Optimal control of discrete-time bilinear systems with applications to switched linear stochastic systems

This paper aims at characterizing the most destabilizing switching law for discrete-time switched systems governed by a set of bounded linear operators. The switched system is embedded in a special class of discrete-time bilinear control systems. This allows us to apply the variational approach to the bilinear control system associated with a Mayer-type optimal control problem, and a second-order necessary optimality condition is derived. Optimal equivalence between the bilinear system and the switched system is analyzed, which shows that any optimal control law can be equivalently expressed as a switching law. This specific switching law is most unstable for the switched system, and thus can be used to determine stability under arbitrary switching. Based on the second-order moment of the state, the proposed approach is applied to analyze uniform mean-square stability of discrete-time switched linear stochastic systems. Numerical simulations are presented to verify the usefulness of the theoretic results.     

Stability analysis of discrete-time switched systems: a switched homogeneous Lyapunov function method

This paper addresses the stability issue of discrete-time switched systems with guaranteed dwell-time. The approach of switched homogeneous Lyapunov function of higher order is formally proposed. By means of this approach, a necessary and sufficient condition is established to check the exponential stability of the considered system. With the observation that switching signal is actually arbitrary if the dwell time is one sample time, a necessary and sufficient condition is also presented to verify the exponential stability of switched systems under arbitrary switching signals. Using the augmented argument, a necessary and sufficient exponential stability criterion is given for discrete-time switched systems with delays. A numerical example is provided to show the advantages of the theoretical results.     

Model order-reduction for discrete-time switched linear systems

This article describes the problem of model order-reduction for a class of hybrid discrete-time switched linear systems composed of linear discrete-time invariant subsystems with a switching rule. This article investigates two novel approaches to model order-reduction. The first approach consists in evaluating the error approximation performance; the problems are solved using the robust stability results of the switched systems. The second approach presents the reachability and observability Gramians of the switched systems, which allows a balanced truncation model reduction procedure. A numerical example shows the effectiveness of the proposed approaches.     

Root-mean-square gains of switched linear systems: A variational approach

We consider the problem of computing the root-mean-square (RMS) gain of switched linear systems. We develop a new approach which is based on an attempt to characterize the “worst-case” switching law (WCSL), that is, the switching law that yields the maximal possible gain. Our main result provides a sufficient condition guaranteeing that the WCSL can be characterized explicitly using the differential Riccati equations (DREs) corresponding to the linear subsystems. This condition automatically holds for first-order SISO systems, so we obtain a complete solution to the RMS gain problem in this case.     

Stability analysis for discrete-time switched time-delay systems

The stability analysis problem is studied in this paper for a class of discrete-time switched time-delay systems. By using a newly constructed Lyapunov functional and the average dwell time scheme, a delay-dependent sufficient condition is derived for the considered system to be exponentially stable. The obtained results provide a solution to one of the basic problems in discrete-time switched time-delay systems, that is, to find a switching signal for which the switched time-delay system is exponentially stable. Two illustrative examples are given to demonstrate the effectiveness of the proposed results.     

Control discrete-time switched singular systems with state delays under asynchronous switching

This article is concerned with the problem of state feedback control for a class of discrete-time switched singular systems with time-varying state delays under asynchronous switching. The asynchronous switching considered here means that the switching instants of the candidate controllers lag behind those of the system modes. The concept of mismatched control rate is introduced. By using the multiple Lyapunov function approach and the average dwell time technique, a sufficient condition for the existence a stabilising switching law is first derived to guarantee the regularity, causality and exponential stability of the closed-loop system in the presence of asynchronous switching. The stabilising switching law is characterised by a upper bound on the mismatched control rate and a lower bound on the average dwell time. Then, the corresponding solvability condition for a set of mode-dependent state feedback controllers is established by using the linear matrix inequality (LMI) technique. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method.     

一类非线性离散切换系统基于观测器的指数镇定

对一类离散非线性切换系统, 考虑了基于观测器的指数镇定问题. 借助微分中值定理(DMVT), 将非线性切换系统转化为线性参数(LPV)切换系统. 当状态变量不完全可获得时, 基于多Lyapunov函数方法, 给出系统在基于观测器的输出反馈控制器下指数镇定的充分条件. 所设计的滞后切换规则能够避免产生滑动模态. 并且将结果推广到系统方程含有不确定性的情况. 最后, 仿真例子说明了设计方法的有效性.     

