Improved Results on Delay-Dependent Robust Control of Uncertain Neutral Systems with Mixed Time-Varying Delays

Abstract

In this paper, the problem of the delay-dependent robust control for a class of uncertain neutral systems with mixed time-varying delays is studied. Firstly, a robust delay-dependent asymptotic stability criterion is shown by linear matrix inequalities (LMIs) after introducing a new Lyapunov–Krasovskii functional (LKF). The LKF including vital terms is expected to obtain results of less conservatism by employing the technique of various efficient convex optimization algorithms and free matrices. Then, based on the obtained criterion, analyses for uncertain systems and controller design are presented. Moreover, on the analysis of the state-feedback controller, different from the traditional method which multiplies the matrix inequality left and right by some matrix and its transpose, respectively, we can obtain the state-feedback gain directly by calculating the LMIs through the toolbox of MATLAB in this paper. Finally, the feasibility and validity of the method are illustrated by examples.

1. Introduction

Strictly speaking, some degrees of time delays are frequently encountered in almost all practical engineering systems such as [1–3], in which the existence of the delays may lower the performances of these engineering control systems and even result in the instability of such control systems. Then, it is an essential requirement to investigate the stability for systems with time delays. And quantities of results on stability for time-delay systems in actual applications have been obtained in the near past [4–6]. The stability analyses of the systems can be classified into delay-dependent ones and delay-independent ones. The stability analyses criteria in delay-dependent systems are always less conservative than the delay-independent ones. In addition, the stability criteria are developed by applying the LKF methodology. Otherwise, for some time-delay systems in practice such as air-feed system [7] and power converters system [8], the controllers are designed to make the systems more practical by employing advanced methods for the stability analyses. Though the time delay can be constant or varying, the former is a special case of the latter which is always more complex and typical. Thus, the stabilization problem of dynamical systems with time-varying delays has been taken into consideration.

On the other hand, there are also many typical practical models of time-delay neutral dynamical systems such as partial element equivalent circuit model [9] and population genetics and community ecology [10], which are described by neutral functional differential equations containing time delays not only in the state but also in its derivative. Obviously, studying neutral systems with mixed delays (NSMDs) is more challenging than the system with a delay only in the state. Results about stability analysis for NSMD have been obtained in recent years. In the following, some classical methods are shown for NSMD. In [11], delay-dependent conditions were improved via utilizing the LMI method. In [12], by applying a candidate LKF, less conservative criteria were proposed. Besides, to reduce conservatism, an approach of piece-wise delay was expressed in [13]. Advanced integral inequalities were used in [14] to get the delay-dependent stability conditions, which were less conservative. NSMD with uncertainties were observed, and then the robust stability analyses were discussed in [15]. As we can see in [16], compared with an inequality which might not cause a less conservative result, a tighter inequality based on a suitable LKF could get the advantage we want. From [11–16], we can see that the methods of LKF and integral terms are the mainstream ones. By constructing an LKF with inequality terms and applying the LMI method, sufficient conditions are obtained.

As known, uncertainties can lead to poor performance in many time-delay systems. Therefore, much attention has been paid to the analysis of stability and stabilization for systems with parametric uncertainties. And robust control can be used for the analysis and design of the systems including uncertainties which are complex to handle in dynamical models. The problem of robust stabilization for a class of uncertain neutral time-delay dynamical systems with the uncertain matrix coefficients of both the delayed state and derivative of the delayed state is considered so that we can ensure the discussed systems stable under specific conditions. In addition, to overcome the external disturbances of actual operations in engineering models such as in [17, 18] and make them stable artificially, control is also an important issue. In [17], robust control was designed for PFC rectifiers. For the cooperative driving system with external disturbances and communication delays in the vicinity of traffic signals, robust control was studied in [18]. As presented in [19], the issue on robust control was discussed. Besides, control of neutral systems has been expressed in many papers up to now. In [20], the problem of robust control with state feedback was solved in the uncertain neutral system. For an uncertain neutral system in [21], performance for Markovian uncertain systems via the sliding mode control method was considered. Moreover, delay-dependent filtering conditions were proposed for uncertain neutral stochastic systems with Markovian switching and mode-dependent time delays [22]. By applying a state-feedback controller in [23], a neutral-type time-delay system was studied. And in another paper [24], the feedback controller was used to obtain the results.

