FINITE-DIMENSIONAL REALIZATIONS OF DISCRETE-TIME WEIGHTING PATTERNS*

The extensive use of weighting patterns in the analysis and synthesis of linear, finite-dimensional systems relies heavily on a realizability result involving a decomposition of the weighting pattern $W( t,\tau )$ into an inner product of two vector terms $\psi ( t )$ and $\beta ( \tau )$. This result, as stated for the continuous-time case, depends on the invertibility of the transition matrix $\Phi ( t,\tau )$ for the general case. Hence, the realizability condition must be modified for the discrete-time case, where the transition matrix is not invertible in general.This paper presents and proves a discrete-time realizability condition which closely parallels the continuous-time results. At the same time, some insight is conveyed regarding the differences between the continuous- and discrete-time cases. Finally, auxiliary results are proved along established lines.