Measures and LMIs for optimal control of piecewise-affine systems

This paper considers the class of deterministic continuous-time optimal control problems (OCPs) with piecewise-affine (PWA) vector field, polynomial Lagrangian and semialgebraic input and state constraints. The OCP is first relaxed as an infinite-dimensional linear program (LP) over a space of occupation measures. This LP is then approached by an asymptotically converging hierarchy of linear matrix inequality (LMI) relaxations. The relaxed dual of the original LP returns a polynomial approximation of the value function that solves the Hamilton-Jacobi-Bellman (HJB) equation of the OCP. Based on this polynomial approximation, a suboptimal policy is developed to construct a state feedback in a sample-and-hold manner. The results show that the suboptimal policy succeeds in providing a suboptimal state feedback law that drives the system relatively close to the optimal trajectories and respects the given constraints.

[1]  Jonathan P. How,et al.  Output feedback controller synthesis for piecewise-affine systems with multiple equilibria , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[2]  Mikael Johansson,et al.  Piecewise linear control systems - a computational approach , 2002, Lecture notes in control and information sciences.

[3]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[4]  Stephen P. Boyd,et al.  Quadratic stabilization and control of piecewise-linear systems , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).

[5]  Luis Rodrigues,et al.  Controller synthesis for piecewise affine slab differential inclusions: A duality-based convex optimization approach , 2007, 2007 46th IEEE Conference on Decision and Control.

[6]  P. Julián,et al.  High-level canonical piecewise linear representation using a simplicial partition , 1999 .

[7]  Didier Henrion,et al.  GloptiPoly 3: moments, optimization and semidefinite programming , 2007, Optim. Methods Softw..

[8]  L. Rodrigues.  Dynamic output feedback controller synthesis for piecewise-affine systems , 2002 .

[9]  Didier Henrion,et al.  Nonlinear optimal control synthesis via occupation measures , 2008, 2008 47th IEEE Conference on Decision and Control.

[10]  R. W. Brockett,et al.  Asymptotic stability and feedback stabilization , 1982 .

[11]  Emmanuel Trélat,et al.  Nonlinear Optimal Control via Occupation Measures and LMI-Relaxations , 2007, SIAM J. Control. Optim..

[12]  Luis Rodrigues.  Stability of sampled-data piecewise-affine systems under state feedback , 2007, Autom..

[13]  Didier Henrion,et al.  Convex Computation of the Region of Attraction of Polynomial Control Systems , 2012, IEEE Transactions on Automatic Control.

[14]  J. Cortés.  Discontinuous dynamical systems , 2008, IEEE Control Systems.

[15]  Mato Baotić,et al.  Optimal control of piecewise affine systems - a multi-parametric approach , 2005 .

[16]  J. Lasserre.  Moments, Positive Polynomials And Their Applications , 2009 .

[17]  John N. Tsitsiklis,et al.  Complexity of stability and controllability of elementary hybrid systems , 1999, Autom..

[18]  Romain Bourdais,et al.  Practical stabilization for piecewise-affine systems: A BMI approach , 2012 .

[19]  Anders Rantzer,et al.  Piecewise linear quadratic optimal control , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[20]  M. Johansson,et al.  Piecewise Linear Control Systems , 2003 .

[21]  Luis Rodrigues,et al.  Extension of local linear controllers to global piecewise affine controllers for uncertain non-linear systems , 2008, Int. J. Syst. Sci..

[22]  Yu. S. Ledyaev,et al.  Asymptotic controllability implies feedback stabilization , 1997, IEEE Trans. Autom. Control..

[23]  Luis Rodrigues,et al.  A New methodology for piecewise affine models using Voronoi partitions , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.