Sufficient and necessary conditions for the stability of second-order switched linear systems under arbitrary switching

Many practical systems can be modelled as switched systems, whose stability problem is challenging even for linear subsystems. In this article, the stability problem of second-order switched linear systems with a finite number of subsystems under arbitrary switching is investigated. Sufficient and necessary stability conditions are derived based on the worst-case analysis approach in polar coordinates. The key idea of this article is to partition the whole state space into several regions and reduce the stability analysis of all the subsystems to analysing one or two worst subsystems in each region. This article is an extension of the work for stability analysis of second-order switched linear systems with two subsystems under arbitrary switching.     

A second-order maximum principle for discrete-time bilinear control systems with applications to discrete-time linear switched systems

A powerful approach for analyzing the stability of continuous-time switched systems is based on using optimal control theory to characterize the “most unstable” switching law. This reduces the problem of determining stability under arbitrary switching to analyzing stability for the specific “most unstable” switching law. For discrete-time switched systems, the variational approach received considerably less attention. This approach is based on using a first-order necessary optimality condition in the form of a maximum principle (MP), and typically this is not enough to completely characterize the “most unstable” switching law. In this paper, we provide a simple and self-contained derivation of a second-order necessary optimality condition for discrete-time bilinear control systems. This provides new information that cannot be derived using the first-order MP. We demonstrate several applications of this second-order MP to the stability analysis of discrete-time linear switched systems.     

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.     

Observer-based stabilization of switching linear systems

In this paper, we present a “deep pole assignment method” to study the observer-based stabilization of switching linear systems where the dynamics of each mode are known a priori but the switching times of modes are arbitrary. The design can be used for both finite and infinite switched linear systems. We emphasize our paper on the case where the switchings of the observer and controller do not coincide with those of the system.     

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.     

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.     

On linear co-positive Lyapunov functions for sets of linear positive systems

In this paper we derive necessary and sufficient conditions for the existence of a common linear co-positive Lyapunov function for a finite set of linear positive systems. Both the state dependent and arbitrary switching cases are considered. Our results reveal an interesting characterisation of “linear” stability for the arbitrary switching case; namely, the existence of such a linear Lyapunov function can be related to the requirement that a number of extreme systems are Metzler and Hurwitz stable. Examples are given to illustrate the implications of our results.     

Stability analysis and control synthesis of switched impulsive systems

In this paper, we investigate the stability analysis problem of switched impulsive nonlinear systems and several stabilization problems of switched discrete‐time linear systems are studied. First, sufficient conditions ensuring globally uniformly asymptotically stability of switched nonlinear impulsive system under arbitrary and DDT (dynamical dwell time which defines the length of the time interval between two successive switchings) switching are derived, respectively. In the DDT switching case, we first consider the switched system composed by stable subsystems, then we extend the results to the case where not all subsystems are stable. The stabilizations of switched discrete‐time linear system under arbitrary switching, DDT switching and asynchronous switching are investigated respectively. Based on the stability analysis results, the control synthesis consists of controller design for each subsystem and state impulsive jumping generators design at switching instant. With the aid of the state impulsive jumping generators at switching instant, the ‘energy’ produced by switching can be minimized, which leads to less conservative results. Several numerical examples are given to illustrate the proposed results within this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

Common quadratic Lyapunov-like functions with associated switching regions for two unstable second-order LTI systems

In this paper we establish quadratic Lyapunov-like functions for qualitative analysis of switched systems. Specifically, for a class of switched systems consisting of two unstable second-order LTI (linear time-invariant) subsystems, we explore in detail some necessary and sufficient conditions for the existence of common (weakly) quadratic Lyapunov-like functions with associated switching regions in R 2 plane. The existence conditions and the construction of such quadratic Lyapunov-like functions are established by using the conic switching laws.     

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.     

General Solution of Stability Problem for Plane Linear Switched Systems and Differential Inclusions

Characterization and control of stability of switched dynamical systems and differential inclusions have attracted significant attention in the recent past. The most of the current results for this problem are obtained by application of the Lyapunov function method which provides sufficient but frequently over conservative stability conditions. For planar systems, practically verifiable necessary and sufficient conditions are found only for switched systems with two subsystems. This paper provides explicit necessary and sufficient conditions for asymptotic stability of switched systems and differential inclusions with arbitrary number of subsystems; these conditions turned out to be identical for the both classes of systems. A precise upper bound for the number of switching points in a periodic solution, corresponding to the break of stability, is found. It is shown that, for a switched system, the break of stability may also occur on a solution with infinitely fast switching (chattering) between some two subsystems.     

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.     

Switched convergence of second-order switched nonlinear systems

In this paper, the problem of stabilizing second-order switched nonlinear systems consisting of two unstable subsystems is studied. We present a method for determining if a switched system can be stabilized. Important results from Differential Geometry have been needed in order to study the stabilization in the nonlinear case. Hence, it is established if there exists a switching law under which the solution of a switched system for a given initial condition converges to the origin. In order to illustrate the results we present several numerical examples. Furthermore, the results are applied to a kind of switched systems of higher dimension.     

