Computer Systems Science & Engineering DOI:10.32604/csse.2021.016475
Article

Output Feedback Robust H∞ Control for Discrete 2D Switched Systems

Wenjun Gao1, Yang Yang1,*, Hongtao Duan1, Mingsheng Li1, Jinrong Ma1 and Songwei Jin2

1School of Electronic Information and Engineering, Changchun University of Science and Technology, Changchun, 130000, China
2Department of Computer Science and Engineering, University of California, Riverside, CA92521, USA
*Corresponding Author: Yang Yang. Email: cloneyang@126.com
Received: 03 January 2021; Accepted: 17 February 2021

Abstract: The two-dimensional (2-D) system has a wide range of applications in different fields, including satellite meteorological maps, process control, and digital filtering. Therefore, the research on the stability of 2-D systems is of great significance. Considering that multiple systems exist in switching and alternating work in the actual production process, but the system itself often has external perturbation and interference. To solve the above problems, this paper investigates the output feedback robust H∞ stabilization for a class of discrete-time 2-D switched systems, which the Roesser model with uncertainties represents. First, sufficient conditions for exponential stability are derived via the average dwell time method, when the system’s interference and external input are zero. Furthermore, in the case of introducing the external interference, the weighted robust H∞ disturbance attenuation performance of the underlying system is further analyzed. An output feedback controller is then proposed to guarantee that the resulting closed-loop system is exponentially stable and has a prescribed disturbance attenuation level γ. All theorems mentioned in the article will also be given in the form of linear matrix inequalities (LMI). Finally, a numerical example is given, which takes two uncertain values respectively and solves the output feedback controller’s parameters by the theorem proposed in the paper. According to the required controller parameter values, the validity of the theorem proposed in the article is compared and verified by simulation.

Keywords: 2-D systems; robust H∞ control; LMI; output feedback; switched systems; Roesser model

1  Introduction

The issue of stability analysis and controller synthesis is a hot research topic. Reference Medvedeva et al. [1–3] investigated the stability of one-dimensional (1-D) continuous-time or discrete-time systems. Considering the complexity of many manufacturing processes and physical phenomena, a 2-D continuous-time or discrete-time system that depends on two independent variables has its irreplaceable application area. 2-D systems have attracted considerable research attention in control theory and practice over the past few decades due to their wide applications. Reference Du et al. [4–7] showed multi-dimensional digital filtering, linear image processing, signal processing, and process control. Different models such as the Roesser model and Fornasini–Marchesini model can represent 2-D systems, and the stability issues concerning these two models can be found [8,9].

On the other hand, considerable interest has been devoted to the research of switched systems during the recent decades. A switched system comprises a family of subsystems described by continuous or discrete-time dynamics and a switching law that specifies the active subsystem at each instant of time. The switching strategy improves control performance [10–13] and arise many engineering applications, such as in motor engine control, constrained robotics, and satellite image control systems [14]. As far as time-dependent switching is concerned, the average dwell time (ADT) switching is employed in most references owing to its flexibility [15,16].

However, perturbations and uncertainties widely exist in practical systems. In some cases, the perturbations can be merged into the disturbance, which can be bounded in the appropriate norms. The main advantage of robust H∞ control is that its performance specification considers the system’s worst-case performance in terms of energy gain. This is more appropriate for system robustness analysis and robust control under modeling disturbances than other performance specifications. Recently, the problems of robust H∞ control and filtering for 2-D systems have been studied by many researchers [17–20], and so do the same problems of switched systems [21–23]. However, to the best of our knowledge, the output feedback robust H∞ control problem of 2-D switched systems in the Roesser model with uncertainties has not yet been thoroughly investigated, which motivates this present study.

In this paper, we confine our attention to the robust H∞ control problem of discrete 2-D switched systems described by the Roesser model with uncertainties. The main theoretical contributions are threefold: (1) We contribute to the development of stabilization for a class of 2-D switched systems that are exponentially stable, which the Roesser model with uncertainties represents. (2) A sufficient condition is presented to ensure a 2-D switched system’s exponential stability at a given disturbance attenuation level robust weighted H∞. (3) Based on the above two points, this article further designs the output feedback controller of the closed-loop system.

The paper is organized as follows. According to the current research results, we first study the exponential stability of 2-D switched systems described by the Roesser model with uncertainties. Further, when the system contains perturbations, we analyze the robust H∞ performance index of the system. Finally, we design an output feedback controller for the open-loop system, and an example is given to illustrate the effectiveness of the proposed method.

