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LMI approach to decentralized exponential stability of linear large-scale systems with interval non-differentiable time-varying delays

Manlika Rajchakit1, Piyapong Niamsup23 and Grienggrai Rajchakit1*

Author Affiliations

1 Division of Mathematics and Statistics, Faculty of Science, Maejo University, Chiangmai, 50290, Thailand

2 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiangmai, 50200, Thailand

3 Center of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok, 10400, Thailand

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Advances in Difference Equations 2013, 2013:332  doi:10.1186/1687-1847-2013-332


The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/332


Received:1 August 2013
Accepted:14 October 2013
Published:19 November 2013

© 2013 Rajchakit et al.; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper addresses decentralized exponential stability problem for a class of linear large-scale systems with time-varying delay in interconnection. The time delay is any continuous function belonging to a given interval, but not necessarily differentiable. By constructing a suitable augmented Lyapunov-Krasovskii functional combined with Leibniz-Newton’s formula, new delay-dependent sufficient conditions for the existence of decentralized exponential stability are established in terms of LMIs. Numerical examples are given to show the effectiveness of the obtained results.

Keywords:
decentralized exponential stability; large-scale systems; uncertain systems; interval time-varying delay; Lyapunov function; linear matrix inequalities

1 Introduction

The theory and applications of functional differential equations form an important part of modern non-linear dynamics. Such equations are natural mathematical models for various real life phenomena where the after-effects are intrinsic features of their functioning. In recent years, functional differential equations have been used to model processes in different areas such as population dynamics and ecology, physiology and medicine, economics, and other natural sciences. Stability analysis of linear systems with time-varying delays <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M1">View MathML</a> is fundamental to many practical problems and has received considerable attention [1-4]. Most of the known results on this problem are derived assuming only that the time-varying delay <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M2">View MathML</a> is a continuously differentiable function, satisfying some boundedness condition on its derivative: <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M3">View MathML</a>. In delay-dependent stability criteria, the main concern is to enlarge the feasible region of stability criteria in a given time-delay interval. Interval time-varying delay means that a time delay varies in an interval in which the lower bound is not restricted to being zero. By constructing a suitable augmented Lyapunov functional and utilizing free weight matrices, some less conservative conditions for asymptotic stability are derived in [5-10] for systems with time delay varying in an interval. However, the shortcoming of the method used in these works is that the delay function is assumed to be differential and its derivative is still bounded: <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M4">View MathML</a>.

On the other hand, there has been a considerable research interest in large-scale interconnected systems. A typical large-scale interconnected system such as a power grid consists of many subsystems and individual elements connected together to form a large, complex network capable of generating, transmitting and distributing electrical energy over a large geographical area. In general, a large-scale system can be characterized by a large number of variables representing the system, a strong interaction between subsystem variables, and a complex interaction between subsystems. The problem of decentralized control of large-scale interconnected dynamical systems has been receiving considerable attention, because there is a large number of large-scale interconnected dynamical systems in many practical control problems, e.g., transportation systems, power systems, communication systems, economic systems, social systems, and so on [11-14]. The operation of large-scale interconnected systems requires the ability to monitor and stabilize in the face of uncertainties, disturbances, failures and attacks through the 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 a large-scale 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 [15-18].

To the best of our knowledge, there has been no investigation on the exponential stability of large-scale systems with time-varying delays interacted between subsystems. In fact, this problem is difficult to solve; particularly, when the time-varying delays are interval, non-differentiable and the output is subjected to such time-varying delay functions. The time delay is assumed to be any continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, but the delay function is bounded but not necessarily differentiable. This allows the time-delay to be a fast time-varying function and the lower bound is not restricted to being zero. It is clear that the application of any memoryless feedback controller to such time-delay systems would lead to closed loop systems with interval time-varying delays. The difficulties then arise when one attempts to derive exponential stability conditions. Indeed, the existing Lyapunov-Krasovskii functional and associated results in [11,14,15,18-35] cannot be applied to solve the problem posed in this paper as they would either fail to cope with the non-differentiability aspects of the delays, or lead to very complex matrix inequality conditions, and any technique such as matrix computation or transformation of variables fails to extract the parameters of the memoryless feedback controllers. This has motivated our research.

