2 3 66
 7 7 64 7 5




2 6 1 cos 14 t 16 3 46 4 1 1

R

ki

cos 4 t

2 3 6
 77 6 6 7 4 5




8k

3 7 7 7 5 3 sin t 77 7 5 1 1 1 2

1 4

3

sin 4 t

i





CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

The objective is to investigate the exponential stabilizability for the uncertain FTCSMP in the mean square sense. The proposed algorithm is used. As per Corollary . the success of the stability test algorithm to give positive-denite solutions is necessary and sucient for the exponential stabilization in the mean square. First of all the given parameter uncertainties can be represented as 2 6 F1 t
 F2 t
 6 6 4

2 6 H1  21 6 64



1 4



1 2

3 2 77 6 75
E1  32 664

1 4







1 sin 4 

t




1

cos 4

2 3 7 6 7
H2  12 6 6 7 4 5

t


1 2

2 3

1 3



3 7 7 7 5

3 2 7 6 7
E2  12 6 6 7 5 4





1 2



3 7 7 7 5

CASE
System with parameter uncertainties As per Corollary . the existence of symmetric positive-denite solutions implies that the uncertain FTCSMP is exponentially stabilizable. In this case the algorithm converges to steady-state positive solutions at 1  2  . The solutions Pki
are summarized in Table .
and are shown in Figure . for FTCSMP with norm bounded uncertainties while Pki for certain FTCSMP are summarized in Table . and shown in Figure . . Note that the solutions of the Riccati-like matrix equations for the uncertain FTCSMP over-bound those for the certain FTCSMP. This result can be justied as follows. Recall that the additional term associated with the parameter uncertainties is i PiDi Pi  1 iGi this term is symmetric positive-denite for symmetric positivedenite Pki . Assign the following identities Quki  Qki  i Pki Di Pki  i Gi
uncertain F T CSMP Qcki  Qki


certain F T CSMP




.. STABILIZATION OF FTCSMP WITH PARAMETER UNCERTAINTIES

Table .
 Solutions of Riccati-like matrix equations for uncertain FTCSMP. 2 66 2 4172 0 8307 P11 = 6 4

3 7 7 7 5

2 6 6 2 0284 0 6855 P12 = 6 4

3 7 7 7 5

2 66 1 7890 0 8683 P21 = 6 4

3 7 7 7 5

2 6 6 1 5154 0 7178 P22 = 6 4

3 7 7 7 5

0 8307 0 6784

0 8683 0 8309

0 6855 0 5677

0 7178 0 6913

Table . Solutions of Riccati-like matrix equations for certain FTCSMP. 2 66 2 0776 0 6073 P11 = 6 4

3 7 7 7 5

2 6 6 1 7936 0 5283 P12 = 6 4

3 7 7 7 5

2 66 1 4703 0 6158 P21 = 6 4

3 7 7 7 5

2 6 6 1 2930 0 5449 P22 = 6 4

3 7 7 7 5

0 6073 0 3662

0 6158 0 4711

0 5283 0 3442

0 5449 0 4427

Solutions of Riccati-like eqns.

P11


2.5 2 1.5 1 0.5

0

10

20 30 Iterations

40

P21


2 1.8 1.6 1.4 1.2 1 0.8 0.6

0

10

20 30 Iterations

40

P12


2.5 2 1.5 1 0.5 0

50

0

Solutions of Riccati-like eqns.

Solutions of Riccati-like eqns.

Solutions of Riccati-like eqns.

CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

10

20 30 Iterations

40

50

40

50

P22
1.5

1

0.5

50

0

10

20 30 Iterations

Solutions of Riccati-like eqns.

P11


2.5 2 1.5 1 0.5 0

0

10

20 30 Iterations

40

50

P12


2 1.5 1 0.5 0

0

10

20 30 Iterations

P21
Solutions of Riccati-like eqns.

Solutions of Riccati-like eqns.

Figure . Positive-denite solutions Pki  for uncertain FTCSMP.

Solutions of Riccati-like eqns.





1.5

1

0.5 0

10

20 30 Iterations

40

50

40

50

40

50

P22
1.4 1.2 1 0.8 0.6 0.4

0

10

20

30

Iterations

Figure .  Positive-denite solutions Pki  for certain FTCSMP.




.. STABILIZATION OF FTCSMP WITH PARAMETER UNCERTAINTIES

since Quki Qcki following similar arguments used in Remark  in Section .. we can show that Pkiu Pkic  that is the positive-de nite solutions for the uncertain FTCSMP over-bound those of the certain FTCSMP. Consequently the above result implies that more control e ort is needed to stabilize the FTCSMP with parameter uncertainties. The state feedback control law gains are shown in Table .. Table . State feedback control laws for uncertain and certain FTCSMP. F T CSMP Norm Bounded Uncertainty

Certain

T K11

T K12

T K21

T K22

2 6 2
4172 6 6 4

3 7 7 7 5

2 6 2
7045 6 6 4

3 7 7 7 5

2 6 1
7890 6 6 4

3 7 7 7 5

2 6 2
0206 6 6 4

3 7 7 7 5

2 6 2
0776 6 6 4

3 7 7 7 5

2 6 2
3914 6 6 4

3 7 7 7 5

2 6 1
4703 6 6 4

3 7 7 7 5

2 6 1
7239 6 6 4

3 7 7 7 5

0
8307

0
6073

0
9140

0
7044

0
8683

0
6158

0
9570

0
7266

CASE
Stability of uncertain FTCSMP versus constants i The selection of the constants i is a key factor for the success of the stabilization algorithm. A necessary condition for the success of the algorithm is the existence of small positive constants i. With this condition the stabilization algorithm turns out to be necessary and su cient for the exponential stabilization of the uncertain




CHAPTER . FTCSMP WITH ACTUATOR SATURATION AND PARAMETER UNCERTAINTIES

4 x 10

P11 

Solutions of Riccati-like eqns.

2 1.5 1 0.5

0 0.7

x 10 14

0.8 6

0.9

1

P21 

12 10 8 6 4 2 0

0.7

0.8

0.9

1

Solutions of Riccati-like eqns.

Solutions of Riccati-like eqns.

Solutions of Riccati-like eqns.

FTCSMP. In the following the behavior of the FTCSMP with norm-bounded uncertainties is examined for a range of values of 1 2  as illustrated in Figure . . As can be seen solutions are bounded for small and become unbounded for large . In this example the uncertain FTCSMP is stabilizable for

. 10

P12 

4 x 10

8 6 4 2 0 0.7

0.8

0.9

1

P22 

3000 2500 2000 1500 1000 500 0

0.7

0.8

0.9

1

Figure .  Positive-denite solutions P  versus . ki

i

. Chapter Summary In this chapter the stabilization of FTCSMP with practical constraints has been considered. The case where the FTCSMP is controlled by actuators with potential saturation has been addressed in Section . . Actuators are assumed to have non-linear characteristics they operate linearly within certain limits and saturate to xed levels if the limits are exceeded. This chapter derives su cient conditions for the exponential stability in the mean square which involve the solution of Riccati-like matrix equations. An algorithm to investigate the stability of the FTCSMP has been constructed. Sta-

.. CHAPTER SUMMARY




bility of other dynamical systems such as JLS has been established as an interesting special case of this work. Section . considered the eect of parameter uncertainties on the behavior of FTCSMP. A comprehensive study has been completed to characterize the exponential stabilization of the FTCSMP with parameter uncertainties. Parameter uncertainties are assumed to be time-varying unknown-but-bounded. A norm bounded uncertainty form was assumed an upper bound has been used and a stabilization algorithm has been constructed. The algorithm provides the necessary and sucient condition for exponential stabilization in the mean square of the uncertain FTCSMP. It has been concluded that the stability of the certain FTCSMP is necessary for the stability of the uncertain FTCSMP. The conditions involve the solution of Riccati-like matrix equations. JLS and FTCS without parameter uncertainties have been shown to be special cases from this result.

