symbolic model checking for real time systems abstract we describe finite state programs over real numbered time in a guarded command language with real valued clocks or equivalently as finite automata with real valued clocks model checking answers the question which states of a real time program satisfy a branching time specification given in an extension of ctl with clock variables we develop an algorithm that computes this set of states symbolically as a fixpoint of a functional on state predicates without constructing the state space for this purpose we introduce a calculus on computation trees over real numbered time unfortunately many standard program properties such as response for all nonzeno execution sequences during which time diverges cannot be characterized by fixpoints we show that the expressiveness of the timed calculus is incomparable to the expressiveness of timed ctl fortunately this result does not impair the symbolic verification of implementable real time programs those whose safety constraints are machine closed with respect to diverging time and whose fairness constraints are restricted to finite upper bounds on clock values all timed ctl properties of such programs are shown to be computable as finitely approximable fixpoints in a simple decidable theory

reach controllability of single input affine systems on a simplex we study the reach control problem rcp for a single input affine system with a simplicial state space we extend previous results by exploring arbitrary triangulations of the state space particularly allowing the set of possible equilibria to intersect the interior of simplices in the studied setting it is shown that closed loop equilibria nevertheless only arise on the boundary of simplices this allows to define a notion of reach controllability which quantifies the effect of the control input on boundary equilibria using reach controllability we obtain necessary and sufficient conditions for solvability of rcp by affine feedback

blaschke oscillatory equations of the form f a z f 0 abstract we study the zero sequences of the non trivial solutions of f a z f 0 where a z is analytic in the unit disc we offer several aspects illustrating the fact that it is not so uncommon for these zero sequences to be blaschke sequences the typical results can be divided into two categories 1 we search for conditions on a z under which the zero sequences of solutions of are blaschke sequences 2 for given blaschke sequences satisfying certain conditions we construct an analytic function a z of minimal growth such that these blaschke sequences are the zero sequences of certain solutions of this discussion is a continuation of the recent paper j heittokangas solutions of f a z f 0 in the unit disc having blaschke sequences as the zeros comput methods funct theory 5 2005 49 63

discrete abstractions of nonlinear systems based on error propagation analysis this paper proposes a computational method for the feasibility check and design of discrete abstract models of nonlinear dynamical systems first it is shown that a given discrete time dynamical system can be transformed into a finite automaton by embedding a quantizer into its state equation under this setting a sufficient condition for approximate bisimulation in infinite steps of time between the concrete model and its discrete abstract model is derived the condition takes the form of a set of linear inequalities and thus can be checked efficiently by a linear programming solver finally the iterative refinement algorithm which generates a discrete abstract model under a given error specification is proposed the algorithm is guaranteed to terminate in finite iterations

finite approximations of switched homogeneous systems for controller synthesis in this note we demonstrate the use of a control oriented notion of finite state input output approximation to synthesize correct by design controllers for hybrid plants under sensor limitations specifically we consider the problem of designing stabilizing switching controllers for a pair of unstable homogeneous second order systems with binary output feedback in addition to yielding a deterministic finite state approximate model of the hybrid plant our approach allows one to efficiently establish a useable upper bound on the quality of approximation and leads to a discrete optimization problem whose solution immediately provides a certified finite state controller for the plant the resulting controller consists of a deterministic finite state observer and a corresponding full state feedback control law

stability analysis for switched boolean networks under arbitrary switching signals this note investigates the stability of switched boolean networks sbns under arbitrary switching signal by using the tool of semi tensor product of matrices first an algebraic expression is given for the sbn via the semi tensor product based on it a necessary and sufficient condition is presented for the switching point reachability of the system second using the switching point reachability the pointwise stabilizability is studied and a necessary and sufficient condition is obtained finally the global stability of sbns under arbitrary switching signal is investigated and an easily verifiable necessary and sufficient condition is presented two illustrative examples are presented to show that the main results are effective in analyzing the stability of sbns

