City Research Online

Abstract

The thesis presents results for quadratic linear systems related to two fundamental control problems, namely controllability and pole assignment under state feedback. First, the controllability condition for second-order systems is formulated in low dimensions involving only a rank condition on a second-order matrix pencil. Second, a new necessary and sufficient condition for complete controllability for symmetric quadratic linear dynamical systems is derived. The proof is based on the notion of compound matrices and uses the Binet–Cauchy formula. Next, an algorithmic method is proposed for the solution of the pole assignment problem associated with symmetric quadratic dynamical systems that are completely controllable. This problem is shown to be equivalent to two subproblems, one linear and the other multilinear. Solutions of the linear problem must be decomposable vectors, i.e. they must lie in an appropriate projective variety known as the Grassmann variety. The proposed method computes a reduced and independent set of quadratic Plücker relations with only three terms, each of which completely describes the specific Grassmann variety. The advantage of this approach is that the complete set of feedback solutions is obtained, over which further optimisation can be carried out if desired. The method is then extended to higher-order linear dynamical systems. Next, an alternative approach to the solution of the pole assignment problem under output feedback is developed, which relies on necessary and sufficient controllability and observability conditions. A methodology for designing static output feedback controllers for linear time-invariant systems is presented so that the control system meets performance specifications set by the designer. Moreover, a study of the solution space and the properties of homogeneous and non-homogeneous regular generalized linear systems is developed. The case of both consistent and non-consistent initial conditions is addressed, and solutions exhibiting impulsive behaviour are fully analysed. Following this, the solution space and properties of homogeneous and non-homogeneous generalized linear discrete-time systems are presented. Using strict equivalence transformations, the form of these systems is simplified and the resulting solution is easier to obtain. It is shown that, under the assumption of regularity of the corresponding matrix pencils, the solution of the system is unique. The theoretical results presented in the work are illustrated by several numerical examples based on concrete linear-algebraic algorithms that have been developed and analysed.

Publication Type:	Thesis (Doctoral)
Subjects:	Q Science > QA Mathematics T Technology > TK Electrical engineering. Electronics Nuclear engineering
Departments:	School of Science & Technology > Department of Engineering School of Science & Technology > School of Science & Technology Doctoral Theses Doctoral Theses

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