$H_{\infty }$ control for singular systems with interval time-varying delays via dynamic feedback controller

Abstract

The issue of dynamic feedback $H_{\infty }$ control in this paper is carefully revisited for singular systems with interval time-varying delays. A Lyapunov-Krasovskii functional, made up of state decomposition components and relatively few decision variables, is proposed. Further, a delay-dependent admissible criterion (conforming to non–impulsiveness, regularity, and stability) with a given $H_{\infty }$ performance index for the unforced nominal singular systems is founded by some linear matrix inequalities. Subsequently, a dynamic feedback controller for the closed–loop system is designed and the corresponding admissibility conditions are obtained. By means of a state decomposition recombination method, the desired dynamic feedback controller parameters are clearly determined by solving each decomposition component of the dynamic feedback controller. Interestingly, our results can enhance the previous results and the method proposed has greater flexibility. By numerical examples, some comparisons are shown to reveal the superiority and feasibility of our method.

Introduction

The integration and analysis of singular systems (SSs) (also called generalized state–space systems, differential–algebraic systems, semi–state systems, or descriptor systems, and so on) have attracted long attention from a large number of scholars, especially in the control or mathematical circles [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. These types of systems have unique characteristics. For example, they contain algebraic equations in addition to differential equations, which means their states can not be completely determined by their dynamic [11], [12], [13], [14], [15]. Therefore, the research on SSs is much more intricate than that of regular ones. In addition to studying the stability of the systems, we also need to think about their regularity and non–impulsiveness. Furthermore, the effect of time-delay in the practical use of control systems is unavoidable. It may lead to the decline of system performance and even the instability of the systems in serious cases. As a result, for SSs with time delays, in the light of the challenge of their research, numerous rich research results and applications have been surging out [16], [17], [18], [19], [20].

In recent decades, the application of feedback control has been one of the hot directions in the analysis and synthesis of SSs. It is controlled according to deviation. Meanwhile, it has the ability to suppress the influence of any internal and external disturbance on the controlled quantity, and has high control accuracy. Many valuable research results have been developed [21], [22], [23], [24], [25], [26], [27], [28]. For example, Chen et al. [21] studied the mixed $H_{\infty }$ and passive control problem based on the static out feedback for SSs with time delays. The dissipative fault–tolerant control with slow state feedback was discussed for nonlinear singular perturbed systems in [22]. The stabilization of neutral singular Markovian jump systems (SMJSs) was explored under state feedback control in [23], [24]. The output feedback control problem for SMJSs with uncertain transition rates was studied in [25]. We find that the feedback control discussed in the above literature is either state feedback control or static output feedback. However, the performance of dynamic feedback (DF) control is much better than that of static feedback control, mainly because its control response and control gain have greater degrees of freedom. So, the DF control issue has attracted more attention [29], [30], [31], [32], [33], [34], [35], [36]. Zhang et al. [29] considered the dynamic output–feedback stabilization of singular LPV systems and acquired the necessary and sufficient conditions for stability. Lin et al. [30] studied the resilient $H_{\infty }$ dynamic output–feedback controller design problem. Park et al. [31], by using the linear matrix inequality (LMI) approach, concerned the dynamic output–feedback control for singular interval–valued fuzzy system. Long and zhong in [32] designed a DF controller for a class of SSs and explored the $H_{\infty }$ control issue simultaneously. To sum up, we find that to obtain the desired feedback controller parameters, the usual processing method is to convert the obtained conditions containing nonlinear matrix inequality conditions into LMI conditions through some matrix transformation techniques or other technical means. Nevertheless, the linearization of some matrix terms including singular matrix or system coefficient matrix increases the computational complexity. Naturally, there will be a problem is how to better reduce the computational complexity and conservatism in such a processing process? Recently, a state decomposition recombination (SDR) approach in [37] to deal with the $H_{\infty }$ filtering for SSs has aroused our great interest. The main idea is to decompose and reorganize the closed–loop systems with a full-order filter through the characteristics of a singular matrix, obtain the equivalent transformation systems, and then analyze their admissibility. One of the main superiority of this method is that it can deal with decision variables more flexibly and remove the number of redundant decision variables to abate the math-based complex difficulty.

Inspired by the above statement, the DF $H_{\infty }$ control issue for SSs in this paper is revisited. First, by constructing a novel state decomposition augmented Lyapunov-Krasovskii functional (LKF), the admissibility criteria with $H_{\infty }$ performance index for unforced nominal SSs are derived. Second, with the help of the established admissibility criteria, some delay dependence sufficient conditions for the closed–loop systems with a DF controller are obtained by employing an SDR method. Finally, through three numerical examples, the advantages and feasibility of the presented results will be reflected. But here is what we know, few literature have tried to use the SDR method to study the DF issue for SSs. Thus, this article fills this gap. The main innovations are :

• In this paper, one of the differences from previous literature work on DF control in [29], [30], [31], [32], [33], [34], [35], [36] is the construction of a new state decomposition augmented LKF. More specifically, we construct LKF from the perspective of state decomposition components rather than from the overall state. According to the characteristics of the singular matrix and combined with the tighter integral inequality technique, the established LKF can decrease the conservatism and computational complexity.

• The delay–dependent conditions of the admissibility (according to non–impulsiveness, regularity, and stability) for the closed–loop SSs applying the DF controller are given by using the SDR method. In contrast, the results derived are less conservative than those in [7], [32], [37]. In particular, the desired DF controller parameters can be obtained more accurately and flexibly by discussing each decomposition component of the studied controller parameters.

$\mathbf{Notations}:$ $\nabla >0$ is a positive definite matrix. The symbols $T$ and $-1$ are transpose and inverse, respectively. $I_d,I_{d\times f},0_d,0_{d\times f}$ are $d\times d,d\times f$ identity matrices and $d\times d,d\times f$ zero matrices, respectively. $\left[\begin{array}{cc}\nabla & \varnothing \\ ★& ▵\end{array}\right]$ stands for $\left[\begin{array}{cc}\nabla & \varnothing \\ {\varnothing }^T& ▵\end{array}\right]$. ${\parallel x\left(t\right)\parallel }_{\overline{d}}=\underset{-\overline{d}\le t\le 0}{sup}\parallel x\left(t\right)\parallel$. $col\{{▵}_1,{▵}_2,\dots ,{▵}_d\}={[{▵}_1^T,{▵}_2^T,\dots ,{▵}_d^T]}^T$. $sym\left\{\nabla \right\}=\nabla +{\nabla }^T$. $det\left(\nabla \right)$ is the determinant of $\nabla$. ${\mathbb{R}}^{d\times f}$ and ${\mathbb{R}}^d$ are the set of $d\times f$ real matrices and $d-$dimensional Euclidean space, respectively.