Critical dwell time of switched linear systems

In this paper, we consider the relation between the switching dwell time and the stabilization of switched linear control systems. First of all, a concept of critical dwell time is given for switched linear systems without control inputs, and the critical dwell time is taken as an arbitrary given positive constant for a switched linear control systems with controllable switching models. Secondly, when a switched linear system has many stabilizable switching models, the problem of stabilization of the overall system is considered. An on-line feedback control is designed such that the overall system is asymptotically stabilizable under switching laws which depend only on those of uncontrollable subsystems of the switching models. Finally, when a switched system is partially controllable （While some switching models are probably unstabilizable）, an on-line feedback control and a cyclic switching strategy are designed such that the overall system is asymptotically stabilizable if all switching models of this uncontrollable subsystems are asymptotically stable. In addition, algorithms for designing switching laws and controls are presented.     

Asynchronously switched control of switched linear systems with average dwell time

This paper concerns the asynchronously switched control problem for a class of switched linear systems with average dwell time (ADT) in both continuous-time and discrete-time contexts. The so-called asynchronous switching means that the switchings between the candidate controllers and system modes are asynchronous. By further allowing the Lyapunov-like function to increase during the running time of active subsystems, the extended stability results for switched systems with ADT in nonlinear setting are first derived. Then, the asynchronously switched stabilizing control problem for linear cases is solved. Given the increase scale and the decrease scale of the Lyapunov-like function and the maximal delay of asynchronous switching, the minimal ADT for admissible switching signals and the corresponding controller gains are obtained. A numerical example is given to show the validity and potential of the developed results.     

Robust stability and controller synthesis of discrete linear switched systems with dwell time

Sufficient conditions are derived for the robust stability of discrete-time, switched, linear systems with dwell time in the presence of polytopic type parameter uncertainty. A Lyapunov function, in quadratic form, is assigned to each of the subsystems. This function is allowed to be time-varying and piecewise linear during the dwell time and it becomes time invariant afterwards. Asymptotic stability conditions are obtained in terms of linear matrix inequalities for the nominal set of subsystems. These conditions are then extended to the case where the subsystems encounter polytopic type parameter uncertainties. The developed method is applied to l ₂-gain analysis where a bounded real lemma is derived, and to H _∞ control and estimation, both for the nominal and the uncertain cases.     

Stability analysis of switched systems under dynamical dwell time control approach

This article investigates the stability of a class of switched systems using dynamical dwell time approach. First, the condition for stability of switched systems whose subsystems are stable are presented with dynamical dwell time approach, which is shown to be less conservative in switching law design than dwell time approach. Then the proposed approach is extended to the switched systems with both stable and unstable subsystems. Finally, some numerical examples are given to illustrate the effectiveness of the proposed results.     

Stabilizing switching design for switched linear systems: A state-feedback path-wise switching approach

This paper addresses the problem of switching stabilization for discrete-time switched linear systems. Based on the abstraction-aggregation methodology, we propose a state-feedback path-wise switching law, which is a state-feedback concatenation from a finite set of switching paths each defined over a finite time interval. We prove that the set of state-feedback path-wise switching laws is universal in the sense that any stabilizable switched linear system admits a stabilizing switching law in this set. We further develop a computational procedure to calculate a stabilizing switching law in the set.     

Stabilization and optimization of switched linear systems

In this paper, we establish the equivalence among switched convergency, asymptotic stabilizability, and exponential stabilizability for force-free switched linear systems, and discuss the implication to the infinite-time horizon optimal switching problem. We show that, for a general cost function under mild assumptions, the finiteness of the optimal cost is equivalent to the asymptotic stabilizability of the switched linear system. Finally, we prove the equality between the optimal costs for the switched system and for the relaxed differential inclusion.     

Analysis and synthesis of switched linear control systems

Switched linear systems have a long history of interest in the control community, and have attracted considerable attention recently because they are not only practically relevant, but also tangible with the rich results in the linear system theory. Rapid progress in the field has generated many new ideas and powerful tools. This paper provides a concise and timely survey on analysis and synthesis of switched linear control systems, and presents the basic concepts and main properties of switched linear systems in a systematic manner. The fundamental topics include (i) controllability and observability, (ii) system structural decomposition, (iii) feedback controller design for stabilization, and (iv) optimal control.     

Stability of linear discrete switched systems with delays based on average dwell time method

This paper deals with the problem of exponential stability for a class of linear discrete switched systems with constant delays.The switched systems consist of stable and unstable subsystems.Based on the average dwell time method, some switching signals will be found to guarantee exponential stability of these systems.The explicit state decay estimation is also given in the form of the solutions of linear matrix inequalities(LMIs).An example relating to networked control systems(NCSs) illustrates the effect...     

Reachability analysis of constrained switched linear systems

In this note, we investigate the reachability of switched linear systems with switching/input constraints. We prove that, under a mild assumption of the feasible switching signals, the reachability set is the reachable subspace of the unconstrained system. We also address the local reachability for switched linear systems with input constraints and present a complete criterion for a general class of switched linear systems.     

Output regulation of switched linear multi-agent systems: an agent-dependent average dwell time method

The output regulation problem of switched linear multi-agent systems with stabilisable and unstabilisable subsystems is investigated in this paper. A sufficient condition for the solvability of the problem is given. Owing to the characteristics of switched multi-agent systems, even if each agent has its own dwell time, the multi-agent systems, if viewed as an overall switched system, may not have a dwell time. To overcome this difficulty, we present a new approach, called an agent-dependent average dwell time method. Due to the limited information exchange between agents, a distributed dynamic observer network for agents is provided. Further, a distributed dynamic controller based on observer is designed. Finally, simulation results show the effectiveness of the proposed solutions.     

