brief paper identification of dynamic systems using piecewise affine basis function models piecewise affine pwa basis function autoregressive exogenous bpwarx models are proposed in this paper for nonlinear black box identification a bpwarx model is a weighted sum of pwa basis bpwa functions which are the minimum or maximum of n 1 affine functions in n dimensions since the bpwa functions have a universal representation capability for continuous pwa functions the bpwarx models provide better accuracy than the hinging hyperplane arx hharx models with the same number of parameters and the same order of computational complexity when using a modified gauss newton algorithm to build the models from input output data

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

brief equivalence of hybrid dynamical models this paper establishes equivalences among five classes of hybrid systems mixed logical dynamical mld systems linear complementarity lc systems extended linear complementarity elc systems piecewise affine pwa systems and max min plus scaling mmps systems some of the equivalences are established under rather mild additional assumptions these results are of paramount importance for transferring theoretical properties and tools from one class to another with the consequence that for the study of a particular hybrid system that belongs to any of these classes one can choose the most convenient hybrid modeling framework

hybrid model predictive control application towards optimal semi active suspension the optimal control problem of a quarter car semi active suspension has been studied in the past considering that a quarter car semi active suspension can either be modelled as a linear system with state dependent constraint on control of actuator force input or a bi linear system with a control of variable damping coefficient saturation the seemingly simple problem poses several interesting questions and challenges does the saturated version of the optimal control law derived from the corresponding un constrained system i e clipped optimal remain optimal for the constrained case as suggested in some previous publications or should the optimal deviate from the clipped optimal as suggested in other publications if the optimal control law of the constrained system does deviate from its unconstrained counter part how different are they what is the structure of the optimal control law does it retain the linear state feedback form as the unconstrained case in this paper we attempt to an

efficient online solution of multi parametric mixed integer quadratic problems this paper considers model predictive control of a class of piecewise affine systems and introduces a method to reduce the complexity of the optimisation problem which is solved online the optimisation problem is a mixed integer quadratic program which is parameterised by the initial state of the system dynamics the complexity of the problem solved online is reduced by moving part of the computational burden offline in the offline computations the structure of the mixed integer problem is explored to determine a polytopic partition of the set of initial states each polytope of the partition is then considered and it is determined which integer combinations can cannot be optimal given that the initial state is contained in the current polytope the result of the offline computations is that each polytope is associated with the set of potentially optimal integer combinations the online solution of the mixed integer problem consists of a two step procedure first it is determined which polytope conta

optimal control of a class of linear hybrid systems with saturation we consider a class of first order linear hybrid systems with saturation a system that belongs to this class can operate in several modes or phases in each phase each state variable of the system exhibits a linear growth until a specified upper or lower saturation level is reached and after that the state variable stays at that saturation level until the end of the phase a typical example of such a system is a traffic signal controlled intersection we develop methods to determine optimal switching time sequences for first order linear hybrid systems with saturation that minimize criteria such as average queue length worst case queue length average waiting time and so on first we show how the extended linear complementarity problem elcp which is a mathematical programming problem can be used to describe the set of system trajectories of a first order linear hybrid system with saturation optimization over the solution set of the elcp then yields an optimal switching time sequence although this method yields globally optimal switching time sequences it is not feasible in practice due to its computational complexity therefore we also present some methods to compute suboptimal switching time sequences furthermore we show that if there is no upper saturation then for some objective functions the globally optimal switching time sequence can be computed very efficiently we also discuss some approximations that lead to suboptimal switching time sequences that can be computed very efficiently finally we use these results to design optimal switching time sequences for traffic signal controlled intersections

equivalent piecewise affine models of linear hybrid automata in this technical note we examine the relationship between linear hybrid automata lha and piecewise affine pwa systems while a lha is an autonomous non deterministic model a pwa is a deterministic model with inputs through the key idea of modeling the uncertainty associated with lha transitions as input disturbances in a pwa model by extending continuous time pwa models to include the dynamics of discrete states and resets we show in a constructive way that a lha can be equivalently represented as a pwa system where equivalent means that the two systems generate the same trajectories besides filling in a missing theoretical link between the lha modelling framework and the pwa modelling framework the result has the practical advantage of enabling the use of several existing control theoretical tools developed for pwa models to a wider class of hybrid systems