optimization of lyapunov invariants in verification of software systems the paper proposes a control theoretic framework for verification of numerical software systems and puts forward software verification as an important application of control and systems theory the idea is to transfer lyapunov functions and the associated computational techniques from control systems analysis and convex optimization to verification of various software safety and performance specifications these include but are not limited to absence of overflow absence of division by zero termination in finite time absence of dead code and certain user specified assertions central to this framework are lyapunov invariants these are properly constructed functions of the program variables and satisfy certain properties analogous to those of lyapunov functions along the execution trace the search for the invariants can be formulated as a convex optimization problem if the associated optimization problem is feasible the result is a certificate for the specification

positive invariance of constrained affine dynamics and its applications to hybrid systems and safety verification motivated by long time dynamic analysis of hybrid systems and safety verification problems this paper addresses fundamental positive invariance issues of an affine dynamical system on a general polyhedron and their applications necessary and sufficient algebraic conditions are established for the existence of a positively invariant set of an affine system on a polyhedron using the tools of lexicographic relation and long time oscillatory dynamic analysis a linear program based algorithm is proposed to verify these conditions and its computational complexity is analyzed the positive invariance results are applied to obtain an explicit characterization of global switching behaviors of piecewise affine systems further the positive invariance techniques developed in this paper are exploited to show the decidability of safety verification of a class of affine dynamics on semialgebraic sets

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 necessity of the invariance conditions for reach control on polytopes abstract we study the reach control problem rcp to make the solutions of an affine system defined on a polytopic state space reach and exit a prescribed facet of the polytope in finite time without first leaving the polytope so called invariance conditions are used to prevent solutions from leaving the polytope through facets which are not designated as the exit facet these conditions are known to be necessary for solvability of the rcp on polytopes by continuous state feedback we study whether the invariance conditions are also necessary for solvability of the rcp on polytopes by open loop controls we show by way of a counterexample that surprisingly the answer is negative we identify a suitable class of polytopes for which the invariance conditions remain necessary conditions

ltl receding horizon control for finite deterministic systems this paper considers receding horizon control of finite deterministic systems which must satisfy a high level rich specification expressed as a linear temporal logic formula under the assumption that time varying rewards are associated with states of the system and these rewards can be observed in real time the control objective is to maximize the collected reward while satisfying the high level task specification in order to properly react to the changing rewards a controller synthesis framework inspired by model predictive control is proposed where the rewards are locally optimized at each time step over a finite horizon and the optimal control computed for the current time step is applied by enforcing appropriate constraints the infinite trajectory produced by the controller is guaranteed to satisfy the desired temporal logic formula simulation results demonstrate the effectiveness of the approach

finite order solutions of linear differential equations in the unit disc abstract necessary and sufficient conditions for the analytic coefficients of the complex linear differential equation f k a k 1 z f k 1 a 1 z f a 0 z f a k z are found such that all solutions satisfy f lim sup r 1 log t r f log 1 r moreover estimates for the number of linearly independent solutions of maximal growth are found in terms of the growth of the coefficients in addition sufficient conditions for the coefficients such that the zero sequence z n of any non trivial solution f of with a k 0 satisfies 1 z n 1 are found several non trivial examples are given in order to show that the established results are in a sense sharp

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

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

improved algorithm for mixed integer quadratic programs and a computational study in a recent paper 6 we suggested an algorithm for solving complicated mixed integer quadratic programs based on an equivalent formulation that employs a nonsingular transformation of variables the objectives of the present paper are two first to present an improved version of this algorithm which reduces substantially its computational requirements second to report on the results of a computational study with the revised algorithm

monotonic reach control on polytopes the technical note studies the problem of making the trajectories of an affine system defined on a polytopic state space reach a prescribed facet of the polytope in finite time without first leaving the polytope the focus is on solvability by continuous piecewise affine feedback and we formulate a variant of the problem in which trajectories exit in a monotonic sense this allows to obtain necessary and sufficient conditions for solvability in certain geometric situations

