In this paper, we review two types of doubling algorithm and some techniques for analyzing them. We then use the techniques to study the doubling algorithm for three different nonlinear matrix equations in the critical case. We show that the convergence of the doubling algorithm is at least linear with rate 1/2. As compared to earlier work on this topic, the results we present here are more general, and the analysis here is much simpler.
We propose two new methods for solution of the eigenvalue assignment problem associated with the second-order control system Specifically, the methods construct feedback matrices F, and F, such that the closedloop quadratic pencil P,(g = I'M + I(D + BF,) + (K+ BF,)has a desired set of eigenvalues and the associated eigenvectors are well conditioned. Method 1 is a modification of the singular value decomposition-based method proposed by Juang and Maghami which is a second-order adaptation of the wellknown robust eigenvalue assignment method by Kautsky et al. for first-order systems. Method 2 is an extension of the recent non-modal approach of Datta and Rincon for feedback stabilization of second-order systems. Robustness to numerical round-off errors is achieved by minimizing the condition numbers of the eigenvectors of theclosed-loop second-order pencil. Control robustness to large plant uncertainty will not be explicitly considered in this paper. Numerical results for both the two methods are favourable. Acomparative study of the methods is included in the paper.
The vibration of fast trains is governed by a quadratic palindromic eigenvalue problemAccurate and efficient solution can only be obtained using algorithms which preserve the structure of the eigenvalue problem. This paper reports on the successful application of the structure-preserving doubling algorithms.
From the necessary and sufficient conditions for complete reachability and observability of periodic time-varying descriptor systems, the symmetric positive semi-definite reachability/observability Gramians are defined. These Gramians can be shown to satisfy some projected generalized discrete-time periodic Lyapunov equations. We propose a numerical method for solving these projected Lyapunov equations, and an illustrative numerical example is given. As an application of our results, the balanced realization of periodic descriptor systems is discussed.
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