2  Notation

The following notations are used throughout the paper: the superscript “T” denotes the transpose, and the notation X ≥ Y (X > Y) means that matrix X − Y is positive semi-definite (positive definite, respectively). ‖⋅‖ denotes the Euclidean norm. I represents the identity matrix. Diag {ai} denotes a diagonal matrix with the diagonal elements ai, i = 1, 2,…,n. X-1 denotes the inverse of X. Rn denotes the n dimensional vector. Z+ represents the set of all non-negative integers. The l2 norm of a 2-D signal w(i,j) is given by ‖w‖2=∑i=0∞∑j=0∞‖w(i,j)‖2 , where w(i,j) belongs to l2{[0,∞),[0,∞)} .

3  Problem Formulation and Preliminaries

The uncertain Roesser model for a 2-D switched system G:u→y is given by the following state equation:

x¯(k,l)=Aδ(k,l)(Δ)x(k,l)+Bδ(k,l)w(k,l)z(k,l)=Cδ(k,l)x(k,l)+Dδ(k,l)w(k,l),k,l=0,1,2⋯ (1)

with

where Gδ(k,l)(k,l) is an unknown uncertain matrix and satisfies the norm bounded condition GTδ(k,l)(k,l)Gδ(k,l)(k,l)≤I , and where xh(k,l)∈Rn1 and xv(k,l)∈Rn2 denote the system horizontal state and vertical state, respectively. Furthermore, x(k,l) is the whole state in Rn with n=n1+n2 , and w(k,l)∈Rq is the interference input which belongs to w(k,l)∈l2{[0,∞),[0,∞)} . u(k,l)∈Rp and z(k,l)∈Rq are control input and control output respectively; k and l are integers in Z. δ(k,l):Z+×Z+→N={1,2,3⋯,N} is the switching signal. N is the number of subsystems. δ(k,l)=i,i∈N denotes that the ith subsystem is activated., Ai, Bi, Ci, Di, Fi, Gi, Hi, are constant matrices with appropriate dimensions.

The boundary condition satisfies:

X(0)=[xhT(0,0),xhT(0,1),⋯,xvT(0,0),xvT(1,0),⋯]T (2)

It is easy to know from the above formula ‖X(0)‖2<∞ .

Remark 1. “In this paper, it is assumed that switching occurs only at each sampling point of k or l. The switching sequence can be described as

((k0,l0),δ(k0,l0)),⋯,((kk,lk),δ(kk,lk)),⋯ (3)

with (kk,lk) denoting the kth switching instantly. It should be noted that the value of δ(k,l) only depends on k+l [24,25].

Definition 1. System (1) is said to be exponentially stable under δ(k,l) if for a given j ≥ 0, there exist positive constants c and f, such that

∑k+l=D‖x(k,l)‖2≤fe−c(D−j)∑k+l=j‖x(k,l)‖r2 (4)

holds for all D ≥ j [26].

Remark 2. From Definition 1, it is easy to see that when j is given, ∑k+l=j‖x(k,l)‖r2 will be bounded, and ∑k+l=D‖x(k,l)‖r2 will tend to be zero exponentially as D goes to infinity, which also means that ‖x(k,l)‖ will tend to be zero exponentially.

Definition 2. For a given scalar γ > 0, system (1) is said to have a weighted disturbance attenuation level γ under switching signal δ(k,l) if it satisfies the following conditions [4]:

(1) when w(k,l) =0, system (1) is asymptotically stable or exponentially stable;

(2) under the zero-boundary condition, we have

∑k=0∞∑l=0∞(ϕk+l‖z‖22)<γ2∑k=0∞∑l=0∞‖w‖22),∀0≠w(k,l)∈l2{[0,∞),[0,∞)} (5)

where 0 < ϕ < 1 and the l2 -norm of 2D discrete signal z(k,l) and w(k,l) are defined as

‖z‖22=‖z(k+1,l)‖22+‖z(k,l+1)‖22,‖w‖22=‖w(k+1,l)‖22+‖w(k,l+1)‖22 (6)

Definition 3. For any k+l=D≥j=kz+lz , let Nδ(j,D) denote the switching number of δ(⋅) on an interval [z,D) . If

Nδ(j,D)≤N0+D−jτa (7)

holds for given N0≥0 and τa≥0 , then the constant τa is called the average dwell time and N0 is the chatter bound [27].

Lemma 1. For a given matrix S=[S11S12S21S22] , where S11 and S22 are square matrices, the following conditions are equivalent [28].

(i)S<0 ;

(ii)S11<0 , S22−S12TS11−1S12<0 ;

(iii)S22<0 , S11−S12S22−1S12T<0 .

