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37
A Survey of Computational Complexity Results in Systems and Control
, 2000
"... The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fi ..."
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Cited by 116 (21 self)
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The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fields. We begin with a brief introduction to models of computation, the concepts of undecidability, polynomial time algorithms, NPcompleteness, and the implications of intractability results. We then survey a number of problems that arise in systems and control theory, some of them classical, some of them related to current research. We discuss them from the point of view of computational complexity and also point out many open problems. In particular, we consider problems related to stability or stabilizability of linear systems with parametric uncertainty, robust control, timevarying linear systems, nonlinear and hybrid systems, and stochastic optimal control.
Observability and Controllability of Piecewise Affine and Hybrid Systems
 IEEE Transactions on Automatic Control
, 1999
"... In this pap e we prove in a constructive way, the ee ale b e we e pie a#ne syste and a broad class of hybridsyste de e d by inte line dynamics, automata, and propositional logic. By focusing our inveon the forme class, we show through countethat obse ability and controllability prope rtie cannot b ..."
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Cited by 93 (14 self)
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In this pap e we prove in a constructive way, the ee ale b e we e pie a#ne syste and a broad class of hybridsyste de e d by inte line dynamics, automata, and propositional logic. By focusing our inveon the forme class, we show through countethat obse ability and controllability prope rtie cannot be e asilydely from those of the comp tline subsyste Inste we propose practical nume te base onmixe te line programming. Keywords Hybrid syste controllability,obse ability, pie line syste pie a#ne syste mixe teline programming I. Introducti In recent yearsb oth control and computer science haveb een attractedb y hybridsystem [1], [2], [23], [25], [26],b ecause they provide a unified framework fordescribgARB( cesses evolving accordingto continuous dynamics, discrete dynamics, and logic rules. The interest is mainly motivatedb y the large variety of practical situations, for instance realtime systems, where physical processes interact with digital controllers. Several modelingformalisms h...
Effective Synthesis of Switching Controllers for Linear Systems
, 2000
"... In this work we suggest a novel methodology for synthesizing switching controllers for continuous and hybrid systems whose dynamics are defined by linear differential equations. We formulate the synthesis problem as finding the conditions upon which a controller should switch the behavior of the sys ..."
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Cited by 78 (8 self)
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In this work we suggest a novel methodology for synthesizing switching controllers for continuous and hybrid systems whose dynamics are defined by linear differential equations. We formulate the synthesis problem as finding the conditions upon which a controller should switch the behavior of the system from one "mode" to another in order to avoid a set of bad states, and propose an abstract algorithm which solves the problem by an iterative computation of reachable states. We have implemented a concrete version of the algorithm, which uses a new approximation scheme for reachability analysis of linear systems.
A Clustering Technique for the Identification of Piecewise Affine Systems
, 2001
"... We propose a new technique for the identification of discretetime hybrid systems in the PieceWise Affine (PWA) form. This problem can be formulated as the reconstruction of a possibly discontinuous PWA map with a multidimensional domain. In order to achieve our goal, we provide an algorithm that ..."
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Cited by 49 (7 self)
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We propose a new technique for the identification of discretetime hybrid systems in the PieceWise Affine (PWA) form. This problem can be formulated as the reconstruction of a possibly discontinuous PWA map with a multidimensional domain. In order to achieve our goal, we provide an algorithm that exploits the combined use of clustering, linear identification, and pattern recognition techniques. This allows to identify both the affine submodels and the polyhedral partition of the domain on which each submodel is valid avoiding gridding procedures. Moreover, the clustering step (used for classifying the datapoints) is performed in a suitably defined feature space which allows also to reconstruct different submodels that share the same coefficients but are defined on different regions. Measures of confidence on the samples are introduced and exploited in order to improve the performance of both the clustering and the final linear regression procedure.
