First Advisor

Pieter Frick

Date of Publication


Document Type


Degree Name

Doctor of Philosophy (Ph.D.) in Electrical and Computer Engineering


Electrical Engineering




Automatic control, Control theory, System analysis



Physical Description

4, viii, 148 leaves: ill. 28 cm.


This dissertation addresses the design of controllers for multi variable finite-dimensional, linear, autonomous dynamical systems with distinct sets of slow and fast dynamics, which thus display multiple time scale behavior. It seeks, specifically, to compose overall controllers from lower-order dynamic output compensators, which are designed separately for slow and fast approximating models of the given plant. Reduction of the dynamic order of the design problem and the avoidance of numerically ill-conditioned interaction between modes of disparate orders of magnitude are among the patent advantages which pertain to such a design. As is well known, the explicit singularly perturbed systems, as a class, possess the multiple time scale property, while the broader class of implicit singularly perturbed systems and the multiple time scale systems are partially overlapping system classes. A composite state feedback controller scheme for the explicit singularly perturbed system has long been known. In connection with dynamic output controllers, however, only the case of the explicit singularly perturbed system with, restrictively, open-loop-stable fast dynamics has so far received attention in the literature. The dissertation, in providing a composite dynamic controller design suitable as well to the implicit singularly perturbed multiple time scale system, which furthermore is permitted to exhibit fast (or "parasitic") as well as slow (or "normal") open-loop instabilities, thus presents a more comprehensive dynamic controller strategy for this system than so far reported in the literature. Working with multivariable transfer functions, the dissertation applies certain fractional representation techniques of modem Algebraic System Theory to the frequency domain study of the multiple time scale system. Following the work of D. W. Luse and H. K. Khalil, we replace the transfer matrix of the multiple time scale system with two or more lower-order transfer functions, each of which has validity, in its own respective frequency range, as an approximation to the first. Following the work. of M. Vidyasagar, we write the rational transfer function of each of these approximating lower-order subsystems as a "fraction" over the Ring of proper and stable rational matrices. Parametrizations, in terms of "free" matrices belonging to this Ring, of the sets of stabilizing controllers for the lower-order subsystems :md the corresponding achievable stable closed-loop behaviors then enable the relevant design syntheses to be achieved. In this development we exploit, specifically, a Theorem proved by Luse and Khalil concerning the relation of the closed-loop poles in a feedback configuration of multiple time scale systems to the poles in corresponding lower-order closed-loop systems. The dissertation's novel contribution thus resides in (i) interpreting the Theorem of Luse and Khalil as the outline for a possible separate and-composite dynamic output controller strategy, and in (ii) adapting algebraic techniques derivative from Vidyasagar for actually realizing the putative strategy as a set of concretely implementable design procedures. Three specific design procedures are developed and formalized in the dissertation: the first for achieving mere stabilization of the multiple time scale system, the second for the placement of slow and fast poles within specified subregions of the Complex Plane, and the third for achieving entirely arbitrary pole placement. Since these procedures derive, methodologically, from Vidyasagar's fractional representations, they are intrinsically multivariable in character. Since the procedures are validated by the Theorem of Luse and Khalil, they are applicable, in principle, to the very broadest class of linear autonomous multiple time scale systems. The dissertation presents its three design procedures in high-Ievel-algorized form. For application to the explicit singularly perturbed system, the three procedures entail no further matrix operations than addition, multiplication, inversion and the determination of linear constant controller and observer gains, the most basic of operations in any available Control software. In connection with the implicitly singularly perturbed multiple time scale system, their concrete application requires some further computational development pertaining to the attainment of coprime factorizations of general rational matrices, a topic of independent active interest in the current literature on Control. Elsewhere in the literature, singularly perturbed discrete time, distributed, multidimensional, time-varying and nonlinear systems have been studied. Such systems have been studied, furthermore, in several contexts involving optimal and stochastic control, but nearly always from the time domain point of view. The dissertation tackles only the problems of robust stabilization and pole placement in the finite-dimensional, linear, autonomous case. Future work will attempt to extend its results on stabilization and pole-placement, appropriately, to some of the other general multiple time scale system classes. Further frequency domain investigations, related to the dissertation, may as well explore other problems pertaining to the multiple time scale systems which so far have received treatment only in the available time domain literature.


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