# I³MS - Mönningmann Seminar

## Prof. Dr.-Ing. Martin Mönnigmann - To optimize or not to optimize: event-based predictive control

Automatic Control and System Theory, Ruhr University Bochum

### Abstract

Model predictive control is a theoretically sound and practically relevant method for the control of complex systems. Among other aspects, MPC has been acclaimed for its ability to handle constraints. Ultimately, MPC boils down to solving constrained optimization problems. It is usually assumed these problems must be solved numerically. As a result, MPC defines a control law only implicitly, or point-by-point: Solving the optimization problem for the current state x(t_k)∈R^n (a point in state space) results in an optimal input signal u(〖x(t〗_k))∈R^m (a point in input space). By periodically repeating the optimization for the evolving state x(t_1 ),x(t_2 ),x(t_3 ),…, the optimal control law x→u(x) is sampled, resulting in a sequence of points u(x(t_1 ),u(x(t_2 )),u(〖x(t〗_3)),… . Many research projects have been devoted to tailoring optimization algorithms for MPC. Instead, the present talk advocates investigating conditions under which the optimal control law x→u(x) can be characterized more elegantly than by solving optimization problems point-by-point. The central idea is as follows: The solution to the linear-quadratic MPC problem at a point x(t_k) does not only yield the point u(〖x(t〗_k)), but it defines an affine control law u(x)=Kx+b. This affine control law is not the global solution to the MPC problem, but it provides the optimal u(x) on a full-dimensional polytope in state space that contains x(t_k). Consequently, no MPC optimization problem needs to be solved at all, as long as the system stays in the same polytope. A computationally simple event-triggered MPC algorithm can be devised that, loosely speaking, uses the affine control law as long as possible, and triggers computing the next affine control law whenever the current polytope is left. We introduce the central idea for linear MPC, discuss its computational aspects and implications for networked MPC, and give an outlook on extensions to nonlinear MPC.