A new algorithm for generalized Sylvester-observer equation and its application to state and velocity estimations in vibrating systems

We propose a new algorithm for block-wise solution of the generalized Sylvesterobserver equation XA − FXE = GC, where the matrices A,E, and C are given, the matrices X,F , and G need to be computed, and the matrix E may be singular. The algorithm is based on an orthogonal decomposition of the triplet (A,E,C) to the observer-Hessenberg-triangular form. It is a natural generalization of the widelyknown 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 and velocity estimations of second-order control systems modeling a wide variety of vibrating structures. For dense un-structured data, the proposed algorithm is more efficient than the recently proposed SVD-based algorithm of the authors, it is heavily composed of BLAS-3 (Level 3 Basic Linear Algebra Subprograms) operations, which make the algorithm an ideal candidate for high-performance computing.