Stabilization of Discrete-Time Piecewise Affine Systems in Implicit Representation

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

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.

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.

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.

This paper dealt with the stabilization problem of a discrete-time piecewise affine (PWA) systems with time-delay. An H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> state feedback controller design is proposed, and the PWA system was described as ellipsoid which characterized by a set of vector inequalities..thereby the constraint of linear matrix inequalities (LMIs) was released. In terms of LMIs, the time-delay PWA system was stabilized in Lyapunov theory, and the minimum problem of the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> performance index can be cast as a convex optimization problem with LMIs constraints. The simulation results verified the effectiveness of the proposed method.

Non-synchronized Observer-based Control of Discrete-time Piecewise Affine Systems: an LMI Approach

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

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

This article is concerned with the adaptive fault‐tolerant tracking control (FTTC) issue with performance constraint for piecewise affine (PWA) systems subject to actuator failures. The mode switching of PWA systems bring out a challenge to guarantee the predefined performance. To conquer this obstacle, a novel adaptive FTTC strategy with time‐varying performance guarantees for PWA systems is devised. First, a piecewise generalized restricted potential function for PWA systems is introduced. Then, by using tracking error, the performance function and a set of matrices, an adaptive FTTC scheme is developed. With the aid of a time‐varying positive definite Lyapunov function, it is proved that the tracking error can be constrained within predefined performance bounds by the developed method. Finally, the simulation results further illustrate the proposed theoretical findings.

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.

In this paper we present an approach for designing of reference tracking controllers for constrained, discrete-time piecewise affine systems. The approach follows the idea of reference governor techniques where the desired set-point is filtered by a system called the “reference governor”. Based on the system current state, set-point, and prescribed constraints, the reference governor computes a new set-point for a low-level controller so that the state and input constraints are satisfied and convergence to the original set-point is guaranteed.

This paper proposes a MIMO Explicit Generalized Predictive Control (EGPC) for minimizing payload oscillation of a Gantry Crane System subject to input and output constraints. In order to control the crane system efficiently, the traditional GPC formulation, based on online Quadratic Programming (QP), is rewritten as a multiparametric quadratic programming problem (mp-QP). An explicit Piecewise Affine (PWA) control law is obtained and holds the same performance as online QP. To test effectiveness, the proposed method is compared with two GPC formulations: one that handle constraints (CGPC) and another that does not handle constraints (UGPC). Results show that both EGPC and CGPC have better performance, reducing the payload swing when compared to UGPC. Also both EGPC and CGPC are able to control the system without constraint violation. When comparing EGPC to CGPC, the first is able to calculate (during time step) the control action faster than the second. The simulations prove that the overall performance of EGPC is superior to the other used formulations.

Combining Truncated Binary Search Tree and Direct Search for Flexible Piecewise Function Evaluation for Explicit MPC in Embedded Microcontrollers

FPGA Implementation of Explicit Model Predictive Control for Closed Loop Control of Depth of Anesthesia