Internally Hankel $k$-Positive Systems

Abstract

There has been an increased interest in the variation diminishing properties of controlled linear time-invariant (LTI) systems and time-varying linear systems without inputs. In controlled LTI systems, these properties have recently been studied from the external perspective of $k$-positive Hankel operators. Such systems have Hankel operators that diminish the number of sign changes (the variation) from past input to future output if the input variation is at most $k-1$. For $k=1$, this coincides with the classical class of externally positive systems. For linear systems without inputs, the focus has been on the internal perspective of $k$-positive state-transition matrices, which diminish the variation of the initial system state. In the LTI case and for $k=1$, this corresponds to the classical class of (unforced) positive systems. This paper bridges the gap between the internal and external perspectives of $k$-positivity by analyzing internally Hankel $k$-positive systems, which we define as state-space LTI systems where controllability and observability operators as well as the state-transition matrix are $k$-positive. We show that the existing notions of external Hankel and internal $k$-positivity are subsumed under internal Hankel $k$-positivity, and we derive tractable conditions for verifying this property in the form of internal positivity of the first $k$ compound systems. As such, this class provides new means to verify external Hankel $k$-positivity, and lays the foundation for future investigations of variation diminishing controlled linear systems. As an application, we use our framework to derive new bounds for the number of over- and undershoots in the step responses of LTI systems. Since our characterization defines a new positive realization problem, we also discuss geometric conditions for the existence of minimal internally Hankel $k$-positive realizations.