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

a control problem for affine dynamical systems on a full dimensional polytope given an affine system on a full dimensional polytope the problem of reaching a particular facet of the polytope using continuous piecewise affine state feedback is studied necessary conditions and sufficient conditions for the existence of a solution are derived in terms of linear inequalities on the input vectors at the vertices of the polytope special attention is paid to affine systems on full dimensional simplices in this case the necessary and sufficient conditions are equivalent and a constructive procedure yields an affine feedback control law that solves the reachability problem under consideration

numerical experience with lower bounds for miqp branch and bound the solution of convex mixed integer quadratic programming miqp problems with a general branch and bound framework is considered it is shown how lower bounds can be computed efficiently during the branch and bound process improved lower bounds such as the ones derived in this paper can reduce the number of quadratic programming qp problems that have to be solved the branch and bound approach is also shown to be superior to other approaches in solving miqp problems numerical experience is presented which supports these conclusions

symbolic control of stochastic systems via approximately bisimilar finite abstractions symbolic approaches for control design construct finite state abstract models that are related to the original systems then use techniques from finite state synthesis to compute controllers satisfying specifications given in a temporal logic and finally translate the synthesized schemes back as controllers for the original systems such approaches have been successfully developed and implemented for the synthesis of controllers over non probabilistic control systems in this paper we extend the technique to probabilistic control systems modelled by controlled stochastic differential equations we show that for every stochastic control system satisfying a probabilistic variant of incremental input to state stability and for every given precision e 0 a finite state transition system can be constructed which is e approximately bisimilar to the original stochastic control system moreover we provide results relating stochastic control systems to their corresponding finite state transition systems in terms of probabilistic bisimulation relations known in the literature we demonstrate the effectiveness of the construction by synthesizing controllers for stochastic control systems over rich specifications expressed in linear temporal logic our technique enables automated correct by construction controller synthesis for stochastic control systems which are common mathematical models employed in many safety critical systems subject to structured uncertainty

brief paper tracking and synchronisation for a class of pwa systems in this paper the tracking problem for a class of discontinuous piecewise affine pwa systems is addressed we propose an observer based output feedback control design consisting of a feedforward a piecewise affine feedback law and a model based observer solving the tracking problem these synthesis results can also be employed to tackle the master slave synchronisation problem for pwa systems it is shown that for certain classes of pwa systems the design is characterised in terms of linear matrix inequalities the results are illustrated by application to mechanical systems with discontinuous friction characteristics

safety controller synthesis using human generated trajectories this paper focuses on the task of safety controller synthesis that is designing a controller that will take a system from any point within a compact set of initial states to a point inside a set of acceptable goal states while never entering any state that is deemed unsafe to do this we use a human generated trajectory based approach we introduce the control autobisimulation function which is the analog of the control lyapunov function for approximate bisimulation we consider a class of hybrid systems and use this function to determine a set of admissible feedback control laws that guarantee trajectory robustness for underlying dynamics that are linear affine feedback linearizable and differentially flat this property ensures that any trajectory of the closed loop system that is initialized within some neighborhood of a nominal trajectory will stay within some tube of the nominal trajectory when given the same input this feedback control and input can be used as the controller for some subset of the initial states we demonstrate how to combine multiple trajectories into a synthesized controller that satisfies the safety problem

symbolic models for networks of control systems in this note we propose symbolic models for networks of discrete time nonlinear control systems if each subsystem composing the network admits an incremental input to state stable lyapunov function and if some small gain theorem type conditions are satisfied a network of symbolic models each one associated with each subsystem composing the network is proposed and shown to be approximately bisimilar to the original network with any desired accuracy

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

on algorithmic analysis of transcriptional regulation by ltl model checking studies of cells in silico can greatly reduce the need for expensive and prolonged laboratory experimentation the use of model checking for the analysis of biological networks has attracted much attention recently the practical limitations are still the size of the model and the time needed to generate the state space this paper is focused on the model checking approach for analysis of piecewise linear deterministic models of genetic regulatory networks firstly the qualitative simulation algorithm of de jong et al that builds the heart of genetic network analyzer gna is revisited and its time complexity is studied in detail secondly a novel algorithm that reduces the state space generation time is introduced the new algorithm is developed as an abstraction of the original gna algorithm finally a fragment of linear time temporal logic for which the provided abstraction is conservative is identified efficiency of the new algorithm when implemented in the parallel model checking environment is demonstrated on a set of experiments performed on randomly modified biological models in general the achieved results bring a new insight into the field of qualitative simulation emerging in the context of systems biology

