Concepts of strict positive realness and the absolute stability problem of continuous-time systems

After the conditions for absolute stability and positive realness are reviewed, it is shown that the current definition of positive realness is inadequate to establish less conservative criteria for absolute stability. In this paper, the concepts of wide positive realness (WPR) and wide strict positive realness (WSPR) are proposed. Both WPR and WSPR functions may have poles in the open left and right half-planes, and WPR functions may have single poles on the imaginary axis including ± j ∞. Using the concepts of WPR and WSPR, less conservative criteria, including algebraic and frequency domain criteria, are presented for absolute stability of slope-restricted multiple nonlinearity feedback systems. Refined results for non-slope-restricted non-linearity feedback systems are also proposed. It is pointed out that there is no spectral factorization for WPR and WSPR functions if they have poles at both j ∞ and - j ∞, this can be used to explain the deficiencies of algebraic criteria presented in previous publications.