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

adaptive neural control for a class of switched nonlinear systems in this paper adaptive neural control is presented for a class of switched nonlinear systems with switching jumps and uncertainties in both system models and switching signals condition on limiting variation of the lyapunov function is given for input to state asymptotic stability of switched systems with switching jumps the control objective is achieved uniformly with respect to a class of switching signals the coupled difficulties from the discrepancy between control gains and switching jumps are overcome by discontinuous adaptive neural control combined with the classical adaptive control smooth approximations of the discontinuous controls are then presented for a systematic design procedure

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

uniform stabilization of nonlinear systems with arbitrary switchings and dynamic uncertainties we solve the problem of global uniform input to state stabilization of nonlinear switched systems with time varying and periodic dynamics with dynamic uncertainties and with external disturbances the switching signal is assumed to be unknown and the dynamics of the known components of the state vector is equivalent to the general triangular form gtf with non invertible input output maps in our first and most general result we prove that if the dynamic uncertainty is treated as external disturbance then the general triangular form system can be stabilized with arbitrarily small gain w r t the dynamic uncertainty by means of a switching independent smooth and periodic feedback hence using a suitable extension of the well known small gain theorem to our case of switched systems with arbitrary switchings we obtain the uniform input to state stabilization of the entire interconnected system the second part of the paper addresses a more special case of triangular form tf switched systems with right invertible input output i o maps with unknown switchings and with dynamic uncertainties we show that the design becomes simpler and more constructive and the controllers become time invariant if the dynamics is autonomous in this special case finally we consider an example with explicit design of the stabilizing controllers

input output to state stability this work explores lyapunov characterizations of the input output to state stability ioss property for nonlinear systems the notion of ioss is a natural generalization of the standard zero detectability property used in the linear case the main contribution of this work is to establish a complete equivalence between the ioss property and the existence of a certain type of smooth lyapunov function as corollaries one shows the existence of norm estimators and obtains characterizations of nonlinear detectability in terms of relative stability and of finite energy estimates

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

relaxed conditions for the stability of switched nonlinear triangular systems under arbitrary switching this paper addresses the stability of a class of nonlinear switching systems under arbitrary switching it focuses on switching systems whose modes are made of a cascade of nonlinear scalar systems it relaxes the stability conditions proposed by angeli and liberzon by relying on the strong iiss property which is known to be preserved under cascade interconnection its applicability is illustrated by the study of switched bilinear systems in triangular form

barrier lyapunov functions for the output tracking control of constrained nonlinear switched systems abstract in this paper we investigate the output tracking control problem of constrained nonlinear switched systems in lower triangular form first when all the states are subjected to constraints we employ a barrier lyapunov function blf which grows to infinity whenever its arguments approach some finite limits to prevent the states from violating the constraints based on the simultaneous domination assumption we design a continuous feedback controller for the switched system which guarantees that asymptotic output tracking is achieved without transgression of the constraints and all closed loop signals remain bounded provided that the initial states are feasible then we further consider the case of asymmetric time varying output constraints by constructing an appropriate blf finally the effectiveness of the proposed results is demonstrated with a numerical example

a characterization of integral input to state stability the notion of input to state stability iss is now recognized as a central concept in nonlinear systems analysis it provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite l sup 2 gains it plays a central role in recursive design coprime factorizations controllers for nonminimum phase systems and many other areas in this paper a newer notion that of integral input to state stability iiss is studied the notion of iiss generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states in that sense generalizing the linear h sup 2 theory it allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance we obtain several necessary and sufficient characterizations of the iiss property expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations

uniform stability properties of switched systems with switchings governed by digraphs characterizations of various uniform stability properties of switched systems described by differential inclusions and whose switchings are governed by a digraph are developed these characterizations are given in terms of stability properties of the system with restricted switchings and also in terms of lyapunov functions

quantile regression under random censoring censored regression models have received a great deal of attention in both the theoretical and applied econometric literature most of the existing estimation procedures for either cross sectional or panel data models are designed only for models with fixed censoring in this paper a new procedure for adapting these estimators designed for fixed censoring to models with random censoring is proposed this procedure is then applied to the clad and quantile estimators of powell j econom 25 1984 303 32 1986a 143 to obtain an estimator of the coefficients under a mild conditional quantile restriction on the error term that is applicable to samples exhibiting fixed or random censoring the resulting estimator is shown to have desirable asymptotic properties and performs well in a small scale simulation study