Lemma 2. Assuming that x∈Rp,y∈Rq and U,V,W are a suitable dimension matrix, then inequality xTUVWy+yTWTVTUTx≤ε1xTUUTx+ε2yTWWTy is true for any VTV≤I , if and only if there exist positive scalars ε1,ε2 and ε1≥ε2 .

Proof.

4  Exponential Stability Analysis

This section focuses on the exponential stability analysis of the 2D switched systems. The following theorem presents sufficient conditions that can guarantee that system (1) is exponentially stable.

Theorem 1. Consider 2D discrete switched system (1) with w(k,l)=0 , for a given positive constant ϕ < 1, if there exist a set of positive-definite symmetric matrices Pi∈Rn×n,i∈N and two positive scalars ε1,ε2 , such that

[−ϕPiAiTPi0ε1HiTPiAi−PiPiFi00FiTPi−ε2I0ε1Hi00−ε1I]<0 (8)

ε1≥ε2 (9)

Then, the system is exponentially stable for any switching signal with the average dwell time satisfying

τa>τa∗=ln⁡μ−ln⁡ϕ (10)

where μ ≥ 1 satisfies

Pi≤μPj,∀i,j∈N (11)

Proof. Without loss of generality, we assume that the ith subsystem is active. For the ith subsystem, we consider the following Lyapunov function candidate:

Vi(x(k,l))=xT(k,l)Pix(k,l) (12)

where Pi is an n×n positive-definite matrix for any i∈N , and thus Vi(x(k,l))>0,∀x(k,l)≠0 and Vi(k,l)=0 only when x(k,l)=0 . Then we have Vi(x¯(k,l))−ϕVi(x(k,l))=xT(k,l)(AiT(Δ)PiAi(Δ)−ϕPi) .

Using Lemma 1 to (8), we can get the equivalent inequality as follows:

⇒[−ϕPiAiTPiPiAi−Pi]+ε1[HiT0][Hi0]+1ε2[0PiFi][0FiTPi]<0 (13)

Then using Lemma 2 to (13), we can get

[−ϕPiAiTPiPiAi−Pi]+[HaiT0]GiT[0FiTPi]+[0PiFi]Gi[Hai0]<0 (14)

Using Lemma 1 to (14), we can get

AiT(Δ)PiAi(Δ)−ϕPi<0 (15)

Form (15), we know

Vi(x¯(k,l))≤ϕVi(x(k,l)) (16)

The equality holds only if Vi(x¯(k,l))=Vi(x(k,l))=0 .

It follows from (16) that

∑k+l=N+1Vi(k,l)≤ϕ∑k+l=NVi(k,l)≤ϕN−N0+1∑k+l=N0Vi(k,l) (17)

Now, let n = Nδ(j,D) denote the switching number of δ(⋅) on an interval [j,D) , and let mk−n+1<mk−n+2<⋯<mk−1<mk denote the switching points of δ(⋅) over the interval [j,D) , thus, for D∈[mk,mk+1) , we have from (16)

∑k+l=DVδ(mk)(k,l)<ϕD−mk∑k+l=mkVδ(mk)(k,l) (18)

Using (11) and (12), at switching instant mk=k+l , we have

∑k+l=mkVδ(mk)(k,l)≤μ∑k+l=mkVδ(mk−1)(k,l) (19)

Also, according to Definition 3, it follows that

n=Nδ(j,D)≤N0+D−jτa (20)

Therefore, the following inequality can be obtained easily:

∑k+l=DVδ(mk)(k,l)<ϕD−mk∑k+l=mkVδ(mk)(k,l)≤μϕD−mk∑k+l=mk−Vδ(mk−1)(k,l)<μϕD−mk∑k+l=mk−1Vδ(mk−1)(k,l)ϕmk−mk−1≤⋯<μn−1ϕD−mk−n+1∑k+l=mk−n+1Vδ(mk−n+1)(k,l)≤μnϕD−mk−n+1∑k+l=mk−n+1−Vδ(j)(k,l)<μnϕD−j∑k+l=jVδ(j)(k,l) (21)

Combining (20), inequality (21) can be written as follows:

∑k+l=DVδ(mk)(k,l)≤e−(−ln⁡μτa−ln⁡ϕ)(D−j)∑k+l=jVδ(j)(k,l) (22)

There exist two positive constants a and b (a ≤ b) such that

∑k+l=D‖x(k,l)‖2≤bae−(−ln⁡μτa−ln⁡ϕ)(D−j)∑k+l=j‖x(k,l)‖2 (23)

By Definition 1, we know that if −ln⁡μτa−ln⁡ϕ >0, that is τa>τa∗=ln⁡μ−ln⁡ϕ , the 2D discrete switched system is exponentially stable.

