(257e) On Multiparametric/Explicit NMPC for Quadratically Constrained Problems

Providing exact solutions to the multiparametric nonlinear programming problem (mpNLP) has been proven a challenging task. Even though there has been encouraging work [1-3], most of the major subsequent efforts have focused on providing approximate solutions to the problem. For example, [4], an outer-approximation of the mpNLP is created through the linearization of the nonlinear terms of the objective function and the constraints. In [5,6] a quadratic approximation to the objective function and linear approximations to the constraints are obtained and the mpNLP is approximated by a mpQP, where an explicit solution can be found. An algorithm for the solution of nonlinear parametric optimization of polynomial functions subject to polynomial constraints based on cylindrical algebraic decomposition was presented in [7]. More recently, [8] proposed the decomposition of a mpMINLP into a series of mpQPs using quadratic approximations of the objective function, while [9,10] focused on multiparametric polynomial programming (mpPP) and based on symbolic calculations the analytical solution of the optimality conditions is obtained.

We present the expansion of the Basic Sensitivity Theorem to a second order Taylor expansion approach and the implications to explicit model predictive control of quadratically constrained systems. The expansion enables the derivation of an algorithm for the analytical solution of convex multiparametric quadratically constraint programming (mpQCQP) problems and explicit quadratically constrained NMPC problems. We derive the analytical parametric expressions of the control actions for a quadratically constrained MPC problem and its corresponding critical regions. We also expand the case to problems with non-convex quadratic constraints. We finally show the piecewise non-linear form of the solution and closed-loop validation of the results.

References

[1] Fiacco, A., Sensitivity analysis for nonlinear programming using penalty methods (1976) Mathematical Programming, 10(1), 287-311.

[2] Fiacco, A., Sensitivity analysis and stability analysis in nonlinear programming (1983).

[3] Fiacco, A., Kyparisis, J., Convexity and concavity properties of the optimal value function in parametric nonlinear programming (1986) Journal of optimization theory and applications, 48(1), 95-126.

[4] Dua, V., Pistikopoulos, E. N., Algorithms for the solution of multiparametric mixed-integer nonlinear optimization problems (1999) Industrial and Engineering Chemistry Research, 38(10), 3976-3987.

[5] Johansen, T., On multi-parametric nonlinear programming and explicit nonlinear model predictive control (2002) Volume 3, 2768-2773.

[6] Grancharova, A., Johansen, T., Tondel, P., Computational aspects of approximate explicit nonlinear model predictive control (2007) Lecture Notes in Control and Information Sciences, 358(1), 181-192.

[7] Fotiou, I., Rostalski, P., Parrilo, P., Morari, M., Parametric optimization and optimal control using algebraic geometry methods (2006) International Journal of Control, 79(11), 1340-1358.

[8] Dominguez, L., Pistikopoulos, E.N. A quadratic approximation-based algorithm for the solution of multiparametric mixed-integer nonlinear programming problems (2013) AIChE Journal, 59(2), 483-495.

[9] Dua, V. Mixed integer polynomial programming (2015) Computers and Chemical Engineering, 72, 387-394.

[10] Charitopoulos, V., Dua, V., Explicit model predictive control of hybrid systems and multiparametric mixed integer polynomial programming (2016) AIChE Journal, 62(9), 3441-3460.