Passivity Matrices: Theory and Applications

• Passivity matrices are matrix-valued certificates that quantify and generalize passivity indices for multi-input multi-output systems by capturing input-output channel couplings.
• They are derived from nonlinear dissipativity theory and LTI system analysis using LMIs and the KYP Lemma to ensure robust stability and facilitate controller synthesis.
• Applications include gain scheduling, electrical networks, and wave physics, demonstrating reduced conservatism and enhanced structural insights over traditional scalar indices.

A passivity matrix is a matrix-valued certificate quantifying the passivity properties of a dynamical or operator-theoretic system, generalizing classic scalar passivity indices to the multi-input multi-output (MIMO) case. Passivity matrices arise naturally in nonlinear dissipativity theory, linear system analysis, robust control, network engineering, passivation design, and wave physics. They provide structural, computational, and physical insights unavailable from scalar indices, enabling both certification and synthesis of complex passive systems.

1. Formal Definitions and Frameworks

The passivity matrix generalizes the scalar passivity index to encapsulate passivity deficiency or excess along all input-output channels and their couplings. For a dissipative control-affine system

$\dot{x} = f(x,u), \quad y = h(x,u)$

with storage function $V(x) \geq 0$ and the canonical quadratic supply rate $s(u, y) = u^{\top} y$, the system is called input-feedforward/output-feedback (IF–OFP) passive with passivity matrices $(\Phi, \Xi) \in S^m$ if

$$\dot V(x) \leq u^{\top} y - u^{\top} \Phi u - y^{\top} \Xi y$$

for all state-input pairs. The augmented supply-rate matrix

$$Q := \begin{pmatrix} -\Phi & \tfrac12 I \ \tfrac12 I & -\Xi \end{pmatrix}$$

characterizes the dissipativity inequality in quadratic form. The matrix $\Phi$ quantifies input-feedforward passivity (IFPM), $\Xi$ output-feedback passivity (OFPM) (Ru et al., 8 Jan 2026).

For linear time-invariant (LTI) systems, the frequency-domain passivity matrix is built from the Hermitian part

$$H(\omega) = \frac{1}{2}\bigl[G(j\omega) + G(j\omega)^{H}\bigr]$$

where $G(s)$ is the transfer function. Any constant $\Phi \preceq H(\omega)$ for all $\omega$ is a valid IFPM. The Kalman–Yakubovich–Popov (KYP) Lemma relates the existence of a storage matrix $P>0$ (often called the passivity matrix in LTI state-space) to a matrix LMI (Beattie et al., 2018, Ru et al., 8 Jan 2026).

2. Variational, Geometric, and Stability Properties

Under the matrix-valued framework, passivity indices correspond to the curvature (Hessian) of the dissipation functional

$$J_T[u] = \frac{1}{T}\int_0^T \bigl[ u^{\top}(t)y(t) - \dot V(x(t)) \bigr] dt$$

about a nominal trajectory. Its second variation yields a kernel operator on $L^2[0,T]$ whose spectral decomposition identifies principal passivity directions and their strengths. For LTI systems, in the infinite horizon limit, the Fourier transform recovers $H(\omega)$ as the frequency-domain “Hessian” (Ru et al., 8 Jan 2026). This formalism reveals that passivity for MIMO systems is fundamentally anisotropic: both the direction and the intensity of passivity deficiency or excess matter.

Representative matrices can be selected by criteria such as maximizing trace or minimum eigenvalue of $\Phi$ subject to $\Phi \preceq H(\omega)$, each of which is solvable as an LMI. The Loewner partial order organizes admissible passivity matrices and their optimal representatives (Ru et al., 8 Jan 2026).

For feedback-interconnection stability, sharp conditions generalize classic scalar margins: if subsystems possess passivity matrices $(\Phi_1, \Xi_1)$ and $(\Phi_2, \Xi_2)$, negative-feedback interconnection is $L_2$-stable iff

$$\Phi_1 + \Xi_2 \succ 0, \quad \Phi_2 + \Xi_1 \succ 0$$

which is less conservative than requiring all scalar indices to be strictly positive (Ru et al., 8 Jan 2026).

3. Passivity Matrices in LTI System Certification and Hamiltonian Structures

A canonical application of passivity matrices is the KYP-LMI for minimal LTI systems: $$W(P) = \begin{pmatrix} -PA - A^H P & C^H - PB \ C - B^H P & D + D^H \end{pmatrix} \succeq 0,$$ where $P = P^H \succ 0$ is the storage/passivity matrix. Existence of such a $P$ certifies strict (or standard) passivity. The solution set is convex and often highly non-unique: all port-Hamiltonian realizations correspond to different $P$ (Beattie et al., 2018, Bankmann et al., 2019).

Robustness of passivity (passivity radius) is quantified by the minimum perturbation to system matrices that destroys $W(P) \succeq 0$. The analytic center $P_{\mathrm{ac}}$ of the feasible set, maximizing $\log\det W(P)$, yields maximal passivity radius and optimal symmetry properties for the underlying port-Hamiltonian representation. Numeric algorithms (Newton, steepest-ascent) efficiently compute $P_{\mathrm{ac}}$ for high-dimensional problems (Beattie et al., 2018, Bankmann et al., 2019).