The proof is completed.

Remark 3. Note that when μ=1 in (10), (11) turns out to be Pi=Pj,∀i,j∈N . In this case, we have

τa>τa∗=ln⁡μ−ln⁡ϕ , which means that the switching signal can be arbitrary.

5  Robust H∞ Performance Analysis

This section focuses on the Robust H∞ stabilization problem for a class of discrete-time 2D switched systems represented by a Roesser model with uncertainties. The following theorem presents sufficient conditions that can guarantee that system (1) is exponentially stable and has a prescribed weighted H∞ disturbance attenuation level γ .

Theorem 2. For given positive scalars γ and ϕ < 1, there exist symmetric and positive-definite matrices Pi∈Rn×n,i∈N , and two positive scalars ε1,ε2 ,such that

[−ϕPi0AiTPi0CiTε1HiT0−γ2BiTPi0DiT0PiAiPiBi−PiPiFi0000FiPi−ε2I00CiDi00−I0ε1Hi0000−ε1I]<0 (24)

ε1≥ε2 (25)

Then, 2D switched system (1) is exponentially stable and has a prescribed weighted H∞ disturbance attenuation level γ for any switching signals with average dwell time satisfying (10), where μ ≥ 1 satisfies (11).

Proof. It is an obvious fact that (24) implies that inequality (8) holds. By Lemma 2, we can find that system (1) is exponentially stable when w(k,l) = 0. Now we are able to prove that system (1) has a prescribed weighted H∞ performance γ for any nonzero w(k,l)∈l2{[0,∞),[0,∞)} .

To establish the weighted H∞ performance, we choose the same Lyapunov functional candidate as in (12) for the system (1). Following the proof line of Theorem 1, we can get

if

Φ=[AiT(Δ)PiAi(Δ)−ϕPi+CiTCiAiT(Δ)PiBi+CiTDiBiTPiAi(Δ)+DiTCiBiTPiBi−γ2+DiTDi]<0 (26)

Using Lemma 1 to (26), we can get the equivalent inequality as follows:

[−ϕPi+CiTCiCiTDiAiT(Δ)DiTCi−γ2+DiTDiBiTAi(Δ)Bi−(Pi)−1]<0⇔[−ϕPi+CiTCiCiTDiAiTDiTCi−γ2+DiTDiBiTAiBi−(Pi)−1]+[HaiT00]GiT[00FiT]+[00Fi]Gi[Hai00]<0 (27)

Pre- and post-multiplying (24) by diag{II(Pi)−1III} , we obtain

[−ϕPi0AiT0CiTε1HiT0−γ2BiT0DiT0AiBi−(Pi)−1Fi0000Fi−ε2I00CiDi00−I0ε1Hi0000−ε1I]<0 (28)

Using Lemma 1 to (28), we can get

[−ϕPi+CiTCi+ε1HaiTHaiCiTDiAiTDiTCi−γ2+DiTDiBiTAiBi−(Pi)−1+1ε2FiFiT]<0 (29)

Then using Lemma 2, we find that (29) is equivalent to (27).

Thus it can be obtained from (24) that

Vi(x¯(k,l))−ϕVi(x(k,l))+γ2wTw−zTz<0 (30)

Then we have

Vi(x¯(k,l))≤ϕVi(x(k,l))+γ2wTw−zTz (31)

Let

F(k+l)=‖z‖22−γ2‖w‖22=‖zh(k,l)zv(k,l)‖22−γ2‖wh(k,l)wv(k,l)‖22 (32)

Summing up both sides of (31) from (D-1) to 0 with respect to l and 0 to (D-1) with respect to k, respectively, and applying the zero-boundary condition, one gets

∑k+l=DVδ(mk)(k,l)<ϕD−1∑k+l=D−1Vδ(mk)(k,l)−∑k+l=D−1F(k,l)<ϕD−mk∑k+l=mkVδ(mk)(k,l)−∑m=mkD−1∑k+l=mϕD−1−k−lF(k,l)≤μϕD−mk∑k+l=mk−Vδ(mk−1)(k,l)−∑m=mkD−1∑k+l=mϕD−1−k−lF(k,l)<μϕD−(mk−−1)∑k+l=mk−−1Vδ(mk−1)(k,l)−μϕD−mk∑k+l=mk−−1F(k,l)−∑m=mkD−1∑k+l=mϕD−1−k−lF(k,l)<μNδ(k+l,D)ϕD−(mk−−1)∑k+l=mk−−1Vδ(mk−1)(k,l)−μNδ(k+l,D)∑m=mk−−1D−1∑k+l=mϕD−1−k−lF(k,l)<μNδ(k+l,D)ϕD−(mk−−1)∑k+l=mk−1Vδ(mk−1)(k,l)−μNδ(k+l,D)∑m=mk−1D−1∑k+l=mϕD−2−k−lF(k,l)≤μNδ(k+l−1,D)ϕD−mk−1∑k+l=mk−1−Vδ(mk−2)(k,l)−∑m=mk−1−D−1∑k+l=mμNδ(k+l+1,D)ϕD−1−k−lF(k,l)⋯<∑k+l=0μNδ(k+l,D)ϕDVδ(1)(k,l)−∑m=0D−1∑k+l=mμNδ(k+l,D)ϕD−1−k−lF(k,l) (33)

