Robust Model Predictive Control for Piecewise Affine Systems

Abstract

In this paper, we study a robust model predictive control (MPC) strategy for piecewise affine (PWA) systems with uncertainty that is described as a set of polytopic parameter-varying models in a polytope corresponding to each partition of the PWA systems. First, an infinite horizon MPC technique for guaranteeing robust stability is developed for uncertain PWA systems. According to the condition of the PWA system states, the sequence of piecewise linear feedback controller at each sampling time is derived on-line by solving a convex optimization problem involving linear matrix inequalities. The feasible PWA control law design can robustly stabilize the uncertain PWA systems. However, the on-line optimization problems may lead to a computational burden. Then we further propose an improved robust MPC algorithm. When the current state is outside of the region of PWA systems containing the origin, the proposed on-line robust MPC algorithm is utilized; once the current state enters the region with the origin, sequence attraction domains where the origin is included are constructed off-line one within another, and the explicit control laws corresponding to different attraction domains can drive the state to the origin. The two algorithms are illustrated with a numerical example. The simulation results show that both controller design methods can stabilize the PWA systems with polytopic uncertainty, and the improved algorithm can reduce the on-line computation cost.

Complexity reduction of Robust Model Predictive Controller for uncertain Piecewise Affine Systems

“…In [6] a RMPC for UPWA systems with polytopic parameter uncertainty is presented. The paper considers the case of unconstrained PWA systems.…”

“…The developed algorithm aims at reducing the computational load. It consists of two different parts; the first based on an on-line optimization problem, and then in the second part, when the state enters a computed attraction domains [6], an explicit feedback control law is applied. The algorithm reduces the computational time, but in the first part of the algorithm an on-line heavy computational load still exists.…”

“…Figure 3 shows the results for some different initial states, where each of those initial states, for visibility purpose, is covered by the reachable regions at the 20 th depth level ( As mentioned in the introduction, the approach presented in [6] can not be applied to our example, as the considered set-point differs from the origin, and also the two-tanks example has constraints over the control signals. Thus we can not compare our approach to this technique.…”

Dynamic output‐feedback fuzzy MPC for Takagi‐Sugeno fuzzy systems under event‐triggering–based try‐once‐discard protocol

“…(4) How to formulate an online optimization problem in terms of the dynamic output-feedback FMPC and obtain the sufficient conditions such that the states of the closed-loop system converge to a bounded region near the origin? Motivated by the above challenges, the main contributions can be highlighted as follows: (1) a novel protocol, event-triggering-based TOD protocol, is developed for the T-S fuzzy system, where the characteristics of such a protocol such as the time-varying threshold and the priority are both taken into consideration; (2) to handle system nonlinearities in the context of the T-S fuzzy model, a "min-max" problem of the objective function over the infinite-time horizon is formulated, which is based on the dynamic output-feedback FMPC with respect to the immeasurable states; (3) for the purpose of overcoming the obstacles accompanied with the proposed event-triggering-based TOD protocol, the influence of the TOD protocol is reflected into the controller design in terms of the S-procedure technique; meanwhile, the impact of the estimation error is taken into account by using the quadratic boundedness methodology; and (4) a certain upper bound of the objective is put forward to establish an auxiliary optimization problem for the solvability. Then, based on such an online problem, sufficient conditions are derived for ensuring the input-to-state stability of the closed-loop system.…”

H∞ model predictive control for constrained discrete‐time piecewise affine systems

“…In the work of Zou and Li, RMPC for piecewise linear (PWL) systems with uncertainty are studied where the sequence of PWL feedback controllers is derived online by solving a convex optimization problem involving LMIs, which can lead to a broad computational burden. In order to increase efficiency, presents a constrained infinite horizon RMPC for uncertain PWL systems using LMIs.…”