A unified approach to decentralised control based on the exterior algebra and algebraic geometry methods

The aim of the thesis is to provide a unifying framework and tools for the study of a number of control theory problems arising in Decentralised control. These problems are known as frequency assignment problems and they include the decentralised problem of pole assignment by state, output feedback and zero assignment by decentralised squaring-down. It is further shown that decentralised dynamic problems where the dynamic complexity of the controller is fixed , may also be reduced to the same formulation given for the constant Determinantal Assignment Problem(DAP). The unifying mathematical problem studied here is the Decentralised - Determinantal Assignment Problem (D- DAP) which is a special form of the general DAP defined for multivariable systems. The study of D-DAP involves the use of tools from exterior algebra in an essential way and also tools from classical algebraic geometry, specifically the theory of Grassmann varieties. In this thesis the mathematical framework of D-DAP is fully developed and used for the study of pole, zero assignment problems by decentralised constant controllers and new solvability conditions are given. The mathematical framework of D-DAP also allows the study of structural properties of Decentralised controllers. In fact new invariants for decentralised control are introduced in the form of Pliicker matrices and Decentralised Indices and a new characterisation of fixed modes and fixed zeros is given based on tools from exterior algebra. The classical notion of fixed modes, fixed zeros is extended to almost fixed modes, almost fixed zeros for certain families of systems. Finally it is shown that the general framework of D-DAP is suitable for the study of dynamic problems such as pole assignment by decentralised classical controllers of the P-D, P-I, P-I-D type