conewise linear systems non zenoness and observability conewise linear systems are dynamical systems in which the state space is partitioned into a finite number of nonoverlapping polyhedral cones on each of which the dynamics of the system is described by a linear differential equation this class of dynamical systems represents a large number of piecewise linear systems most notably linear complementarity systems with the p property and their generalizations to affine variational systems which have many applications in engineering systems and dynamic optimization the challenges of dealing with this type of hybrid system are due to two major characteristics mode switchings are triggered by state evolution and states are constrained in each mode in this paper we first establish the absence of zeno states in such a system based on this fundamental result we then investigate and relate several state observability notions short time and t time or finite time local global observability for the short time observability notions constructive finitely verifiable algebraic both sufficient and necessary conditions are derived due to their long time mode transitional behavior which is very difficult to predict only partial results are obtained for the t time observable states nevertheless we completely resolve the t time local observability for the bimodal conewise linear system for finite t and provide numerical examples to illustrate the difficulty associated with the long time observability

controllability and stabilizability of a class of continuous piecewise affine dynamical systems this paper studies controllability and stabilizability of continuous piecewise affine dynamical systems which can be considered as a collection of ordinary finite dimensional linear input state output systems together with a partition of the product of the state space and input space into full dimensional polyhedral regions each of these regions is associated with one particular linear system from the collection the main results of the paper are popov belevitch hautus type necessary and sufficient conditions for both controllability and stabilizability of such systems

quantum mechanics on the light cone ii the spin 1 2 case for pt i see ibid vol 25 p 4673 1992 the authors present 1 a set of spin 1 2 poincare group operators acting on the past light cone 2 a continuity equation resulting from the hamiltonian with a positive definite density 3 the spin 1 2 basis states in the non relativistic limit the hamiltonian is equivalent to the pauli schrodinger equation they also consider the hamiltonian modified by a radial potential

discovering multiple lyapunov functions for switched hybrid systems in this paper we analyze local asymptotic stability of switched hybrid systems whose subsystems have polynomial vector fields by discovering multiple lyapunov functions in quadratic forms we start with an algebraizable sufficient condition for the existence of quadratic multiple lyapunov functions then since different discrete modes are considered we apply real root classification together with a projection operator to underapproximate this sufficient condition step by step arriving at a set of semialgebraic sets which only involve the coefficients of the pre assumed multiple lyapunov function note that for each step we additionally use the information on the different discrete modes to optimize our intermediate computation results afterward we compute a sample point in the corresponding semialgebraic set for the coefficients resulting in a multiple lyapunov function finally we test our approach on some examples using a prototypical implementation and compare it with the generic quantifier e

strongly regular differential variational systems a differential variational system is defined by an ordinary differential equation ode parameterized by an algebraic variable that is required to be a solution of a finite dimensional variational inequality containing the state variable of the system this paper addresses two system theoretic topics for such a nontraditional nonsmooth dynamical system namely non zenoness and local observability of a given state satisfying a blanket strong regularity condition for the former topic which is of contemporary interest in the study of hybrid systems we extend the results in our previous paper where we have studied zeno states and switching times in a linear complementarity system lcs as a special case of the differential variational inequality dvi the lcs consists of a linear time invariant ode and a linear complementarity problem the extension to a nonlinear complementarity system ncs with analytic inputs turns out to be non trivial as we need to use the lie derivatives of analytic functions in order to arrive at an expansion of the solution trajectory near a given state further extension to a differential variational inequality is obtained via its equivalent karush kuhn tucker formulation for the second topic which is classical in system theory we use the non zenoness result and the recent results in a previous paper pertaining to the b differentiability of the solution operator of a nonsmooth ode to obtain a sufficient condition for the short time local observability of a given strongly regular state of an ncs refined sufficient conditions and necessary conditions for local observability of the lcs satisfying the p property are obtained

uncertainty quantification in hybrid dynamical systems uncertainty quantification uq techniques are frequently used to ascertain output variability in systems with parametric uncertainty traditional algorithms for uq are either system agnostic and slow such as monte carlo or fast with stringent assumptions on smoothness such as polynomial chaos and quasi monte carlo in this work we develop a fast uq approach for hybrid dynamical systems by extending the polynomial chaos methodology to these systems to capture discontinuities we use a wavelet based wiener haar expansion we develop a boundary layer approach to propagate uncertainty through separable reset conditions we also introduce a transport theory based approach for propagating uncertainty through hybrid dynamical systems here the expansion yields a set of hyperbolic equations that are solved by integrating along characteristics the solution of the partial differential equation along the characteristics allows one to quantify uncertainty in hybrid or switching dynamical systems the above methods are demonstrated on example problems

