Maximum Achievable Passivity Index

• Maximum Achievable Passivity Index is a measure that determines the largest excess passivity (or smallest shortage) a dynamical system can showcase under prescribed supply rates.
• It employs rigorous methodologies including SOS programming, frequency-domain analysis, and convex optimization to compute passivity indices for nonlinear and LTI systems.
• Its applications influence controller synthesis, passivization, and robust design across domains such as power systems and polarimetric media.

A maximum achievable passivity index quantifies the largest passivity surplus (or smallest passivity shortage) that can be certified, realized, or imposed on a dynamical system (continuous, discrete, linear, or nonlinear) according to a specified supply rate—most commonly, input-feedforward passivity (IFP) or output-feedback passivity (OFP) indices—with or without control or structural constraints. It serves as a system-level metric for characterizing dissipativity and guides controller synthesis, passivization, and robustness analysis across nonlinear, LTI, data-driven, and domain-specific contexts such as power systems or polarimetric media.

1. Formal Definitions and Mathematical Framework

The IFP and OFP passivity indices $\nu$ and $\rho$ are strictness parameters in storage-function dissipation inequalities:

• IFP index $\nu$: $\dot V(x) \leq y^T u - \nu u^T u$
• OFP index $\rho$: $\dot V(x) \leq y^T u - \rho y^T y$

Given a system $\dot x = f(x,u),\ y = h(x,u)$, the largest admissible $\nu^*$ (respectively, $\rho^*$) is the supremum of $\nu$ for which a storage function $V$ exists satisfying the corresponding inequality for all $(x,u)$ in a prescribed set (typically determined by operational limitations) (Zakeri et al., 2019). For LTI systems, frequency-domain characterizations are available: for a closed-loop system $H(s)$,

$$\text{IFP-index:}\quad \nu^* = \frac{1}{2}\inf_\omega \lambda_{\min}(G(j\omega)+G(j\omega)^*),\qquad \text{OFP-index:}\quad \rho^* = \frac{1}{2}\inf_\omega \lambda_{\min}(G(j\omega)^{-1}+G(j\omega)^{-1*})$$

for $G(s)$ stable (Su et al., 2019).

Maximum achievable passivity indices appear in broader settings including:

• Nonlinear systems (with operational bounds)
• Static or dynamic I/O transformations
• Data-driven or on-line estimation frameworks
• Constrained or structured controller synthesis
• Physical domains (e.g., polarimetric Mueller matrices)

2. Computational Approaches and Algorithms

2.1. Nonlinear Systems: SOS and Polynomial Approximation

For nonlinear systems, the largest passivity index over sets $X \subset \mathbb{R}^n$ and $U \subset \mathbb{R}^m$ is computed via the following SOS program:

• Decision variables: storage $V(x) \geq 0$ on $X$, index $\nu$
• Constraints: $$\frac{\partial V}{\partial x}f(x,u) - [h(x,u)^T u - \nu\,u^T u] \le 0$$, enforced $\forall (x,u)\in X\times U$
• Nonpolynomial dynamics are approximated by Taylor or Bernstein polynomial expansions with explicit error bounding, handled by auxiliary variables and SOS multipliers
• Maximization is with respect to $\nu$ (or $\rho$ for OFP) (Zakeri et al., 2019)

2.2. LTI Systems: Frequency-Domain and KYP-LMI Characterization

For linear SISO systems, the SISO OFP-maximum can be computed as: $$\gamma_{\max} = \inf_{\omega: G(j\omega)\neq 0} \Re\{1/G(j\omega)\}$$ i.e., the infimum of the real part of the inverse Nyquist locus. For MIMO, the maximum index is obtained by solving an LMI over the frequency axis: $$G+G^H \succeq 2G\gamma G^H\ ,\quad \gamma=\gamma^* I$$ and then $$\gamma_{\max} = \min_\omega \lambda_{\min}\left( (G+G^H)/(2GG^H) \right )$$ (Peng et al., 15 Jan 2026). For fixed-structure controller synthesis, the maximal index is achieved by convex optimization (SDP) with affine dependence on controller parameters via SOS matrix certificates (Su et al., 2019).

2.3. On-Line, Data-Driven Estimation

Estimates of maximal passivity indices can be performed from streaming input-output data. For the IFP index, one computes

$$\hat\nu^*(t_f) = \min_{t_0 < t_f} \frac{ \int_{t_0}^{t_f} u^T y\,dt + K_s (t_f-t_0) }{ \int_{t_0}^{t_f} u^T u\,dt }$$

Typically, the search over $(t_0,t_f)$ can be reduced to a 1-D search, and with suitable conditions, the minimum is achieved in the limit $t_1 \to t_0$ (Welikala et al., 2022).