Under the zero-initial condition, we have

∑k+l=0μNδ(k+l,D)ϕDVδ(1)(k,l)=0 (34)

Thus, we have

∑m=0D−1∑k+l=mμNδ(k+l,D)ϕD−1−k−lF(k,l)<−∑k+l=DVδ(mk)(k,l) (35)

Multiplying both sides of (35) by μ−Nδ(0,D) , we can get the following inequality:

∑m=0D−1∑k+l=mμ−Nδ(0,k+l)ϕD−1−k−l‖z‖22<γ2∑m=0D−1∑k+l=mμ−Nδ(0,k+l)ϕD−1−k−l‖w‖22 (36)

Noting Nδ(0,k+l)≤k+lτa , and using (10), we have

μ−Nδ(0,k+l)=eμ−Nδ(0,k+l)ln⁡μ≥e(k+l)ln⁡ϕ (37)

Thus

∑m=0D−1∑k+l=me(k+l)ln⁡ϕϕD−1−k−l‖z‖22<γ2∑m=0D−1∑k+l=mμ−Nδ(0,k+l)ϕD−1−k−l‖w‖22⇒∑m=0D−1∑k+l=mϕD−1‖z‖22<γ2∑m=0D−1∑k+l=mϕD−1−k−l‖w‖22⇒∑m=0D−1∑k+l=mϕD−1−k−lϕk+l‖z‖22<γ2∑m=0D−1∑k+l=mϕD−1−k−l‖w‖22⇒∑m=0∞∑k+l=mϕk+l‖z‖22<γ2∑m=0∞∑k+l=m‖w‖22⇒∑k=0∞∑l=0∞ϕk+l‖z‖22<∑k=0∞∑l=0∞γ2‖w‖22 (38)

According to Definition 3, we can see that system (1) is exponentially stable and has a prescribed weighted robust H∞ disturbance attenuation level γ.

The proof is completed.

6  Robust H∞ Control Problem

This subsection will deal with the Robust H∞ control problem of 2D switched systems via dynamic output feedback. Our purpose is to design a dynamic output feedback controller such that the closed-loop system is exponentially stable and has a specified robust weighted H∞ disturbance attenuation level γ.

Consider the following discrete 2D switched systems in the Roesser model with uncertainties:

x¯(k,l)=A1δ(k,l)(Δ)x(k,l)+B1δ(k,l)w(k,l)+B2δ(k,l)u(k,l)z(k,l)=C1δ(k,l)x(k,l)+D11δ(k,l)w(k,l)+D12δ(k,l)u(k,l),k,l=0,1,2⋯y(k,l)=C2δ(k,l)x(k,l)+D21δ(k,l)w(k,l) (39)

where x(k,l)∈Rn,w(k,l)∈Rnw,u(k,l)∈Ru,z(k,l)∈Rz and y(k,l)∈Ry are, respectively, the state, the disturbance input, the control input, the controlled output, and the measurement output of the plant, k and l are integers in Z+ . A1i,B1i,B2i,C1i,C2i,D11i,D12i and D21i with i∈N are constant matrices with appropriate dimensions. We do not assume the disturbance input signal’s statistics w(k,l) other than that its energy is bounded, i.e., ‖w‖2<∞ .

Introduce the following output feedback controller of order nc :

x¯c(k,l)=Acδ(k,l)xc(k,l)+Bcδ(k,l)y(k,l)u(k,l)=Ccδ(k,l)xc(k,l)+Dcδ(k,l)y(k,l) (40)

where

The closed-loop system consisting of the plant (39) and the controller (40) is of the form

x^˙(k,l)=A¯δ(k,l)x^(k,l)+B¯δ(k,l)w(k,l)z¯(k,l)=C¯δ(k,l)x^(k,l)+D¯δ(k,l)w(k,l) (41)