continuous selections of trajectories of hybrid systems we present results on existence of continuous selections of trajectories of hybrid systems evolving according to lipschitz nonlinear inclusions first we utilize the skorohod metric to define both a family of pseudo metrics and a metric on the set of trajectories accepted by the hybrid automaton we show that under a non zeno condition the latter metric is complete second the existence of continuous selections with respect to the family of pseudo metrics is proved under a relatively mild assumption of transversality

robust stability of hybrid limit cycles with multiple jumps in hybrid dynamical systems for a broad class of hybrid dynamical systems we establish results for robust asymptotic stability of hybrid limit cycles with multiple jumps per period hybrid systems are given in terms of differential and difference equations with set constraints hybrid limit cycles are given by compact sets defined by periodic solutions that flow and jump under mild assumptions we show that asymptotic stability of such hybrid limit cycles is not only equivalent to asymptotic stability of a fixed point of the associated poincare map but also robust to perturbations specifically robustness to generic perturbations which capture state noise and unmodeled dynamics and to inflations of the flow and jump sets are established in terms of mathcal k mathcal l bounds a two gene network with binary hysteresis is presented to illustrate the notions and results throughout the paper

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

robust trajectory tracking for a class of hybrid systems an internal model principle approach this paper deals with the asymptotic tracking of periodic trajectories for hybrid systems having linear dynamics in each operating mode and isolated discrete switching events switching systems parametric uncertainties are considered and the dimension of the state vector is allowed to vary among modes to deal with the hybrid nature of the system and the possible discontinuities of its solutions at switching times a properly amended tracking control problem is defined and a feedback control law based on a discontinuous version of the classical internal model principle is proposed the innovative design of the discrete time dynamics of the compensator guarantees the robust existence of a steady state response giving zero tracking error in the controlled output and local convergence to it

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

relaxed persistent flow jump conditions for uniform global asymptotic stability for hybrid systems sufficient conditions are derived for the uniform global asymptotic stability of a given closed set these conditions are written in terms of a lyapunov function candidate and assume a semiglobal practical persistent flow resp persistent jump property of the solutions to the hybrid system the use of the new conditions is illustrated via the stability analysis of a physically inspired example and of an event triggered control algorithm

robust stabilization of chained systems via hybrid control the problem of global robust exponential stabilization of nonholonomic chained systems is addressed and solved by means of a hybrid state feedback control law it is shown that the control law yields global exponential stability and global robustness against a class of small measurement errors actuator noises and exogenous disturbances

a common framework for complete and incomplete attitude synchronization in networks with switching topology in this paper we study attitude synchronization for elements in the unit sphere in mathbb r scriptscriptstyle scriptscriptstyle 3 and for elements in the three dimensional 3 d rotation group for a network with switching topology the agents angular velocities are assumed to be the control inputs and a switching control law for each agent is devised that guarantees synchronization provided that all elements are initially contained in a region which we identify later in the paper the control law is decentralized and it does not require a common orientation frame among all agents we refer to synchronization of unit vectors in mathbb r scriptscriptstyle 3 as incomplete synchronization and of 3 d rotation matrices as complete synchronization our main contribution lies in showing that these two problems can be analyzed under a common framework where all agents dynamics are transformed into unit vectors dynamics on a sphere of appropriate dimension

solutions to hybrid inclusions via set and graphical convergence with stability theory applications motivated by questions in stability theory for hybrid dynamical systems we establish some fundamental properties of the set of solutions to such systems using the notion of a hybrid time domain and general results on set and graphical convergence we establish under weak regularity and local boundedness assumptions that the set of solutions is sequentially compact and upper semicontinuous with respect to initial conditions and system perturbations the general facts are then used to establish several results for the behavior of hybrid systems that have asymptotically stable compact sets these results parallel what is already known for differential inclusions and difference inclusions for example the basin of attraction for a compact attractor is relatively open the attractivity is uniform from compact subsets of the basin of attraction and asymptotic stability is robust with respect to small perturbations

output to state stability for hybrid systems output to state stability oss is a dual notion of input to state stability for dynamical systems this paper presents lyapunov and asymptotic characterizations of oss for hybrid dynamical systems emphasizing that a globally detectable i e nonuniformly oss hybrid system admits a smooth oss lyapunov function

