Resilient Dynamic Output Feedback Control for ...

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Resilient Dynamic Output Feedback Control for Discrete-Time Descriptor Switching Markov Jump Systems and Its Applications Jimin Wang · Shuping Ma

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Abstract This paper investigates the resilient dynamic output feedback (DOF) control problem for discrete-time descriptor switching Markov jump systems for the first time, where the time-varying transition probabilities (TPs) are described by a piecewise-constant matrix and a high-level signal subject to average dwell time switching. The controllers to be designed can tolerate additive gain perturbations. Firstly, by constructing a stochastic Lyapunov functional and using an average dwell time method, a sufficient condition is given such that the resultant closed-loop systems are stochastically admissible and have a H∞ noise attenuation performance. Then, based on the matrix inequality decoupling technique, a novel linear matrix inequality (LMI) condition is presented such that the resultant closed-loop systems are stochastically admissible with a H∞ noise attenuation performance. When the uncertain parameters exist not only in plant matrices but also in controller gain matrices, the resilient DOF controller is developed in terms of LMIs, which can be of full-order or reduced-order. Compared with the previous ones, the proposed design methods don’t impose extra constraints on system matrices or slack variables, which show less conservatism. Finally, numerical examples are given to illustrate the superiority and applicability of the new obtained methods. Keywords DOF control · Resilient controllers · Discretetime descriptor Markov jump systems · Average dwell time (ADT) switching J. Wang School of Mathematics, Shandong University, Jinan, 250100, China E-mail: [email protected] S. Ma School of Mathematics, Shandong University, Jinan, 250100, China E-mail: [email protected] Shuping Ma is the corresponding author.

1 Introduction Many dynamical systems may happen some unexpected structural conversions, such as random faults, abrupt failures, and so forth, which are suitable to be modeled by Markov jump systems (MJSs) [1]. In the past few decades, the rapid development on such system has been witnessed due to its huge application in many real-word systems. The analysis and design of MJSs have attracted wide publicity, such as stability analysis, controller design, H∞ filtering problem, and so forth [2-11]. It should be mentioned that the most dominant factor of MJSs is the TPs of the jump process, which determines the behaviors of the system. Most of the aforementioned results were concerned with time-homogeneous MJSs, and lots of controller design results were obtained based on the assumption that all the terms of the TPs are time-invariant, including completely known TPs, partly known TPs, completely unknown TPs [6-11]. For example, the analysis and synthesis for MJSs with incomplete TPs were studied in [6]; the control and filtering issues for MJSs with uncertain TPs were given in [7-8]. Recently, a polytope set was used to describe the characteristics of time-varying TP-based uncertainties for MJSs in [9-11]. What should be pointed out is that, the most interesting research topic is to study piecewise homogeneous MJSs due to the elegant application in our real life, the networked control systems are taken as a typical example. As is known to all, the stochastic signals can be used to model the packet dropouts and networkinduced delays in the network environment, which may occur frequently in different periods [12]. In [12], authors considered the robust control synthesis of networked control systems, where random delays existing in both controller-toactuator and sensor-to-controller communication links are modeled as stochastic processes, the results on MJSs with one Markov chain are not applicable. So, it is significant

2

and necessary to study a more general jump model with two stochastic signals. In response to this challenge, a preliminary study of this topic has been carried out, the arbitrary switching signal and stochastic switching signal in the higher layer of time-varying TPs have been investigated in [13,14]. Recently, the high-level deterministic switching affected by ADT constraint was studied in [15,16,17,18], which only reflects on the TP matrix. In order to satisfy the real engineering challenges, the switching MJSs were introduced in [19], where the high-level ADT switching affected not only the TP matrix but also the system state. Recently, the stability analysis and controller design for switching MJSs have been studied in [20-24]. In [24], the problem of finite-time analysis and control for switching Markov jump neural networks with additive time varying delays was studied. However, all these results were concerned with the regular switching MJSs. As far as we know, this type of descriptor system has not been addressed yet, which is one of the motivations in this paper. On another research front, descriptor systems, also mean singular systems, differential-algebraic systems, generalized state-space systems or semi-state systems, appear in many practical systems, such as biological systems, network control systems, economic systems, power systems, and so on. They have been widely investigated in the past decades [25, 26,27]. When the abrupt structural changes occur in the descriptor plants, they can be modeled by the general descriptor MJSs (DMJSs). Such models have been well found in the practical engineering systems, such as electrical systems, biological systems and so on. As a result, the analysis and design theories for DMJSs have been developed greatly, such as stability analysis, controller design, and H∞ filtering problem [28-35]. Especially, the static output feedback (SOF) control issue for DMJSs was involved in [29,33-35]. Unfortunately, in order to design the controllers via the existing LMI methods, special structures must be imposed on the system output matrices or slack variables in these references. Practically, these restrictions may largely increase conservativeness. How to avoid these constraints is one important idea of this paper. More recently, for DMJSs with time-varying TPs, the stability analysis and controller design were discussed in [36-39]. In [39], with the ADT conditions, the finite-time boundedness and state feedback stabilization for nonlinear DMJSs with time-varying TPs were studied. Up to now, the DOF controller design problem for such systems has not been investigated, especially when the parameter uncertainties exist in both plant matrices and controller matrices, which is another motivation behind this study. In this paper, we design the resilient DOF controller for a type of discrete-time switching DMJSs with the ADT methods, where the uncertainties exist in both plant matrices and controller matrices. Firstly, based on a stochastic

Jimin Wang, Shuping Ma

Lyapunov functional and a ADT condition, a sufficient condition is given such that the resultant closed-loop systems are stochastically admissible with a H∞ noise attenuation performance index. Then by using the matrix inequality decoupling technique, the DOF controller is developed in terms of LMIs, where either the rank constraint on output matrices or special structures of a few key matrices in the existing literatures are overcome. Last, two numerical examples are provided to show the efficiency of the proposed methods. Notations: Throughout this paper, the following notations are used. Table 1 The notations and descriptions Notations

Descriptions

X ≥ 0 (X > 0)

The semi-positive definite (positive definite) matrix

I

The identity matrix with appropriate dimensions

0

The zero matrix with appropriate dimensions

(·)T

The transpose of a matrix

diag{· · · }

A block-diagonal matrix

∥x∥

Euclidean norm of the vector x

E[·]

The mathematical expectation

∗

An ellipsis for the terms that are introduced by symmetry

sym(X )

X + XT

⋆

Matrices that are not relevant in the discussion

l2 [0, ∞)

The space of summable infinite sequence over [0, ∞)