the algorithmic analysis of hybrid systems we present a general framework for the formal specification and algorithmic analysis of hybrid systems a hybrid system consists of a discrete program with an analog environment we model hybrid systems as finite automata equipped with variables that evolve continuously with time according to dynamical laws for verification purposes we restrict ourselves to linear hybrid systems where all variables follow piecewise linear trajectories we provide decidability and undecidability results for classes of linear hybrid systems and we show that standard program analysis techniques can be adapted to linear hybrid systems in particular we consider symbolic model checking and minimization procedures that are based on the reachability analysis of an infinite state space the procedures iteratively compute state sets that are definable as unions of convex polyhedra in multidimensional real space we also present approximation techniques for dealing with systems for which the iterative procedures do not converge

symbolic models for nonlinear control systems without stability assumptions finite state models of control systems were proposed by several researchers as a convenient mechanism to synthesize controllers enforcing complex specifications most techniques for the construction of such symbolic models have two main drawbacks either they can only be applied to restrictive classes of systems or they require the exact computation of reachable sets in this paper we propose a new abstraction technique that is applicable to any nonlinear sampled data control system as long as we are only interested in its behavior in a compact set moreover the exact computation of reachable sets is not required the effectiveness of the proposed results is illustrated by synthesizing a controller to steer a vehicle

stabilizing polynomial approximation of explicit mpc a given explicit piecewise affine representation of an mpc feedback law is approximated by a single polynomial computed using linear programming this polynomial state feedback control law guarantees closed loop stability and constraint satisfaction the polynomial feedback can be implemented in real time even on very simple devices with severe limitations on memory storage

realization theory for linear hybrid systems the paper develops realization theory for linear hybrid systems i e hybrid systems in continuous time without guards whose continuous dynamics is determined by linear control systems and whose discrete dynamics is determined by a finite state automaton we will formulate necessary and sufficient conditions for the existence of a realization we will show that minimality is equivalent to observability and span reachability and that minimal systems are isomorphic in turn observability and span reachability can be characterized by rank conditions

brief paper reachability and stabilization of discrete time affine systems with disturbances this paper studies the fundamental problems whether an affine system affected by additive disturbances is robustly transferable from a source set simplex to a target set polytope and whether it is robustly stabilizable with its state constrained in a simplex first a necessary and sufficient condition is derived for the existence of affine feedback control that solves the robust reachability problem further investigation is provided for two situations relying on whether the union of the source set and the target set is convex or non convex for the former one a necessary and sufficient condition is obtained in the form of linear inequalities while for the latter several computationally feasible sufficient conditions are found second we show that robust stabilization subject to a state constraint is equivalent to find a feasible solution to a linear equation once it is known that either of the problems has a solution by checking the derived conditions design of control laws is then straightforward

brief paper on a reachability problem for affine hypersurface systems on polytopes this paper studies the problem for an affine hypersurface system with n 1 inputs to reach a polytopic target set starting from inside a polytope in the state space we present a solution which begins with a characterization of solvability by open loop control and concludes with a procedure to synthesize a feedback control our emphasis is on methods of subdivision triangulation and covers which explicitly account for the capabilities of the control system in contrast with the previous literature the partition methods are guaranteed to yield a correct feedback synthesis assuming the problem is solvable by open loop control

symbolic models for nonlinear control systems affected by disturbances in the last few years there has been a growing interest in the use of symbolic models for the formal verification and control design of purely continuous or hybrid systems symbolic models are abstract descriptions of continuous systems where one symbol corresponds to an aggregate of continuous states in this article we face the problem of deriving symbolic models for nonlinear control systems affected by disturbances the main contribution of this article is in proposing symbolic models that can be effectively constructed and that approximate nonlinear control systems affected by disturbances in the sense of alternating approximate bisimulation