Chapter SYNTHESIS OF FAULT TOLERANT CONTROL LAWS .

Introduction

The FTCSMP can be modelled by stochastic di erential equations and can be viewed as a general hybrid system. The research in this area can be broadly classied into two categories The rst concerns with deriving conditions for the existence of controllers. The second deals with properties of FTCSMP such as stability controllability and observability. The major portion of the book has been focused on studying system properties in particular the stochastic stability. The exponential stability in the mean square was studied in  . Stochastic stability of FTCSMP in noisy environment was considered in   for multiple failure process in   and with parameter uncertainties in  . In this chapter a state feedback controller for FTCSMP will be designed in both noise-free and noisy environments. The matrix maximum principle  is used to minimize an equivalent deterministic cost function. Firstly a fault tolerant control law is designed in noise-free environment. Three scenarios have been considered. The rst scenario assumes that both the failure and the FDI processes are accessible for the controller. In the second scenario the controller is recongured based on the decisions of the FDI algorithm only and does not need accessible failure process. The case where the FDI process itself is not M.M. Mahmoud, J. Jiang, Y. Zhang: Active Fault Tolerant Control Systems, LNCIS 287, pp. 148−183, 2003.  Springer-Verlag Berlin Heidelberg 2003

.. INTRODUCTION




able to provide any decision on time due to excessive computational delays is the third scenario. In these scenarios control laws are developed to reduce the risk of losing system stability. A computational algorithm is proposed to calculate the corresponding control law gains. Secondly the chapter also investigates the synthesis of control laws in noisy environment. To be consistent with the model of FTCSMP with noise dened in Chapter  three types of noise are considered state-dependent control-dependent and purely additive Gaussian noise. In particular conditions for the existence of a control law in the nite time horizon are derived. The limiting behaviors of the cost function the Riccati-like and the covariance-like di erential equations have been studied. The conditions that guarantee the niteness of the cost function and the existence of steady-state solutions for both Riccati-like and covariance-like di erential equations are stated and veried. Similar to the noise-free case a computational algorithm is proposed. It is shown that under certain conditions the algorithm converges to constant control gains. The theoretical results are illustrated by numerical examples. It has been emphasized that the optimization approach has often been used to achieve a systematic synthesis for fault tolerant control law  . As a matter of fact optimality is not the essential concern in AFTCS. This fact is consistent with the major characteristics of AFTCS dened in Chapter  for which AFTCS should accept degraded performance in the presence of system faults the main objective of AFTCS is to maintain system stability with graceful performance degradation. Furthermore it is pointed out that the optimality collapses in the event of imperfect FDI performance   similar results have been obtained in Chapter . In this chapter and without loss of generality the optimization problem will provide optimal controllers only if system stochastic parameters are known a priori and under the perfect FDI performance. Otherwise the optimization problem is merely a method




CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

to synthesize a fault tolerant control law.

. Design Approaches to Fault Tolerant Control Laws In this section a fault tolerant control law will be synthesized for FTCSMP. Three scenarios will be considered. The rst reects the nature of the FTCSMP which uses two separate processes to represent the random faults and the decisions of the FDI process. Unfortunately the resulting controller is not realizable in practice. In the second scenario the controller is recongured based on the results of the FDI process and does not need to have access to the failure process. Therefore this scenario is closer to practical FTCSMP. In some situations the FDI process is not able to make any decision. This situation may occur due to a fault in the data acquisition hardware or the excessive amount of time required for the statistical tests to converge to a decision. This is the third scenario where a control law will be developed to reduce the risk of losing stability. To design a control law for the second and the third scenarios the probability distributions for the failure and the FDI processes are used. Depending on the complexity of the system and the assumed distribution functions the probabilities are determined either analytically using stochastic modeling or numerically using Monte-Carlo simulations. A control law is designed in both the nite and the innite time horizon framework. Necessary conditions which guarantee the existence of such controllers are derived and veried.

.. The problem statement Consider the linear FTCSMP    xt

_෈ ෉ =

Ax t

෈ ෉ +

u x t
t
t ෈

෈ ෉ ඵ෈ ෉

B t u x t
t
t

෉ =

෈ ෈ ෉෉

෈

෈ ෉ ඵ෈ ෉

ᝇK

t xt

෈ඵ෈ ෉෉

෈ ෉

෉

. 

.. DESIGN APPROACHES TO FAULT TOLERANT CONTROL LAWS




To simplify the notation an indicator function ki
SථR will be used to indicate that the failure process
t
is currently in state k and the FDI process 
t is in state i. The objective here is to synthesize a fault tolerant control law for the FTCSMP .  that minimizes the following quadratic performance index



Ztf T J  E
fx
tQ

t

tx
t  uT
tR

t

tu
tg dt
 to

.

where Q

t

t and R

t

t are semi-positive and positive-de nite matrices respectively. Denote Qki and Rki when  k S
  i R. If the cost function is de ned as V
x
t

t

t
t  min u J

. 

Then the optimal control problem becomes

  tf R T T min E fx
tQ

t

tx
t  u
tR

t

tu
tg dt u

to

Subject to

.

x
t  Ax
t  B

tK

tx
t

De ne covariance-like matrices as Xki
t  Efx
tx
tT j
t  k

t  ig  Efx
tx
tT kig

.


Then considering the dynamical evolution of Xki
t
the optimization problem . can be transformed to an equivalent deterministic problem. In this case the covariancelike matrices will be regarded as the system state and the feedback gain matrices Ki
t
as the control. The equivalent deterministic cost function can be stated in Lemma . .




CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

Lemma . Let the current state of the FDI process be i R and the actuator failure process is in state k S . If
to


to
are independent of x to
then for the set of matrices Qki  KiT Rki Ki
the cost function is J d1 

Zt trfX f

to



ki t
Qki  KiT Rki Ki
gdt

Proof The quadratic cost function to be minimized is J

ZE t fx t T Q
t f




t x t
 uT t
R
t
 t

u t
gki dt





to

Under the state feedback u t

K J

ZE t x t T fQ
t f






t






to

Zt E x t T fQ

KT

t x t
 the cost function becomes





t R
t
 t

K







t gx t
ki dt





Z

tf T T T 
ki  Ki Rki Ki gx t
ki dt  Eftrfx t
fQki  Ki Rki Ki gx t
gki gdt to to tf tf Eftrf Qki  KiT Rki Ki
x t
x t
T ki ggdt  trf Qki  KiT Rki Ki
Xki t
gdt .  to to f



.

Z

Z

Note that tr AB
 tr BA
 therefore J

Zt trfX f

T ki t
Qki  Ki Rki Ki
gdt

to

.

The proof is complete. The time-derivatives of Xki t
can be calculated as X ki



A
Bk Ki
Efx t
x t
T ki g  Efx t
x t
T ki g A
Bk Ki
T

Xjk hEfx t x t T jig
Efx t x t T kigi j S X= qjik hEfx t x t T kj g
Efx t x t T kigi



2



















j k 6



j 2R =

j i 6







.




.. DESIGN APPROACHES TO FAULT TOLERANT CONTROL LAWS

re-arranging terms Xki
A ᝇ Bk Ki Xki  Xki A ᝇ Bk Ki T 

X

j 2S




jk Xji ᝇ Xki 

=

X

j 2R

qjik Xkj ᝇ Xki




=

j k

j i

6

6

.  De ne Aki
A ᝇ Bk Ki ᝇ  I

 X jk ᝇ  I qjik j 2S j 2R X

=

. 

=

j k

j i

6

6

we have

Xki
Aki Xki  Xki ATki 

X

j 2S

jk Xji 

=

X

j 2R

qjik Xkj

. 

