Stability and Stabilization of Linear Systems with Time-varying Delays

Abstract

The dissertation contributes to stability and stabilization problems of linear systems subject to unknown, possibly time-varying delays. The issues under investigation are mainly: 1. How to determine the stability of systems when they contain time-varying delays? 2. What is the largest range of delay such that there exists a feedback controller that can stabilize all linear plants subject to delays within the range? Concerning the first issue, the dissertation sets out to cast the stability problem as one in robust stability analysis, and develop L2 and L∞ type stability conditions reminiscent of robust stability bounds typically found in robust control theory. The development is built on the well-known conventional robust stability analysis. Other than their conceptual appeal, these conditions can be checked using standard robust control toolboxes. For the stabilization problem, drawing upon analytic interpolations and rational approximation techniques, the dissertation develops fundamental bounds on the delay margin of linear time-invariant (LTI) systems subject to constant delays, within which the delayed plant is guaranteed to be stabilizable by a single LTI controller. For single-input single-output systems, these bounds can be computed efficiently, requiring computing only the largest real eigenvalue of a constant matrix. For multi-input multi-output systems, estimates on the variation ranges of multiple delays can be obtained by solving LMI problems, and further, by computing the radius of delay variations. Furthermore, the bounds and estimates can be extended to systems with time-varying delays. When specialized to more specific cases, e.g., to plants with one unstable pole and one nonminimum phase zero, our results give rise to analytical expressions exhibiting explicit dependence of the bounds and estimates on the pole and the zero, thus demonstrating how fundamentally unstable poles and nonminimum phase zeros may limit the range of delays over which a plant can be stabilized.

Date of Award	6 Oct 2015
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Jie CHEN (Supervisor)