synthesis of reactive switching protocols from temporal logic specifications we propose formal means for synthesizing switching protocols that determine the sequence in which the modes of a switched system are activated to satisfy certain high level specifications in linear temporal logic ltl the synthesized protocols are robust against exogenous disturbances on the continuous dynamics and can react to possibly adversarial events both external and internal finite state approximations that abstract the behavior of the underlying continuous dynamics are defined using finite transition systems such approximations allow us to transform the continuous switching synthesis problem into a discrete synthesis problem in the form of a two player game between the system and the environment where the winning conditions represent the high level temporal logic specifications restricting to an expressive subclass of ltl formulas these temporal logic games are amenable to solutions with polynomial time complexity by construction existence of a discrete switching strategy for the discrete synthesis problem guarantees the existence of a switching protocol that can be implemented at the continuous level to ensure the correctness of the nonlinear switched system and to react to the environment at run time

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

a topological obstruction in a control problem abstract one of the important discoveries in control theory is a topological obstruction to continuous feedback stabilization for general nonlinear control systems in this note we describe another topological obstruction arising from a very different control problem called the reach control problem motivated by a classical topological obstruction for extending continuous maps on spheres we introduce the problem of extending continuous maps on simplices it is shown that the same condition as in the sphere case gives rise to the obstruction

reach control problem for linear differential inclusion systems on simplices this technical note concerns the reach control problem for the dynamical systems represented by linear differential inclusions the goal is to derive conditions for the trajectories of a differential inclusion defined on a full dimensional simplex to reach exit facets in finite time using affine feedback to achieve this goal an invariance condition is firstly derived for lipschitz differential inclusions as an application of the obtained invariance condition the reach control problem rcp is solved for two classes of differential inclusions norm bounded linear differential inclusion and polytypic linear differential inclusion in the sense of strong and weak reachability respectively

propositional logic in control and monitoring problems by using linear constraints on integer variables logical inference can be incorporated within model predictive control and model based detection problems resulting in mixed integer quadratic or mixed integer linear programming problems qualitative knowledge can be used to prioritize control objectives or to improve performance of detection schemes this article demonstrates some capabilities of combining logic using integer variables with quantitative models

realization theory of discrete time linear switched systems the paper presents realization theory of discrete time linear switched systems we present necessary and sufficient conditions for an input output map to admit a discrete time linear switched system realization in addition we present a characterization of minimality of discrete time linear switched systems in terms of reachability and observability further we prove that minimal realizations are unique up to isomorphism we also discuss algorithms for converting a linear switched system to a minimal one and for constructing a state space representation from input output data the paper uses the theory of rational formal power series in non commutative variables

directed sets and cofinal types we show that 1 s 1 x x xl and l are the only cofinal types of directed sets of size sl but that there exist many cofinal types of directed sets of size continuum a partially ordered set d is directed if every two elements of d have an upper bound in d in this note we consider some basic problems concerning directed sets which have their origin in the theory of moore smith convergence in topology 12 3 19 9 one such problem is to determine all essential kind of directed sets needed for defining the closure operator in a given class of spaces 3 p 47 concerning this problem the following important notion was introduced by j tukey 19 two directed sets d and e are cofinally similar if there is a partially ordered set c in which both can be embedded as cofinal subsets he showed that this is an equivalence relation and that d and e are cofinally similar iff there is a convergent map from d into e and also a convergent map from e into d the equivalence classes of this relation are called cofinal types this concept has been extensively studied since then by various authors 4 13 7 8 already from the first introduction of this concept it has been known that 1 1 x x 1 and x1 represent different cofinal types of directed sets of size btl but no more than five such types were known the main result of this paper shows that 1 1 x x 1 and x1 are the only cofinal types of convergence in spaces of character 81 which can be constructed without additional set theoretic assumptions on the other hand we shall construct many different cofinal types of directed sets of size continuum this gives a solution to problem 1 of j isbell 7 the paper also contains several results about the structure of the class of all cofinal types as well as a result about decomposing arbitrary partially ordered sets into directed sets the results of this note were proved in february march 1982 and presented to the asl in january 1983 1 a decomposition theorem in this section we show that an arbitrary partially ordered set can be decomposed into a number of its directed subsets depending on the sizes of its antichains this result is connected with an unpublished problem of f galvin concerning the dilworth decomposition theorem 6 and it generalizes a similar result of e milner and k prikry 11 the transitivity condition of a partial ordering is not used in our proof so we state our result so as to apply to an arbitrary received by the editors december 3 1984 1980 mathpmaties subjeet classification primary 03e05 03e35 secondary 06a10 18b35 54a15 c1985 american mathematical society 0002 9947 85 1 00 25 per page

