robust almost global splay state stabilization of pulse coupled oscillators this paper deals with the problem of asymptotically stabilizing the splay state configuration of a network of identical pulse coupled oscillators through the design of the their phase response function the network of pulse coupled oscillators is modeled as a hybrid system the design of the phase response function is performed to achieve almost global asymptotic stability of a set wherein oscillators phases are evenly distributed on the unit circle to establish such a result a novel lyapunov function is proposed robustness with respect to frequency perturbation is assessed finally the effectiveness of the proposed methodology is shown in an example

gossip coverage control for robotic networks dynamical systems on the space of partitions future applications in environmental monitoring delivery of services and transportation of goods motivate the study of deployment and partitioning tasks for groups of autonomous mobile agents these tasks may be achieved by recent coverage algorithms based upon the classic methods by lloyd these algorithms however rely upon critical requirements on the communication network information is exchanged synchronously among all agents and long range communication is sometimes required this work proposes novel coverage algorithms that require only gossip communication i e asynchronous pairwise and possibly unreliable communication which robot pair communicates at any given time may be selected deterministically or randomly a key innovative idea is describing coverage algorithms for robot deployment and environment partitioning as dynamical systems on a space of partitions in other words we study the evolution of the regions assigned to each agent rather than the evolution of the agents positions the proposed gossip algorithms are shown to converge to centroidal voronoi partitions under mild technical conditions our treatment features a broad variety of results in topology analysis and geometry first we establish the compactness of a suitable space of partitions with respect to the symmetric difference metric second with respect to this metric we establish the continuity of various geometric maps including the voronoi diagram as a function of its generators the location of a centroid as a function of a set and the widely known multicenter function studied in facility location problems third we prove two convergence theorems for dynamical systems on metric spaces described by deterministic and stochastic switches

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

attitude estimation with intermittent measurements we propose a framework for attitude estimation on the special orthogonal group so 3 using intermittent body frame vector measurements we consider the case where the vector measurements are synchronously intermittent all measurements are received at the same time and the case where the vector measurements are asynchronously intermittent not all measurements are received at the same time the proposed observers have a measurement triggered structure where the attitude is predicted using the continuously measured angular velocity when the vector measurements are not available and adequately corrected upon the arrival of the vector measurements a hybrid framework is proposed to capture the behaviour of the closed loop system by extending the state with timers that are reset at each jump of the observer state almost global asymptotic stability is shown using rigorous lyapunov techniques for hybrid systems

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

stability and completion of zeno equilibria in lagrangian hybrid systems this paper studies lagrangian hybrid systems which are a special class of hybrid systems modeling mechanical systems with unilateral constraints that are undergoing impacts this class of systems naturally display zeno behavior an infinite number of discrete transitions that occur in finite time leading to the convergence of solutions to limit sets called zeno equilibria this paper derives simple conditions for stability of zeno equilibria utilizing these results and the constructive techniques used to prove them the paper introduces the notion of a completed hybrid system which is an extended hybrid system model allowing for the extension of solutions beyond zeno points a procedure for practical simulation of completed hybrid systems is outlined and conditions guaranteeing upper bounds on the incurred numerical error are derived finally we discuss an application of these results to the stability of unilaterally constrained motion of mechanical systems under perturbations that violate the constraint

the geometrical properties of planar lattice trails the authors investigate the winding angle and mean square end to end distance of lattice trails on a square lattice by both exact enumeration and monte carlo methods they find similar behaviour to that of self avoiding walks to the extent that the complete distribution is found to agree with the recent exact calculation of the winding angle distribution function for self avoiding walks by duplantier and saleur 1988 who found a gaussian distribution the exponent nu was found to be indistinguishable from 3 4

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

on path lifting mechanisms and unwinding in quaternion based attitude control the unit quaternion is a pervasive representation of rigid body attitude used for the design and analysis of feedback control laws because the space of unit quaternions constitutes a double cover of the rigid body attitude space quaternion based control laws are often by design inconsistent i e they do not have a unique value for each rigid body attitude inconsistent quaternion based control laws require an additional mechanism that uniquely converts an attitude estimate into its quaternion representation however conversion mechanisms that are memoryless e g selecting the quaternion having positive scalar component have a limited domain where they remain injective and when used globally introduce discontinuities into the closed loop system we show through an explicit construction and lyapunov analysis that such discontinuities can be hijacked by arbitrarily small measurement disturbances to stabilize attitudes far from the desired attitude to remedy this limitation we propose a hybrid dynamic algorithm for smoothly lifting an attitude path to the unit quaternion space we show that this hybrid dynamic mechanism allows us to directly translate quaternion based controllers and their asymptotic stability properties obtained in the unit quaternion space to the actual rigid body attitude space we also show that when quaternion based controllers are not designed to account for the double covering of the rigid body attitude space by a unit quaternion parameterization they can give rise to the unwinding phenomenon which we characterize in terms of the projection of asymptotically stable sets finally we employ the main results to show that certain hybrid feedbacks can globally asymptotically stabilize the attitude of a rigid body

