Multi-parametric Programming for Model Predictive Control

Abstract

Model predictive control (MPC) solves a quadratic optimization problem to generate control law in each step. The usual methods of solution for quadratic optimization problem are interior point method, active set method etc. But most of the techniques are computationally heavy to perform the job in small amount of time. So a method is required where on-line computation is less. In multi-parametric quadratic programming (mp-QP) method an off-line computation is done a prior and a binary search tree is prepared. The on-line computation mainly involves a search through the binary-tree. The mp-QP is suitable for the class of optimization problem, where the objective function is to minimize or maximize a performance criterion subject to a given set of constraints where some of the parameter vary between lower and upper bounds. Also mp-QP is suitable for multi-objective optimization, where multi criteria problems can be reformulated as multi-parametric programming problems and a parametrized optimal solution is obtained. Multi-parametric programming is a technique for obtaining: (i) the objective and optimization variable as functions of the varying parameters and (ii) the regions in the space of the parameters where these functions are valid. The newly developed convex optimization solver CVXGEN is utilized successfully for off-line calculations which involves of dividing the parameter space into different polyhedral regions.In each one, the objective function has a constant value. The process involves another kind of optimization problem. For CVXGEN, worst case solving time is in milliseconds, even for a large problem.Thus, the use of CVXGEN minimizes the off-line calculation in mp-QP technique. In this work, an input constraint MPC problem is chosen from existing literature. The problem is solved for both two step prediction and three step prediction cases.The control input and states are ploted for both the MPC problems, and the results are compared.