=

j k

j i

6

6

Xki can be solved forward in time with the initial conditions Xki to 
Efxto xto T kig. Therefore the equivalent deterministic optimization problem to be solved becomes t Rf min trfXki tQki  KiT Rki Ki gdt Ki to subject to

Xki
AkiXki  Xki ATki 

.  P

j 2S =

j k 6

jk Xji 

qk Xkj j 2R ji P

=

j i 6

.. A control law on nite time horizon In the following necessary conditions for the existence of a fault tolerant control law for the FTCSMP .  are developed. A control law is derived for each of the three scenarios using the equivalent deterministic cost function. Scenario
Both failure and FDI processes are accessible

The model of FTCSMP involves two separate processes
t to represent random faults in the system and t to represent decisions of the FDI process. Therefore a control law for the FTCSMP can naturally lead to a controller as a function of both




CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

t and ඵ෈t෉. In the following a controller will be synthesized based on the technique of matrix maximum principle . ෈ ෉

Theorem . The control law gain matrices can be obtained from Rki Kki Xki෈t෉ ᝇ BkT Pki ෈t෉Xki ෈t෉ = 0 where Pki ෈t෉ are the solutions of the Riccati-like matrix equations


P_ki ෈t෉ + A^Tki Pki ෈t෉ + Pki ෈t෉A^ki +

P
kj Pji t P qk Pkj t

j 2S j

෈ ෉+

=k

ij

j 2R j

6

=i

෈ ෉+

KkiT RkiKki + Qki = 0

6

Pki ෈tf ෉ = 0 A^ki is dened as A^ki = A
Bk Ki
21 I

X


j 2S j

=k

kj
2 I 1

X qk j

6

ij

j 2R =i 6

and Xki ෈t෉ are the solutions of the following covariance-like dierential equations

P

X_ ki ෈t෉ = A~ki Xki ෈t෉ + Xki ෈t෉A~Tki +
jk Xji ෈t෉ + j 2S j

P qk Xkj t

j 2R

=k

j

6

=i

ji

෈ ෉

6

Xki ෈to෉ = Efx෈to ෉x෈to ෉T j ථki g A~ki is dened in .

Proof Recall that the equivalent deterministic optimization problem has been stated in  .  . Applying the techniques of matrix maximum principle  the associated Hamiltonian is H = trfXki ෈t෉෈Qki + KkiT Rki Kki ෉g +

trf෈A~ki Xki ෈t෉ + Xki ෈t෉A~Tki +

=k

ki

Aki Xkiද ෈t෉ + Xkiද ෈t෉A~Tki +

j 2R ji j

6

The state optimality condition is = ~

+

j 2S j

H X_ kiද = P

 . 

P
jk Xji P qk Xkj P T g X


j 2S j

=k 6

෉

=i

ki

6

jk Xji + ද

X qk X

j 2R j

=i 6

ද

ji kj

 .






.. DESIGN APPROACHES TO FAULT TOLERANT CONTROL LAWS

The co-state optimality condition is Pkiද _

=

=

ᝇH Xki ᝇfKkiදT Rki Kkiද Qki ATki Pkiද t Pkiද t Aki + ^

+

෈ ෉ ^

෈ ෉ +

X


X

ද k ද kj Pji + qij Pkj g .  j 2S j 2R

+

j

=k

j

6

=i 6

and 0 =

H Kki

RkiKkiද Xkiද t

=

෈ ෉ +

BkT Pkiද t Xkiද t ෈ ෉

෈ ෉

. 

Equations .
 .  and .  provide the conditions stated in Theorem . . Note that for the non-singular Xki the optimal control gain is uniquely determined by the solutions of the Riccati-like di erential equations Pki t . In other words the Riccati-like and the covariance-like di erential equations are not coupled through Kki . Therefore this case is not a boundary value problem. Similar result was obtained in Section . using the principle of dynamic programming. ෈ ෉

Scenario
Only the decision from the FDI process is accessible This is the second scenario where only the decision from the FDI is available for control recon guration. It has been illustrated in the previous chapter that errors in fault detection and identi cation may lead to a degraded system performance or even complete loss of the system stability 
. To reduce the risk of losing system stability the fault probability distribution for the actuator has to be considered. Let the actuator failure process  t have a Markovian transition probability de ned in . and a probability distribution as ෈ ෉

vk t

෈ ෉ =

Pr f t

෈ ෉ =

kg

. 

where vk t
෈ ෉

0

 k  S

Ps vk t

k=1

෈ ෉ = 1

. 




CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

The initial condition is assumed to be a random variable with .

Efx to
g   Efx to
xT to
g  X to
From .  the covariance-like matrices will be Xi t


Xs v

k=1

. 

k t
Xki t


Theorem . A necessary condition for the existence of a control law ui t
 Kix t
i
R is that the gains Ki satisfy the following Riccati-like matrix equations Pki t
 ATki Pki t
 Pki t
Aki 

P
kj Pji t P qk Pkj t


j 2S =

j 2R ij




=

j k

KiT Rki Ki  Qki  

j i

6

6

 Pki tf
  Aki is dened in Theorem . with

X ki t
 Aki Xki t
 Xki t
ATki  P
jk Xji t
 P qjik Xkj t


Ps vk to Xo

 Xki to


k=1




j 2S =

j k 6

j 2R =

j i 6

Aki is dened in .

and the control gains are obtained from

Xs hR

k=1

T ki Ki vk t
Xki t
Bk vk t
Pki t
Xki t


i



Proof The proof can be obtained using similar arguments as in Theorem . . Therefore only the cost function to be minimized will be stated.

Lemma . Let the actuator failure process  t
have a probability distribution v  v1 v2  vs . Assume that  to
is independent from x to
 If the current state of the FDI process is i
R then for some matrices Qki  KiT Rki Ki
the cost function is



J d2 

Zt Xs trf v f

to

k=1

k t
Xki t
Qki  KiT Rki Ki
gdt




.. DESIGN APPROACHES TO FAULT TOLERANT CONTROL LAWS

The Riccati-like matrix equations are coupled with the covariance-like dierential equations through the gain matrices i in a non-linear way. This case requires the solution of a boundary value problem to obtain the optimal control gain matrices. Fortunately these calculations can be done o-line and only controller selection is completed on-line. K

Scenario
Neither the failure nor the FDI processes are accessible There are cases where the decision of the FDI process is not available in time. One possible situation arises due to large amount of time required by the FDI algorithm to arrive at a decision. This issue was emphasized by several works in the area of FTCS  
   . Another possible case is malfunctions to the data acquisition hardware from which the FDI algorithm acquires data. To synthesize a control law the actuator failure process
is assumed to have the probability distribution in

.  whereas the conditional probability distribution for the FDI process is
t

k i

w

t




where w

k i

t


Pr

Pr



i=1

f k i

w

t


t




i

j


t









k

k

g

S

i

.



R

Then the covariance-like matrix becomes X t




Xr Xs

i=1 k=1

w

k i

t
v

k

t
X

ki

. 

t


The cost function for this scenario is dened in Lemma . .

Lemma . Let the actuator failure process 1

v w

t

o


2

v

t

o


s to




v

k k i t o
  w1 t o


x t

o


k

2

w

t

o


have a probability distribution o
 and the FDI process has a conditional probability distribution k o
and  o
are independent from r o
. Assume that



w

Then for some matrices d3

J



t

tr

to

v t


t

Q

T ki  Ki Rki Ki


Z Xr Xs f tf


t


i=1 k=1

w

k i

t
v

k

t
X

ki

t

the cost function to be minimized is t
Q

T ki  Ki Rki Ki
gdt

.




CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

Theorem . states the necessary conditions for which a control law exists.

