On new stability criterion for delay-differential systems of neutral type

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Abstract

In this paper, a novel stability criterion is presented for neutral differential systems. Based on the Lyapunov method, the stability criterion is derived in terms of linear matrix inequalities which can be easily solved by efficient convex optimization algorithms. Numerical examples are included to show the effectiveness of the proposed method.

Introduction

Time delay arises naturally in connection with system process and information flow for different part of dynamical systems. Therefore, considerable efforts are concentrated on the stability analysis for systems including time delays, since the delay phenomenon is often encountered in various systems, and is frequently a source of instability. For more information of delay systems, see Refs. [1], [2].

Recently, the stability analysis of neutral differential systems, which contain delays both in its state and in the derivatives of its states, has been widely investigated by many researcher [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. Some stability criteria [3], [4] formulated in terms of matrix measure and matrix norm or spectral radius have been presented based on frequency-domain methods. Also, since the linear matrix inequalities (LMI) technique combined with Lyapunov functional approach has been presented by Park and Won [5], the LMI technique has been extensively utilized to obtain less conservative criteria for asymptotic stability of several class of neutral systems [6], [7], [8], [9], [10], [11], [12], [13]. The developed stability criteria in this literature are classified often into two categories according to their dependence on the size of the delays, namely, delay-independent criteria and delay-dependent criteria. In general, the delay-dependent stability criterion is less conservative than delay-independent one when the size of time-delay is small.

In this paper, we present a novel delay-dependent stability criterion for neutral delay differential systems based on a new model transformation technique. The proposed methods employs free weighting matrices to obtain less conservative stability criterion. The criterion is derived in terms of LMIs, which can be easily solved by using various convex optimization algorithms [16]. Four numerical examples are given to show the superiority of the present result to those in the literature.

Notations

Rn is the n-dimensional Euclidean space. Rm×n denotes the set of m×n real matrix. ★ denotes the symmetric part. X>0 (X⩾0) means that X is a real symmetric positive definitive matrix (positive semi-definite). I denotes the identity matrix with appropriate dimensions. diag{…} denotes the block diagonal matrix. Cn,h=C([−h,0],Rn) denotes the Banach space of continuous functions mapping the interval [−h,0] into Rn, with the topology of uniform convergence.

Section snippets

Main results

Consider the neutral systems described by the following state equation:ẋ(t)=Ax(t)+A1x(t−h)+A2ẋ(t−h),x(s)=φ(s),s∈[−h,0],where x(t)∈Rn is the state vector, A,A1,A2Rn×n are known real parameter matrices, h>0 is a constant delay, φ(s)∈Cn,h is a given continuous vector valued initial function.

Now, define an operator D(xt):Cn,hRn asD(xt)=x(t)+∫tt−hGx(s)ds−A2x(t−h),where xt=x(t+s),s∈[−h,0] and G∈Rn×n is a constant matrix which will be chosen to make the system asymptotically stable. With the above

Numerical examples

Example 1

Consider the following neutral systems:ddtx(t)−0.2100.2x(t−h)=−20.10−1x(t)+00.50.50x(t−h).

Using the approaches in [3], [5], [7], [8], [9], no conclusion can be made since their stability condition for certain operators are not satisfied. However, by applying Theorem 1, one can see that the system is asymptotically stable for h<1.256.

For reference, the solutions of LMIs of Theorem 1 for h=1.2559 with α=3.24 are as follows:X=865.3813−87.7497−87.7497137.1341,W=103×1.1259−0.0609−0.06090.0046,Y=

Conclusions

In this paper, we present a novel stability criterion for asymptotic stability of a class of neutral systems. Utilizing the operator which has free weighting matrix, we transform the original system to the equivalent neutral system. Then, the delay-dependent stability criteria is derived in terms of LMIs by establishing the Lyapunov functional which have free weighting matrices. Through numerical examples, we showed the derived criterion is less conservative than those in other literature.

References (18)

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