Abstract:In recent years, a great interest has been shown towards Krylov subspace techniques applied to model order reduction of large-scale dynamical systems. A special interest has been devoted to single-input single-output (SISO) systems by using moment matching techniques based on Arnoldi or Lanczos algorithms. In this paper, we consider multiple-input multiple-output (MIMO) dynamical systems and introduce the rational block Arnoldi process to design low order dynamical systems that are close in some sense to the original MIMO dynamical system. Rational Krylov subspace methods are based on the choice of suitable shifts that are selected a priori or adaptively. In this paper, we propose an adaptive selection of those shifts and show the efficiency of this approach in our numerical tests. We also give some new block Arnoldi-like relations that are used to propose an upper bound for the norm of the error on the transfer function.

In this paper, we consider the balanced truncation method for model reductions in large-scale linear and time-independent dynamical systems with multi-inputs and multioutputs. The method is based on the solutions of two large coupled Lyapunov matrix equations when the system is stable or on the computation of stabilizing positive and semi-definite solutions of some continuous-time algebraic Riccati equations when the dynamical system is not stable. Using the rational block Arnoldi, we show how to compute approximate solutions to these large Lyapunov or algebraic Riccati equations. The obtained approximate solutions are given in a factored form and used to build the reduced order model. We give some theoretical results and present numerical examples with some benchmark models.

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