Theorem . A necessary condition for the existence of a control law u෈t෉ = ᝇKx෈t෉ is that the constant gain K satis
es the following Riccati-like matrix equations

P_ki෈t෉ + A^Tki Pki ෈t෉ + Pki෈t෉A^ki +

P

kj Pji ෈t෉ +

j 2S j

=k

P qk Pkj t

j 2R j

6

=i

ij

෈ ෉+

K T Rki K + Qki = 0

6

8 Pki ෈tf ෉ = 0

with

X_ ki ෈t෉ = A~ki Xki ෈t෉ + Xki ෈t෉A~Tki + P jk Xji ෈t෉ + P qjik Xkj ෈t෉ j 2S

8 Xki ෈to ෉ =

=k

Pr Ps wk to vk to Xo j

෈ i=1 k=1 i

෉

෈

6

j 2R j

=i 6

෉

and the control gain is obtained from

Xr Xs hR i=1 k=1

i

ki Kwik ෈t෉vk ෈t෉Xki ෈t෉ ᝇBkT wik ෈t෉vk ෈t෉Pki ෈t෉Xki ෈t෉ = 0

Similar to the previous theorems the three optimality conditions provide the covariance-like Riccati-like and the gain equations respectively. Theorem . is similar to Theorem . in the sense that to obtain an optimal control law a boundary value problem must be solved. The couplings between the Riccati-like and the covariance-like dierential equations are non-linear. The nonlinearity will result in increased computation time. Fortunately gain calculations can be carried out o-line.

.. Control laws on innite time horizons As tf approaches in nity the behavior of the optimization problem needs to be carefully examined. The cost function itself may become in nite therefore the solution may not exist. The limiting behavior for the solutions of the dierential equations is not well de ned. In this section these di culties will be examined with and the




.. DESIGN APPROACHES TO FAULT TOLERANT CONTROL LAWS

conditions for which an optimal control law exists for the innite time horizon are derived. Since the three scenarios considered in the last section have a unied form of solution the development of this section is not restricted to any particular scenario.

I. The cost function for in
nite time horizon

For an optimization problem to be solvable as f
the cost function must be nite for some control and associated state trajectories. Under certain conditions this niteness can be guaranteed through the following lemma. t

Lemma . If the FTCSMP
. is exponentially stable in the mean square a non-empty set of control and trajectory pairs for each

u t


x t


exists for which
. is nite

o
.

x t

Proof Let the FTCSMP .  be exponentially stable in the mean square then for some

a

b






  

Efjj jj2 g  jj ojj2 f  o g x t


a x

exp

b t

t

.





however

Efjj jj2 g Ef T x t




x

t
x t


 t


 t

U

g

. 

Let Q  t


 t



U

T

 t


. 

 t



by Schwarz inequality

Ef T x

t
Q

ki x

t


g Ef T 

x

T ki Uki x

t
U

t


g Efjj 

x t
U

jj g  Efjj

T 2

x t


k2k T jj2g U

. 

then

Ef T x

t
Q

ki x

t


g  jj ojj2jj jj2 f  o g a x

U

exp

b t

t




. 




CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

Similarly let R t

 t

 W T t

 t

W t

 t



.

we have

EfuT t
Rki u t
g  EfxT KkiT t
Rki Kki x t
g ajjxo jj2 jjKki jj2 jjW jj2 expf
b t
to
g .  Therefore the cost function to be minimized is t Z lim E fxT 
Q f

 t !1 f

t o =o tf

Z lim E

 t !1 f lim

t o =o tf

Z

tf !1 t o =o



 

x 
 uT 
R 

 

u 
g d

t Z 

 

x 
d  lim E

xT 
Q

tf !1

f

to =o

uT 
R 

 

u 
d

ajjxo jj2  jjF jj2  jjKki jj2 jjW jj2  expf
b 1
o
g  

. 

The proof is completed.

II. Steady-state solutions for Pki It was proved that if the FTCSMP .  is exponentially stable in the mean square then there exist steady-state solutions for the Riccati-like matrix equations  .

Lemma . If the system
. is exponentially stable in the mean square then the Riccati-like dierential equations have unique positive-denite steady-state solutions

Pki
obtained from the following algebraic matrix equations ATki Pki  Pki Aki 

Xkj Pji  X qijk Pkj  KiT RkiKi  Qki 

j 2S =

j k 6

j 2R

.

=

j i 6

III. Steady-state solutions for Xki The limiting behavior for the covariance-like matrices is the key factor for the cost function to be nite. In the following the existence of steady-state solutions for




.. DESIGN APPROACHES TO FAULT TOLERANT CONTROL LAWS

the covariance-like dierential equations will be studied. Theorem . provides the condition under which these solutions exist.

Theorem . If the system
. is deterministically stable then the covariance-like dierential equations have steady-state solutions Xki obtained from the following algebraic matrix equations

A
ki Xki  Xki A
Tki 

X


j 2S j

jk Xji 

=k

X qk X

j 2R j

6

ji kj  Xki to   

.

=i 6

Proof The FTCSMP . under the state feedback is x t  A ᝇ Bk Ki xt  Aki xt

.

Let the fundamental transition matrix associated with Aki be  ki t

to   expAki t ᝇ to

.


Then the state trajectory due to initial conditions xto   is 6

xt   kit to xto 

.

and the covariance-like matrices are Xki t   kit to Xki to  Tki t to 

.

Substituting it into .  we have X ki t  A
ki  kit to Xki to  Tki t to   ki t

P

to Xki to   Tkit to A
Tki
jk  ji t to Xji to   Tji t to  j 2S j



=k 6

P qk  kj t toXkj to  T t to

j 2R ji j

=i 6

kj

.




CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

If the feedback gain matrix stable we have

i

K

is chosen such that the system is deterministically

lim !1
ki 

t

o   8 k 2 S

t t

i

.

2R

Taking the limit in .   we have t

 lim !1 ki     X

.

t

That is steady-state solutions for the covariance-like di erential equations exist as ! 1. t

IV. Steady-state control gains The existence of steady-state solutions for the Riccati-like and covariance-like di erential equations leads to constant gain matrices i . The control gains in the innite time horizon are obtained by Theorem .. K

Theorem . A feedback control gain that minimizes
. is obtained from the solution of one of the following equations


T
ki ki  ki ᝇ k ki  ki  


 Ps Ps T     ki i k  ki ᝇ k k ki ki
k=1 k=1

i
r s h
k  ᝇ T k    PP R

K

X

R

K v X

i=1 k=1

B

P

X

B

ki K wi vk Xki

R

v P

X

k wi vk Pki Xki

B

Scenario Scenario


.

Scenario 

where ki and ki are the solutions of the algebraic equations
. and
. respectively. P

X

V. A computational algorithm To synthesize a control law for the FTCSMP .  Riccati-like and covariance-like di erential equations coupled by gain equations must be solved. The complexity of the solution for the di erent scenarios considered in this section is di erent. The




.. DESIGN APPROACHES TO FAULT TOLERANT CONTROL LAWS

following unied computational algorithm can be used to nd an optimal control law for the FTCSMP.

