New relations between norms of system transfer functions

It is well-known that the H2H2-norm and H∞H∞-norm of a transfer function can differ arbitrarily since both norms reflect fundamentally different properties. However, if the pole structure of the transfer function is known it is possible to bound the H∞H∞-norm from above by a constant multiple of the H2H2-norm. It is desirable to compute this constant as tightly as possible. In this article we derive a tight bound for the H∞H∞-norm given knowledge of the H2H2-norm and the poles of a transfer function. We compute the bound in closed form for multiple input multiple output transfer functions in continuous and discrete time. Furthermore we derive a general procedure to compute the bound given a weighted L2L2-norm.