basic properties and characterizations of incremental stability prioritizing flow time for a class of hybrid systems abstract this paper introduces incremental stability notions for a class of hybrid dynamical systems given in terms of differential equations and difference equations with state constraints the specific class of hybrid systems considered are those that do not have consecutive jumps nor zeno behavior the notion of incremental asymptotic stability is used to describe the behavior of the distance between every pair of solutions to the system having stable behavior incremental stability and approaching zero asymptotically incremental attractivity a version of this notion that is uniform in hybrid time with respect to initial conditions is also introduced these notions prioritize flow time and are illustrated in examples basic properties of the class of systems are considered and those implied by the new notions are revealed an equivalence characterization of the uniform notion is provided in terms of a k l function moreover sufficient and necessary conditions under which asymptotic stability implies the new incremental notions are provided we consider the case when the original hybrid system has an asymptotically stable compact set and also the case when an auxiliary hybrid system which has twice the dimension of the original system has a diagonal like set asymptotically stable

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

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

dynamic pricing of seasonal goods with spot and forward purchase demands this paper deals with the problem of jointly determining the order size and dynamic prices for a perishable inventory system over a finite time planning horizon when firms simultaneously provide customers with a prompt delivery option and a delivery schedule option customers are segmented into two types namely spot purchase customers and the forward purchase customers demands for both types of customers are assumed to be time and price dependent the decision maker of the inventory system is assumed to apply pricing policies to stimulate demand to improve revenues under the condition that customers with forward purchases may cancel their orders a mathematical model is developed to find the optimal number of price settings the optimal dynamic prices and the order quantity a solution procedure is found to determine the optimal decisions

differential automata and their discrete simulators on presente la notion d automate differentiel pour la formulation d une classe de problemes aux valeurs initiales sur des champs vectoriels hysteretiques

combined effects of drivers disturbance risk preference heterogeneity and the nearest following vehicle headway on traffic flow instability analytical studies abstract driver s disturbance risk preference is defined as the driver s reaction degree to the disturbance which is a kind of the driver s attributes and can be expressed by the disturbance risk preference coefficient this paper proposes a new car following model and firstly attempts to analytically investigate the combined effects of drivers disturbance risk preference heterogeneity and the nearest following vehicle headway on traffic flow instability under some specific traffic scenarios solution of the model yields many insights including the following 1 when drivers disturbance risk preference increases from the following driver to the preceding driver traffic flow instability increases and the bigger the disturbance risk preference coefficient ratio of the preceding driver s coefficient to the following driver s coefficient the bigger the increasing effect and vice versa at this scenario the nearest following vehicle headway increases traffic flow instability 2 when drivers disturbance risk preference is homogeneous drivers disturbance risk preference and the nearest following vehicle headway have no effects on traffic flow instability 3 when drivers disturbance risk preference reduces from the following driver to the preceding driver for the general traffic situation traffic flow instability reduces and the smaller the disturbance risk preference coefficient ratio of the preceding driver s coefficient to the following driver s coefficient the bigger the reducing effect and vice versa at this scenario if the coefficient ratio reduces from the following to the preceding or is homogeneous the nearest following vehicle headway increases traffic flow instability otherwise the effects of the nearest following vehicle headway on traffic flow instability cannot be confirmed

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

a general stability criterion for time varying systems using a modified detectability condition uniform asymptotic stability for general nonlinear time varying systems is investigated from the output to state viewpoint a unified criterion is proposed using a modified detectability condition and certain integral inequalities the criterion extends the krasovskii lasalle theorem to nonperiodic time varying systems an example taken from the tracking control of mobile robots is given to illustrate that much as the invariance principle of lasalle is used in studying the stability of time invariant systems our results can be used to study the stability of nonperiodic time varying systems by checking the modified detectability condition

nonpathological lyapunov functions and discontinuous caratheodory systems differential equations with discontinuous right hand side and solutions intended in caratheodory sense are considered for these equations sufficient conditions which guarantee both lyapunov stability and asymptotic stability in terms of nonsmooth lyapunov functions are given an invariance principle is also proven