Taking all the discussed elements into consideration, a representative system named the teleoperation system was introduced in [25]. The teleoperation system is composed of a master manipulated by a human operator and a slave operating in a remote environment through a communication channel. Time-varying delays existed in the long-range or flexible communication channel [26]; thus, the problem of stability of teleoperation systems for time-varying delays by neutral LMI techniques was discussed in [25]. Besides, due to that no real system is pure linear, time-varying model uncertainties considered as perturbations existed in real implementation of bilateral teleoperation [27], and then the study on robust control design for teleoperation systems was done in [28]. Moreover, the stability criteria above were all given in the form of LMIs, which can be used to compute the maximum values of delays. And in order to obtain the controller parameters directly by solving LMIs, state-feedback control law was found to stabilize the system with guaranteeing performance in [29]. Above all, in our paper, both time-varying delays and uncertainties are considered in the neutral model to analysis the robust control with state-feedback controller by the technique of LMIs. Different from the general delay system such as [16, 23], this paper also considers delays in the neutral terms. And unlike the delays in [11–15, 20, 24], the delays shown in both discrete and distributed forms in this article are all varying and different instead of constant or the same delay function. Then, the model derived from the actual application is more complex but more representative. Moreover, appropriate Lyapunov function is constructed to deal with the complicated model terms technically and get the control gain directly without applying the traditional method in [30].

Hence, motivated by the discussions above, the problem of delay-dependent robust control of uncertain neutral systems with mixed time-varying delays is explored in this article. After introducing an improved LKF and using the LMI method, a sufficient condition for the asymptotic stability of the system is established. Analyses on uncertainties and control are made then. Numerical examples are given to verify the results finally.

The main contributions of this paper are summered as follows:(1)In the LKF newly constructed, all terms are taken into account to obtain a larger delay bound.(2)The methods including not only free-weighting matrices but also efficient convex optimization algorithms are applied technically to get results of less conservatism.(3)Sufficient conditions are given all by LMIs, and the solutions can be solved by MATLAB Toolbox directly.(4)Though similar papers concerning the delay-dependent robust control of uncertain neutral systems with mixed time-varying delays have already been many, there are some differences between the current submission and latest works. To the extent of the authors’ knowledge, none of the existing works including all the terms of robust control, neutral systems, uncertainties, and mixed time-varying delays derived from actual engineering problems. Besides, the Lyapunov functional proposed in this article can get better conservatism with larger delay bounds when compared with these in [31–33], which is shown in Example 3. Finally, different from the traditional method which multiplies the matrix inequality left and right by some matrix and its transpose, respectively, such as in [30], the controller gain illustrated in Theorem 4 of this paper can be obtained more easily and simply.

1.1. Notation

The notations throughout this paper are fairly standard. and denote, respectively, the matrix inverse and transpose of . and are the n-dimensional Euclidean space and the set of real matrices. means that is positive definite. is a unit matrix with compatible dimensions. denotes . Let , . expresses the Euclidean norm of function . refers to the space of square integral vectors.

2. Description of the Problem

Consider the NSMD with uncertainties:with , , , and as the state vector, the control input, the disturbance which belongs to , and the controlled output, respectively; and are given continuous initial condition functions. Continuous functions and are time-varying delays which satisfywhere are constants. and are matrices with uncertainties satisfyingwhere and are known constant matrices with suitable dimensions. The time-varying unknown matrices , and satisfy the following form:where and are known constant matrices with appropriate dimensions and as an unknown matrix function satisfies for any .

For system (1), the nominal form is given as follows:

For system (5), a state-feedback controller is designed bywhere is a constant matrix to be designed.

Combining (1) with (3) and the controller expression (6), we can get the closed-loop system of (1):

Remark 1. The mixed time-varying delays, respectively, distribute in the state and the derivative of the state, and the system could not always be stable with all values of delays. Then, we must find the bound of each time-varying delay, the larger the boundary and the better conservative system indicating that the broader time scope can be got to keep the system stable.

