In this paper, the reliable H∞ control problem is investigated for discrete-time piecewise linear systems with time delays and actuator failures. The time delays are assumed to be infinitely distributed in the discrete-time domain, and the possible failure of each actuator is described by a variable varying in a given interval. The aim of the addressed reliable H∞ control problem is to design a controller such that, for the admissible infinite distributed delays and possible actuator failures, the closed-loop system is exponentially stable with a given disturbance attenuation level γ. The controller gain is characterized in terms of the solution to a linear matrix inequality that can be easily solved by using standard software packages. A simulation example is exploited in order to illustrate the effectiveness of the proposed design procedures

Feng

Gang Feng

Guoliang Wei

Kuang

Veillette

Wang

Yang

Zidong Wang

Automatica

English

Wang, Z

Wei, G

Feng, G

Brunel University Research Archive

ReliableH∞Control forDiscrete-TimePiecewiseLinearSystemswith InfiniteDistributedDelays ⋆Zidong Wang a,b, Guoliang Wei b, Gang Feng caDepartment of Information Systems and Computing, Brunel University, Uxbridge, Middlesex, UB8 3PH, U.K.bSchool of Information Science and Technology, Donghua University, Shanghai 200051, China.cDepartment of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong.AbstractIn this paper, the reliable H∞ control problem is investigated for discrete-time piecewise linear systems with time delays andactuator failures. The time delays are assumed to be infinitely distributed in the discrete-time domain, and the possible failureof each actuator is described by a variable varying in a given interval. The aim of the addressed reliable H∞ control problemis to design a controller such that, for the admissible infinite distributed delays and possible actuator failures, the closed-loopsystem is exponentially stable with a given disturbance attenuation level γ. The controller gain is characterized in terms ofthe solution to a linear matrix inequality that can be easily solved by using standard software packages. A simulation exampleis exploited in order to illustrate the effectiveness of the proposed design procedures.Key words: Piecewise linear systems; H∞ control; reliable control; infinite distributed delays; actuator failure.1 IntroductionThe control systems with piecewise components, suchas dead-zone, saturation and relays, are called piecewiselinear (PWL) systems that have become an importantarea of research. It has also been shown that the PWLsystems could approximate nonlinear ones in an efficientway. Therefore, over the past decade, The stability anal-ysis, control and filtering synthesis problems for PWLsystems have been extensively investigated, see e.g. [1,7].In particular, for the discrete-time PWL systems, theanalysis and synthesis problems have been dealt withby some researchers [1, 7]. On the other hand, reliablecontroller problems for linear or nonlinear systems havegained considerable attention and a number of resultshave been reported in the literature, see, example, [5,8].Recently, there has been significant progress on boththe analysis and synthesis issues for linear/nonlinearsystems with various types of delays, see e.g. [2, 4]. Itis worth pointing out that the distributed delay occursvery