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Studies in Systems, Decision and Control 268 Shanling Dong Zheng-Guang Wu Peng Shi Control and Filtering of Fuzzy Systems with Switched Parameters Studies in Systems, Decision and Control Volume 268 Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the ﬁelds of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output ** Indexing: The books of this series are submitted to ISI, SCOPUS, DBLP, Ulrichs, MathSciNet, Current Mathematical Publications, Mathematical Reviews, Zentralblatt Math: MetaPress and Springerlink More information about this series at http://www.springer.com/series/13304 Shanling Dong Zheng-Guang Wu Peng Shi • • Control and Filtering of Fuzzy Systems with Switched Parameters 123 Shanling Dong National Laboratory of Industrial Control Technology Institute of Cyber-Systems and Control, Zhejiang University Hangzhou, Zhejiang, China Zheng-Guang Wu National Laboratory of Industrial Control Technology Institute of Cyber-Systems and Control, Zhejiang University Hangzhou, Zhejiang, China Peng Shi School of Electrical and Electronic Engineering University of Adelaide Adelaide, SA, Australia Victoria University Melbourne, VIC, Australia ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-35565-4 ISBN 978-3-030-35566-1 (eBook) https://doi.org/10.1007/978-3-030-35566-1 © Springer Nature Switzerland AG 2020 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland This research monograph is dedicated to our parents Preface The past decades have witnessed an increasing popularity in the Takagi–Sugeno (T–S) fuzzy model since it has powerful capabilities in exactly modeling complicated nonlinear systems via transforming them into a family of local linear subsystems An overall T–S fuzzy model can be seen as a “combination” of those local linear subsystems with membership functions and IF-THEN rules Many related works have been reported and the theoretic results have been applied to various ﬁelds, such as communication networks, mechanical systems, and power electronic systems On the other hand, practical systems often experience abrupt variations resulting from stochastic component failures, unexpected environment disturbances, changes between subsystems, and so on Random switching mechanisms have been used to model these complicated phenomena, such as the Markov jump principle In the past decades, fuzzy systems with switched parameters have received considerable attention and a large number of results have been published This book presents the recent advances in analysis and synthesis of fuzzy systems with switched parameters Chapter provides an overview of recent developments of fuzzy systems with switched parameters Chapter investigates the reliable control problem for T–S fuzzy systems with switched actuator failures and quantization Chapter is concerned with the non-fragile guaranteed cost control problem for T–S fuzzy systems with Markov jump parameters and time-varying delays Based on the hidden Markov model, Chap studies the asynchronous quantized control problem for T–S fuzzy Markov jump systems Chapter focuses on the asynchronous dissipative control problems for both continuous-time and discrete-time systems By the static output feedback control method, Chap addresses the extended-dissipative control problem for T–S fuzzy systems with asynchronous modes and intermittent measurements Chapter considers the asynchronous sliding mode control problem for T–S fuzzy Markov jump systems with matched uncertainties The H1 and L2 −L1 ﬁltering problems are discussed in Chap for T–S fuzzy switched systems with quantization, respectively Chapters and 10 deal with the reliable ﬁltering problems