synthesis for constrained nonlinear systems using hybridization and robust controllers on simplices in this technical note we propose an approach to controller synthesis for a class of constrained nonlinear systems it is based on the use of a hybridization that is a hybrid abstraction of the nonlinear dynamics this abstraction is defined on a triangulation of the state space where on each simplex of the triangulation the nonlinear dynamics is conservatively approximated by an affine system subject to disturbances except for the disturbances this hybridization can be seen as a piecewise affine hybrid system on simplices for which appealing control synthesis techniques have been developed in the past decade we extend these techniques to handle systems subject to disturbances by synthesizing and coordinating local robust affine controllers defined on the simplices of the triangulation we show that the resulting hybrid controller can be used to control successfully the original constrained nonlinear system our approach though conservative can be fully automated and is computationally tractable to show its effectiveness in practical applications we apply our method to control a pendulum mounted on a cart

span reachability and observability of bilinear hybrid systems the paper presents an algebraic characterization of observability and span reachability of bilinear hybrid systems without guards i e hybrid systems whose continuous dynamics are determined by bilinear control systems and whose discrete transitions are triggered externally the proposed characterization provides necessary and sufficient conditions for observability and span reachability in terms of rank conditions for generalized observability and controllability matrices this characterization is useful for the system theory of bilinear hybrid systems in particular it is used in the realization theory of bilinear hybrid systems petreczky 2006

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

on reachability and controllability of switched boolean control networks this paper investigates the reachability and controllability of switched boolean control networks bcns by using the semi tensor product method first a kind of switching input state transfer graph is defined based on which and the algebraic expression of switched bcns we thus propose a new kind of switching input state incidence matrix that contains the complete information of the switching input state transfer graph second by the switching input state incidence matrix several necessary and sufficient conditions are obtained for the reachability and controllability of switched bcns finally an algorithm is established to find a switching sequence and a control scheme to realize the reachability of switched bcns in the shortest time the study of an illustrative example shows that the new results obtained in this paper are very effective in investigating the reachability and controllability of switched bcns

time varying affine feedback for reach control on simplices abstract we study the reach control problem for affine systems on simplices and the focus is on cases when it is known that the problem is not solvable by continuous state feedback in previous work we used the reach control indices to construct a discontinuous piecewise affine control that solves the problem here we investigate the extent to which time varying feedbacks can be used a simple time varying affine feedback is proposed and shown to solve the problem

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

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

linear differential equations with coefficients in weighted bergman and hardy spaces complex linear differential equations of the form with coefficients in weighted bergman or hardy spaces are studied it is shown for example that if the coefficient a j z of belongs to the weighted bergman space a 1 k j where a 0 for all j 0 k 1 then all solutions are of order of growth at most a measured according to the nevanlinna characteristic in the case when a 0 all solutions are shown to be not only of order of growth zero but of bounded characteristic conversely if all solutions are of order of growth at most a 0 then the coefficient aj z is shown to belong to a pj for all p j 0 1 k j and j 0 k 1 analogous results when the coefficients belong to certain weighted hardy spaces are obtained the non homogeneous equation associated to is also briefly discussed

an obstruction to solvability of the reach control problem using affine feedback this paper studies the reach control problem rcp using affine feedback on simplices the contributions of this paper are threefold first we identify a new obstruction to solvability of the rcp using affine feedback and provide necessary and sufficient conditions for occurrence of such an obstruction second for two input systems these conditions are formulated in terms of scalar linear inequalities third computationally efficient necessary conditions are proposed for checking the obstruction for multi input systems as feasibility programs in terms of linear inequalities in contrast to the previous work in the literature no assumption is imposed on the set of possible equilibria so the results are applicable to the general rcp