the effect of coupling conditions on the stability of bimodal systems in r3 abstract this paper investigates the global asymptotic stability of a class of bimodal piecewise linear systems in r 3 the approach taken allows the vector field to be discontinuous on the switching plane in this framework verifiable necessary and sufficient conditions are proposed for global asymptotic stability of bimodal systems being considered it is further shown that the way the subsystems are coupled on the switching plane plays a crucial role on global asymptotic stability along this line it is demonstrated that a constant which is called the coupling constant in the paper can be changed without changing the eigenvalues of subsystems and this change can make bimodal system stable or unstable

lyapunov theory for zeno stability zeno behavior is a dynamic phenomenon unique to hybrid systems in which an infinite number of discrete transitions occurs in a finite amount of time this behavior commonly arises in mechanical systems undergoing impacts and optimal control problems but its characterization for general hybrid systems is not completely understood the goal of this paper is to develop a stability theory for zeno hybrid systems that parallels classical lyapunov theory that is we present lyapunov like sufficient conditions for zeno behavior obtained by mapping solutions of complex hybrid systems to solutions of simpler zeno hybrid systems defined on the first quadrant of the plane these conditions are applied to lagrangian hybrid systems which model mechanical systems undergoing impacts yielding simple sufficient conditions for zeno behavior finally the results are applied to robotic bipedal walking

continuous piecewise affine dynamical systems do not exhibit zeno behavior in the context of continuous piecewise affine dynamical systems we study the zeno behavior i e infinite number of mode transitions in finite time interval in this note the main result reveals that piecewise affine dynamical systems do not exhibit zeno behavior a direct benefit of the main result is that one can apply smooth ordinary differential equations theory in a local manner for the analysis of piecewise affine systems

higher order moreau s sweeping process mathematical formulation and numerical simulation in this paper we present an extension of moreau s sweeping process for higher order systems the dynamical framework is carefully introduced qualitative dissipativity stability existence regularity and uniqueness results are given the time discretization of these nonsmooth systems with a time stepping algorithm is also presented this differential inclusion can be seen as a mathematical formulation of complementarity dynamical systems with arbitrary dimension and arbitrary relative degree between the complementary slackness variables applications of such high order sweeping processes can be found in dynamic optimization under state constraints and electrical circuits with ideal diodes

preasymptotic stability and homogeneous approximations of hybrid dynamical systems hybrid dynamical systems are systems that combine features of continuous time dynamical systems and discrete time dynamical systems and can be modeled by a combination of differential equations or inclusions difference equations or inclusions and constraints preasymptotic stability is a concept that results from separating the conditions that asymptotic stability places on the behavior of solutions from issues related to existence of solutions in this paper techniques for approximating hybrid dynamical systems that generalize classical linearization techniques are proposed the approximation techniques involve linearization tangent cones homogeneous approximations of functions and set valued mappings and tangent homogeneous cones where homogeneity is considered with respect to general dilations the main results deduce preasymptotic stability of an equilibrium point for a hybrid dynamical system from preasymptotic stability of the equilibrium point for an approximate system further results relate the degree of homogeneity of a hybrid system to the zeno phenomenon that can appear in the solutions of the system

on the regularization of zeno hybrid automata fundamental properties of hybrid automata such as existence and uniqueness of executions are studied particular attention is devoted to zeno hybrid automata which are hybrid automata that take infinitely many discrete transitions in finite time it is shown that regularization techniques can be used to extend the zeno executions of these automata to times beyond the zeno time different types of regularization may however lead to different extensions a water tank control problem and a bouncing ball system are used to illustrate the results

uniqueness of solutions of linear relay systems conditions are given for uniqueness of solutions of linear time invariant systems under relay feedback from a hybrid dynamical point of view this entails the deterministic specification of the discrete transition rules the results are based on the formulation of relay systems as complementarity systems and use the constructive theory of the linear complementarity problem

zeno stability of the set valued bouncing ball hybrid dynamical systems consist of both continuous time and discrete time dynamics a fundamental phenomenon that is unique to hybrid systems is zeno behavior where the solution involves an infinite number of discrete transitions occurring in finite time as best illustrated in the classical example of a bouncing ball in this note we study the hybrid system of the set valued bouncing ball for which the continuous time dynamics has a set valued right hand side this system is typically used for deriving bounds on the solution of nonlinear single valued hybrid systems in a small neighborhood of a zeno equilibrium point in order to establish its local stability we utilize methods of lyapunov analysis and optimal control to derive a necessary and sufficient condition for zeno stability of the set valued bouncing ball system and to obtain a tight bound on the zeno time as a function of initial conditions

group and total dissipativity and stability of multi equilibria hybrid automata complex systems which consist of different interdependent and interlocking subsystems typically have multiple equilibrium points associated with different set points of each operation mode these systems are usually interpreted as hybrid systems this paper studies the conditions for dissipativity and some stability properties of a class of hybrid systems with multiple co existing equilibrium points modelled as nonlinear hybrid automata a classification of equilibria for hybrid automata is proposed the objective is to identify dissipative components as groups of discrete locations within the hybrid automaton formed according to existing equilibria an example is provided