often in reality which has drawn increasing researchattention, see e.g. [3, 4]. However, almost all existingworks on distributed delays have exclusively focused oncontinuous-time systems that are described in the formof either finite or infinite integral. With the increasingapplication of digital control systems, the distributeddelays may appear digitally (i.e. in a discrete-time man-ner) and therefore it becomes desirable to study thediscrete-time systems with distributed delays. Very re-cently, in [4], some pioneering work has been carried out⋆ This work was supported in part by the LeverhulmeTrust Visiting Fellowship of the U.K., an International JointProject sponsored by the Royal Society of the U.K., and theAlexander von Humboldt Foundation of Germany. Corre-sponding author Guoliang Wei.Email address: Guoliang.Wei1973@gmai.com (GuoliangWei).on the formulation of discrete-time distributed delays.Up to date, to the best of the authors’ knowledge, theresearch on discrete-time piecewise linear systems withdiscrete-time distributed delays is still an open problemthat deserves further investigation.Motivated by the above discussion, in this paper, weconsider the reliable H∞ control problem for a class ofdiscrete-time PWL systems with infinite distributed de-lays and actuator failures. The objective is to design areliableH∞ controller such that, in the presence of timedelays as well as actuator failure, the closed-loop PWLcontrol system is exponentially stable and also satisfiesa prescribed H∞ disturbance attenuation index.2 Problem FormulationConsider the following discrete-time piecewise linear sys-tems with infinite distributed delays:x(k + 1)=Alx(k) +Dl∞∑d=1µdx(k − d)+Blu(k) + Elv(k), (1)z(k) =Cl(k)x(k), forx ∈ Sl, l = 1, . . . , L, (2)x(k) = φ(k), ∀ k ∈ Z−,where x(k) ∈ Rn is the state, {Sl}l∈L denotes a partitionof the state space into a number of closed polyhedralsubspaces, L is the index set of subspaces, z(k) ∈ Rq isthe controlled output vector, u(k) ∈ Rm is the controlinput, v(k) ∈ l2[0,∞) is the disturbance input, φ(k)(∀ k ∈ Z−) is the initial state, and Al, Bl, Cl, Dl and Elare all constant matrices with appropriate dimensionscorresponding to the l-th local model of the systems.When the state of the system transits from one region toanother at the time k, the dynamics is governed by thelocal model of the former one. µd ≥ 0 is the convergenceconstants that satisfy the following condition:Preprint submitted to Automatica 16 September 2009µ¯ :=∞∑d=1µd ≤∞∑d=1dµd < +∞. (3)As in [1], for presentation convenience, we define Ω :={l, j|x(k) ∈ Sl, x(k + 1) ∈ Sj , j 6= l} to represent allpossible transitions from one region to itself or another.Remark 1 Distributed time-delays have been widelyrecognized and intensively studied for continuous-timesystems, see e.g. [3, 4]. However, the corresponding re-sults for discrete-time systems have been very few duemainly to the difficulty in formulating the distributeddelays in a discrete-time domain. In (1)-(2), we intro-duce the distributed delay term∑∞d=1 µdx(k − d) thatcan be regarded as the discretization of the infinite in-tegral form∫ t−∞k(t − s)x(s)ds for the continuous-timesystem, and a special inequality is employed to facilitatethe subsequent mathematical analysis.When the actuators experience failures, we use uF (k) todescribe the control signal sent from