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 H 2 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 H 2 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 H 2 norm, both in the continuous and discrete time cases. It is also shown that, in continuous time, the H 2 norm can be bounded by a function of the H ∞ norm and the bandwidth of the system.