Complexity of Stability and Controllability of Elementary Hybrid Systems
, 1997
"... this paper, weconsider simple classes of nonlinear systems and provethatbasic questions related to their stabilityandcontrollabilityare either undecidable or computationally intractable (NPhard). As a special case, weconsider a class of hybrid systems in which the state space is partitioned into tw ..."
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Cited by 34 (10 self)
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this paper, weconsider simple classes of nonlinear systems and provethatbasic questions related to their stabilityandcontrollabilityare either undecidable or computationally intractable (NPhard). As a special case, weconsider a class of hybrid systems in which the state space is partitioned into two halfspaces, and the dynamics in eachhalfspace correspond to a differentlinear system
Stabilizing Model Predictive Control of Hybrid Systems
"... Abstract—In this note, we investigate the stability of hybrid systems in closedloop with model predictive controllers (MPC). A priori sufficient conditions for Lyapunov asymptotic stability and exponential stability are derived in the terminal cost and constraint set fashion, while allowing for dis ..."
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Cited by 29 (25 self)
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Abstract—In this note, we investigate the stability of hybrid systems in closedloop with model predictive controllers (MPC). A priori sufficient conditions for Lyapunov asymptotic stability and exponential stability are derived in the terminal cost and constraint set fashion, while allowing for discontinuous system dynamics and discontinuous MPC value functions. For constrained piecewise affine (PWA) systems as prediction models, we present novel techniques for computing a terminal cost and a terminal constraint set that satisfy the developed stabilization conditions. For quadratic MPC costs, these conditions translate into a linear matrix inequality while, for MPC costs based on 1,norms, they are obtained as norm inequalities. New ways for calculating low complexity piecewise polyhedral positively invariant sets for PWA systems are also presented. An example illustrates the developed theory. Index Terms—Hybrid systems, Lyapunov stability, model predictive control (MPC), piecewise affine systems. I.
OptimizationBased Verification and Stability Characterization of Piecewise Affine and Hybrid Systems
 In Hybrid Systems: Computation and Control
, 2000
"... In this paper, we formulate the problem of characterizing the stability of a piecewise affin (PWA) system as a verification problem. The basic idea is to take the whole R^n as the set of initial conditions, and check that all the trajectories go to the origin. More precisely, we test for semiglobal ..."
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Cited by 29 (8 self)
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In this paper, we formulate the problem of characterizing the stability of a piecewise affin (PWA) system as a verification problem. The basic idea is to take the whole R^n as the set of initial conditions, and check that all the trajectories go to the origin. More precisely, we test for semiglobal stability by restricting the set of initial conditions to an (arbitrarily large) bounded set X(0), and label as "asymptotically stable in T steps" the trajectories that enter an in variant set around the origin within a finite time T ,or as "unstable in T steps" the trajectories which enter a (very large) set X_inst . Subsets of X (0) leadin ton2W of the two previous cases are labeled as "nv classifiable in T steps". The domain of asymptotical stability in T steps is a subset of the domain of attraction ofan equilibrium poin t, an has the practicalmeanca of collectin inPv)v convW2xvP from which the settlin time of the system is smaller than T . In addition it can be computed algorithmically i...
Optimal controllers for hybrid systems: Stability and piecewise linear explicit form
 in Proceedings of the 39th IEEE Conference on Decision and Control
, 2000
"... In this paper we propose a procedure for synthesizing piecewise linear optimal controllers for hybrid systems and investigate conditions for closedloop stability. Hybrid systems are modeled in discretetime within the mixed logical dynamical (MLD) framework[8], or, equivalently [7], as piecewise af ..."