actuators. Considerthe actuator failure model [8] with failure parameter F :uF (k) = Fu(k), (4)where 0 ≤ F = diag{f1, · · · , fm} ≤ F = diag{f1, · · · , fm}≤ F¯ = diag{f¯1, · · · , f¯m} ≤ I, in which the variables fi(i = 1, · · · ,m) quantify the failures of the actuators. LetF0 =diag{f01, · · · , f0m} := {F + F¯}/2= diag{{f1+ f¯1}/2, · · · , {fm + f¯m}/2}, (5)F˜ =diag{f˜1, · · · , f˜m} := {F¯ − F}/2= diag{{f¯1 − f1}/2, · · · , {f¯m − fm}/2}. (6)Apparently, f˜i = {f¯i− f i}/2, (i = 1, . . . ,m) and we canrewrite F as follows:F = F0 +∆ = F0 + diag{δ1, · · · , δm}, (7)where |δi| ≤ f˜i, i = 1, . . . ,m.We consider the following controller u(k) = Klx(k) (l =1, . . . , L) where Kl is the controller gain matrix to bedesigned. Then, the closed-loop delayed piecewise linearsystem with actuator failures can be given byx(k + 1)=Alx(k) +Dl∞∑d=1µdx(k − d) + Elv(k), (8)z(k) =Cl(k)x(k), for x ∈ Sl, l = 1, . . . , L, (9)where Al := Al +BlFKl.Definition 1 Given a scalar γ > 0, the piecewise linearsystem (8)-(9) is said to be exponentially stable with dis-turbance attenuation level γ if it is exponentially stableand, under zero initial conditions, ‖z(k)‖2 < γ2‖v(k)‖2holds for all nonzero v(k) ∈ l2[0,∞).The objective of this paper is to design a reliable H∞controller for the discrete-time piecewise linear system(8)-(9) such that the closed-loop system (8)-(9) is expo-nentially stable with disturbance attenuation level γ.3 Main ResultsLemma 1 [4] LetM ∈ Rn×n be a positive semi-definitematrix, xi ∈ Rn and constant ai > 0(i = 1, 2, . . .). If theseries concerned is convergent, then we have(∞∑i=1aixi)TM(∞∑i=1aixi)≤(∞∑i=1ai)∞∑i=1aixTi Mxi.(10)The inequality (10), which is actually an extension ofthe Ho¨lder-Minsowski inequality, plays a key role in es-tablishing the LMI framework. Based on inverse Ho¨lder-Minsowski inequality, we might know how conservativethe relaxation is. It would be interesting to further im-prove (10) so that the resulting conservatism can be re-duced.Theorem 1 Consider the closed-loop piecewise linearsystem (8)-(9) with known actuator failure parametermatrix F and a prescribed H∞ performance index γ > 0.If there exist matrices Xl > 0, Q¯ > 0 and K¯l such thatthe following linear matrix inequalitiesMl :=Θl11 ∗ ∗ ∗ ∗0 − 1µ¯Q¯ ∗ ∗ ∗0 0 −γ2I ∗ ∗Θl41 DlXl El −Xj ∗ClXl 0 0 0 −I< 0, (11)for (l, j) ∈ Ω hold, where Θl11 = −Xl + µ¯Q¯, Θl41 =AlXl + BlFK¯l, then the closed-loop system (8)-(9) isexponentially stable with disturbance attenuation level γ.In this case, the desired controller is given as follows:Kl = K¯lX−1l . (12)Proof : Let Pl = X−1l and define the following Lyapunov-Krasovskii functional candidate:V (k) = xT (k)Plx(k) +∞∑d=1µdk−1∑τ=k−dxT (τ)Qx(τ). (13)Given the set Ω representing possible transitions fromone region to itself or another, the difference of V (k) canbe calculated as follows:∆V (k) = V (k + 1)− V (k)=[Alx(k) +Dl∞∑d=1µdx(k − d) + Elv(k)]TPj×[Alx(k) +Dl∞∑d=1µdx(k − d) + Elv(k)]−xT (k)Plx(k) + xT (k)µ¯Q)x(k)−∞∑d=1µdxT (k − d)Qx(k − d). (14)2From Lemma 1, it can be easily seen that−∞∑d=1µdxT (k − d)Qx(k − d)≤ −1µ¯(∞∑d=1µdx(k − d))TQ(∞∑d=1µdx(k − d)),(15)where µ¯ is defined in (3). It follows from (14) and (15)that∆V (k)≤[Alx(k) +Dl∞∑d=1µdx(k − d) + Elv(k)]TPj×[Alx(k) +Dl∞∑d=1µdx(k − d) + Elv(k)]−1µ¯(∞∑d=1µdx(k − d))TQ(∞∑d=1µdx(k − d))−xT (k)Plx(k) + µ¯xT (k)Qx(k). (16)Denoting the following matrix variables:ξ(k) =[xT (k)∞∑d=1µdxT (k − d) vT (k)]T,ζ(k) =[xT (k)∞∑d=1µdxT (k − d)]T, (17)it follows from (16) that ∆V (k) ≤ ζT (k)Πl1ζ(k), whereΠl1 :=[−Pl + µ¯Q+ATl PjAl ∗DTl PjAl −1µ¯Q+DTl PjDl].With the given controller parameters in (12), it followsthat (11) is true. Therefore, we know from Schur comple-ment lemma that ∆V (k) < 0. It can now be concludedfrom Lemma 1 of [6] that the discrete-time piecewise lin-ear systems (8)-(9) with v(k) = 0 is exponentially stable.Let us now prove the H∞ performance. Assuming zeroinitial condition, we have from (16) that:JN =N∑k=0[zT (k)z(k)− γ2vT (k)v(k)]≤N∑k=0[zT (k)z(k)− γ2vT (k)v(k) + ∆V (k)]≤ ξT (k)Πl2ξ(k), (18)whereΠl2 :=Πl211 ∗ ∗DTl PjAl −1µ¯Q+DTl PjDl ∗ETl PjAl ETl PjDl −γ2I + ETl PjEl .with Πl211 := −Pl + µ¯Q + ATl PjAl + CTl Cl. Pre- andpost-multiplying (11) by diag{Pl, Pl, I, I, I}, we canconclude from Schur complement that JN < 0. Basedon Definition 1, the proof is complete.It is worth mentioning that, when F ≡ I in Theorem 1,the corresponding results for the “no-fault” case can beobtained easily. Furthermore, when F is unknown butsatisfies the constraints (4)-(7), we have the followingtheorem.Theorem 2 Consider the closed-loop discrete-timepiecewise linear system (8)-(9). For a prescribed con-stant γ > 0, if there exist positive definite matricesXl > 0, Q¯ > 0, a diagonal matrix R > 0 and a matrixK¯l such that the following linear matrix inequalitiesΘl11 ∗ ∗ ∗ ∗ ∗ ∗0 − 1µ¯Q¯ ∗ ∗ ∗ ∗ ∗0 0 −γ2I ∗ ∗ ∗ ∗Θ¯l41 DlXl El −Xj ∗ ∗ ∗0 0 0 RBTl −R ∗ ∗ClXl 0 0 0 0 −I ∗K¯l 0 0 0 0 0 −RF˜−2< 0, (19)for (l, j) ∈ Ω hold, where Θ¯l41 := AlXl + BlF0K¯l andΘl11 has been defined in Theorem 1, then the resultingdiscrete-time piecewise linear system (8)-(9) is exponen-tially stable with disturbance attenuation γ. In this case,the parameters of the desired controller are given in (12).Proof : Noticing (7), we know that Ml in (11) can berewritten asMl :=Ml0 + [0, 0, 0, BTl , 0]T∆[K¯l, 0, 0, 0, 0]+[K¯l, 0, 0, 0, 0]T∆[0, 0, 0, BTl , 0], (20)whereMl0 :=−Xl + µ¯Q¯ ∗ ∗ ∗ ∗0 − 1µ¯Q¯ ∗ ∗ ∗0 0 −γ2I ∗ ∗AlXl +BlF0K¯l DlXl El −Xj ∗ClXl 0 0 0 −I.From (7) and the elementary inequality xT y + yTx ≤εxTx+ ε−1yT y, we haveMl ≤Ml0 + [0, 0, 0, BTl , 0]TR[0, 0, 0, BTl , 0]+[K¯l, 0, 0, 0, 0]TR−1F˜ 2[K¯l, 0, 0, 0, 0]=Θ˜l11 ∗ ∗ ∗ ∗0 − 1µ¯Q¯ ∗ ∗ ∗0 0 −γ2I ∗ ∗Θ¯l41 DlXl El −Xj +BlRBTl ∗ClXl 0 0 0 −I,3where Θ˜l11 := −Xl + µ¯Q¯+ K¯Tl R−1F˜ 2K¯l.Pre- andPost-Multiplying (19) by diag{I, I, I, I, R−1, I,I} and by Schur complement, we have Ml < 0 forl, j = 1, . . . , L and therefore we can know from Theorem1, (19) and (12) that the system (8)-(9) is exponentiallystable with the prescribed disturbance attenuation levelγ and given controller parameters in (12).Remark 2 As in [1] and the references therein, theregion partition information {Sl}l∈L is assumed to beknown. In the case that the set Ω can be determined inthe analysis, the number of LMIs involved in Theorem 1can be reduced. However, for PWL systems, it is gener-ally difficult to determine the set before the control lawis designed. A possible solution is to design the controllaw and switching set simultaneously, and this gives riseto a challenging topic for future research.Remark 3 The kernel information µd for the dis-tributed delays is included in the conditions in Theorem1. Since the distributed delays considered are infinite, it istheoretically difficult to make our main results