Partial enumeration MPC: Robust stability results and application to an unstable CSTR

Abstract We describe a partial enumeration (PE) method for fast computation of a suboptimal solution to linear MPC problems [1] with robust stability properties. Given that the suboptimal PE-based control law is non-unique (that is, a set-valued map) and (possibly) discontinuous, we treat the closed-loop system, appropriately augmented, as a difference inclusion. We derive novel robust exponential stability results for difference inclusions of this type. In particular we show that Strong Robust Exponential Stability (SRES) holds, for any sufficiently small but otherwise arbitrary perturbation. Such approach allows us to show SRES of the closed-loop system under PE-based MPC. Application to a simulated open-loop unstable CSTR with separation unit and recycle is presented to show performance and timing results for PE-based MPC, as well as to highlight its robustness to process/model mismatch, disturbances and measurement noise.

[1]  G. Goodwin,et al.  Global analytical model predictive control with input constraints , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[2]  D. Q. Mayne,et al.  Suboptimal model predictive control (feasibility implies stability) , 1999, IEEE Trans. Autom. Control..

[3]  Gabriele Pannocchia,et al.  Disturbance models for offset‐free model‐predictive control , 2003 .

[4]  Alberto Bemporad,et al.  A survey on explicit model predictive control , 2009 .

[5]  Manfred Morari,et al.  Real-Time Suboptimal Model Predictive Control Using a Combination of Explicit MPC and Online Optimization , 2011, IEEE Trans. Autom. Control..

[6]  Andrew R. Teel,et al.  Examples when nonlinear model predictive control is nonrobust , 2004, Autom..

[7]  Andrew R. Teel,et al.  Smooth Lyapunov functions and robustness of stability for difference inclusions , 2004, Syst. Control. Lett..

[8]  Stephen P. Boyd,et al.  Fast Model Predictive Control Using Online Optimization , 2010, IEEE Transactions on Control Systems Technology.

[9]  Michael A. Demetriou,et al.  Proceedings of the 46th IEEE Conference on Decision and Control , 2007, IEEE Conference on Decision and Control.

[10]  Claudio De Persis,et al.  Proceedings of the 41st IEEE Conference on Decision and Control , 2002 .

[11]  Stephen J. Wright,et al.  Suboptimal MPC and partial enumeration: robust stability and computational performance , 2010 .

[12]  Andrew R. Teel,et al.  Nominally Robust Model Predictive Control With State Constraints , 2007, IEEE Transactions on Automatic Control.

[13]  Stephen J. Wright,et al.  Application of Interior-Point Methods to Model Predictive Control , 1998 .

[14]  H. J. Ferreau,et al.  An online active set strategy to overcome the limitations of explicit MPC , 2008 .

[15]  Stephen J. Wright,et al.  Fast, large-scale model predictive control by partial enumeration , 2007, Autom..

[16]  W. P. M. H. Heemels,et al.  Predictive control of hybrid systems: Input-to-state stability results for sub-optimal solutions , 2009, Autom..

[17]  Frank J. Christophersen,et al.  Controller complexity reduction for piecewise affine systems through safe region elimination , 2007, 2007 46th IEEE Conference on Decision and Control.

[18]  D. Limón,et al.  Input-to-state stable MPC for constrained discrete-time nonlinear systems with bounded additive uncertainties , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[19]  Mato Baotic,et al.  On the computation of linear model predictive control laws , 2010, Autom..

[20]  Alberto Bemporad,et al.  Efficient On-Line Computation of Constrained Optimal Control , 2008, SIAM J. Control. Optim..

[21]  Stephen J. Wright,et al.  Efficient Cooperative Distributed MPC using Partial Enumeration* *This work was supported by National Science Foundation (Grant CTS-0456694) , 2009 .

[22]  Hans Joachim Ferreau,et al.  Efficient Numerical Methods for Nonlinear MPC and Moving Horizon Estimation , 2009 .

[23]  W. P. M. H. Heemels,et al.  Lyapunov Functions, Stability and Input-to-State Stability Subtleties for Discrete-Time Discontinuous Systems , 2009, IEEE Transactions on Automatic Control.

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

[25]  Stephen J. Wright,et al.  Conditions under which suboptimal nonlinear MPC is inherently robust , 2011, Syst. Control. Lett..