4. Passivity Matrices and Convex Optimization: Selection and Radius

Passivity matrices admit a convex optimization interpretation. In LTI and port-Hamiltonian systems, the feasible set for $P$ is a convex domain, generally defined by bounding LMI, with the analytic center providing an optimally robust certificate. Key metrics:

• The minimum eigenvalue of $W(P)$, $\alpha(P)$, and of $P^{-1}W(P)P^{-1}$, $\beta(P)$, provide margin measures;
• The passivity radius $\rho(P)$ relates directly to the product $\alpha(P)\beta(P)$;
• The analytic center maximizer $P_{\mathrm{ac}}$ simultaneously places the “center” of the feasible set and achieves large robustness margins (Beattie et al., 2018, Bankmann et al., 2019).

These structural properties underpin robust passivation and less conservative controller design.

5. Applications: Gain Scheduling, Electrical Networks, and Wave Systems

Gain-Scheduled Control with Passivity Matrices:

In VSP (very strictly passive) controller gain-scheduling, scheduling matrices $\Phi_i(t)\in \mathbb{R}^{n\times n}$ replace scalar signals, increasing design freedom. Passivity is preserved provided all $\Phi_i(t)$ are uniformly bounded and the system is strongly active (at least one full-rank $\Phi_i(t)$ at all times). The composite controller’s passivity matrix structure determines closed-loop $L_2$-stability under the classical passivity theorem. Empirically, matrix scheduling achieves order-of-magnitude reductions in regulation error in MIMO robotic benchmarks (Moalemi et al., 2024).

Passivity in Electrical Networks:

For three-phase transmission networks represented in D-Q (rectangular) coordinates, the passivity matrix is the Hermitian part $\Re\{Y(j\omega)\}$ of the D-Q admittance. Positivity of this matrix at all frequencies certifies passivity, but can be violated by converter droop dynamics at low frequencies. Reformulating in polar variables yields new passivity matrices whose positive (semi)definiteness can be restored by appropriate control design, allowing decentralized verification (Dey et al., 2021).

Wave Physics and Scattering:

In scattering systems, the set of admissible transmission, reflection, and absorption matrices is fully determined by passivity, energy conservation, and the classical Horn eigenvalue-sum inequalities. The resulting polyhedral set of feasible eigenvalue tuples generalizes the notion of the passivity constraint to wave physics, connecting matrix passivity concepts with algebraic geometry and convex polyhedra (Guo et al., 2024).

6. Shifted and Generalized Passivity Matrices

In port-Hamiltonian systems, shifted passivity (around nonzero operating points) introduces state-dependent passivity matrices $M(s)$,

$$M(s) = \nabla[\mathcal{F}(s)s^*] + \nabla[\mathcal{F}(s)s^*]^\top - 2R^*$$

whose monotonicity (negative semidefiniteness) over the co-energy range certifies shifted passivity and stability. For quadratic–affine systems, $M$ is constant. Output feedback can adjust $M$ to ensure passivation if the open-loop matrix condition fails (Monshizadeh et al., 2017).

7. Large-Scale Systems, Computational Aspects, and Extensions

For high-order or large-input/output systems, direct Hamiltonian spectral tests (passivity matrix eigenvalue analysis) become computationally prohibitive. Hierarchical adaptive sampling and piecewise frequency-warped algorithms efficiently identify regions where passivity matrix positivity fails, retaining reliability while reducing complexity by orders of magnitude (Stefano et al., 2020).

Applications extend to decompositions in polarization optics (Mueller matrices), where scalar matrix inequalities and convex decompositions into pure passive components play a similar role, and provide necessary and sufficient experimental validation tools (José et al., 2019).

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Summary Table: Passivity Matrices—Key Structures and Criteria

Context	Passivity Matrix (Form)	Passivity Criterion
General dissipative sys.	$(\Phi, \Xi)$ (input/output)	$$\dot V \leq u^{\top}y - u^{\top}\Phi u - y^{\top}\Xi y$$
LTI state-space	$P$ (storage, KYP-LMI)	$W(P) \succeq 0$ (block LMI)
Scattering/Wave physics	$T, R, A$ (Hermitian)	$T+R+A=I$, $T,R,A \succeq 0$ plus Horn inequalities
Power system (D-Q)	$\Re\{Y(j\omega)\}$	$\Re\{Y(j\omega)\} \succeq 0$ for all $\omega$
Shifted pH systems	$M(s)$	$M(s) \preceq 0$ over co-energy range

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Significance:

Passivity matrices provide a comprehensive, geometric, and computationally tractable formalism for certifying, analyzing, and designing passive behavior in complex MIMO systems. The migration from scalar to matrix-valued indices dramatically reduces conservatism, exposes underlying channel couplings, enables robust controller synthesis (via optimized matrix selection), and connects energy-based system theory with operator inequalities, convex optimization, and the spectral theory of structured matrices. These tools underpin not only classical control and circuit design but also contemporary advances in robotics, energy systems, wave transport, and large-scale network science (Ru et al., 8 Jan 2026, Beattie et al., 2018, Moalemi et al., 2024, Guo et al., 2024, Bankmann et al., 2019, Stefano et al., 2020, Dey et al., 2021, Monshizadeh et al., 2017, José et al., 2019).