on the existence of control lyapunov functions and state feedback laws for hybrid systems for a class of hybrid systems given in terms of constrained differential and difference equations inclusions we study the existence of control lyapunov functions when compact sets are asymptotically stable as well as the stabilizability properties guaranteed when control lyapunov functions exist an existence result asserting that asymptotic stabilizability of a compact set implies the existence of a smooth control lyapunov function is established when control lyapunov functions are available conditions guaranteeing the existence of stabilizing continuous state feedback control laws are provided

dynamical properties of hybrid automata hybrid automata provide a language for modeling and analyzing digital and analogue computations in real time systems hybrid automata are studied here from a dynamical systems perspective necessary and sufficient conditions for existence and uniqueness of solutions are derived and a class of hybrid automata whose solutions depend continuously on the initial state is characterized the results on existence uniqueness and continuity serve as a starting point for stability analysis lyapunov s theorem on stability via linearization and lasalle s invariance principle are generalized to hybrid automata

a recurrence principle for stochastic difference inclusions the invariance principle is extended to a recurrence principle and is developed for stochastic difference inclusions for these systems random solutions are not unique under appropriate lyapunov like conditions it is established for every random solution that almost every complete sample path converges to the largest weakly totally recurrent set contained in a level set of the lyapunov like function such a set is not larger and is sometimes smaller than the largest weakly invariant set contained in the level set the principle is useful for establishing robust uniform asymptotic stability in probability or robust uniform strong recurrence under weak lyapunov conditions for stochastic discrete time control systems that employ discontinuous feedback laws examples demonstrate the achieved results

an integral invariance principle for differential inclusions with applications in adaptive control the byrnes martin integral invariance principle for ordinary differential equations is extended to differential inclusions on bbb r n the extended result is applied in demonstrating the existence of adaptive stabilizers and servomechanisms for a variety of nonlinear system classes

asymptotic stability in hybrid systems via nested matrosov functions a theorem on nested matrosov functions is extended to time varying hybrid systems it provides sufficient conditions for uniform global asymptotic stability of a compact set an application to parameter identification with state resets is made and illustrated on an example

on the stability and stabilization of quaternion equilibria of rigid bodies we study attitude control of rigid bodies on quaternion coordinates under three mathematically different perspectives depending on how the system dynamics are assumed to evolve in the first case we suppose that one equilibrium point is chosen a priori and a continuous controller is used under the assumption that the rigid body always spins in the same direction in the second case we relax the assumption that the sense of rotation is constant finally a third scenario is considered in which hybrid switching control is used to choose the direction in which to spin that is both equilibria are continuously considered with regard to less energy consumption it is showed that each of three scenarios must be treated in a different theoretical setting a comparative study in simulations is also provided

quaternion based hybrid feedback for robust global attitude synchronization we apply recent results on robust global asymptotic stabilization of the attitude of a single rigid body to the problem of globally synchronizing the attitude of a network of rigid bodies using a decentralized strategy the proposed hybrid feedback scheme relies on the communication of a binary logic variable between each pair of neighboring rigid bodies that determines the orientation of a torque component acting to reduce their relative error through a hysteretic switch of this logic variable the hybrid feedback achieves global synchronization under the assumption that the network is connected and acyclic the hysteresis eliminates chattering while preventing the unwinding phenomenon apparent in some quaternion based attitude control schemes the results are exercised in a numerical example

an invariance principle for nonlinear hybrid and impulsive dynamical systems in this paper we develop an invariance principle for dynamical systems possessing left continuous flows specifically we show that left continuity of the system trajectories in time for each fixed state point and continuity of the system trajectory in the state for every time in some dense subset of the semi infinite interval are sufficient for establishing an invariance principle for hybrid and impulsive dynamical systems as a special case of this result we state and prove new invariant set stability theorems for a class of nonlinear impulsive dynamical systems namely state dependent impulsive dynamical systems these results provide less conservative stability conditions for impulsive systems as compared to classical results in the literature and allow us to address the stability of limit cycles and periodic orbits of impulsive systems

