Abstract
This paper deals with the problems of delay-dependent stability and performance for uncertain neutral systems with time-varying delays, and nonlinear perturbations. The time-varying delays are neutral, discrete, and distributed time-varying delays that the upper bounds for the delays are available. The restrictions on the derivatives of the discrete and distributed time-varying delays are removed, which mean that a fast discrete time-varying delay is allowed. The uncertainties under consideration are nonlinear time-varying parameter perturbations and norm-bounded uncertainties, respectively. Firstly, by applying a novel Lyapunov-Krasovskii functional approach, Wirtinger-based integral inequality, Peng-Park’s integral inequality, decomposition technique of constant matrix, descriptor model transformation, Leibniz Newton formula and utilization of zero equation, and improved delay-dependent bounded real lemmas (BRL) for systems are established in terms of linear matrix inequalities (LMIs). Then, based on the obtained BRL, some less conservative delay-dependent stability criteria of uncertain neutral systems with mixed time-varying delays and nonlinear perturbations are obtained and improved performance criterion with the framework of LMIs is introduced. Finally, some numerical examples are given to illustrate that the presented method is effective.
1. Introduction
Time delay is frequently a source of instability and a source of generation of oscillation in many dynamic systems such as hybrid systems (and practical application) [1, 2], networked control systems, biological systems, mechanical systems, and chemical or process control systems [3]. Thus, analysis and synthesis problem for systems with time-varying delay have become an important issue and large varieties of problems have been researched since the nineties by several researchers [4–8], performance [9–11].
In some physical system, the system models can be described by functional differential equation of neutral type, in which the models depend on the state delay but also depend on the state derivatives, are often encountered in various fields, such as population ecology [12], distributed networks containing lossless transmission lines [13], heat exchangers, and robots in contact with rigid environments [14]. For interesting research methods, stability criteria for application neutral stochastic systems and neural networks have been discussed in [15–19]. On the one hand, some LMI criteria on robust stability for uncertain ones have been deeply derived in [20–23]. Very recently, improved performance analysis and stability for uncertain systems with time-varying delays were proposed in [10, 11, 24]. However, there are rooms for further improvements in the feasible region of criteria for performance and stability.
Stability criteria for time-delay systems are generally divided into two classes: delay-independent one and delay-dependent one. Delay-independent stability criteria tend to be more conservative, especially for small size delay, and such criteria do not give any information on the size of the delay. On the other hand, delay-dependent stability criteria are concerned with the size of the delay and usually provide a maximal delay size. Generally speaking, the latter ones are less conservative than the former ones when the time-delay values are small. Much time and efforts have been put into the development of some techniques and new Lyapunv-Krasovskii functional because how to choose Lyapunov-Krasovskii functional and estimate an upper bound of time-derivative of Lyapunov-Krasovskii functional play key roles to improve the feasible region of stability criteria. Delay-dependent stability criteria for these systems are established in terms of linear matrix inequalities (LMIs).
With above motivations, based on Lyapunov stability theory, improved performance criteria and stability analysis for uncertain neutral systems with mixed time-varying delays and nonlinear perturbations are derived by the framework of LMIs which will be introduced in Theorem 11. Some numerical examples are given to illustrate that the presented method is effective.
2. Problem Formulation
We introduce some notations and lemmas that will be used throughout the paper. denotes the set of all real nonnegative numbers; denotes the -dimensional space with the vector norm ; denotes the Euclidean vector norm of ; denotes the set real matrices; denotes the transpose of the matrix ; is symmetric if ; denotes the identity matrix; matrix is called semipositive definite () if , for all ; is positive definite () if for all ; matrix is called seminegative definite () if , for all ; is negative definite () if for all ; means ; means ; denotes the space of all continuous vector functions mapping into when , ; ; represents the elements below the main diagonal of a symmetric matrix.