dependenton the individual time-delays. Nevertheless, if we con-sider bounded distributed delay described by∑d˜d=1 x(k−d) and construct appropriate Lyapunov functional suchas V (k) = xT (k)Plx(k) +∑d˜d=1∑k−1τ=k−d xT (τ)Qx(τ),delay-dependent conditions could be established, therebyreducing the conservatism with respect to the time-delays.4 Numerical ExampleConsider a piecewise linear systems (1)-(2) with L ,{1, 2}, where l = 1 if x1(k) ≥ 0 and l = 2 if x1(k) < 0.The model parameters are given as follows:A1 =0.81 0.3 0.5−0.2 0 −0.30.1 0 0.81 , A2 =0.5 0.2 0.2−0.2 0 −0.40.2 0 0.5 ,D1 =0.1 0 0.10.1 0 0.030 0 0.17 , D2 =0.1 0 0.10.1 0 0.30.1 0.4 0.17 ,Bi =[0.5 0.5 0.5]T, Ei =[0.5 0 0.5]T, (i = 1, 2)C1 = [0.36 0.15 0.25], C2 = [0.36 0.5 0.25],γ2 = 1, F = 0.1, F¯ = 0.9.According to Theorem 2, by using the LMI toolbox, thecontroller parameters can be calculated as follows:K1 = [−0.9058 − 0.4009 − 1.0941],K2 = [−0.8734 − 0.2880 − 0.7357].Fig. 1 depicts the state responses for the uncontrolledsystems, which are apparently unstable. Fig. 2 gives thesimulation results of the responses of the closed-looppiecewise linear systems, which confirm that the closed-loop system is indeed stable.References[1] G. Feng, Stability analysis of piecewise discrete-time linearsystems, IEEE Trans. Automatic Control, vol. 47, no. 7, pp.1108-1112, 2002.0 50 100−2.5−2−1.5−1−0.500.511.52kx 1(k)0 50 100−0.6−0.4−0.200.20.40.60.8kx 2(k)0 50 100−1.5−1−0.500.51kx 3(k)Fig. 1. The state evolution x(k) of uncontrolled systems0 50 100−3−2.5−2−1.5−1−0.500.511.5kx 1(k)0 50 100−1.5−1−0.500.511.522.5kx 2(k)0 50 100−2−1.5−1−0.500.51kx 3(k)Fig. 2. The state evolution x(k) of controlled systems[2] H. Gao, C. Wang and J. Wang, A delay-dependent approachto robust H∞ filtering for uncertain discrete-time state-delayed systems, IEEE Transactions on Signal Processing,vol. 52, no. 6, pp. 1631-1640, 2004.[3] Y. Kuang, H. Smith, and R. Martin, Global stabilityfor infinite-delay, dispersive Lotka-Volterra systems: weaklyinteracting populations in nearly identical patches, Journal ofDynamics and Differential Equations, vol. 3, no. 3, pp. 339-360, 1991.[4] Y. Liu, Z. Wang, J. Liang and X. Liu, Synchronizationand state estimation for discrete-time complex networkswith distributed delays, IEEE Trans. Systems, Man, andCybernetics - Part B, vol. 38, no. 5, pp. 1314-1325, 2008.[5] R. J. Veillette, J. V. Medanic, and W. R. Perkins, Designof reliable control systems, IEEE Trans. Automatic Control,vol. 37, pp. 293–304, 1992.[6] Z. Wang, F. Yang, D. W. C. Ho and X. Liu, RobustH∞ filtering for stochastic time-delay systems with missingmeasurements, IEEE Trans. Signal Processing, vol. 54, no. 7,pp. 2579-2587, Jul. 2006.[7] J. Xu and L. Xie, Non-synchronized H∞ estimation ofpiecewise linear systems, In: Proc. 1st IEEE Conferenceon Industrial Electronics and Applications, Singapore, May,2006, pp. 1-6.[8] G. H. Yang, J. L. Wang, and Y. C. Soh, Reliable H∞controller design for linear systems, Automatica, vol. 37,pp. 717–725, 2001.4

Stability analysis of piecewise discrete-time linear systems,

Reliable H∞ control for discrete-time piecewise linear systems with infinite distributed delays

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Reliable H∞ control for discrete-time piecewise linear systems with infinite distributed delays

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