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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 304824, 12 pages

http://dx.doi.org/10.1155/2014/304824

## Resilient Robust Finite-Time Controller Design for Uncertain Neutral System with Mixed Time-Varying Delays

School of Electrical Engineering and Automation, Anhui University, Hefei 230601, China

Received 29 January 2014; Accepted 7 April 2014; Published 13 May 2014

Academic Editor: Hao Shen

Copyright © 2014 Xia Chen and Shuping He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The delay-dependent resilient robust finite-time control problem of uncertain neutral time-delayed system is studied. The disturbance input is assumed to be energy bounded and the time delays are time-varying. Based on the Lyapunov function approach and linear matrix inequalities (LMIs) techniques, a state feedback controller is designed to guarantee that the resulted closed-loop system is finite-time bounded for all uncertainties and to satisfy a given constraint condition. Simulation results illustrate the validity of the proposed approach.

#### 1. Introduction

Dynamical systems with time delays and uncertain parameters have been of considerable interest over the past decades. In fact, time delays are always the important source of system instability and poor performance [1–4]. As a special class of time-delay systems, the neutral type time-delayed system has also received some attention in recent years. This time-delayed system contains time delays both in its state and in the derivative of its states. Moreover, neutral time-delayed systems are frequently encountered in many dynamics, such as automatic control, distributed network system containing lossless transmission line, heat exchangers, and population ecology. Various analysis approaches have been utilized to find stability criteria and control design conditions for asymptotic stability of neutral time delays [5–10].

It is now worth pointing out that the control performances mentioned above concern the desired behavior of control dynamics over an infinite-time interval and it always deals with the asymptotic property of system trajectories. For controlling a dynamical system, it can meet the requirements of asymptotic stability, but it will not reflect the transient characteristics. Asymptotic stability is unable to satisfy the transient requirements of industrial production if there exists large amount of overshoot, oscillation change, and nonlinear disturbance within a finite-time interval. To deal with this transient performance of control dynamics, Dorato gave the concept of finite-time stability [11] (or short-time stability) in the early 1960s. Then, the relevant concepts of finite-time bounded (FTB) [12], finite-time stabilization [13], finite-time control [14], and finite-time [15] control have been revisited in form of linear matrix inequalities (LMIs) techniques. And this transient performance is widely applied to time-delay systems, uncertain systems, nonlinear systems, stochastic systems, and so forth. However, to the best of our knowledge, very few results in the literature consider the related control problems of neutral time-varying delays in the finite-time interval.

On the other hand, the - performance has attracted considerable attention as an important performance evaluation index when it was first proposed in 1989 [16]. In engineering practice, although the study of the impact of noise and delay on the system performance is important, the extremum problem of the controlled output cannot be ignored, because the controlled output should be controlled within a certain range. In control theory and engineering application, the - control has very important significance that lies in its performance index which can control the output value minimization. Unfortunately, up to now, the theme of - control design of uncertain neutral systems with time-varying delays has received little attention.

Motivated by the above discussion, this paper focuses on the problem of finite-time - controller design for a class of neutral systems with mixed time-varying delays and uncertainties. By constructing a suitable Lyapunov function, the sufficient conditions are derived that closed-loop controlled system is FTB and satisfies the given finite-time interval induced - norm of the operator from the unknown disturbance to the output. We also show that the - controller designing problem can be dealt with by solving a set of coupled LMIs. Finally, a numerical example illustrates the effectiveness of the developed techniques.

#### 2. Problem Statement

Consider the following neutral time-delayed system with uncertainties: where is the state, is the controlled input, is the controlled output, and is the disturbance input that belongs to and for a given positive number and constant time , the following form is satisfied: and are time-varying delays and satisfy where , , , and are constant scalars. is the continuous initial function. , , , , and are known constant matrices, and , , , , and are unknown time-variant matrices representing the norm-bounded parameter uncertainties and satisfy the following form: where , , , , , and are known real matrices with suitable dimension and is an unknown real and possibly time-varying matrix with Lebesgue measurable elements satisfying

In this paper, we consider the state feedback controller as follows: where is the unknown controller gain and is the time-varying controller gain which satisfies

Then, we can get the following closed-loop control system: where , , , , , , , , and .