2 Preliminaries Consider the following discrete-time switching DMJSs:  Ex(k + 1) = [A(rk , θk ) + ∆A(rk , θk , k)]x(k)     +[B(rk , θk ) + ∆B(rk , θk , k)]u(k)     +F (rk , θk )ω(k),  z(k) = [C(rk , θk ) + ∆C(rk , θk , k)]x(k) (1)    +[D(r , θ ) + ∆D(r , θ , k)]u(k) k k k k     +Dω (rk , θk )ω(k),   y(k) = H(rk , θk )x(k) + Hω (rk , θk )ω(k), where x(k) ∈ Rn is the system state, u(k) ∈ Rp is the control input, ω(k) ∈ Rv is the disturbance signal that belongs to l2 [0, ∞); Switching law {rk , k ≥ 0} is a discretetime Markov stochastic process taking values in a finite state space J = {1, 2, ..., J}, the evolution of {rk , k ≥ 0} is governed by the following TPs: Pr{rk+1 = j|rk = i} = µθijk ,

Resilient Dynamic Output Feedback Control for Discrete-Time Descriptor Switching Markov Jump Systems and Its Applications

where µθijk ≥ 0, ∀ i, j ∈ J with

J ∑ j=1

µθijk = 1. Here,

µθijk is a function of θk , θk is a high-level ADT switching signal to determine the time-varying property. Furthermore, θk is assumed to take values independently in another finite state space M = {1, 2, ..., M }. The time sequence k0 < k1 < · · · kl < kl+1 < · · · represents every switching instant of θk and k0 is also denoted as initial time, θk is continuous from right everywhere. When k ∈ [kl , kl+1 ), it means that θkl th transition probability matrix is active. The matrix E ∈ Rn×n is singular with rank(E) = r ≤ n. For every i ∈ J , α ∈ M, rk = i, θk = α, A(rk , θk ) = Ai,α , B(rk , θk ) = Bi,α , F (rk , θk ) = Fi,α , H(rk , θk ) = Hi,α , Hω (rk , θk ) = Hωi,α , C(rk , θk ) = Ci,α , D(rk , θk ) = Di,α , Dω (rk , θk ) = Dωi,α , ∆A(rk , θk , k) = ∆Ai,α , ∆B(rk , θk , k) = ∆Bi,α , ∆C(rk , θk , k) = ∆Ci,α , ∆D(rk , θk , k) = ∆Di,α . Ai,α , Bi,α , Fi,α , Hi,α , Hωi,α , Ci,α , Di,α , Dωi,α are known compatible dimension constant matrices. The uncertainties are considered with the following form: [ ] [ ] [ ] ∆Ai,α ∆Bi,α M1i,α = Γi,α N1i,α N2i,α , (2) ∆Ci,α ∆Di,α M2i,α where M1i,α , M2i,α , N1i,α , N2i,α are known real constant matrices with appropriate dimensions, Γi,α is an unknown T Γi,α ≤ I. The TP matrix is further real matrix satisfying Γi,α defined by:   (θ ) (θ ) (θ ) µ11k µ12k · · · µ1Jk  (θk ) (θk ) (θ )   µ21 µ22 · · · µ2Jk  Π (θk ) =  (3) .. .. ..   .. .  . . . .  (θ )

(θ )

(θ )

µJ1k µJ2k · · · µJJk

Remark 1. As pointed out in [19-24], the proposed timevarying TPs have the characteristic of a hierarchical structure, where a Markov chain rk is in the lower layer and a switching signal θk is in the higher layer. Such hierarchical structure provides the potential regulation ability to satisfy the real engineering challenges, for example, a DC motor device [23,39], neural networks [24]. It should be noted that when the switching signal in the higher layer disappears, then the problem studied in this paper reduces to the general DMJSs studied in [28,29,31,33,35]. In this paper, we design the resilient DOF controllers as follows:  c x (k + 1) = [Ac (rk , θk ) + ∆Ac (rk , θk , k)]xc (k)    +[B c (rk , θk ) + ∆B c (rk , θk , k)]y(k), u(k) = [C c (rk , θk ) + ∆C c (rk , θk , k)]xc (k)    +[Dc (rk , θk ) + ∆Dc (rk , θk , k)]y(k), (4) where xc (k) ∈ Rm is the controller state with m ≤ n. c c c Aci,α , Bi,α , Ci,α and Di,α are designed to be the parameter c c matrices of appropriate dimensions. ∆Aci,α , ∆Bi,α , ∆Ci,α ,

3

c ∆Di,α represent the additive type of gain variations with the following form: [ ] [ ] c ] [ ∆Aci,α ∆Bi,α M3i,α c = Γi,α N3i,α N4i,α , (5) c c ∆Ci,α ∆Di,α M4i,α

where M3i,α , M4i,α , N3i,α , N4i,α are known real constant c matrices with appropriate dimensions, Γi,α is an unknown c T c real matrix satisfying (Γi,α ) Γi,α ≤ I. Remark 2. The resilient DOF controller developed in (4) can be of full-order or reduced-order. It is of full-order when m = n, while reduced-order when m < n. In addition, note that controller (4) reduces to SOF controller when m = 0, c c in this case, there is only Di,α + ∆Di,α . Especially, when m = 0 and the high-level switching signal disappears, the problem discussed in the paper reduces to the SOF control issue for discrete-time DMJSs given in [29,33,35]. Define [ ]T , (6) η(k) = xT (k) (xc (k))T applying controller (4) into system (1), then the corresponding closed-loop system can be written as { b Eη(k + 1) = Ai,α η(k) + Bi,α ω(k), (7) z(k) = Ci,α η(k) + Di,α ω(k), b Ai,α , Bi,α , Ci,α , Di,α are defined in the following where E, form (see (8)). Remark 3. The resilient control problem considered in this paper means that the uncertain parameters coexist in all plant matrices and controller gains, which are more complicated than the ones studied in [14,27]. Further, if the controller to be designed has no additive gain perturbations, the problem studied in this paper reduces to the general robust controller design problem studied in [4, 28, 31]. Note that, employing the technique presented in [5], the problem for designing a DOF controller gain of order m is reformulated as the search for a SOF controller gain. In the following, define [ ] [ ] Ai,α + ∆Ai,α 0 0 I Ai,α = , Hi,α = , 0 0] [ [ Hi,α ]0 0 Bi,α + ∆Bi,α Fi,α Bi,α = , Fi,α = , I 0 0 [ ] [ ] 0 Hωi,α = , Ci,α = Ci,α + ∆Ci,α 0 , H ωi,α [ ] Di,α = 0 Di,α + ∆Di,α , Dωi,α = Dωi,α . Then, the matrices in (8) are rewritten as [ ] b = E 0 , Ai,α = Ai,α + Bi,α Ki,α Hi,α , E 0 I Bi,α = Fi,α + Bi,α Ki,α Hωi,α , Ci,α = Ci,α + Di,α Ki,α Hi,α , Di,α = Dωi,α + Di,α Ki,α Hωi,α ,