2.4. Static I/O Transformations and Cone Conditions

For SISO systems with passivity shortage $(\rho,\nu)$, the region of possible passivity indices after static I/O transformation is characterized via projective quadratic inequalities (PQI), yielding double-cone regions. The supremal input or output passivity index achievable by linear static transformations is expressible in closed form: $\mu^* = \frac{1 - \sqrt{1-4\rho\nu}}{2\rho}$ for valid $\rho\nu < 1/4$, with explicit SISO and MIMO extension via cone or S-lemma characterizations (Sharf et al., 2019, Sharf et al., 2019).

3. Domain-Specific Passivity Indices

3.1. Mueller Matrices in Polarimetry

In polarization optics, the maximal passive scaling (passivity index) of a Mueller matrix $M$ corresponds to the maximal achievable output transmittance such that $M$ remains passive. The index is given by

$T_{\max} = \frac{1}{1+\max(\|D\|,\|P\|)}$

where $D$ and $P$ are the diattenuation and polarizance vectors derived from $M$. This index quantifies the maximal physically admissible intensity transmission compatible with passivity (José et al., 2019).

3.2. Power Electronics and Converter Control

For dispatchable virtual oscillator controlled grid-forming converters (dVOC), the explicit OFP passivity index can be obtained as

$$\delta_k^{\max} = -\,\frac{p_k^\star\cos\varphi + q_k^\star\sin\varphi}{v_{\max}^2} \;-\;\alpha_{\min}$$

where $p_k^*,q_k^*$ are node setpoints, $v_{\max}$ terminal voltage, $\alpha_{\min}$ the smallest admissible amplitude-regulation gain, and $\varphi$ a network parameter. Maximizing $\delta_k$ involves operating at $v_{\max}$, minimizing $\alpha_k$, and selecting moderate setpoints (He et al., 2023).

4. Operational Limitations and Sensitivity

Operational bounds on state and input variables sharply constrain the achievable passivity index. As the admissible set $X \times U$ is expanded, the provable passivity surplus necessarily decreases, possibly becoming negative at a critical radius. All maximal-index guarantees are only local within the prescribed bounds (Zakeri et al., 2019).

The parameter dependence is often non-monotonic and can exhibit critical thresholds, e.g., for scalar systems, the maximal $\rho(r)$ remains nearly constant until a threshold radius and then decays rapidly.

5. Geometric and Graphical Interpretation

The Positive Damping Region (PD region) framework gives a geometric characterization: for SISO systems, the Nyquist locus must remain in an index-dependent disk in the complex plane for output-feedback passivizability with index $\gamma$: $$P_{PD}^{(SISO)}(\gamma) = \left\{ z \in \mathbb{C}: |z - (1/(2\gamma))| \leq 1/(2\gamma) \right\}$$ The maximum passivity index is realized when the locus is tangent to the boundary (Peng et al., 15 Jan 2026). For MIMO, a similar disk in the Rayleigh quotient plane applies. This geometric metric also quantifies the waterbed effect: greater passivity surplus restricts the frequency bandwidth over which passivity can be ensured.

6. Illustrative Examples and Applications

• Nonlinear system subject to quartic storage search yields $\rho^* \approx 0.35$ locally; beyond a critical state bound, passivity is lost (Zakeri et al., 2019).
• LTI plant $G(s) = 1/(Ts + k)$ has maximum output-feedback index $\gamma_{\max} = k$, as all $1/G(j\omega)$ are $k$ real (Peng et al., 15 Jan 2026).
• In polarimetry, a partial diattenuator with $\|D_0\|=0.6$ achieves $T_{\max}=0.625$ (José et al., 2019).
• In passivation synthesis, the supremal passivity index achieved by static I/O transformations is characterized via explicit matrix inequalities and often can be made unbounded (in the absence of further constraints) (Sharf et al., 2019).

7. Limitations, Robustness, and Open Directions

The maximum achievable passivity index is inherently local if the supply rate or dissipation inequality is only enforced over a bounded region. Nonlinear systems require careful polynomial approximation and complexity tradeoffs (e.g., Taylor vs Bernstein expansions). For LTI settings, while geometric and LMI characterizations are sharp, frequency discretization can mask pathological points.

In on-line settings, estimator accuracy depends on the Lipschitz bound $K_s$ and richness of the observed data; insufficient excitation or over-conservative bounds can lead to overly optimistic indices (Welikala et al., 2022).

A plausible implication is that quantitative passivity indices provide not only a pass/fail metric but enable systematic controller design trade-offs (e.g., via the waterbed effect) and adaptive tuning in real time. Their integration with data-driven methods, sector bounds, and distributed robust control remains a key subject of ongoing research.