geometry of the limit sets of linear switched systems this paper is concerned with asymptotic stability properties of linear switched systems under the hypothesis that all the subsystems share a nonstrict quadratic lyapunov function we provide a large class of switching signals for which a large class of switched systems are asymptotically stable for this purpose we define what we call nonchaotic inputs which generalize the different notions of inputs with dwell time next we turn our attention to the behavior for possibly chaotic inputs finally we give a sufficient condition for a system composed of a pair of hurwitz matrices to be asymptotically stable for all inputs

uniform stability of switched linear systems extensions of lasalle s invariance principle 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

adaptive high gain λ tracking with variable sampling rate it is well known that proportional output feedback control can stabilize any relative degree one minimum phase system if the sign of the feedback is correct and the proportional gain is high enough moreover there exists simple adaptation laws for tuning the proportional gain the so called high gain adaptive controllers which are not based on system identification or plant parameter estimation algorithms or injection of probing signals if tracking of signals is desired then these simple controllers are also applicable without invoking an internal model if the tracking error is not necessarily supposed to converge to zero but towards a ball around zero of arbitrarily small but prespecified radius 0 in this note we consider a sampled version of the high gain adaptive tracking controller the motivation for sampling arises from the possibility that the output of a system may not be available continuously but only at discrete time instants the problem is that the stiffness of the system increases as the proportional gain is increased our result shows that adaptive sampling tracking is possible if the product hk of the decreasing sampling rate h and the increasing proportional gain k decreases at a rate proportional to 1 logk

an extension of lasalle s invariance principle for switched systems in this paper we address invariance principles for a certain class of switched nonlinear systems we provide an extension of lasalle s invariance principle for these systems and state asymptotic stability criteria we also present some related results on the compactness of the trajectories of these switched systems

adaptive sampled data tracking for input constrained exothermic chemical reaction models we consider digital input constrained adaptive and non adaptive output feedback control for a class of nonlinear systems which arise as models for controlled exothermic chemical reactors our objective is set point control of the temperature of the reaction with pre specifiedasymptotic tracking accuracy set by the designer our approach is basedon tracking controllers but in a context of piecewise constant sampled data output feedbacks and possibly adapted sampling periods the approach does not require any knowledge of the systems parameters does not invoke an internal model is simple in its design copes with noise corruptedoutput measurements andrequires only a feasibility assumption in terms of the reference temperature andthe input constraints

on the convergence of linear switched systems this paper investigates sufficient conditions for the convergence to zero of the trajectories of linear switched systems we provide a collection of results that use weak dwell time dwell time strong dwell time permanent and persistent activation hypothesis the obtained results are shown to be tight by counterexample finally we apply our result to the three cell converter

a refinement of matrosov s theorem for differential inclusions this note presents a refinement of matrosov s theorem for a class of differential inclusions whose set valued map is defined as a closed convex hull of finitely many vector fields this class of systems may arise in the analysis of switched nonlinear systems when stability with arbitrary switching between the given vector fields is considered assuming uniform global stability of a compact set it is shown that uniform global attractivity of the set can be verified by tailoring matrosov functions to individual vector fields this refinement of matrosov s theorem is an extension of the existing matrosov results which may be easier to apply to certain differential inclusions than existing results as demonstrated by an example

a converse lyapunov theorem for nonlinear switched systems 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

stability of a class of linear switching systems with applications to two consensus problems in this paper we first establish a stability result for a class of linear switched systems involving kronecker product the problem is interesting in that the system matrix does not have to be hurwitz at any time instant this class of linear switched systems arises in the control of multi agent systems under switching network topology as applications of this stability result we give the solvability conditions for both the leaderless consensus problem and the leader following consensus problem for general marginally stable linear multi agent systems under switching network topology in contrast with some existing results our results only assume that the dynamic graph is uniformly connected

