<i>H</i>₂Analysis of LTI Discrete-Time Systems via Conversion to Externally Positive Systems

In this paper, we explore H2 analysis techniques of general, not necessarily positive, discrete-time linear time-invariant (LTI) systems by means of positive system theory. While externally positive systems are a particular class of systems whose impulse responses take only nonnegative values, we often encounter such impulse responses even in the analysis of general LTI systems. A typical example is the computation of the H2 norm where we focus on squared impulse responses. In view of this observation, we first clarify that we can construct an SISO externally positive system whose impulse response is given by the square of the impulse response of a given SISO LTI system. This system conversion allows us to reduce the H2 norm computation problem of a general LTI system to the l∞-induced norm computation problem (or l1 problem in short) of the resulting externally positive system. It also turns out that this problem reduction enables us to derive various alternative formulas for the H2 norm of general discrete-time LTI systems.