Algorithm  Choose an initial guess for the control gain matrices K The chosen gain matrices should guarantee both the deterministic and the stochastic stability for the system. Theorem
. in Chapter
can be used to investigate stochastic stability. o i

 For the selected initial gain solve the boundary value problem. Denote the obtained solutions by P and X . o ki

o ki

 Substitute the calculated P and X in the control gain equation to compute K 1. o ki

o ki

i

 Use K 1 in Step  to calculate P 1 and X 1 . i

ki

 Iterate to generate a list for K  P itr i

ki

itr ki

and X . itr ki


 Stop when a specied tolerance
 is achieved. That is kK

itr i

K ᝇ1k

. itr i

.. A numerical example Consider the following system with one possible actuator fault. The system and the related design parameters are given as A
  B1
  B2
 

The actuator failure rates are assumed to be 12   21  




CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

The conditional transition rates of the FDI process are assumed to be 1

q ij

3 7 7
7 5

2 66 ᝇ

 64


 ᝇ


q

2 6 ᝇ
 
 6 6 4

2

ij


 ᝇ


3 7 7 7 5

The weighting matrices are chosen to be f



fQ11
Q12
Q21
Q22 g


  
g









Rki

i
j


f
g

The objective is to synthesize a control law for the given FTCSMP. The designed controller gain must be able to stabilize the unstable open-loop system and to maintain the closed-loop stochastic stability when the system is either fault-free or with actuator fault. All three scenarios presented in this section are considered. The proposed algorithm is used to calculate the corresponding steady-state control gains. Scenario
Both failure and FDI processes are accessible

Steady-state control gains are shown in Figure . . Whereas Riccati-like and covariancelike solutions for  iterations are shown in Figures . and . respectively. 1.5

K11

1.7

K12

1.48

Fault Tolernat control law

Fault Tolernat control law

1.75

1.65 1.6 1.55

1.46 1.44 1.42 1.4 1.38 1.36

1.5

0

5

10

15

20

25

1.34

30

0

5

10

15 Iterations

20

25

30

10

15 Iterations

20

25

30

Iterations 4.45

K21

4.5

Fault Tolernat control law

Fault Tolernat control law

4.55

4.45 4.4 4.35 4.3

0

5

10

15 20 Iterations

25

30

K22

4.4 4.35 4.3 4.25

4.2

Figure . The control gains

0

Kki

5

 in scenario .




.. DESIGN APPROACHES TO FAULT TOLERANT CONTROL LAWS

Riccati-like equations solutions

1.7 1.65 1.6 1.55 1.5

0

5

22.6

Riccati-like equations solutions

1.5

P11

10

15 20 Iterations

25

22.2 22.0 21.8 21.6 21.4

0

5

1.46 1.44 1.42 1.4 1.38 1.36 0

22.2

P21

22.4

P21

1.48

1.34

30

Riccati-like equations solutions

Riccati-like equations solutions

1.75

10

15 20 Iterations

25

10

15 20 Iterations

25

30

10

15 20 Iterations

25

30

P22

22.1 22.0 21.9 21.8 21.7 21.6 21.5 21.4 21.3 21.2

30

5

0

5

Figure . The solutions of Riccati-like equations P  in scenario . ki

Covariance-like equations solutions

0.35 0.3 0.25 0.2

0

5

10

15 20 Iterations

25

30

X21

0.17

Covariance-like equations solutions

0.13

X11

0.4

0.16

0.14 0.13 0.12 0.11 0

5

0.11 0.1 0.09 0.08 0.07 0.06 0.05

0

0.17

0.15

0.1

X12

0.12

Covariance-like equations solutions

Covariance-like equations solutions

0.45

10

15 20 Iterations

25

30

0.16

5

10

15 20 Iterations

25

30

X22

0.15 0.14 0.13 0.12 0.11 0.1 0

5

10

15 20 Iterations

25

30

Figure .  The solutions of covariance-like equations X  in scenario . ki

Scenario
Only FDI process is accessible

For this case there are two steady-state gain matrices one for the normal system and the other for the case when the actuator fault has occurred as shown in Figure . .





CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

K1

K2

4.6 4.4

Fault Tolernat Control Law

Fault Tolernat Control Law

4.0

3.5

3.0

2.5

4.2 4.0 3.8 3.6 3.4 3.2

0

5

10

15 20 Iterations

25

0

30

5

10

15 20 Iterations

25

30

Figure . The control gains K  in scenario . i

The solutions of the Riccati-like equations are shown in Figures . . Whereas Figure .
shows the solutions of the covariance-like equations.

P11

2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6

P12

2.5

Riccati-like equation solutions

Riccati-like equation solutions

2.5

2.4 2.3 2.2 2.1 2.0 1.9

0

5

10

15

20

25

30

0

5

10

Iterations

P21

28 27 26 25 24 23 22 21

0

5

10

15 20 Iterations

25

30

25

15 20 Iterations

25

30

P22

23.5

Riccati-like equation solutions

Riccati-like equation solutions

29

15 20 Iterations

23.0 22.5 22 21.5 21.0

0

5

10

30

Figure .  The solutions of Riccati-like equations P  in scenario . ki

Scenario
Neither the failure nor the FDI processes are accessible

This scenario is the most severe one in which the FDI process is unable to provide information on the fault status of the system. In this case there is only one steadystate gain matrix to be calculated as shown in Figure . .

.. DESIGN APPROACHES TO FAULT TOLERANT CONTROL LAWS 0.18

X1

Covariance-like equation solutions

Covariance-like equation solutions

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0

5

10

15 20 Iterations

25




X2

0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02

30

0

5

10

15 20 Iterations

Figure .
 The solutions of covariance-like equations 3.476

30

X  in scenario . i

K

3.475

Fault Tolerant Control Law

25

3.474 3.473 3.472 3.471 3.47 3.469 3.468 3.467

0

5

10

15

20

25

30

Iterations

Figure . Steady-state control gain

P11

2.15

2.146 2.144 2.142 2.14 2.138 0

5

10

15 Iterations

20

25

22.68 22.66

22.62 22.6 22.58 22.56 0

5

10

2.076 2.074 2.072 2.07 2.068 0

5

15 Iterations

20

25

30

10

15 Iterations

20

25

30

15 Iterations

20

25

30

P22

22.26

22.64

22.54

2.08 2.078

2.066

30

P21

22.7

Riccati-like equation solutions

2.082

2.148

2.136

P12

2.084

Riccati-like equation solutions

2.152

Riccati-like equation solutions

Riccati-like equation solutions

2.154

K  scenario .

22.24 22.22 22.2 22.18 22.16 22.14 22.12

0

5

10

Figure . The solutions of Riccati-like equations

P

ki

 in scenario .




CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

Similarly Figures . and . show the steady-state solutions of the Riccati-like and the covariance-like equations respectively. 0.11

X

Covariance-like equations solution

0.1098 0.1096 0.1094 0.1092 0.109 0.1088 0.1086 0.1084 0.1082 0

5

10

15

20

25

30

Iterations

Figure . The solution of covariance-like equation  in scenario . X

The steady-state controller gains for these three scenarios are summarized in Table . . Table .  Steady-state controller gains in the three scenarios. Kki

ᝇ

Scenario

Kk

ᝇ

S cenario

K11

  


K12

    

K1

 
 


K21

   

K2

 

K22

 






K

ᝇ

K



S cenario

   






.. A FAULT TOLERANT CONTROL LAW IN NOISY ENVIRONMENT

. A Fault Tolerant Control Law in Noisy Environment In this section conditions for the existence of a control law in nite time horizon will be derived for FTCSMP in the presence of noise. The optimization problem is solved using the matrix maximum principle applied to an equivalent deterministic cost function. The limiting behaviors of the cost function Riccati-like and covariance-like dierential equations are then studied. Conditions under which the cost function is nite and the steady-state solutions exist for both Riccati-like and covariancelike dierential equations are derived. The calculation of the control law requires the solution of a set of non-linear coupled dierential equations in the form of a boundary value problem. Therefore an algorithm is proposed to design a controller iteratively.

.. The problem statement A FTCSMP in the presence of noise as dened in  . is rewritten here for convenience xt

_෈ ෉ =

Atxt ෈ ෉

෈ ෉ +

B t u x t
t
t ෈ ෈ ෉෉

D x t
t
t W1 t

+

෈

෈ ෉

u x t
t
t ෈

෈ ෉ ඵ෈ ෉

෈ ෉

෉ =

_

෉

ᝇK

෈

෈ ෉ +

෈ ෉ ඵ෈ ෉

෉

E u t
t
t W2 t ෈

෈ ෉

෈ ෉

෉

_

෈ ෉ +

.

