Sensitivity Functions for Linear Discrete Systems with Applications

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}}{partialalpha_{Jl}partialalpha_{Jl-1}cdotspartialalpha_{J2}partialalpha_{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.