Model Predictive Control (MPC) for Constrained Nonlinear Systems

Citation

De Oliveira, Simone Loureiro (1996) Model Predictive Control (MPC) for Constrained Nonlinear Systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/MXYW-TB71. https://resolver.caltech.edu/CaltechETD:etd-12192007-112031

Abstract

This thesis addresses the development of stabilizing model predictive control algorithms for nonlinear systems subject to input and state constraints and in the presence of parametric and/or structural uncertainty, disturbances and measurement noise.

Our basic model predictive control (MPC) scheme consists of a finite horizon MPC technique with the introduction of an additional state constraint which we have denoted contractive constraint. This is a Lyapunov-based approach in which a Lyapunov function chosen a priori is decreased, not continuously, but discretely; it is allowed to increase at other times (between prediction horizons). We will show in this work that the implementation of this additional constraint into the on-line optimization makes it possible to prove rather strong stability properties of the closed-loop system. In the nominal case and in the absence of disturbances, it is possible to show that the presence of the contractive constraint renders the closed-loop system exponentially stable. We will also examine how the stability properties weaken as structural and/or parametric model/plant mismatch, disturbances and measurement noise are considered.

Another important aspect considered in this work is the computational efficiency and implement ability of the algorithms proposed. The MPC schemes previously proposed in the literature which are able to guarantee stability of the closed-loop system involve the solution of a nonlinear programming problem at each time step in order to find the optimal (or, at least, feasible) control sequence. Nonlinear programming is the general case in which both the objective and constraint functions may be non-linear, and is the most difficult of the smooth optimization problems.

Due to the difficulties inherent to solving nonlinear programming problems and since MPC requires the optimal (or feasible) solution to be computed on-line, it is important that an alternative implementation be found which guarantees that the problem can be solved in a finite number of steps. It is well-known that both linear and quadratic programming (QP) problems satisfy this requirement.

If a standard quadratic objective function is used and the input/state constraints are linear in the decision variables, then the contractive constraint (which is originally a quadratic constraint) can be implemented in such a way that the optimization problem to be solved in the prediction step of the MPC algorithm is reduced to a QP. Having linear input/state constraints means that a linear approximation of the original nonlinear system has to be used in the prediction as well as in the computation of the contractive constraint. Thus, in order to make the algorithm more easily implementable we introduce the difficulty of having to handle the mismatch between the real nonlinear system and its linear approximation which is used for prediction. In other words, we now have a robust MPC control problem at hand. In this case, it is the contractive constraint which comes to the rescue and allows the MPC controller to stabilize the closed-loop system spite of the linear/nonlinear mismatch, for certain choices of the contractive parameter (the parameter which defines how much "shrinkage" of the states is required during one prediction horizon).

In summary, this thesis is an application of contractive principles to model predictive control and it is dedicated to robust stability analysis, design and implementation of state and output feedback "contractive" MPC schemes.

Thesis Files