Elsevier

Systems & Control Letters

Volume 112, February 2018, Pages 42-50
Systems & Control Letters

Stability of two-dimensional descriptor systems with generalized directional delays

https://doi.org/10.1016/j.sysconle.2017.12.003Get rights and content

Abstract

This paper is concerned with the problem of stability analysis of two-dimensional (2-D) descriptor systems described by a Roesser-type model with generalized directional time-varying delays. By constructing an improved 2-D Lyapunov–Krasovskii functional candidate and utilizing zero-type free matrix equations, new delay-dependent conditions are derived in terms of linear matrix inequalities (LMIs) to ensure that the system under consideration is regular, causal and internally stable. The obtained results are shown to extend the existing literature by numerical examples.

Introduction

Descriptor model is commonly used to describe dynamics of various practical phenomena such as electrical circuit networks, power systems, multibody mechanics, aerospace engineering and chemical and physical processes [[1], [2]]. The systems of this type are also known as singular systems [3], implicit systems [[4], [5]] or differential/difference-algebraic equations [[6], [7]]. In such a system, the state variables are subject to both dynamical equations and algebraic constraints which result a number of different features from classical systems such as impulsive behaviors in the state response, non-properness of transfer matrix or non-causality between input/output and states. These characteristic properties make the study of descriptor systems much more complicated and challenging than classical systems. On the other hand, as an inherent characteristic, time-delay is ubiquitously encountered in engineering systems which has various effects on the system performance [8]. Thus, the study of qualitative behavior of time-delay systems plays an important role in applied models which has received significant research attention in the last two decades (see, e.g. [[9], [10], [11], [12], [13]]). In particular, a great deal of effort from researchers has been devoted to the problems of stability analysis and control of singular systems with delays and many results have been reported in the literature. To mention a few, we refer the reader to [[14], [15], [16], [17], [18]] for the problem of stability analysis and [[19], [20], [21], [22], [23], [24]] for some other control issues related to singular delayed systems.

Two-dimensional systems can be used to describe dynamics of many practical models where the information propagation occurs in each of the two independent directions [[25], [26]]. Recently, due to their widespread applications in circuit analysis, image processing, seismographic data transmission or multi-dimensional digital filtering, the theory of 2-D systems has attracted considerable research attention (see, e.g. [[27], [28], [29], [30]] and the references therein). There have been a few papers concerning the problems of stability and stabilization of 2-D descriptor systems. For example, in [31], the stability problem was studied for 2-D linear singular systems in general model. Sufficient conditions were derived in a type of Lyapunov matrix inequalities to ensure that a 2-D system is acceptable (see Definition 1) and asymptotically stable. In [32], the problems of stability and stabilization via state feedback controllers were investigated for a class of delay-free 2-D singular Roesser systems. By decomposing the system into slow- and fast-subsystems, and based on the Lyapunov function method, sufficient conditions in terms of linear matrix inequalities (LMIs) were derived to design a stabilizing state feedback controller. The problem of H control was also considered in [[33], [34]] for 2-D singular Roesser models with constant delays. By using the bounded real lemma approach, delay-independent LMI-based conditions were derived for the design of state feedback controllers that make the closed-loop system to be acceptable and stable with a prescribed H performance level. However, the proposed method of [[33], [34]] cannot be extended to 2-D singular systems with time-varying delays which are encountered in many practical systems, for instance, in 2-D models of networked control systems. Looking at the literature so far, apart from [35], it is clear that there has been no reported work on stability of 2-D singular systems with time-varying delays. Based on the Lyapunov–Krasovskii functional (LKF) method and by employing a Jensen-type discrete inequality to manipulate the difference of a LKF candidate, delay-dependent stability conditions were first derived in [35] for a class of 2-D singular Roesser systems with state-varying delays.

It is noted also that practical models in real world applications produce different types of delays because of variable networks transmission conditions. Since the properties of these delays may not be identical, it is not reasonable to lump all the delays into one type [18]. Thus, it is interesting and important to study 2-D singular models with different types of delays. Besides, the use of Jensen-type inequalities usually produces undesired conservatism in the derived stability conditions. Therefore, reducing the conservativeness of stability conditions is always an important issue in applications of control engineering which needs further investigation.