(9)

4

Jimin Wang, Shuping Ma

[

] c c c c Ai,α + ∆Ai,α + (Bi,α + ∆Bi,α )(Di,α + ∆Di,α )Hi,α (Bi,α + ∆Bi,α )(Ci,α + ∆Ci,α ) Ai,α = , c c (Bi,α + ∆Bi,α )Hi,α Aci,α + ∆Aci,α [ ] [ ] c c b = E 0 , Bi,α = Fi,α + (Bi,α + ∆Bi,α )(Di,α + ∆Di,α )Hωi,α , E c c (Bi,α + ∆Bi,α )Hωi,α ] [0 I c c c c ) , + ∆Ci,α + ∆Di,α )Hi,α (Di,α + ∆Di,α )(Ci,α Ci,α = Ci,α + ∆Ci,α + (Di,α + ∆Di,α )(Di,α c c )Hωi,α . Di,α = Dωi,α + (Di,α + ∆Di,α )(Di,α + ∆Di,α

with

[ Ki,α =

]

E{

∞ ∑

∥ η(k) ∥2 | η(k0 ), rk0 , θk0 } ≤ T (η(k0 ), rk0 , θk0 ).

k=k0

(iii) System (7) with ω(k) = 0 is said to be stochastically admissible if it is regular, causal and mean-square stochastically stable. Definition 2 For given a prescribed level of noise attenuation γ > 0, system (7) is said to be stochastically admissible with a guaranteed H∞ performance γ, if system (7) with ω(k) = 0 is stochastically admissible, and under zero initial condition, for all uncertainties satisfying (2) and (5), the following H∞ performance holds } { ∞ ∞ ∑ ∑ T ω T (k)ω(k). (10) z (k)z(k) < γ 2 E k=k0

(ii) there exist real matrices W , X and Y such that

c c Aci,α + ∆Aci,α Bi,α + ∆Bi,α . c c c c Ci,α + ∆Ci,α Di,α + ∆Di,α

Definition 1 [25,36] (i) System (7) with ω(k) = 0 is said to be regular and causal, b Ai,α ) are regular and causal. if the pairs (E, (ii) System (7) with ω(k) = 0 is said to be mean-square stochastically stable for any initial state η(k0 ) and initial mode rk0 , θk0 , there exists a scalar T (η(k0 ), rk0 , θk0 ) > 0 such that the following inequality holds:

W < 0 and W + X T Y + Y T X < 0. Lemma 2 [14] Let X, Y and Σ be any real matrices with appropriate dimensions and Σ T Σ ≤ I, then XΣY + Y T Σ T X T ≤ ϵ−1 XX T + ϵY T Y, ∀ϵ > 0.

3 Main results In this section, the resilient DOF control problem for system (1) is investigated. To this end, we first present a sufficient condition such that system (7) with uncertainties c = 0, is stochastically admissible and has Γi,α = 0 and Γi,α a H∞ noise attenuation performance. Theorem 1 System (7) with uncertainties Γi,α = 0 and c = 0 is stochastically admissible with a H∞ noise Γi,α attenuation performance index, if for given scalars µ > 1, 0 < δ < 1, there exist symmetric matrix Pbi,α > 0, matrices Mi,α , Ni,α , Qi,α , Ti,α , ∀i, j ∈ J , α, β ∈ M, and a scalar γ > 0, such that 11 12 T  Wi,α Wi,α Mi,α Bi,α Ci,α 22  ∗ Wi,α Ni,α Bi,α 0    T  < 0,  ∗ ∗ −γ 2 I Di,α ∗ ∗ ∗ −I



k=k0

Definition 4 [23,39] For any time T2 > T1 ≥ 0, let Nσ (T1 , T2 ) denotes the number of switching of σk over 1 (T1 , T2 ). If Nσ (T1 , T2 ) ≤ N0 + T2τ−T holds for N0 ≥ a 0, τa > 0, then τa is called the average dwell-time, and N0 is the chattering bound. The objective of the paper is to design the resilient DOF controller in the form of (4) for system (1) such that system T (7) with any uncertainties satisfying Γi,α Γi,α ≤ I and c T c (Γi,α ) Γi,α ≤ I, is stochastically admissible and has a H∞ noise attenuation performance. Before ending this section, the following useful lemmas are introduced. Lemma 1 [40] The following conditions involving real scalar ε and real matrices W , X, Y and Z are equivalent: (i) there exist real scalar ε and real matrices W , X, Y and Z such that [ ] W Y T − εX T Z T < 0; ∗ εZ + εZ T

(8)

Pbi,α ≤ µPbi,β ,

(11)

(12)

if (11)-(12) are feasible, then the ADT of the high-level switching signal θk satisfies: µ τa > τa∗ = − ln ln δ ,

(13)

where 11 b +E b T Xi,α E b − δE b T Pbi,α E, b Wi,α = sym(Mi,α (Ai,α − E)) 12 T T T T T b N +E b Xi,α + Q S , Wi,α = −Mi,α + (Ai,α − E) i,α i,α i,α 22 T T T Wi,α = −Ni,α − Ni,α + Xi,α + Si,α Ti,α + Ti,α Si,α , J ∑ α b Xi,α = µij Pj,α , j=1

Si,α ∈ R(m+n)×(n−r) is the arbitrary matrix satisfying b T Si,α = 0 and rank (Si,α ) = n − r. E

Resilient Dynamic Output Feedback Control for Discrete-Time Descriptor Switching Markov Jump Systems and Its Applications

Proof. Firstly, we prove that system (7) with uncertainc ties Γi,α = 0 and Γi,α = 0 is regular and causal. From (11), it follows that [ 11 12 ] Wi,α Wi,α (14) 22 < 0, ∗ Wi,α we rewrite (14) as the following equivalent form: ei,α Aei,α + sym{AeT Sei,α Q e i,α } − δ E e T Pei,α E e < 0, AeTi,α X i,α (15) where [ ] [ ] b b0 E I E e= E , Aei,α = b −I , 0 0 Ai,α − E [ ] [ ] Qi,α Ti,α Pbi,α 0 e Pei,α = , Qi,α = , T T Mi,α Ni,α [ 0 0] Si,α 0 . Sei,α = 0 I ei,α = Since X

J ∑ j=1

e b e µα ij Pj,α and Pi,α > 0, we have Xi,α ≥ 0,

together with (15), it yields that e < 0. e i,α } − δ E e T Pei,α E sym{AeTi,α Sei,α Q