for T–S fuzzy switched systems in discrete-time and continuous-time domains, respectively The dissipative asynchronous ﬁlter design problem is solved in Chap 11 for continuous-time T–S fuzzy vii viii Preface Markov jump systems with the hidden Markov model The networked fault detection problem is investigated in Chap 12 for fuzzy Markov jump systems It is our hope that this book will be a helpful reference for people working in the ﬁeld of systems and control by fuzzy modeling techniques with switched parameters Hangzhou, China Hangzhou, China Adelaide/Melbourne, Australia September 2019 Shanling Dong Zheng-Guang Wu Peng Shi About This Book This book presents recent advancements on control and ﬁltering design for Takagi– Sugeno (T–S) fuzzy systems with switched parameters The T–S fuzzy model has received a great deal of attention from people working in the ﬁeld of control science and engineering since it has powerful ability in transforming complicated nonlinear systems into a set of linear subsystems Typical applications of T–S fuzzy systems include communication networks, mechanical systems, power electronic systems, and so on Practical systems often experience abrupt variations in their parameters or structures led by outside disturbance, component failures, and so on Random switching mechanisms have been used to model these stochastic changes, such as the Markov jump principle In the past decades, a plenty of results on fuzzy systems with switched parameters have been reported In general, there are three kinds of the controller/ﬁlter for fuzzy Markov jump systems, namely, the mode-independent controller/ﬁlter, the mode-dependent controller/ﬁlter, and the asynchronous one Compared with the mode-dependent case, mode-independence does not focus on whether modes are accessible, ignores partially useful mode information, and thus results in some conservatism The mode-dependent design approach needs us to timely acquire complete and correct information about the mode from the studied plant In fact, factors like component failures and data dropouts often make it difﬁcult to obtain exact mode message, which further let the mode-dependent controllers/ﬁlters less useful Recently, to overcome these issues, researchers devote themselves to studying the asynchronous technique Modes of asynchronous controllers/ﬁlters are accessed by observing the original systems based on certain probabilities In this book, we will investigate the problems of the mode-independent, mode-dependent, and asynchronous controller/ ﬁlter design In our study, some networked constraints, such as data dropouts and time delays, are also considered Based on Lyapunov function and matrix inequality techniques, performances of the targeted systems are analyzed, including the stochastic stability, dissipativity, H1 , and so on This book not only shows how these approaches solve the control and ﬁltering problems effectively, but also gives the potential ix x About This Book and meaningful research directions and ideas, which will help the readers understand this ﬁeld thoroughly The book covers many ﬁelds including continuous-time and discrete-time Markov processes, fuzzy systems, robust control, and ﬁlter design problems Investigation on these aspects is meaningful both from the theoretical and practical points of view It is primarily intended for the researchers in system and control theory It may also function as a valuable reference to lead graduate students and undergraduate students to an active and interesting control research ﬁeld The book provides many cases of fuzzy control problems which may be valuable materials for scientists, engineers, and researchers in the ﬁeld of