Hz analysis of LTI systems via conversion to positive systems

Motivated by recent drastic advances in the study of linear-time invariant (ETI) positive systems, we explore analysis techniques of general, not necessarily positive ETI systems using positive system theory. Even though a positive system is characterized by its peculiar property that its impulse response is nonnegative, we often deal with nonnegative impulse responses even in general ETI system analysis. A typical example is the computation of the H2 norm where we focus on squared impulse responses. To deal with such products of impulse responses in a systematic fashion, in this paper, we first establish a construction technique of an ETI system whose impulse response is given by the product of impulse responses of two different ETI systems. Then, as the main result, we reduce the H2 norm computation problem of a general ETI system into the L∞-induced norm computation problem (or L1 problem in short) of a positive system, by which we can derive a closed-form formula for the H∞ norm computation.