The main purpose of this paper is to design an appropriate resilient state feedback controller (7), such that the closed-loop control system is finite-time bounded and satisfies the given performance index constraints.

Before proceeding with the study, we give the relevant definitions and lemmas first.

*Definition 1. *For given positive scalars , , and and a symmetrical positive determined matrix , the closed-loop system is robust finite-time bounded (FTB) with respect to , if there exists a positive constant with , such that, for all the external disturbances satisfying condition (2), the following formula is satisfied:

*Remark 2. *If the disturbance input is not present in the closed-loop system, that is, , the concept of FTB will reduce into finite-time stability (FTS). It is worth mentioning that Lyapunov stability and finite-time stability are two different concepts. The former is largely known to the control characteristic in infinite-time interval, but the latter concerns the boundedness analysis of the controlled states within a finite-time interval. Obviously, a finite-time stable system may not be Lyapunov stochastically stable and vice versa.

*Definition 3. *The state feedback controller in the form of (7) is considered as a robust finite-time - controller for the closed-loop system , if the system is FTB with respect to and under the zero initial condition, there exist two positive scalars and for all disturbance which satisfy condition (2), such that
where , .

Lemma 4 (see [17]). *For any real positive scalars , (where ) and a positive definite symmetric matrix , then the following inequality holds for a vector function which can let the integrals converge:
*

Lemma 5 (see [17]). *For any positive scalar and positive definite symmetric matrix , the following inequality is satisfied:
*

Lemma 6 (see [15]). *For any given appropriate dimension matrix and , if there exists a matrix which satisfied and a scalar , then
*

#### 3. Main Results

In this section, our main purpose is to solve the design problem of a resilient robust finite-time - controller for a class of uncertain neutral systems with mixed time-varying delays.

Theorem 7. *Given positive scalars , , , and , positive definite symmetric matrix , and time-delay parameters , , , and , the closed-loop system is FTB with respect to , if there exist positive scalars , , and symmetric positive definite matrices , , and , such that
**
where
*

*Proof. *Construct a positive definite Lyapunov function as follows:
where
We take the time derivative of along the trajectory of system and it yields the following:

For any symmetric positive definite matrices , the following equations are satisfied according to Leibniz-Newton lemma:
where

According to (20)-(21), we can obtain
Since , , , and are positive definite symmetric matrices, we have
where

Recalling formula (24) and Lemmas 4 and 5 and using Schur complement, we can get
that is,
Pre- and postmultiplying (27) by , we have
Then integrating the aforementioned inequality from 0 to , where , it yields
Considering condition (2), (29) can be simplified as
On the other hand,

Then, formula (27) can be written as
which can be guaranteed by condition (16). This completes the proof.

According to Theorem 7, we will obtain the resilient robust finite-time - controller for a class of uncertain neutral system with mixed time-varying delays.

Theorem 8. *Given positive scalars , , , and , positive definite symmetric matrix , and time-delay parameters , , , and , the closed-loop neutral system is FTB with respect to and satisfies the cost function (11) for all admissible disturbance , if there exist positive scalars and and symmetric positive definite matrices , , , such that conditions (15) and (16) and the following LMI hold:
*

*Proof. *Similar to the proof of Theorem 7, (29) can be rewritten as
Then, we have

From (33), we can obviously get
Considering system , we have
Combining (35)–(37), we can obtain
that is,

Letting , we have . This completes the proof.

Theorem 9. *Given positive scalars , , , and , positive definite symmetric matrix , and time-delay parameters , , , and , the closed-loop neutral system is FTB with respect to , satisfies the cost function (11) for all admissible disturbance , and exists as a state feedback controller in the form of (7) with , if there exist positive scalars , , , , and , , and symmetric positive definite matrices , , , , , , , , , and , such that the following LMIs are feasible:
*

*where*

*Proof. *Replacing , , , and in (15) with , , , , , and , respectively, we have
wherebring formulas (4) and (8) into , and according to Lemma 6, we have
whereConsidering
(49) can be guaranteed by
Using Schur complement, equality (54) can be rewritten as
where

Letting , , , , , , , , , , , , , ,