intelligent control Also, the contents of this book can be used for advanced courses focusing on fuzzy modeling, analysis, and control Chapter 12 Networked Fault Detection for Fuzzy MJSs 12.1 Introduction In realistic control systems, faults are significant obstacles to obtain a better performance, higher reliability and safety [1–3] They would likely appear due to unexpected variations in signals and components, sudden changes of working conditions, environmental noises, parameter shifting, etc It is important to detect them timely and accurately for avoiding the degradation of performance and instability The model-based fault detection (FD) method has been proposed, whose primary goals are to construct an appropriate filter/observer, generate the residual signal and evaluate the produced residual signal A fault alarm will be sent out once the residual evaluation value is larger than the predefined threshold Recently, plenty of dynamical systems have adopted this approach to detect faults and guarantee the system’s normal operation Through the delta operator method, the work in [4] has analyzed the FD issue for T–S fuzzy systems with time-varying delays The FD filtering problem has been addressed in [5] for nonlinear switched stochastic systems For T–S fuzzy systems with unknown bounded noises and sensor faults, the work in [6] has considered the FD observer design as a multi-objective H− /H∞ performance index An event-triggered method has been used in [7] to investigate the FD problem for nonlinear networked systems In this chapter, we investigate the dissipative asynchronous FD problem for T–S fuzzy MJSs with network data losses We assume that there are imperfect communication links between the plant and the designed FD filter, described by Bernoulli process Besides, the HMM theory is applied to describe the non-synchronization between two Then based on the fuzzy inference, a fuzzy FD filter is devised to produce a residual signal and the FD issue is transformed as a filtering problem Via the mode-dependent and fuzzy-basis-dependent Lyapunov function technique, a sufficient condition is developed to ensure the stochastic stability and the strict dissipativity of the FD system Two approaches are proposed to obtain FD filter gains, which can be solved by using Matlab Toolbox © Springer Nature Switzerland AG 2020 S Dong et al., Control and Filtering of Fuzzy Systems with Switched Parameters, Studies in Systems, Decision and Control 268, https://doi.org/10.1007/978-3-030-35566-1_12 197 198 12 Networked Fault Detection for Fuzzy MJSs 12.2 Preliminary Analysis Consider the following discrete-time T–S fuzzy systems with Markov jump: Plant r ule i: IF μ1k is φi1 , μ2k is φi2 , , and μ pk is φi p , THEN xk+1 = Aδk i xk + Bδk i u k + Cδk i wk + Dδk i f k , yk = E δk i xk + Fδk i u k + G δk i wk + Hδk i f k , (12.1) where xk ∈ R n x is the state variable; u k ∈ R n u is the given input; wk ∈ R n w is the external disturbance which belongs to l2 [0, +∞); f k ∈ R n f is the known fault signal; and yk ∈ R n y is the measured output System matrices are known with appropriate dimensions The variable i (i ∈ I = {1, 2, , r }) means the ith fuzzy rule and r is the total sum of rules φi j ( j ∈ J = {1, 2, , p}) is the fuzzy set and μ jk is the premise variable δk is adopted to represent a discrete-time Markov jump process, taking values in V = {1, 2, , V } δk is subject to the transition probability matrix Υ = [λab ] with (12.2) Pr{δk+1 = b|δk = a} = λab , a, b ∈ V, V λab = where λab ≥ and b=1 With δk = a, the overall fuzzy systems can be inferred as xk+1 = Aah xk + Bah u k + Cah wk + Dah f k , yk = E ah xk + Fah u k + G ah wk + Hah f k , where r Aah = r h i (μk )Aai , Bah = i=1 r Cah = h i (μk )Bai , i=1 r h i (μk )Cai , Dah = i=1 r E ah = h i (μk )Dai , i=1 r h i (μk )E , Fah = i=1 r G ah = h i (μk )Fai , i=1 r h i (μk )G , Hah = i=1 (12.3) h i (μk )Hai , i=1 μk = μ1k , μ2k , , μ pk , h i (μk ) = p j=1 r i=1 φi j (μ jk ) p j=1 φi j (μ jk ) φi j (μ jk ) denotes the grade of membership of μ jk in φi j and we assume that p j=1 φi j (μ jk ) ≥ Accordingly, it is observed easily that h i (μk ) ≥ and r i=1 h i (μk ) = In the following, we represent h i (μk ) as h i for convenient expression 12.2 Preliminary Analysis 199 Owing to communication link constraints between the plant and an FD filter to be designed, data losses may happen stochastically We adopt the following model to represent the random data dropout phenomenon: y f k = βk yk , (12.4) where βk represents Bernoulli process βk = means that data are transmitted successfully Otherwise, there will be transmission failure if βk = We suppose that βk is subject to Pr{βk = 1} = β, Pr{βk = 0} = − β (12.5) It easily follows that E{βk } = β Define β¯k = βk − β, and we have E{β¯k } = 0, E{β¯k2 } = β(1 − β) = β¯ (12.6) √ with β¯ = β(1 − β) and β ∈ (0, 1] In this chapter, we are interested in designing an FD filter to generate the residual signal that is sensitive to faults Besides, it is supposed that premise variables of the designed filter are the same as those of the plant Applying the PDC approach, we devise the following fuzzy filter: Filter r ule i: IF μ1k is φi1 , μ2k is φi2 , , and μ pk is φi p , THEN xˆk+1 = Aˆ ρk i xˆk + Bˆ ρk i y f k , rk = Eˆ ρk i xk + Fˆρk i y f k , (12.7) where xˆk ∈ R n xˆ is the state variable; rk ∈ R nr is the residual signal; Filter matrices ( Aˆ ρk i , Bˆ ρk i , Eˆ ρk i , Fˆρk i ) are to be solved The variable ρk is introduced to observe the plant mode, which is subject to the conditional transition probability matrix Φ = [ϕas ] (ρk ∈ L = {1, 2, , L}) with Pr{ρk = s|δk = a} = ϕas , where ϕas > and filter as L s=1 (12.8) ϕas = Therefore, when ρk = s, we can represent the xˆk+1 = Aˆ sh xˆk + Bˆ sh y f k , rk = Eˆ sh xˆk + Fˆsh y f k , where Aˆ sh = Eˆ sh = r i=1 r i=1 h i Aˆ si , Bˆ sh = h i Eˆ si , Fˆsh = r i=1 r i=1 h i Bˆ si , h i Fˆsi (12.9) 200 12 Networked Fault Detection for Fuzzy MJSs For FD systems, we introduce a reference model to obtain a better performance, that is, f˜(z) = W (z) f (z), where W (z) is a known stable weighting matrix The minimal state-space realization is expressed as x˜k+1 = A W x˜k + BW f k , f˜k = E W x˜k + FW f k , (12.10) where x˜k ∈ R n x˜ is the state of weighted fault; f˜k ∈ R n f˜ is the weighted fault; And (A W , BW , E W , FW ) are known constant matrices Defining the residual error as ek = rk − f˜k , combining systems (12.3), (12.9) and (12.10), we obtain the following FD system: ζk+1 = ( A˘ 1ash + β¯k A˘ 2ash )ζk + ( B˘ 1ash + β¯k B˘ 2ash )υk , ek = ( E˘ 1ash + β¯k E˘ 2ash )ζk + ( F˘1ash + β¯k F˘2ash )υk , where A˘ 1ash = r r h i h j A˘ 1asi j , A˘ 1asi j = A¯ , β Bˆ s j E¯ Aˆ s j h i h j A˘ 2asi j , A˘ 2asi j = 0 , Bˆ s j E¯ h i h j B˘ 1asi j , B˘ 1asi j = B¯ , ˆ β Bs j F¯ai h i h j B˘ 2asi j , B˘ 2asi j = , ˆ Bs j F¯ai i=1 j=1 A˘ 2ash = r r i=1 j=1 B˘ 1ash = r r i=1 j=1 B˘ 2ash = r r i=1 j=1 E˘ 2ash = r r h i h j E˘ 2asi j , E˘ 2asi j = Fˆs j E¯ , i=1 j=1 F˘1ash = r r h i h j F˘1asi j , F˘1asi j = β Fˆs j F¯ai − F¯W , i=1 j=1 F˘2ash = r r h i h j F˘2asi j , F˘2asi j = Fˆs j F¯ai , i=1 j=1 E˘ 1ash = r r h i h j E˘ 1asi j , F¯W = 0 FW , i=1 j=1 E˘ 1asi j = β Fˆs j E¯ − E¯ W Eˆ s j , E¯ W = E W , AW 0 BW , B¯ = , A¯ = Bai Cai Dai Aai (12.11) 12.2 Preliminary Analysis 201 E¯ = E , F¯ai = Fai G Hai , υk = u kT wkT f kT T , ζk = x˜kT xkT xˆkT T The FD problem of this chapter can be represented