Stabilization of a class of discrete-time switched systems via observer-based output feedback

In this paper, observer-based static output feedback control problem for discrete-time uncertain switched systems is investigated under an arbitrary switching rule. The main method used in this note is combining switched Lyapunov function (SLF) method with Finsler’s Lemma. Based on linear matrix inequality (LMI) a less conservative stability condition is established and this condition allows extra degree of freedom for stability analysis. Finally, a simulation example is given to illustrate the efficiency of the result.     

离散多平衡点正切换系统有限区间稳定与镇定

在切换事件中,外界环境的干扰或者事物自身的发展变化会导致多平衡点现象.此时,多平衡点切换系统模型比传统的切换系统模型更适合描述此类事件.因此本文研究离散多平衡点正切换线性系统在有限时间区间上的稳定性与镇定性.第1,给出离散多平衡点线性切换系统为正的充要条件.第2,提出离散多平衡点正切换线性系统在有限时间区间上稳定的概念.第3,通过构造合适的Lyapunov函数以及合理分配系统的驻留时间与切换次数,针对部分子系统不稳定的离散多平衡点正切换线性系统,建立所考虑的自治系统有限时间稳定的充分条件.第4,给出非自治多平衡点正切换线性系统的控制器设计.最后,仿真例子验证理论结果的正确性.     

Stabilisation of second-order LTI switched positive systems

The stabilisation problem of second-order switched positive systems consisting of two unstable subsystems is considered in this article. By considering the vector fields and geometric characteristics, a necessary and sufficient condition for the stabilisability of second-order switched positive systems with two unstable subsystems is provided. Furthermore, it is shown via this condition that neither second-order switched positive systems consisting of two subsystems with unstable nodes nor second-order switched positive systems consisting of one subsystem with unstable nodes and the other with a saddle point can be stabilised via any switching law.     

Necessary and sufficient conditions for regional stabilisability of second-order switched linear systems with a finite number of subsystems

In this paper, the switching stabilisation problem of second-order switched linear systems is studied. By extending the best case switching signal (BCSS) criteria for second-order switched linear systems with two subsystems to the general case with any finite number of subsystems, necessary and sufficient conditions for regional stabilisability of second-order switched linear systems with any finite number of subsystems are proposed in this paper. The main idea of this paper is to divide all the subsystems in a region into two groups based on their trajectory directions in that region. By comparing pairwise, the most “stable” subsystem for each group in a region can be determined and the BCSS analysis among all the subsystems in that region is reduced to the BCSS analysis between the two most “stable” subsystems in that region.     

Robust stabilizing control laws for a class of second-order switched systems

For a class of second-order switched systems consisting of two linear time-invariant (LTI) subsystems, we show that the so-called conic switching law proposed previously by the present authors is robust, not only in the sense that the control law is flexible (to be explained further), but also in the sense that the Lyapunov stability (resp., Lagrange stability) properties of the switched system are preserved in the presence of certain kinds of vanishing perturbations (resp., nonvanishing perturbations). The analysis is possible since the conic switching laws always possess certain kinds of “quasi-periodic switching operations”. We also propose for a class of nonlinear second-order switched systems with time-invariant subsystems a switching control law which locally exponentially stabilizes the entire nonlinear switched system, provided that the conic switching law exponentially stabilizes the linearized switched systems (consisting of the linearization of each nonlinear subsystem). This switched control law is robust in the sense mentioned above.     

Asynchronous switching output feedback control of discrete-time switched linear systems

In this paper, the problem of dynamic output-feedback control synthesis is addressed for discrete-time switched linear systems under asynchronous switching. The proposed hybrid controller consists of a standard dynamic output-feedback switching control law and an impulsive reset law induced by controller state jumps. Using the average dwell time technique incorporating with multiple quadratic Lyapunov functions, the switching control synthesis conditions for asymptotic stability with guaranteed weighted ?₂-gain performance are derived as a set of linear matrix inequalities (LMIs). The proposed hybrid synthesis scheme advances existing design methods for output-feedback asynchronous switching control of switched linear systems in two important aspects: LMI formulation of the synthesis problem; and arbitrary order of the controller state. A numerical example is used to illustrate the effectiveness and advantages of the proposed design technique.     