Consider the following uncertain neutral system with mixed time-varying delays of the form:where is the state variable, is the disturbance input that belongs to , is the controlled output, is the neutral time-varying delay, and is the discrete and distributed time-varying delays satisfyingwhere and are positive real constants. is initial condition function and , , , and are uncertain matrices. The uncertainties , , and represent the nonlinear parameter perturbations with respect to the current state , the delayed state , and the neutral delayed state , respectively, and are bounded in magnitudewhere and are given positive real constants. We assume that the uncertainties are norm-bounded and can be described aswhere , , , , , , , , , , , , and are real constant matrices with appropriate dimensions. The uncertain matrix satisfiesand is said to be admissible, where is known matrix satisfyingThe uncertain matrix satisfies
Definition 1. Given a scalar , the system (1) is said to be asymptotically stable with the performance level , if it is asymptotically stable and satisfies the -norm constraint , for all nonzero under zero initial condition.
Lemma 2 (see [25]). Suppose that is given by (8)-(10). Let , , and be real constant matrices of appropriate dimension with . Then, the inequalityholds if and only if for any positive real constant ,
Lemma 3 (Jensen’s inequality). For any symmetric positive definite matrix , positive real number , and vector function such that the following integral is well defined, then
Lemma 4 (Wirtinger-based integral inequality, [26]). For any matrix , the following inequality holds for all continuously differentiable functions :where and
Lemma 5 (Peng-Park’s integral inequality, [27, 28]). For any matrix , positive scalars and satisfying , vector function , such that the concerned integrations are well defined; thenwhereand
Lemma 6 (see [11]). For a positive matrix , the following inequality holds:
Lemma 7 (see [11]). For a positive matrix , the following inequality holds:
Lemma 8 (see [29]). For any constant symmetric positive definite matrix , is discrete time-varying delays with (3) and vector function such that the integrations concerned are well defined; then
Lemma 9 (see [29]). For any constant matrices , , is discrete time-varying delays with (3) and vector function such that the integration is well defined, and then
Lemma 10 (see [29]). Let be a vector-valued function with first-order continuous-derivative entries. Then, the following integral inequality holds for any constant matrices , and is discrete time-varying delays with (3):where
3. Main Results
In this section, a new performance and stability analysis for the system (1) will be introduced by using the Lyapunov functional method combining with linear matrix inequality technique. Firstly, we will establish a new version of delay-dependent BRL for the nominal system (1). We can rewrite the nominal system (1) as follows:We introduce the following notations for later use:where , and other terms are 0.where , exceptand other terms are 0.
Theorem 11. For a prescribed scalar , given scalars and , the system (26) is asymptotically stable with the performance ; if , there exist positive definite symmetric matrices , , , , and , , , any appropriate dimensional matrices, , , , , , , and , , , and positive real constants , , and satisfying the following LMIs:
Proof. Under the conditions of the theorem, we first show the asymptotic stability of the system (26). To this end, we consider the nominal system (26) with , that is,From model transformation method, we rewrite the system (36) in the following system:In order to improve of the discrete delay in (26), let us decompose constant matrices and aswhere , , , and are real constant matrices. By utilizing the zero equations, we obtainwhere and will be chosen to guarantee the asymptotic stability of the system (26). By (39)-(42), the systems (37)-(38) can be represented by the formConstruct a Lyapunov-Krasovskii functional candidate for the systems (43)-(44) of the formwhere with The time derivative of along the trajectory of systems (43)-(44) is given byIt is noted that EP is really . Then the time derivative of is calculated as Differentiating , we have Using Lemmas 8 and 10, is calculated asUsing Lemma 4 (Wirtinger-base integral inequality) and Lemma 5 (Peng-Park’s integral inequality), an upper bound of can be obtained as It is from Lemma 9 that we have By Lemma 6, we can obtain as follows: By Lemma 7 and calculating , we have Using Lemma 3 (Jensen’s Inequality) that we have Calculating leads to From (4), (5), and (6), we obtain for any positive real constants , and ,According to (48)-(60), it is straightforward to see thatwhere , , , , and
If the conditions (31)-(34) and hold, then (61) implies that there exists such that Therefore system (26) is asymptotically stable.
