Positive Damping Region: A Graphic Tool for Passivization Analysis with Passivity Index

TL;DR

This work develops a geometric framework for passivization of LTI systems using a positive damping region in the complex plane, enabling OF and IF passivity analyses via Nyquist plots and Rayleigh quotients. A frequency-dependent disk region $\mathcal{P}_{\rm PD}(\sigma)$ governs passivability: $G(j\omega)$ (or its Rayleigh quotient) must reside inside this disk, with bandwidth contraction occurring as the passivity index $\sigma$ increases. The method integrates with Nyquist, Nichols, and generalized-KYP-based design, providing a visual and computational pathway for passivity-based stability and controller tuning, including generalized passivity definitions such as negative imaginariness. The framework is illustrated with power-system stability applications, clarifying fundamental trade-offs between damping strength and passive bandwidth, and offering a practical, extensible tool for engineers.

Abstract

This paper presents a geometric framework for analyzing output-feedback and input-feedforward passivization of linear time-invariant systems. We reveal that a system is passivizable with a given passivity index when the Nyquist plot for SISO systems or the Rayleigh quotient of the transfer function for MIMO systems lies within a specific, index-dependent region in the complex plane, termed the positive damping region. The criteria enable a convenient graphic tool for analyzing the passivization, the associated frequency bands, the maximum achievable passivity index, and the waterbed effect between them. Additionally, the tool can be encoded into classical tools such as the Nyquist plot, the Nichols plot, and the generalized KYP lemma to aid control design. Finally, we demonstrate its application in passivity-based power system stability analysis and discuss its implications for electrical engineers regarding device controller design trade-offs.

Figures (7)

• Figure 1: Feedback interconnected system model. (a): original model. (b): equivalent model with IF passivization and OF passivization, where $\mathcal{R}=I$ for classical passivity definition. The operator is for compatibility with the latter discussions of generalized passivity definitions.
• Figure 2: Illustration of different PD regions $\mathcal{P}_{\rm PD}(\sigma)$ w.r.t. the passivity indices $\sigma$. (a): $\sigma=0$, classical passivity definition. (b): $\sigma=$1. (c): $\sigma=$-1.
• Figure 3: Nichols plot with the PD region. The orange curve is the Nyquist plot of $G_1$ with expressions provided in \ref{['equ: case TF']}, and the PD region $\mathcal{P}_{\rm PD}(0.1)$ is represented by the purple region.
• Figure 4: Application to SISO systems. (a): Nyquist plot (orange curve) of $G_1$ and the PD region $\mathcal{P}_{\rm PD}(1)$ (purple disk) and $\mathcal{P}_{\rm PD}(1/3)$ (red disk). (b): Nyquist plot of $G_2$ and the PD region $\mathcal{P}_{\rm PD}(1/3)$. (c): Nyquist plot of $G_3$ and the PD region $\mathcal{P}_{\rm PD}(1/3)$. (d): phase-angle part of the Bode plot of $G_3$.
• Figure 5: Application to MIMO systems. The orange dashed circle denotes the PD region $\mathcal{P}_{\rm PD}(1/3\cdot I)$. Each colored disk represents the Rayleigh quotient $\rho_G(j\omega)$ at a given frequency $\omega$. For example, the yellow circle represents the boundary of all possible values of the Rayleigh quotient $\rho_{G(j10^{0.7})}(x)$, which intersects the PD region. Here we present the frequencies which $\log(\omega)$ from -3 to 2 with a sampling interval of 0.1.

Theorems & Definitions (27)

• Definition 1: Passivity Hassan_Nonlinear_1996
• Definition 2: Positive Damping Condition
• Definition 3: Input-Feedforward Passivization
• Definition 4: Output-Feedback Passivization
• Theorem 1: MIMO PD Condition
• proof
• Remark 1: Relation with Matrix Pencil and Numerical Range
• Corollary 1: LMI-based MIMO PD Condition
• proof
• Definition 5: OF PD Frequency Band