Piecewise-affine Hybrid Systems Research Articles

Piecewise affine identification of a hydraulic pumping system using evolutionary computation

Identification of piecewise affine hybrid systems is not an easy task since both the parameters determining the discrete modes and the submodels parameters have to be obtained, leading to a non-convex combinatorial optimisation problem. One way around this problem is to solve it in two steps. This work presents an approach for simultaneously estimating the parameters for PieceWise Autoregressive eXogeneous and PieceWise Output Error models. This is accomplished by using evolutionary algorithms for finding the parameters of the discrete modes (Gaussian mixture models) and employing the proposed weighted and extended least squares algorithm to estimate the ARX or ARMAX submodels. The main advantages of the proposed algorithm are: (i) it can be applied to output error models - which correspond to system with measurement noise, (ii) it is less likely to get trapped in local minima and (iii) it is applicable to problems with large datasets. The proposed approach, which is offline, was validated using simulated and experimental data and was compared with another method from the literature. In the case of an experimental plant, parsimonious models with good performance in both dynamic and static regimes were obtained.

In-Block Controllability of Affine Systems on Polytopes

In this technical note, we introduce the study of the in-block controllability (IBC) of affine systems on polytopes. In particular, we study whether all the states in the interior of a given polytope are mutually accessible through its interior by applying uniformly bounded inputs. The proposed study formalizes controllability under state constraints, and it has strong connections to the study of controllability of piecewise affine hybrid systems. We provide easily checkable necessary conditions for IBC. Then, for an important class of polytopes, simplicial polytopes, we show that these conditions are also sufficient.

Control to Facet by Piecewise-Affine Output Feedback

The control-to-facet problem plays an important role in the design of feedback controllers for piecewise-affine hybrid systems on polytopes. In the literature, necessary conditions and sufficient conditions for solvability by static state feedback exist. In this paper, we extend these results to the case of continuous piecewise-affine static output feedback. For the construction of a controller, a triangulation of the output polytope is made which satisfies additional conditions to guarantee compatibility with the induced subdivision of the state polytope. In the state feedback case, the use of this special type of triangulation is not required.

Synthesis for Constrained Nonlinear Systems Using Hybridization and Robust Controllers on Simplices

In this technical note, we propose an approach to controller synthesis for a class of constrained nonlinear systems. It is based on the use of a hybridization, that is a hybrid abstraction of the nonlinear dynamics. This abstraction is defined on a triangulation of the state-space where on each simplex of the triangulation, the nonlinear dynamics is conservatively approximated by an affine system subject to disturbances. Except for the disturbances, this hybridization can be seen as a piecewise affine hybrid system on simplices for which appealing control synthesis techniques have been developed in the past decade. We extend these techniques to handle systems subject to disturbances by synthesizing and coordinating local robust affine controllers defined on the simplices of the triangulation. We show that the resulting hybrid controller can be used to control successfully the original constrained nonlinear system. Our approach, though conservative, can be fully automated and is computationally tractable. To show its effectiveness in practical applications, we apply our method to control a pendulum mounted on a cart.

Event detection occurrence for planar piece-wise affine hybrid systems

Many switched circuits are made of linear components switched by a simple logic unit. In this paper, we use a generic planar Piece-wise Affine Hybrid System (PWAHS) to model this kind of circuit. Usually, simulations are run by analytical methods (adapted to a specific simple model) or by numerical simulations that can miss some events’ occurrences. We propose a generic planar method to simulate PWAHSs with periodic and state dependent events. Using analytical expressions, our approach can reach arbitrary accuracy in event detections without any loss. As a result, we have implemented our method in Scilab and Matlab toolboxes.

Identification of Piecewise Linear Models of Complex Dynamical Systems

AbstractThe paper addresses the realization and identification problem for a subclass of piecewise-affine hybrid systems. The paper provides necessary and sufficient conditions for existence of a realization by a piecewise-affine hybrid system. The paper also proposes an identification algorithm for this subclass of hybrid systems. The considered system class and the identification problem are motivated by applications in systems biology.

