Linear time-varying systems: Algebraic structure, system properties and control

Date

1987-08

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Publisher

Texas Tech University

Abstract

The purpose of this dissertation is two-fold. First, a novel algebraic framework for linear continuous-time time-varying systems based on the fractional representation approach is presented (mainly from C14]). The main structure of this realization and control theory is a module of signals aver a skew polynomial ring consisting of time-varying differential polynomials. The realization problem based on the formal Kalman input-output map (as right module homomorphism) is considered. This leads to the Hankel matrix introduced by Kalman (see C35]). It is shown that reachability and observability can be characterized by coprime polynomial factorization conditions. The relation between the formal impulse response and Rosenbrock's system matrix is also closely investigated. Secondly, several systemtheoretic properties such as stabilizability and detectability and their relations for the class of constant rank systems are explored. One of the primary tools used in this development is an improved version of Dolezal's theorem. It is shown that stabilizability (resp. detectability) and asymptotic controllability (resp. asymptotic observability) are equivalent. The concept of open-loop stabilizability is introduced and its relation to the steady state solution of the matrix Riccati equation is examined. It is also shown that four notions of detectability for constant rank systems are eguivalent to each other.

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