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

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

on estimating distribution functions and quantiles from survey data using auxiliary information summary ratio and difference estimators of a population distribution function under a general sampling design are obtained using auxiliary population information the relative mean errors and relative mean square errors of these estimators and a model based estimator of chambers dunstan 1986 are compared through a simulation study the advantages of the design based estimators over the model based estimator under model misspecifications especially for large samples are demonstrated ratio and difference estimators of a population quantile are also studied

switching behavior of solutions of ordinary differential equations with abs factorable right hand sides abstract we consider nonsmooth dynamic systems that are formulated as the unique solutions of ordinary differential equations odes with right hand side functions that are finite compositions of analytic functions and absolute value functions various non zenoness results are obtained for such solutions in particular any absolute value function in the ode right hand side can only switch between its two linear pieces finitely many times on any finite duration even when a discontinuous control input is included these results are extended to obtain numerically verifiable necessary conditions for the emergence of valley tracing modes in which the argument of an absolute value function is identically zero for a nonzero duration such valley tracing modes can create theoretical and numerical complications during sensitivity analysis or optimization we show that any valley tracing mode must begin either at the initial time or when another absolute value function switches between its two linear pieces

cardinality constrained linear quadratic optimal control as control implementation often incurs not only a variable cost associated with the magnitude or energy of the control but also a setup cost we consider a discrete time linear quadratic lq optimal control problem with a limited number of control implementations termed in this technical note the cardinality constrained linear quadratic optimal control cclq we first derive a semi analytical feedback policy for cclq problems using dynamic programming dp due to the exponential growth of the complexity in calculating the action regions however dp procedure is only efficient for cclq problems with a scalar state space recognizing this fact we develop then two lower bounding schemes and integrate them into a branch and bound bnb solution framework to offer an efficient algorithm in solving general cclq problems adopting the devised solution algorithm for cclq problems we can solve efficiently the linear quadratic optimal control problem with setup costs

brief on solution concepts and well posedness of linear relay systems in this paper we study the well posedness existence and uniqueness of solutions of linear relay systems with respect to two different solution concepts filippov solutions and forward solutions we derive necessary and sufficient conditions for well posedness in the sense of filippov of linear systems of relative degree one and two in closed loop with relay feedback to be precise uniqueness of filippov and also forward solutions follows in this case if the first non zero markov parameter is positive by means of an example it is shown that this intuitively clear condition is not true for systems with relative degree larger than two the influence of the zeno phenomenon an infinite number of relay switching times in a finite length time interval on well posedness is highlighted and although linear relay systems form a rather limited subclass of hybrid dynamical systems the consequences of the presence of the zeno behaviour is typical for many other classes of non smooth and hybrid systems

solution dependence on initial conditions in differential variational inequalities in the first part of this paper we establish several sensitivity results of the solution x t to the ordinary differential equation ode initial value problem ivp dx dt f x x 0 as a function of the initial value for a nondifferentiable f x specifically we show that for xi t equiv x t xi 0 0 leq t leq t a if f is b differentiable on xi t then so is the solution operator x t at 0 b if f is semismooth on xi t then so is x t at 0 c if f has a linear newton approximation on xi t then so does x t at 0 moreover the linear newton approximation of the solution operator can be obtained from the solution of a linear differential inclusion in the second part of the paper we apply these ode sensitivity results to a differential variational inequality dvi and discuss a the existence uniqueness and lipschitz dependence of solutions to subclasses of the dvi subject to boundary conditions via an implicit function theorem for semismooth equations and b the convergence of a nonsmooth shooting method for numerically computing such boundary value solutions

linear passive systems and maximal monotone mappings this paper deals with a class of dynamical systems obtained from interconnecting linear systems with static set valued relations we first show that such an interconnection can be described by a differential inclusions with a maximal monotone set valued mappings when the underlying linear system is passive and the static relation is maximal monotone based on the classical results on such differential inclusions we conclude that such interconnections are well posed in the sense of existence and uniqueness of solutions finally we investigate conditions which guarantee well posedness but are weaker than passivity

on the matrix equation xa ax xp abstract we study the matrix equation xa ax x p in m n k for 1 p n it is shown that every matrix solution x is nilpotent and that the generalized eigenspaces of a are x invariant for a being a full jordan block we describe how to compute all matrix solutions combinatorial formulas for a m x l x l a m and ax l are given the case p 2 is a special case of the algebraic riccati equation

on the existence uniqueness and nature of caratheodory and filippov solutions for bimodal piecewise affine dynamical systems abstract in this paper we deal with the well posedness in the sense of existence and uniqueness of solutions and nature of solutions for discontinuous bimodal piecewise affine systems in a differential inclusion setting first we show that the conditions guaranteeing uniqueness of filippov solutions in the context of general differential inclusions are quite restrictive when applied to bimodal piecewise affine systems later we present a set of necessary and sufficient conditions for uniqueness of filippov solutions for bimodal piecewise affine systems we also study the so called zeno behavior possibility of infinitely many switchings within a finite time interval for filippov solutions

