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 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 may be singular. The algorithm is based on an orthogonal decomposition of the triplet (A, E, C) into the 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 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. Copyright © 2010 John Wiley & Sons, Ltd.