sanov results for glauber spin glass dynamics in this paper we prove a sanov result i e a large deviation principle ldp for the distribution of the empirical measure for the annealed glauber dynamics of the sherrington kirkpatrick spin glass without restrictions on time or temperature we prove a full ldp for the asymmetric dynamics and the crucial upper large deviations bound for the symmetric dynamics in the symmetric model a new order parameter arises corresponding to the response function in sozi83 in the asymmetric case we show that the corresponding rate function has a unique minimum given as the solution of a self consistent equation the key argument used in the proofs is a general result for mixing of what is known as large deviation systems lds with measures obeying an independent ldp

model predictive control for uncertain max min plus scaling systems in this paper we extend the classical min max model predictive control framework to a class of uncertain discrete event systems that can be modelled using the operations maximization minimization addition and scalar multiplication and that we call max min plus scaling mmps systems provided that the stage cost is an mmps expression and considering only linear input constraints then the open loop min max model predictive control problem for mmps systems can be transformed into a sequence of linear programming problems hence the min max model predictive control problem for mmps systems can be solved efficiently despite the fact that the system is non linear a min max feedback model predictive control approach using disturbance feedback policies is also presented which leads to improved performance compared to the open loop approach

observability and controllability of piecewise affine and hybrid systems we prove in a constructive way the equivalence between piecewise affine systems and a broad class of hybrid systems described by interacting linear dynamics automata and propositional logic by focusing our investigation on the former class we show through counterexamples that observability and controllability properties cannot be easily deduced from those of the component linear subsystems instead we propose practical numerical tests based on mixed integer linear programming

optimal complexity reduction of polyhedral piecewise affine systems this paper focuses on the np hard problem of reducing the complexity of piecewise polyhedral systems e g polyhedral piecewise affine pwa systems the results are fourfold firstly the paper presents two computationally attractive algorithms for optimal complexity reduction that under the assumption that the system is defined over the cells of a hyperplane arrangement derive an equivalent polyhedral piecewise system that is minimal in the number of polyhedra the algorithms are based on the cells and the markings of the hyperplane arrangement in particular the first algorithm yields a set of disjoint non overlapping merged polyhedra by executing a branch and bound search on the markings of the cells the second approach leads to non disjoint overlapping polyhedra by formulating and solving an equivalent and well studied logic minimization problem secondly the results are extended to systems defined on general polyhedral partitions and not on cells of hyperplane arrangements thirdly the paper proposes a technique to further reduce the complexity of piecewise polyhedral systems if the introduction of an adjustable degree of error is acceptable fourthly the paper shows that based on the notion of the hyperplane arrangement pwa state feedback control laws can be implemented efficiently three examples including a challenging industrial problem illustrate the algorithms and show their computational effectiveness in reducing the complexity by up to one order of magnitude

stabilizing dynamic controllers for hybrid systems a hybrid control lyapunov function approach this paper proposes a dynamic controller structure and a systematic design procedure for stabilizing discrete time hybrid systems the proposed approach is based on the concept of control lyapunov functions clfs which when available can be used to design a stabilizing state feedback control law in general the construction of a clf for hybrid dynamical systems involving both continuous and discrete states is extremely complicated especially in the presence of non trivial discrete dynamics therefore we introduce the novel concept of a hybrid control lyapunov function which allows the compositional design of a discrete and a continuous part of the clf and we formally prove that the existence of a hybrid clf guarantees the existence of a classical clf a constructive procedure is provided to synthesize a hybrid clf by expanding the dynamics of the hybrid system with a specific controller dynamics we show that this synthesis procedure leads to a dynamic controller that can be implemented by a receding horizon control strategy and that the associated optimization problem is numerically tractable for a fairly general class of hybrid systems useful in real world applications compared to classical hybrid receding horizon control algorithms the proposed approach typically requires a shorter prediction horizon to guarantee asymptotic stability of the closed loop system which yields a reduction of the computational burden as illustrated through two examples

computational techniques for reachability analysis of max plus linear systems this work discusses a computational approach to reachability analysis of max plus linear mpl systems a class of discrete event systems widely used in synchronization and scheduling applications given a set of initial states we characterize and compute its reach tube namely the collection of set of reachable states regarded step wise as reach sets by an alternative characterization of the mpl dynamics we show that the exact computation of the reach sets can be performed quickly and compactly by manipulations of difference bound matrices and further derive worst case bounds on the complexity of these operations the approach is also extended to backward reachability analysis the concepts and results are elucidated by a running example and we further illustrate the performance of the approach by a numerical benchmark the technique comfortably handles twenty dimensional mpl systems i e with twenty continuous state variables and as such it outperforms the state of the art alternative approaches in the literature

