A new block algorithm for generalized Sylvester-observer equation and application to state estimation of vibrating systems

We propose a new block algorithm for the generalized Sylvester-observer equation : XA - FXE = GC, where the matrices A, E, and C are given, the matrices X, F, and G need to be computed, and matrix E could be singular. The algorithm is based on an orthogonal decomposition of the triplet (A, E, C) to observer-Hessenberg-triangular form. It is a natural generalization of the widely-known observer-Hessenberg algorithm for the Sylvester-observer equation XA - FX = GC, which arises in state estimation of a standard first-order state-space control system. An application of the proposed algorithm is made to state estimation of second order control systems modeling a wide variety of vibrating structures. For dense un-structured data, the algorithm is more efficient than the recently proposed SVD-based algorithm of the authors, numerically reliable and heavily composed of Basic Linear Algebra Subprograms -Level 3 (BLAS 3) operations, which make it an ideal candidate for high-performance computing.