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

representations of symmetric linear dynamical systems the purpose of this paper is to study static symmetries in linear time invariant differential dynamical systems the main result is a representation theorem which brings the symmetry strongly into evidence this result is then applied to a number of examples involving permutations and rotations we close by proving a general result on the representation of compact groups on the ring of unimodular polynomial matrices

on robust stability of switched systems in the context of filippov solutions abstract we consider the robust stability problem of a class of nonlinear switched systems defined on compact sets with state dependent switching instead of the caratheodory solutions we study the more general filippov solutions this encapsulates solutions with infinite switching in finite time and sliding modes in the neighborhood of the switching surfaces in this regard we formulate a lyapunov like stability theorem based on the theory of differential inclusions additionally we extend the results to switched systems with simplical uncertainty we also demonstrate that for the special case of polynomial switched systems defined on semi algebraic sets stability analysis can be checked based on sum of squares programming techniques

a class of estimators of the population mean in survey sampling using auxiliary information summary for estimating the mean of a finite population using information on an auxiliary variable a class of estimators is defined as a function of the ratio of sample mean to population mean and the ratio of sample variance to population variance of the auxiliary variable asymptotic expressions for bias and mean squared error are obtained a condition is given for the minimum mean squared error of estimators of this class to be smaller than those which use only the ratio of sample mean to population mean of the auxiliary variable

on the control of uncertain impulsive systems approximate stabilization and controlled invariance the problems of stabilization and controlled invariance of a fairly wide class of uncertain hybrid systems is considered uncertainty enters in the form of a disturbance input that can affect both the continuous and the discrete dynamics a method for designing piecewise constant feedback controllers for this class of systems is developed in the case of controlled invariance the controller ensures that the state of the system remains arbitrarily close to a desired set over an arbitrarily long time horizon in the case of stabilization the controller ensures approximate exponential convergence of the runs of the closed loop system to the zero level set of a lyapunov function

non zenoness of a class of differential quasi variational inequalities the zeno phenomenon of a switched dynamical system refers to the infinite number of mode switches in finite time the absence of this phenomenon is crucial to the numerical simulation of such a system by time stepping methods and to the understanding of the behavior of the system trajectory extending a previous result for a strongly regular differential variational inequality this paper establishes that a certain class of non strongly regular differential variational inequalities is devoid of the zeno phenomenon the proof involves many supplemental results that are of independent interest specialized to a frictional contact problem with local compliance and polygonal friction laws this non zenoness result is of fundamental significance and the first of its kind

on the existence and uniqueness of poincare maps for systems with impulse effects the poincare map is widely used to study the qualitative behavior of dynamical systems for instance it can be used to describe the existence of periodic solutions the poincare map for dynamical systems with impulse effects sies was introduced in the last decade and mainly employed to study the existence of limit cycles periodic gaits for the locomotion of bipedal robots we investigate sufficient conditions for the existence and uniqueness of poincare maps for dynamical sies evolving on a differentiable manifold we apply the results to show the existence and uniqueness of poincare maps for systems with multiple domains

robust non zenoness of piecewise affine systems with applications to linear complementarity systems piecewise affine systems pass constitute an important class of nonsmooth switching dynamical systems subject to state dependent mode transitions arising from control and dynamic optimization a fundamental issue in dynamics analysis of switching systems pertains to the possible occurrence of infinitely many switchings in finite time referred to as the zeno behavior there has been a growing interest in characterization of zeno free switching systems different from the recent non zeno analysis of switching systems the present paper studies non zeno properties of pass subject to system parameter and or initial state perturbations inspired by sensitivity and uncertainty analysis of pass specifically by exploiting the geometry of polyhedral subdivisions and dynamical system techniques this paper establishes a uniform bound on the number of mode switchings for a family of lipschitz pass under mild uniform conditions on system parameters and associated polyhedral subdivisions this result is employed t