direct and indirect model reference adaptive control for multivariable piecewise affine systems this article proposes direct and indirect model reference adaptive control strategies for multivariable piecewise affine systems which constitute a popular tool to model hybrid systems and approximate nonlinear systems a chosen reference model which can be linear or also piecewise affine describes the desired closed loop system behavior that is to be achieved by the adaptive controllers for unknown system dynamics each subsystem acquires its own set of control gains which is tuned under careful consideration of the switching behavior in the indirect approach the use of dynamic gain adjustment avoids singularities in the certainty equivalence principle it is shown for both algorithms that the state of the reference model is tracked asymptotically given a common lyapunov function for the switched reference model is available furthermore parameter convergence in both the direct and indirect approach is proven for sufficiently rich reference signals finally both algorithms are evaluated in numerical simulations and their advantages and disadvantages are discussed

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

technical communique an efficient algorithm for optimal control of pwa systems with polyhedral performance indices we present an algorithm for the computation of explicit optimal control laws for piecewise affine pwa systems with polyhedral performance indices the algorithm is based on dynamic programming dp and represents an extension of ideas initially proposed in kerrigan and mayne 2003 optimal control of constrained piecewise affine systems with bounded disturbances in proceedings of the 41st ieee conference on decision and control las vegas nevada usa december and baotic et al 2003 a new algorithm for constrained finite time optimal control of hybrid systems with a linear performance index in proceedings of european control conference cambridge uk september specifically we show how to exploit the underlying geometric structure of the optimization problem in order to significantly improve the efficiency of the off line computations an extensive case study is provided which clearly indicates that the algorithm proposed in this paper may be preferable to other schemes published in the literature

toward unified analysis and controller synthesis for a class of hybrid systems abstract this work defines a new class of hybrid systems called state based switched sbs systems that have numerous important engineering applications the characterizing feature of these systems is that the discrete event dynamics are associated with the continuous time state making a specific function be equal to zero the choice of this function is application specific and for the closed loop sbs systems defined in this paper it is related to the execution of a desired set of tasks from a pre specified mission plan for this broad class of sbs systems the paper presents a unified analysis and controller synthesis methodology based on lyapunov theory depending on the details of the mission plan the closed loop hybrid system will be divided into two subclasses sequential and non sequential the controller design procedure for both subclasses consists of the same two steps finding a control law and finding a stabilizing switching rule for static state and output feedback of sequential hybrid systems the paper proposes a new hybrid sequential sliding mode controller it is proven that the control mission can be accomplished for sequential hybrid systems under static state and output feedback using this new controller a similar framework is investigated for the more complex class of nonsequential hybrid systems and a systematic procedure for designing the switching rule is presented for some specific instances of these systems

efficient conversion of mixed logical dynamical systems into an equivalent piecewise affine form for hybrid systems described by switched linear difference equations linear threshold conditions automata and propositional logic conditions described in mixed logical dynamical form this note describes two algorithms for transforming such systems into an equivalent piecewise affine form where equivalent means that for the same initial conditions and input sequences the trajectories of the system are identical the proposed techniques exploit ideas from mixed integer programming and multiparametric programming

explicit robustness and fragility margins for linear discrete systems with piecewise affine control law in this paper we focus on the robustness and fragility problem for piecewise affine pwa control laws for discrete time linear system dynamics in the presence of parametric uncertainty of the state space model a generic geometrical approach will be used to obtain robustness fragility margins with respect to the positive invariance properties for pwa control laws defined over a bounded region in the state space it is shown that these margins can be described in terms of polyhedral sets in parameter space the methodology is further extended to the fragility problem with respect to the partition defining the controller finally several computational aspects are presented regarding the transformation from the theoretical formulations to explicit representations vertex halfspace representation of polytopes of these sets

