A spline-based technique for optimal set point regulation through pseudo-inversion of nonminimum phase linear systems

Abstract

This paper considers the optimal output set-point regulation for MIMO, non minimum phase sampled data systems. The usually proposed methods are based on stable model inversion whose exact solution is approximated through preview based implementation schemes. The new approach proposed here considers the meaningful practical situation of plants with a given, possibly uncertain, initial state, that can not be modified through pre-actuation. The structure of the optimal control input is ”a priori” assumed to be given by a smoothing spline function. In this way a twofold objective is achieved: a smooth behavior of the control input and its derivatives can be imposed, a very accurate tracking performance can be obtained by reducing the mesh size of spline [1]. Given the desired transient output response between two fixed set points, the spline coefficients are determined as the least-squares solution of the over determined system of linear equations obtained imposing that the spline function assumed as control input yields the specified output. In this way an optimal least square approximation of the desired output trajectory is obtained avoiding the stable explicit model inversion. Rather, this operation is implicitly approximately performed solving for the spline coefficients, the over-determined system of linear equations carrying the information on the model to be inverted and on the desired output. An interesting feature of this new method is that it also works for linear systems which are not required to be either square or right invertible.