Students: Dissertation: Michael Schilmoeller

Michael Schilmoeller

ABSTRACT
Linear, time-invariant models are used to represent diverse dynamical systems, both natural and man-made. For example, electric power transmission control systems, the human body’s regulation of blood sugar, and the federal reserve’s efforts to the control of inflation can be represented as multivariate linear control systems.

This dissertation addresses the equivalence of linear, time-invariant dynamical systems under output feedback. These systems have the following state-space description:

where x, u, and y are vectors in the state space, input space, and output space, respectively, and A, B, and C are matrices of appropriate dimension. The ordered triple of matrices (A,B,C) completely determine such a system. Choice of output feedback corresponds to picking a matrix K representing a linear transformation of the output vector

so that

Two systems are taken to be equivalent if one can be transformed into the other by some combination of state space, input space, or output space change of basis or by change under output feedback. If (A,B,C) and (A’,B’,C’) represent two systems, asking whether their associated systems are equivalent corresponds to asking whether there exist a matrix K and non-singular matrices T,G,H, such that

The question of the equivalence of two systems is relevant when an engineer is trying to determine whether two systems can be made to achieve the same dynamical behavior through some choice of transformation under output feedback. Equivalence classes are the sets of all triples that are equivalent under the preceding transformation.

Invariants can distinguish between equivalence classes of systems. For simple problems they can facilitate direct calculation of equivalent systems. A complete invariant provides a unique label for each equivalence class. Discovering a complete invariant is an important step toward classifying all linear systems under change in output feedback. Complete invariants may also provide insights that would lead to discovering a canonical form for the triples of these systems.

The equivalence classes of triples (A,B,C) are orbits under an algebraic group action on an affine variety. The first major result of this dissertation is that for any algebraic group action on an affine variety, we find there exists a unique list of polynomials that identify the orbits. That is, the rule for constructing these lists is a complete invariant of the orbits. The polynomials are the reduced Groebner bases of radical ideals for the unique sequence of varieties that define the orbits. The polynomials describe a sequence the varieties, decreasing with respect to set containment, which alternately contain and exclude points of the orbit. The second major result of this dissertation is a set of new mathematical constructions that reduce the complexity of the output feedback classification problem and assist the computation of the list of polynomials that identify the equivalence classes.

Tuesday, June 13, 2000
DISSERTATION COMMITTEE
Joyce O’Halloran, Chair
Gerardo Lafferriere
George G. Lendaris
Serge Preston
Richard Tymerski, Graduate Studies Rep.