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This paper is concerned with the problem of practical stability of linear time-varying delay systems in the presence of bounded disturbances. Based on some comparison techniques associated with positive systems, explicit delay-independent conditions are derived for determining a neighborhood of the origin which ultimately bounds all state trajectories of the system.
Ha Noi Metroplolitan University 12 PRACTICAL STABILITY OF LINEAR TIMETIME-VARYING DELAY SYSTEMS Le Van Hien Hanoi National University of Education Abstract: This paper is concerned with the problem of practical stability of linear timevarying delay systems in the presence of bounded disturbances Based on some comparison techniques associated with positive systems, explicit delay-independent conditions are derived for determining a neighborhood of the origin which ultimately bounds all state trajectories of the system Keywords: Practical stability, time-varying delay, Metzler matrix Email: hienlv@hnue.edu.vn Received 29 July 2018 Accepted for publication 15 October 2018 INTRODUCTION In practical systems, there usually exists an interval of time between a stimulation and the system response [1] This interval of time is often known as time delay of a system Since time-delay unavoidably occurs in engineering systems and usually is a source of poor performance, oscillations or instability [2], the problem of stability analysis and control of time-delay systems is essential and of great importance for theoretical and practical reasons [3] This problem has attracted considerable attention from the mathematics and control communities, see, for example, [4−10] When considering long-time behavior of a system, the framework of Lyapunov stability theory and its extensions for time-delay systems, the Lyapunov-Krasovskii and Lyapunov-Razumikhin methods, have been extensively developed [3] However, realistic systems usually exhibit characteristics for which theoretical definitions in the sense of Lyapunov can be quite restrictive [11] Namely, the desired state of a system may be mathematically unstable in the sense of Lyapunov, but the response of the system oscillates close enough to this state for its performance to be considered as acceptable Furthermore, in many control problems, especially for systems that may lack an equilibrium point due to the presence of disturbances or constrained states, the aim is to bring those states close to Scientific Journal − No27/2018 13 certain sets rather than to a particular state [12-16] In such situations, the concept of practical stability is more suitable and meaningful Practical stability, also referred to as ultimate boundedness with a fixed bound [17], was first proposed in [18], retaken and systematically introduced in [19] to address some potential practical limitations of Lyapunov stability These stability notions not only provide information on the stability of the system, but also characterize its transient behavior with estimations of the bounds on the system trajectories During last decades, considerable research attention has been devoted to study the practical stability of dynamical systems To mention a few, we refer the reader to recent papers [11, 14-16, 20-26] and the references therein It is worth to mention here that, in most of the existing results, the framework of Lyapunov and its variants, have been suitably developed as the main approach to derive conditions for some specific types of practical stability Particularly, in [21] some results parallel to the Lyapunov results have been proposed for the strict practical stability of a general class of delay differential systems Using the idea of perturbing Lyapunov functions combining with the comparison principle, the authors in [23] established sufficient conditions for various types of strict practical stability of nonlinear impulsive delay-free systems In [15] some results on practical stability of nonlinear delay-free switched systems without a common equilibrium and under a time-dependent switching signal were given by employing the idea of direct method proposed in [22] Some interesting applications of practical stability to realistic systems were investigated in [24] for brake model of a bike and in [25] for a congestion control model in computer networks by using some Lyapunov-like functions In [14] and [16], practical stabilization and extended Lyapunov methodology was developed for some classes of nonlinear control systems where the measurement is sampled and possibly delayed In [11], based on the Lyapunov-Krasovskii functional (LKF) proposed in [4], the authors derived sufficient practical stability and stabilizability conditions for LTI systems with constant delays in terms of feasible linear matrix inequalities (LMIs) As discussed by the authors, the approach allows one to constructively obtain bounds on the practical stability region By extending this approach to the case of neutral systems, the authors in [26] proposed a set of bilinear matrix inequalities (BMIs) for practical exponential convergence of a class of nonlinear neutral systems with multiple constant delays and bounded disturbances