Analysis of Frequency-Robust Multivariable Dynamical Systems under Parametric Perturbations1

A problem of studying the sensitivity of ellipsoidal frequency estimates of quality of multivariable dynamical systems to parameter variations is considered. The apparatus of sensitivity functions of extreme elements of singular values of real-valued transfer matrices is used for solving of this problem. Combined use of the frequency sensitivity apparatus and the state space method makes it possible to construct sensitivity models which can be used to define ellipsoidal estimates of the frequency sensitivity functions for the state, output and error of linear multivariable continuous systems in the form of a majorant and a minorant of these functions. The calculations use a singular value decomposition of matrices composed of frequency parametric sensitivity functions. The resulting ellipsoidal estimates have the property of minimum sufficiency due to the meaningfulness of the singular values decomposition of matrix. The method allows to select in the state, output and error spaces the subspaces characterized at each frequency value by the largest and the smallest normal variation, using the elements of the left singular basis corresponding to the extreme singular values. The use of the right singular basis permits to identify in the parameter space the subspaces that produce the largest and the smallest normal variation of the amplitude frequency response. This approach allows to solve the problem of “optimal nominal”, that is, the problem of choosing the nominal value of the vector of plant parameters, which delivers the lowest value of ellipsoidal estimates of frequency sensitivity functions to multivariable controlled processes and to compare the course of multivariable controlled processes by ellipsoidal estimates of parameter frequency sensitivity.