(197b) Simultaneous Constrained Moving Horizon State Estimation and Model Predictive Control by Multi-Parametric Programming

Explicit/multi-parametric Model Predictive Control (mp-MPC) solves the on-line optimization problem in a traditional MPC framework with multi-parametric programming techniques and by this derives the governing control laws for the system at hand as a set of explicit functions of the system states. The main advantage of explicit/multi-parametric MPC is that it replaces the online optimization-based implementation of traditional MPC with simple function evaluations making real-time application possible [1], [2]. A further advantage is that the feasible region of the optimization problem is known in advance and the controller can be adopted accordingly. The implementation of any MPC relies on the assumption that the state values are readily available. In reality however, the available measurements do not produce this information directly and the state information needs to be inferred from those measurements. Moving Horizon Estimation (MHE) obtains the system state estimates and the disturbance estimates by solving a constrained optimization problem [3]. The main advantage of MHE is that system information, such as non?negativity of a concentration, can be incorporated in the formulation of the constrained optimization and hence in the calculation of the estimates.

Traditionally, the estimator and the controller for a system would have been designed independently, following the separation principle ideas. However, since the separation principle does not hold for constrained control and estimation [4], the estimation and the control of the system have to be designed simultaneously. The methodology proposed here addresses this by 1. formulating the dynamics that govern the estimation error, 2. using these dynamics to determine the set that bounds the error, and 3. robustifying against the estimation error [5]. This methodology has previously been applied using the Luenberger observer [5] and the unconstrained MHE [6], [7]. We make the expansion to the constrained case so that the full potential of the MHE can be exploited to improve the control of the system. We present a method for obtaining the estimation error dynamics and the bounding error set for a general formulation of a constrained MHE. We tackle this problem by solving the constrained MHE with multi-parametric programming techniques [1, 8] and then show that the error can be bound by a robustly positive invariant set [9]. This error set is then explicitly accounted for by a robust tube-based MPC to guarantee satisfaction of the system constraints [5].

[1] E. N. Pistikopoulos, ?Perspectives in multiparametric programming and explicit model predictive control,? AIChE Journal, vol. 55, no. 8, pp. 1918?1925, 2009.

[2] A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos, ?The explicit linear quadratic regulator for constrained systems,? Automatica, vol. 38, no. 1, pp. 3?20, Jan. 2002.

[3] C. V. Rao, ?Moving horizon strategies for the constrained monitoring and control of nonlinear discrete-time systems,? Ph.D. dissertation, University of Wisconsin?Madison, 2000.

[4] J. B. Rawlings and D. Q. Mayne, Model Predictive Control: Theory and Design. Cheryl M. Rawlings, Nob Hill Publishing, 2009.

[5] D. Mayne, S. Rakovic, R. Findeisen, and F. Allgöwer, ?Robust output feedback model predictive control of constrained linear systems,? Automatica, vol. 42, no. 7, pp. 1217?1222, July 2006.

[6] A. Alessandri, M. Baglietto, and G. Battistelli, ?Receding-horizon estimation for discrete-time linear systems,? IEEE Transactions on Automatic Control, vol. 48, no. 3, pp. 473?478, 2003.

[7] A. Voelker, K. Kouramas, and E. N. Pistikopoulos, ?Simultaneous state estimation and model predictive control by multi-parametric programming,? in 20th European Symposium on Computer Aided Process Engineering, Ischia, Naples, Italy, June 2010.

[8] M. L. Darby and M. Nikolaou, ?A parametric programming approach to moving-horizon state estimation,? Automatica, vol. 43, no. 5, pp. 885?891, May 2007.

[9] S. Rakovic, E. Kerrigan, K. Kouramas, and D. Mayne, ?Invariant approximations of the minimal robust positively invariant set,? IEEE Transactions on Automatic Control, vol. 50, no. 3, pp. 406?410, 2005.