(16)

e = m + r, there exist two nonsingular Since rank(E) matrices P, V ∈ R(2(m+n))×(2(m+n)) such that ] [ [ ] Ae1i,α Ae2i,α Im+r 0 e e , P EV = , P Ai,α V = e3 e4 0 0 Ai,α Ai,α ] ] [ [ 1 1 e2 Sei,α Pi,α Pei,α −T e −T e −1 , , P Si,α = e2 P Pi,α P = 3 Si,α ∗ Pei,α ] [ e i,α V = Q e2 . e1 Q Q i,α i,α (17) e T Sei,α = 0, it follows that Se1 = 0, namely, From E i,α [ ] 0 2 ∈ R(m+2n−r)×(m+2n−r) , P −T Sei,α = e2 , Sei,α Si,α 2 rank(Sei,α ) = m + 2n − r. (18) T Pre- and post-multiply (16) by V and V , along with (17) and (18), it is obtained that [ ] ⋆ ⋆ < 0, (19) 2 e2 ⋆ sym{(Ae4i,α )T Sei,α Qi,α } 2 e2 which further implies that sym{(Ae4i,α )T Sei,α Qi,α } < 0, 4 then Aei,α is nonsingular. From Definition 1, it follows that e Aei,α ) is regular and causal. Further, based on the pair (E, b − Ai,α )= det (z E e − Aei,α ) for every i ∈ the fact that det(z E

J , α ∈ M, it is obtained that system (7) with uncertainties c Γi,α = 0 and Γi,α = 0 is regular and causal. Next, when ω(k) = 0, we prove that system (7) with c uncertainties Γi,α = 0 and Γi,α = 0 is mean-square

5

stochastically stable. We construct the Lyapunov functional as : b T Pbi,α Eη(k), b V (η(k), rk , θk ) = η T (k)E (20) where Pbi,α > 0. We have E[V (η(k + 1), rk+1 , θk )|rk , θk ] b T Xi,α Eη(k b = η T (k + 1)E + 1). b b Let y(k) = Eη(k + 1) − Eη(k), one has E[V (η(k + 1), rk+1 , θk )|rk , θk ] b b = (Eη(k) + y(k))T Xi,α (Eη(k) + y(k)).

(21)

b b b Using the equations y(k) = Eη(k +1)− Eη(k) and Eη(k + 1) = Ai,α η(k), it is obtained that b (Ai,α − E)η(k) − y(k) = 0,

(22)

then it follows that [ ] T b {(Ai,α − E)η(k) − y(k)}, 0 = 2 η T (k) y T (k) Ti,α (23) ] [ T T Ni,α . For any matrix Qi,α ∈ where Ti,α = Mi,α R(n−r)×(m+n) , Si,α ∈ R(m+n)×(n−r) is any full column b T Si,α = 0, along with y(k) = rank matrix satisfying E b b Eη(k + 1) − Eη(k), one has 0 = 2y T (k)Si,α [Qi,α η(k) + Ti,α y(k)].

(24)

Adding (23) and (24) into (21), it is obtained that ei,α ϑT (k), E[V (η(k + 1), rk+1 , θk )|rk , θk ] = ϑ(k)Θ (25) where [ ] ϑ(k) = η T (k) y T (k) , ] [ 11 12 e ei,α = Θi,α Wi,α , Θ 22 ∗ Wi,α 11 e b +E b T Xi,α E. b Θi,α = sym{Mi,α (Ai,α − E)} From (11), it follows that E[V (η(k + 1), rk+1 , θk )|rk , θk ] ≤ δE[V (η(k), rk , θk )], summing up both sides of (23) from kl to k and taking expectations, one has E[V (η(k), rk , θk )|rkl , θkl ] ≤ δ k−kl E[V (η(kl ), rkl , θkl )]. (26) On the other hand, based on (12), at switching instants kl , it follows E[V (η(kl ), rkl , θkl )] ≤ µE[V (η(kl ), rkl , θkl−1 )]. Then, based on this, together with (26), it is obtained that E[V (η(k), rk , θk )|rkl , θkl ] ≤ δ k−kl µE[V (η(kl ), rkl , θkl−1 )],

6

Jimin Wang, Shuping Ma

one can continue the iterative procedure to obtain E[V (η(k), rk , θk )|rk0 , θk0 ] ≤ δ k−kl µE[V (η(kl ), rkl , θkl−1 )|rk0 , θk0 ] ≤ δ k−kl µ × δ kl −kl−1 E[V (η(kl−1 ), rkl−1 , θkl−1 )|rk0 , θk0 ], ≤ · · · ≤ δ k−k0 µNσ (k0 ,k) E[V (η(k0 ), rk0 , θk0 )]

thus when ω(k) = 0, system (7) with uncertainties Γi,α = 0 c and Γi,α = 0 is mean-square stochastically stable. Finally, let Jzω (k) = E[V (η(k + 1), rk+1 , θk )|rk , θk ] −δV (η(k), rk , θk ) + z T (k)z(k) −γ 2 ω T (k)ω(k).

k−k0

= δ k−k0 µN0 + τa E[V (η(k0 ), rk0 , θk0 )] 1 = µN0 (δµ τa )k−k0 E[V (η(k0 ), rk0 , θk0 )]. (27) For k = k0 , k0 + 1, · · · , k0 + Nσ (k0 , k), summing up both sides of (27) gives rise to E[

Nσ ∑ (k0 ,k)

= µN0

(28)

N (k ,k)+1

1−β1 σ 0 1−β1

Jzω (k) = φ(k)Πi,α φT (k),

(32)

where

V (η(k), rk , θk )|η(k0 ), rk0 , θk0 ]

k=k0

= µN0 (1 + β1 + β12 + · · · N (k ,k) +β1 σ 0 )E[V (η(k0 ), rk0 , θk0 )]

By a similar derivation as before, when ω(k) ̸= 0, from (20)-(25), it follows that

E[V (η(k0 ), rk0 , θk0 ],

[ ] φ(k) = η T (k) y T (k) ω T (k) ,  11  T 12 T Wi,α + Ci,α Ci,α Wi,α Mi,α Bi,α + Di,α Ci,α 22 , ∗ Wi,α Ni,α Bi,α Πi,α =  2 T ∗ ∗ −γ I + Di,α Di,α

1

where β1 = δµ τa , and from (13), it is obtained that 0 < β1 < 1. Now, set [ ] [ ] ζ1 (k) Im+r 0 −1 b ζ(k) = = I η(k), Z EI = , ζ2 (k) 0 0 ] [ 1 Ai,α A2i,α , Z Ai,α I = A3i,α[ A4i,α ] 1 b2 Pi,α Pbi,α −T b −1 . Z Pi,α Z = ∗ Pb3 i,α

