(65b) A New Approach To Explicit Mpc

The nullspace method of self-optimizing control is here used as an alternative approach to obtain explicit MPC state feedback control laws. The insights obtained from this approach is used to propose an alternative local method for identifying the critical regions.

Introduction

For a quadratic cost function and linear constraints on states and inputs it is known that the optimal control law is piece-wise affine in the states, u = K_i x + G_i (Bemporad et al., 2002). This explicit MPC solution can be found using parametric programming. The solution of a parametric program consists of a set of critical regions where the optimal set of active constraints remains unchanged. The critical regions are polyhedra in the parameter space. In each region i there is an optimal affine control law.

A key issue in implementation is to identify the region i that contains the current state. This is called the point location problem (de Berg et al., 2000). When this is known the optimal control can be implemented. Previous work includes application of a binary search tree (Tøndel et al., 2003) and reachability analysis (Spjøtvold et al., 2006). Typically these methods require significant on-line storage space and computing time.

Alternative approach: Self-optimizing control and the nullspace method

Self-optimizing control is when acceptable operation under all conditions is achieved with constant setpoints for the controlled variables. As long as the set of active constraints remains unchanged then for a QP problem we can find an optimal linear measurement combination that yields zero loss for the case with no implementation error (nullspace method, Alstad and Skogestad, 2007). Since the set of active constraints remains unchanged throughout each region the null-space method can be used to find linear combinations c that yield zero loss for each region.

Let c_i be the self-optimizing variable for region i, and let c_J be self-optimizing variables for neighboring regions, and let j be in the index set J. Since c_j is zero in neighboring region j, it will optimally be different from zero in region i, since the optimal active set is different. Due to continuity it will however be zero on the boundary. This means that by tracking variables c_J for neighboring regions it is optimal to switch to region j when c_j changes sign. This insight may be used as an alternative method for identifying the critical regions.

Details on the application to explicit MPC

The discrete form of a MPC problem with quadratic cost function and linear constraints can be written as a QP problem (Rawlings and Muske, 1993). The optimal solution is a control law of the state feedback form u = K_i x + G_i. This implies that the affine combination c_i = u - (K_i x + G_i) is optimally zero inside region i, whilst in a neighboring region other K and G matrices yield zero loss. Given the set of active constraints for each region, the same control laws can be found applying the null-space method for each region with the measurement combination y = [u x], where x is the state at time k. Hence, can be understood as a special case of self-optimizing control. The main assumption is that the present state contain all the information about the disturbances.

Assuming all states are known at time t, it is therefore optimal to:

1.) In each region apply control such that c_i = u - (K_i x + G_i) is zero.

2.) Track the neighboring feedback laws c_J = u - (K_J x + G_J).

3.) When one of the neighboring c_j's changes sign, switch controller such that c_j = u - (K_j x + G_j) becomes zero.

It is implicitly assumed that the process and disturbances are such that the state can only move into a neighboring region between each time instant.

Examples

To illustrate ideas examples of both SISO and MIMO control problems will be presented.

Discussion

In this work we present a method for using the same variables both for control and switching. After the parametric program has been solved there is no need to store the hyperplanes separating the regions or other geometrical information, as this is implicitly contained in the state feedback law for the neighboring regions. The nullspace method can easily be extended to cover output feedback.

References

A. Bemporad, M. Morari, V. Dua, E. N. Pistikopoulos, "The explicit linear quadratic regulator for constrained systems," Automatica vol. 38 pp. 3-20, 2002

M. de Berg, M. van Kreveld, M. Overmars, O. Schwarzkopf, "Computational Geometry, Algorithms and Applications," Second Edition, Springer, Berlin 2000

P. Tøndel, T. A. Johansen, and A. Bemporad, ?Computation of piecewise affine control via binary search tree,? Automatica, vol. 39, no. 5, pp. 945?950, 2003.

J. Spjøtvold, S. V. Rakovic, P. Tøndel and T. A. Johansen, "Utilizing Reachability Analysis in Point Location Problems," In Proc of the IEEE Conf. Decision and Control, San Diego, 2006

V. Alstad and S. Skogestad, "Null Space Method for Selecting Optimal Measurement Combinations as Controlled Variables," Ind. Eng. Chem. Res. 2007, 46, 846-853

Kenneth R. Muske and James B. Rawlings, "Model predictive control with linear models," AIChE J., 39(2):262-287, February 1993.