Analysis and minimization of L2-sensitivity for linear systems and two-dimensional state-space filters using general controllability and observability Gramians

A novel expression for the evaluation of L₂-sensitivity is developed for the cases of linear discrete-time systems, linear continuous-time systems, and two-dimensional (2-D) state-space digital filters. This is accomplished by introducing the concept of general controllability and observability Gramians in each case. Moreover, the L₂-sensitivity measures obtained here contain the conventional L₁/L₂-sensitivity measures as a special case. An iterative procedure for constructing the optimal coordinate transformation matrix that minimizes the L₂-sensitivity measure is then presented in each case. This procedure is advantageous since the initial estimate and the estimate at each iteration can be calculated analytically. Finally, three numerical examples are given to illustrate the utility of the proposed techniques.