Frequency-Domain Stability Conditions for Discrete-Time Switched Systems

We consider discrete-time switched systems with switching of linear time-invariant right-hand parts. The notion of a connected discrete switched system is introduced. For systems with the connectedness property, we propose necessary and sufficient frequency-domain conditions for the existence of a common quadratic Lyapunov function that provides the stability for a system under arbitrary switching. The set of connected switched systems contains discrete control systems with several time-varying nonlinearities from the finite sectors, considered in the theory of absolute stability. We consider the case of switching between three linear subsystems in more details and give an illustrative example.     

Decentralized switching control for hierarchical systems

In this paper, a decentralized switching scheme is introduced for uncertain interconnected systems with a structure which is “approximately hierarchical”, and where the switching controller acts on the control agents independently of each other. In this problem, it is assumed that the plant at any point of time can be described by a finite set of linear time-invariant (LTI) finite-dimensional models, but that the switching controller does not require any knowledge of these plant models; the only requirement made is that there exists a known finite set of decentralized controllers, containing at least one controller which can stabilize and regulate the actual physical plant at any time. Simulation results obtained for the proposed decentralized switching controller are given.     

Design of two-layer switching rule for stabilization of switched linear systems with mismatched switching

A two-layer switching architecture and a two-layer switching rule for stabilization of switched linear control systems are proposed, under which the mismatched switching between switched systems and their candidate hybrid controllers can be allowed. In the low layer, a state-dependent switching rule with a dwell time constraint to exponentially stabilize switched linear systems is given; in the high layer, supervisory conditions on the mismatched switching frequency and the mismatched switching ratio are presented, under which the closed-loop switched system is still exponentially stable in case of the candidate controller switches delay with respect to the subsystems. Different from the traditional switching rule, the two-layer switching architecture and switching rule have robustness, which in some extend permit mismatched switching between switched subsystems and their candidate controllers.     

Supervision of integral-input-to-state stabilizing controllers

The subject of this paper is hybrid control of nonlinear systems with large-scale uncertainty. We describe a high-level controller, called a “supervisor”, which orchestrates logic-based switching among a family of candidate controllers. We show that in this framework, the problem of controller design at the lower level can be reduced to finding an integral-input-to-state stabilizing control law for an appropriate system with disturbance inputs. Employing the recently introduced “scale-independent hysteresis” switching logic, we prove that in the case of purely parametric uncertainty with unknown parameters taking values in a finite set the switching terminates in finite time and state regulation is achieved.     

Stabilization of switched continuous-time systems with all modes unstable via dwell time switching

Stabilization of switched systems composed fully of unstable subsystems is one of the most challenging problems in the field of switched systems. In this brief paper, a sufficient condition ensuring the asymptotic stability of switched continuous-time systems with all modes unstable is proposed. The main idea is to exploit the stabilization property of switching behaviors to compensate the state divergence made by unstable modes. Then, by using a discretized Lyapunov function approach, a computable sufficient condition for switched linear systems is proposed in the framework of dwell time; it is shown that the time intervals between two successive switching instants are required to be confined by a pair of upper and lower bounds to guarantee the asymptotic stability. Based on derived results, an algorithm is proposed to compute the stability region of admissible dwell time. A numerical example is proposed to illustrate our approach.     

Stabilization of second-order LTI switched systems

This paper studies and solves the problem of asymptotic stabilization of switched systems consisting of unstable secondorder linear time-invariant (LTI) subsystems. Necessary and sufficient conditions for asymptotic stabilizability are first obtained. If a switched system is asymptotically stabilizable, then the conic switching laws proposed in the paper are used to construct a switching law that asymptotically stabilizes the system. Switched systems consisting of two subsystems with unstable foci are studied first and then the results are extended to switched systems with unstable nodes and saddle points. The results are applicable to switched systems that consist of more than two subsystems.     

Stability and Stabilization of Two‐Dimensional LTI Switched Systems with Potentially Unstable Focus

This paper investigates stability and stabilization of two‐dimensional switched linear time‐invariant (LTI) systems with potentially unstable focus. For the case that the origin is a single common focus of all subsystems, we first give continuous positive definite functions related only to the elements of subsystems' state matrices. Then, based on the continuous positive definite functions obtained, this paper proposes several sufficient conditions of stability/asymptotic stability/instability of the kind of switched LTI systems. By means of the stability results proposed, global asymptotic stabilizing controls (GASC), global asymptotic stabilizing switching paths (GASSP) and corresponding algorithms are designed for two‐dimensional switched LTI systems with focus. Finally, two illustrative examples and numerical simulations demonstrate the effectiveness of the new stability and stabilization results obtained in this paper.