Abstract
Methods reductions of large scale linear time invariant systems are numerous, include among these which are based on the projection onto the Krylov subspace (Arnoldi, Lanczos, Arnoldi rational, Lanczos rational, Adaptive rational Arnoldi, rational Krylov) and methods based on singular values decomposition. Against, the reduction approaches of large scale switched linear systems are very limited (LMI, Arnoldi). In this chapter, two model reductions algorithms for approximation of large-scale linear switched systems are proposed, which are based on the Krylov subspace on the one hand and on the singular value decomposition on the other hand. At first the principle of the Dual rational Krylov based method is presented, based on this method for presenting at first the iterative dual rational Krylov approach that constructs a union of Krylov subspaces to generate two projection matrices. The iterative dual rational Krylov is low in cost, numerically efficient but the stability of reduced linear switched system is not always guaranteed. In the second part, the iterative SVD-Dual Rational Krylov approach is presented. This method is a combining of two sided-projections, one side is generated by the dual Rational Krylov-based model reduction techniques and the other side is generated by the SVD model reduction techniques, while the SVD-side depends on the observability gramian. This method is numerically efficient, minimize the \( H_{\infty } \) error between the original switched system and reduced one and preserve the stability of reduced systems. A simulation two examples are considered in order to take a performance study of these proposed approaches.
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Mohamed, K., Mehdi, A., Mami, A. (2015). Iterative Dual Rational Krylov and Iterative SVD-Dual Rational Krylov Model Reduction for Switched Linear Systems. In: Zhu, Q., Azar, A. (eds) Complex System Modelling and Control Through Intelligent Soft Computations. Studies in Fuzziness and Soft Computing, vol 319. Springer, Cham. https://doi.org/10.1007/978-3-319-12883-2_15
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