On The Complexity of Bounded Time Reachability for Piecewise Affine Systems (original) (raw)
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Reachability for piecewise affine systems is known to be undecidable, starting from dimension 2. In this paper we investigate the exact complexity of several decidable variants of reachability and control questions for piecewise affine systems. We show in particular that the region-to-region bounded time versions leads to NP-complete or co-NP-complete problems, starting from dimension 2. We also prove that a bounded precision version leads to P SP ACE-complete problems.
Reachability analysis of continuous-time piecewise affine systems
Automatica, 2005
This paper proposes an algorithm for the characterization of reachable sets of states for continuous-time piecewise affine systems. Given a model of the system and a bounded set of possible initial states, the algorithm employs a linear matrix inequality approach to compute both upper and lower bounds on reachable regions. Rather than performing computations in the state-space, this method uses impact maps to find the reachable sets on the switching surfaces of the system. This tool can then be used to deduce safety and performance results about the system.
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Piecewise affine systems are powerful models for describing both non-linear and hybrid systems. One of the key problems in controlling these systems is the inherent computational complexity of controller synthesis and analysis, especially if constraints on states and inputs are present. This paper illustrates how reachability analysis based on multi-parametric programming may serve to obtain controllers of low complexity. Specifically, two different controller computation schemes are presented. In addition, a method to obtain stability guarantees for general receding horizon control of PWA systems is given.
Reachability Problems in Piecewise FIFO Systems
ACM Transactions on Computational Logic, 2012
Systems consisting of several finite components that communicate via unbounded perfect FIFO channels (i.e., FIFO systems) arise naturally in modeling distributed systems. Despite well-known difficulties in analyzing such systems, they are of significant interest as they can describe a wide range of communication protocols. In this article, we study the problem of computing the set of reachable states of a FIFO system composed of piecewise components. This problem is closely related to calculating the set of all possible channel contents, that is, the limit language, for each control location. We present an algorithm for calculating the limit language of a system with a single communication channel. For multichannel systems, we show that the limit language is piecewise if the initial language is piecewise. Our construction is not effective in general; however, we provide algorithms for calculating the limit language of a restricted class of multichannel systems in which messages are not passed around in cycles through different channels. We show that the worst case complexity of our algorithms for single-channel and important subclasses of multichannel systems is exponential in the size of the initial content of the channels.
A Computatuional Analysis of the Reachability Problem for a Class of Hybrid Dynamical Systems
1996
Hybrid systems possess continuous dynamics de ned within regions of state spaces and discrete transitions among the regions. Many practical control veri cation and synthesis tasks can be reduced to reachability problems for these systems that decide if a particular state-space region is reachable from an initial operating region. In this paper, we present a computational analysis of the face reachability problem for a class of three-dimensional dynamical systems whose state spaces are de ned by piecewise constant vector elds and whose trajectories never return to a state-space region once they exit the region. These systems represent a restricted class of control systems whose dynamics results from a juxtaposition of piecewise parameterized vector elds. We had previously developed a computational algorithm for synthesizing the desired dynamics of a system in phase space by piecing together vector elds geometrically. We demonstrate in this paper that the reachability problem for this class of systems is decidable while the computation is provably intractable (i.e., PSPACE-hard). We prove the intractability via a reduction of satis ability of quanti ed boolean formulas to this reachability problem. This result sheds light on the computational complexity of phase-space based control synthesis methods and extends the work of Asarin, Maler, and Pnueli 2] that proves computational undecidability for three-dimensional constant-derivative systems.
Deciding reachability for planar multi-polynomial systems
Lecture Notes in Computer Science, 1996
In this paper we i n vestigate the decidability of the reachability problem for planar non-linear hybrid systems. A planar hybrid system has the property that its state space corresponds to the standard Euclidean plane, which is partitioned into a nite number of (polyhedral) regions. To each of these regions is assigned some vector eld which g o verns the dynamical behaviour of the system within this region. We p r o ve the decidability of point to point and region to region reachability problems for planar hybrid systems for the case when trajectories within the regions can be described by polynomials of arbitrary degree.
A Computational Analysis of the Reachability Problem for a Class of Hybrid Dynamical Systems
1997
Hybrid systems possess continuous dynamics defined within regions of state spaces and discrete transitions among the regions. Many practical control verification and synthesis tasks can be reduced to reach ability problems for these systems that decide if a particular state-space region is reachable from an initial operating region. In this paper, we present a computational analysis of the face reachability problem for a class of three-dimensional dynamical systems whose state spaces are defined by piecewise constant vector fields and whose trajectories never return to a state-space region once they exit the region. These systems represent a restricted class of control systems whose dynamics results from a juxtaposition of piecewise parameterized vector fields. We had previously developed a computational algorithm for synthesizing the desired dynamics of a system in phase space by piecing together vector fields geometrically. We demonstrate in this paper that the reachability problem for this class of systems is decidable while the computation is provably intractable (i.e., PSPACE-hard). We prove the intractability via a reduction of satisfiability of quantified boolean formulas to this reachability problem. This result sheds light on the computational complexity of phase-space based control synthesis methods and extends the work of Asarin, Maler, and Pnueli [2] that proves computational undecidability for three-dimensional constant-derivative systems.
On the geometric properties of reachable and controllable sets for linear discrete systems
Journal of optimization theory and applications, 2004
A linear time-invariant discrete system with a single bounded control input is considered. Various types of constraints imposed on the control input are investigated. The set of system states that can be reached from the origin (reachable set) is studied. The set of system states from where the origin can be reached (nullcontrollable set) is studied as well. Various examples are presented.
IFAC Proceedings Volumes, 2004
This paper is concerned with analysis of the reachable sets of piecewise affine systems with polytopic uncertainties and energy-bounded disturbances. We derive a linear matrix inequality condition which explicitly provides an outer approximation of the reachable set in terms of a piecewise quadratic form. Moreover, we apply the analysis result to analysis of Takagi-Sugeno fuzzy systems.