Mathematical Problems in Engineering

Volume 2014, Article ID 184292, 15 pages

http://dx.doi.org/10.1155/2014/184292

## Stability Analysis and Output Tracking Control for Linear Systems with Time-Varying Delays

^{1}School of Electrical Engineering, Chungbuk National University, 52 Naesudong-ro, Cheongju 361-763, Republic of Korea^{2}Department of Biomedical Engineering, School of Medicine, Chungbuk National University, 52 Naesudong-ro, Cheongju 361-763, Republic of Korea

Received 16 April 2014; Revised 28 May 2014; Accepted 30 May 2014; Published 29 June 2014

Academic Editor: Yuxin Zhao

Copyright © 2014 K. H. Kim et al. 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 problem of stability analysis and output tracking control for linear systems with time-varying delays is studied. First, by construction of a newly augmented Lyapunov-Krasovskii functional, a delay-dependent stability criterion for nominal systems with time-varying delays is established in terms of linear matrix inequalities (LMIs). Second, based on the sense, the proposed method is extended to solve the problem of designing an output tracking controller to track the output of a given reference model. Finally, three examples are included to show the validity and effectiveness of the presented delay-dependent stability and the output tracking controller design.

#### 1. Introduction

The problem of output tracking control has received a great deal of attention since this issue is an important requirement in many systems [1–5]. In detail, output tracking controller design methods are utilized in robot systems [1, 2], flight systems [3, 4], and so on. Moreover, in [5], the problem of output tracking control is derived in sense for time-delayed systems with nonlinear perturbations. The main purpose of output tracking control is to design a closed-loop feedback controller such that the output of plant tracks the output of a given reference model as close as possible. Moreover, disturbances should be considered in designing output tracking control since disturbances can lead to adverse effects on the performance of systems. To minimize the effects of the disturbances on systems, one possible approach is to design a tracking controller in sense. Since control which has been used to minimize the effects of the disturbances was firstly introduced by Zames [6], the study on controller design has been attracted by many researchers [7–10]. Similarly, output tracking controller has an objective of designing a controller such that the closed-loop system is asymptotically stable and the tracking error by the effects of disturbances does not exceed a prescribed level.

On the other hand, time-delay is one of the sources of instability and poor performance in various systems such as physical and chemical systems and industrial and engineering systems. The reason is that time-delays frequently occur in various systems due to the finite capabilities of information processing and transmission. For this reason, during the past several decades, stability analysis for time-delayed systems has been widely studied by many researchers [11–23]. One of the aims of analysis for time-delayed systems is to establish a less conservative stability criterion which can find maximum upper bounds of time-delays for guaranteeing the asymptotic stability of systems. Stability analysis for time-delayed systems can be classified into delay-independent criteria and delay-dependent ones. Since delay-dependent stability criteria use information of time-delays such as lower bounds and upper bounds of delays while delay-independent ones do not have them, delay-dependent stability criteria are generally less conservative than delay-independent ones especially when the sizes of time-delays are small.

In line with this thinking, according to Zhang and Yu [5], it has been well acknowledged that output tracking control design for time-delayed system is more general and more difficult than stabilization. However, most existing results on output tracking control have focused on systems without time-delays. From the practical point of view, it is worth designing output tracking controllers for time-delayed systems since it is well known that there exist time-delays in many practical systems. Therefore, to establish less conservative stability criteria and design output tracking controllers for time-delayed systems are still challenging.

Motivated by the matters mentioned above, this paper investigates the problem of stability analysis and tracking controller designing for linear systems with time-varying delays and disturbances. First, in Theorem 5, by constructing a newly augmented Lyapunov-Krasovskii functional, an improved delay-dependent stability criterion is derived by utilizing reciprocally convex approach [15] and some new zero equalities with LMI framework [24] which can be formulated as convex optimization algorithms. Second, based on the results of Theorem 5, an tracking controller design method for linear systems with time-varying delays and disturbances will be proposed in Theorem 7. Finally, through three examples, validity and effectiveness of the proposed theorem will be verified.