From (28), we have lim

E

[ Nσ (k0 ,k) ∑

Nσ (k0 ,k)→∞

] ζ1T (k)ζ1 (k)|η(k0 ), rk0 , θk0

k=k0

≤ M (η(k0 ), rk0 , θk0 ) < ∞, (29) where N0

µ M (η(k0 ), rk0 , θk0 ) = τ 1−β V (η(k0 ), rk0 , θk0 ), 1 1 −1 τ = ( min λmin Pbi,α ) .

by using Schur complement, from (11), it follows that Jzω (k) < 0, that is E[V (η(k + 1), rk+1 , θk )|rk , θk ] (33) < δV (η(k), rk , θk ) − z T (k)z(k) + γ 2 ω T (k)ω(k), similar to the proof process of (27), from (33), we have E[V (η(k), rk , θk−1 )] < δE[V (η(k − 1), rk−1 , θk−1 )] − E[z T (k − 1)z(k − 1)] +γ 2 ω T (k − 1)ω(k − 1) < δµE[V (η(k − 1), rk−1 , θk−2 )] − E[z T (k − 1)z(k − 1)] +γ 2 ω T (k − 1)ω(k − 1) < δ 2 µE[V (η(k − 2), rk−2 , θk−2 )] −δµE[z T (k − 2)z(k − 2)] − E[z T (k − 1)z(k − 1)] +γ 2 {δµω T (k − 2)ω(k − 2) + ω T (k − 1)ω(k − 1)} < ··· < δ k−k0 µNσ (k0 ,k) E[V (η(k0 ), rk0 , θk0 )] k−1 ∑ k−l−1 Nσ (l,k) T −E{ δ µ z (l)z(l)} +γ 2

i∈J ,α∈M

l=k0 k−1 ∑

δ k−l−1 µNσ (l,k) ω T (l)ω(l).

l=k0

Further, system (7) with ω(k) = 0 is decomposed as the following equivalent form ζ1 (k + 1) = A1i,α ζ1 (k) + A2i,α ζ2 (k), 0 = A3i,α ζ1 (k) + A4i,α ζ2 (k).

(30)

From the second equation in (30), it follows that ζ2 (k) = −(A4i,α )−1 A3i,α ζ1 (k), along with (29), it is obtained that lim Nσ (k0 ,k)→∞

E

[ Nσ (k0 ,k) ∑

(34) Based on this, under the zero initial condition, it is obtained that 0 ≤ E[V (η(k), rk , θk−1 )] k−1 ∑ k−l−1 Nσ (l,k) T < −E{ δ µ z (l)z(l)} (35) l=k0 k−1 ∑ +γ 2 δ k−l−1 µNσ (l,k) ω T (l)ω(l), l=k0

] which implies that

η T (k)η(k)|η(k0 ), rk0 , θk0

k=k0

≤ T (η(k0 ), rk0 , θk0 ) < ∞, (31)

E{

k−1 ∑

l=k0

β1k−l z T (l)z(l)} < γ 2

k−1 ∑ l=k0

β1k−l ω T (l)ω(l), (36)

Resilient Dynamic Output Feedback Control for Discrete-Time Descriptor Switching Markov Jump Systems and Its Applications

from (36), it is further obtained that ∞ ∑


1, 0 < δ < 1, β, there exist symmetric matrix Pbi,α > 0, matrices Mi,α , Ni,α , Qi,α , Ti,α , Li,α , Si,α , ∀i, j ∈ J , α, β ∈ M, and a scalar γ > 0, such that (12) and  11 12  13 14 15 Θi,α Θi,α Θi,α Θi,α Θi,α 22 23 25  ∗ Wi,α  Θi,α 0 Θi,α   2 34 T T  Θi,α =  ∗ ∗ −γ I Θ H S i,α ωi,α i,α    ∗ ∗ ∗ −I βDi,α − βDi,α Li,α  ∗

∗

∗

∗

7

−βLi,α − βLTi,α < 0, (38)

11 b + Bi,α Si,α Hi,α ) Θi,α = sym(Mi,α (Ai,α − E) T b T b b b b +E Xi,α E − δ E Pi,α E, T 12 T T b Θi,α = −Mi,α + (Ai,α − E) Ni,α + QTi,α Si,α b T Xi,α + H T S T B T , +E i,α i,α i,α 13 Θi,α = Mi,α Fi,α + Bi,α Si,α Hωi,α , T T T T 14 , Si,α Di,α + Hi,α = Ci,α Θi,α 15 T T Θi,α = β(Mi,α Bi,α − Bi,α Li,α ) + Hi,α Si,α , 23 Θi,α = Ni,α Fi,α + Bi,α Si,α Hωi,α , 25 = βNi,α Bi,α − βBi,α Li,α , Θi,α 34 T T T T Θi,α = Dωi,α + Hωi,α Si,α Di,α ,

Si,α ∈ R(m+n)×(n−r) is the arbitrary matrix satisfying b T Si,α = 0 and rank (Si,α ) = n − r. Then the modeE dependent DOF controller is given by ] [ c c Ai,α Bi,α = L−1 Ki,α = c c i,α Si,α . Ci,α Di,α c = Proof. Firstly, under the uncertainties Γi,α = 0 and Γi,α 0, substituting (9) into inequality (11), it follows that  11 12  13 Πi,α Πi,α Πi,α (Ci,α + Di,α Ki,α Hi,α )T 22 23  ∗ Wi,α  Πi,α 0   2 T  < 0,  ∗ ∗ −γ I (Dωi,α + Di,α Ki,α Hωi,α ) ∗ ∗ ∗ −I (39) where 11 b + Bi,α Ki,α Hi,α )) Πi,α = sym(Mi,α (Ai,α − E T T b b Xi,α E b − δE b Pbi,α E, +E 12 T b Πi,α = −Mi,α + (Ai,α − E + Bi,α Ki,α Hi,α )T Ni,α T T T b Xi,α , +Qi,α Si,α + E 13 Πi,α = Mi,α (Fi,α + Bi,α Ki,α Hωi,α ), 23 Πi,α = Ni,α (Fi,α + Bi,α Ki,α Hωi,α ),

then, decoupling some product terms, (39) is equivalent to the following inequality (see (40)). Further, (40) is equivalent to T + L T Y + Y T L < 0, (41) where



  11 12 13 14  T T Θi,α Θi,α Θi,α Θi,α Hi,α Ki,α 22 23  ∗ Wi,α   Θi,α 0  0 T   T = = T T  2 34  , L  ∗  Hωi,α Ki,α ∗ −γ I Θi,α 0 ] [ ∗ T ∗ T ∗ T−I T T T Ni,α − Li,α Bi,α 0 Di,α − LTi,α Di,α Y = Y1 Bi,α , T T T Y1 = Bi,α Mi,α − LTi,α Bi,α .

