Integral constraints on sensitivity vectors for multivariable linear systems
In his celebrated work [Network analysis and feedback amplifier design, Van Nostrand, New Jersey (1945)] \textit{H. W. Bode} proved that for any minimum phase SISO system with relative degree greater than one, a reduction in sensitivity in one frequency band should be necessarily accompanied by an increase in sensitivity in other frequency bands. This result formulated in the form of integral constraints on sensitivity in feedback systems has been extended to open-loop unstable systems, systems with right half plane zero, and MIMO systems by \textit{J. S. Freudenberg} and \textit{D. P. Looze} [Frequency domain properties of scalar and multivariable feedback systems, Springer, Heidelberg (1988; Zbl 0637.93035)] and refined in several other works. The main difference between the results obtained by Gomez and Goodwin and the previous ones is a vectorial form of the integral constraints. Instead of the scalar form used previously, this publication exploits the inherent directionality of multivariable systems leading to both spatial and frequency tradeoffs between different outputs and inputs and between different frequency bounds. The main results are contained in two theorems given the integral vectorial constraints on the complementary sensivity function \(T(s)\) and sensitivity function \(S(s)\) for system with lumped parameters. These results are then extended to time-delay systems and used to study the ``cost'' of diagonal decoupling. The authors discuss also the impact of near pole-zero cancellation. The paper is clearly written and endowed with numerical examples. The considerations exploit coprime fractional representations of control systems and the use of Blaschke products.
scientific article; zbMATH DE number 885860
scientific article