Selecting state variables to minimize eigenvalue sensitivity of multivariable systems

Abstract

It often happens that linear, time invariant, multivariable systems must be simulated or synthesized using digital and analog computers. The simulation may require accurate reproduction of the simulated system, and the synthesis may demand faithful representation of a mathematical model. Because of the degradation introduced by such factors as the finite wordlength constraints of digital computers and the temperature variation of amplifier gains in analog computers, the computer realization will differ from its original model. In this paper a simple method is presented for minimizing the deviation in the eigenvalues of the computer realization when all elements of the system matrix are subject to simultaneous variations. This procedure involves the selection of a suitable set of state variables, the choice depending simply on the location of the system eigenvalues. When the system is realized using the state vector obtained by this method, greater component variations and shorter computer wordlengths can be tolerated before stability is sacrified and satisfactory behavior is compromised. As a result, utilization of the algorithm discussed in this paper can result either in relaxation of computer tolerances, and hence reducing the cost of realization, or in more faithful reproduction of the original system.