reachability and control synthesis for piecewise affine hybrid systems on simplices in this paper we consider the synthesis of control laws for piecewise affine hybrid systems on simplices the construction is based on the solution to the control to facet problem at the continuous level and on dynamic programming at the discrete level the construction is given as an explicit algorithm using only linear algebra and reach set computations for automata no numerical integration is required the method is conservative in that it may fail to find a control law where one exists but one cannot hope for a sharp algorithm for control synthesis since reachability for piecewise affine hybrid systems is undecidable

the growth oscillation and fixed points of solutions of complex linear differential equations in the unit disc abstract we consider the complex differential equations of the form a k z f k a k 1 z f k 1 a 1 z f a 0 z f f z where a 0 0 a 1 a k and f are analytic functions in the unit disc d z c z 1 some results on the finite iterated order and the finite iterated convergence exponent of zero points in d of meromorphic analytic solutions are obtained the fixed points of solutions of differential equations are also investigated in this paper

control to facet by piecewise affine output feedback the control to facet problem plays an important role in the design of feedback controllers for piecewise affine hybrid systems on polytopes in the literature necessary conditions and sufficient conditions for solvability by static state feedback exist in this paper we extend these results to the case of continuous piecewise affine static output feedback for the construction of a controller a triangulation of the output polytope is made which satisfies additional conditions to guarantee compatibility with the induced subdivision of the state polytope in the state feedback case the use of this special type of triangulation is not required

lyapunov stability of a class of discrete event systems discrete event systems des are dynamical systems which evolve in time by the occurrence of events at possibly irregular time intervals logical des are a class of discrete time des with equations of motion that are most often nonlinear and discontinuous with respect to event occurrences recently there has been much interest in studying the stability properties of logical des and several definitions for stability and methods for stability analysis have been proposed here the authors introduce a logical des model and define stability in the sense of lyapunov and asymptotic stability for logical des then the authors show that more conventional analysis of stability which employs appropriate lyapunov functions can be used for logical des the authors provide a general characterization of the stability properties of automata theoretic des models petri nets and finite state systems furthermore the lyapunov stability analysis approach is illustrated on a manufacturing system that processes batches of n different types of parts according to a priority scheme to prove properties related to the machine s ability to reorient itself to achieve safe operation and a load balancing problem in computer networks to study the ability of the system to achieve a balanced load to minimize underutilization

flow functions control flow functions and the reach control problem the paper studies the reach control problem rcp to make trajectories of an affine system defined on a polytopic state space reach and exit a prescribed facet of the polytope in finite time without first leaving the polytope we introduce the notion of a flow function which provides the analog of a lyapunov function for the equilibrium stability problem a flow function comprises a scalar function that decreases along closed loop trajectories and its existence is a necessary and sufficient condition for closed loop trajectories to exit the polytope it provides an analysis tool for determining if a specific instance of rcp is solved without the need for calculating the state trajectories of the closed loop system results include a variant of the lasalle principle tailored to rcp an open problem is to identify suitable classes of flow functions we explore functions of the form v x max v i x and we give evidence that these functions arise naturally when rcp is solved using continuous piecewise affine feedbacks next we introduce the notion of a control flow function it is shown that the artstein sontag theorem of control lyapunov functions has direct analogies to rcp via control flow functions

temporal logic analysis of gene networks under parameter uncertainty the lack of precise numerical information for the values of biological parameters severely limits the development and analysis of models of genetic regulatory networks to deal with this problem we propose a method for the analysis of genetic regulatory networks under parameter uncertainty we consider models based on piecewise multiaffine differential equations dynamical properties expressed in temporal logic and intervals for the values of uncertain parameters the problem is then either to guarantee that the system satisfies the expected properties for every possible parameter value the corresponding parameter set is then called valid or to find valid subsets of a given parameter set the proposed method uses discrete abstractions and model checking and allows for efficient search of the parameter space however the abstraction process creates spurious behaviors in the abstract systems along which time does not progress consequently the verification of liveness properties expressing that something will eventually happen and implicitly assuming progress of time often fails a solution to this second problem is proposed using the notion of transient regions this approach has been implemented in a tool for robust verification of gene networks and applied to the tuning of a synthetic network built in e coli