$H_2$ Analysis of LTI Systems via Conversion to Externally Positive Systems

Motivated by recent advances in the study of linear time-invariant (LTI) positive systems, we explore analysis techniques of general, not necessarily positive, LTI 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 LTI system analysis. A typical example is the computation of the H
₂
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 LTI system whose impulse response is given by the product of impulse responses of two different LTI systems. Then, as the main result, we reduce the H
₂
norm computation problem of a general LTI system into the L∞-induced norm computation problem (or L
₁
problem in short) of a positive system, by which we can derive various formulas for the H
₂
norm computation.