Relating H₂ and H_∞ bounds for finite-dimensional systems

Abstract

For a linear time invariant system, the infinity-norm of the transfer function can be used as a measure of the gain of the system. This notion of system gain is ideally suited to the frequency domain design techniques such as H∞ optimal control. Another measure of the gain of a system is the H2 norm, which is often associated with the LQG optimal control problem. The only known connection between these two norms is that, for discrete time transfer functions, the H2 norm is bounded by the H∞ norm. It is shown in this paper that, given precise or certain partial knowledge of the poles of the transfer function, it is possible to obtain an upper bound of the H∞ norm as a function of the H2 norm, both in the continuous and discrete time cases. It is also shown that, in continuous time, the H2 norm can be bounded by a function of the H∞ norm and the bandwidth of the system. Show more