shape restricted smoothing splines via constrained optimal control and nonsmooth newton s methods shape restricted smoothing splines receive considerable attention motivated by many important applications in science and engineering in this paper we consider smoothing splines subject to general linear dynamics and control constraints and formulate them as finite horizon constrained linear optimal control problems with unknown initial state and control by exploring techniques from functional and variational analyses optimality conditions are developed in terms of variational inequalities due to the control constraints the optimality conditions give rise to a nonsmooth b differentiable equation of an optimal initial condition whose unique solution completely determines the shape restricted smoothing spline a modified nonsmooth newton s algorithm with line search is used to solve this equation detailed convergence analysis of the proposed algorithm is presented using techniques from nonsmooth analysis and polyhedral theory we show the global convergence of the algorithm for shape restricted smoothing splines subject to general polyhedral control constraints

ratio methods to the mean estimation with known quantiles we address the problem of estimating the finite population mean in survey sampling by exploiting any available auxiliary information in order to increase the precision of classical estimators the idea is to use any population quantiles of the available auxiliary variables which are known in many real situation from census administrative files etc this is achieved using these known quantities in the construction of the estimators by modifying the usual ratio estimation methods and afterwards defining a general class of exponentiation ratio estimators the advantages of the proposed estimators are demonstrated using theoretical asymptotic tools and through a simulation study

switching relay and complementarity systems a tutorial on their well posedness and relationships this paper reviews the relationships between switching systems defined from a partition of the state space into convex cells and relay or complementarity dynamical systems which are other classes of discontinuous systems well posedness results i e results on the existence and the uniqueness of solutions for different classes of relay and complementarity systems which are also switching systems are presented the reverse issue when can a switching system be rewritten equivalently as a relay or a complementarity system is also tackled many examples from mechanics circuits and biology illustrate the developments throughout the paper the paper focuses on systems with continuous solutions i e with no state jumps convexity is the central property

well posedness conditions for bimodal piecewise affine systems abstract this paper considers well posedness the existence and uniqueness of the solutions of bimodal piecewise affine systems in r n it is assumed that both modes are observable but only one of the modes is in observable canonical form this allows the vector field to be discontinuous when the trajectories change mode necessary and sufficient conditions for well posedness are given as a set of algebraic conditions and sign inequalities it is shown that these conditions induce a joint structure for the system matrices of the two modes this structure can be used for the classification of well posed bimodal piecewise affine systems furthermore it is also shown that under certain conditions well posed bimodal piecewise affine systems in r n may have one or two equilibrium points or no equilibrium points

fixing zeno gaps in computer science fixpoints play a crucial role most often least and greatest fixpoints are sufficient however there are situations where other ones are needed in this paper we study on an algebraic base a special fixpoint of the function f x a x that describes infinite iteration of an element a we show that the greatest fixpoint is too imprecise special problems arise if the iterated element contains the possibility of stepping on the spot e g skip in a programming language or if it allows zeno behaviour we present a construction for a fixpoint that captures these phenomena in a precise way the theory is presented and motivated using an example from hybrid system analysis

deadness and how to disprove liveness in hybrid dynamical systems what if we designed a tool to automatically prove the dynamical properties of systems for which analytic proof is difficult or impossible to obtain such a tool would represent a significant advance in the understanding of complex dynamical systems with nonlinearities this is precisely what this paper offers a solution to the problem of automatically proving some dynamic stability properties of complex systems with multiple discontinuities and modes of operation modelled as hybrid dynamical systems for this purpose we propose a reinterpretation of some stability properties from a computational viewpoint chiefly by using the computer science concepts of safety and liveness however these concepts need to be redefined within the framework of hybrid dynamical systems in computer science terms here we consider the problem of automatically disproving the liveness properties of nonlinear hybrid dynamical systems for this purpose we define a new property which we call deadness this is a dynamically aware property of a hybrid system which if true disproves the liveness property by means of a finite execution we formally define this property and give an algorithm which can derive deadness properties automatically for a type of liveness property called inevitability we show how this algorithm works for three different examples that represent three classes of hybrid systems with complex behaviours

asymptotic behavior of waves abstract the asymptotic conjugation relation lim t g t m e itp m g m e itp 2 0 is established for all l 2 r n under mild assumptions on and g where m h f 1 h f denotes fourier multiplication the asymptotic estimate lim t j u t u j 2 0 for finite energy solutions u of the wave equation is deduced from along with generalizations to a class of first order symmetric hyperbolic systems of partial differential equations that are homogeneous and constant coefficient and a weakened version for the klein gordon equation also deduced from is the fact that for a free schrodinger particle the probability of being in the set ta at time t tends to the probability that the velocity is in a as t

well posedness analysis of switch driven piecewise affine systems the well posedness problem existence and uniqueness of solutions of a class of multi modal piecewise affine systems is addressed where binary switches individually act under autonomous switching first a new transition rule on the discrete state called the switch based transition rule is introduced and some relations with the mode based transition rule are discussed next a sufficient condition for such a multi modal system to be well posed for all external inputs is derived in terms of well posedness of its subsystems of lower complexity bimodal systems finally an easily checkable condition for the bimodal system to be well posed for all external inputs is given which consequently allows us to algebraically determine well posedness of the multi modal systems in question

