Singular perturbations and time-scale methods in control theory: Survey 1976-1983

In an earlier note it was shown that designing a nonstrictly proper output feedback compensator using a model that neglects parasitic elements might lead to an unstable closed-loop system even when the neglected modes are asymptotically stable. The purpose of this note is to show that this instability phenomenon might take place even when the reduced-order design model is strictly proper.

A further note on the robustness of output feedback control methods to modeling errors

The mathematical treatment of systems with multiple time scales has traditionally employed the use of singular perturbation theory for differential equations. This paper treats such systems from a frequency-domain viewpoint. Previously existing two-frequency-scale results, involving approximation and algebraic characterization, are extended. In addition, an exact frequency-scale decomposition is presented.

Frequency domain results for systems with multiple time scales

This paper discusses typical applications of singular perturbation techniques to control problems in the last fifteen years. The first three sections are devoted to the standard model and its convergence, stability and controllability properties. The next two sections deal with linear-quadratic optimal control and one with cheap (near-singular) control. Then the composite control and trajectory optimization are considered in two sections, and stochastic control in one section. The last section returns to the problem of modeling, this time in the context of large scale systems. The bibliography contains more than 250 titles.

Singular perturbation techniques in control theory

It is pointed out that the study of systems evolving at multiple time-scales is simplified by studying reduced-order models of these systems valid at specific time-scales. The present investigation is concerned with an extension of results on the time-scale decomposition of autonomous systems to that of input-output systems. The results are employed to study conditions under which positive realness of a transfer function is preserved under singular perturbation. Attention is given to the perturbation theory for linear operators, the multiple time-scale structure of autonomous linear systems, the input-output description of two time-scale linear systems, the positive realness of two time-scale systems, and multiple time-scale linear systems.

Input-output description of linear systems with multiple time-scales†

Many systems to which modern automatic control techniques are applied exhibit multiple time scale behavior Recent work has shown that the control task can be simplified through model reduction or decomposition methods which rely on such behavior Application of modern control methods to complicated system models can lead to complicated or oversensitive controllers The authors show why decomposition of control problems into slow and fast portions is justified and present some theorems relating to the problem Most decomposition and reduction methods employ singular perturbation theory as their mathematical basis This paper approaches the problem from a frequency-domain viewpoint

D.W. Luse

Papers

A Frequency Domain Approach for Systems with Slow and Fast Modes