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.