A dwell time approach to the stability of switched linear systems based on the distance between eigenvector sets

In this article, a sufficient condition on the minimum dwell time that guarantees the stability of switched linear systems is given. The proposed method interprets the stability of switched linear systems through the distance between the eigenvector sets of subsystem matrices. Thus, an explicit relation in view of stability is obtained between the family of the involved subsytems and the set of admissible switching signals.     

Finite‐time stability of switched positive linear systems

This brief paper addresses the finite‐time stability problem of switched positive linear systems. First, the concept of finite‐time stability is extended to positive linear systems and switched positive linear systems. Then, by using the state transition matrix of the system and copositive Lyapunov function, we present a necessary and sufficient condition and a sufficient condition for finite‐time stability of positive linear systems. Furthermore, two sufficient conditions for finite‐time stability of switched positive linear systems are given by using the common copositive Lyapunov function and multiple copositive Lyapunov functions, a class of switching signals with average dwell time is designed to stabilize the system, and a computational method for vector functions used to construct the Lyapunov function of systems is proposed. Finally, a concrete application is provided to demonstrate the effectiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Stability analysis for a class of switched linear systems

This paper investigates the stability of a class of switched linear systems, and proposes a number of new results on the stability analysis. A novel analysis method is developed by using the 2‐norm technique, and then several stability results are obtained based on the new analysis method. It is shown that the main results obtained in this paper not only guarantee the stability of the systems under arbitrary switching, but also provide an algorithm to find the minimum dwell time (MDT) with which switches make the switched systems stable. Finally, several illustrative examples with numerical simulations are studied by using the results obtained in this paper. The study of examples shows that our analysis method works very well in analyzing the stability of some classes of switched linear systems and, moreover, it has some advantages over the common quadratic Lyapunov function method in some cases. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

On stability of switched linear systems with perturbed switching paths

1 Introduction The study of switched and hybrid dynamical systems has attracted much attention since 1990s. A switched linear sys- tem is a hybrid system which consists of several linear time- invariant subsystems and a switching path that orchestrates the switching among them. The importance of the switched linear systems scheme stems from the facts that i) the frame- work represents a wide class of practical systems, ii) the two-level system structure provides an effective multiple- controll…     

Dwell time and average dwell time methods based on the cycle ratio of the switching graph

A new method to find an upper bound on dwell time and average dwell time for switched linear systems is proposed. The method is based on computing the maximum cycle ratio and the maximum cycle mean of the directed graph that governs switchings. For planar switched systems, an upper bound for dwell time and average dwell time can be estimated by considering only the cycles of length two.     

Exponential H∞ filtering for uncertain discrete‐time switched linear systems with average dwell time: A µ‐dependent approach

In this paper, the problem of exponential H_∞ filter problem for a class of discrete‐time polytopic uncertain switched linear systems with average dwell time switching is investigated. The exponential stability result of the general discrete‐time switched systems using a discontinuous piecewise Lyapunov function approach is first explored. Then, a new µ‐dependent approach is proposed, which means the analysis or synthesis of the underlying systems is dependent on the increase degree µ of the piecewise Lyapunov function at the switching instants. A mode‐dependent full‐order filter is designed such that the developed filter error system is robustly exponentially stable and achieves an exponential H_∞ performance. Sufficient existence conditions for the desired filter are derived and formulated in terms of a set of linear matrix inequalities, and consequently the minimal average dwell time and the corresponding filter are obtained from such conditions for a given system decay degree. A numerical example is presented to demonstrate the potential and effectiveness of the developed theoretical results. Copyright © 2007 John Wiley & Sons, Ltd.     

On the stabilization of switched linear stochastic systems with unobservable switching laws

This paper is concerned with the stabilization problem of switched linear stochastic systems with unob- servable switching laws. In this paper the system switches among a finite family of linear stochastic systems. Since there are noise perturbations, the switching laws can not be identified in any finite time horizon. We prove that if each individual subsystem is controllable and the switching duration uniformly has a strict positive lower bound, then the system can be stabilized by using a controller that uses online state estimation.     

On the stabilization of switched linear stochastic systems with unobservable switching laws

1 Introduction Consider the following switched linear stochastic system dx(t) = Aθtx(t)dt +Bθtu(t)dt + Fθtdwt, E x0 2 < ∞ (1) where x0 = x(0). The system state x(t) ∈ Rn is completely observable and u(t) ∈ Rr is the input. The system noise {wt} is an l-dimensional standard Wiener process, which is independent of {x(s),s t}. Aθt, Bθt, Fθt are coefficient matrices with suitable dimensions. The switching law (or signal) θt : [0, ∞) → Θ is a piecewise constant function of time. I…     

Design of switching sequences for controllability realization of switched linear systems

In this paper, we study the problem of designing switching sequences for controllability of switched linear systems. Each controllable state set of designed switching sequences coincides with the controllable subspace. Both aperiodic and periodic switching sequences are considered. For the aperiodic case, a new approach is proposed to construct switching sequences, and the number of switchings involved in each designed switching sequence is shown to be upper bounded by d(d-d₁+1). Here d is the dimension of the controllable subspace, , where (A_i,B_i) are subsystems. For the periodic case, we show that the controllable subspace can be realized within d switching periods.