Construction of Externally Positive Systems for Discrete-Time LTI System Analysis

This paper is concerned with analysis techniques of discrete-time LTI systems via construction of relevant externally positive systems. Recently, the author established a construction method of an externally positive system whose impulse response is given by the square of the original discrete-time LTI SISO system to be analyzed. This externally positive system allows us to characterize the H₂ norm of the original system by means of the l_∞-induced norm characterization of externally positive systems. It is nonetheless true that, for the original system of order n, the order of the resulting externally positive system is n² and hence this leads to the increase of associated computational burden. With this important issue in mind, in this paper, we show that the order can be reduced down to n(n + 1)/2 by using the elimination and duplication matrices that are intensively studied by Jan R. Magnus in the 80's. In addition to the computational complexity reduction for the aforementioned H₂ analysis, we show that such construction of externally positive systems with reduced order is quite effective in semidefinite-programming-based peak value analysis of impulse responses of general LTI systems.