Parallel and large-scale matrix computations in control: some ideas

Abstract

The design and analysis of time-invariant linear control systems give rise to a variety of interesting linear-algebra problems. Numerically effective methods now exist for several of these problems. However, algorithms for large-scale computations and efficient parallel algorithms for these problems are virtually nonexistent. In this paper, we propose several efficient general-purpose parallel algorithms for single-input and multiinput eigenvalue assignment problems. A desirable feature of these algorithms is that they are composed of simple linear-algebraic operations such as matrix-vector multiplication, solution of a linear system, and computations of the eigensystem and singular values of a symmetric matrix, for which efficient parallel algorithms have already been developed and parallel software libraries are being built based on these algorithms. The proposed algorithms thus have potential for implementations on some existing and future parallel processors. We also propose a numerical method for the Sylvester matrix equation arising in the construction of a Luenberger observer. The method does not need reduction to “condensed” forms and is thus suitable for large and sparse matrices. The method also exhibits a certain parallelism.