event triggered adaptive control for a class of uncertain nonlinear systems in this technical note the problem of event trigger based adaptive control for a class of uncertain nonlinear systems is considered the nonlinearities of the system are not required to be globally lipschitz since the system contains unknown parameters it is a difficult task to check the assumption of the input to state stability iss with respect to the measurement errors which is required in most existing literature to solve this problem we design both the adaptive controller and the triggering event at the same time such that the iss assumption is no longer needed in addition to presenting new design methodologies based on the fixed threshold strategy and relative threshold strategy we also propose a new strategy named the switching threshold strategy it is shown that the proposed control schemes guarantee that all the closed loop signals are globally bounded and the tracking stabilization error exponentially converges towards a compact set which is adjustable

finite population sampling with multivariate auxiliary information abstract this article examines strategies that are approximately design unbiased and nearly optimal assuming a large sample survey and a regression superpopulation model a new class of predictors is proposed to link certain features of optimal design unbiased and model unbiased predictors generalized regression predictors are shown to pervade the subclass of asymptotically design unbiased adu predictors generalized regression predictors are combined with model based stratification to construct highly efficient adu strategies

the asymptotic localization of position probability in quantum mechanics the method of stationary phase is discussed in a mathematically rigorous way a resulting lemma is used to derive the asymptotic localization of position probability under the evolution operator ut f 1 exp i t f where is a continuous function of the wave vector k k1 kn n 1 2 with continuous first and second derivatives the measurement of velocity is discussed and the interpretation of the self adjoint operator vi f 1 i f as representing the ith component of velocity is also discussed

the rational complementarity problem abstract an extension of the linear complementarity problem lcp of mathematical programming is the so called rational complementarity problem rcp this problem occurs if complementarity conditions are imposed on input and output variables of linear dynamical input state output systems the resulting dynamical systems are called linear complementarity systems since the rcp is crucial both in issues concerning existence and uniqueness of solutions to complementarity systems and in time simulation of complementarity systems it is worthwhile to consider existence and uniqueness questions of solutions to the rcp in this paper necessary and sufficient conditions are presented guaranteeing existence and uniqueness of solutions to the rcp in terms of corresponding lcps using these results and proving that the corresponding lcps have certain properties we can show uniqueness and existence of solutions to linear mechanical systems with unilateral constraints electrical networks with diodes and linear dynamical systems subject to relays and or coulomb friction

a unified framework for hybrid control model and optimal control theory we propose a very general framework that systematizes the notion of a hybrid system combining differential equations and automata governed by a hybrid controller that issues continuous variable commands and makes logical decisions we first identify the phenomena that arise in real world hybrid systems then we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems evolving on continuous variable state spaces and subject to continuous controls and discrete transitions the model captures the identified phenomena subsumes previous models yet retains enough structure to pose and solve meaningful control problems we develop a theory for synthesizing hybrid controllers for hybrid plants in all optimal control framework in particular we demonstrate the existence of optimal relaxed and near optimal precise controls and derive generalized quasi variational inequalities that the associated value function satisfies we summarize algorithms for solving these inequalities based on a generalized bellman equation impulse control and linear programming

finite valued feedback laws and piecewise classical solutions abstract often in engineering literature we find control systems in which the open loop inputs are piecewise constant and take values in a finite set such open loop inputs cause the system to have fairly regular solutions on the other hand when acting in closed loop feedback laws taking values in a finite set may not be reinterpreted as open loop inputs of the considered type in fact pathological behaviours such as the accumulation of discontinuities may appear zeno phenomenon we give some conditions which can be used as tools for building finite valued feedback laws not causing such pathological behaviours

metrics for weighted transition systems axiomatization and complexity abstract simulation distances are essentially approximations of simulation which provide a measure of the extent by which behaviors in systems are inequivalent in this paper we consider the general quantitative model of weighted transition systems where transitions are labeled with elements of a finite metric space we study the so called point wise and accumulating simulation distances which provide extensions to the well known boolean notion of simulation on labeled transition systems we introduce weighted process algebras for finite and regular behavior and offer sound and approximate complete inference systems for the proposed simulation distances we also settle the algorithmic complexity of computing the simulation distances

complementarity modeling of hybrid systems a complementarity framework is described for the modeling of certain classes of mixed continuous discrete dynamical systems the use of such a framework is well known for mechanical systems with inequality constraints but we give a more general formulation which also applies for instance to switching control systems the main theoretical results in the paper are concerned with uniqueness of smooth continuations the solution of this problem requires the construction of a map from the continuous state to the discrete state a crucial technical tool is the so called linear complementarity problem from mathematical programming and we introduce various generalizations of this problem