Although practical stability provides a more relaxed concept of stability, there are very few studies especially for time-varying delay systems Furthermore, when dealing with time-varying systems, the developed methodologies such as LKF and its variants either lead to matrix Riccati differential equations (RDEs) [27] or indefinite LMIs So far, there Ha Noi Metroplolitan University 14 has been no efficient computational tool available to solve RDEs or indefinite LMIs In addition, the constructive approaches proposed in the aforementioned works are inapplicable to time-varying systems Therefore, an alternative and efficient approach to address the problem of practical stability of time-varying systems with delays is obviously necessary This motivates us in the present research In this paper, we address the problem of practical stability of linear time-varying systems with time-varying delay and bounded disturbances We present a constructive approach based on some techniques developed for positive systems which we have successfully applied to linear time-varying systems with delays [28, 29] New explicit delay-independent conditions are derived for determining a neighborhood of the origin which attracts exponentially all state trajectories of the system In addition, our conditions also guarantee the Lyapunov exponential stability of the system in the absence of input disturbances The remainder of this paper is organized as follows In Section 2, we present the problem statement, review some background results and introduce some notations that will be used throughout this paper The main results are presented in Section Illustrative examples and a conclusion are given in Section and Section 5, respectively PRELIMINARIES Notation ℝ and ℕ denote the set of real numbers and natural numbers, respectively For a given n ∈ ℕ , n≜{1,2,…, n} ℝ n is the n-dimensional vector space endowed with the norm ‖x‖ = max |xi | for x = (xi) ∈ ℝ n The non-negative orthant of ℝ n will be ∈ denoted by ℝ n+ By int(X), we denote the interior of the subset X ⊂ ℝ n Let ℝ m×n be the set of all m × n real matrices For a matrix A ∈ ℝ m×n , ri ( A) ∈ ℝ1×m , denotes the ith row of A Inequalities between vectors will be understood componentwise Specifically, for u = (ui) and = ( i) in ℝ n , u ≥ write u instead ≫ means of int ( ℝ n+ ) = { x ∈ ℝ n : x ≫ 0} Denote for all ∈ ≥ > In and if particular, = ∈ i = if and only if > 0, otherwise for all ∈ then we ℝ = {x ∈ ℝ : x ≥ 0} n + then v = (vi ) ∈ int ( ℝ n+ ) We also specifically use the notation , that means > n and > for any vector = max{ , 0} for real number = Consider the following linear time-varying system with delay Scientific Journal − No27/2018 15 xɺ (t ) = A(t ) x (t ) + B (t ) x(t − τ (t )) + d (t ), t ≥ 0, x(t ) = φ (t ), t ∈ [−τ max , 0], = where ! ∈ ℝ n is the system state vector and " (1) = " ! ∈ ℝ n is unknown input disturbance vector, A(t ) = (aij (t )) ∈ ℝ n×n and B(t ) = (bij (t )) ∈ ℝ n×n are timevarying system matrices whose elements are assumed to be continuous on ℝ + , τ (t ) is a time-varying delay and φ (.) ∈ C ([−τ max , 0], ℝ n ) is the vector-valued initial function specifying the initial | φi | = sup | φi (t ) | and φ −τ max ≤t ≤ state ∞ of the system, φ (t ) = (φi (t )) ∈ ℝ n Let us denote = max i∈n | φi | Remark 2.1 In this paper, the time delay # is assumed to be continuous in time, ≤ # %& , for all ≥ 0, where the not necessarily differentiable, and satisfies ≤ # upper bound # %& is a known constant We not impose any restriction on the rate of (such as slowly time-varying condition #( ≤ #) 0, which allows a larger range of delay as τ < 12.4292 CONCLUSION This paper has addressed the problem of practical stability of linear time-varying systems with a time-varying delay and bounded disturbances New explicit conditions have been derived for determining a µ -neighborhood and a finite transient time T guaranteeing that all state trajectories of the system converge exponentially to the µ -neighborhood after the transient time T REFEENCES V Kolmanovskii and A Myshkis (1992), “Applied Theory of Functional Differential Equations’’, Dordrecht: Kluwer AP S.I Niculescu (2001), “Delay Effects 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nghiên cứu tính ổn định thực hành lớp hệ tuyến tính khơng dừng với trễ biến thiên nhiễu bị chặn Dựa số kĩ thuật so sánh lý thuyết hệ dương, điều kiện hiển, độc lập với độ trễ, thiết lập cho việc xác định lân cận compact điểm cân lí tưởng (điểm gốc) bao chung quỹ đạo trạng thái hệ Từ khóa: khóa Tính ổn định thực hành, trễ biến thiên, ma trận Metzler ... conditions for various types of strict practical stability of nonlinear impulsive delay- free systems In [15] some results on practical stability of nonlinear delay- free switched systems without a common... allows a larger range of delay as τ < 12.4292 CONCLUSION This paper has addressed the problem of practical stability of linear time-varying systems with a time-varying delay and bounded disturbances... problem of practical stability of linear time-varying systems with time-varying delay and bounded disturbances We present a constructive approach based on some techniques developed for positive systems