Stability and Stabilizability of Switched Linear Systems: A Short Survey of Recent Results
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paper_title: Control of Systems Integrating Logic, Dynamics, and Constraints

paper_content:

This paper proposes a framework for modeling and controlling systems described by interdependent physical laws, logic rules, and operating constraints, denoted as mixed logical dynamical (MLD) systems. These are described by linear dynamic equations subject to linear inequalities involving real and integer variables. MLD systems include linear hybrid systems, finite state machines, some classes of discrete event systems, constrained linear systems, and nonlinear systems which can be approximated by piecewise linear functions. A predictive control scheme is proposed which is able to stabilize MLD systems on desired reference trajectories while fulfilling operating constraints, and possibly take into account previous qualitative knowledge in the form of heuristic rules. Due to the presence of integer variables, the resulting on-line optimization procedures are solved through mixed integer quadratic programming (MIQP), for which efficient solvers have been recently developed. Some examples and a simulation case study on a complex gas supply system are reported.
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paper_title: Differential equations with discontinuous right-hand sides☆

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Many phenomena from physics, control theory, and economics have been successfully described by differential equations with discontinuous functions. This in turn leads to a great need to develope a theory for such differential equations. However, as far as existence and qualitative behavior are concerned, such a theory is far from being complete. Even existence results are available only for very special discontinuous function f (as the discontinuous part in equation), e.g., see [S] for piecewise continuous f; [9] for f having one discontinuous point, [S, 121 for f of bounded variation. In this paper, we are going to establish some existence results for differential equations with discontinuous right-hand sides with great generality. Some of the results in this paper can be stated and proved for high order ordinary and partial differential equations. For example, we may consider the following problem
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paper_title: Observability and Controllability of Piecewise Affine and Hybrid Systems

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We prove, in a constructive way, the equivalence between piecewise affine systems and a broad class of hybrid systems described by interacting linear dynamics, automata, and propositional logic. By focusing our investigation on the former class, we show through counterexamples that observability and controllability properties cannot be easily deduced from those of the component linear subsystems. Instead, we propose practical numerical tests based on mixed-integer linear programming.
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paper_title: Basic Problems in Stability and Design of Switched Systems

paper_content:

By a switched system, we mean a hybrid dynamical system consisting of a family of continuous-time subsystems and a rule that orchestrates the switching between them. The article surveys developments in three basic problems regarding stability and design of switched systems. These problems are: stability for arbitrary switching sequences, stability for certain useful classes of switching sequences, and construction of stabilizing switching sequences. We also provide motivation for studying these problems by discussing how they arise in connection with various questions of interest in control theory and applications.
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paper_title: Controllability and reachability criteria for switched linear systems

paper_content:

Complete geometric criteria are presented for controllability and reachability of switched linear systems
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paper_title: Switching in Systems and Control

paper_content:

I. INTRODUCTION 1. Basic Concepts II . STABILITY OF SWITCHED SYSTEMS 2. Stability under Arbitrary Switching 3. Stability under Constrained Switching III. SWITCHING CONTROL 4. Systems not Stabilizable by Continuous Feedback 5. Systems with Sensor or Actuator Constraints 6. Systems with Large Modeling Uncertainty IV. SUPPLEMENTARY MATERIAL A. Stability B. Lie Algebras Notes and References Bibliography Index
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paper_title: Perspectives and results on the stability and stabilizability of hybrid systems

paper_content:

This paper introduces the concept of a hybrid system and some of the challenges associated with the stability of such systems, including the issues of guaranteeing stability of switched stable systems and finding conditions for the existence of switched controllers for stabilizing switched unstable systems. In this endeavour, this paper surveys the major results in the (Lyapunov) stability of finite-dimensional hybrid systems and then discusses the stronger, more specialized results of switched linear (stable and unstable) systems. A section detailing how some of the results can be formulated as linear matrix inequalities is given. Stability analyses on the regulation of the angle of attack of an aircraft and on the PI control of a vehicle with an automatic transmission are given. Other examples are included to illustrate various results in this paper.
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paper_title: Piecewise Linear Control Systems

