Zero dynamics assignment and its applications to the stabilization of linear time-varying systems

Abstract

An algorithm to select an output for desired vector relative degree, which is not restricted to be uniform (the same for all outputs), and zero dynamics in linear time-varying (LTV) multi-input multi-output (MIMO) systems is proposed. The proposed algorithm is developed under reasonable assumptions of uniform (with respect to time) and lexicography-fixed controllability (with some choice of fixed controllability indices) and is inspired by a generalization of the Ackermann’s formula. One application of the proposed algorithm is the context of higher-order sliding-mode control for perturbed LTV MIMO systems, achieving finite-time stability. The developed algorithm also provides an extension of the classical Ackermann–Utkin formula for outputs of uniform relative degree one and includes the design of flat outputs for LTV MIMO systems. Hence other application is the feedback controller design for trajectory tracking task. Finally, a complete HOSM methodology is provided and tested in an LTV model of the non-holonomic car.