Optimal complexity reduction of polyhedral piecewise affine systems (original) (raw)

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.

A New Technique for Translating Discrete Hybrid Automata into Piecewise Affine Systems

Mathematical and Computer Modelling of Dynamical Systems, 2004

The paper proposes a new translation algorithm that translates a hybrid system described as a discrete hybrid automaton (DHA) into an equivalent piecewise affine (PWA) system. The translation algorithm exploits, among others, a new technique for cell enumeration in hyperplane arrangement, all proposed in this paper. The new translation technique enables the transfer of several analysis and synthesis tools developed for PWA systems to a DHA class of hybrid systems.

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.

Perturbations in polyhedral partitions and the related fragility of piecewise affine control

Bulletin Mathematique De La Societe Des Sciences Mathematiques De Roumanie, 2018

The control design techniques for linear or hybrid systems under constraints lead often to off-line state-space partitions with non-overlapping convex polyhedral regions. This corresponds to a piecewise affine (PWA) state feedback control laws associated to polyhedral partition of the state-space. The aim of this paper is to consider perturbation in the representation of the vertices of such polyhedral regions. Another idea behind this study is to perform a quantization operation on the representation of the state-space regions and the associated PWA control laws in order to reduce the hardware requirements in terms of processor speed and memory unit. The quantized state-space partitions lose some of the important properties of the explicit controllers: non-overlapping, convexity and invariant characterization. The major contribution of this work is to analyze to what extend the non-overlapping and the invariance characteristics of the PWA controller can be preserved when perturbati...

Qualitative Abstraction of Piecewise Affine Systems

Qualitative or symbolic abstractions of hybrid systems re- ceived considerable interest recently to solve problems of hybrid systems estimation, control and verification symboli- cally. To abstract a hybrid system one has to slice the continu- ously valued input/output/state-space into a (finite) set of par- titions. The number of partitions potentially grows exponen- tially with the dimension of the space. As a consequence, one has to divide the spaces carefully in order to obtain a manage- able abstraction. This paper presents a systematic procedure to partition the state-space of piecewise affine (PWA) systems into qualitatively distinct regions. As a consequence, we ob- tain a moderately large set of partitions that characterises the hybrid dynamics of the PWA system. The abstraction scheme helps also to keep the number of so called spurious behaviors of qualitative simulation small, in particular when compared to the typically used grid-based abstractions.

On The Complexity of Bounded Time Reachability for Piecewise Affine Systems

Lecture Notes in Computer Science, 2014

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.

Reconfiguration in Hierarchical Control of Piecewise-Affine Systems

Hybrid Systems: Computation and Control, 2002

The In this paper the problem of reconfiguration in hierarchical control of piecewise-affine systems in discrete time is considered as the choice of input constraints applied to the low-level control. It is shown how such reconfiguration can provide fault-tolerance to actuator faults while reducing the computational complexity of low-level control. The approach is based on partitioning the state space while taking into account multiple possibilities for the inputs available to low-level control. A so-called "reconfiguration database" is computed at design-time which determines the input constraints that provide for reachability between regions of a state-space partition. This database is used as a basis for reconfiguration decisions at runtime.

On simplifying and classifying piecewise-linear systems

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 2002

A basic methodology to understand the dynamical behavior of a system relies on its decomposition into simple enough functional blocks. In this work, following that idea, we consider a family of piecewise-linear systems that can be written as a feedback structure. By using some results related to control systems theory, a simplifying procedure is given. In particular, we pay attention to obtain equivalent state equations containing both a minimum number of nonzero coefficients and a minimum number of nonlinear dynamical equations (canonical forms). Two new canonical forms are obtained, allowing to classify the members of the family in different classes. Some consequences derived from the above simplified equations are given. The state equations of different electronic oscillators with two or three state variables and two or three linear regions are studied, illustrating the proposed methodology.

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.