Title:

Algebraic synthesis methods for linear multivariable control systems

The mathematical formulation of various control synthesis problems (such as Decentralized Stabilization Problem (DSP), Total Finite Settling Time Stabilization for discrete time linear systems (TFSTS), Exact Model Matching Problem (EMMP), Decoupling and Noninteracting Control Problems) via the algebraic framework of Matrix Fractional Representation (MFR)  i.e. the representation of the transfer matrices of the system as matrix fractions over the ring of interest  results to the study of matrix equations over rings, such as : A . X + B . Y = C , (X. A + Y . B =C) (1) A· X = B , (y. A = B) (2) A·X·B = C (3) A·X + Y·B = C, X·A + B·Y = C, A·X·B + C·Y·D = E (4). The main objective of this dissertation is to further investigate conditions for existence and characterization of certain types of solutions of equation (1) ; develop a unifying algebraic approach for solvability and characterization of solutions of equations (1)  (4), based on structural properties of the given matrices, over the ring of interest. The standard matrix Diophantine equation (1) is associated with the TFSTS for discrete time linear systems and issues concerning the characterization of solutions according to the Extended McMillan Degree (EMD) (minimum EMD, or fixed EMD) of the stabilizing controllers they define, are studied. A link between the issues in question and topological properties of certain families of solutions of (1) is established . Equation (1) is also studied in association with the DSP and Diagonal DSP (DDSP) for continuous time linear systems. Conditions for characterizing block diagonal solutions of (1) (which define decentralized stabilizing controllers) are derived and a closed form description of the families of diagonal and two blocks diagonal decentralized stabilizing controllers is introduced. The set of matrix equations (1)  (4) is assumed over the field of fractions of the ring of interest , ℛ , (mainly a Euclidean Domain (ED) and thus a Principal Ideal Domain (PID)) and solvability as well as parametrization of solutions over ℛ is investigated under the unifying algebraic framework of extended non square matrix divisors, projectors and annihilators of the known matrices over ℛ . In practice the ring of interest is either the ring of polynomials ℝ [s] , or the rings of proper ℝ_pr(s) and especially proper and stable rational functions R_op(s). The importance of R_op(s) is highlighted early in the thesis and further computational issues arising from its structure as an ED are considered.