linear complementarity systems zeno states a linear complementarity system lcs is a hybrid dynamical system defined by a linear time invariant ordinary differential equation coupled with a finite dimensional linear complementarity problem lcp the present paper is the first of several papers whose goal is to study some fundamental issues associated with an lcs specifically this paper addresses the issue of zeno states and the related issue of finite number of mode switches in such a system the cornerstone of our study is an expansion of a solution trajectory to the lcs near a given state in terms of an observability degree of the state on the basis of this expansion and an inductive argument we establish that an lcs satisfying the p property has no strongly zeno states we next extend the analysis for such an lcs to a broader class of problems and provide sufficient conditions for a given state to be weakly non zeno while related mode switch results have been proved by brunovsky and sussmann for more general hybrid systems our analysis exploits the special structure of the lcs and yields new results for the latter that are of independent interest and complement those by these two and other authors

linear complementarity systems we introduce a new class of dynamical systems called linear complementarity systems the time evolution of these systems consists of a series of continuous phases separated by events which cause a change in dynamics and possibly a jump in the state vector the occurrence of events is governed by certain inequalities similar to those appearing in the linear complementarity problem of mathematical programming the framework we describe is suitable for certain situations in which both differential equations and inequalities play a role for instance in mechanics electrical networks piecewise linear systems and dynamic optimization we present a precise definition of the solution concept of linear complementarity systems and give sufficient conditions for existence and uniqueness of solutions

asymptotic stabilization of minimum phase nonlinear systems how a class of multivariable nonlinear systems can be stabilized about an equilibrium via smooth state feedback is shown more precisely conditions under which for every compact set of initial states it is possible to design a feedback law which drives to the equilibrium all initial states in this compact set are described the theory includes the development of globally defined transformations of the system equations to their global normal form

containment control in mobile networks in this paper the problem of driving a collection of mobile robots to a given target destination is studied in particular we are interested in achieving this transfer in an orderly manner so as to ensure that the agents remain in the convex polytope spanned by the leader agents while the remaining agents only employ local interaction rules to this aim we exploit the theory of partial difference equations and propose hybrid control schemes based on stop go rules for the leader agents non zenoness liveness and convergence of the resulting system are also analyzed

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

analysis of zeno behaviors in a class of hybrid systems this note investigates conditions for existence of zeno behaviors where a system undergoes an unbounded number of discrete transitions in a finite length of time in a class of hybrid systems zeno behavior occurs for example when a controller unsuccessfully attempts to satisfy an invariance specification by switching the system among different configurations faster and faster two types of zeno systems are investigated 1 strongly zeno systems where all runs of the system are zeno and 2 weakly zeno systems where only some runs of the system are zeno for constant rate and bounded rate hybrid systems and some nonlinear generalizations necessary and sufficient conditions for both zenoness and strong zenoness are derived the analysis is based on studying the trajectory set of a certain equivalent continuous time system that is associated with the dynamic equations of the hybrid system the relation between the possibility of existence of zeno behaviors in a system and the problem of existence of non zeno safety controllers that keep the system in a specified region of its operating space is also examined it is shown that in certain zeno systems a minimally interventive safety controller may not exist even if a safety controller exists disproving a conjecture made earlier in the literature

characterization of well posedness of piecewise linear systems one of the basic issues in the study of hybrid systems is the well posedness existence and uniqueness of solutions problem of discontinuous dynamical systems the paper addresses this problem for a class of piecewise linear discontinuous systems under the definition of solutions of caratheodory the concepts of jump solutions or of sliding modes are not considered here in this sense the problem to be discussed is one of the most basic problems in the study of well posedness for discontinuous dynamical systems first we derive necessary and sufficient conditions for bimodal systems to be well posed in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions next its extensions to the multimodal case are discussed as an application to switching control in the case that two state feedback gains are switched according to a criterion depending on the state we give a characterization of all admissible state feedback gains for which the closed loop system remains well posed

convergence of time stepping method for initial and boundary value frictional compliant contact problems beginning with a proof of the existence of a discrete time trajectory this paper establishes the convergence of a time stepping method for solving continuous time boundary value problems for dynamic systems with frictional contacts characterized by local compliance in the normal and tangential directions our investigation complements the analysis of the initial value rigid body model with one frictional contact encountering inelastic impacts by stewart arch ration mech anal 145 1998 pp 215 260 and the recent analysis by anitescu optimization based simulation for nonsmooth rigid multibody dynamics argonne national laboratory argonne il 2004 using the framework of measure differential inclusions in contrast to the measure theoretic approach of these authors we follow a differential variational approach and address a broader class of problems with multiple elastic or inelastic impacts applicable to both initial and affine boundary value problems our main convergence result pertains to the case where the compliance in the normal direction is decoupled from the compliance in the tangential directions and where the friction coefficients are sufficiently small