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

stability analysis for stochastic hybrid systems this survey addresses stability analysis for stochastic hybrid systems shs which are dynamical systems that combine continuous change and instantaneous change and that also include random effects we re emphasize the common features found in most of the models that have appeared in the literature which include stochastic switched systems markov jump systems impulsive stochastic systems switching diffusions stochastic impulsive systems driven by renewal processes diffusions driven by levy processes piecewise deterministic markov processes general stochastic hybrid systems and stochastic hybrid inclusions then we review many of the stability concepts that have been studied including lyapunov stability lagrange stability asymptotic stability and recurrence next we detail lyapunov based sufficient conditions for these properties and additional relaxations of lyapunov conditions many other aspects of stability theory for shs like converse lyapunov theorems and robustness theory are not fully developed hence we also formulate some open problems to serve as a partial roadmap for the development of the underdeveloped pieces

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

stability via hybrid event time lyapunov function and impulsive stabilization for discrete time delayed switched systems this paper proposes a new stability analysis method called the hybrid event time lyapunov function htlf approach for discrete time delayed switched systems ddss two types of stability notions guas and event guas are proposed to reflect the effects on stability from events a basic stability result is derived a ddss has event guas if and only if there exists an htlf which is strictly decreasing and converges to zero moreover some sufficient conditions expressed by htlf razumikhin type stability theorems are established the issue of impulsive stabilization for ddss is studied it is proved that an unstable ddss can be stabilized by impulsive control and the impulsive stabilization to input to state stability of ddss is also achieved

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

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

finite time stability of sets for hybrid dynamical systems abstract notions and tools for finite time stability of closed sets for hybrid dynamical systems modeled as hybrid inclusions are introduced finite time stability of a closed set is defined as the following two properties lyapunov stability namely the property that solutions that start close to the set stay close to it and finite time convergence in the latter property the amount of time required to converge to the set of interest is captured by a settling time function that depends on hybrid time namely the time elapsed during flows and the number of jumps or events of the hybrid system various sufficient conditions for such properties to hold for a given closed set are established conditions involving lyapunov like functions that strictly decrease during flows that strictly decrease during jumps and that strictly decrease along both regimes are proposed these functions are only required to be locally lipschitz a link between non finite time asymptotic stability and the proposed notion is also established in addition sufficient conditions for finite time attractivity involving properties at jumps of the system the second derivative of a lyapunov like function and a property of nested sets are provided throughout the paper examples exercise the results in addition an application to finite time parameter estimation is provided and results are applied to it

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

dynamical properties of hybrid systems simulators this paper analyzes the dynamical properties of a class of hybrid systems simulators a hybrid system is a dynamical system with a state that can both flow and jump its simulator attempts to generate its solutions approximately the paper presents mild regularity conditions on the hybrid system and its simulator to guarantee that simulated solutions are close to actual solutions on compact hybrid time intervals and that asymptotically stable compact sets are preserved in a semiglobal practical sense under simulation in fact it is established that asymptotically stable compact sets are continuous in the integration step size parameter of the simulator that is as the step size of the simulator converges to zero the asymptotically stable set observed in simulations approaches the asymptotically stable compact set of the true hybrid system examples are used to illustrate concepts and results

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

follow the bouncing ball global results on tracking and state estimation with impacts in this paper we formulate tracking and state estimation problems of a translating mass in a polyhedral billiard as a stabilization problem for a suitable set due to the discontinuous trajectories arising from the impacts we use hybrid systems stability analysis tools to establish the results using a novel concept of mirrored images of the target mass we prove that 1 a tracking control algorithm and 2 an observer algorithm guarantee global exponential stability results for specific classes of polyhedral billiards including rectangles moreover we combine these two algorithms within dynamic controllers that guarantee global output feedback tracking the results are illustrated via simulations

