Structure of linear systems: Geometric and transfer matrix approaches
Christian Commault, Jean-François Lafay, M. Malabre · Czech digital mathematics library · 1991
The aim of this communication is to show how, depending on the type of the control law (static or dynamic), some fine structures (internal or input-output ones) have to be known precisely, since they completely characterize the solvability of control problems like decoupling, disturbance decoupling or model matching, ....These structures mainly describe zeros (finite and at infinity) and kernel indices.Both geometric and transfer matrix approaches are used in accordance with internal and external points of view.Up to the sixties, the analysis and the control design of linear systems was performed in the frame of the transfer matrix approach (Bode, Black, Nyquist, ...).Then, from 1960 to 1970, the notion of state, which gave rise to the famous break-through in the study of multivariable systems, became so popular that it often received the label "modern approach".From 1970 to 1980 all approaches have been developed giving rise to the transfer matrix approach, to the polynomial approach and to the geometric approach and many control problems have been solved within each approach but with specific tools and, most often, the given conditions were approachdependent in the sense that these specific tools were explicitly used inside.Then, it became more clear that, as far as linear systems only were concerned, all approaches were somewhat equivalent: each new result in one direction could almost systematically be obtained in another approach.After the description of some common bridges between the tools of one approach and another, a deeper global vision has finally been reached, since the beginning of the eighties, with the help of structural information.Indeed, some authors have tried to exploit more intensively the fine structure of linear systems, breaking free from particular tools such as invariant subspaces or specific factorizations and, thanks to this structural frame have provided structural solutions to control problems like model matching or decoupling.These structural informations are invariants as controllability and observability indices, finite and infinite zeros, Morse's invariants, Kernel indices, essential orders, .... Now, this way of tackling systems, within a structural approach, appears to be particularly efficient.The aim of this communication is to give a short review of the structures that control people should precisely know and, depending on the type of control law (in essence: static or dynamic), to explain why internal or external structures play a key role.The paper is organized as follows: Sections 1 (Introduction) and 2 (Basic concepts and notation) describe, in more details, the different control laws which are frequently used and the basic geometric tools.Section 3 is devoted to geometric and transfer matrix characterization of internal and external structures like zeros or kernel indices.Then, applications of this structural approach are given in Section 4 in the context of the decoupling and model matching problems.Section 5 is devoted to concluding remarks.•