Robust pseudo-inversion of polytopic systems using second order cone programming

Abstract

This paper deals with the almost exact set point regulation for Linear Time-Invariant (LTI) continuous-time polytopic systems. To this purpose, a novel robust pseudo-inversion based feedforward control is proposed. The method is based on a two Degrees of Freedom (2DoF) control scheme where the output $r(t)$ of a pseudo-inverting feedforward filter $\Sigma_{ff}$ is used as input forcing a closed-loop system $\Sigma_{f}. \Sigma_{f}$ is the feedback connection of an LTI continuous time polytopic plant $\Sigma_{p}$ with an LTI robustly stabilizing controller $\Sigma_{c}$ . In steady-state condition, an exact tracking is achieved endowing $\Sigma_{c}$ of the internal model of constant signals. In the transient state an optimal tracking is obtained computing the transient component $r_{t}(t)$ of $r(t)$ through the on-line minimization of a worst case quadratic cost functional of the transient tracking error. To improve the numerical efficiency of the optimization procedure, the transient input $r_{t}(t)$ is modeled as a B-spline. These functions are universal approximators which admit a parsimonious parametric representation so that the minimization of the worst case cost functional can be formulated as a robust estimation problem. In turn, this problem can be recast as a Second Order Cone Program (SOCP) which can be efficiently solved using primal and dual point methods.