Applying Lemma 1 to (38), then it is obtained that (41) holds, which further implies (11) holds. Therefore, based on Theorem 1, if (12) and (38) hold, system (7) with c uncertainties Γi,α = 0 and Γi,α = 0 is stochastically

8

Jimin Wang, Shuping Ma



  11 12 T  T T Ξi,α Ξi,α Mi,α Fi,α Ci,α ( Hi,α Ki,α ) 22  ∗ Wi,α  [ T ] Ni,α Fi,α 0  0 T T T T   + sym   Bi,α Mi,α Bi,α Ni,α 0 Di,α < 0, T T  T  ∗   Hωi,α Ki,α ∗ −γ 2 I Dωi,α 0 ∗ ∗ ∗ −I where

(40)

11 b b +E b T Xi,α E b − δE b T Pbi,α E, = sym(Mi,α (Ai,α − E)) Ξi,α 12 T T T T T b b Ξi,α = −Mi,α + (Ai,α − E) Ni,α + Qi,α Si,α + E Xi,α ,

c Θi,α + sym(O1 Γi,α P1 ) + sym(O2 Γi,α P2 ) + sym(O3 Γi,α P3 ) + sym(O2 Γi,α | {z } |

[

0 N2i,α

{z

YT

W

]

[ ] M c Li,α M3i,α Γi,α P3 ) < 0, }| 4i,α {z } X

(42)

admissible with a H∞ noise attenuation performance index. ⊓ ⊔ Remark 5. By using the matrix inequality decoupling technique, the mode-dependent DOF controller design problem for discrete-time switching DMJSs is solved in Theorem 2 in the LMIs frame, where the order of the desired controllers is less than or equal to that of the plants. When E = I, the problem studied in this paper reduces to the DOF controller design problem for discrete-time switching MJSs. It should be pointed out that similar problem was investigated in [14], however, the order of the designed DOF controllers therein must be equal to that of the plants (i.e., the order cannot be less than that of the plants). Remark 6. It is noted that, the complete access for the jumping mode is assumed in [14], but the ideal assumption may be limited by several factors such as cost, physical constraints or difficulty of measuring. In this case, it is appropriate to design mode-independent controllers. When we set Li,α = L, Si,α = S in (38), our results become the mode-independent DOF controller, which are useful from the practical point of view. Considering the above, the results in this paper are more powerful and desirable than [14]. Remark 7. A useful application of Theorem 2 is to provide the SOF gains by setting m = 0. In the following, we study the resilient DOF controller design in the form of (4) for uncertain system (1) such that T system (7) with any uncertainties satisfying Γi,α Γi,α ≤ I c T c and (Γi,α ) Γi,α ≤ I, is stochastically admissible with a H∞ noise attenuation performance index. Based on Theorem 2, replacing Ai,α +∆Ai,α , Bi,α +∆Bi,α , Ci,α +∆Ci,α , c c c c Di,α + ∆Di,α , Aci,α + ∆Aci,α , Bi,α + ∆Bi,α , Ci,α + ∆Ci,α c c c and Di,α + ∆Di,α in (38) with Ai,α , Bi,α , Ci,α , Di,α , Ai,α , c c c Bi,α , Ci,α and Di,α , respectively. From (2), (5), (38), it follows that the following inequality holds (see (42))

where

[ ] ] [ M1i,α M1i,α M i,α      [ 0 ] [ 0 ]      Ni,α M1i,α   M1i,α     , O1 =  0 0  , O2 =       0 0        M2i,α M2i,α  0 0   0   0     0  , O3 =   0  [ ]  M3i,α  Li,α M4i,α 

and

[[ ] [ ]] P1 = [ N1i,α 0 0[ 0 0 β N ] 2i,α 0] , P2 = [ 0 0 0 0 −β ]N2i,α 0 Li,α , P3 = [ P31 0 P33 0 0] , [ ] P31 = N3i,α N4i,α Hi,α , P33 = N3i,α N4i,α Hωi,α .

From Lemma 1, (42) holds if the following inequality is satisfied : [ 11 ] 12 Υi,α Υi,α < 0, (43) ∗ −ϱV − ϱV T where 11 Υi,α = Θi,α + sym(O1 Γi,α P1 ) + sym(O2 Γi,α P2 ) c +sym(O3 Γi,α P3 ), [ ]T [ ] 12 c T M3i,α Υi,α = ϱO2 Γi,α 0 N2i,α Li,α + P3T (Γi,α ) M4i,α V T .

We rewrite (43) as [ ] [ ] [ ] Θi,α 0 O1 + sym( Γi,α P1 0 ) T ∗ −ϱV − ϱV 0 [ ] [ [ ] ] O2 +sym( Γi,α P2 ϱ 0 N2i,α Li,α ) (44) 0 [ ] [ O3 ] Γ c [ P 0 ]) < 0, +sym( M 3 i,α V M3i,α 4i,α

Resilient Dynamic Output Feedback Control for Discrete-Time Descriptor Switching Markov Jump Systems and Its Applications

            

Θi,α

0

∗ −ϱV − ϱV T ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

O1 ζ1 P1T O2 0

0

0

−ζ1 I 0 0 ∗ −ζ1 I 0 ∗ ∗ −ζ2 I ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

ζ2 P[2T

ζ2 ϱLTi,α

0 T N2i,α

0 0 0 −ζ2 I ∗ ∗

]

T  ] ζ3 P3 M V M3i,α 0   4i,α  0 0   0 0   < 0. 0 0   0 0   −ζ3 I 0 

9

[ O3

then, by using Lemma 2, there exist scalars ζ1 > 0, ζ2 > 0, ζ3 > 0, such that (44) fulfills if the following inequality holds (see (45)). Theorem 3 System (7) is stochastically admissible with a H∞ noise attenuation performance index, if for given scalars µ > 1, 0 < δ < 1, ζ1 , ζ2 , ζ3 , ϱ, β, there exist symmetric matrix Pbi,α > 0, matrices Mi,α , Ni,α , Qi,α , Ti,α , Li,α , Si,α , V, ∀i, j ∈ J , α, β ∈ M, and a scalar γ > 0, such that (12) and (45) hold. If (12) and (45) are feasible, then the ADT of the high-level switching signal θk satisfies (13). Furthermore, the mode-dependent DOF controller is given by ] [ c c Ai,α Bi,α = L−1 Ki,α = c c i,α Si,α . Ci,α Di,α Remark 8. By using the ADT method and matrix inequality decoupling technique, considering the uncertainties in both plant matrices and controller matrices, the resilient DOF controller for discrete-time switching DMJSs is designed in the frame of strict LMIs. It should be noted that when the considering systems are with uncertainties in both plant matrices and controller matrices, it is more difficult to deal with than the case of a single uncertain problem, since[ the uncertainty coupling terms as ] [ ] M c O2 Γi,α 0 N2i,α Li,α M3i,α P appears in (42). HowΓ i,α 3 4i,α ever, by using the matrix inequality decoupling technique, the trouble is solved perfectly in this paper. Remark 9. From Theorem 3, a minimum H∞ performance γ can be obtained by solving the following optimisation problem: Minimize γ Subject to LMIs (12)and(45), ∀ i, j ∈ J , α, β ∈ M and γ = γ 2 . Therefore, the corresponding optimal H∞ √ noise attenuation performance is γmin = γ min . For comparison purpose, we consider the SOF control issue for descriptor MJSs, which is a special case of system (7) with m = 0 and the high-level switching signal disappeared. The relevant discrete-time DMJSs are given by { Ex(k + 1) = A(rk )x(k) + B(rk )u(k), y(k) = H(rk )x(k),