formal analysis of piecewise affine systems through formula guided refinement we present a computational framework for identifying a set of initial states from which all trajectories of a piecewise affine pwa system with additive uncertainty satisfy a linear temporal logic ltl formula over a set of linear predicates in its state variables our approach is based on the construction and refinement of finite abstractions of infinite systems we derive conditions guaranteeing the equivalence of an infinite system and its finite abstraction with respect to a specific ltl formula and propose a method for the construction of such formula equivalent abstractions while provably correct the overall method is conservative and expensive a tool for pwa systems implementing the proposed procedure using polyhedral operations and analysis of finite graphs is made available examples illustrating the analysis of pwa models of gene networks are included

control of systems integrating logic dynamics and constraints this paper proposes a framework for modeling and controlling systems described by interdependent physical laws logic rules and operating constraints denoted as mixed logical dynamical mld systems these are described by linear dynamic equations subject to linear inequalities involving real and integer variables mld systems include linear hybrid systems finite state machines some classes of discrete event systems constrained linear systems and nonlinear systems which can be approximated by piecewise linear functions a predictive control scheme is proposed which is able to stabilize mld systems on desired reference trajectories while fulfilling operating constraints and possibly take into account previous qualitative knowledge in the form of heuristic rules due to the presence of integer variables the resulting on line optimization procedures are solved through mixed integer quadratic programming miqp for which efficient solvers have been recently developed some examples and a simulation case study on a complex gas supply system are reported

dynamic programming for constrained optimal control of discrete time linear hybrid systems in this paper we study the solution to optimal control problems for constrained discrete time linear hybrid systems based on quadratic or linear performance criteria the aim of the paper is twofold first we give basic theoretical results on the structure of the optimal state feedback solution and of the value function second we describe how the state feedback optimal control law can be constructed by combining multiparametric programming and dynamic programming

on hybrid systems and closed loop mpc systems the following five classes of hybrid systems were recently proven to be equivalent linear complementarity extended linear complementarity mixed logical dynamical systems piecewise affine systems and max min plus scaling systems some of the equivalences were obtained under additional assumptions such as boundedness of certain system variables in this paper for linear or hybrid plants in closed loop with a model predictive control mpc controller based on a linear model fulfilling linear constraints on input and state variables and utilizing a quadratic cost criterion we provide a simple and direct proof that the closed loop system is a subclass of any of the former five classes of hybrid systems this result is of extreme importance as it opens up the use of tools developed for the mentioned hybrid model classes such as robust stability and safety analysis tools to study closed loop properties of mpc

lyapunov method for boundedness of solutions of nonlinear impulsive functional differential equations sufficient conditions are investigated for boundedness of solutions of nonlinear impulsive functional differential equations with impulses at fixed moments the main tools are lyapunov functions and gronwall type of inequality

equivalence of lcp and pls it is our purpose here to show that two prototype models of complementary pivot and fixed point theory and the corresponding path following solution methods are conceptually equivalent

on the input output representation of piecewise affine state space models this paper addresses the conversion of discrete time piecewise affine pwa state space models into input output form necessary and sufficient conditions for the existence of equivalent input output representations of a given pwa state space model are derived connections to the observability properties of pwa models are investigated under a technical assumption it is shown that every finite time observable pwa model admits an equivalent input output representation when an equivalent input output model exists a constructive procedure is presented to derive its equations several examples illustrate the proposed results

nonlinear regulation the piecewise linear approach this paper approaches nonlinear control problems through the use of discrete time piecewise linear systems these are systems whose next state and output maps are both described by pl maps i e by maps which are affine on each of the components of a finite polyhedral partition various results on state and output feedback observers and inverses standard for linear systems are proved for pl systems many of these results are then used in the study of more general both discrete and continuous time systems using suitable approximations

input to state stability and interconnections of discontinuous dynamical systems in this paper we will extend the input to state stability iss framework to continuous time discontinuous dynamical systems dds adopting piecewise smooth iss lyapunov functions the main motivation for investigating piecewise smooth iss lyapunov functions is the success of piecewise smooth lyapunov functions in the stability analysis of hybrid systems this paper proposes an extension of the well known filippov s solution concept that is appropriate for open systems so as to allow interconnections of dds it is proven that the existence of a piecewise smooth iss lyapunov function for a dds implies iss in addition a small gain iss interconnection theorem is derived for two dds that both admit a piecewise smooth iss lyapunov function this result is constructive in the sense that an explicit iss lyapunov function for the interconnected system is given it is shown how these results can be applied to construct piecewise quadratic iss lyapunov functions for piecewise linear systems including sliding motions via linear matrix inequalities