In this paper, we consider a class of large-scale linear systems with interval time-varying delays in interconnections. Compared to the existing results, our result has its own advantages. (i) Stability analysis of previous papers reveals some restrictions: The time delay was proposed to be either time-invariant interconnected or the lower delay bound is restricted to being zero, or the time delay function should be differential and its derivative is bounded. In our result, the above restricted conditions are removed for the large-scale systems. In addition, the time delay is assumed to be any continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, but the delay function is bounded but not necessarily differentiable. This allows the time-delay to be a fast time-varying function, and the lower bound is not restricted to being zero. (ii) The developed method using new inequalities for lower bounding cross terms eliminates the need for over-bounding and provides larger values of the admissible delay bound. We propose a set of new Lyapunov-Krasovskii functionals, which are mainly based on the information of the lower and upper delay bounds. (iii) The conditions will be presented in terms of the solution of LMIs that can be solved numerically in an efficient manner by using standard computational algorithms [36].

The paper is organized as follows. Section 2 presents definitions and some well-known technical propositions needed for the proof of the main results. Main result for decentralized exponential stability of large-scale systems is presented in Section 3. Numerical examples showing the effectiveness of the obtained results are given in Section 4. The paper ends with conclusions and cited references.

2 Preliminaries

The following notations are used in this paper. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M5">View MathML</a> denotes the set of all real non-negative numbers; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M6">View MathML</a> denotes the n-dimensional space with the scalar product <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M7">View MathML</a> and the vector norm <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M8">View MathML</a>; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M9">View MathML</a> denotes the space of all matrices of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M10">View MathML</a>-dimensions; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M11">View MathML</a> denotes the transpose of matrix A; A is symmetric if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M12">View MathML</a>; I denotes the identity matrix; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M13">View MathML</a> denotes the set of all eigenvalues of A; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M14">View MathML</a>; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M15">View MathML</a> denotes the set of all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M6">View MathML</a>-valued differentiable functions on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M17">View MathML</a>; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M18">View MathML</a> stands for the set of all square-integrable <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M19">View MathML</a>-valued functions on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M20">View MathML</a>. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M21">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M22">View MathML</a>; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M23">View MathML</a> denotes the set of all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M24">View MathML</a>-valued continuous functions on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M25">View MathML</a>; matrix A is called semi-positive definite (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M26">View MathML</a>) if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M27">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M28">View MathML</a>; A is positive definite (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M29">View MathML</a>) if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M30">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M31">View MathML</a>; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M32">View MathML</a> means <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M33">View MathML</a>. ∗ denotes the symmetric term in a matrix.

Consider a class of linear large-scale systems with interval time-varying delays composed of N interconnected subsystems <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M34">View MathML</a> of the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M35">View MathML</a>

(2.1)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M36">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M37">View MathML</a>, is the state vector, the system matrices <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M38">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M39">View MathML</a> are of appropriate dimensions.

The time delays <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M40">View MathML</a> are continuous and satisfy the following condition:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M41">View MathML</a>

and the initial function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M42">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M43">View MathML</a>, with the norm

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M44">View MathML</a>

Definition 2.1 Given <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M45">View MathML</a>. The zero solution of system (2.1) is α-exponentially stable if there exists a positive number <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M46">View MathML</a> such that every solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M47">View MathML</a> satisfies the following condition:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M48">View MathML</a>

We end this section with the following technical well-known propositions, which will be used in the proof of the main results.

Proposition 2.1For any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M49">View MathML</a>and positive definite matrix<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M50">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M51">View MathML</a>

Proposition 2.2 (Schur complement lemma [37])

Given constant matricesX, Y, Zwith appropriate dimensions satisfying<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M52">View MathML</a>. Then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M53">View MathML</a>if and only if

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M54">View MathML</a>

Proposition 2.3[38]

For any constant matrix<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M55">View MathML</a>and scalarh, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M56">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M57">View MathML</a>such that the following integrations are well defined, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M58">View MathML</a>

3 Main results

In this section, we investigate the decentralized exponential stability of linear large-scale system (2.1) with interval time-varying delays. It will be seen from the following theorem that neither free-weighting matrices nor any transformation are employed in our derivation. Before introducing the main result, the following notations of several matrix variables are defined for simplicity

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M59">View MathML</a>

The following is the main result of the paper, which gives sufficient conditions for the decentralized exponential stability of linear large-scale system (2.1) with interval time-varying delays. Essentially, the proof is based on the construction of Lyapunov Krasovskii functions satisfying the Lyapunov stability theorem for a time-delay system [37].