∗

(45)

−ζ3 I

and the SOF controller to be designed is u(k) = Dc y(k), thus the resultant closed-loop systems can be written as: Ex(k + 1) = Ai x(k),

(46)

where Ai = Ai + Bi D Hi . Based on Theorem 2, a sufficient condition is obtained in the following corollary, such that system (46) is stochastically admissible. Corollary 1 System (46) is stochastically admissible, if for given a scalar β, there exist symmetric matrices Pi > 0, Mi , Ni , Qi , Ti , L, S, ∀i, j ∈ J , such that   11 12 Ui13 Ui Ui  ∗ Ui22 βNi Bi − βBi L  < 0, (47) ∗ ∗ −βL − βLT c

where Ui11 = sym(Mi (Ai − E) + Bi SHi ) + E T Xi −E T Pi E, 12 Ui = −Mi + (Ai − E)T NiT + QTi S T +HiT S T BiT + E T Xi , 13 Ui = β(Mi Bi − Bi L) + HiT S T , Ui22 = −Ni − NiT + Xi + STi + TiT S T , Xi =

J ∑

µij Pj , S ∈ Rn×(n−r) is the arbitrary matrix

j=1

satisfying E T S = 0 and rank (S) = n − r. Then the modeindependent SOF controller is given by Dc = L−1 S. Remark 10. In Corollary 1, a sufficient condition on the SOF control issue for discrete-time DMJSs is proposed. It should be noted that similar problems were studied in [29, 33, 35], where special structures on freedom matrices Mi , Ni in [33] and rank constraint on output matrix C in [29, 35] were needed. These assumptions are very idealistic and restrictive. However, by using the matrix inequality decoupling technique, the hard constraints are overcome perfectly in this paper. 4 Examples In this section, two numerical examples are given to show the advantage and practicability of the proposed methodology. Firstly, we show that the obtained conditions in this

10

Jimin Wang, Shuping Ma

paper are less conservative than the existing ones, and then the proposed methodology is implemented on an electrical circuit model borrowed from [39], which illustrates the practicability of the proposed method. Example 1. Consider the SOF control issue for DMJSs with the following parameters:  0.3273 0.5944 0.8595 A1 =  −0.4755 −0.4438 0.39  , 0.5160  −0.1300 −1.3225  0.1334 0.3595 B1 =  −0.0795 −0.6715  , ] [ 0.2031 0.0241 −7.8616 −19.4388 0.14147 , F1 =  8.6016 15.2036 −7.3950 0.6743 0.2822 −0.4024 A2 =  0.2594 −1.2487 0.0508  , −0.5722  −0.0145 −1.1567  −0.5662 −0.1161 B2 =  −0.1638 0.6096  , [ 0.5675 0.1072 ] −2.3072 5.7088 −6.7980 F2 = . 5.8012 10.7396 5.6042

and introducing the following normalized variables, { τ = t/T, L = L/T, Lrτ = Lrt /T, Lrt = { R1 if rt = 1 Rrτ = Rrt /T, Rrt = , R2 if rt = 2

L1 if rt = 1 , L2 if rt = 2



The singular matrix and TPs are given by  [ ] 24 0 0.5 0.5 E =  0 1 0.5  , [µij ]2×2 = . 0.9 0.1 0 1 0.5

then, the differential equations for the boost chopper with RC load are given as: uv (τ ) = Li˙ L (τ ) + (1 − Q1 )iR (τ )Rrτ + uw (τ ), uv (τ ) = Li˙ L (τ ) + (1 − Q1 )i˙ Lc (τ )Lrτ + uw (τ ), 0 = i˙ Lc (τ )Lrτ − iR (τ )Rrτ ,

(48)

where Q1 takes values 0 or 1. [ ]T Denote xT = iL (τ ) iR (τ ) iLc (τ ) , w = uw (τ ), u = uv (τ ). Then system (48) can be transformed into 

   L0 0 0 −(1 − Q1 )Rrτ 0  L 0 (1 − Q1 )Lrτ  x˙ =  0 0 0x R 00 Lrτ rτ  0  0 −1 1 +  1  u +  −1  w. 0 0



[ ]T Choosing S = 0 2 −2 , β = 0.08. Solving LMI (47) in Corollary 1, it is feasible, and the stabilizing controller gain is given by [ ] 0.0013 0.0025 c D = . 0.0025 0.0010

Additionally, the resistor is given by the following expreskn kn sion: Rrτ = Rrτ + ∆Rrτ , where Rrτ is known and ∆Rrτ ≤ 0.02, note in this case that the system state matrices are affected by ∆Rrτ . Then, the DC chopper circuit system can be described by switching DMJSs (1). Let the sampling time Ts = T /8, the acquired continuoustime system mode can be discretized. Based on the above discussion, the parameters of this system can be set as: – Q1 takes value 0 and L1 operates

However, if we use the method in [33] (Lemma 1), it is infeasible. Therefore, from the above analysis, it shows that Corollary 1 in our paper is less conservative than the result given in [33] (Lemma 1). Example 2. In order to illustrate the applicability of the theoretical results, the DC chopper circuit is introduced. From Fig. 1, it shows that the switch “Q1 ” is governed by a pulse-width-modulation device and can switch instantly within each cycle T . When “Q1 ” is turned on or turned off, the load circuit can get different voltages. L is the inductor, R is the resistor, V is the source voltage, and W is the disturbance signal. By adjusting the duty cycle, appropriate average output voltage can be achieved. So the on-off of “Q1 ” can represent the switching signals. In the load circuit, the switch of “S1 ” occupies two positions in a random way, which follows a Markov chain. Based on Kirchhoff’s laws