Motivated from the above discussion, in this paper, we study the problem of stability analysis of a general class of 2-D descriptor systems described by the Roesser model with delays. The novelties of this paper are three points.

  • The stability problem is studied for 2-D descriptor systems with interval discrete and distributed time-varying delays in both horizontal and vertical directions which encompass the 2-D descriptor models considered in the existing literature as some special cases.

  • An improved 2-D LKF candidate which comprises of some quadratic terms and summation terms in single, double and triple forms, is constructed.

  • The technique of zero-type free matrix equations is utilized to further reduce conservativeness of the proposed stability conditions.

On the basis of these features, new delay-dependent conditions are formulated in terms of LMIs to ensure that the system under consideration is regular, causal and internally stable. The obtained results are shown to extend the existing results in the literature by numerical examples.

Notation. Z denotes the set of integers, Z[a,b]{a,a+1,,b} for a,bZ, ab. Rn×m denotes the set of n×m real matrices and diag(A,B)A00Bfor two matrices A,B of appropriate dimensions. Sym(A)A+A for ARn×n. A matrix MRn×n is semi-positive definite, M0, if xMx0, xRn; M is positive definite, M>0, if xMx>0, xRn, x0.

Section snippets

Preliminaries

Consider a class of 2-D descriptor systems with mixed directional time-varying delays described by the following Roesser model (2-D DRM) Exh(i+1,j)xv(i,j+1)=Axh(i,j)xv(i,j)+Aτxh(iτh(i),j)xv(i,jτv(j))+Adl=1dh(i)xh(il,j)l=1dv(j)xv(i,jl),i,jZ+,where xh(i,j)Rnh and xv(i,j)Rnv are the horizontal state vector and the vertical state vector, respectively, E,A,Aτ,AdRn×n (n=nh+nv) are given real matrices, where E is singular with rank(E)=rn. τh(i), dh(i) and τv(j), dv(j) are respectively the

Main results

We are now in a position to derive LMI-based conditions ensuring that system (1) is admissible. For the brevity, we denote the block matrix I(α,β)=diag(αInh,βInv) for scalars α,β and the following augmented vectors x(i,j)=xh(i,j)xv(i,j),x(i+1,j+1)=xh(i+1,j)xv(i,j+1),xτ(i,j)=xh(iτh(i),j)xv(i,jτv(j)),xτM(i,j)=xh(iτhM,j)xv(i,jτvM),xτm(i,j)=xh(iτhm,j)xv(i,jτvm),xd(i,j)=l=1dh(i)xh(il,j)l=1dv(j)xv(i,jl),zh(i,j)=xh(i+1,j)xh(i,j),zv(i,j)=xv(i,j+1)xv(i,j),η(i,j)=x(i,j)xτ(i,j)xτm(i,j)xτM(i

Numerical examples

In this section, we give some numerical examples to illustrate the effectiveness and least conservativeness of our obtained results in this paper.

Example 1

Many thermal process in reactors, heat exchanger or pipe furnaces can be described by Darboux partial differential equations [25]. In this example, we consider a thermal process which is described by the following delayed partial differential equation T(s,t)x+T(x,t)t=a0T(x,t)+a1T(xτx,t)+a2T(x,tτt)+c10xfT(xs,t)ds+c20tfT(x,tθ)dθ

Concluding remarks

This paper has dealt with the problem of stability analysis of 2-D descriptor systems with mixed directional time-varying delays. Based on an improved 2-D LKF and utilizing the technique of zero-type free matrix equations, improved delay-dependent conditions have been formulated in terms of tractable LMIs to ensure that the underlining system is regular, causal and internally stable. The effectiveness and advantages of the proposed method in this paper have been demonstrated by the given three

Acknowledgment

This work was supported by the NAFOSTED of Vietnam under Grant 101.01-2018.05.

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