Singular Perturbation Margin Assessment for LTI System with Zero Dynamics

Abstract

Singular perturbation approach is an advanced method for nonlinear system stability analysis and control system design. In recent years, the notion of singular perturbation margin has been introduced which, when specialized to Single-Input, Single-Output (SISO) Linear Time-Invariant (LTI) systems, has been found to have a bijective relationship with the phase margin for all-pole systems. A simple yet effective method for assessing the singular perturbation margin in terms of phase margin of the (unperturbed) nominal system using time-scale or bandwidth-scale separation has also been reported for all-pole LTI systems, which can be used as a design tool for Multiple-Timescale-Nested-Loops (MTNL) systems. In this paper, we will extend the earlier results to LTI systems with stable zero-dynamics (non-minimum phase zeros). First, a bijective relationship between the Singular Perturbation Margin (SPM) ε <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</inf> the PM of the nominal system is established with an approximation error on the order of O(ε <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). To prove this result, relationships between the singular perturbation parameter ε, PM of the perturbed system, PM of the nominal system with the net phase change due to phase lead by zeros, and phase lag by poles of the fast system is revealed. Finally, an easy yet practically significant method for estimating the nominal systems PM loss is obtained. Two academic examples are offered to demonstrate the correctness and applications of the theoretical results. The theoretical results have been applied to 3-DOF trajectory tracking control design of wheeled ground vehicles and validated on a full-scale Kia Soul EV, which is also presented.