Finite-Time Stability of Large-Scale Systems with Interval Time-Varying Delay in Interconnection

We investigate finite-time stability of a class of nonlinear large-scale systems with interval time-varying delays in interconnection. Time-delay functions are continuous but not necessarily differentiable. Based on Lyapunov stability theory and new integral bounding technique, finite-time stability of large-scale systems with interval time-varying delays in interconnection is derived. The finite-time stability criteria are delays-dependent and are given in terms of linear matrix inequalities which can be solved by various available algorithms. Numerical examples are given to illustrate effectiveness of the proposed method.


Introduction
It is well known that several real-world processes often depend on time-delay; namely, the present state depends on the past states.Consequently, time-delay systems have been investigated extensively in the past decades.Time-delay is often the main source of instability and poor performance of the systems (see [1][2][3][4][5]).Therefore, it is important to study time-delay systems, especially those with time-varying delays.
There are two types of stability of time-varying delays systems, namely, delay-dependent and delay-independent ones.The delay-dependent conditions are usually less conservative than delay-independent ones, especially when the time-delays are relatively small.One of the main purposes of delay-dependent stability criteria is to estimate the maximum possible allowable delay bounds.
For stability analysis of time-delay systems, a common approach used is by choosing an appropriate Lyapunov-Krasovskii functional and then properly estimating upper bounds of its derivative along trajectories of solutions of the system (see [1,6]).In order to estimate such upper bounds of derivative of Lyapunov-Krasovskii functional, various mathematical tools have been used such as the Jensen inequality [3,4,7], lower bound lemma for reciprocal convexity [8][9][10][11], delay partitioning method [11], and freeweighting matrix variables method [6,12,13].Recently, a new integral inequality which improves Jensen inequality, namely, Wirtinger-based integral inequality, has been proposed in [14].In [15], a novel integral inequality has been proposed which has been shown to be less conservative than Jensen inequality.In [16,17], some new bounding techniques have been developed and enhanced stability criteria for time-delay systems have been derived using these bounding techniques which are shown to be less conservative than some existing results which used Wirtinger-based or Jensen inequalities.In [18], by using Wirtinger-based integral inequality combined with the reciprocally convex approach, some less conservative stability criteria for time-delay systems are derived.
Large-scale interconnected systems have been the subject of considerable research (see [19][20][21][22][23][24][25]), which can be characterized by a large number of variables representing the systems, a strong interaction between subsystem variables, and a complex interaction between subsystems.The large-scale interconnected system can be found in many practical control problems such as transportation systems, electrical power systems, communication systems, and economic systems.The operation of large-scale interconnected systems requires the capability of monitoring and stabilizing in the face of uncertainties, disturbances, failures, and attacks through the 2 Complexity utilization of internal system states.However, even with the assumption that all the state variables are available for feedback control, the task of effective controlling of a largescale interconnected system using a global (centralized) state feedback controller is still not easy as there is a necessary requirement for information transfer between the subsystems [22].
Generally, studies of dynamical systems are involved with stability analysis which is defined over an infinite-time interval.Nevertheless, in many real-world applications, the main aim is concerned with the behavior of the system over a fixed finite-time interval, for instance, the problem of sending a rocket from the neighborhood of point  to the neighborhood of point  over a fixed time interval [26].In this example, the concept of finite-time stability (FTS) is proposed.The problem of FTS has been revisited using linear matrix inequality (LMI) technique which allows us to find feasible condition guaranteeing FTS (see [26][27][28][29]).Some interesting results on stability and stabilization in the context of time-delay systems have been obtained by using the Lyapunov-Krasovskii functional technique (see [7,11,30]).Most existing work assumed that the timedelays are either constant or differentiable.In [20,24], delaydependent sufficient conditions for stability of large-scale system with time-varying delay were studied.However, the approach used there cannot be applied to systems with interval nondifferentiable time-varying delays.Moreover, to the best of our knowledge, there are few results for FTS of large-scale interconnected systems, especially systems with interval nondifferential time-varying delay.
Motivated by the above discussions, we shall derive new FTS criteria for large-scale systems with interval time-varying delays in interconnection.The main contributions of our studies are as follows.(i) The time-delay functions are only required to be continuous but not necessarily differentiable.(ii) By employing an improved integral inequality in [18], we derive new and less conservative FTS for large-scale interconnected systems with time-varying delays in terms of LMIs.
The rest of this paper is organized as follows.Section 2 presents notations, definitions, and auxiliary lemmas required for the proof of the main results.In Section 3, the FTS criteria for large-scale interconnected system with time-varying delays are obtained.Illustrative numerical examples are presented in Section 4. Section 5 gives the conclusion of the paper.

Problem Formulation and Preliminaries
Under the practical constraint of decentralized information coupled with the facts that the measurement of all the states and the real-time knowledge of the interval time-varying delays are not available, we consider the following nonlinear large-scale interconnected system: where () = [ and ℎ  () are time-delay functions which are continuous and satisfy for all  ≥ 0 and all ,  = The segment of the trajectory of () denoted by   is defined by   = {( + ) :  ∈ [−ℎ 2 , 0]} with the norm defined by Definition 1 (see [29]).For given positive numbers  1 ,  2 ,  and a symmetric positive definite matrix , the nonlinear large-scale system (1) is finite-time stable (FTS) with respect to ( 1 ,  2 , , ) if the following condition holds: Proposition 2 (see [22] Schur complement lemma).Given matrices , , , where  =   > 0 and  =   ,  +    −1  < 0 if and only if [     − ] < 0.