as the following two steps: (1) Produce the residual signal: for T–S fuzzy MJSs (12.3), design a suitable dissipative fuzzy FD filter (12.7) to generate the residual signal Furthermore, the designed filter can guarantee that the FD system (12.11) is stochastically stable with strict dissipative performance; (2) Construct an FD measure: calculate the residual evaluation value and the threshold via a chosen evaluation function If the threshold is smaller than the residual evaluation value, a fault alarm is sent out In this chapter, we select the evaluation function J(r ) and the threshold Jth (r ) as k0 +k riT ri , Jth (r ) = J(r ) = i=k0 sup w=0,u=0, f =0 J(r ), (12.12) where k0 is the initial evaluation time Via the following test, the occurrence of fault can be detected: J(r ) ≤ Jth (r ) =⇒ no faults, (12.13) J(r ) > Jth (r ) =⇒ faults =⇒ alarm 12.3 Main Results In this section, a sufficient condition is developed to ensure the stochastic stability and the strict dissipativity for the FD system (12.11) on the supposition that the fuzzy FD filter matrices in (12.7) are known Then, two asynchronous FD design approaches are given Theorem 12.1 The FD system (12.11) is stochastically stable and strictly (Q, S, R)α-dissipative if there exist n-dimensional matrices Pai > and Masi > (n = n x˜ + n x + n xˆ ) for any a ∈ V, s ∈ L and i, j, t ∈ I subject to L ϕas Masi < Pai , (12.14) Γasiit < 0, (12.15) Γasi jt + Γas jit < 0, i < j, (12.16) s=1 where 202 12 Networked Fault Detection for Fuzzy MJSs ⎤ −Γ11 Γ11 Γ13 Γ11 Γ14 ⎢ ∗ −Γ22 Γ231 Γ23 Γ231 Γ24 ⎥ ⎥, =⎢ T ⎦ ⎣ ∗ ∗ −Masi − E˘ 1asi jS ∗ ∗ ∗ Γ44 = diag{ P¯at , P¯at }, Γ23 = diag{Q− , Q− }, A˘ B˘ = ¯ ˘1asi j , Γ14 = ¯ ˘1asi j , β A2asi j β B2asi j ⎡ Γasi jt Γ11 Γ13 E˘ 1asi j F˘1asi j = , Γ , 24 β¯ E˘ 2asi j β¯ F˘2asi j T T ˘ = − F˘1asi j S − S F1asi j − R + α I, Γ23 = Γ44 V P¯at = λab Pbt , Γ22 = diag{I, I } b=1 Proof Define r r Mash = h i Masi , Pah = i=1 r h+ t Pbt , Pbh + = h i Pai , i=1 t=1 (12.17) V P¯ah + = + λab Pbh + , h = h(μk+1 ) b=1 According to the fuzzy principle, (12.11) and (12.14)–(12.16), we have L ϕas Mash < Pah , (12.18) s=1 and r r r h+ t h i h j Γasi jt Γash = t=1 i=1 j=1 r r −1 r h+ t = t=1 h i2 Γasiit i=1 (12.19) r + h i h j (Γasi jt + Γas jit ) < 0, i=1 j=i+1 where ⎤ −Γ11h + Γ11h + Γ13h Γ11h + Γ14h 1 ⎢ ∗ −Γ22 Γ23 Γ23h Γ23 Γ24h ⎥ ⎥, =⎢ T ⎣ ∗ ∗ −Mash − E˘ 1ash S⎦ ∗ ∗ ∗ Γ44h ⎡ Γash 12.3 Main Results 203 Γ11h + = diag{ P¯ah + , P¯ah + }, A˘ B˘ Γ13h = ¯ ˘1ash , Γ14h = ¯ ˘1ash , β A2ash β B2ash ˘ E F˘ Γ23h = ¯ ˘1ash , Γ24h = ¯ ˘1ash , β E 2ash β F2ash T Γ44h = − F˘1ash S − S T F˘1ash − R + α I T Q− , we have Applying Schur Complement to (12.19), and considering −Q = Q− T Γ11h + Γ13h − Mash < 0, Γ13h T T Γ134h Γ11h + Γ134h − Γ234h Γ231 Γ234h + Γ44h < 0, where (12.20) Γ134h = Γ13h Γ14h , Γ234h = Γ23h Γ24h , T S −Mash − E˘ 1ash Γ23 = diag{Q, Q}, Γ44h = ∗ Γ44h Furthermore, from (12.18), it follows that Δi < 0, i = 1, 2, where (12.21) L T ϕas Γ13h Γ11h + Γ13h − Pah , Δ1 = s=1 L T T ϕas Γ134h Γ11h + Γ134h − Γ234h Γ231 Γ234h + Γ44h , Δ2 = s=1 Γ44h = T S −Pah − E˘ 1ash ∗ Γ44h We choose the Lyapunov function as Vk = ζkT Pδk h ζk , (12.22) where Pδk h = ri=1 h i Pδk i , and Pδk i is mode-dependent and fuzzy-basis-dependent It is assumed that at time k, δk = a At the next time k + 1, δk+1 = b and the Lyapunov function matrix turns to be Pbh + Along the trajectory of system (12.11) with υk = 0, we have T Pbh + ζk+1 } − E{ζkT Pah ζk } E{ΔVk } = E{ζk+1 = ζkT Δ1 ζk < (12.23) 204 12 Networked Fault Detection for Fuzzy MJSs It is clearly concluded that system (12.11) is stochastically stable Now, we are going to establish the strict dissipativity performance for system (12.11) The energy supply function