Passivity-Based Lyapunov Techniques

• Passivity-Based Lyapunov Techniques are methods that use energy-like storage functions to certify stability, synchronization, or convergence in nonlinear and adaptive systems.
• They exploit inherent dissipativity by constructing quadratic or energy-based Lyapunov functions that ensure robustness and modularity across distributed and networked architectures.
• Applications span adaptive control, sliding-mode strategies, optimization dynamics, and safety-critical human–robot interactions, providing scalable and resilient control designs.

Passivity-based Lyapunov techniques are a foundational class of analysis and synthesis tools that leverage input–output passivity properties to construct Lyapunov functions certifying stability, synchronization, or convergence of nonlinear, networked, or adaptive systems. These methods systematically exploit the inherent energy-like structure of dynamical systems by selecting storage functions (typically quadratic or energy-based) and showing their non-increasing nature along trajectories, in combination with network interconnections, uncertainty, or structural adaptation. Passivity-based Lyapunov approaches unify and extend classical Lyapunov direct methods by encoding dissipativity properties that are robust to perturbations, modular under interconnection, and applicable to a broad range of heterogeneous and complex dynamical architectures.

1. Foundational Principles of Passivity and Storage Functions

A dynamical system $\dot{x} = f(x, u),\; y = h(x, u)$ is called passive if there exists a continuously differentiable storage function $V(x) \geq 0$ with $V(0)=0$ that satisfies the dissipation inequality

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

for all $(x, u)$. Strict passivity further adds a positive definite dissipation term. The storage $V$ serves as a Lyapunov (or Lyapunov-like) function: if $\dot{V}(x) < 0$ except at equilibria, stability follows. For linear time-invariant (LTI) systems, passivity is equivalent to strict positive realness (SPR) of the input–output transfer, and the Kalman–Yakubovich–Popov (KYP) lemma guarantees the existence of a quadratic storage function by solving a Lyapunov equation

$A^\top P + P A = -Q, \qquad Q \succ 0,$

where $V(x) = x^\top P x$ (Cho et al., 2023, Wafi et al., 2 Mar 2026).

2. Structured Lyapunov Synthesis in Passivity-Based Adaptive Control

In adaptive and multi-agent settings, passivity-based Lyapunov constructions are designed to certify not only stability but convergence of synchronization errors and parameter estimates. A prototypical synthesis proceeds as follows:

• SPR Regime: A filtered error $\vartheta_u$ and a parameter error $V(x) \geq 0$0 are defined. The candidate Lyapunov function is

$V(x) \geq 0$1

where $V(x) \geq 0$2 solves the Lyapunov equation for the nominal error dynamics. Adaptive laws are tailored to cancel cross-terms, yielding strict decrease:

$V(x) \geq 0$3

Parameter convergence is ensured under persistent excitation (PE), and global synchronization or tracking is guaranteed if the graph is suitably connected (Wafi et al., 2 Mar 2026).

• Non-SPR Regime and Frequency Shaping: For systems lacking a globally SPR channel, frequency shaping techniques are used to recover passivity. Output or regressor shaping creates an SPR transfer, and a frequency-shaped Lyapunov function (using matrices provided by the Meyer–Kalman–Yakubovich lemma) certifies stability in the shaped coordinates:

$V(x) \geq 0$4

with $V(x) \geq 0$5 solving

$V(x) \geq 0$6

Standard dissipation arguments yield guarantees analogous to the SPR case (Wafi et al., 2 Mar 2026).

3. Passivity-Based Lyapunov Techniques in Distributed, Networked, and Constrained Systems

Passivity-based Lyapunov methods extend naturally to large-scale, interconnected, and networked settings:

• Consensus and Output Agreement: For agents interconnected over digraphs, submanifold-constrained storage functions $V(x) \geq 0$7 vanish only on the agreement submanifold $V(x) \geq 0$8. Lyapunov candidates of the form

$V(x) \geq 0$9

decrease strictly along the projections onto the disagreement subspace, ensuring output agreement via LaSalle's invariance (Yue et al., 29 Aug 2025).

• Optimization Dynamics: In continuous-time optimization, the use of passivity-friendly state representations (e.g., Brayton–Moser form) enables Krasovskii-type quadratic storage functions in the velocities:

$V(0)=0$0

which, under strict convexity, guarantees strict dissipation and convergence to equilibrium (optimality) (Kosaraju et al., 2017). For inequality-constrained flows, piecewise quadratic storages indexed by the active constraints ensure non-increasing composite Lyapunov functions under switching.

• Hybrid and Safety-Critical Control: In human–robot interaction, composite storage functions built from physically motivated quadratic terms in force, velocity, and attitude errors provide strict proofs that the robotic system remains output strictly passive and thus energetically safe (Ding et al., 2022).

4. Extensions: Adaptive, Sliding-Mode, and Data-Driven Passivity-Based Lyapunov Design

Advanced passivity-based Lyapunov methodologies span several domains:

• Adaptive Control: Adaptive laws leveraging passivity yield Lyapunov functions that couple tracking errors with parameter errors, e.g.,

$V(0)=0$1

Cancellation of mixed terms via adaptive tuning rules ensures monotonic decrease and robust convergence, without depth invertibility or restrictive parametrization (Wang, 2015).

