Explicit Model Predictive Control for Higher Order Systems
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The idea of viewing an optimal control problem as a parametric program has introduced new areas of use for control schemes such as Model Predictive Control (MPC). Some classes of multi-parametric programs have the desirable property of the solution being a piecewise affine (PWA) function which applies for multi-parametric linear programming (mpLP), multi-parametric quadratic programming (mpQP) as well as their mixed integer counterparts mpMILP and mpMIQP. This property gives us the ability to finitely represent and compute the explicit solution for an MPC problem. Moreover, a PWA function is well-suited for real-time applications with fast dynamics, i.e. having small sampling time, as the computational cost and required time for evaluating a PWA function is low. This thesis treats topics within solving a multi-parametric quadratic programming. The main motivation of our work on multi-parametric programming has been to extend the applicability of this approach for higher order systems where the number of state variables is large. This includes modifying both offline and online algorithms existing in literature for solving a multi-parametric quadratic programming as well as for retrieving the computed solutions for online use in the form of a look-up table. The thesis is therefore structured as below: Chapter 1 This chapter briefly introduces the model predictive for control design as well as its advantages and its implementation challenges. The concept of multi-parametric quadratic programming and the transition from a linear MPC to an mpQP is also discussed. Chapter 2 This Chapter reviews some of the most successful algorithms in literature for 1) solving the resulting mpQP offline and 2) retrieving the control action during the online evaluation of the solution. Chapter 3 Chapter 3 presents a new approach for exploring the combinatorial tree in a combinatorial algorithm for non-degenerate mpQPs. The objective is to eliminate, as many as possible, the feasible combinations of active constraints from the combinatorial tree. A simple downward and upward algorithm is suggested which guarantees enumeration of all optimal active sets in non degenerate cases. Chapter 4 The problem of degeneracy is addressed in Chapter 4 where two alternative treatments are suggested for degenerate problems. The first alternative uses geometric operations while the second one is based on developing theoretical properties of adjacent critical regions which show degeneracy on their common facets. Chapter 5 In Chapter 5, the online evaluation of the obtained MPC solution is addressed and a region-free point-location algorithm is suggested using the steps of the qpOASES (Online Active SEt Strategy for quadratic programming).