for system (12.11) is denoted as Fdis (ek , υk ) = ekT Qek + 2ekT Sυk + υkT Rυk , (12.24) where matrices Q, S and R are known with RT = R And Q is a negative semiT Q− definite matrix, which implies that −Q = Q− From (12.21), it follows that E{ΔVk − Fdis (ek , υk ) + αυkT υk } = ζk υk T Δ2 ζk < υk (12.25) Summing up the above inequality from k = to K yields that K K VK +1 − V0 − E{Fdis (ek , υk )} + α k=0 υkT υk < (12.26) k=0 With the zero initial condition, namely, V0 = 0, it follows that K K E{Fdis (ek , υk )} > α k=0 K υkT υk + VK +1 > α k=0 υkT υk (12.27) k=0 We can obtain that system (12.11) is strictly dissipative, which completes the proof Based on Theorem 12.1, we focus on designing an (n x˜ + n x )-dimensional asynchronous FD filter for T–S fuzzy MJSs with data losses in the following Theorem 12.2 If there exist n-dimensional matrices Pai = P1ai P2ai > 0, Masi = ∗ P3ai M1asi M2asi (n = 2(n x˜ + n x )), (n x˜ + n x )-dimensional matrices N1s , N2s , X s , Aˆs j , ∗ M3asi Bˆ s j , Eˆs j , and Fˆs j for any a ∈ V, s ∈ L and i, j, t ∈ I subject to L ϕas Masi < Pai , (12.28) Λasiit < 0, (12.29) Λasi jt + Λas jit < 0, i < j, (12.30) s=1 where 12.3 Main Results 205 ⎡ Λasi jt Λ11 ⎤ −Λ11 0 Λ14 Λ15 ⎢ ∗ −Λ11 Λ24 Λ25 ⎥ ⎢ ⎥ ⎢ ∗ −Λ33 Λ34 Λ35 ⎥ =⎢ ∗ ⎥, ⎣ ∗ ∗ ∗ −Masi Λ45 ⎦ ∗ ∗ ∗ ∗ Λ55 T ¯ T ¯ P − N1s − N1s P2at − X s − N2s = 1at T , ¯ ∗ P3at − X s − X s Λ14 = N1s A¯ + β Bˆ s j E¯ Aˆs j , N2s A¯ + β Bˆ s j E¯ Aˆs j Λ15 = N1s B¯ + β Bˆ s j F¯ai , Λ33 = diag{I, I }, N2s B¯ + β Bˆ s j F¯ai Λ24 = β¯ Bˆ s j E¯ β¯ Bˆ s j F¯ai , , Λ25 = ˆ ¯ ¯ β¯ Bˆ s j F¯ai β Bs j E Λ34 = Q− (β Fˆs j E¯ − E¯ W ) Q− Eˆs j , ¯ − Fˆs j E¯ βQ Λ35 = Q− (β Fˆs j F¯ai − F¯W ) , ¯ − Fˆs j F¯ai βQ Λ45 = −(β Fˆs j E¯ − E¯ W )T S , −Eˆ T S sj Λ55 = − (β Fˆs j F¯ai − F¯W )T S − S T (β Fˆs j F¯ai − F¯W ) − R + α I, there exists an (n x˜ + n x )-dimensional dissipative FD filter in the form of (12.7) with (n xˆ = n x˜ + n x ), which can guarantee that the FD system (12.11) is stochastically stable with strict dissipativity Moreover, we can obtain the filter gains as follows: Aˆ s j = X s−1 Aˆs j , Eˆ s j = Eˆs j , Bˆ s j = X s−1 Bˆ s j , Fˆs j = Fˆs j (12.31) −1 Proof Pre-multiplying diag{Γ11 Γ11 , I, I, I } and post-multiplying its transpose to (12.15) with Γ11 = diag{Ns , Ns }, we have ⎤ −1 T 1 Γ11 (Γ11 ) Γ11 Γ13 Γ11 Γ14 −Γ11 1 ⎢ ∗ −Γ22 Γ23 Γ23 Γ23 Γ24 ⎥ ⎥ < =⎢ T ⎣ ∗ ∗ −Masi − E˘ 1asii S⎦ ∗ ∗ ∗ Γ44 ⎡ Γasiit (12.32) Considering P¯at > 0, it follows that ( P¯at − Ns ) P¯at−1 ( P¯at − Ns )T > Furthermore, we have 206 12 Networked Fault Detection for Fuzzy MJSs P¯at − Ns − NsT > −Ns P¯at−1 NsT (12.33) Accordingly, it yields that 1 T −1 T − (Γ11 ) > −Γ11 Γ11 (Γ11 ) Γ11 − Γ11 (12.34) From (12.32), it follows that ⎤ 1 T 1 − (Γ11 ) Γ11 Γ13 Γ11 Γ14 Γ11 − Γ11 1 ⎢ ∗ −Γ22 Γ23 Γ23 Γ23 Γ24 ⎥ ⎥ < =⎢ T ⎣ ∗ ∗ −Masi − E˘ 1asii S ⎦ ∗ ∗ ∗ Γ44 ⎡ Γasiit (12.35) On the other hand, define Aˆs j = X s Aˆ s j , Eˆs j = Eˆ s j , Bˆ s j = X s Bˆ s j , Fˆs j = Fˆs j , i = j, (12.36) and Ns = N1s X s N2s X s (12.37) Then based on (12.29), we can clearly find that Λasiit = Γasiit (12.38) Based on (12.32), (12.34) and (12.38), it is easy to find that we can obtain (12.15) from (12.29) Adopting the similar method to (12.16) and (12.30), we can also achieve (12.16) from (12.30) And the filter matrices can be derived from (12.36) The proof is completed On the basis of Theorem 12.2, another filter design method is developed by using Finsler’s Lemma P P Theorem 12.3 If there exist n-dimensional matrices Pai = 1ai 2ai > 0, Masi = ∗ P3ai M1asi M2asi (n = 2(n x˜ + n x )), (n x˜ + n x )-dimensional matrices X s , Aˆs j , Bˆ s j , Eˆs j , ∗ M3asi and Fˆs j for any a ∈ V, s ∈ L and i, j, t ∈ I subject to L ϕas Masi < Pai , (12.39) s=1 