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Quantitatively hyper-positive real rational functions III

Daniel Alpay, Izchak Lewkowicz

Abstract

Hyper-Positive Real, matrix-valued, rational functions are associated with absolute stability (the Lurie problem). Here, quantitative subsets of Hyper-positive functions, related through nested inclusions, are introduced. Structurally, this family of functions turns out to be matrix-convex and closed under inversion. A state-space characterization of these functions through a corresponding Kalman-Yakubovich-Popov Lemma, is given. Technically, the classical Linear Matrix Inclusions, associated with passive systems, are here substituted by Quadratic Matrix Inclusions.

Quantitatively hyper-positive real rational functions III

Abstract

Hyper-Positive Real, matrix-valued, rational functions are associated with absolute stability (the Lurie problem). Here, quantitative subsets of Hyper-positive functions, related through nested inclusions, are introduced. Structurally, this family of functions turns out to be matrix-convex and closed under inversion. A state-space characterization of these functions through a corresponding Kalman-Yakubovich-Popov Lemma, is given. Technically, the classical Linear Matrix Inclusions, associated with passive systems, are here substituted by Quadratic Matrix Inclusions.
Paper Structure (18 sections, 129 equations, 5 figures)

This paper contains 18 sections, 129 equations, 5 figures.

Table of Contents

  1. Introduction and Main Results
  2. Background
  3. Rational Functions in Quadratic Form
  4. An analogy with $\mathbb{C}$
  5. A Useful Lemma
  6. Proof of Theorem \ref{['Tm:Set_Of_HP_Functions']}
  7. Realization of Quantitatively Hyper-Positive Real Rational Functions
  8. Proof of item (i) of Theorem \ref{['Tm:Kyp_Hyper_Pos_W']}
  9. Proof of items (ii), (iii) of Theorem \ref{['Tm:Kyp_Hyper_Pos_W']}
  10. Two kinds of Inversion
  11. $n, m$ Matrix-Convexity of Realizations of $\mathcal{HP}_{\color{blue}\beta}$ Functions
  12. Preliminaries
  13. Proof of Proposition \ref{['Pn:Sets_Of_Realizations']}
  14. Two kinds of Matrix-convexity
  15. Balanced Truncation of a Convex-hull of Functions
  16. ...and 3 more sections

Figures (5)

  • Figure 1: Sub-Unit Disks and their Image under the Cayley Transform
  • Figure 2: Three $\mathcal{HB}_{\color{blue}\beta}$ associated with the same $\mathcal{HP}_{\color{blue}\beta}$ function.
  • Figure 3: The Nyquist plots of: ${\color{red}f_1}(s)=\frac{3}{5}+\frac{8a^2}{5(s+a)^2}$${\color{blue}f_2}(s)=\frac{41}{15}-\frac{8b^2}{5(s+b)^2}$ and $f_3(s)=\frac{3s+\frac{1}{3}c}{s+c}$ .
  • Figure 4: ${\rm\bf ~Z}_{\rm\bf in}(s)=R_1+\frac{\frac{1}{C}}{s+\frac{1}{R_2C}}$
  • Figure 5: ${\rm\bf\color{red}\hat{Z}}=R_1+ {\frac{sR_2}{s+\frac{R_2}{L}}}_{|_{L=\frac{1}{C}}}=R_1+\frac{sR_2}{s+CR_2}$