Generation of sensitivity functions for linear systems using low-order models

Abstract

New proofs are given for the recently demonstrated total symmetry and complete simultaneity properties for the companion canonic form for single-input linear time-invariant controllable systems. These proofs result in a convenient closed-form expression for the complete simultaneity property. The use of these properties to generate by one n th-order sensitivity model all the sensitivity functions \frac{\partialx_{i}}{\partialv_{j}}|_v^{0}, i=1,...,n, j=1,...,r, for a single-input linear time-invariant controllable n th-order system which depends on r different parameters is reviewed. This method represents an improvement over known methods for generating the sensitivity functions, which generally require a composite dynamic system of order n(r+1) . This result is then extended to the case of multi-input normal linear systems, where, at most, 2m-1 dynamic n th-order systems are needed in addition to the system to generate all the sensitivity functions of the system state with respect to any number of parameters ( m is the dimension of u ). It is shown that the algebraic calculations that must be made in the m -input case are much less than m times the calculations needed for the single-input case. The implications of these results for the computer aided sensitivity analysis of systems are discussed.