Next, we shall establish the performance of system (26) under zero initial condition. To this end, we introduceNoting zero initial condition, it can be shown that for any nonzero and . This can be written as where is define in (45). After some algebraic manipulations, we obtain where , , , , and
If the condition (35) holds, we have . Thus,Integrating both sides of (65) from to , we obtainThen, letting t and under zero initial condition, we have and , which leads toTherefore is satisfied for any nonzero
The proof of theorem is complete.
According to Theorem 11, we can obtain delay-dependent robust asymptotic stability criteria with the performance level of system (1). We introduce the following notations for later use
Theorem 12. For a prescribed scalar , given scalars and , the system (1) is robustly asymptotically stable with the performance ; if , there exist positive definite symmetric matrices , , , , and , , , any appropriate dimensional matrices, , , , , , and , , , and positive real constants , , , and satisfying the following LMIs:
Proof. Replacing , and in (35) with , , , and , respectively, we find that condition (35) is equivalent to the following condition:By using Lemma (1), we can find that (74) is equivalent to the LMIs as follows: where is a positive real constant. From Theorem (26) and conditions (69)-(73), system (1) is robustly asymptotically stable and satisfies . The proof of theorem is complete.
4. Numerical Examples
Example 1. Consider the following uncertain neutral system with mixed time-varying delays (26). We consider robust asymptotic stability with performance of system (26) by using Theorem 11. The system (26) is specified as follows:Decompose the matrices and , whereIt is easy to see that , , , , , and . By using LMI Toolbox in MATLAB, we use (31)-(35) in Theorem 11. This example shows that the solutions of LMIs are given as follows:
Example 2. Consider system (26) with , , , and , which means neutral system with time-varying delaywith the parameters Decompose matrix and as follows: and , where Table 1 lists the comparison of the upper bounds delays for asymptotic stability of system (79) by different methods. And the derivative of the discrete time-varying delay is unknown, because we removed the derivative of the discrete time-varying delay. We can see from Table 1 that our results are superior to those in [6, 30–33].
Example 3. Consider system (26) with and ,with the parameters Decompose matrix and as follows: and , where Table 2 lists the comparison of the upper bounds delays for asymptotic stability of system (82) by different methods. And the derivative of the discrete time-varying delay is unknown, because we removed the derivative of the discrete time-varying delay. It is clear that our results are superior to those in [22, 32, 34, 35].
Example 4. We consider system (82) with the parameters Decompose matrix and as follows: and , where Table 3 shows the comparison of the upper bounds delay allowed obtained for asymptotic stability of system (82) by other methods. And the derivative of the discrete time-varying delay is unknown, because we removed the derivative of the discrete time-varying delay. It can be found from Table 3 that our results are significantly better than those in [22, 34].
5. Conclusions
The problem of robust stability and performance for uncertain neutral systems with time-varying delays was studied. The restriction on the derivative of the discrete time-varying delay is removed. The uncertainties under consideration are nonlinear time-varying parameter perturbations and norm-bounded uncertainties, respectively. By applying a novel Lyapunov-Krasovskii functional approach, Wirtinger-based integral inequality and Peng-Park’s integral inequality, decomposition technique of constant matrix, descriptor model transformation, Leibniz-Newton formula and utilization of zero equation, and improved delay-dependent bounded real lemmas (BRL) for systems are established in terms of linear matrix inequalities (LMIs). Then, based on the obtained BRL, some less conservative delay-dependent stability criteria of uncertain neutral systems with mixed time-varying delays and nonlinear perturbations are obtained and improved performance criterion with the framework of LMIs is introduced. Numerical examples have shown significant improvements over some existing results.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This work was supported by the Thailand Research Fund (TRF), the Office of the Higher Education Commission (OHEC), Khon Kaen University (Grant no. MRG6080042), Research and Academic Affairs Promotion Fund, Faculty of Science, Khon Kaen University, Fiscal year 2018, and National Research Council of Thailand and Khon Kaen University, Thailand (Grant no. 6100060).