Simultaneous reduced-order control and fault detection for discrete-time switched systems with relaxed persistent dwell time regularity

This paper focuses on the problem of simultaneous control and fault detection (FD) for discrete-time switched systems. The main goal is to develop a control/detection unit (CDU) associated with a switching law to control the system and detect faults simultaneously. When the switched systems are with partial measurable states, we directly use these states to construct partial control and FD signals. Next, the reduced-order CDU is designed to generate the other control and FD signals. Compared with existing results based on full-order CDU, the proposed results lead to less conservatism and reduce design complexity. The switching law is constructed in the frame of persistent dwell time (PDT) switching. A novel switching number constraint condition is introduced, which further relaxes the restrictions on the dwell time of switching processes of PDT. The less restriction on dwell time degrades the performance requirement of each subsystem and upgrades the degree of freedom for switching law design. Based on the proposed results, a class of nonweighted performance indexes is introduced to characterize the fault sensitivity and robustness. Finally, the effectiveness of the proposed method is illustrated by an example.     

Passivity analysis and control for discrete-time two-dimensional switched systems in Roesser model

This paper investigates the passivity issues of discrete-time two-dimensional switched systems in Roesser model under a state-dependent switching law. Conditions for passivity of discrete-time two-dimensional switched systems are obtained without the requirement of passivity of subsystems. When the system states are completely available, state feedback controllers and switching laws are designed. In the case of partial state measurements, dynamic output feedback controllers and switching laws depending only on the state of the output feedback controllers are constructed to guarantee passivity of the closed-loop switched system. Finally, numerical examples are provided to illustrate the efficiency of the results.     

Robust stability analysis of uncertain switched linear systems with unstable subsystems

The problem of robust stability for switched linear systems with all the subsystems being unstable is investigated. Unlike the most existing results in which each switching mode in the system is asymptotically stable, the subsystems may be unstable in this paper. A necessary condition of stability for switched linear systems is first obtained with certain hypothesis. Then, under two assumptions, sufficient conditions of exponential stability for both deterministic and uncertain switched linear systems are presented by using the invariant subspace theory and average dwell time method. Moreover, we further develop multiple Lyapunov functions and propose a method for constructing multiple Lyapunov functions for the considered switched linear systems with certain switching law. Several examples are included to show the effectiveness of the theoretical findings.     

Stability of switched positive nonlinear systems

This paper considers the stability problem for a class of switched positive nonlinear systems (SPNSs), which includes switched positive linear systems as a special case. We derive a necessary and sufficient condition for stability of continuous‐time SPNSs defined by homogeneous and cooperative vector fields under average dwell time switching. A corresponding necessary and sufficient condition is also given for stability of discrete‐time SPNSs defined by homogeneous and order‐preserving vector fields under average dwell time switching. The stability results for switched positive linear systems, which have been studied in the literature, can be easily obtained. We also extend the results to general switched linear systems. Finally, a numerical example is provided to illustrate the effectiveness of our results. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability of discrete-time switched systems with admissible edge-dependent switching signals

The problem of stability is studied in this paper for a class of discrete-time switched systems with unstable subsystems. Two new definitions of slow switching and fast switching on the basis of admissible edge-dependent average dwell time are proposed, respectively. Some conditions are established by using multiple Lyapunov function method to guarantee the global uniform exponential stability of discrete-time switched systems. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed results.     

Adaptive output tracking of linear uncertain discrete-time networked systems over switching topologies

ABSTRACT

In this paper, we study the asymptotic output tracking problem for a class of minimum phase uncertain linear discrete-time multi-agent systems over jointly connected digraphs. As the systems contain arbitrarily large uncertain parameters, the robust feedback control technique does not work. We develop a distributed adaptive control law composed of a distributed observer and a purely decentralised adaptive control law. We first establish a stability result for a class of linear switched systems, which extends the existing results. This stability result leads to a novel distributed observer for the leader system which estimates the leader's signal using the output of the leader system locally and passes the estimated leader's signal to each follower. We then further show that the distributed adaptive control law solves our problem under some mild conditions. An example is used to illustrate the effectiveness and generality of our approach.     

Properties of linear switching time-varying discrete-time systems with applications

We study two discrete-time, linear switching time-varying (LSTV) structures, each of which consists of a periodic switch connected to several linear time-invariant (LTI) systems. Such structures can be used to represent any linear periodically time-varying (LPTV) systems. We give basic properties associated with the LSTV structures in terms of their LTI building blocks, and then apply the results to solve a general approximation problem: How to optimally approximate an LPTV system with period p by an LPTV system with period ? The optimality is measured using norms. The study is extended to general multirate periodic systems.