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Cited by 28 (7 self)
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In this paper we propose a procedure for synthesizing piecewise linear optimal controllers for hybrid systems and investigate conditions for closedloop stability. Hybrid systems are modeled in discretetime within the mixed logical dynamical (MLD) framework[8], or, equivalently [7], as piecewise affine (PWA) systems. A stabilizing controller is obtained by designing a model predictive controller (MPC), which is based on the minimization of a weighted 1/∞norm of the tracking error and the input trajectories over a finite horizon. The control law is obtained by solving a mixedinteger linear program (MILP) which depends on the current state. Although efficient branch and bound algorithms exist to solve MILPs, these are known to be NPhard problems, which may prevent their online solution if the samplingtime is too small for the available computation power. Rather than solving the MILP on line, in this paper we propose a different approach where all the computation is moved off line, by solving a multiparametric MILP (mpMILP). As the resulting control law is piecewise affine, online computation is drastically reduced to a simple linear function evaluation. An example of piecewise linear optimal control of the heat exchange system [16] shows the potential of the method.
Identification of piecewise affine systems via mixedinteger programming
 Automatica
, 2004
"... This paper addresses the problem of identification of hybrid dynamical systems, by focusing the attention on hinging hyperplanes (HHARX) and Wiener piecewise affine (WPWARX) autoregressive exogenous models. In particular, we provide algorithms based on mixedinteger linear or quadratic programming ..."
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Cited by 22 (4 self)
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This paper addresses the problem of identification of hybrid dynamical systems, by focusing the attention on hinging hyperplanes (HHARX) and Wiener piecewise affine (WPWARX) autoregressive exogenous models. In particular, we provide algorithms based on mixedinteger linear or quadratic programming which are guaranteed to converge to a global optimum. For the special case where switches occur only seldom in the estimation data, we also suggest a way of trading off between optimality and complexity by using a change detection approach. 1
Deciding Stability and Mortality of Piecewise Affine Dynamical Systems
, 2001
"... In this paper we studyproblJ: such as: given a discrete timedynamical system of the form x(t +1)=f(x(t)) where f : R n #R n is a piecewise a#ne function, decide whetheral trajectories converge to 0. We show in our main theorem that this AttractivityProblc isundecidabl as soon as n2. The same is ..."
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Cited by 22 (1 self)
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In this paper we studyproblJ: such as: given a discrete timedynamical system of the form x(t +1)=f(x(t)) where f : R n #R n is a piecewise a#ne function, decide whetheral trajectories converge to 0. We show in our main theorem that this AttractivityProblc isundecidabl as soon as n2. The same is true of tworelkMI problI+J Stabil+J (is thedynamical systemglJH #RI asymptotical# stablto andMortal#M (do al trajectories go through 0?). We then show that Attractivity andStabilI: becomedecidabl in dimension 1 for continuous functions. c # 2001El1/JkR Science B.V.Al rights reserved. Keywords: Discretedynamical systems; Piecewise a#ne systems; Piecewiselecew systems; Hybrid systems;Mortal/JM Stabil/JM Decidabilk: 1.IP141 In this paper we studyproblJ+ such as: given a discrete timedynamical system of the form x(t +1)=f(x(t)) where f : R n #R n is a(possibl discontinuous) piecewise # This research waspartl carried outwhil Bllkk was visitingTsitsiklJ at MIT (Cambridge) and Koiran at ENS (Lyon). This research was supported by the ARO under grant DAAL0392G0115, by the NATO under grant CRG961115 and by the European Commission under the TMR(AlMkI;/z network contract ERBFMRXCT960074. # Corresponding author. Email addresses: blmCppCpA/J#JM:/zRkJ; (V.D.BlD./kIH Ol./kIH:J/zRkJ;/lkJ;/l (O. Bournez), pascal),/;MJMI/zRkJ;/ll (P. Koiran), christos@cs.berkel/ll (C.H. Papadimitriou), jnt@mit.edu (J.N. TsitsiklM#/ 03043975/01/$  see front matter c # 2001El1/kRk Science B.V.Al rights reserved. PII: S03043975(00)003996 688 V.D. Blondel et al. / Theoretical Computer Science 255 (2001) 687696 a#ne function, decide whetheral trajectories converge to 0. We show in our main theorem (Theorem 2) that this AttractivityProblc isundecidabl as soon as n2. The same is true of t...