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130
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 187 (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 139 (21 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 108 (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.
Timing Analysis of Asynchronous Circuits using Timed Automata
 in P.E. Camurati, H. Eveking (Eds.), Proc. CHARME'95, LNCS 987
, 1995
"... Abstract. In this paper we present a method for modeling asynchronous digital circuits by timed automata. The constructed timed automata serve as\mechanical " and veri able objects for asynchronous sequential machines in the same sense that (untimed) automata do for synchronous machines. Th ..."
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Cited by 64 (16 self)
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Abstract. In this paper we present a method for modeling asynchronous digital circuits by timed automata. The constructed timed automata serve as\mechanical &quot; and veri able objects for asynchronous sequential machines in the same sense that (untimed) automata do for synchronous machines. These results, combined with recent results concerning the analysis and synthesis of timed automata provide for the systematic treatment of a large class of problems that could be treated by conventional simulation methods only in an adhoc fashion. The problems that can be solved due to the results presented in this paper include: the reachability analysis of a circuit with uncertainties in gate delays and input arrival times, inferring the necessary timing constraints on input signals that guaranteeaproper functioning of a circuit and calculating the delay characteristics of the components required inorder to meet some given behavioral speci cations. Notwithstanding the existence of negative theoretical results concerning the worstcase complexity of timed automata analysis algorithms, initial experimentation with the Kronos tool for timing analysis suggest that timed automata derived from circuits might not be so hard to analyze in practice. 1
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 56 (9 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
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 43 (9 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.
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 40 (9 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...
On the decidability of the reachability problem for planar differential inclusions
 In HSCC’2001, number 2034 in LNCS
, 2001
"... Abstract. In this paper we develop an algorithm for solving the reachability problem of twodimensional piecewise rectangular differential inclusions. Our procedure is not based on the computation of the reachset but rather on the computation of the limit of individual trajectories. A key idea is ..."
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Cited by 39 (15 self)
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Abstract. In this paper we develop an algorithm for solving the reachability problem of twodimensional piecewise rectangular differential inclusions. Our procedure is not based on the computation of the reachset but rather on the computation of the limit of individual trajectories. A key idea is the use of onedimensional affine Poincaré maps for which we can easily compute the fixpoints. As a first step, we show that between any two points linked by an arbitrary trajectory there always exists a trajectory without selfcrossings. Thus, solving the reachability problem requires considering only those. We prove that, indeed, there are only finitely many “qualitative types ” of those trajectories. The last step consists in giving a decision procedure for each of them. These procedures are essentially based on the analysis of the limits of extreme trajectories. We illustrate our algorithm on a simple model of a swimmer spinning around a whirlpool. 1
Efficient computation of reachable sets of linear timeinvariant systems with inputs
 in HSCC’06, vol. 3927 in LNCS
, 2006
"... Abstract. This work is concerned with the problem of computing the set of reachable states for linear timeinvariant systems with bounded inputs. Our main contribution is a novel algorithm which improves significantly the computational complexity of reachability analysis. Algorithms to compute over ..."
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Cited by 37 (7 self)
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Abstract. This work is concerned with the problem of computing the set of reachable states for linear timeinvariant systems with bounded inputs. Our main contribution is a novel algorithm which improves significantly the computational complexity of reachability analysis. Algorithms to compute over and underapproximations of the reachable sets are proposed as well. These algorithms are not subject to the wrapping effect and therefore our approximations are tight. We show that these approximations are useful in the context of hybrid systems verification and control synthesis. The performance of a prototype implementation of the algorithm confirms its qualities and gives hope for scaling up verification technology for continuous and hybrid systems. 1