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

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 th-order sensitivity model all the sensitivity functions for a single-input linear time-invariant controllable th-order system which depends on 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 . This result is then extended to the case of multi-input normal linear systems, where, at most, dynamic 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 ( is the dimension of ). It is shown that the algebraic calculations that must be made in the -input case are much less than times the calculations needed for the single-input case. The implications of these results for the computer aided sensitivity analysis of systems are discussed.