Comparison of Peak and RMS Gains for Discrete-Time Systems

S. Boyd and J. Doyle

Systems and Control Letters, 9(1):1-6, June 1987.

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, the authors establish that the peak gain of a system is not more than factor (2n+1) times more than the RMS gain of a system (i.e., the maximum magnitude of its requency response), where n is the dimension of the system. 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.