Remark 2. The two kinds of both discrete and distributed delays exist in the actual systems. And the large delay always causes the instability of the system. And either of the large delay can cause the instability of the system. Thus, considering the bounds of the discrete and distributed delays is a must.

Lemma 1. Schur complement [34]: given matrices with , then we have

Lemma 2. Jensen’s inequality [35]: for any scalar and any constant matrix , , the following inequality holds

Lemma 3. Cauchy matrix inequality [36]: let be a positive define matrix, then

Lemma 4 (see [37]). For any constant matrix , and free-weighting matrix , we havewhere

Lemma 5 (see [38]). Given matrices with suitable dimensions and satisfying , then for any constant , the inequality holds

3. Main Results

In this section, a new sufficient condition criterion of asymptotic stability for system (14) is presented by using the LKF method and LMI technique:

Theorem 1. For the given constants , and , system (14) is asymptotically stable if there exist positive-definite matrices and matrices with appropriate dimensions satisfying the following LMI:

Proof. Choose an LKF candidate aswherewithThe time derivative of is given byBy applying Lemma 4,Similarly,By using Lemma 2, we can getBased on the Newton–Leibniz formula, we haveAccording to Lemmas 2 and 3, we can getThen, we can obtainFrom system (14), it is easy to getCombing formulas (19) to (30), we haveBy Lemma 1, we can obtain . Obviously, system (14) is ensured to be asymptotically stable.

Remark 3. In the field of analysis of delay-dependent stability, the main purpose is to find the maximum delay bounds for guaranteeing stability. In the discussed LKF , the three terms , , and play an important role in reducing the conservatism of our results. More specifically, by taking the states , , as variables, more information can be sufficiently utilized on state variables, which can yield less conservatism. And as mentioned in [37], in order to derive an upper bound of in each intervals by Lemma 4, two different free-weighting matrices and are introduced at each interval. In addition, the integral terms in devote to a less result with Jensen’s inequality presented.

Remark 4. The LMI problem illustrated in Theorem 1 is a convex problem in relation to solution variables. And various convex optimization algorithms can solve the problem. In this paper, Matlab LMI Toolbox, as a useful software, for solving convex optimization problem, is used in which the interior-point algorithm was implemented.

Remark 5. The proposed criteria in Theorem 1 may give larger stability regions in compassion with some of the existing criteria. This claim is due to the following characteristics of the proposed stability analysis scheme: in [16], the relationship between the augmented LKF and the tightness of inequalities was explored. It has been shown that a tighter inequality does not always lead to a less conservative criterion. Nevertheless, a suitable augmented LKF and a tighter inequality, both of which are indispensable, can effectively reduce conservatism. In this paper, an augmented LKF combined with more cross terms can be well dealt with Lemma 4 technically. In addition, double- and triple-integral terms help to further increase the stability region.

Remark 6. (i) The system modeled in (1) is more representative than the ones in [11, 16, 36, 37]. For example, if , the neutral items are linear, then system (1) here can reduce to the neutral model without uncertainties in [11]. Besides, when , the system becomes the general uncertain delay time model in [16, 37]. If , then the general delay time systems without uncertainties in [16, 36] are got. And if , the varying delay terms are turned into constant delay ones. (ii) In the system of this article, all the coefficient matrices of the states and the derivative are functional matrices with uncertainties, different from the known constant matrices or few uncertain matrices in the existing literature. In the majority of the papers including all the cited papers in this paper, uncertainties are neglected more or less. However, in this article, uncertainties are all considered in the states and the derivative vector. (iii) Most articles about the content of neutral time-delay systems are concerned with the stability problem such as [9, 11–15], and our article not only solves the stability problem but also contains the control strategies.

Remark 7. If and , we can get a special case of system (14):By applying Theorem 1, solutions of system (32) and the upper delay bounds comparing with other results are shown in Example 1.

Theorem 2. For a prescribed scalar and given constants , system (5) with is robustly asymptotically stable with performance if there exist positive-definite matrices ,