Q 1T Λ¯ asiit Q < 0, (12.40) 12.3 Main Results 207 O T Λ¯ asiit O < 0, (12.41) Q 2T (Λ¯ asi jt + Λ¯ as jit )Q < 0, i < j, (12.42) O T (Λ¯ asi jt + Λ¯ as jit )O < 0, i < j, (12.43) where ⎡ Λ¯ asi jt Λ¯ 11 −Λ¯ 11 0 Λ¯ 14 ⎢ ∗ −Λ¯ 11 Λ24 ⎢ =⎢ ∗ −Λ33 Λ34 ⎢ ∗ ⎣ ∗ ∗ ∗ −Masi ∗ ∗ ∗ ∗ ¯ ¯ P − Xs P , = 1at ¯ 2at ∗ P3at − X s − X sT ⎤ Λ¯ 15 Λ25 ⎥ ⎥ Λ35 ⎥ ⎥, Λ45 ⎦ Λ55 β Bˆ s j E¯ Aˆs j β Bˆ s j F¯ai Λ¯ 14 = , Λ¯ 15 = , ˆ ˆ ¯ β Bs j E As j β Bˆ s j F¯ai Q 11 Q 112 Q 11 Q 212 , Q2 = , O= , Q 22 Q 22 O12 ⎡ ⎤ ⎡ ⎤ 0 0 Aai Bai = ⎣ I ⎦ , Q 112 = ⎣0 0 0 ⎦ , 000 0 ⎡ Aai +Aa j B +B ⎤ 000 a j = ⎣0 0 0 ⎦ , 000 0 Q1 = Q 11 Q 212 Q 22 = diag{I, I, I, I, I, I }, O12 = diag{I, I, I, I, I }, there exists an (n x˜ + n x )-dimensional dissipative FD filter with (n xˆ = n x˜ + n x ) in the form of (12.7), which can guarantee that the FD system (12.11) is stochastically stable with strict dissipativity Moreover, we can obtain the filter gains via solving (12.31) Proof Inequalities (12.29) and (12.30) can be rewritten as Λasiit = Λ¯ asiit + O¯ T Ns Q¯ i + Q¯ iT Ns T O¯ < 0, and where Λasi jt + Λas jit = Λ¯ asi jt + Λ¯ as jit + O¯ T Ns ( Q¯ i + Q¯ j ) + ( Q¯ iT + Q¯ Tj )Ns T O¯ < 0, (12.44) (12.45) 208 12 Networked Fault Detection for Fuzzy MJSs −I 0 0 A¯ B¯ , Q¯ i = 0 −I 0 0 0 ⎡ ⎡ ⎤ I 00000000 N1s ⎢ I 0 0 0 0⎥ ⎢ N2s ⎢ ⎥ O¯ = ⎢ ⎣ 0 I 0 0 0⎦ , N s = ⎣ 0 000I 00000 ⎤ 0 ⎥ ⎥ N1s ⎦ N2s At the same time, Q , Q and O are orthogonal complements of Q¯ iT , Q¯ iT + Q¯ Tj and O¯ T , respectively Hence, via Finsler’s Lemma, we find that the feasibility of (12.44) (or (12.29)) is equivalent to the feasibility of (12.40)–(12.41) It is easy to conclude that the relationship between (12.45) (or (12.30)) and (12.42)–(12.43) is equivalent by similar proof The proof is completed Remark 12.4 From Theorem 12.1, it can be easily observed that there are some product terms between Lyapunov matrix P¯at and system matrices like A˘ 1asi j and B˘ 1asi j Via introducing the slack matrix Ns , these couplings are eliminated, as shown in Theorem 12.2 Nonlinear matrix inequalities in Theorem 12.1 become linear Then based on Theorem 12.2, we utilize Finsler’s Lemma to remove unnecessary matrices N1s and N2s , and achieve Theorem 12.3, which further reduce the number of unknown variables However, the number of LMIs increases by 0.5V Lr (r + 1), which is a trade-off between less conservatism and computational burden 12.4 Conclusion In this chapter, we have studied the dissipative asynchronous FD problem for nonlinear MJSs with data dropouts via the T–S fuzzy technique A sufficient condition has been developed to ensure the stochastic stability and the strictly dissipative performance of FD systems by applying the Lyapunov function approach We have established two LMI-based methods for the existence of the dissipative asynchronous FD filter, which can be cast into a convex optimization problem References Wei, Y., Qiu, J., Karimi, H.R.: Reliable output feedback control of discrete-time fuzzy affine systems with actuator faults IEEE Trans Circuits Syst I: Regul Pap 64(1), 170–181 (2017) Kommuri, S.K., Defoort, M., Karimi, H.R., Veluvolu, K.C.: A robust observer-based sensor fault-tolerant control for PMSM in electric vehicles IEEE Trans Ind Electron 63(12), 7671– 7681 (2016) Rathinasamy, S., Karimi, H.R., Joby, M., Santra, S.: Resilient sampled-data control for Markovian jump systems with adaptive fault-tolerant mechanism IEEE Trans Circuits Syst II: Express Briefs 64(11), 1312–1316 (2017) References 209 Li, H., Gao, Y., Wu, L., Lam, H.K.: Fault detection for T-S fuzzy time-delay systems: delta operator and input-output methods IEEE Trans Cybern 45(2), 229–241 (2015) Su, X., Shi, P., Wu, L., Song, Y.