This thesis is devoted to a number of structural and behavioural problems in linear multivariable control system theory. The first problem addresses the subject of determination of the finite and infinite frequency structure of a rational matrix. A novel method is proposed that determines the finite and infinite frequency structure of any rational matrix. Some neat and numerically stable algorithms are developed to implement this method. The second problem concerns the resol vent decompositions of a regular polynomial matrix and solutions of regular polynomial matrix descriptions (PMDs). Regarding these fundamental is'sues, three contributions are made therein. Firstly, based on a general resolvent decomposition a complete solution of regular PMDs is presented that takes into account both the non-zero initial conditions of the pseudo state and the non-zero initial conditions of the input. Secondly, two special resolvent decompositions are proposed, both of which are applied to formulate the solution of the regular PMDs. The first one is formulated in terms of the finite, infinite, and the generalised infinite Jordan pairs, which is a refinement of the results given by Gohberg et al. [74] and Vardulakis [25]. The second resolvent decomposition is proposed on the Weierstrass canonical form of the generalised companion matrix of the polynomial matrix. Thirdly, a new characterization of the impulsive free initial conditions of regular PMDs is given and the relationship between the finite and infinite frequency structure of a regular polynomial matrix and its generalised companion matrix is determined. In the third problem a generalization of the chain-scattering representation for general plants is presented. Through the notion of input-output consistency, the conditions under which the generalised chain-scattering representation and the dual generalised chain-scattering representation exist are proposed. Some algebraic system properties of the GCSRs and DGCSRs are studied. The fourth problem is devoted to a new notion of realization of behaviour. We introduce a notion realization of behavior which is shown to be a generalization of the classical concept of a realization of transfer function. By using this approach, the input-output structures of the generalized chain-scattering representations and the dual generalized chain-scattering representations are investigated in a behavioral theory context. The last problem is devoted to the subjects of system wellposedness and internal stability. We present certain generalisations to the classical concepts of wellposedness and internal stability. The input consistency and output uniqueness of the closed-loop system in the standard control feedback configurations are discussed. Based on this, a number of notions are introduced such as fully internal wellposedness, externally internal wellposedness, and externally internal stability, which characterize the rich input-output and stability features of the general control systems in a general setting. On the basis of these notions the extended JL control problem is defined in a general setting.

A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. The minimum rank of a sign pattern matrix A is the minimum of the rank of the real matrices whose entries have signs equal to the corresponding entries of A. It is conjectured that the minimum rank of every sign pattern matrix can be realized by a rational matrix. The equivalence of this conjecture to several seemingly unrelated statements are established. For some special cases, such as when A is entrywise nonzero, or the minimum rank of A is at most 2, or the minimum rank of A is at least n - 1,(where A is mxn), the conjecture is shown to hold.Connections between this conjecture and the existence of positive rational solutions of certain systems of homogeneous quadratic polynomial equations with each coefficient equal to either -1 or 1 are explored. Sign patterns that almost require unique rank are also investigated.