Theorem 3.1Given<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M45">View MathML</a>. System (2.1) isα-exponentially stable if there exist symmetric positive definite matrices<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M61">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M62">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M63">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M64">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M65">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M34">View MathML</a>, and matrices<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M67">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M34">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M69">View MathML</a>, such that the following LMI holds:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M70">View MathML</a>

(3.1)

Moreover, the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M71">View MathML</a>of the system satisfies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M72">View MathML</a>

Proof We consider the following Lyapunov-Krasovskii functional for system (2.1):

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M73">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M74">View MathML</a>

It is easy to verify that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M75">View MathML</a>

(3.2)

Taking the derivative of V in t along the solution of system (2.1), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M76">View MathML</a>

Applying Proposition 2.3 and the Leibniz-Newton formula

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M77">View MathML</a>

we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M78">View MathML</a>

Note that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M79">View MathML</a>

Using Proposition 2.3 gives

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M80">View MathML</a>

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M81">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M82">View MathML</a>

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M83">View MathML</a>

Similarly, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M84">View MathML</a>

Note that when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M85">View MathML</a> or <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M86">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M87">View MathML</a>

respectively. Besides, using Proposition 2.3 again, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M88">View MathML</a>

Hence,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M89">View MathML</a>

Therefore, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M90">View MathML</a>

(3.3)

By using the following identity relation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M91">View MathML</a>

we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M92">View MathML</a>

(3.4)

Adding all the zero items of (3.4) into (3.3), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M93">View MathML</a>

Applying Proposition 2.1, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M94">View MathML</a>

(3.5)

Therefore, applying inequalities (3.5) and noting that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M95">View MathML</a>

we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M96">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M97">View MathML</a>.

By condition (3.1), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M98">View MathML</a>

(3.6)

Integrating both sides of (3.6) from 0 to t, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M99">View MathML</a>

Furthermore, taking condition (3.2) into account, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M100">View MathML</a>

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M101">View MathML</a>

This completes the proof of the theorem. □

Remark 3.1 Theorem 3.1 provides sufficient conditions for linear large-scale system (2.1) in terms of the solutions of LMIs, which guarantees the closed-loop system to be exponentially stable with a prescribed decay rate α. The developed method using new inequalities for lower bounding cross terms eliminates the need for over-bounding and provides larger values of the admissible delay bound. Note that the time-varying delays are non-differentiable; therefore, the methods proposed in [11,14,15,18-35] are not applicable to system (2.1). LMI condition (3.2) depends on parameters of the system under consideration as well as the delay bounds. The feasibility of the LMIs can be tested by the reliable and efficient Matlab LMI Control Toolbox [36].

4 Numerical examples

In this section, we give a numerical example to show the effectiveness of the proposed result.

Example 4.1 This example is a large-scale model composed of two machine subsystems as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M102">View MathML</a>

where the absolute rotor angle and angular velocity of the machine in each subsystem are denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M103">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M104">View MathML</a>, respectively; the ith system coefficient <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M38">View MathML</a>; the modulus of the transfer admittance <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M39">View MathML</a>; the initial input <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M107">View MathML</a>; the time-varying delays <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M108">View MathML</a> between the two machines in the subsystem:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M109">View MathML</a>

It is worth nothing that the delay functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M110">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M111">View MathML</a> are non-differentiable; therefore, the methods in [11,14,15,18-35] are not applicable to this system. By using LMI Toolbox in Matlab [36], LMIs (3.1) is feasible with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M112">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M113">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M114">View MathML</a>, and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M115">View MathML</a>

According to Theorem 3.1, the system is exponentially stable. Finally, the solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M71">View MathML</a> of the system satisfies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M117">View MathML</a>

Figure 1 shows the trajectories of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M118">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M119">View MathML</a> of the closed loop system with the initial conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M120">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/332/mathml/M121">View MathML</a>.

thumbnailFigure 1. The trajectories of a solution of the linear large-scale system.

The trajectories of a solution of the linear large-scale system are shown in Figure 1, respectively.

5 Conclusion

In this paper, the problem of the decentralized exponential stability for large-scale time-varying delay systems has been studied. The time delay is assumed to be a function belonging to a given interval, but not necessarily differentiable. By effectively combining an appropriate Lyapunov functional with the Newton-Leibniz formula and free-weighting parameter matrices, this paper has derived new delay-dependent conditions for the exponential stability in terms of linear matrix inequalities, which allow simultaneous computation of two bounds that characterize the exponential stability rate of the solution. The developed method using new inequalities for lower bounding cross terms eliminates the need for over-bounding and provides larger values of the delay bound. Numerical examples are given to show the effectiveness of the obtained result.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally and significantly in writing this paper. The authors read and approved the final manuscript.

Acknowledgements

This work was supported by the Thailand Research Fund Grant, the Commission for Higher Education and Faculty of Science, Maejo University, Thailand. The second author is supported by the Center of Excellence in Mathematics, Thailand, and Commission for Higher Education, Thailand. The authors thank anonymous reviewers for valuable comments and suggestions, which allowed us to improve the paper.

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