limited feedback for temporally correlated mimo channels with other cell interference limited feedback improves link reliability with a small amount of feedback from the receiver to the transmitter in cellular systems the performance of limited feedback will be degraded in the presence of other cell interference when the base stations have limited or no coordination this paper establishes the degradation in sum rate of users in a cellular system due to uncoordinated other cell interference and feedback delay a goodput metric is defined as the rate when the bits are successfully received at the mobile station and used to derive an upper bound on the performance of limited feedback systems with delay this paper shows that the goodput gained from having delayed limited feedback decreases doubly exponentially as the delay increases the analysis is extended to precoded spatial multiplexing systems where it is shown that the same upper bound can be used to evaluate the decay in the achievable sum rate to reduce the effects of interference zero forcing interference nulling is applied at the receiver where it is shown that the effect of the interference on the achievable sum rate can be suppressed by nulling out the interferer numerical results show that the decay rate of the goodput decreases when the codebook quantization size increases and when the doppler shift in the channel decreases

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

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

locating omega limit sets using height functions this work deals with the problem of locating the limit set of a bounded solution of a given autonomous vector field f on a riemannian manifold assuming to know that the limit set is contained in an embedded submanifold s and using an auxiliary function that we call height function w for f and s we show how to obtain a better estimate of the location of under mild assumptions several consequences and an application to a type of polynomial vector fields are presented

lasalle yoshizawa corollaries for nonsmooth systems in this technical note two generalized corollaries to the lasalle yoshizawa theorem are presented for nonautonomous systems described by nonlinear differential equations with discontinuous right hand sides lyapunov based analysis methods that achieve asymptotic convergence when the candidate lyapunov derivative is upper bounded by a negative semi definite function in the presence of differential inclusions are presented a design example illustrates the utility of the corollaries

lyapunov based sufficient conditions for exponential stability in hybrid systems lyapunov based sufficient conditions for exponential stability in hybrid systems are presented the focus is on converting non strict lyapunov conditions having certain observability properties into strict lyapunov conditions for exponential stability both local and global results are considered the utility of the results is illustrated through an example

global asymptotic stabilization for a class of bilinear systems by hybrid output feedback this technical note deals with the global asymptotic stabilization problem for a class of bilinear systems a state feedback controller solving this problem is obtained uniting a local controller having an interesting behavior in a neighborhood of the origin and a constant controller valid outside this neighborhood the approach developed is based on the use of a hybrid loop and more precisely a hybrid state feedback this result is extended to the case where the state of the plant is not fully available and only the measured output can be used for control purposes in this case a dynamical controller constituted by an observer and a state feedback is built by means of hybrid output feedback framework in both cases the conditions are expressed by a set of linear matrix inequalities

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

formation control using binary information in this paper we study the problem of formation keeping of a network of strictly passive systems when very coarse information is exchanged we assume that neighboring agents only know whether their relative position is larger or smaller than the prescribed one this assumption results in very simple control laws that direct the agents closer or away from each other and take values in finite sets we show that the task of formation keeping while tracking a desired velocity and rejecting matched disturbances is still achievable under the very coarse information scenario in contrast with other results of practical convergence with coarse or quantized information here the control task is achieved exactly

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

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

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

stability theorems for delay differential inclusions this technical note addresses stability problems of delay differential inclusions a chain rule is proposed for a wide class of lyapunov krasovskii functionals which are not necessarily invariantly differentiable we also obtain an invariance like theorem where the derivative of the candidate functional is bounded above by a continuous negative semidefinite function several examples are given to show the effectiveness of the results

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

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

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

hybrid systems with delayed jumps asymptotic stability via robustness and lyapunov conditions hybrid systems subject to delayed jumps form a class of dynamical systems with broad applications this article develops sufficient conditions for robust asymptotic stability of hybrid systems in the presence of delayed jumps more precisely given a delay free hybrid system we introduce a higher order delayed system parametrized by the length of delays we show that when the delay parameter is set to zero the higher order model captures the solutions of the delay free system under mild conditions it is shown that when the delay free system has an asymptotically stable compact set for small enough delays solutions of the delayed system converge to a neighborhood of a set of interest related to the aforementioned compact set then lyapunov functions for the delay free system are used to develop sufficient conditions for asymptotic stability in the presence of delays unlike prior work in the literature the results pertaining to these notions of stability hold for systems possessing zeno solutions with time varying delays importantly the required conditions are expressed in finite dimensional space and depend primarily on the data of the delay free system the practical stability result is validated numerically through the hybrid system model of a controlled boost converter circuit with state triggered switches and zeno solutions the higher order model and the derived lyapunov conditions are utilized to obtain quantitative bounds on maximum allowable delays for switched systems and sampled data control

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