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

invariance principles for switched systems with restrictions in this paper we consider switched nonlinear systems with otherwise arbitrary compact index sets and subjected to different switching constraints we present invariance principles for these systems and derive by using observability like notions some convergence and asymptotic stability criteria these results enable us to analyze the stability of solutions of switched systems with restrictions originating from the timing of the switchings state dependent constrained switching and switching whose logic has memory i e the active subsystem can only switch to a prescribed subset of subsystems

invariance principles for hybrid systems with connections to detectability and asymptotic stability this paper shows several versions of the lasalle s invariance principle for general hybrid systems the broad framework allows for nonuniqueness of solutions zeno behaviors and does not insist on continuous dependence of solutions on initial conditions instead only a mild structural property involving graphical convergence of solutions is posed the general invariance results are then specified to hybrid systems given by set valued data further results involving invariance as well as observability detectability and asymptotic stability are given

an extension of lasalle s invariance principle and its application to multi agent consensus in the paper an extension of lasalle s invariance principle to a class of switched linear systems is studied one of the motivations is the consensus problem in multi agent systems unlike most existing results in which each switching mode in the system needs to be asymptotically stable this paper allows that the switching modes are only lyapunov stable under certain ergodicity assumptions an extension of lasalle s invariance principle for global asymptotic stability is obtained then it is used to solve the consensus reaching problem of certain multi agent systems in which each agent is modeled by a double integrator and the associated interaction graph is switching and is assumed to be only jointly connected

brief paper on the algebraic characterization of invariant sets of switched linear systems in this paper a suitable lasalle principle for continuous time linear switched systems is used to characterize invariant sets and their associated switching laws an algorithm to determine algebraically these invariants is proposed the main novelty of our approach is that we require no dwell time conditions on the switching laws by not focusing on restricted control classes we are able to describe the asymptotic properties of the considered switched systems observability analysis of a flying capacitor converter is proposed as an illustration

global stability results for switched systems based on weak lyapunov functions in this paper we study the stability of nonlinear and time varying switched systems under restricted switching we approach the problem by decomposing the system dynamics into a nominal like part and a perturbation like one most stability results for perturbed systems are based on the use of strong lyapunov functions i e functions of time and state whose total time derivative along the nominal system trajectories is bounded by a negative definite function of the state however switched systems under restricted switching may not admit strong lyapunov functions even when asymptotic stability is uniform over the set of switching signals considered the main contribution of the current paper consists in providing stability results that are based on the stability of the nominal like part of the system and require only a weak lyapunov function these results may have wider applicability than results based on strong lyapunov functions the results provided follow two lines first we give very general global uniform asymptotic stability results under reasonable boundedness conditions on the functions that define the dynamics of the nominal like and the perturbation like parts of the system second we provide input to state stability iss results for the case when the nominal like part is switched linear time varying we provide two types of iss results standard iss that involves the essential supremum norm of the input and a modified iss that involves a power type norm

brief paper input to state stability of switched systems and switching adaptive control in this paper we prove that a switched nonlinear system has several useful input to state stable iss type properties under average dwell time switching signals if each constituent dynamical system is iss this extends available results for switched linear systems we apply our result to stabilization of uncertain nonlinear systems via switching supervisory control and show that the plant states can be kept bounded in the presence of bounded disturbances when the candidate controllers provide iss properties with respect to the estimation errors detailed illustrative examples are included

an extension of lasalle s invariance principle for a class of switched linear systems in this paper lasalle s invariance principle for switched linear systems is studied unlike most existing results in which each switching mode in the system needs to be asymptotically stable in this paper the switching modes are allowed to be only lyapunov stable under certain ergodicity assumptions an extension of lasalle s invariance principle for global asymptotic stability of switched linear systems is proposed provided that the kernels of derivatives of a common quadratic lyapunov function with respect to the switching modes are disjoint except the origin

analyzing the stability of switched systems using common zeroing output systems this paper introduces the notion of common zeroing output systems czos to analyze the stability of switched systems the concept of czos allows one to verify weak zero state detectability it characterizes a common behavior of any individual subsystem when the output signal for each subsystem is approaching zero heuristically speaking it removes the effect of switching behavior and thus enables one to analyze stability properties in systems with complex switching signals with the help of czos the krasovskii lasalle theorem can be extended to switched nonlinear time varying systems with both arbitrary switching and more general restricted switching cases for switched nonlinear time invariant systems the needed detectability condition is further simplified leading to several new stability results particularly when a switched linear time invariant system is considered it is possible to generate a recursive method which combines a krasovskii lasalle result and a nested matrosov result to find a czos if it exists the power of the proposed czos is demonstrated by consensus problems in literature to obtain a stronger convergence result with weaker conditions

