Analysis of frequency-robust multivariable dynamical systems

We consider the problem of studying the sensitivity of ellipsoidal frequency estimates of quality of multivariable dynamic systems to parameter variations. To solve the problem, we use the apparatus of sensitivity functions of extreme elements of singular value decomposition of real-valued transfer matrices. The joint usage of the apparatus of frequency sensitivity with the method of state space allowed us to construct the models of sensitivity. On the basis of the obtained models, the ellipsoidal estimates of the frequency sensitivity functions for the state, output and error of linear multivariable continuous systems in the form of the majorant and minorant of these functions have been determined. The singular value decomposition of matrices composed of frequency parametric sensitivity functions has been applied to the calculations. The obtained ellipsoidal estimates have the property of minimum sufficiency due to the substantial possibilities of the singular value decomposition of matrices. This approach made it possible to use the elements of the left singular basis corresponding to the extreme singular values, to select in the state, output, and error spaces the subspaces characterized for each frequency value by the largest and smallest normal variation of the amplitude-frequency response. Using the right singular basis made it possible to identify the subspaces in the parameter space which produce the largest and the smallest normal variation of the amplitude-frequency response. The proposed approach has solved the problem of the “optimal nominal” — the choice of the nominal value of the vector of primary physical parameters of the control object aggregates that deliver the smallest value of ellipsoidal estimates of the frequency sensitivity functions to the multivariable controlled process. Such parameters include: dimensions of various parts and characteristics of their manufacturing accuracy, physical properties of materials as well as various values determining their design. The approach made it possible to compare the course of multidimensional controlled processes by ellipsoidal estimates of the frequency parameter sensitivity.