Predavanje Function Approximation Techniques in Modelling and Control

In this talk we review two techniques for approximating a given nonlinear function by a different function of desired properties. The properties we are interested in are the fixed complexity of the approximate function and optimality of the approximation, expressed as the minimization of the integrated squared error. The first method allows to approximate arbitrary smooth nonlinear functions by piecewise affine (PWA) functions with a given number of regions. Parameters of the optimal approximation are searched for by solving a nonlinear optimization problem. We will also illustrate how this technique can be extended to derive optimal PWA approximations of dynamical systems whose vector fields are nonlinear, hence obtaining a well-performing, yet simple model that can be used to accelerate simulations and to simplify control synthesis. The second method allows to search for simple PWA functions which approximates a different, more complex PWA function optimally. We illustrate how this approach can be used to derive simple feedback laws that provide guarantees of recursive feasibility and closed-loop stability, yet have lower complexity compared to traditionally used explicit MPC approaches. We show that the search for the optimal approximation in this case boils down to solving a series of convex optimization problems.