Guaranteed cost control for impulsive nonlinear Itô stochastic systems with mixed delays☆
Introduction
Delay differential equations have become a significant subject with their extensive application in various industrial fields in recent decades, such as biological engineering, chemical systems, nuclear reactor. The delay phenomenon is considered to be one of the most important factors affecting system performance, and has been widely concerned by researchers. The main research work is the influence of time delay on the stability and other control problems. For example, the stability of delay differential systems [1], [2], [3], [4], [5], [6], [7], controllability issue [8], control schemes for networked systems [9], [10], admissible output consensualization control for singular multi-agent systems [11], the drive-response H∞ synchronization [12], to name but a few.
In addition, the nonlinearity and stochasticity of the systems are also generally regarded as two considerable factors leading to the complexity of systems. Therefore, some techniques have been proposed to deal with the control problems of nonlinear stochastic systems in order to make the system meet the performance requirements and be stable. For instance, for the results of uncertain nonlinear stochastic systems, one can refer to [13], [14], [15]. In [16], Zhu first solved the event-triggered feedback control problem for stochastic nonlinear delay systems with exogenous disturbances. The H∞ control problem was studied to stabilize the nonlinear stochastic systems by using Hamilton-Jacobi equation in [17].
It should be noted that the impulsive control is a valid way to stabilize complex systems, even if the behavior of the systems may not be available to the design of the controller [18]. In past several dozens years, different impulsive control strategies were proposed to deal with the stability in various fields, including stochastic systems [19], [20], [21], [22], [23]. Several sufficient conditions of the exponential stability of impulsive neural networks with delays and stochastic noise were obtained on the basis of the Lyapunov functional approach in [19]. The drive-response synchronization of Riemann–Liouville fractional order uncertain complex valued neural networks with delays and impulses is studied and the global stability was obtained by applying the Lyapunov functional in [20]. Literature [21] obtained several sufficient conditions to ensure the global asymptotical stability of switched impulsive stochastic genetic regulatory networks by using Lyapunov functional and linear matrix inequality (LMI) technique.
An excellent control system should not only satisfy the stability, but also ensure a better performance. The guaranteed cost control is a frequently used algorithm to provide an upper bound on a given cost function for closed-loop systems and it can ensure that the system performance degradation caused by delays or other uncertainty factors is less than this bound [24], [25]. The problem of guaranteed cost control has attracted widely research interest, but up to now, only the linear system has established a relative systematically framework. And when it comes to other complex systems, there is still a considerable research demands, such as multi-agent systems [11], [26], [27]. The robust finite-time stabilization with guaranteed cost control for a class of neural networks with time-varying delay was studied in [28]. The authors in [29] studied the problem of guaranteed cost control for Hopfield neural networks with interval multiple nondifferentiable time-varying delays. In [29], a guaranteed cost controller was designed via memoryless state feedback control by constructing a set of augmented Lyapunov–Krasovskii functionals combined with Newton–Leibniz formula and some sufficient conditions for the state feedback of system performance were given by using the LMI.
In the literature of the existing control theories, almost all controller construction methods are based on the Lyapunov functional and LMI technique [21], [28], [29]. However, the computation complexity of these methods will be greatly increased for complex nonlinear systems with more factors. Therefore, the traditional LMI is no longer suitable for nonlinear systems with the impact of complex factors. A new approach is to use Hamilton–Jacobi inequality (HJI) to implement the controller design for complex nonlinear systems considering the effects of nonlinear, impulsive, stochastic phenomena, delays, etc. Specially, in [2], a state feedback controller was designed to ensure that the system was asymptotically stable in probability and an upper bound was guaranteed on a prespecified quadratic cost function in terms of the HJI for nonlinear stochastic systems. However, for impulsive nonlinear Itô stochastic systems with mixed delays (INISS-MDs), there has been no works to investigate the guaranteed cost control problem. Considering the influence of impulse, mixed delays and stochastic noises, the INISS-MDs have a wider application. This inspires us to do the present work.
Motivated by the above analysis and discussions, we are concerned with the guaranteed cost control problem for INISS-MDs in this paper. The system considered in this paper is represented by nonlinear Itô stochastic differential equations, and the consideration of impulsive perturbation and mixed delays adds great difficulties to the design of the controller. A novel sufficient condition is derived to ensure the asymptotic stability in probability for INISS-MDs by applying the HJI, Lyapunov–Krasovskii function and stochastic analysis theories. In addition, the given cost function is guaranteed to have an upper bound, and so the results are more easily validated and applied for complex systems in practice. The main contributions are list below.
(i) The state feedback cost guaranteed controller is designed for the first time for INISS-MDs. It covers a wider range and may have better application prospects.
(ii) A sufficient condition is obtained for INISS-MDs by means of an HJI. This proposed method is simpler than the LMI technique. Moreover, the traditional LMI technique is no longer suitable for the proposed problem.
(iii) Furthermore, the impulse disturbance is analyzed to help the system reflect the actual situation better.
The rest of the paper is organized as follows. The problem description and several preliminaries are presented in Section 2. Section 3 gives the main results. A sufficient condition is obtained to ensure the asymptotic stability in probability for INISS-MDs and can guarantee the given cost function to have an upper bound. In Section 4, a simulation result is presented to illustrate the effectiveness of the theory. Finally, the conclusion is drawn in Section 5.
Section snippets
Preliminaries
Now we introduce the following notations. Let be the set of all positive real numbers and be the n-dimensional Euclidean space. stands for the n × m real matrices and denotes the set of all n × n symmetric matrices. Denote (A) as the trace of matrix A and I as the identity matrix. The superscript T represents the transpose operation. For symmetric matrices X and Y, the notation X ≥ Y (X > Y) stand for is positive-semi-definite (positive definite) matrix. λmin( · ) (λmax( · ))
Main results
For system (2.1), a set of sufficient conditions of the cost guaranteed control is denoted in this section. Theorem 3.1 For system (2.1) with cost function (2.2), if there exist a functional with and positive definite matrices such thatwhere for
An example
To illustrate the validity of Theorem 3.1, we present an example. Example 4.1 Considering the following INISS-MD:where the time-delays satisfy and . Set the η1 and η2 as the first and second entries of η. Let and give the positive definite matrices as Take
Conclusion
The problem of guaranteed cost control for INISS-MDs has been investigated. A sufficient condition is derived to ensure the asymptotic stability in probability for INISS-MDs and the given cost function is guaranteed to have an upper bound by applying the terms of the HJI, Lyapunov–Krasovskii functional and stochastic analysis theories. Moreover, our results reveal that impulses effects can accelerate the convergence rate of the equilibrium point. We consider many factors such as impulsive
Declaration of Competing Interest
We declare that we have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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This work was supported by the National Natural Science Foundation of China (61773217), the Natural Science Foundation of Hunan Province(S2020JJMSXM1232), the Scientific Research Fund of Hunan Provincial Education Department (18A013), Hunan Provincial Science and Technology Project Foundation (2019RS1033), Hunan Normal University National Outstanding Youth Cultivation Project (XP1180101) and the Construct Program of the Key Discipline in Hunan Province.