towards a stability theory of general hybrid dynamical systems in recent work we proposed a general model for hybrid dynamical systems whose states are defined on arbitrary metric space and evolve along some notion of generalized abstract time for such systems we introduced the usual concepts of lyapunov and lagrange stability we showed that it is always possible to transform this class of hybrid dynamical systems into another class of dynamical systems with equivalent qualitative properties but defined on real time r 0 the motions of this class of systems are in general discontinuous this class of systems may be finite or infinite dimensional for the above discontinuous dynamical systems and hence for the above hybrid dynamical systems we established the principal lyapunov stability theorems as well as lagrange stability theorems for some of these we also established converse theorems we demonstrated the applicability of these results by means of specific classes of hybrid dynamical systems in the present paper we continue the work described above in doing so we first develop a general comparison theory for the class of hybrid dynamical systems resp discontinuous dynamical systems considered herein making use of stability preserving mappings we then show how these results can be applied to establish some of the principal lyaponov stability theorems for the latter we also state and prove a converse theorem not considered previously finally to demonstrate the applicability of our results we consider specific examples throughout the paper

lyapunov based hybrid loops for stability and performance of continuous time control systems we construct hybrid loops that augment continuous time control systems we consider a continuous time nonlinear plant in feedback with a possibly non stabilizing given nonlinear dynamic continuous time state feedback controller the arising hybrid closed loops are guaranteed to follow the underlying continuous time closed loop dynamics when flowing and to jump in suitable regions of the closed loop state space to guarantee that a positive definite function v of the closed loop state and or a positive definite function v p of the plant only state is non increasing along the hybrid trajectories sufficient conditions for the construction of these hybrid loops are given for the nonlinear case and then specialized for the linear case with the use of quadratic functions for the linear case we illustrate specific choices of the functions v and v p which allow for the reduction of the overshoot of a scalar output the proposed approaches are illustrated on linear and nonlinear examples

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

smooth lyapunov functions for hybrid systems part i existence is equivalent to robustness hybrid systems are dynamical systems where the state is allowed to either evolve continuously flow on certain subsets of the state space or evolve discontinuously jump from other subsets of the state space for a broad class of such systems characterized by mild regularity conditions on the data we establish the equivalence between the robustness of stability with respect to two measures and a characterization of such stability in terms of a smooth lyapunov function this result unifies and generalizes previous results for differential and difference inclusions with outer semicontinuous and locally bounded right hand sides furthermore we give a description of forward completeness of a hybrid system in terms of a smooth lyapunov like function

construction of synergistic potential functions on so 3 with application to velocity free hybrid attitude stabilization we propose a systematic and comprehensive procedure for the construction of synergistic potential functions which are instrumental in hybrid control design on text so 3 a new map via angular warping on text so 3 is introduced to generate a central family of potential functions allowing an explicit determination of the critical points and the synergistic gap some optimization results on the synergistic gap are also provided the proposed synergistic potential functions are used for the design of a global velocity free hybrid attitude stabilization scheme relying solely on inertial vector measurements

hybrid output feedback for attitude tracking on mathbb so 3 we present two solutions to the global exponential attitude tacking control problem using only attitude information on mathbb so 3 the first approach is based on a combined hybrid observer hybrid controller structure we derive a hybrid angular velocity observer for rigid body systems designed directly on mathbb so 3 times mathbb r 3 guaranteeing global exponential stability of the zero estimation error thereafter we propose an observer based attitude tracking control scheme guaranteeing global exponential stability the second approach consists of introducing an auxiliary dynamical system that generates the necessary damping in the absence of the velocity we prove that the proposed auxiliary system based hybrid controller guarantees global exponential stability as well

global stabilization of spherical orientation by synergistic hybrid feedback with application to reduced attitude tracking for rigid bodies in this paper we propose a hybrid feedback based on a synergistic potential function that achieves global asymptotic stabilization of a desired orientation on the n sphere with a nominal robustness to measurement disturbances a task that is not possible by classical feedback be it smooth nonsmooth periodic or any combination thereof due to the topological structure of the sphere we extend this basic result to a tracking controller for the reduced attitude or pointing direction of a rigid body and provide a method to remove jumps in the controlled torque by backstepping the proposed hybrid feedback is compared with a similar smooth feedback in simulation where it is illustrated that the hybrid feedback overcomes performance limitations inherent to the smooth feedback we provide two examples of a synergistic potential function one defined on a general sphere and the other on the unit circle