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This thesis treats analysis and design of piecewise linear control systems. Piecewise linear systems capture many of the most common nonlinearities in engineering systems, and they can also be used for approximation of other nonlinear systems. Several aspects of linear systems with quadratic constraints are generalized to piecewise linear systems with piecewise quadratic constraints. It is shown how uncertainty models for linear systems can be extended to piecewise linear systems, and how these extensions give insight into the classical trade-offs between fidelity and complexity of a model. Stability of piecewise linear systems is investigated using piecewise quadratic Lyapunov functions. Piecewise quadratic Lyapunov functions are much more powerful than the commonly used quadratic Lyapunov functions. It is shown how piecewise quadratic Lyapunov functions can be computed via convex optimization in terms of linear matrix inequalities. The computations are based on a compact parameterization of continuous piecewise quadratic functions and conditional analysis using the S-procedure. A unifying framework for computation of a variety of Lyapunov functions via convex optimization is established based on this parameterization. Systems with attractive sliding modes and systems with bounded regions of attraction are also treated. Dissipativity analysis and optimal control problems with piecewise quadratic cost functions are solved via convex optimization. The basic results are extended to fuzzy systems, hybrid systems and smooth nonlinear systems. It is shown how Lyapunov functions with a discontinuous dependence on the discrete state can be computed via convex optimization. An automated procedure for increasing the flexibility of the Lyapunov function candidate is suggested based on linear programming duality. A Matlab toolbox that implements several of the results derived in the thesis is presented.
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paper_title: Analysis and synthesis of switched linear control systems

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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.
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paper_title: Qualitative analysis of discrete-time switched systems

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We investigate some qualitative properties for time-controlled switched systems consisting of several linear discrete-time subsystems. First, we study exponential stability of the switched system with commutation property, stable combination and average dwell time. When all subsystem matrices are commutative pairwise and there exists a stable combination of unstable subsystem matrices, we propose a class of stabilizing switching laws where Schur stable subsystems are activated arbitrarily while unstable ones are activated in sequence with their duration time periods satisfying a specified ratio. For more general switched system whose subsystem matrices are not commutative pairwise, we show that the switched system is exponentially stable if the average dwell time is chosen sufficiently large and the total, activation time ratio between Schur stable and unstable subsystems is not smaller than a specified constant. Secondly, we use an average dwell time approach incorporated with a piecewise Lyapunov function to study the /spl Lscr//sub 2/ gain of the switched system.
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paper_title: Persistent disturbance attenuation properties for networked control systems

paper_content:

In this paper, both the asymptotic stability and l/sup /spl infin// persistent disturbance attenuation issues are investigated for a class of networked control systems (NCSs) under bounded uncertain access delay and packet dropout effects. The basic idea is to formulate such NCSs as discrete-time switched systems with arbitrary switchings. Then the NCSs' stability and performance problems can be reduced to the corresponding problems of such switched systems. The asymptotic stability problem is considered first, and a necessary and sufficient condition is derived for the NCSs' asymptotic stability based on robust stability techniques. Secondly, the NCSs' l/sup /spl infin// persistent disturbance attenuation properties are studied, and an algorithm is proposed to calculate the l/sup /spl infin// induced gain of the NCSs. The decidability issue of the algorithm is also discussed.
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paper_title: A singularity test for the existence of common quadratic Lyapunov functions for pairs of stable LTI systems

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In this paper we derive algebraic conditions that are both necessary and sufficient for a pair of stable third order linear time-invariant (LTI) systems to have a common quadratic Lyapunov function (CQLF). We also show how these results can be extended to derive necessary and sufficient conditions for the existence of a CQLF for pairs of LTI systems of arbitrary order.
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paper_title: Switching in Systems and Control

paper_content:

