Improved results on reachable set estimation of singular systems

Highlights

This paper considers the problem of reachable set estimation for linear singular systems. The contribution of this paper lies in the following three aspects:

• A new inequality scaling method is used to deal with Lyapunov functional with e index and the conservatism of the result is reduced.

• Different from the Lyapunov functional construction in the comparison literatures, the candidate Lyapunov functional in this paper contains more time-delay information based on time-delay segmentation method.

• A practical example shows the effectiveness of the proposed method.

Abstract

This paper investigates the problem of reachable set estimation for a general class of continuous-time singular linear systems with time-varying delay. By employing a new Lyapunov functional candidate, a criterion with the help of linear matrix inequalities (LMIs) is developed utilizing the new inequality scaling method, free-weighting matrix and the time-delay segmentation method. The reachable set of the underlying system is bounded by the intersection of ellipsoids. The effectiveness of our results are demonstrated by the examples.

Introduction

Compared with state-space systems, modeling of some physical systems which can be better described by the singular systems, and in some fields the singular systems are also called generalized state-space systems [1], [2], [3], [4]. There are a lot of research with many successful application on the singular systems, such as economic systems, aircraft modeling, circuit systems and chemical processes [5], [6], [7], [8]. In addition to their existing research on practical significance, the concept and result of some singular systems are mainly inspired by the state-space system. For example, the admissibilization of singular interval value T-S fuzzy systems based on mismatch membership function is considered in [9]; the adaptive sliding mode control method is applied to the singular semi-Markov jump systems, an adaptive sliding mode control law based on an observer is designed to adapt to nonlinear uncertain singular semi-Markov jump systems [10]; the new sliding mode control design methodology for fuzzy singularly perturbed systems is considered in [11]; the problem of singular stochastic Markov systems with the sliding mode control has been studied in [12]; the state feedback control for singular system by applying equivalent sets technique is considered in [13]; a class of discrete-time singular Markov jump systems with time-varying delay is analyzed in [14]. As the main factor affecting system stability, time-delay has been widely considered in many practical engineering modeling [15], for instance the aircraft systems and networked control systems [16]. The reachable set estimation of time-delay system has been applied to interference suppression [17] and the regional control method for time-delay system has been used in saturated actuators [18]. The problem of disturbance rejection for a class of linear singular systems in the presence of external disturbances and time-varying delay is considered in [19], by designing the state feedback repetitive controller, the regularity, impulse free and asymptotic stability of the closed-loop singular modified repetitive control system are guaranteed, and the optimal upper bound of time-varying delay is given. The problem of reachable set bounding for a class of linear systems with both discrete and distributed delays is investigated in [20]. In reference [21], a class of uncertain master and slave neural networks with mixed time delays is studied by establishing an exponential H_∞ synchronization method.

The reachable set contains all the possible states that exist in a system under a specific constraint, which has been applied to many systems and models since it was proposed in 1960s. Such as [22] considered the general case where K ellipsoids are involved at each step; the multiple control objectives are considered in [23]; the online track planning for small drones is considered in [24] and the safe pilot maneuvers is considered in [25]; some results on state estimation have been given in [26], [27], [28], [29], [30], [31], [32], [33], [34], and so on. At present, more and more attention has been paid to the determination of the boundary of the systems reachable set and some results have been achieved [35]. The delay-dependent conditions which contains the reachable set with time-varying delay of the linear system is proposed firstly in [36]. An improved condition is obtained in [37], by employing delay-partitioning approach and augmented Lyapunov functional. The problem of estimation of reachable set for time-delay systems with peak inputs and polytopic uncertainties is solved using the maximal Lyapunov-Krasovskii functional [38]. A modified condition has been derived for the reachable set estimation in [39]. The estimation of reachable sets for singular system is proposed for the first time in [40], where fast and slow subsystems decomposition method is used. Based on the idea in [40], the improved results is proposed in [41]. Now singular systems [42] and reachable set estimation [25] has been applied to many practical fields. Many circuit systems can be modeled as singular systems. By studying the reachable set estimation of singular systems, the current and voltage in the circuit can be controlled within a safe range and the whole system can be kept in a safe running state. Right now, there are few literatures on the estimation of reachable set for the singular systems and there is ample room to improve the existing results, which are the motivation of this paper. The main motivation of this paper is to reduce the conservatism of previous results. The new inequality scaling method [43] can deal directly with the Lyapunov function with e index, which is better than the previous method of directly substituting the upper and lower interval. The new time-delay segmentation method [44] satisfies a larger time-delay interval by a tunable parameter. To this end, the new inequality scaling method and time-delay segmentation method are used to study the problem of reachable set estimation for the singular systems with time-varying delay.

To reduce the conservatism, the idea of time-delay segmentation and the inequality scaling technique are applied. Firstly, a time-delay segmentation method is used to establish the Lyapunov function and the new inequality is used to reduce the conservatism. Secondly, a sufficient condition in terms of LMIs is established utilizing the new inequality scaling method. Finally, the effectiveness and superiority of the method are presented by simulation results, which show our results are less conservative than the research in [40], [41].

Notations: Throughout the paper, the notation used is standard. ${\mathbb{R}}^n$ denotes the n-dimensional Euclidean space; the matrix P is symmetric positive define (negative definite) for P > 0 ( < 0); the superscript ‘T’ and ‘$-1$’ represent the matrix transpose and inverse; ‘L’ is the identity matrix with the compatible dimensions; * stands for the symmetric matrix, $A+A^T$ is defined as sym{A}; the norm of the Euclidean vector norm is ‖ · ‖ and $x_t\left(\theta \right)=x(\theta +t),(\theta \in [-{\tau }_M,0])$. The matrix assumptions which not stated in this paper are compatible for algebraic operations.