Comparison of peak and RMS gains for discrete-time systems
A convolution system can have a frequency response which is small for all frequencies, yet still greatly amplify the peaks of signals passing through it. For finite-dimensional systems, however, we establish the simple bound \(\| h\|_ 1\leq (2n+1)\| h\|_{H^{\infty}}\), where \(\| h\|_ 1\) is the peak gain of the system, \(\| h\|_{H^{\infty}}\) is the maximum frequency response of the system, and n its dimension. The same result for continuous-time systems is due to Gohberg and Doyle and it is mentioned in ``Robust control of multivariable and large scale systems'', Final Tech. Report, Honeywell Systems and Research Center, Minneapolis, MN (1985). The bound implies that \(H^{\infty}\)-optimal controllers, which minimize the maximum of some disturbance-to-error transfer function, cannot have very large peak gains from the disturbance to error.
scientific article; zbMATH DE number 4051471
scientific article