A numerically reliable solution for the squaring-down problem in system design
The authors consider the squaring-down problem for a linear time-invariant system with an unequal number of inputs and outputs, i.e. the problem of transforming the initial system into an invertible square system having an equal number of inputs and outputs. The aim of the present paper is to investigate the squaring-down problem by static and dynamic compensators for the linear time-invariant system, based on matrix pencil theory. The special concern of the authors is to achieve a numerically reliable method for computing pre- and post-compensators. Using only orthogonal transformations, the paper obtains a ``condensed fort'' for decomposing the matrices of the system, which can be computed in a numerically reliable manner and it is essential to derive explicit formulae for the desired pre- and post-compensators. The resulted squared-down system succeeds to preserve the key fundamental properties, namely stabilizability, detectability, minimum phaseness, and the infinite-zero structure of the original system. If the original system is stabilizable and dynamic compensators are employed, then the induced additional invariant zeros are assignable to the open left-half plane. Furthermore, the designed post- and pre-compensators are stable, and two numerical examples compare the accuracy of the ``special coordinate basis'' form of \textit{A. Saberi} and \textit{P. Sannuti} [Int. J. Control 45, 1655--1704 (1987; Zbl 0623.93014); IEEE Trans. Autom. Control 33, No. 4, 358--365 (1988; Zbl 0642.93027)] with the authors' ``condensed form'', illustrating the numerical superiority of the proposed method.
decomposing techniques of system matrices
scientific article; zbMATH DE number 2116192
scientific article