離散時間線形時不変システム解析のための外部非負システムの構成と低次元化

<p>This paper is concerned with the analysis of discrete-time LTI systems via construction of associated externally positive systems. Recently, the authors 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 <i>H</i>₂norm of the original system by means of the closed-form <i>l_∞</i>-induced norm characterization of externally positive systems. It is nonetheless true that, for the original system of order <i>n</i>, the order of the resulting externally positive system is <i>n</i>², incurring a drastic increase in computational burden of computer-aided analysis and synthesis. With this important issue in mind, in this paper, we show that the order can be reduced down to <i>n</i>(<i>n</i>+1)<i>/</i>2 by using the elimination and duplication matrices that are intensively studied by J. R. Magnus in the 80's. In addition to the computational complexity reduction for the aforementioned <i>H</i>₂analysis, 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.</p>