SYNTHESIS OF LINEAR MULTIVARIABLE FEEDBACK SYSTEMS IN INFINITE INDEX NORM
Doctor of Philosophy
The deficiency of the widely used LQG method is that it depends heavily on the precision of plant parameters and the noise spectrum. The robustness problem can be formalized in singular value analysis. With the help of operator theory a new method for the synthesis of linear multivariable feedback systems in H(,(INFIN)) norm is developed from the singular value analysis. The positive feature of H(,(INFIN)) norm synthesis is the transparency for robustness conditions, the weighting functions are directly related to the specifications of design requirements. In this dissertation the LQG problem is restated as an interpolation problem in H(,2) space. The interpolation problem in simplest case can be solved by an explicit formula. The H(,(INFIN)) optimal norm can be obtained from the consideration of the ratio of two H(,2) norms. The close relations and similarities between H(,(INFIN)) and H(,2) are brought out. The total H(,(INFIN)) optimal solutions can be constructed by the unitary dilation from the interpolation space. The explicit formulas in s domain for these purposes are given, including the repeated zeros case and the degenerate case. The optimal solutions must belong to the degenerate case, in this case the problem can be solved by separating the singular part of Pick matrix from the regular part by a Cholesky decomposition. These results are also developed in a recursive version for repeated zeros. The zeros and interpolation condition vectors of a system can be determined numerically by an algorithm to solve eigenvalues and eigenvectors of a pencil. To convert the two-sided problem to a one-sided problem and to convert the nonsquare problem to a square problem are related to the spectral factorization which is discussed in detail. The optimal solutions of the nonsquare problem need not be all-pass, which is related to the existence of critical point. The theory applied to the sensitivity design problem can be considered as an extension of the classical lead-lag design method from SISO to MIMO with more profound mathematical background. The robust stability problem can also be formalized and solved in the framework. The robust sensitivity design introduces a new type of mathematical problem, which can be approximated in our framework in certain situations. The regulation, tracking, filtering and optimal controller design problem under the inexactly known noise spectrum can be solved in the general model in H(,(INFIN)) space by introducing proper weighting functions.