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A Geometric Method for Passivation and Cooperative Control of Equilibrium-Independent Passivity-Short Systems

Miel Sharf, Anoop Jain, Daniel Zelazo

TL;DR

This paper addresses the problem of passivating equilibrium-independent passive-short (EIPS) systems that lack monotone steady-state I/O relations by introducing a geometric, PQI-based monotonicity framework. It develops a linear transformation that monotonizes the steady-state I/O relation and, consequently, passivizes the system with respect to all forced equilibria; this transformation is realizable via a sequence of simple gain and feedback blocks. The approach yields a network-optimization perspective for diffusively-coupled EI-IOP systems, including conditions for maximal monotonicity (MEIP) through the notion of cursive relations and a method to guarantee convergence in transformed networks. Two case studies illustrate the method on linear time-invariant systems and networks of gradient systems with non-convex potentials, demonstrating both passivation and improved network-level behavior. The framework unifies classical passivity with EI-short systems and offers a constructive path toward extending passivation and network analysis to broader nonlinear settings, potentially including MIMO extensions in future work.

Abstract

Equilibrium-independent passive-short (EIPS) systems are a class of systems that satisfy a passivity-like dissipation inequality with respect to any forced equilibria with non-positive passivity indices. This paper presents a geometric approach for finding a passivizing transformation for such systems, relying on their steady-state input-output relation and the notion of projective quadratic inequalities (PQIs). We show that PQIs arise naturally from passivity-shortage characteristics of an EIPS system, and the set of their solutions can be explicitly expressed. We leverage this connection to build an input-output mapping that transforms the steady-state input-output relation to a monotone relation, and show that the same mapping passivizes the EIPS system. We show that the proposed transformation can be implemented through a combination of feedback, feed-through, post- and pre-multiplication gains. Furthermore, we consider an application of the presented passivation scheme for the analysis of networks comprised of EIPS systems. Numerous examples are provided to illustrate the theoretical findings.

A Geometric Method for Passivation and Cooperative Control of Equilibrium-Independent Passivity-Short Systems

TL;DR

This paper addresses the problem of passivating equilibrium-independent passive-short (EIPS) systems that lack monotone steady-state I/O relations by introducing a geometric, PQI-based monotonicity framework. It develops a linear transformation that monotonizes the steady-state I/O relation and, consequently, passivizes the system with respect to all forced equilibria; this transformation is realizable via a sequence of simple gain and feedback blocks. The approach yields a network-optimization perspective for diffusively-coupled EI-IOP systems, including conditions for maximal monotonicity (MEIP) through the notion of cursive relations and a method to guarantee convergence in transformed networks. Two case studies illustrate the method on linear time-invariant systems and networks of gradient systems with non-convex potentials, demonstrating both passivation and improved network-level behavior. The framework unifies classical passivity with EI-short systems and offers a constructive path toward extending passivation and network analysis to broader nonlinear settings, potentially including MIMO extensions in future work.

Abstract

Equilibrium-independent passive-short (EIPS) systems are a class of systems that satisfy a passivity-like dissipation inequality with respect to any forced equilibria with non-positive passivity indices. This paper presents a geometric approach for finding a passivizing transformation for such systems, relying on their steady-state input-output relation and the notion of projective quadratic inequalities (PQIs). We show that PQIs arise naturally from passivity-shortage characteristics of an EIPS system, and the set of their solutions can be explicitly expressed. We leverage this connection to build an input-output mapping that transforms the steady-state input-output relation to a monotone relation, and show that the same mapping passivizes the EIPS system. We show that the proposed transformation can be implemented through a combination of feedback, feed-through, post- and pre-multiplication gains. Furthermore, we consider an application of the presented passivation scheme for the analysis of networks comprised of EIPS systems. Numerous examples are provided to illustrate the theoretical findings.

Paper Structure

This paper contains 19 sections, 15 theorems, 34 equations, 9 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Contributions
  4. Background
  5. Maximal Equilibrium-Independent Passivity
  6. Equilibrium-Independent Shortage of Passivity
  7. Diffusively-Coupled Network Model
  8. Motivation and Problem Formulation
  9. Monotonization of I/O Relations by Linear Transformations: A Geometric Approach
  10. From Monotonization to Passivation and Implementation
  11. Finite $\mathcal{L}_2$-gain and Input-Output Passivity
  12. Finite $\mathcal{L}_2$-gain and Input-Output $(\rho,\nu)$-Passivity
  13. Equilibrium-Independent Passive Shortage and Linear and Time-Invariant Systems
  14. Maximality of Input-Output Relations and the Network Optimization Framework
  15. Case Studies
  16. ...and 4 more sections

Key Result

Theorem 1

A relation $k$ is maximally monotone if and only if there exists a convex function $\Phi$ such that the subgradient $\partial \Phi$ is equal to $k$. Moreover, $\Phi$ is unique up to an additive constant. The function $\Phi$ is called the integral function of $k$.

Figures (9)

  • Figure 1: A diffusively-coupled network.
  • Figure 2: Steady-state relations of the system in Example \ref{['exam.InputOutputShortage']}.
  • Figure 3: A double cone (in blue), and the associated symmetric section $S$ (in solid red). The parts of $\mathbb{S}^1$ outside $S$ are presented by the dashed red line
  • Figure 4: The transformed system $\tilde{\Sigma}$ after the linear transformation $T$. If $T=\left[abcd\right]$, then $\delta_A = b/a, \delta_B =d-\frac{b}{a}c, \delta_C = c$ and $\delta_D = a$.
  • Figure 5: Monotonization, passivation and convexification by the transformation $T$. For general output-passive short systems, convexification is equivalent to passivation. For EI-IOP($\rho,\nu$) systems, integral functions do not necessarily exist, so monotonization of the steady-state relation is equivalent to passivation.
  • ...and 4 more figures

Theorems & Definitions (52)

  • Definition 1
  • Definition 2
  • Theorem 1: Rockafellar1997
  • Definition 3: Burger2014
  • Definition 4
  • Remark 1
  • Definition 5
  • Remark 2
  • Remark 3
  • Remark 4
  • ...and 42 more