F t W3 t ෈ ෈ ෉෉

_

෈ ෉

t xt

෈ඵ෈ ෉෉

෈ ෉

The FTCSMP . is assumed to satisfy both the growth and uniform Lipschitz conditions. The indicator function ki
SථR dened in Section ..  will be used to specify that the failure process t
is in the state k
and the FDI process t
is in the state i. ථ

෈ ෉

ඵ෈ ෉




CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

The stochastic optimal control problem is then dened as min E u

Rtf to

fxT

t
Q t

 t

x

t
 uT

 t

R t

 t

u  t

g dt

Subject to




.

x t
 A t
x t
B t

K  t

x t
D x t

t

t
W 1 t
 E u t

t

t
W 2 t
 F

t

W 3 t


where Q t

 t

and R t

 t

are semi-positive and positive-denite matrices

respectively. Denote Qki and Rki when  k
S
  i
R.

.. A control law on nite time horizon In practical FTCSMP only the FDI process is measurable scenario  proposed in Section .. is reconsidered here in noisy environment. In this section the stochastic optimization problem is also converted into an equivalent deterministic problem and then solved using the technique of matrix maximum principle .

Lemma . Let t
be in state k
S
and  t
be in state i
R. If to

 to
are independent from x to

then for the set of matrices Qki  KiT Rki Ki

the cost function is

Ztf

Jd  trfXki t
Qki  KiT Rki Ki
gdt to

where Xki t
are as dened in ..

Proof Proof is detailed in Section ...




.. A FAULT TOLERANT CONTROL LAW IN NOISY ENVIRONMENT

The time derivatives of Xki t
are X ki  A ᝇ Bk Ki
Xki  Xki A ᝇ Bk Ki
T  

X

j 2R

X

j 2S

jk Xji ᝇ Xki

=




j k


qjik Xkj ᝇ Xki  Fk FkT   Xki
  Xki
6

.

=

j i 6

where Dkl and Ekl are dened in .  and    Xki
 tr Dkl Xki Dkc T h

n

oi




 Xki
 KiT tr Ekl Xki Ekc T h

n



 oi

l
c  
  
n Ki


.

l
c  
  
m

Dene Aki  A
Bk Ki
 I

 X jk
I qjik j 2S j 2R X

=

j k 6

.


=

j i 6

Then X X X ki  AkiXki  Xki ATki  jk Xji  qjik Xkj  Fk FkT   Xki
  Xki
. j 2S j 2R

=

=

j k

j i

6

6

Xki are solved forward in time with the initial conditions Xki to
 Efx to
x to
T j kig

Therefore the equivalent deterministic optimization problem to be solved becomes t Rf min trfXki t
Qki  KiT Rki Ki
gdt Ki to subject to Aki Xki  Xki ATki 

P

j 2S =

j k 6

jk Xji 

P k q Xkj  Fk FkT   Xki
  Xki

X ki   j 2R ji

=

j i 6

. 




CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

Using the probability distribution of the actuator failure process dened in .  the covariance-like matrices becomes Xi t


s X k=1

. 

vk t
Xki t


A control law for the FTCSMP .  in the presence of Gaussian noise in a nite time horizon is given in Theorem ..

Theorem . A necessary condition for the existence of a control law ui t
 ᝇKix t
i R is that the gains Ki satisfy Pki t
 ATki Pki t
 Pki t
Aki 

P

j 2S j



=k

kj Pji t


P

j 2R j

6

=i

qijk Pkj t
  Pki
  Pki


6

Qki  KiT Rki Ki
 
Pki tf
 

Aki is de
ned in Theorem . with 



Pki
 tr




Dkl

Pki
 KiT tr

T




Ekl

Pki Dkc T



Pki Ekc





l
c 
  
n



. 

Ki

l
c 
  
m

and X ki t
 A ki Xki t
 Xki t
A Tki 

P

j 2S j

=k

jk Xji t


P k q Xkj t
 Fk FkT j 2R ji j

6

=i



Xki
  Xki



6

Xki to
 Efx to
x to
T j ki g 

Xki
and  Xki
are de
ned in .. The controller gains are obtained from s X



k=1

Rki   ki Pki
Kiද vk t
Xkiද t


s X k=1

BkT vk t
Pkiද t
Xkiද t
 


where ki Pki
 tr






Ekl

T

Pki Ekc





l
c 
  
m

. 

.. A FAULT TOLERANT CONTROL LAW IN NOISY ENVIRONMENT




From . the Hamiltonian for the optimization problem .  is s s X X H trf vk
tXki
t
Qki  KiT Rki Kig  trf vk
t
Aki Xki  Xki ATki k=1 k=1 X X T k .  jk Xji  qji Xkj  Fk Fk  
Xki   
Xki PkiT g j 2R j 2S

Proof

j

=k

j

6

=i 6

The proof can be completed by satisfying the optimality conditions of the matrix maximum principle technique  similar to those in Section ... Note that to solve the optimization problem and to nd a control law a boundary value problem has to be solved. This boundary value problem is more complex when compared with the case of noise-free environment in Section . . The increased non-linearity arises because of the control-dependent noise term  ki
Pki .

.. A control law on innite time horizon In this section the limiting behaviors of the cost function and the covariance-like dierential equations in the innite time horizon are studied. In Section . it was mentioned that the limiting behavior is not well treated for hybrid systems in general and for FTCSMP in particular. The results revealed that a boundary value problem has to be solved. Therefore without loss of generality to  tf for the equations to be solved forward in time while to
tf  for the equations to be solved backward in time. The quadratic cost function to be minimized is J

1R min trfXki
t
Qki  KiT Rki Ki gdt K o

. 

i

Since
Qki  KiT Rki Ki  are positive-denite matrices the behavior of the cost function for large t is determined by the limiting behavior of the covariance-like matrices Xki
t. Recall that the FTCSMP .  can be written as dx
t

fA
t
B


tK

tg x
tdt  D
x
t

t tdW1 E
u
t

t tdW2  F


t dW3
t

.




CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

Let ki
t to   exp
A ᝇ Bk Ki 
t ᝇ to be the fundamental matrix of . when

t  k 
t  i the state trajectory solution becomes

Zt

x
t 

Zt

ki
t to x
to   to

ki
t



ki
t

 D
x
  kdW1
 

Zt

 E
ui
  kdW2
  

to

ki
t

.

 Fk dW3
 

to

If the Wiener processes Wi
t i    are assumed to be white Gaussian noise i
t with i
tdt  dWi i    

.


Efi
 g  Efi
tiT
 g  Vi 
t
 

The covariance-like matrices are Xki  Efx
tx
tT j kig E

Rt 

to



Rt to




ki
t to x
to  

ki
t ki
t

 Fk dW3
 

Rt to




ki
t

 D
x
  kdW1
  

ki
t to x
to  

 E
ui
  kdW2
  

Rt to

Rt to

ki
t

Rt to

ki
t

 E
ui
  kdW2
 

 D
x
  kdW1
 

T   ki
t  Fk dW3
  

Expanding and rearranging the terms we have Xki 

T ki
t to Xki
to  ki
t to  



Z!1

t

o

ki
t

Z!1

t

o

ki
t

 Fk V3FkT

 Dk V1 Xki DkT  Ek V2Ki Xki KiT EkT

T
t ki T
t ki

 d

 d

.

The rst term in . is due to the initial conditions. For deterministically stabilizable system we have t

lim !1

T ki
t to Xki
to  ki
t to  

. 

.. A FAULT TOLERANT CONTROL LAW IN NOISY ENVIRONMENT




The last term in . is a contraction integral if the pair A Bk
is stabilizable by a constant feedback Ki and for suciently small noise intensity 
. That is tZ!1 0

ki t

Dk V1 Xki DkT Tki t

d


tZ!1 0

ki t

Ek V2 Ki Xki KiT EkT Tki t

d
! 