a characterization of integral iss for switched and time varying systems most of the existing characterizations of the integral input to state stability iiss property are not valid for time varying or switched systems in cases where converse lyapunov theorems for stability are not available this paper provides a characterization that is valid for switched and time varying systems and shows that natural extensions of some of the existing characterizations result in only sufficient but not necessary conditions the results provided also pinpoint suitable iiss gains and relate these to supply functions and bounds on the function defining the system dynamics

characterizations of input to state stability for hybrid systems abstract we study input to state stability iss for a broad class of hybrid systems which are combinations of a differential equation on a constraint set and a difference equation on another constraint set for this class of hybrid systems we establish the equivalence of iss nonuniform iss and the existence of a smooth iss lyapunov function by additionally assuming that the right hand side of the differential equation has a convex property with respect to inputs moreover we demonstrate by examples that the equivalence may fail when such a convexity assumption is removed

input to state stability of impulsive and switching hybrid systems with time delay this paper investigates input to state stability iss and integral input to state stability iiss of impulsive and switching hybrid systems with time delay using the method of multiple lyapunov krasovskii functionals it is shown that even if all the subsystems governing the continuous dynamics in the absence of impulses are not iss iiss impulses can successfully stabilize the system in the iss iiss sense provided that there are no overly long intervals between impulses i e the impulsive and switching signal satisfies a dwell time upper bound condition moreover these impulsive iss iiss stabilization results can be applied to systems with arbitrarily large time delays conversely in the case when all the subsystems governing the continuous dynamics are iss iiss in the absence of impulses the iss iiss properties can be retained if the impulses and switching do not occur too frequently i e the impulsive and switching signal satisfies a dwell time lower bound condition several illustrative examples are presented with their numerical simulations to demonstrate the main results

stability and robustness analysis for switched systems with time varying delays a new technique is presented for the stability and robustness analysis of nonlinear switched time varying systems with uncertainties and time varying delays the delays are allowed to be discontinuous but are required to be piecewise continuous and arbitrarily long with known upper bounds the technique uses an adaptation of halanay s inequality and a trajectory based technique and is used for designing switched controllers to stabilize linear time varying systems with time varying delays

uniform input to state stability for switched and time varying impulsive systems we provide a lyapunov function based method for establishing different types of uniform input to state stability iss for time varying impulsive systems the method generalizes to impulsive systems with inputs the well established philosophy of assessing the stability of a system by reducing the problem to that of the stability of a scalar system given by the evolution of the lyapunov function on the system trajectories this reduction is performed in such a way that the resulting scalar system has no inputs novel sufficient conditions for iss are provided which generalize existing results for time invariant and time varying switched and nonswitched impulsive and nonimpulsive systems in several directions

on the representation of switched systems with inputs by perturbed control systems this paper provides representations of switched systems described by controlled differential inclusions in terms of perturbed control systems the control systems have dynamics given by differential equations and their inputs consist of the original controls together with disturbances that evolve in compact sets their sets of maximal trajectories contain as a dense subset the set of maximal trajectories of the original system several applications to control theory dealing with properties of stability with respect to inputs and of detectability are derived as a consequence of the representation theorem 2004 elsevier ltd all rights reserved

a converse lyapunov theorem for nonuniform in time global asymptotic stability and its application to feedback stabilization lyapunov like characterizations for the concepts of nonuniform in time robust global asymptotic stability and input to state stability for time varying systems are established the main result of our work enables us to derive 1 necessary and sufficient conditions for feedback stabilization for affine in the control systems and 2 sufficient conditions for the propagation of the input to state stability property through integrators

effect of delayed impulses on input to state stability of nonlinear systems abstract this paper studies the input to state stability iss and integral input to state stability iiss of nonlinear systems with delayed impulses by using lyapunov method and the analysis technique proposed by hespanha et al 2005 some sufficient conditions ensuring iss iiss of the addressed systems are obtained those conditions establish the relationship between impulsive frequency and the time delay existing in impulses and reveal the effect of delayed impulses on iss iiss an example is provided to illustrate the efficiency of the obtained results

