A Nonlinear Sum-of-Squares Model Predictive Control Approach

This note presents a novel model predictive control (MPC) strategy for input-saturated nonlinear systems having a polynomial structure. The results here proposed are a significant generalization w.r.t. similar existing algorithms which are tailored only for linearized or multi-model plant descriptions. A first key aim is to present sum-of-squares (SOS) conditions under which off-line one-step controllable sets for nonlinear polynomial systems can be derived. A second relevant contribution is to describe an on-line MPC strategy that leads to less conservative performance w.r.t. most existing methods based on global linearization approaches. An illustrative example is finally provided in order to show the effectiveness of the proposed SOS-based MPC algorithm.

[1]  David Q. Mayne,et al.  Reachability analysis of discrete-time systems with disturbances , 2006, IEEE Transactions on Automatic Control.

[2]  Riccardo Scattolini,et al.  A stabilizing model-based predictive control algorithm for nonlinear systems , 2001, Autom..

[3]  Graziano Chesi,et al.  Computing output feedback controllers to enlarge the domain of attraction in polynomial systems , 2004, IEEE Transactions on Automatic Control.

[4]  Mayuresh V. Kothare,et al.  Efficient scheduled stabilizing output feedback model predictive control for constrained nonlinear systems , 2003, Proceedings of the 2003 American Control Conference, 2003..

[5]  M. Morari,et al.  Parametric optimization and optimal control using algebraic geometry methods , 2006 .

[6]  F. Bruzelius.  Linear Parameter-Varying Systems - an approach to gain scheduling , 2004 .

[7]  Basil Kouvaritakis,et al.  Efficient robust predictive control , 2000, IEEE Trans. Autom. Control..

[8]  Andrew Packard,et al.  Stability Region Analysis Using Polynomial and Composite Polynomial Lyapunov Functions and Sum-of-Squares Programming , 2008, IEEE Transactions on Automatic Control.

[9]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[10]  Pablo A. Parrilo,et al.  Nonlinear control synthesis by convex optimization , 2004, IEEE Transactions on Automatic Control.

[11]  Alberto Bemporad,et al.  The explicit linear quadratic regulator for constrained systems , 2003, Autom..

[12]  Franco Blanchini,et al.  Set-theoretic methods in control , 2007 .

[13]  J. Lasserre,et al.  Solving nonconvex optimization problems , 2004, IEEE Control Systems.

[14]  Frank Allgöwer,et al.  Nonlinear Model Predictive Control and Sum of Squares Techniques , 2006 .

[15]  Graziano Chesi,et al.  On the Gap Between Positive Polynomials and SOS of Polynomials , 2007, IEEE Transactions on Automatic Control.

[16]  Carsten W. Scherer,et al.  Matrix Sum-of-Squares Relaxations for Robust Semi-Definite Programs , 2006, Math. Program..

[17]  Mayuresh V. Kothare,et al.  An e!cient o"-line formulation of robust model predictive control using linear matrix inequalities (cid:1) , 2003 .

[18]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[19]  Graziano Chesi,et al.  Homogeneous Lyapunov functions for systems with structured uncertainties , 2003, Autom..

[20]  David Q. Mayne,et al.  Constrained model predictive control: Stability and optimality , 2000, Autom..

[21]  B. Reznick,et al.  Sums of squares of real polynomials , 1995 .

[22]  Edoardo Mosca,et al.  An ellipsoidal off-line MPC scheme for uncertain polytopic discrete-time systems , 2008, Autom..

[23]  J. Lasserre,et al.  Detecting global optimality and extracting solutions in GloptiPoly , 2003 .

[24]  Ufuk Topcu,et al.  Local stability analysis using simulations and sum-of-squares programming , 2008, Autom..

[25]  A. Vicino,et al.  On convexification of some minimum distance problems , 1999, 1999 European Control Conference (ECC).

[26]  Stephen J. Wright,et al.  Nonlinear Predictive Control and Moving Horizon Estimation — An Introductory Overview , 1999 .

[27]  Manfred Morari,et al.  Robust constrained model predictive control using linear matrix inequalities , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[28]  Graziano Chesi,et al.  Estimating the domain of attraction via union of continuous families of Lyapunov estimates , 2007, Syst. Control. Lett..

[29]  Moritz Diehl,et al.  Approximate robust dynamic programming and robustly stable MPC , 2006, Autom..