Variation diminishing linear time-invariant systems. (February 2022)

• Record Type: Journal Article
• Authors: Grussler, Christian; Sepulchre, Rodolphe
• Is Part Of: Automatica. Volume 136(2022)
• Publication Date: 2022-02
• Subjects: Positive system -- Total positivity -- Variation diminishing -- n-dimensional systems -- Time-invariant -- Convex optimization; Automatic control -- Periodicals; Automation -- Periodicals; 629.805
• DOI: 10.1016/j.automatica.2021.109985
• Languages: English
• ISSNs: 0005-1098

Abstract

Abstract: This paper studies the variation diminishing property of k -positive linear time-invariant (LTI) systems, which diminish the number of sign changes (variation) from input to output, if the input variation is at most k − 1 . We characterize this property for the discrete-time Toeplitz and Hankel operators of finite-dimensional causal systems. Our main result is that these operators have a dominant approximation in the form of series or parallel interconnections of k first order positive systems. This is shown by expressing the k -positivity of a LTI system as the external positivity (that is, 1-positivity) of k compound LTI systems . Our characterization generalizes well known properties of externally positive systems ( k = 1 ) and totally positive systems ( k = ∞ ; also known as relaxation systems in case of the Hankel operator). All results readily extend to continuous-time systems by considering sampled impulse responses.