*Notation. * is the -dimensional Euclidean space and denotes the set of real matrix. For symmetric matrices and , (resp., ) means that the matrix is positive definite (resp., nonnegative). denotes the transposition of . denotes the identity matrix. and denote the zero matrix and the zero matrix, respectively. refers to the Euclidean vector norm and the induced matrix norm. denotes the block diagonal matrix, respectively. represents the elements below the main diagonal of a symmetric matrix. is the space of square integrable vector. means that the elements of the matrix include the value of ; for example, .

#### 2. Problem Statements

Consider the following system with time-varying delays: where is the state vector, is the control input, is the disturbance input which belongs to , is the vector of controlled output, and , , , , , , and are known real constant matrices. means a time-delay satisfying time-varying continuous function as follows: where and are known scalars.

Designing the output tracking control has an objective to design a controller such that the output of the system tracks a reference signal. Therefore, the reference signal is assumed to be generated by reference model as follows: where is the reference signal which has the same dimension as , is the reference state, and is the energy bounded reference input. and are known real constant matrices with the assumption that the matrix is Hurwitz. The state-feedback controller is considered as the following form: where and are gain matrices of the state-feedback controller. Let us define , , and . Then the following augmented systems can be obtained as where

From the sense, let us consider the two requirements of the output tracking controller as follows.(i)With , the closed-loop system (5) with control input is asymptotically stable.(ii)The effects of and on the tracking error are attenuated below a desired level in the sense; that is, where is a prescribed scalar. Then, if the obtained controller is satisfied with two requirements, then it is said to be an output tracking controller.

The objective of this paper is to design a state-feedback controller (4) such that system (5) achieves the output tracking below a prescribed level and the output tracking performance is minimized.

Before proceeding further, the following lemmas will be utilized in deriving main results.

Lemma 1 (see [17]). *For a positive matrix , scalars such that the integrations are well defined; then
*

Lemma 2 (see [15]). *For a scalar in the interval , a given matrix , two matrices and , for all vector , let one define a function given by
**
Then, if there exists a matrix in such that , then the following inequality holds
*

Lemma 3 (Finsler’s lemma [25]). *Let , , and such that . The following statements are equivalent:*(1)*. *(2)*, where is a right orthogonal complement of .*

Lemma 4 (see [21]). *For a positive matrix , a symmetric matrix , and a matrix , the following two statements are equivalent:*(1)*;
*(2)*There exists a matrix of appropriate dimension such that
*

#### 3. Main Results

This section consists of two subsections. The first section introduces an improved stability criterion for system (1) with and as Theorem 5. The second subsection will investigate the problem of an output tracking controller design method for the augmented system (5) based on the results of Theorem 5.

##### 3.1. Stability Analysis

Let us consider the system (1) with and given by In this subsection, a delay-dependent stability criterion for system (12) is derived. For simplicity of matrix and vector representation, are defined as block entry matrices which will be used in Theorem 5. For example, and . The other notations are defined as where is the right orthogonal complement of .

Now, a delay-dependent stability of the system (12) is given as Theorem 5.

Theorem 5. *For given scalars and , the system (12) is asymptotically stable for and , if there exist positive definite matrices , , , , , , and , symmetric matrices , and any matrices , , and , satisfying the following LMIs:
**
where are the four vertices of with the bounds of and , that is, and when , and when , and when , and and when .*

*Proof. *For positive definite matrices , , , , , and , let us consider the following Lyapunov-Krasovskii functional candidate as
where
with , and .

Now, calculating the time-derivative of yields
Also, can be represented as
The time-derivative of can be given as follows:
Calculating gives
Inspired by the work in [22], the following four zero equalities with symmetric matrices , , , and are considered:
By adding (21) into (20), can be represented as follows:
From Lemma 1, an upper-bound of can be calculated as
where which satisfies . It should be noted that if is zero, then the term of and are zero. Also, if is , then the term of and are zero. Thus, inequality (24) still holds.

By using Lemma 2 with which is defined in (13), a new upper-bound of can be obtained as

The is calculated asApplying Lemma 1 to (26), the upper-bound of can be derived as
where satisfies . Note that when is zero, the terms of and are zero, and when is , the terms of and are zero. Thus, inequality (27) still holds.

By the use of Lemma 2 to (27) with defined in (13), the following inequality can be obtained:
The time-derivative of is obtained as follows:
Using Lemma 1 to (29) yields a new upper-bound of as