I. INTRODUCTION 1. Basic Concepts II . STABILITY OF SWITCHED SYSTEMS 2. Stability under Arbitrary Switching 3. Stability under Constrained Switching III. SWITCHING CONTROL 4. Systems not Stabilizable by Continuous Feedback 5. Systems with Sensor or Actuator Constraints 6. Systems with Large Modeling Uncertainty IV. SUPPLEMENTARY MATERIAL A. Stability B. Lie Algebras Notes and References Bibliography Index
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paper_title: Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for two stable second order linear time-invariant systems

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In this paper, necessary and sufficient conditions are derived for the existence of a common quadratic Lyapunov function for two stable second order linear time-invariant systems.
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paper_title: Lie-Algebraic Stability Criteria for Switched Systems

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It was recently shown that a family of exponentially stable linear systems whose matrices generate a solvable Lie algebra possesses a quadratic common Lyapunov function, which implies that the corresponding switched linear system is exponentially stable for arbitrary switching. In this paper we prove that the same properties hold under the weaker condition that the Lie algebra generated by given matrices can be decomposed into a sum of a solvable ideal and a subalgebra with a compact Lie group. The corresponding local stability result for nonlinear switched systems is also established. Moreover, we demonstrate that if a Lie algebra fails to satisfy the above condition, then it can be generated by a family of stable matrices such that the corresponding switched linear system is not stable. Relevant facts from the theory of Lie algebras are collected at the end of the paper for easy reference.
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paper_title: Criteria of asymptotic stability of differential and difference inclusions encountered in control theory

paper_content:

Abstract For a class of differential inclusions, to which many of the practically important control systems can be reduced, necessary and sufficient conditions for asymptotic stability of the zero solution are established by the method of Lyapunov functions. Similar results are obtained for the corresponding class of difference inclusions.
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paper_title: A necessary and sufficient condition for robust asymptotic stability of time-variant discrete systems

paper_content:

A necessary and sufficient condition for the stability of time-variant interval matrices is presented. This condition allows stability to be tested by checking only products of vertex (extreme) matrices. The implementation of the test in the form of an algorithm and two illustrative examples are provided. >
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paper_title: A common Lyapunov function for stable LTI systems with commuting A-matrices

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The paper demonstrates that a common quadratic Lyapunov function exists for all linear systems of the form x/spl dot/=A/sub i/x, i=1,2,/spl middot//spl middot//spl middot/,N, where the matrices A/sub i/ are asymptotically stable and commute pairwise. This in turn assures the exponential stability of a switching system x/spl dot/(t)=A(t)x(t) where A(t) switches between the above constant matrices A/sub i/. >
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paper_title: Common Lyapunov functions and gradient algorithms

paper_content:

This note is concerned with the problem of finding a quadratic common Lyapunov function for a large family of stable linear systems. We present gradient iteration algorithms which give deterministic convergence for finite system families and probabilistic convergence for infinite families.
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paper_title: A converse Lyapunov theorem for nonlinear switched systems

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Abstract In this paper we present a converse Lyapunov theorem for uniform asymptotic stability of switched nonlinear systems. Its proof is a simple consequence of some results on converse Lyapunov theorems for systems with bounded disturbances obtained by Lin et al. (SIAM J. Control Optim. 34 (1996) 124–160), once an association of the switched system with a nonlinear system with disturbances is established.
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paper_title: A Converse Lyapunov Theorem for a Class of Dynamical Systems which Undergo Switching

paper_content:

The authors investigate the stability of a system in which the dynamics at any instant in time will follow one of a fixed set of vector fields. They allow switching between members of the family of vector fields to be completely random. The focus of this paper is to prove a converse Lyapunov theorem for this class of systems.
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paper_title: Basic Problems in Stability and Design of Switched Systems

paper_content:

By a switched system, we mean a hybrid dynamical system consisting of a family of continuous-time subsystems and a rule that orchestrates the switching between them. The article surveys developments in three basic problems regarding stability and design of switched systems. These problems are: stability for arbitrary switching sequences, stability for certain useful classes of switching sequences, and construction of stabilizing switching sequences. We also provide motivation for studying these problems by discussing how they arise in connection with various questions of interest in control theory and applications.
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paper_title: Multiple Lyapunov Functions and Other Analysis Tools for Switched and Hybrid Systems

paper_content:

