Necessary and sufficient conditions for regional stabilisability of generic switched linear systems with a pair of planar subsystems

In this article, the regional stabilisability issues of a pair of planar LTI systems are investigated through the geometrical approach, and easily verifiable necessary and sufficient conditions are derived. The main idea of the article is to characterise the best case switching signals based upon the variations of the constants of the integration of the subsystems. The conditions are generic as all possible combinations of the subsystem dynamics are considered.     

Necessary and sufficient condition for stabilizability of discrete-time linear switched systems: A set-theory approach

In this paper, the stabilizability of discrete-time linear switched systems is considered. Several sufficient conditions for stabilizability are proposed in the literature, but no necessary and sufficient. The main contributions are the necessary and sufficient conditions for stabilizability based on the set-theory and the characterization of a universal class of Lyapunov functions. An algorithm for computing the Lyapunov functions and a procedure to design the stabilizing switching control law are provided, based on such conditions. Moreover, a sufficient condition for non-stabilizability for switched system is presented. Several academic examples are given to illustrate the efficiency of the proposed results. In particular, a Lyapunov function is obtained for a system for which the Lyapunov–Metzler condition for stabilizability does not hold.     

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.     

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.     

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.     

Necessary and sufficient conditions of stationary average consensus for second-order multi-agent systems

This paper investigates the stationary average consensus problem for second-order discrete-time multi-agent systems (SDMAS). A stationary consensus problem is to find a control algorithm that brings the state of a group of agents to a common constant value which is called the collective decision. We introduce the concept of stationary average consensus of SDMAS and propose a consensus algorithm. Based on the polynomial stability and the graph theory, we obtain two necessary and sufficient conditions of stationary average consensus of SDMAS. The last theorem provides an algebraic criterion of stationary average consensus, and can help us to determine the parameters in the consensus algorithm. Furthermore, in this consensus algorithm, only the states of the agents are transferred among the agents. Therefore, this algorithm can not only solve the stationary average consensus problem but also reduce the amount of transferred data. A numerical example is provided to illustrate the efficiency of our results.     

Disturbance decoupling of switched linear systems

In this paper, we consider disturbance decoupling problems for switched linear systems. We will provide necessary and sufficient conditions for three different versions of disturbance decoupling, which differ based on which signals are considered to be the disturbance. In the first version, the exogenous input is considered as the disturbance, in the second, the switching signal and in the third both of them are considered as disturbances. All three versions of disturbance decoupling have direct counterparts for linear parameter-varying (LPV) systems, while the latter instance of the problem is relevant for disturbance decoupling of piecewise linear systems, as we will show. The solutions of the three disturbance decoupling problems will be based on geometric control theory for switched linear systems and will entail both mode-dependent and mode-independent static state feedback.     

Robust control for a class of state-constrained high-order switched nonlinear systems with unstable subsystems

This paper addresses the issue of robust stabilisation for a class of state-constrained high-order switched nonlinear systems whose subsystems are allowed to be all unstabilizable. A novel robust control scheme is established based on effective coupling of the technique of adding a power integrator and the method of multiple barrier Lyapunov functions. Explicitly, a sufficient condition for robustly stabilising the resulting closed-loop system in a domain is presented, which meanwhile can also guarantee that the violation of the constraints does not happen. Moreover, in the switching law design, we do not require the connection of barrier Lyapunov functions for two adjacent subsystems, and even a certain amount of jump is also allowed. Finally, the effectiveness of the provided control strategy is demonstrated via a simulation example.     

A control design method for a class of switched linear systems

In this note we consider the stability properties of a system class that arises in the control design problem of switched linear systems. The control design we are studying is based on a classical pole-placement approach. We analyse the stability of the resulting switched system and develop analytic conditions which reduce the complexity of the stability problem. We further consider two special cases for which strongly simplified conditions are obtained that support the analytic controller design.     

Componentwise ultimate bound and invariant set computation for switched linear systems

We present a novel ultimate bound and invariant set computation method for continuous-time switched linear systems with disturbances and arbitrary switching. The proposed method relies on the existence of a transformation that takes all matrices of the switched linear system into a convenient form satisfying certain properties. The method provides ultimate bounds and invariant sets in the form of polyhedral and/or mixed ellipsoidal/polyhedral sets, is completely systematic once the aforementioned transformation is obtained, and provides a new sufficient condition for practical stability. We show that the transformation required by our method can easily be found in the well-known case where the subsystem matrices generate a solvable Lie algebra, and we provide an algorithm to seek such transformation in the general case. An example comparing the bounds obtained by the proposed method with those obtained from a common quadratic Lyapunov function computed via linear matrix inequalities shows a clear advantage of the proposed method in some cases.     

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.     

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.     

Realization theory of discrete-time linear switched systems

The paper presents realization theory of discrete-time linear switched systems. We present necessary and sufficient conditions for an input–output map to admit a discrete-time linear switched system realization. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observability. Further, we prove that minimal realizations are unique up to isomorphism. We also discuss algorithms for converting a linear switched system to a minimal one and for constructing a state-space representation from input–output data. The paper uses the theory of rational formal power series in non-commutative variables.     

Finite data-rate feedback stabilization of switched and hybrid linear systems

We study the problem of asymptotically stabilizing a switched linear control system using sampled and quantized measurements of its state. The switching is assumed to be slow enough in the sense of combined dwell time and average dwell time, each individual mode is assumed to be stabilizable, and the available data rate is assumed to be large enough but finite. Our encoding and control strategy is rooted in the one proposed in our earlier work on non-switched systems, and in particular the data-rate bound used here is the data-rate bound from that earlier work maximized over the individual modes. The main technical step that enables the extension to switched systems concerns propagating over-approximations of reachable sets through sampling intervals, during which the switching signal is not known; a novel algorithm is developed for this purpose. Our primary focus is on systems with time-dependent switching (switched systems) but the setting of state-dependent switching (hybrid systems) is also discussed.     

Closed curve solutions and limit cycles in a class of second-order switched nonlinear systems

This paper studies the problem of finding the initial states for which the solution of a class of switched systems consisting of unstable second-order nonlinear subsystems is convergent. A method is described and applied to establish the regions in the plane where it is possible to define a switching law such that the solution of a class of switched nonlinear systems converges to the origin. We prove that, under certain conditions, these regions are delimited by closed curve solutions of the switched system. Furthermore, a sufficient condition for the closed curve solution to be a limit cycle is presented. Finally, a numerical example is included in order to illustrate the results.     

Controllability and reachability criteria for switched linear systems

This paper investigates the controllability and reachability of switched linear control systems. It is proven that both the controllable and reachable sets are subspaces of the total space. Complete geometric characterization for both sets is presented. The switching control design problem is also addressed.     

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.     

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…     

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.     

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.