I. Necoara, B. De Schutter, W.P.M.H. Heemels, S. Weiland, M. Lazar, and T.J.J. van den Boom, "Control of PWA systems using a stable receding horizon method: Extended report," Tech. rep. 04-019a, Delft Center for Systems and Control, Delft University of Technology, Delft, The Netherlands, 26 pp., Oct. 2004. A short version of this report has been published in the Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, July 2005. Paper 2794/Tu-E21-TO/2.
In this paper we derive stabilization conditions for the class of PWA systems using the linear matrix inequality (LMI) framework. We consider the class of piecewise affine feedback controllers and the class of piecewise quadratic Lyapunov functions that guarantee stability of the closed-loop system. We take into account the piecewise structure of the system and therefore the matrix inequalities that we solve are less conservative. We prove that the infinite-horizon quadratic cost is bounded if certain LMIs are satisfied. Using the upper bound of the infinite-horizon quadratic cost as a terminal cost and constructing also a convex terminal set we show that the receding horizon control stabilizes the PWA system. We derive also an algorithm for enlarging the terminal set based on a backward procedure; therefore, the prediction horizon can be chosen shorter, removing some computations off-line.