Sensitivity Functions for Linear Discrete Systems with Applications

Abstract

z-transform techniques are employed to establish general symmetry and simultaneity properties of the first sensitivity functions of the phase-canonical form of single-input, nth-order, linear, constant, discrete-time, controllable systems. It is demonstrated that computation of the first sensitivity function requires one nth-order model in addition to the system model. This simultaneity property is extended to arbitrary single-input, nth-order, linear, constant, discrete systems. In complete analogy with results presented for continuous systems, symmetry and simultaneity properties may be established for the computation of the /th sensitivity function \begin{equation*}^{l}\beta^{y} \triangleq \frac{\partial^{l}y_{i}}{\partial\alpha_{Jl}\partial\alpha_{Jl-1}\cdots\partial\alpha_{J2}\partial\alpha_{J1}}|_{\alpha=\alpha_{0}} {\rm for} \substack{i = 1,2,\ldots,n J_{k} = 1,2,\ldots,n k = 1,2,\ldots,l.}\end{equation*} Extension of these results to multi-input systems is also mentioned. The usefulness of the simultaneity property is illustrated by applying the results to the design of a low-sensitivity optimal control law.