data based identification and control of nonlinear systems via piecewise affine approximation the piecewise affine pwa model represents an attractive model structure for approximating nonlinear systems in this paper a procedure for obtaining the pwa autoregressive exogenous arx autoregressive systems with exogenous inputs models of nonlinear systems is proposed two key parameters defining a pwarx model namely the parameters of locally affine subsystems and the partition of the regressor space are estimated the former through a least squares based identification method using multiple models and the latter using standard procedures such as neural network classifier or support vector machine classifier having obtained the pwarx model of the nonlinear system a controller is then derived to control the system for reference tracking both simulation and experimental studies show that the proposed algorithm can indeed provide accurate pwa approximation of nonlinear systems and the designed controller provides good tracking performance

identification of piecewise affine systems based on statistical clustering technique this paper is concerned with the identification of a class of piecewise affine systems called a piecewise affine autoregressive exogenous pwarx model the pwarx model is composed of arx sub models each of which corresponds to a polyhedral region of the regression space under the temporary assumption that the number of sub models is known a priori the input output data are collected into several clusters by using a statistical clustering algorithm we utilize support vector classifiers to estimate the boundary hyperplane between two adjacent regions in the regression space in each cluster the parameter vector of the sub model is obtained by the least squares method it turns out that the present statistical clustering approach enables us to estimate the number of sub models based on the information criteria such as caic and mdl the estimate of the number of sub models is performed by applying the identification procedure several times to the same data set after having fixed the number of sub models to different values finally we verify the applicability of the present identification method through a numerical example of a hammerstein model

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 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

convex relaxations for l 2 gain analysis of piecewise affine polynomial systems this paper proposes some sufficient conditions based on the computation of polynomial and piecewise polynomial storage functions for gain analysis of discrete time piecewise affine or piecewise polynomial systems the computation of such storage functions is performed by means of convex optimisation techniques via the sum of squares decomposition of multivariate polynomials

on optimization of stochastic maxmin plus scaling systemsan approximation approach a large class of discrete event and hybrid systems can be described by a maxmin plus scaling mmps model i e a model in which the main operations are maximization minimization addition and scalar multiplication accordingly optimization of mmps systems appears in different problems defined for discrete event and hybrid systems for a stochastic mmps system this optimization problem is computationally highly demanding as often numerical integration has to be used to compute the objective function the aim of this paper is to decrease such computational complexity by applying an approximation method that is based on the moments of a random variable and that can be computed analytically

mpc for continuous piecewise affine systems abstract a large class of hybrid systems can be described by a max min plus scaling mmps model i e using the operations maximization minimization addition and scalar multiplication first we show that continuous piecewise affine systems are equivalent to mmps systems next we consider model predictive control mpc for these systems in general this leads to nonlinear nonconvex optimization problems we present a new mpc method for mmps systems that is based on canonical forms for mmps functions in case the mpc constraints are linear constraints in the inputs only this results in a sequence of linear optimization problems such that the mpc control can often be computed in a much more efficient way than by just applying nonlinear optimization as was done in previous work

a global piecewise smooth newton method for fast large scale model predictive control in this paper the strictly convex quadratic program qp arising in model predictive control mpc for constrained linear systems is reformulated as a system of piecewise affine equations a regularized piecewise smooth newton method with exact line search on a convex differentiable piecewise quadratic merit function is proposed for the solution of the reformulated problem the algorithm has considerable merits when applied to mpc over standard active set or interior point algorithms its performance is tested and compared against state of the art qp solvers on a series of benchmark problems the proposed algorithm is orders of magnitudes faster especially for large scale problems and long horizons for example for the challenging crude distillation unit model of pannocchia rawlings and wright 2007 with 252 states 32 inputs and 90 outputs the average running time of the proposed approach is 1 57 ms

inverse parametric optimization with an application to hybrid system control we present a number of results on inverse parametric optimization and its application to hybrid system control we show that any function that can be written as the difference of two convex functions can also be written as a linear mapping of the solution to a convex parametric optimization problem we exploit these results in application to the control of systems with piecewise affine dynamics and show that it is possible to model such systems as optimizing processes optimal control problems for such systems can be remodeled as bilevel optimization problems and solved with existing techniques