Finite-Time Stability of Nonlinear Large-Scale Systems
In this section, we derive FTS criteria for large-scale interconnected system with time-varying delays.For the sake of simplicity, the following notations will be used: For , ,  = 1, . . ., ,  ̸ =  and  ̸ = , we let ; The other matrices Θ  , and Ω  , which do not appear above would be defined to be zero matrices with appropriate dimensions.Theorem 6.For given positive numbers  1 ,  2 ,  and a symmetric positive definite matrix , if there exist symmetric positive definite matrices   ,   ,   ,   and matrices  1 , . . .,  4 of appropriate dimensions such that the conditions Complexity 5 hold, then system ( 1) is FTS with respect to ( 1 ,  2 , , ).
Proof.We choose the following Lyapunov-Krasovskii functional: where We first show that From we have where  1 = min =1,...,  min (  ).Thus, (16) holds.Similarly, we obtain the following estimation: Taking the derivative of (,   ) with respect to  along the trajectory of solution of the system (1), we obtain From Proposition 4, we have ( From Proposition 3 and Newton-Leibniz formula, we obtain for  = 1, 2. It follows that From Lemma 5, we have Thus, from (26), we obtain Hence, we obtain the estimation of V(,   ) as From ( 1) and Proposition 4, we get Adding inequality (29) into the left-hand side of ( 28) and from the fact that we have From assumption (12), it follows from Proposition 2 that Ω  < 0, ∀ = 1, . . ., .Therefore, we have Multiply both sides of (32) by  − ; we obtain Integrating both sides of (33) from 0 to , we have Hence, from ( 16) and (19), it follows that From ( 13), we have Therefore, system (1) is FTS with respect to ( 1 ,  2 , , ).
Remark 7. The obtained FTS criteria ( 12) are delay-dependent and are formulated in terms of solutions of LMIs which can be easily solved by various available algorithms such as MATLAB LMI Toolbox [31].
Remark 8.In Theorem 6, the chosen Lyapunov-Krasovskii functional depends only on the lower and upper bounds of time-delay functions to remove restrictions on the differentiability of time-delay functions.In several existing results on stability of large-scale systems, for example, in [20,24], the restrictions on the differentiability of time-delay functions are required.As a result, the methods proposed in [20,24] are not applicable to system (1).
Remark 9.In [6,12,13], the free-weighting matrix variables have been introduced to obtain stability conditions for timedelay systems.On the other hand, we have not introduced free-weighting matrix variables in order to derive stability criteria which results in less matrix variables regarding computational burdens.
Remark 10.In Theorem 6, we have used Proposition 3 (Jensen inequality) to estimate some integrals and to derive FTS criterion for large-scale interconnected systems with time-varying delays.Nonetheless, Proposition 3 is rather restrictive and further improvement can be made if new refined Jensen-based inequalities reported in the recent literature, for example, in [16,17], are used.
Remark 11.In (1), if   (⋅) ≡ 0, then we have the following linear large-scale system with interval time-varying delays in interconnection: Then, from Theorem 6, we obtain FTS criteria for linear largescale system (37) as in the following corollary.
Corollary 12.For given ,  > 0,  2 >  1 > 0, and let  be a symmetric positive definite matrix.If there exist symmetric positive definite matrices   ,   ,   ,   and matrices  1 , . . .,  4 of appropriate dimensions such that the conditions and other matrices Γ  , which do not appear above would be defined to be zero matrices with appropriate dimensions, then system (37) is FTS with respect to ( 1 ,  2 , , ).

Numerical Examples
In this section, we provide numerical examples to illustrate the effectiveness of our theoretical results.

Conclusion
In this paper, new delay-dependent FTS criteria of nonlinear large-scale interconnected systems with time-varying delay have been derived in terms of solutions of LMIs which could be solved by various available algorithms.By choosing an appropriate Lyapunov-Krasovskii functional and then by using an improved integral inequality, it has been shown by two numerical examples that the obtained FTS criteria are effective and less conservative than some existing results.

Notations
R  : Th e-dimensional Euclidean space R × : Th es e to fa l l ×  real matrices N: The set of all positive integers diag {⋅}: Th eb l o c kd i a g o n a lm a t r i x   : The transposition of matrix   ≥ 0 ( > 0):  is a semipositive definite matrix (positive)  ≥  ( > ):  −  ≥ 0 ( −  > 0)  max (): max{Re() :  is eigenvalue of }  min (): min{Re() :  is eigenvalue of } ‖ ⋅ ‖: The usual Euclidean norm (or the induced matrix norm).

Table 1 :
[32]mum allowable upper bounds ℎ 2 of the time-varying delay for different values of the lower bounds ℎ 1 in example of[32].