An illustration of a quasi-interpolation driven technique for feedback synthesis

This article illustrates an offline approximate feedback synthesis algorithm on one of the simplest unconstrained optimal control problems — the continuous-time linear quadratic regulator (LQR). The driving engine behind the approximation algorithm is a special kind of quasi-interpolation technique based on the theory of approximate approximations. The explicit feedback control is approximated via the quasi-interpolation scheme on a uniform-cardinal grid over the phase-space. This method provides theoretical guarantees for the uniform i.e. L∞ convergence of the approximated feedback law to the actual optimal feedback, up to some saturation error, which can be reduced to an arbitrary number. Although the analytical expression for the LQ feedback is available, we present the technique in this simple case to serve as an illustrative test bed for the algorithm. We provide a couple of numerical examples to illustrate our results. The algorithm has the potential to be employed for tight uniform approximation of optimal feedback maps under constraints and will be reported subsequently.