On The Complexity of Bounded Time Reachability for Piecewise Affine Systems (original) (raw)
On the complexity of bounded time and precision reachability for piecewise affine systems
Theoretical Computer Science, 2016
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.
Low complexity control of piecewise affine systems with stability guarantee
2004
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 Analysis of Multi-affine Systems
Springer eBooks, 2006
We present a computationally attractive technique to study the reachability of rectangular regions by trajectories of continuous multi-affine systems. The method is iterative. At each step, finer partitions and finite quotients that over-approximate the reachability properties of the initial system are produced. We exploit some convexity properties of multi-affine functions on rectangles to show that the construction of the quotient at each step requires only the evaluation of the vector field at the set of all vertices of all rectangles in the partition and finding the roots of a finite set of scalar affine functions. This methodology can be used for formal analysis of biochemical networks, aircraft and underwater vehicles, where multi-affine models are widely used.
Formal Analysis of Discrete-Time Piecewise Affine Systems
IEEE Transactions on Automatic Control, 2010
In this technical note, we study temporal logic properties of trajectories of discrete-time piecewise affine (PWA) systems. Specifically, given a PWA system and a linear temporal logic formula over regions in its state space, we attempt to find the largest region of initial states from which all trajectories of the system satisfy the formula. Our method is based on the iterative computation and model checking of finite quotients. We illustrate our method by analyzing PWA models of two synthetic gene networks.
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.
Temporal Logic Control of Discrete-Time Piecewise Affine Systems
IEEE Transactions on Automatic Control, 2012
We consider the problem of controlling a discretetime piecewise affine (PWA) system from a specification given as a Linear Temporal Logic (LTL) formula over linear predicates in its state variables. We present a computational framework for finding initial states and feedback control strategies guaranteeing the satisfaction of such a specification by all the trajectories of the closed loop system. Our solution is based on abstracting the system to a finite transition system and on controlling the abstraction from an LTL specification.
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.
Stabilizing low complexity feedback control of constrained piecewise affine systems
Automatica, 2005
Piecewise affine (PWA) 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. In addition, few results are available which address the issue of computing stabilizing controllers for PWA systems without placing constraints on the location of the origin. This paper first introduces a method to obtain stability guarantees for receding horizon control of discrete-time PWA systems. Based on this result, two algorithms which provide low complexity state feedback controllers are introduced. Specifically, we demonstrate how multi-parametric programming can be used to obtain minimum-time controllers, i.e., controllers which drive the state into a pre-specified target set in minimum time. In a second segment, we show how controllers of even lower complexity can be obtained by separately dealing with constraint satisfaction and stability properties. To this end, we introduce a method to compute PWA Lyapunov functions for discrete-time PWA systems via linear programming. Finally, we report results of an extensive case study which justify our claims of complexity reduction.
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.
… and Control, 2004. …, 2004
Piecewise affine (PWA) systems are useful models for describing non-linear and hybrid systems. One of the key problems in designing controllers for these systems is the inherent computational complexity of controller synthesis and analysis. These problems are amplified in the presence of state and input constraints and additive but bounded disturbances. In this paper, we exploit set invariance and parametric programming to devise an efficient robust time optimal control scheme. Specifically, the state is driven into the ...
A symbolic approach to controlling piecewise affine systems
49th IEEE Conference on Decision and Control (CDC), 2010
We present a computational framework for automatic synthesis of a feedback control strategy for a piecewise affine (PWA) system from a specification given as a Linear Temporal Logic (LTL) formula over an arbitrary set of linear predicates in its state variables. First, by defining partitions for its state and input spaces, we construct a finite abstraction of the PWA system in the form of a control transition system. Second, we develop an algorithm to generate a control strategy for the finite abstraction. While provably correct and robust to small perturbations in both state measurements and applied inputs, the overall procedure is conservative and expensive. The proposed algorithms have been implemented and are available for download. Illustrative examples are included.
Parameter Synthesis for Piecewise Affine Systems from Temporal Logic Specifications
Lecture Notes in Computer Science
In this paper, we consider discrete-time continuous-space Piecewise Affine (PWA) systems with parameter uncertainties, and study temporal logic properties of their trajectories. Specifically, given a PWA system with polytopal parameter uncertainties, and a Linear Temporal Logic (LTL) formula over linear predicates in the states of the system, we attempt to find subsets of parameters guaranteeing the satisfaction of the formula by all trajectories of the system. We illustrate our method by applying it to a PWA model of a two-gene network.
Algorithms for Computing Reachable Sets and Control Sets
IFAC Proceedings Volumes, 2001
Recently, considerable progress has been made in the numerical computation of reachable sets and control sets. It is the purpose of this paper to survey some of these developments. In general, computation of reachable sets and control sets is a very difficult problem, since the objects which are to be computed have full dimension in the state space. In turns out that certain reformulated problems which in many cases give the desired objects are numerically easier to handle.
Formal analysis of piecewise affine systems through formula-guided refinement
Automatica, 2013
We present a computational framework for identifying a set of initial states from which all trajectories of a piecewise affine (PWA) system 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 (i.e. systems where states can take infinitely many values). We derive conditions guaranteeing the equivalence of an infinite system and its finite abstraction with respect to a specific temporal logic formula and propose methods aimed at the construction of such formula-equivalent abstractions. We show that the proposed procedure can be implemented using polyhedral operations and analysis of finite graphs. While provably correct, the overall method is conservative and expensive. The proposed algorithms have been implemented as a software tool that is available for download. Illustrative examples for the PWA models of two gene networks are included.