PhD Thesis of Wim Michiels

Defence

Place

Auditorium Oude Molen (MOLE 00.00)
Kasteelpark Arenberg 50
3001 Heverlee (Leuven)
Belgium

Date

Wednesday May 22, 2002 at 10.30

Title

Stability and stabilization of time-delay systems

Promotors

Prof.Dr.ir. D. Roose and Prof.Dr.ir. R. Sepulchre

Jury members

Prof. Dr. ir. J. Berlamont, chairman (K.U. Leuven, Dept. Civil Engineering)
Prof. Dr. ir. D. Roose, promotor (K.U. Leuven, Dept. Computer Science)
Prof. Dr. ir. R. Sepulchre, promotor (University of Liège- Montéfioré Institute)
Prof. Dr. ir. J. Vandewalle (K.U. Leuven, Dept. Electrical Engineering)
Prof. Dr. ir. S. Vandewalle (K.U. Leuven, Dept. Computer Science)
Prof. Dr. ir. B. De Moor (K.U. Leuven, Dept. Electrical Engineering)
Prof. Dr. ir. S.-I. Niculescu (University of Compiègne, Heudiasyc)

Some pictures

Slides of the presentation (in Dutch)

Abstract

Time-delays are important phenomena in industrial processes, economical and biological systems. For instance, they appear as transportation and communication lags and also arise as feedback delays in control loops. Because time-delays have a major influence on the stability of such dynamical systems, it is important to include them in the mathematical description, leading to a modelling with delay differential equations. In this thesis we consider the stability and stabilization of systems described by this type of differential equations. It consists of two parts:

In the first part we describe constructive eigenvalue based methods for both the stabilization and the robust stabilization of linear time-delay systems. First we develop a new numerical stabilization procedure, which extends the classical pole placement method for ordinary differential equations. Unlike methods based on finite spectrum assignment, this procedure does not render the closed loop system finite dimensional but consists of controlling the rightmost eigenvalues only. Because these are moved to the left half plane in a (quasi-)continuous way, we call our method continuous pole placement. We explain the method and discuss its theoretical properties by means of the stabilization of a linear finite dimensional single input system in the presence of an input delay, using static state feedback. Generalizations to dynamic state feedback and multiple input, multiple output systems are provided. Next, we discuss various extensions and applications of delayed state feedback and the continuous pole placement method. We perform a numerical case-study on the effect of input delays on the stabilizability of second order linear system with static state feedback, thereby completely characterizing the class of stabilizable systems. The approach relies on a successive application of the continuous pole placement procedure and makes use of elements from numerical continuation and bifurcation analysis. We also discuss the design of modified Smith predictors, applicable to a class of unstable systems, and the stabilization of integrator chains with both input delays and input constraints. Finally, we consider the robust stabilization of linear time-delay systems and present a numerical procedure. We assume static perturbations on the system matrices and express the robustness of the stability in terms of stability radii. In the numerical procedure, these stability radii are maximized as a function of the controller parameters. This corresponds to a H-infty synthesis problem, which is solved by a quasi-continuous shaping of some frequency response plots. The structure of the iterative procedure is similar to the structure of the continuous pole placement algorithm.

In the second part of the thesis we study the stability of the series interconnection of nonlinear time-delay systems.We first analyze the effect of bounded input perturbations on the stability of a class of nonlinear globally asymptotically stable delay differential equations which are affine in the input. We investigate under which conditions all solutions of the perturbed system are bounded and, if this is not possible, whether semi-global boundedness results can be achieved by controlling the size or shape of the perturbations. Second, we use these boundedness results to study the stabilization of partially linear cascade systems with partial state feedback. In this study, the peaking phenomenon plays a prominent role. As a mathematical tool, both Lyapunov based and trajectory based proof techniques are dealt with.

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