   −1 1 −1.6 2 1 A1,1 =  0.8 −1 1  , B1,1 =  1.9 2  , −1.3 0.9 −0.6 1 0.8  T   0.4 −0.5 H1,1 =  0.6  , F1,1 =  0.6  , Hω11 = −1.2, [ 0.2 ]0 [ ] 0.12 0.006 0.03 0.8 1 C1,1 = , D1,1 = , −0.24 0.12 0.15 −0.2 0  [ ] [ ] 0.3 0.1 0   Dω1,1 = , M111 = 0 , M211 = , 1.7 0.1 0  T [ ]T 0.5 0.1   N111 = −0.4 , N211 = , 0 0

Resilient Dynamic Output Feedback Control for Discrete-Time Descriptor Switching Markov Jump Systems and Its Applications

11

S1

– Q1 takes value 0 and L2 operates L



   −1 0.96 −1.4 1 0.1 Q1 A1,2 =  0.6 −0.67 1.1  , B1,2 =  0.7 1.2  , L1 L2 R1 R2 V 0.84 0.7 −1.2 1.6 1  T   −0.6 −0.1 W H1,2 =  0.2  , F1,2 =  0.8  , Hω12 = 1.2, boost chopper ] 0.7 [ ] [ 0.2 0.14 0.04 0.06 0 0 C1,2 = , D1,2 = , Fig. 1 Boost chopper with RC load: descriptor switching Markov jump −0.04 0 0.08  system  −1.24 2.48 ] ] [ [ 0.015 −0.06 0.2 , , M112 =  0  , M212 = Dω1,2 = 0 0.1 0.01 1.5  T [ ]T η1 0 0.1 η2 N112 =  −0.2  , N212 = , 1 η3 0 η4 0 η5

0.5

– Q1 takes value 1 and L1 operates 





0.8 0.7 −1.78 0.95 −1.5    A2,1 = 0.7 −1 0.8 , B2,1 = 0.5 0.5  , 1.9 0.4 1.3 1 −0.6  T   −0.7 0.4 H2,1 =  −0.9  , F2,1 =  0.5  , Hω21 = 1.3, [ −0.1 ]0.2 [ ] 0.05 0 −0.025 0 0.186 C2,1 = , D2,1 = , 0 0.002 0.01 0 0.124  [ ] [ ] −0.04 0.2 0.06 Dω2,1 = , M121 =  0  , M221 = , 0.1 0 0 T  [ ]T 0 0.1   , N221 = , N121 = −0.1 0 0

0 η(k)



−0.5

−1

−1.5

−2

0

10

20

30

40

50

40

50

time k

Fig. 2 State responses of the closed-loop system

2

– Q1 takes value 1 and L2 operates    −1.4 0.8 −1 0.2 2 A2,2 =  0.5 −1.3 0.3  , B2,2 =  2 0.8  , 0 0.8 −0.4 0.9 −0.2  T   −0.3 0.2 H2,2 =  1  , F2,2 =  0.9  , Hω22 = −1.5, [ 0.1 ] 1 [ ] 0.04 0.02 0 0.08 0 C2,2 = , D2,2 = , −0.014 0.04 0.06   −0.2 0.4 [ ] −0.03 0.19 Dω2,2 = , M122 =  0  , 0 0  T [ ] 0 0 M222 = , N122 =  −0.06  , −0.04 0 [ ]T 0.1 N222 = . 0 

1

0

10

Fig. 3 Jumping mode

20

30

12

Jimin Wang, Shuping Ma

matrices are given by 

2

K1,1

K2,1

1

K1,2 0

10

20

30

40

50

K2,2

Fig. 4 high-level switching signal

In the controller (4), the gain variations are considered to be [

] [ ] [ ] 0.4 0.02 0 M311 = , M312 = , M321 = , 0.7 [ 0 ] [0 ] [ ] −0.06 0 0.03 M322 = , M411 = , M412 = , 0 [ 0 ] [ 0.1 ] 0.01 −0.01 M421 = , M422 = , 0 [ 0 ] N311 = N [ 321 = ] 0.03 0 , [ ] N312 = 0 0.06 , N322 = 0 0.3 , N411 = 0.12, N412 = N421 = 0, N422 = 0.05.

The singular matrix and piecewise-constant TPs are given by 

3 2 E =  0 −1 0 0

[ [µ1ij ]2×2 =

 0 0, 0

] [ ] 0.5 0.5 0.8 0.2 , [µ2ij ]2×2 = . 0.4 0.6 0.32 0.68 

 0 Choosing S11 = S12 = S21 = S22 =  0  , δ = 0.99, −2 ζ1 = 1.09, ζ2 = 0.8, ζ3 = 3, β = 0.02, ϱ = 0.02, µ = 1.02. Solving LMIs (12), (45), it is obtained that γ = 7.9373, and the mode-dependent DOF controller gain

−0.0588  0 =  0.0454 0.0377  0.0069  0 =  −0.1375 0.3890  0  −0.0030 =  0.0018 0.0036  0.0081  0 =  0.0594 −0.0717

 −0.0315 −0.0040  0 0 , 0.0309 0.0295  0.0262 −0.0478  −0.0002 0.0038  0.0140 0 , −0.0013 −0.1113  0.0039 0.1172  0 0 −0.0023 0.0090  , −0.0022 0.0231  −0.0034 −0.0172  0.0002 0.0036  0.0140 0 . 0.0012 −0.0052  −0.0009 0.0462

According to (13), the minimum ADT can be calculated as τa∗ = 1.97, thus we regulate the switching signal to satisfy τa > τa∗ . Then we can set twenty-two times switches in 50s interval, which is shown in Fig. 4. For simulation, taking exogenous disturbance ω(k) = e−k , the simulation of state responses of the closed-loop system is presented in Fig. 2. From Fig. 2, it is obtained that the closed-loop system is stochastically admissible.

5 Conclusions In this paper, the resilient DOF controller design problem for a type of discrete-time switching DMJSs has been investigated, where the underlying systems contain the discretetime DMJSs as a special case. Additive gain perturbations to reflect the imprecision in controller implementation have been considered. By constructing a stochastic Lyapunov functional and using the ADT technique, sufficient conditions are presented such that the corresponding closed-loop systems are stochastically admissible with a prescribed H∞ performance level. Then by applying the matrix inequality decoupling technique, the desired controller design parameters have been computed in the strict LMIs frame from a new perspective, which can be of full-order or reducedorder. Finally, numerical examples reveal less conservatism in comparison to the previous methods.

6 Acknowledgments This work is supported by National Natural Science Foundation of China (61473173). The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of this paper.

Resilient Dynamic Output Feedback Control for Discrete-Time Descriptor Switching Markov Jump Systems and Its Applications

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Jimin Wang, Shuping Ma