input constrained adaptive tracking with applications to exothermic chemical reaction models we consider input constrained adaptive output feedback control for a class of nonlinear systems which are prototype models for controlled exothermic chemical reactions our objective is set point control of the output i e the temperature of the reaction in the context of chemical reactions practical considerations lead us to work in the presence of input constraints we adopt an approach based on modified lambda tracking controllers whereby prespecified asymptotic tracking accuracy quantified by lambda 0 set by the designer is ensured the adaptive control strategy does not require any knowledge of the system s parameters and does not invoke an internal model only a feasibility assumption in terms of the reference temperature and the input constraints is assumed

global optimal feedback control for general nonlinear systems with nonquadratic performance criteria abstract optimal control of general nonlinear nonaffine controlled systems with nonquadratic performance criteria that permit state and control dependent time varying weighting parameters is solved classically using a sequence of linear quadratic and time varying problems the proposed method introduces an approximating sequence of riccati equations asre to explicitly construct nonlinear time varying optimal state feedback controllers for such nonlinear systems under very mild conditions of local lipschitz continuity the sequences converge globally to nonlinear optimal stabilizing feedback controls the computational simplicity and effectiveness of the asre algorithm is an appealing alternative to the tedious and laborious task of solving the hamilton jacobi bellman partial differential equation so the optimality of the asre control is studied by considering the original nonlinear nonquadratic optimization problem and the corresponding necessary conditions for optimality derived from pontryagin s maximum principle global optimal stabilizing state feedback control laws are then constructed this is compared with the optimality of the asre control by considering a nonlinear fighter aircraft control system which is nonaffine in the control numerical simulations are used to illustrate the application of the asre methodology which demonstrate its superior performance and optimality

a simple selftuning controller for stably invertible systems in this paper we construct a new trajectory following adaptive control algorithm for linear time invariant finite dimensional systems that possess a stable inverse the exponential stability of the control scheme is demonstrated

a switching algorithm for global exponential stabilization of uncertain chained systems this note deals with chained form systems with strongly nonlinear unmodeled dynamics and external disturbances the objective is to design a robust nonlinear state feedback law such that the closed loop system is globally kexponentially stable we propose a novel switching control strategy involving the use of input state scaling and integrator backstepping the new features of our controllers include the ability to achieve lyapunov stability exponential convergence and robustness to a set of uncertain drift terms

uniform asymptotic stability of switched nonlinear time varying systems and detectability of reduced limiting control systems this paper is concerned with the study of both local and global uniform asymptotic stability for switched nonlinear time varying nltv systems through the detectability of output maps with this aim the notion of reduced limiting control systems for switched nltv systems whose switchings verify time state dependent constraints and the concept of weak zero state detectability for those reduced limiting systems are introduced necessary and sufficient conditions for the global uniform asymptotic stability of families of trajectories of the switched system are obtained in terms of this detectability property these sufficient conditions in conjunction with the existence of multiple weak lyapunov functions yield a criterion for the global uniform asymptotic stability of families of trajectories of the switched system this criterion can be seen as an extension of the classical krasovskii lasalle theorem an interesting feature of the results is that no dwell time assumptions are made moreover they can be used for establishing the global uniform asymptotic stability of the switched nltv system under arbitrary switchings the effectiveness of the proposed results is illustrated by means of various interesting examples including the stability analysis of a semiquasi z source inverter

invariance principles for switching systems via hybrid systems techniques invariance principles and sufficient conditions for asymptotic stability for switching systems are given multiple lyapunov like functions are used and dwell time persistent dwell time and weak dwell time switching signals are considered the invariance principles are derived from general invariance principles for hybrid systems asymptotic stability is concluded under observability assumptions or common bounds on the lyapunov like functions