-D.: Fault detection filtering for nonlinear switched stochastic systems IEEE Trans Autom Control 61(5), 1310–1315 (2016) Chadli, M., Abdo, A., Ding, S.X.: H− /H∞ fault detection filter design for discrete-time TakagiSugeno fuzzy system Automatica 49(7), 1996–2005 (2013) Li, H., Chen, Z., Wu, L., Lam, H.-K., Du, H.: Event-triggered fault detection of nonlinear networked systems IEEE Trans Cybern 47(4), 1041–1052 (2017) Index A Actuator failures, 15 Asynchronous controllers/filters, Asynchronous output feedback controller, 96 Filtering error systems, 142, 160, 189 Fuzzy-basis-dependent, 38 Fuzzy-basis-dependent Lyapunov function, 139 Fuzzy switched systems, 103, 139 B Bernoulli distribution, 13 Bounded cost, 58 Bounded region, 124 G Guaranteed cost control, 31 Guaranteed cost index, 34 C Cluster observation, 50 Complete observation, 50 Conditionally independent, 52 Conditional probability matrix, 51 D Data dropouts, 96 Dissipative performance bound, 71 Dissipative systems, Dissipativity, E Energy supply function, 71, 79 Estimation problem, Evaluation function, 201 Extended dissipative performance, 4, 98 F Fault detection filter, 199 H Hidden Markov Model, H∞ filtering, 3, 143 H∞ performance, 86 Homogenous/nonhomogeneous chain, 160 Markov I Infinite-distributed delays, 11, 12 L Linear matrix inequalities, Logarithmic quantizer, 14, 142 L –L ∞ filtering, L –L ∞ performance, 17, 98 Lyapunov function, Lyapunov–Krasovskii functions, 37, 38 M Markov jump rule, Markov jump systems, © Springer Nature Switzerland AG 2020 S Dong et al., Control and Filtering of Fuzzy Systems with Switched Parameters, Studies in Systems, Decision and Control 268, https://doi.org/10.1007/978-3-030-35566-1 211 212 Mean-square stability, 12 Minimal upper bound, Mode-dependent approach, Mode-dependent controller/filter, Mode-independence, N Non-monotonic Lyapunov function, Normalized fuzzy weighting function, 77 O Optimization problem, 59 Output feedback control, 95 P Parallel distributed compensation, Passivity performance, 86 Piecewise homogenous Markov jump, 3, 160 Q Quantization, 11 Quantization density, 15, 141 R Reference model, 200 Reliable controller/filter, 4, 12, 177 Index Residual error, 200 Residual signal, 199 Right continuous trajectories, 69 S Sector bound method, 15, 52 Sensor failures, 4, 159, 176 Sliding mode control, 115, 117 Sliding mode control law, 123 Sliding mode dynamics, 118, 127 Sliding surface design, 117, 127 Sojourn probabilities, 2, 103 Stochastic logarithmic quantizer, 51 Stochastic stability, 6, 72 Supply rate, 71, 78 Switched parameters, Switched systems, Switching mechanisms, Synchronous controller/filters, T Threshold, 201 Transition probabilities, Transition rate matrix, 69 T–S fuzzy model, W Weak infinitesimal generator, 120 ... 25(1 ), 70–83 (2017) Dong, S ., Wu, Z.-G ., Shi, P ., Su, H ., Lu, R.: Reliable control of fuzzy systems with quantization and switched actuator failures IEEE Trans Syst ., Man, Cybern.: Syst 47(8 ), 2198–2208... (2017) Wu, Z.-G ., Dong, S ., Shi, P ., Su, H ., Huang, T ., Lu, R.: Fuzzy- model-based nonfragile guaranteed cost control of nonlinear Markov jump systems IEEE Trans Syst ., Man, Cybern.: Syst 47(8 ), 2388–2397... Wu, Z.-G ., Shi, P ., Su, H ., Chu, J.: Network-based robust passive control for fuzzy systems with randomly occurring uncertainties IEEE Trans Fuzzy Syst 21(5 ), 966–971 (2013) 13 Liu, Z ., Wang,
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Xem thêm: Control and filtering of fuzzy systems with switched parameters, 1st ed , shanling dong, zheng guang wu, peng shi, 2020 1075 , Control and filtering of fuzzy systems with switched parameters, 1st ed , shanling dong, zheng guang wu, peng shi, 2020 1075