.  The integral due to pure additive noise in . is tZ!1

ki t

Fk V3 FkT Tki t

d


. 

0

For positive-denite Fk V3 FkT and stabilizable pair A Bk
the integral has a unique solution given by


A ᝇ Bk Ki
Xki  Xki A ᝇ Bk Ki
T  Fk V3FkT  

.

Therefore when time approaches innity the covariance-like matrices . should converge to steady-state solutions lim X t
 X ki t!1 ki

.

The non-vanishing steady-state solutions of the covariance-like matrices make the cost function . innite as the time approaches innity. To overcome this diculty a scalar dierential equation is used to force the purely additive noise to diminish as time increases. This does not imply that all environment noises have been dropped. State and control dependent noises still exist and aect the calculation of the gain matrices as per Theorem .. Dene scalar dierential equations such that


lim z t
Fk FkT   t!1 ki

.

where

ᝇzki  jP2Sjk zki j

=k 6



P k qji zki j 2R j =i 6

8zki to
 zkio

.




CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

The initial conditions for covariance-like matrices are set as follows Xki to
 Efx to
xT to
j kig  zkio Xo

The scalar dierential equations enforce the cost function to be nite and X ki will be decreasing as t
. The optimization problem becomes

R1 Ps min trf vk t
Xki t
Qki  KiT Rki Ki
gdt K 0 k=1 i

subject to Aki Xki  Xki ATki 

P

j 2S j

and

P

j 2S j

=k 6

jk zki 

=k

jk Xji 

6

P qk Xkj  zkiFk F T   Xki
  Xki
 X ki 

j 2R ji j

k

=i 6

P qk zki  zki 

j 2R ji j

=i 6

. The following Theorem generalizes the conditions of Theorem ..

Theorem . A necessary condition for the existence of a control law ui t
 Kix t
i  R is that the gains Ki satisfy the following di
erential matrix equations Pki t
 A Tki Pki t
 Pki t
A ki 

P

j 2S j

=k 6

kj Pji t


P qk Pkj t
j 2R ij j

=i 6

 Pki
  Pki
 Qki  KiT Rki Ki

Pki tf


and

P P X ki t
 Aki Xki t
 Xki t
ATki  jk Xji t
 qjik Xkj t
j 2S j 2R j

=k 6

j

=i 6

o Xo zki t
Fk FkT   Xki
  Xki

Xki to
 zki





.. A FAULT TOLERANT CONTROL LAW IN NOISY ENVIRONMENT

where

ᝇzki t


 

P

j 2S j

and

=k

jk zki
t



P qk zki t


j 2R j

6

P

=i

ji




j

wki tf


Xs R

The terms

Proof



=k

j

6

=i 6

  



ki  ki
Pki  Ki vk
tXki
t



Xki

Ps vk t Xki t


k=1

P qk Xkj

j 2R ji j

Xki



=i

zkio

P



Pki






Xs BT v

k=1





zki Fk FkT

 



Qki KiT RkiKi g trf 



Xki




k k
tPki
tXki
t  

Pki and ki are as de
ned as in Theorem ..




The Hamiltonian is

H trf 




 

6

The control gains are calculated from k=1




kj wki
t  qijk wki
t  tracefFkT Pki
tFk g   j 2S j 2R

wki t



zki to






k=1

Xki PkiT g

  




Ps vk t

Aki Xki Xki ATki






trf


P

j 2S j

6

=k







P

j 2S

jk zki

6



=k

jk Xji

P qk zki wT g j

j 2R ji j

6

ki



=i 6

.
 Since we have two equality constraints the Hamiltonian denes two Lagrangian multipliers Pki and wki  Therefore the optimality conditions must be satised for two co-state equations. The rst state optimality condition is H Pki Aki Xkiද t

Xkiද

 




 

Xkiද t ATki
 



P

j 2S j

=k 6

jk Xjiද 

P

qk X ද  zki Fk FkT j 2R ji kj j

=i

Xkiද




Xkiද

  




6

.




CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

while the second state optimality condition is zkiද _

=

H wki

=

X


ᝇ

jk zki

j 2S j

ද

ᝇ

=k

X qk z

j 2R j

6

.

ද

ji ki

=i 6

The rst co-state optimality condition is

ᝇH Xki ᝇfQki KiT RkiKi ATkiPkiද t Pkiද t Aki

Pkiද _

=

=

+ ^

+

෈ ෉+

෈ ෉ ^

+

P
kj P P qk P

+ත෈Pki ෉+ ᝇ෈Pki ෉g ji + j 2R ij kj

ද

j 2S j

=k

j

6

ද

=i 6

.
 The second co-state optimality condition is wkiද _

=

H zki

=

tracefFkT Pkiද t Fk g ෈ ෉

X


j 2S j

The third optimality condition implies 0 =

H Ki

=

Xs

k=1

ki ෈Pki ෉Ki vk ෈t෉Xki ෈t෉ +

~ ᝇ

ද

ද

kj wki ද

=k

j 2R j

6

Xs BT v

k=1

X qk w

ද

ij ki

.


=i 6

k k ෈t෉Pki ෈t෉Xki ෈t෉ = 0 ද

ද

.


The proof is completed. The design of a control law for innite time horizon requires the solution of two additional set of coupled scalar di erential equations. One set zki  is solved forward in time with the initial condition zkio and the second is wki which is solved backward in time with the boundary condition wki tf . Results of Theorem .
dene the necessary condition for the existence of a control law and guarantee the niteness of the cost function at the same time. It has been shown that steady-state solutions for the covariance-like di erential equations exist. Furthermore if the FTCSMP . is exponentially stable in the mean square there exist positive-denite steady-state solutions for the Riccati-like di erential equations  . The existence of steady-state solutions leads to constant steady-state gain matrices Ki as per Theorem .. _

_

෈

෉ = 0




.. A FAULT TOLERANT CONTROL LAW IN NOISY ENVIRONMENT

Theorem . A necessary condition for the existence of constant gain controllers ui ᝇKix is that Ki satisfy the following algebraic equations
A
Tki Pki  Pki A
ki 

X
kj Pji  Xqijk Pkj  Pki  Pki  Qki  KiT RkiKi

j 2S j

and Aki Xki  Xki ATki 

j 2R

=k

j

6

=i 6

X
jkXji  Xqjik Xkj  zkiFkFkT  Xki  Xki  Xki

j 2S j

j 2R

=k

j

6

and zkio 

=i

j 2S =k 6

j 2R j



=i 6

X
kj wki  X qijk wki  tracefFkT PkitFk g

j 2S j

=k 6

j 2R j



6

X
jkzki  Xqjik zki j

and





=i 6

The control gains are obtained by solving

Xs Rki  kiPki KivkXki Xs BkT vkPkiXki

k=1

k=1



Computational algorithm This unied computational algorithm can be used to synthesize an optimal control law. The convergence of the algorithm depends largely on the limiting behavior of the solutions of the Riccati-like and covariance-like as t
 Therefore the conditions derived and stated in previous sections to guarantee the niteness of the cost function and the existence of steady-state solutions for both the Riccati-like and covariance-like dierential equations must be taken into consideration.  Choose an initial guess for the control gain matrices Kio The chosen gain matrices should guarantee both the deterministic and the stochastic stability for the system.




CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

 For the selected initial gains solve the boundary value problem in Theorem
. Theorem
.
for steady-state gains. Denote the obtained solutions by P and X . o ki

o ki

 Substitute the calculated P and X in the control gain equation and compute K 1. o ki

o ki

i

 Use K 1 in Step to calculate P 1 and X 1 . i

ki

ki

 Iterate to generate a list for K  P itr i

itr ki

and X . itr ki

 Stop when a specied tolerance  is achieved. That is kK

itr i

K ᝇ1 k
. itr i

.. A numerical example Consider a scalar system with one possible fault in the actuator. System and other design parameters are given as A

 B1

 B2




The actuator failure rates are assumed to be 12 
 21 



The conditional transition rates of the FDI process are chosen to be q1

ij

2 66 
 
 64







3 7 7  q2 7 5 ij

2 6 
 
 6 6 4




The noise intensities are F1

 F2

 D1

 D2

 E1

 E2






3 7 7 7 5




.. CHAPTER SUMMARY

The weighting matrices are fQ11

Q12 Q21 Q22


 

Rki




i

j

f



g


 
 
 g









f  g

The objective is to synthesize a control law for the FTCSMP in noisy environment. The designed controller gain must be able to stabilize the unstable open-loop system and to maintain its stability even in the presence of actuator fault. The proposed algorithm is used to calculate a control law for innite time horizon. Initial gains used are 1
  and 2
 these gains guarantee both the deterministic and the stochastic stability of the FTCSMP stochastic stability is examined using Theorem . in Chapter . The algorithm converges to constant gain matrices as shown in Figure
. . Steady-state solutions are summarized in Table
. . K




K




Fault Tolerant Control Law

Fault Tolerant Control Law

6 5.5

K1

5 4.5 4 3.5 3 2.5 2 1.5

0

5

10

15 20 25 Iterations

30

35

8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3

K2

0

5

10

Figure
.  The control gains

15 20 25 Iterations

Ki

30

35

.

. Chapter Summary In this chapter a control law has been synthesized for the FTCSMP in noise-free and noisy environments. In noise-free environment three scenarios have been considered

the rst assumes that both the failure and the FDI processes are available for control reconguration. Unfortunately this scenario is not practical. In the second scenario




CHAPTER . SYNTHESIS OF FAULT TOLERANT CONTROL LAWS

Table
. Summary of the steady-state solutions of the Riccati-like covariance-like and control gains equations. Pki

Xki

Ki

P11


 

X11


 

P12


  

X12




K1


 

P21


  

X21




K2


 



P22


 

X22






the controller is recongured based on the decisions of the FDI process only and does not need direct access to the failure process. In practice there are situations where the FDI process itself may not be able to provide any decision. Although these situations are rare they may lead to a complete loss of system stability if not taken care of properly. For these cases the third scenario has been considered. The lack of information from either the failure process or the FDI process has been replaced by the probability distribution for that process. The more realistic case for which the FTCSMP is operating in noisy environment has been considered as well. The FTCSMP was subjected to three types of noises state-dependent control-dependent and purely additive Gaussian noises. For both environments the design of a fault tolerant controller has been completed using an equivalent deterministic cost function with the matrix maximum principle. In particular conditions for the existence of a control law for nite time horizon have been derived. The limiting behavior of the cost function the Riccati-like and the covariance-like dierential equations has been studied. Conditions that guarantee the niteness of the cost function the existence of steady-state solutions for both the Riccati-like and covariance-like dierential equations have been stated and veried.

.. CHAPTER SUMMARY




A computational algorithm has been proposed. The theoretical results have been veried by numerical examples.

Chapter 9 EPILOGUE This book considered a number of important aspects for the analysis and synthesis of fault tolerant control systems (FTCSs). A FTCS is a collection of techniques which provide a systematic approach to maintain system stability and acceptable performance not only during normal system operation, but also in the presence of system component malfunctions. Based on the available resources of system redundancy, the design approaches of FTCSs can be classied into: passive and active. It was the objective of this book to analyze the stochastic behavior of active FTCSs. General guidelines have been provided in the development process to make the approach more applicable to practical systems. As a general conclusion, a number of innovative ideas have been developed, yet they need to be tested in industrial applications. This chapter summarizes the major accomplishments of the work and provides some suggestions for future investigations.

9.1 Summary The following conclusions can be drawn from this book: Chapters 1 and 2 present a comprehensive overview and literature survey of FTCSs. Chapter 1 denes FTCSs, states basic concepts, classies the design approaches, details the literature survey and recent developments in the area of FTCSs. In Chapter 2, a prime focus was given to the major elements in an AFTCS, namely: M.M. Mahmoud, J. Jiang, Y. Zhang: Active Fault Tolerant Control Systems, LNCIS 287, pp. 184−187, 2003.  Springer-Verlag Berlin Heidelberg 2003

9.1. SUMMARY

185

FDI algorithms and control reconguration mechanisms (CRM). For the seek of completeness, classication and modelling of faults in dynamic systems was also discussed. Chapter 3 is an important introduction to the stability concepts of general stochastic dynamic systems. The extension of deterministic stability in the three modes of convergence, dened in probability theory, is outlined and several stochastic concepts have been reviewed. The chapter justied why the Lyapunov second method is used in this book to study the behavior of FTCSMP. Other stochastic tools and properties are stated as well. FTCSMP were dened in Chapter 4. The model of the dynamical system to be controlled, the failure and the FDI processes are dened. An analytical approach to calculate the conditional transition rates of the FDI process was outlined. The chapter concludes by justifying the uniqueness of the proposed FTCSMP representation. Conditions for the stability of the sample solutions of the FTCSMP are dened and stated, with particular interest in exponential stability in the mean square and almost sure asymptotic stability. Important developments to characterize the behavior of FTCSMP in practical environments have been completed in Chapter 5. In particular, the stochastic stability of FTCSMP has been analyzed taking the following into consideration: The e ect of environmental noises in practical control systems and the presence of multiple failures in di erent system components. In Section 5.2, three types of noise have been introduced: state-dependent, control-dependent and independent noise. A testable necessary and sucient condition for the exponential stability in the mean square has been derived. It has been illustrated that the closed-loop system stability of the FTCSMP could be lost if the state and control dependent noises are not considered. Whereas, the independent noise, under the assumption of perfect FDI performance, do not a ect the stability of the FTCSMP. Section 5.3 considered the case where the

186

CHAPTER 9. EPILOGUE

system is subject to random faults in plant components and actuators. A necessary and sucient condition for the exponential stability in the mean square for FTCSMP has been derived. Some existing results were shown to be special cases of this general result. In Chapter 6, practical issues related to the imperfect FDI performance on the stochastic stability of the FTCSMP have been studied. A controller which takes into account of detection errors and delays has been designed. The results revealed that long detection delays and large detection errors may lead to a complete loss of the stochastic stability of the FTCSMP. The case where the FTCSMP is driven by actuators with potential saturation has been addressed in Chapter 7. Sucient conditions for the exponential stability in the mean square have been derived. An algorithm to examine the stability of the FTCSMP has been proposed. Stability conditions for JLS and other hybrid systems has been established as interesting special cases of this work. Furthermore, the chapter considered the eect of inaccurately estimated parameters on the stability of FTCSMP . Inaccuracies in estimated parameters are viewed as uncertainties, which are assumed to be time-varying unknown-but-bounded. A stabilization algorithm, with norm bounded uncertainty measure has been proposed. The algorithm provides the necessary and sucient condition for exponential stabilization in the mean square of the uncertain FTCSMP. Several existing results have been shown to be special cases from this result. The results further con rm that the FDI algorithm and the controller recon guration mechanism must be designed taking into account of practical limitations. In Chapter 8, a fault tolerant control law has been synthesized for the FTCSMP in noise-free and noisy environments. Under the noise-free condition, three scenarios have been considered: the rst assumes that the information on both the failure process and the FDI process is available for control recon guration, the second relies

9.1. SUMMARY

187

only on the information from the FDI process, and the third deals with the most severe cases where the information from the FDI process itself is not available. To move one step closer to practical control systems, a control law in noisy environment has been considered as well. In both cases, the design of a fault tolerant controller has been completed using an equivalent deterministic cost function and applying the matrix maximum principle. In particular, conditions for the existence of a fault tolerant control law have been derived. A computational algorithm has been constructed. The theoretical results have been illustrated by numerical examples.