Adaptive Fault‐Tolerant Tracking Control of Piecewise Affine Systems With Performance Constraint

The main contribution of this paper is to present the systematic and low-complexity translation techniques between a class of hybrid systems referred to as automaton-based DHA and piecewise affine (PWA) systems. As an starting point the general modeling framework of the automaton-based DHA is represented which models the controlled and uncontrolled switching phenomena between linear continuous dynamics including discrete and continuous states, inputs and outputs. The basic theoretical definitions on the state trajectories of the proposed DHA with forward and backward evolutions which yield forward and backward piecewise affine (FPWA and BPWA) systems are given. Next, the well-posedness and equivalency properties are proposed and the sufficient conditions under which the well-posedness property is achieved with the automaton-based DHA and PWA systems are given. It is shown that the graphical structure of the proposed automaton-based DHA makes it possible to obtain analytically the equivalent PWA system with a polynomial complexity in contrast to the existing numerical translation techniques via decomposed structure of the DHA with an exponential complexity. Examples are presented to confirm the effectiveness of the proposed translation techniques.

Piecewise affine (PWA) systems are autonomous systems with discontinuous vector fields which are affine ordinary differential equations at the points of continuity. These systems have applications to many fields of engineering, including systems biology and traffic engineering. We define what it means for a PWA system to be monotone, and we provide a set of sufficient conditions for monotonicity of PWA systems with hyperrectangular invariants. Such sufficient conditions are useful for understanding the dynamics of such PWA systems and for designing controllers for qualitative, reference tracking. We apply these results towards the drug-discovery problem for the cancer-related p53 pathway.

We prove, in a constructive way, the equivalence between hybrid and piecewise affine systems. By focusing our investigation on the latter class, we show through counter-examples that observability and controllability properties cannot be easily deduced from those of the component linear subsystems. Instead, we propose practical numerical tests based on mixed-integer linear programming.

Sequences of Discrete Abstractions for Piecewise Affine Systems

This paper addresses the design of a fault detection and reconfigurable control structure for bimodal piecewise affine (PWA) systems. The PWA bimodal system will be designed to verify input-to-state stability (ISS) in closed loop. The proposed methodology is divided into two parts. First, a Luenberger-based observer structure is proposed to solve the fault detection and identification (FDI) problem for bimodal PWA systems. The unknown value of the fault parameter is estimated by an observer equation, which is derived using a Lyapunov-based methodology. Then, the ISS property is proved for the observer. Second, a fault-tolerant state feedback controller is synthesized for the PWA model. The controller is designed to deal with partial loss of control authority identified by the observer. The ISS property is also proved for the controller. Finally, the ISS property for the interconnection of the controller and the observer-based fault identification mechanism is studied. The design procedure is formulated as a set of linear matrix inequalities (LMIs), which can be solved efficiently using available software packages.

A duality-based convex optimization approach to L 2 -gain control of piecewise affine slab differential inclusions

This paper presents a novel observer-based controller design method for discrete-time piecewise affine (PWA) systems. The basic idea is as follows: at first, a piecewise linear (without affine terms) state feedback controller and a PWA observer are designed separately, and then it is proved that the output feedback controller constructed by the resulting observer and state feedback controller gains can guarantee the stability of the closed-loop system. During the controller design, the piecewise-quadratic Lyapunov function technique is used. Moreover, the region information is taken into account to treat the affine terms, so the controller gains can be obtained by solving a set of linear matrix inequalities, which are numerically feasible with commercially available software. Three simulation examples are given finally to verify the proposed theoretical results. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Formal analysis of piecewise affine systems through formula-guided refinement

This paper considers a human-operated task model based on identification techniques using switched systems expressed by mixed logical dynamical systems. In the system, each subsystem is expressed by a piecewise affine system. For a model to be identified, it is not easy to determine a number of subsystems in advance. In this paper, we propose a simultaneous identification technique which takes into account both a number of subsystems and parameters, so that a piecewise affine model of a human operation can be obtained. In addition, the obtained model is applied to automation. The effectiveness of the proposed method is illustrated by simulations and experiments. © 2011 Wiley Periodicals, Inc. Electr Eng Jpn, 176(1): 46–54, 2011; Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/eej.21079

International audience

The usual invariance results for asymptotic stability for continuous autonomous finite-dimensional dynamical systems involve a positive definite Lyapunov function whose time derivative along the system motions is negative semidefinite (along with certain invariance conditions). This is equivalent to requiring that along the system motions, the Lyapunov function is nonincreasing at all times with increasing time. In this paper we establish an invariance result for asymptotic stability for continuous and discontinuous nonautonomous finite-dimensional dynamical systems, which requires a positive definite Lyapunov function which when evaluated along the system motions is nonincreasing only on certain unbounded discrete time sets E with increasing time. This allows that between the time instants determined by E, the Lyapunov function may increase (i.e., over some finite time intervals, the system may exhibit unstable behavior). We also show that the usual invariance theorem for asymptotic stability reduces to the invariance theorem for continuous dynamical systems presented herein. In addition, we establish a variant to the above result involving estimates of the asymptotic behavior of the system's motions. We apply our results to four examples. One of these involves the stabilization of conservative mechanical systems using energy dissipation intermittently.

This paper discusses the finite-time optimal control problem via sampled-data control of a class of piecewise affine (PWA) systems, where the switching action of the discrete state is determined at each sampling time according to a condition on the continuous state.

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. 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. An illustrative example for a PWA gene network model is included.

We consider the problem of controlling a discrete-time 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.

In this paper, a Lyapunov-based method is provided to study the local asymptotic stability of planar piecewise affine systems with continuous vector fields. For such systems, the state space is supposed to be partitioned into several bounded convex polytopes. A piecewise affine function, not necessarily continuous on the boundaries of the polytopic partitions, is proposed as a candidate Lyapunov function. Then, sufficient conditions for the local asymptotic stability of the system, including a monotonicity condition at switching instants, are formulated as a linear programming problem. In addition, when the problem does not have a feasible solution based on the original partitions of the system, a new partition refinement algorithm is presented. In this way, more flexibility can be provided in searching for the Lyapunov function. Owing to relaxation of the continuity condition imposed on the system boundaries, the proposed method reaches to less conservative results, compared with the previous methods based on continuous piecewise affine Lyapunov functions. Simulation results illustrate the effectiveness of the proposed method.