Feedback stabilization of a second-order system: A nonmodal approach

AbstractA novel approach for feedback stabilization of a second-order model is proposed. Specifically, two nonmodal algorithms are described, and it is shown mathematically that under some mild assumptions on the damping matrix, the feedback matrix obtained by each algorithm indeed stabilizes a closed-loop system. The first algorithm requires the solution of a symmetric positive definite linear system and the inversion of a small matrix. A remarkable feature of this algorithm is that it does not require knowledge of the stiffness and damping for implementation, which makes it very feasible for practical application. The second algorithm requires the solution of a small linear least-squares problem and an estimate of the stability index, which can be obtained just by computing the extremal eigenvalues of the data matrices, which are symmetric for almost all practical applications. These minimal computational requirements make the proposed algorithms suitable for practical implementations, even for large and sparse systems, using the state-of-the-art techniques of matrix computations. This is in sharp contrast with the traditional modal approach ordinarily used by practicing engineers, which requires the solution of a quadratic eigenvalue problem or, equivalently, the solution of a 2n x 2n generalized eigenvalue problem, and therefore may not be practical for very large and sparse problems. Besides the proposed algorithms, the paper also contains some mathematical results on the bounds of the eigenvalues of a second-order pencil, which may be useful for investigating stability of second-order systems, and are of independent interest