stability of switched systems a lie algebraic condition we present a sucient condition for asymptotic stability of a switched linear system in terms of the lie algebra generated by the individual matrices namely if this lie algebra is solvable then the switched system is exponentially stable for arbitrary switching in fact we show that any family of linear systems satisfying this condition possesses a quadratic common lyapunov function we also discuss the implications of this result for switched nonlinear systems c 1999 elsevier science b v all rights reserved

input constrained funnel control with applications to chemical reactor models error feedback control in the presence of input constraints is considered for a class of exothermic chemical reactor models the primary control objective is regulation of a setpoint temperature t with prescribed accuracy given 0 arbitrarily small ensure that for every admissible system and reference setpoint the regulation error e t t is ultimately smaller than that is e t i for all t su4ciently large the second objective is guaranteed transient performance the evolution of the regulation error should be contained in a prescribed performance funnel f around the setpoint temperature t a simple error feedback control with input constraints of the form u t sat u 6 u k t t t t u u an o9set is introduced which achieves the objective in the presence of disturbances corrupting the measurement the gain k t is a function of the error e t t t t and its distance to the funnel boundary the input constraints u 6 u have to satisfy certain feasibility assumptions in terms of the model data and the operating point t

a result on common quadratic lyapunov functions in this note we define strong and weak common quadratic lyapunov functions cqlfs for sets of linear time invariant lti systems we show that the simultaneous existence of a weak cqlf of a special form and the nonexistence of a strong cqlf for a pair of lti systems is characterized by easily verifiable algebraic conditions these conditions are found to play an important role in proving the existence of strong cqlfs for general lti systems

on common quadratic lyapunov functions for pairs of stable lti systems whose system matrices are in companion form in this note the problem of determining necessary and sufficient conditions for the existence of a common quadratic lyapunov function for a pair of stable linear time invariant systems whose system matrices a sub 1 and a sub 2 are in companion form is considered it is shown that a necessary and sufficient condition for the existence of such a function is that the matrix product a sub 1 a sub 2 does not have an eigenvalue that is real and negative examples are presented to illustrate the result

supervisory control of uncertain linear time varying systems we consider the problem of adaptively stabilizing linear plants with unknown time varying parameters in the presence of noise disturbances and unmodeled dynamics using the supervisory control framework which employs multiple candidate controllers and an estimator based switching logic to select the active controller at every instant of time time varying uncertain linear plants can be stabilized by supervisory control provided that the plant s parameter varies slowly enough in terms of mixed dwell time switching and average dwell time switching the noise and disturbances are bounded and small enough in terms of l infinity norms and the unmodeled dynamics are small enough in the input to state stability sense this work extends previously reported works on supervisory control of linear time invariant systems with constant unknown parameters to the case of linear time varying uncertain systems a numerical example is included and limitations of the approach are discussed

on weak invariance principles for nonlinear switched systems in this technical note we develop two weak invariance principles for nonlinear switched systems we first present a union weak invariance principle for switched systems which includes as a special case the integral invariance principle it is shown that the switched solution approaches the largest weakly invariant set of the combined zero loci of the output functions then we extend the union weak invariance principle to an intersection weak invariance principle which greatly reduces the convergence region unlike the existing results in which the constructions of lyapunov functions are inevitable our principles do not require the existence of lyapunov functions numerical examples are presented to demonstrate the feasibility of our principles as well as applications to multi agent consensus problems

on the equivalence relations of detectability and pe conditions this note investigates equivalent relations of detectability and persistency excitation pe for general nonlinear time varying systems three new definitions relating to the detectability are proposed and discussed as a preliminary result we show that these detectability conditions are all equivalent under a mild assumption furthermore two pe conditions that are a nonlinear version extension of the pe proposed in present literature are defined very interestingly it can be shown that these nonlinear version pe conditions are both equivalent to the proposed detectability conditions under the same assumption to the best of our knowledge this is a first result about the connection between detectability and pe in the area of nonlinear time varying systems moreover by combining a newly developed stability criterion with the proposed results an interesting example is given to illustrate how the classic pe conditions for linear systems can be extended and used in nonlinear systems from this example it can also be seen that the uniformly asymptotic stability can be guaranteed either using the detectability conditions or by the pe conditions based on our approach