estimating distribution functions from survey data summary a simple method for estimating population distribution functions and associated quantiles from sample survey data is described and some asymptotic theory for it presented the method assumes a model based approach to survey estimation and allows auxiliary population information to be directly incorporated into the estimation process monte carlo results comparing the proposed method with conventional design based methods are given these suggest that the model based approach offers significant gains when the auxiliary population information is linearly related to the survey variables of interest

brief paper time scaling symmetry and zeno solutions symmetries are often helpful in reducing the complexity of the analysis of dynamical systems here we discuss a symmetry that combines a transformation in space with a scaling of time examples are given of a number of nonsmooth dynamical systems in which a symmetry of this type occurs the time scaling symmetry can be combined with return mapping analysis to achieve a dimension reduction in addition to the one already obtained from considering the return map the method is applied in a study of zeno behavior in linear systems with relay feedback

bounds on the number of switchings for trajectories of piecewise analytic vector fields abstract we study the trajectories of systems x x x where x is a continuous extendably piecewise analytic vector field i e a continuous vector field x such that the domain of admits a locally finite partition i into sets such that for each a i there is a vector field xa which is analytic on a neighborhood of the closure of a and whose restriction to a coincides with that of x we prove that the trajectories are piecewise analytic with a priori bounds on the number of switchings for all trajectories that stay in a fixed compact set and whose duration does not exceed a fixed number t this result implies the existence of a regular synthesis for optimal control problems with a strictly convex lagrangian and a linear dynamics with polyhedral constraints on the controls

regularization of chattering phenomena via bounded variation controls in the control theory the term chattering is used to refer to fast oscillations of controls such as an infinite number of switchings over a finite time interval in this paper we focus on three typical instances of chattering the fuller phenomenon referring to situations where an optimal control features an accumulation of switchings in finite time the robbins phenomenon concerning optimal control problems with state constraints where the optimal trajectory touches the boundary of the constraint set an infinite number of times over a finite time interval and the zeno phenomenon for hybrid systems referring to a trajectory that depicts an infinite number of location switchings in finite time from the practical point of view when trying to compute an optimal trajectory for instance by means of a shooting method chattering may be a serious obstacle to convergence in this paper we propose a general regularization procedure by adding an appropriate penalization of the total variation this produces a family of quasi optimal controls whose associated cost converge to the optimal cost of the initial problem as the penalization tends to zero under additional assumptions we also quantify quasi optimality by determining a speed of convergence of the costs

fast switches in relay feedback systems relays are common in automatic control systems even linear systems with relay feedback are however far from fully understood new results are given about the behavior of these systems via a state space approach it is proved that there exist multiple fast switches if and only if the sign of the first nonvanishing markov parameter of the linear system is positive fast switches are shown to occur as part of stable limit cycles an analysis is developed for these limit cycles that illustrates how they can be predicted

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 for piecewise affine systems via cone copositivity cone copositive piecewise quadratic lyapunov functions pwq lfs for the stability analysis of continuous time piecewise affine pwa systems are proposed the state space is assumed to be partitioned into a finite number of convex possibly unbounded polyhedra preliminary conditions on pwq functions for their sign in the polyhedra and continuity over the common boundaries are provided the sign of each quadratic function is studied by means of cone constrained matrix inequalities which are translated into linear matrix inequalities lmis via cone copositivity the continuity is guaranteed by adding equality constraints over the polyhedra intersections an asymptotic stability result for pwa systems is then obtained by finding a continuous pwq lf through the solution of a set of constrained lmis the effectiveness of the proposed approach is shown by analyzing an opinion dynamics model and two saturating control systems

a model for stochastic hybrid systems with application to communication networks abstract we propose a model for stochastic hybrid systems shss where transitions between discrete modes are triggered by stochastic events much like transitions between states of a continuous time markov chains however the rate at which transitions occur is allowed to depend both on the continuous and the discrete states of the shs based on results available for piecewise deterministic markov process pdps we provide a formula for the extended generator of the shs which can be used to compute expectations and the overall distribution of the state as an application we construct a stochastic model for on off tcp flows that considers both the congestion avoidance and slow start modes and takes directly into account the distribution of the number of bytes transmitted using the tools derived for shss we model the dynamics of the moments of the sending rate by an infinite system of odes which can be truncated to obtain an approximate finite dimensional model this model shows that for transfer size distributions reported in the literature the standard deviation of the sending rate is much larger than its average moreover the later seems to vary little with the probability of packet drop this has significant implications for the design of congestion control mechanisms