impulse differential inclusions a viability approach to hybrid systems impulse differential inclusions are introduced as a framework for modeling hybrid phenomena connections to standard problems in the area of hybrid systems are discussed conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an impulse differential inclusion for sets that violate these conditions methods are developed for approximating their viability and invariance kernels that is the largest subset that is viable or invariant under the action of the impulse differential inclusion the results are demonstrated on examples

stabilization of nonholonomic integrators via logic based switching this paper explains how to stabilize a nonholonomic integrator using a hybrid control law employing switching and logic results concerning asymptotic stability and exponential convergence to the origin are derived the notion of a positively invariant set is extended to hybrid systems and sufficient conditions for invariance are presented the verification of these conditions does not require the computation of the state trajectories and their use goes beyond the analysis of the system presented in this paper

stability and performance of siso control systems with first order reset elements we consider set point regulation and l2 robust stability properties of a class of reset control systems consisting of a minimum phase relative degree one linear siso plant controlled by a novel first order reset element fore these results rely on necessary and sufficient conditions for exponential and l2 finite gain stability of a class of planar reset systems consisting of a scalar linear plant controlled by the novel fore we show that the l2 gain of the planar reset system decreases to zero as the pole and or the gain of the fore are increased to infinity a number of stability results including lyapunov conditions for lp and exponential stability for a larger class of reset and hybrid systems are presented and used to prove our main results

hybrid feedback control and robust stabilization of nonlinear systems in this paper we show for any nonlinear system that is asymptotically controllable to a compact set that a logic based hybrid feedback can achieve asymptotic stabilization that is robust to small measurement noise actuator error and external disturbance the construction of such a feedback hinges upon recasting a stabilizing patchy feedback in a hybrid framework by making it dynamic with a discrete state while insisting on semicontinuity and closedness properties of the hybrid feedback and of the resulting closed loop hybrid system the robustness of stability is then shown as a generic property of hybrid systems having the said regularity properties auxiliary results give uniformity of convergence and of overshoots for hybrid systems and give a characterization of asymptotic stability of compact sets

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

recurrence principles and their application to stability theory for a class of stochastic hybrid systems in this paper we establish results similar in nature to the invariance principle for a class of stochastic hybrid systems modeled by set valued mappings in particular we characterize the set to which bounded random solutions converge in terms of a new concept called weak total recurrence a refinement of the result under the existence of a non increasing on average lyapunov like function is also established a comparison between weak total recurrence and the frequently studied invariance concepts are presented finally application of the main results to establish weak sufficient conditions for certifying stochastic stability properties like uniform global asymptotic stability in probability for compact sets and uniform global recurrence for open bounded sets are discussed

on singular perturbations due to fast actuators in hybrid control systems a stability result is given for hybrid control systems singularly perturbed by fast but continuous actuators if a hybrid control system has a compact set globally asymptotically stable when the actuator dynamics are omitted or equivalently are infinitely fast then the same compact set is semiglobally practically asymptotically stable in the finite speed of the actuator dynamics this result which generalizes classical results for differential equations justifies using a simplified plant model that ignores fast but continuous actuator dynamics even when using a hybrid feedback control algorithm

a nonsmooth hybrid invariance principle applied to robust event triggered design we first propose a nonsmooth hybrid invariance principle with relaxed conditions stemming from the fact that flowing solutions evolve only in the tangent cone and complete jumping solutions cannot jump outside the jump and flow sets we then show an application consisting in the design of event triggered rules to stabilize a class of uncertain linear control systems the event triggering rule depends only on local information that is it uses only the output signals available to the controller the approach proposed combines hybrid and sampled data tools the proposed design conditions are formulated in terms of linear matrix inequalities lmis ensuring global robust asymptotic stability of the closed loop system a tunable parameter is also available to guarantee an adjustable dwell time property of the solutions the effectiveness of the approach is evaluated through an example borrowed from the literature

lyapunov stability theory of nonsmooth systems this paper develops nonsmooth lyapunov stability theory and lasalle s invariance principle for a class of nonsmooth lipschitz continuous lyapunov functions and absolutely continuous state trajectories computable tests based on filipov s differential inclusion and clarke s generalized gradient are derived the primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right hand side such as in nonsmooth dynamic systems or variable structure control