A novel approach to exponential stability of continuous-time Roesser systems with directional time-varying delays
Introduction
Two-dimensional (2D) systems can be used in modeling a large number of practical and physical processes where the information propagation occurs in each of the two independent directions such as thermal processes, gas absorption or water stream heating [1]. During the past few decades, the study of 2D systems both in theory and practice has attracted an increasing attention due to their extensive applications, particularly, in circuit analysis, digital image processing, multi-dimensional digital filtering, repetitive processes or iterative learning control. To mention a few, we refer the reader to [2], [3], [4], [5], [6], [7] and the recent work [8].
On the other hand, it has been well-recognized that time-delay frequently occurs in engineering systems due to many practical reasons such as the finite speed of data processing through a low-rate communication channel or sensor technology that inevitably introduces non-negligible time-delay. In addition, the reaction of realistic systems to exogenous signals is never instantaneous and always infected by time-delay which may degrade the system performance and even causes the system instability. Therefore, the study of stability analysis and control of time-delay systems plays an important role in applied models which has attracted a remarkable attention in recent years [9], [10], [11], [12], [13], [14]. While the study on stability analysis and control of one-dimensional (1D) time-delay systems has been extensively developed, this problem for 2D systems has gained growing attention recently. In [15], [16], [17], based on the Cauchy matrix inequality and the method of using slack matrix variables, delay-dependent stability conditions were derived in the form of linear matrix inequalities (LMIs) for discrete-time 2D systems with time-varying delays in the presence of saturation and quantization nonlinearities. By utilizing the technique of free-weighting matrices, delay-dependent robust stability conditions were derived in [18] for a class of 2D systems described by the second Fornasini–Marchesini model with delays and uncertainties. The problems of control, filtering and state estimation for discrete-time 2D systems with time-delay and disturbances were also studied in [19], [20], [21], [22], [23] by employing the Lyapunov–Krasovskii functional (LKF) method combining with Jensen-based inequalities and free-weighting matrix technique.
Looking at the literature one can realize that while a variety of useful results have been developed for 2D discrete-time systems with and without delays, there have been only a few results devoted to the study of 2D continuous-time systems with delays. On the one hand, many physical processes are actually described by 2D continuous-time systems. For instance, dynamic processes in gas absorption, water stream heating and air drying modeled by Darboux partial differential equations can be directly transformed to 2D continuous-time systems [1]. On the other hand, by a discretization from continuous-time systems, 2D discrete-time systems inherit similar behavior of the continuous ones. Thus, it is relevant and important to study the problem of stability analysis and control of 2D continuous-time systems with delays. Very recently, the LKF method has been applied to derive delay-dependent asymptotic stability conditions and controller design for 2D continuous-time systems with delays [24], [25], [26], [27], [28], [29]. However, the problem of exponential stability analysis of 2D continuous-time systems with delays has not received much attention. In many practical applications such as fast affect control, not only the convergence but also the rate of convergence or the transient decay rate of the system states is a crucial factor to determine. Therefore, the problem of exponential stability analysis is an important issue for time-delay systems. This problem has been well-developed in the context of 1D systems and various approaches have been proposed in the literature such as the use of state transformation [30], [31], weighted LKF [32], [33], [34], [35], [36], [37] or the comparison principle based on differential inequalities [38], [39]. However, the problem of exponential stability analysis of 2D continuous-time systems still poses as a challenging problem. It should be pointed out that the traditional approach based on the LKF method for 1D systems cannot be easily extended to deal with 2D systems. Specifically, for 1D systems, from condition one can easily get which combined with positiveness of functional V(t), named as LKF, gives an exponential estimate with decay rate α for all solutions of the system. Unfortunately, a similar condition , where , , does not give an exponential estimate for 2D systems. Thus, an abstract result or a general scheme for obtaining exponential estimate with prescribed decay rate for 2D continuous-time systems is obviously necessary. So far, there has no result dealt with the problem of exponential estimate for 2D systems with directional time-varying delays which inspires us for the present study.
In this paper, the problem of exponential stability analysis of linear 2D continuous-time systems with directional time-varying delays is investigated. The main contributions of the present paper are as follows.
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An abstract Lyapunov-like theorem which ensures that a linear 2D system with delays is exponentially stable for a prescribed decay rate is first established.
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A systematic approach to the problem of exponential stability of 2D systems with directional time-varying delays based on the abstract theorem is proposed.
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In light of general scheme, and by utilizing 2D weighted integral inequalities proposed in this paper, new delay-dependent exponential stability conditions are derived for a class of 2D continuous-time systems of which the directional delays are time-varying without any restriction on the rate of change. These conditions are derived in terms of tractable matrix inequalities which can be solved by various computational tools to obtain maximum allowable bound of delays and exponential decay rate.
The remaining of this paper is organized as follows. Section 2 presents the problem formulation and some stability concepts. A Lyapunov-like theorem is given in Section 3. New 2D weighted integral inequalities and exponential stability conditions for 2D systems are derived in Section 4. Numerical examples are given in Section 5 to illustrate the effectiveness of the results obtained in this paper.
Notation. and denote the n-dimensional Euclidean space with the vector norm written as and the set of matrices, respectively. For , and denote the transpose and the induced matrix norm of A defined by , . A matrix is symmetric semi-positive definite, write , if , for all , and is positive definite, write , if for all . We denote by the set of symmetric positive definite matrices in . For a subset , denotes the set of -valued continuous functions on Ω. The partial derivatives , , if exist, will be denoted as and , respectively.
Section snippets
Problem formulation
Consider the following Roesser-type continuous-time 2D systemswhere and are the horizontal and vertical state vectors, respectively, () and are given real matrices, τh and τv are directional time-varying delays along horizontal direction and vertical direction, respectively, which satisfywhere and are known constants. In this
Lyapunov-based functional method
To facilitate in presenting our abstract result, in the following we rewrite the problem (1), (2), (3a), (3b) in the operator form. First, splitting the matrices as where and . Then, the operator , , where , defined byis a linear operator. In addition, . The operators and , where
Exponential stability analysis of 2D time-varying delay systems
In this section, we utilize the Lyapunov-based method developed in the preceding section to derive exponential stability conditions for systems (1). First, some new 2D integral inequalities are established. Then, in light of Theorem 1, and by constructing an augmented LKF, delay-dependent conditions that ensure exponential stability of system (1) with a prescribed decay rate are derived in terms of tractable linear matrix inequalities.
Numerical examples
In this section, two numerical examples are given to illustrate the effectiveness of the derived stability conditions in this paper. Example 1 Consider 2D system (1) with the following data and directional time-varying delays , , where are positive constants representing the frequencies of the delays τh and τv. Note at first that is continuous, non-differentiable at arbitrarily many points , . Additionally, in each interval ,
Concluding remarks
The problem of exponential stability analysis with prescribed decay rate has been studied for a class of 2D linear continuous-time systems with time-varying delays. A systematic approach based on an abstract Lyapunov-like scheme and new 2D integral inequalities has been proposed. By utilizing the method presented in this paper, delay-dependent conditions that ensure exponential stability of 2D continuous-time systems with a prescribed decay rate have been derived in terms of tractable matrix
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