uniform asymptotic stability of nonlinear switched systems with an application to mobile robots this paper is concerned with the study of both local and global uniform asymptotic stability for general nonlinear and time varying switched systems two concepts of lyapunov functions are introduced and used to establish uniform lyapunov stability and uniform global stability with the help of output functions an almost bounded output energy condition and an output persistent excitation condition are then proposed and employed to guarantee uniform local and global asymptotic stability based on this result a generalized version of krasovskii lasalle theorem in time varying switched systems is proposed for switched systems with persistent dwell time the output persistent excitation condition is guaranteed to hold under a zero state observability condition it is shown that several existing results in past literature can be covered as special cases using the proposed criteria interestingly as opposed to previous work the main results of this paper are applicable to the situation where some switching systems are not asymptotically stable at the origin the robust practical regulation problem of nonholonomic mobile robots is studied as a way of demonstrating the power of the proposed new criteria a novel switching controller is proposed with guaranteed robustness to orientation error and unknown parameters in mobile robots

a generalization of krasovskii lasalle theorem for nonlinear time varying systems converse results and applications this paper presents a practically applicable characterization of uniform global asymptotic stability uas and ugas for general nonlinear time varying systems under certain output dependent conditions in the spirit of the krasovskii lasalle theorem the celebrated krasovskii lasalle theorem is extended from two directions one is using the weak zero state detectability property associated with reduced limiting systems of the system in question to generalize the condition that the maximal invariance set contained in the zero locus of the time derivative of the lyapunov function is the zero set another one is using an almost bounded output energy condition to relax the assumption that the time derivative of the lyapunov function is negative semi definite then the uas and ugas properties of the origin can be guaranteed by employing these two improved conditions related to certain output function for uniformly lyapunov stable systems the proposed conditions turn out to be also necessary under some mild assumptions and thus give a new characterization of ugas and uas through an equivalence relation the proposed detectability condition can also be verified in terms of usual pe condition to validate the proposed results the obtained stability criteria are applied to a class of time varying passive systems and to revisit a tracking control problem of nonholonomic chained systems for the latter under certain persistency of excitation conditions the k exponential stability is achieved based on our approach

a converse lyapunov theorem for a class of dynamical systems which undergo switching 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

stability and persistent excitation in signal sets persistent excitation pe conditions have been widely used to analyze stability properties of various parameter identification algorithms and to establish uniform global asymptotic stability ugas for a large class of nonlinear time varying systems in order to generalize such conditions to a more general setting a new pe condition is proposed with three basic ingredients a signal set to represent a family of time functions e g trajectories a pseudo distance measure to describe the convergence and some binary relations e g state to output relations closely related to detectability this pe condition is a necessary condition to guarantee ugas under uniform global stability and an integral inequality it becomes a sufficient condition of ugas a novel concept m pair which aims at simplifying the checking of the pe condition is introduced by using m pair it is possible to simplify the structure of the referred signal set in the spirit of the classic krasovskii lasalle theorem and to extend the dimension of the reference signal set similar to the matrosov theorem thus the framework of m pair not only unifies these well known results but also generates more flexibility in checking the pe conditions when applied to nonlinear switched systems three new tools to verify the pe condition are obtained finally an example illustrates that a nonlinear time varying switched system with arbitrary switching can be shown to be ugas without using a common lyapunov function

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

an invariance principle for nonlinear switched systems abstract in this paper we address the problem of extending lasalle invariance principle to switched system we prove an extension of the invariance principle relative to dwell time switched solutions and a second one relative to constrained switched systems

a new asymptotic stability criterion for nonlinear time variant differential equations a new sufficient condition for asymptotic stability of ordinary differential equations is proposed unlike classical lyapunov theory the time derivative along solutions of the lyapunov function may have positive and negative values the classical lyapunov approach may be regarded as an infinitesimal version of the present theorem verification in practical problems is harder than in the classical case an example is included in order to indicate how the present theorem may be applied