Control of Piecewise-Affine Hybrid Systems - Extended abstract

The character of this paper is an extended abstract of the approach developed by the authors for control of piecewise-affine hybrid systems. Control of hybrid systems is a wide ranging research topic with many theoretical problems. The authors have therefore decided early on to restrict attention to piecewise-affine hybrid systems on polytopes. The approach developed includes for affine systems on polytopes sufficient conditions for the problems (1) leaving the polytope in finite time through a prespecified set of facets; (2) remaining inside a particular polytope forever; and (3) converging to a fixed state inside a particular polytope. For piecewise-affine hybrid system on polytopes a procedure for reachability and control synthesis is proposed consisting of (1) for each discrete state the computation of all possible exit facets and exit events; (2) abstraction of the continuous dynamics to an automaton; and (3) the search of a path in the automaton from an initial discrete state to a target discrete state. Open research issues for control of hybrid systems are stated.

The distinguishability of linear control systems

When observabilities of hybrid systems such as piecewise-affine hybrid systems are considered, the distinguishability of subsystems takes a very important role. However, properties and even definitions of such distinguishabilities have not been extensively studied in the literature. This paper will give a definition of the distinguishability of two linear control systems and present the necessary and sufficient condition of the distinguishability, giving an answer for the problem posed by Vidal et al.

Special Section on Symbolic Methods for Complex Control Systems

Special Section on Symbolic Methods for Complex Control Systems

Reachability and Control Synthesis for Piecewise-Affine Hybrid Systems on Simplices

In this paper, we consider the synthesis of control laws for piecewise-affine hybrid systems on simplices. The construction is based on the solution to the control-to-facet problem at the continuous level, and on dynamic programming at the discrete level. The construction is given as an explicit algorithm using only linear algebra and reach-set computations for automata; no numerical integration is required. The method is conservative, in that it may fail to find a control law where one exists, but one cannot hope for a sharp algorithm for control synthesis since reachability for piecewise-affine hybrid systems is undecidable.

Observability of the discrete state for dynamical piecewise hybrid systems

In this paper, we deal with the observability of piecewise-affine hybrid systems. Our aim is to give sufficient conditions to observe the discrete and continuous states, in terms of algebraic and geometrical conditions. Firstly, we will give the algebraic conditions to observe the discrete state based on the switch function reconstruction for linear hybrid systems. Secondly, we will give a geometrical condition based on the transversality concept for nonlinear hybrid systems. Throughout this paper, we illustrate our propositions with examples and simulations.

Towards automatic verification of affine hybrid system stability

Stability is a very important property of feedback control systems. For linear feedback control systems, methods for proving or disproving system stability are well known. They basically amount to eigenvalue computations. For more general classes of feedback control systems, however, these methods are not applicable. A very general method is based on Lyapunov theory [7]. It uses a so-called Lyapunov function which can be regarded as generalized “energy function” of the system under investigation. If energy converges to zero over time then the system state will converge towards an equilibrium state. In the scope of the AVACS project,1 we are concerned with automatically proving hybrid system stability. In a first phase of the project, we focus on a special subclass of hybrid systems, namely piecewise affine hybrid systems. Hybrid systems contain both, discrete states and continuoustime dynamics. Piecewise affine hybrid systems require the continuous-time dynamics to be affine for each discrete state. A powerful method of verifying stability for such systems is based on extensions of Lyapunov theory, as proposed by Pettersson [9], Johansson et. al. [6] or Branicky [4]. These approaches still require system dependent inputs which could – up to now– only be provided by a human proof designer based on his or her intuition and expert knowledge of the problem domain. The work presented in this paper reports on our current status with respect to achieve full automatization of the stability verification task. The goal is to be able to prove stability of piecewise affine hybrid systems “by the push of a button” and without any further knowledge provided by a human user. In the future, we plan to extend and to complement the approach in order to cope with more general non-linear systems, possibly through approximation.

CONTROL OF DISCRETE-TIME PIECEWISE AFFINE SYSTEMS

In this paper we propose a new approach for the control of discrete-time piecewise affine hybrid systems on full-dimensional polytopes. The new strategy is divided into a local and a supervisory control problem. The local problem is to reach and cross one facet of a polytope ensuring that the next sample of the time-discrete trajectory is picked up in the adjacent polytope. Therefore, a local piecewise affine control law is obtained by solving an optimization problem, minimizing the retention period of the trajectory in the polytope or the quadratic sum of the input signal. This procedure is based on system inherent bounds. The supervisory control problem is to find a suitable combination of polytopes and local control strategies that transfers the trajectory to the operating point and keeps it there.

On the use of hybrid system identification methods for max-plus-linear system identification

In recent years, different approaches for identification of hybrid systems and max-plus linear systems that are a special class of discrete event systems, have been developed. This paper introduces a new identification method for piecewise affine hybrid systems and shows that this approach is also easily applicable for identification of max-plus-linear systems.