(12e) Interior Point Solution of Multilevel Qp Problems Arising in Embedded Mpc Formulations

This study examines the use of an interior point strategy to solve multilevel quadratic programming problems that arise from including closed-loop formulations of constrained model predictive control (MPC) within quadratic programming problems. We motivate the formulation through its application to the constraint back-off problem, although the strategy is applicable to several problem types.

The steady-state economic optimum in process plants generally lies at the intersection of two or more constraints. However, in order to avoid constraint violations in the presence of unmeasured disturbances, it is necessary that the operating point be moved some distance from the constraints into the feasible region. The calculation of this constraint back-off may be posed as a dynamic optimization problem in which an economic criterion is optimized subject to the closed-loop trajectories satisfying input and output constraints. This problem may be extended to integrated design and control in which the decision space includes equipment design parameters, control structure and controller tuning.

The majority of studies in the constraint back-off, as well as the more comprehensive integrated plant and control system design problems, have restricted the choice of control system to linear controllers. Baker and Swartz (2004a) considered actuator saturation effects and proposed a formulation that uses complementarity constraints and that was suitable for inclusion in a simultaneous optimization framework. The resulting problem was solved via mixed-integer programming (MIP), and later solved using an interior point strategy (Baker and Swartz, 2004b). In this paper, we consider constrained MPC as the regulatory control system. Since a quadratic programming problem needs to be solved at every sampling period, the resulting problem is multi-level in nature. Here, we focus attention on the constraint back-off problem with a quadratic objective which results in a multilevel quadratic programming problem.

The formulation approach used in this paper is to include the Karush-Kuhn-Tucker (KKT) conditions corresponding to the MPC quadratic programming sub-problems as equality constraints within a single level optimization problem that minimizes the back-off subject to constraints on the closed-loop input and output trajectories. However, the inclusion of complementary constraints within a normal optimization problem results in a mathematical program with complementarity constraints (MPCC), which in general cannot be reliably solved using standard nonlinear programming solvers. The approach followed in this study is to use an interior point implementation tailored to such problems in that the complementarity constraints in the original (primal) problem are treated in the same manner as those arising from the KKT conditions, rather than treating them as general nonlinear constraints. This is a strategy used in IPOPT-C, an algorithm and software implementation developed by Raghunathan and Biegler (2003). An alternative strategy is to rewrite the complementarity constraints using binary variables. This results in the back-off problem with embedded MPC being reformulated as a mixed-integer quadratic program (MIQP).

This back-off formulation is illustrated on two case studies: a continuous stirred tank reactor and a fluid catalytic cracking unit. The efficiency of the interior point and MIQP solution strategies are compared for an increasing number of complementarity constraints (and binary variables) generated by using a sampling period of decreasing duration. For problems with few integer variables, similar CPU times are obtained, but as the number of binary variables is increased, the interior point strategy significantly outperforms the MIQP method. Moreover, in the studies performed, the interior point algorithm always converges to the MIQP solution (known to be globally optimal), suggesting it to be reliable as well as efficient. Other applications that might benefit from this formulation are discussed.

References

Baker, R. and C.L.E. Swartz, ?Rigorous handling of input saturation in the design of dynamically operable plants,? Ind. Eng. Chem. Res., 43(18), 5880-5887 (2004a)

Baker, R. and C.L.E. Swartz, ?Simultaneous solution strategies for inclusion of input saturation in the optimal design of dynamically operable plants,? Optimization and Engineering, 5, 5-24 (2004b)

Raghunathan, A.U. and L.T. Biegler, ?Mathematical programs with equilibrium constraints (MPECs) in process engineering,? Comput. Chem. Eng., 27 (10), 1381-1392 (2003)