Block-decomposition and multi-stage eigenvalue-assignment

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A large-scale system often contains multi-modes of which the simulations and designs are extremely difficult. The large-scale system is usually block-decomposed into a system with either a block-trIangularized system map or a block-diagonalized system map so that the analysis and design of the system can easily be performed. In this thesis, a matrix sign function in conjunction with a geometric approach is utilized to construct the similarity transformation matrices for block-diagonalization and block-triangularization of a system map. Also, the methods to solve the Riccati-type and the Liapunov-type problems via the matrix sign function are described. Furthermore, the eigenvalue injector and the eigenvalue projector of a system map are defined. By utilizing the injector and the projector of a system map, the general properties of a similarity transformation matrix for block-diagonalization or block-triangularization of the system map are investigated. The relationship between block-decomposition and modal aggregation of a system map is discussed, and a multi-stage eigenvalue-assignment algorithm is derived for state feedback designs.

Eigenvalues, Decomposition (Mathematics)