class kl estimates and input to state stability analysis of impulsive switched systems abstract in this paper we investigate input to state stability of impulsive switched systems the goal is to bridge two apparently different but both useful stability notions input to state stability and stability in terms of two measures in the hybrid systems setting based on two class kl function estimates and a comparison theorem for impulsive differential equations two sets of sufficient lyapunov type conditions for input to state stability in terms of two measures are obtained for impulsive switched systems these conditions exploit some nonlinear integral constraints in terms of generalized dwell time conditions to balance the continuous dynamics and impulsive dynamics so that input to state stability is achieved despite possible instability of individual continuous subsystems or destabilizing impulsive effects an illustrative example is presented together with numerical simulations to demonstrate the main results

on converse lyapunov theorems for iss and iiss switched nonlinear systems abstract in this paper we present converse lyapunov theorems for iss and integral input to state stable iiss switched nonlinear systems their proofs are based on existing converse lyapunov theorems for input output to state stable and iiss nonlinear systems and on the association of the switched system with a nonlinear system with inputs and disturbances that take values in a compact set

stochastic input to state stability of switched stochastic nonlinear systems in this paper global asymptotic stability in probability gasip and stochastic input to state stability siss for nonswitched stochastic nonlinear nssnl systems and switched stochastic nonlinear ssnl systems are investigated for the study of gasip the definition which we considered is not the usual notion of asymptotic stability in probability stability in probability plus attractivity in probability it can depict the properties of the system quantitatively correspondingly based on this definition some sufficient conditions are provided for nssnl systems and ssnl systems furthermore the definition of siss is introduced and corresponding criteria are provided for nssnl systems and ssnl systems in the proof of the above results to overcome the difficulties coming with the appearance of switching and the stochastic property at the same time we generalize the past comparison principle and fully use the properties of the functions which we constructed in terms of the average dwell time of the switching laws a sufficient siss condition is obtained for ssnl systems finally some examples are provided to demonstrate the applicability of our results

strong iss implies strong iiss for time varying impulsive systems abstract for time invariant nonimpulsive systems it is already well known that the input to state stability iss property is strictly stronger than integral input to state stability iiss very recently we have shown that under suitable uniform boundedness and continuity assumptions on the function defining system dynamics iss implies iiss also for time varying systems in this paper we show that this implication remains true for impulsive systems provided that asymptotic stability is understood in a sense stronger than usual for impulsive systems

input output to state stability tools for hybrid systems and their interconnections we present results for the analysis of input output properties of a general class of hybrid systems given by a flow set a flow map a jump set a jump map and an output map for this class of systems the notion of input output to state stability is introduced in the first part of the technical note under mild assumptions on the functions and sets defining a hybrid system sufficient conditions for this notion in terms of lyapunov functions are derived equivalences between lyapunov functions for input output to state stability for asymptotic and exponential decay rates are established the sufficient conditions and equivalences are linked to the existence of norm observers for hybrid systems these results are used in the second part of the technical note to study interconnections of hybrid systems an interconnection result in terms of a lyapunov based small gain theorem is also presented examples illustrate the results

lyapunov conditions for input to state stability of impulsive systems this paper introduces appropriate concepts of input to state stability iss and integral iss for impulsive systems i e dynamical systems that evolve according to ordinary differential equations most of the time but occasionally exhibit discontinuities or impulses we provide a set of lyapunov based sufficient conditions for establishing these iss properties when the continuous dynamics are iss but the discrete dynamics that govern the impulses are not the impulses should not occur too frequently which is formalized in terms of an average dwell time adt condition conversely when the impulse dynamics are iss but the continuous dynamics are not there must not be overly long intervals between impulses which is formalized in terms of a novel reverse adt condition we also investigate the cases where i both the continuous and discrete dynamics are iss and ii one of these is iss and the other only marginally stable for the zero input while sharing a common lyapunov function in the former case we obtain a stronger notion of iss for which a necessary and sufficient lyapunov characterization is available the use of the tools developed herein is illustrated through examples from a micro electro mechanical system mems oscillator and a problem of remote estimation over a communication network

input output stability of systems with switched dynamics and outputs this paper provides the extension of results on input to output stability properties of switched systems to switched systems whose dynamics are described by forced differential inclusions and whose outputs are obtained via switched set valued maps lyapunov characterizations of these input output stability properties obtained in terms of certain conceptual output functions are also presented