• Sliding-Mode and Robust Control: In port-Hamiltonian systems, sliding-mode control can be merged with passivity principles by constructing Lyapunov functions of the form

$V(0)=0$2

ensuring both finite-time convergence of the sliding variable and asymptotic stability of the full state under a spectrum of smooth/robust sliding functions (Sakata et al., 2022).

• Neural and Data-Driven Energy Shaping: Neural network parameterizations can be embedded into the Interconnection and Damping Assignment–Passivity Based Control (IDA–PBC) framework, where the NN outputs shape the closed-loop Hamiltonian and damping to achieve Lyapunov decrease and passivity by construction (Sanchez-Escalonilla et al., 2021).
• High-Order and Homogeneous Controllers: Smooth Lyapunov candidates of homogeneous degree, built around plant/integrator interconnections with cross-terms tailored for passivity, allow for the rigorous extension of classical finite-time or ISS controllers (e.g., super-twisting algorithms) to the MIMO and uncertain setting (Garcia-Mathey et al., 2022).

5. Generalization and Interrelationships: Incremental, Shifted, and Krasovskii Passivity

Generalizations of passivity-based Lyapunov techniques include:

• Incremental and Shifted Passivity: By shifting storage functions to nonzero equilibria, passivity-based Lyapunov designs are extended to stabilization around arbitrary operating points. This is formalized via shifted passivity and incremental passivity, wherein the storage $V(0)=0$3 or $V(0)=0$4 satisfy

$V(0)=0$5

This allows tailored Lyapunov proofs for tracking or regulation tasks (Kawano et al., 2019).

• Krasovskii Passivity: By analyzing the prolonged system $V(0)=0$6—where $V(0)=0$7 is also treated as a dynamic state—one can leverage quadratic storages in $V(0)=0$8 (Krasovskii Lyapunov functions) to deduce properties intermediate between differential and shifted passivity. This enables the synthesis of dynamic controllers with systematic Lyapunov guarantees even when direct passivity is difficult to certify.
• Stochastic Weak Passivity: For systems with non-vanishing diffusion at equilibria, stochastic variants of passivity restrict dissipation inequalities to the exterior of small balls, yielding Lyapunov-like criteria for asymptotic weak stability—convergence in distribution and ergodicity—under negative proportional feedback (Fang et al., 2016).

6. Practical Impact, Modular Design, and Future Directions

Passivity-based Lyapunov techniques provide:

• Modular synthesis and analysis tools for distributed, networked, or heterogeneous systems, accommodating time-varying, uncertain, nonlinear, or high-dimensional dynamics, as long as passivity (or recoverable SPR) is certified for the key interconnection channels (Wafi et al., 2 Mar 2026, Yue et al., 29 Aug 2025, Sakata et al., 2022).
• Scalability and compositionality, as each subsystem’s storage and passivity property can be analyzed locally and then combined additively as a global Lyapunov function—especially critical in consensus, synchronization, and optimization on graphs or networks.
• Robustness, including disturbance rejection, ISS, and performance in adaptive or uncertain scenarios, due to the inherent energy-dissipation constraints.
• Flexibility to integrate with data-driven or machine learning techniques (e.g., NN-based Hamiltonian shaping (Sanchez-Escalonilla et al., 2021)), as well as advanced hybrid or multi-modal controllers for safety-critical applications (Ding et al., 2022).

Potential extensions include modular synthesis for time-delays, sampled-data, observer networks, hybrid protocols, and iterative refinement in high-dimensional nonlinear control, communication, and optimization (Wafi et al., 2 Mar 2026, Cho et al., 2023).

• A Passivity-Agnostic Framework for Distributed Adaptive Synchronization under Unknown Leader Dynamics (Wafi et al., 2 Mar 2026)
• A Passivity Analysis for Nonlinear Consensus on Digraphs (Yue et al., 29 Aug 2025)
• A Passivity-Based Method for Accelerated Convex Optimisation (Cho et al., 2023)
• Passivity-based sliding mode control for mechanical port-Hamiltonian systems (Sakata et al., 2022)
• Total Energy Shaping with Neural Interconnection and Damping Assignment -- Passivity Based Control (Sanchez-Escalonilla et al., 2021)
• Stability Analysis of Constrained Optimization Dynamics via Passivity Techniques (Kosaraju et al., 2017)
• Krasovskii and Shifted Passivity Based Control (Kawano et al., 2019)
• Stochastic Weak Passivity Based Stabilization of Stochastic Systems with Nonvanishing Noise (Fang et al., 2016)
• MIMO Super-Twisting Controller using a passivity-based design (Garcia-Mathey et al., 2022)
• Passivity-Based Adaptive Control for Visually Servoed Robotic Systems (Wang, 2015)
• Global Stabilisation of Underactuated Mechanical Systems via PID Passivity-Based Control (Romero et al., 2016)
• A Passivity Based Framework for Safe Physical Human Robot Interaction (Ding et al., 2022)