Basis functions design for the approximation of constrained linear quadratic regulator problems encountered in model predictive control

In earlier work, we proposed to approximate the constrained linear quadratic regulator problem by representing input and state trajectories as linear combinations of basis functions. As a result, a relatively low-dimensional optimization problem is obtained, while at the same time an infinite prediction horizon can be retained, providing benefits regarding computation and closed-loop stability when applied to model predictive control. In the following, we will extend the approach and present an optimization strategy to design basis functions that are tailored to the specific problem instance. We will show on a simulation example that the same approximation quality of the underlying problem can be achieved with fewer basis functions, which reduces the problem dimension and the computational load even further. The optimization of the basis functions can be done offline. We also discuss the constraint-free case and show that, by choosing the basis functions appropriately, the solution of the underlying linear quadratic regulator problem is recovered by the approximation.