Hybrid systems are at the core of most embedded and many other kinds of systems; formal methods for analysis of hybrid systems have made remarkable progress in the last decade and thus provide a strong foundation for assurance in the system core. ::: ::: But there are many systems issues that interact with the hybrid systems core and complicate the overall system design and its assurance case. These include real time and fault tolerance, interaction with human operators, and the relationship between verification and certification. ::: ::: For example, fault tolerance demands multiple redundant sensors, which are themselves prone to faults and inaccuracy, and whose precision degrades as real time progresses from the moment when the sample was taken to that when it is used. Fault tolerance generally also requires multiple independent channels of computation and this raises issues of their synchronization and coordination. ::: ::: There are two broad classes of methods for dealing with these combined issues: one uses architectural means to separate them, so we can reason separately about hybrid control and fault tolerance, for example; the other integrates them, so that a single method is used to reason, for example, about real time and fault tolerance. I decribe some of these methods and sketch some topics for further research. ::: ::: In the larger systems context, the embedded core may be managed by a planning and execution system that uses AI techniques, and/or by a human operator. Both of these may have an imperfect model of the system and incomplete knowledge of its internal state. I outline these topics and some of the interesting research opportunities therein. ::: ::: Finally, many of the systems we consider have the potential to do harm, and thus raise concern for informal or regulated certification. I outline recent developments in this area and their connection to verification. ::: ::: The rich relationship between hybrid systems and everything else suggests a need to reason cooperatively across multiple domains. I sketch a proposal for “an evidential tool bus” to facilitate this.
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paper_title: Perspectives and results on the stability and stabilizability of hybrid systems

paper_content:

This paper introduces the concept of a hybrid system and some of the challenges associated with the stability of such systems, including the issues of guaranteeing stability of switched stable systems and finding conditions for the existence of switched controllers for stabilizing switched unstable systems. In this endeavour, this paper surveys the major results in the (Lyapunov) stability of finite-dimensional hybrid systems and then discusses the stronger, more specialized results of switched linear (stable and unstable) systems. A section detailing how some of the results can be formulated as linear matrix inequalities is given. Stability analyses on the regulation of the angle of attack of an aircraft and on the PI control of a vehicle with an automatic transmission are given. Other examples are included to illustrate various results in this paper.
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paper_title: Stability of switched systems with average dwell-time

paper_content:

It is shown that switching among stable linear systems results in a stable system provided that switching is \slow-on-the-average." In particular, it is proved that exponential stability is achieved when the number of switches in any nite interval grows linearly with the length of the interval, and the growth rate is suciently small. Moreover, the exponential stability is uniform over all switchings with the above property. For switched systems with inputs this guarantees that several input-to-state induced norms are bounded uniformly over all slow-on-the-average switchings. These results extend to classes of nonlinear switched systems that satisfy suitable uniformity assumptions. In this paper it is also shown that, in a supervisory control context, scale-independent hysteresis can produce switching that is slow-on-the-average and therefore the results mentioned above can be used to study the stability of hysteresis-based adaptive control systems.
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paper_title: Stability Theory for Hybrid Dynamical Systems.

paper_content:

This entry provides a short introduction to modeling of hybrid dynamical systems and then focuses on stability theory for these systems. It provides definitions of asymptotic stability, basin of attraction, and uniform asymptotic stability for a compact set. It points out mild assumptions under which different characterizations of asymptotic stability are equivalent, as well as when an asymptotically stable compact set exists. It also summarizes necessary and sufficient conditions for asymptotic stability in terms of Lyapunov functions.
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paper_title: Uniform stability of switched linear systems: extensions of LaSalle's Invariance Principle

paper_content:

This paper addresses the uniform stability of switched linear systems, where uniformity refers to the convergence rate of the multiple solutions that one obtains as the switching signal ranges over a given set. We provide a collection of results that can be viewed as extensions of LaSalle's Invariance Principle to certain classes of switched linear systems. Using these results one can deduce asymptotic stability using multiple Lyapunov functions whose Lie derivatives are only negative semidefinite. Depending on the regularity assumptions placed on the switching signals, one may be able to conclude just asymptotic stability or (uniform) exponential stability. We show by counter-example that the results obtained are tight.
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paper_title: Stabilization of hybrid systems using a min-projection strategy

paper_content:

This paper describes a method of how to stabilize a system consisting of several subsystems. The subsystems are described by nonlinear models with different vector fields. The method is denoted the min-projection strategy, since the vector field associated with the smallest (skew) projection is selected for each state. Conditions are given guaranteeing (exponential) stability. It is also shown how these conditions can be formulated as a nonlinear optimization problem, or, for a pre-determined projection matrix, a linear matrix inequality (LMI) problem. Sliding motions may occur in the basic form of the strategy. However, it is shown how this behavior can be avoided by introducing hysteresis around the switch surfaces, still preserving the stability of the closed-loop hybrid system. Two examples are given to motivate and exemplify the strategy.
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paper_title: Basic Problems in Stability and Design of Switched Systems

paper_content:

By a switched system, we mean a hybrid dynamical system consisting of a family of continuous-time subsystems and a rule that orchestrates the switching between them. The article surveys developments in three basic problems regarding stability and design of switched systems. These problems are: stability for arbitrary switching sequences, stability for certain useful classes of switching sequences, and construction of stabilizing switching sequences. We also provide motivation for studying these problems by discussing how they arise in connection with various questions of interest in control theory and applications.
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paper_title: Stability results for switched controller systems

paper_content:

There are many practical control problems where the control action is determined by switching among a given set of control laws. This paper presents necessary and sufficient conditions to test for quadratic stabilizability and for robust stabilizability with a quadratic storage function for switched controller systems. Algorithms which can be used to construct appropriately stabilizing control laws are also presented.
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paper_title: Quadratic stabilizability of switched linear systems with polytopic uncertainties

paper_content:

In this paper, we consider quadratic stabilizability via state feedback for both continuous-time and discrete-time switched linear systems that are composed of polytopic uncertain subsystems. By state feedback, we mean that the switchings among subsystems are dependent on system states. For continuous-time switched linear systems, we show that if there exists a common positive definite matrix for stability of all convex combinations of the extreme points which belong to different subsystem matrices, then the switched system is quadratically stabilizable via state feedback. For discrete-time switched linear systems, we derive a quadratic stabilizability condition expressed as matrix inequalities with respect to a family of non-negative scalars and a common positive definite matrix. For both continuous-time and discrete-time switched systems, we propose the switching rules by using the obtained common positive definite matrix.
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paper_title: Synthesis of switching rules for switched linear systems through randomized algorithms

paper_content:

In this paper, the design of a stabilizing switching rule for a switched linear system is considered. We first propose a probabilistic algorithm for a known nonconvex condition that employs a multiple Lyapunov function. We prove a probability-one convergence of the algorithm under a new notion of convergence. Then, to reduce its complexity, a modified version of the algorithm is developed. The results are illustrated using two-and three-dimensional systems with multiple switch states.
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paper_title: Synthesis of switched linear systems

paper_content:

This paper treats the following synthesis problem: given a switched linear system, how should the linear vector fields be selected among the possible ones such that the switched system becomes (exponentially) stable? In this paper, this synthesis problem is constructively formulated as a bilinear matrix inequality problem. The starting point for the design is existing stability conditions for switched and hybrid systems using multiple quadratic Lyapunov functions, one for each linear vector field. Using such a general piecewise quadratic Lyapunov function structure, a stabilizing controller is synthesized for a broader class of switched systems than earlier proposed in the literature. One example is given to illustrate the synthesis procedure.
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paper_title: Solution of coupled Lyapunov equations for the stabilization of multimodal linear systems

paper_content:

The paper discusses stabilization of linear multimodal systems using a control law that selects a particular system mode. The systems considered are similar to jump linear systems. In a jump linear system the active mode is selected by a random process. Here, the controller uses a piecewise quadratic Lyapunov function to select the system mode to activate. The paper relates existence of a stabilizing control law to the solution of a pair of coupled Lyapunov equations. Solution of these equations is discussed and related to the location of the eigenvalues of certain matrix operators derived from the component system matrices and the coupling parameter. Relationships to earlier work by the authors (1994) is discussed.
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paper_title: Stability and stabilization of piecewise affine and hybrid systems: an LMI approach

paper_content:

In this paper we present various algorithms both for stability analysis and state-feedback design for discrete-time piecewise affine systems. Our approach hinges on the use of piecewise quadratic Lyapunov functions that can be computed as the solution of a set of linear matrix inequalities. We show that the continuity of the Lyapunov function is not required in the discrete-time case. Moreover, the basic algorithms are made less conservative by exploiting the switching structure of piecewise affine systems and by using relaxation procedures.
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paper_title: Stabilization of second-order LTI switched systems

paper_content:

The problem of asymptotically stabilizing switched systems consisting of second-order LTI subsystems is studied and solved. In a previous paper (1999), the stabilization problem of switched systems consisting of subsystems with unstable foci was addressed. This paper extends the method therein to the stabilization of switched systems consisting of general second-order LTI subsystems. Necessary and sufficient conditions for stabilizability for such systems are obtained. Stabilizing switching control laws are also derived if the system is asymptotically stabilizable. Examples throughout the paper illustrate the approach and results.
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paper_title: Stabilizability and insensitivity of switched linear systems

paper_content:

This note identifies two types of stabilizability for switched systems: the pointwise stabilizability and the consistent stabilizability. Necessary conditions and sufficient conditions for both types are presented for switched linear systems. Insensitivity issues are briefly addressed as well.
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paper_title: Criteria of asymptotic stability of differential and difference inclusions encountered in control theory

paper_content:

Abstract For a class of differential inclusions, to which many of the practically important control systems can be reduced, necessary and sufficient conditions for asymptotic stability of the zero solution are established by the method of Lyapunov functions. Similar results are obtained for the corresponding class of difference inclusions.
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paper_title: A necessary and sufficient condition for robust asymptotic stabilizability of continuous-time uncertain switched linear systems

paper_content:

The main contribution of this paper is a necessary and sufficient condition derived for the existence of asymptotically stabilizing switching laws for a class of switched linear systems with time-variant parametric uncertainties. This result improves upon the sufficient only conditions found in the literature. The method is based on polyhedral Lyapunov-like functions, which represent generalizations of polytopic Lyapunov functions in the classical sense.
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Title: Stability and Stabilizability of Switched Linear Systems: A Short Survey of Recent Results

Section 1: INTRODUCTION
Description 1: Provide an overview of switched systems, their applications, and motivations for studying their stability and stabilizability, including a brief history and scope of the paper.

Section 2: Stability Under Arbitrary Switchings
Description 2: Discuss the conditions under which switched systems remain stable without any restrictions on the switching signals, including concepts like common quadratic Lyapunov functions and necessary and sufficient conditions for stability.

Section 3: Stability Under Restricted Switchings
Description 3: Explore the stability of switched systems when the switching signals are subject to certain constraints, such as dwell time and average dwell time switching, and discuss the use of multiple Lyapunov functions for stability analysis.

Section 4: SWITCHING STABILIZATION
Description 4: Delve into the synthesis of stabilizing switching laws for given dynamical systems, examining methods like quadratic stabilization and multiple Lyapunov functions, and providing necessary and sufficient conditions for switching stabilizability.

Section 5: CONCLUDING REMARKS
Description 5: Summarize the recent developments in the study of stability and stabilizability of switched linear systems and highlight some of the new results reported in the paper.