Passivity-Based Integral Action

• Passivity-based integral action is a control paradigm that fuses integral control with passivity theory to guarantee stability, output regulation, and robustness.
• It leverages storage functions and dissipation inequalities, alongside symplectic discretization, to ensure energy balance and convergence in continuous and discrete-time systems.
• The approach extends to underactuated, nonlinear, and distributed systems, providing robust control in mechanical, optimization, and data-driven contexts with global stability guarantees.

Passivity-based integral action refers to the systematic embedding of integral control within a passivity framework in order to guarantee closed-loop stability, output regulation, and robustness for nonlinear and/or distributed-parameter systems. Unlike classical integral control—which may jeopardize stability if not carefully designed—passivity-based integral action leverages the dissipation inequalities and storage function structure of the system, ensuring that the integral channel interacts with the plant via passive input-output maps. This paradigm has been rigorously developed for finite and infinite-dimensional systems, covers both continuous and discrete-time settings, and extends naturally to underactuated plants, systems on nonlinear manifolds, and convex optimization dynamics.

1. Foundations: Passivity Theory and Integral Action

Passivity theory formalizes energy balance in input-output systems via the existence of a storage function $H(x)$ and a dissipation inequality $\dot{H}(x) \leq u^\top y$, where $u$ is input and $y$ the passive output. Integral action is traditionally introduced to eliminate steady-state error under constant disturbances or time-invariant references. In a passivity-based architecture, the integral term itself is structured as a passive subsystem (typically, $\dot{\xi} = y$), and its interconnection preserves or enhances the overall system passivity.

For finite-dimensional port-Hamiltonian (pH) or input-affine nonlinear systems, this approach dates back to the design of PI/PID passivity-based controllers (PI-PBC/PID-PBC), wherein the plant’s passive output is used as the input to the integrator, and the control law is closed via suitable feedback around the passive output (Aranovskiy et al., 2015, Romero et al., 2016). For distributed-parameter systems, new passivity properties based on integrable port-variables enable the derivation of PI-like boundary controllers with Lyapunov stability guarantees (Kosaraju et al., 2019).

2. Discrete-Time Passivity-Based Integral Action

The preservation of passivity under discretization is nontrivial—standard discretizations may destroy the passivity inequality even for arbitrarily small sampling intervals. This issue is resolved in the globally stable discrete-time PID-PBC of (Moreschini et al., 26 Aug 2025), where both plant and PID controller are discretized via the implicit midpoint (symplectic) rule, which uniquely preserves shifted passivity for arbitrary sampling times. The resulting discrete-time controller: $$u_k = -K_P\,\tilde y_k - \tfrac{1}{2}K_I(\xi_{k+1}+\xi_k) - \tfrac{1}{\delta}K_D\,\mathcal C(x_{k+1}-x_k)$$ ensures lossless/purely dissipative feedback interconnection, and the discrete Lyapunov argument yields global stability. The midpoint-integral term ensures that the integral action (via $\tilde y_k$) accumulates the error needed to drive the output to its reference— guaranteeing zero steady-state error for nonzero setpoints via shifted (incremental) passivity.

3. Structural Integration of Passivity-Based Integral Action in Nonlinear Control

In PI-PBC, the plant is assumed to possess a convex, often separable Hamiltonian, and a known input assignment structure (Aranovskiy et al., 2015). The state is split into actuated/unactuated coordinates, and the passive output is derived from the actuated variables (possibly through a nonlinear map). The design exploits properties such as

$$\dot H(x) \leq u^\top y, \quad \dot U(x) = -Q(x) + e^\top\tilde u$$

with incremental storage $U(x)$, and the controller is implemented as: $$\dot z = -K_I[\Phi(x_a)-\Phi(x_a^*)], \quad u = -K_P[\Phi(x_a)-\Phi(x_a^*)] + z$$ where $z$ is the integral state. Composite Lyapunov functions ensure global stability, and all closed-loop trajectories are proven to converge to the desired equilibrium under mild detectability conditions. The same architecture extends to port-Hamiltonian temperature regulation and general robust PI-PBC stabilization for a broad class of system structures.

4. Passivity-Based Integral Action for Port-Hamiltonian and Mechanical Systems

For fully actuated mechanical pH systems, integral action is introduced to reject constant matched disturbances and provide steady-state regulation. In (Chan-Zheng et al., 2022), the integral state $z$ evolves according to

$\dot z = -K_i\,\bar y$

where $\bar y$ is the shaped error variable derived from shifted coordinates. The overall feedback law is

$$u = G(q)^{-1}\left[\nabla_q V(q) - M_d^{-1}M(q)K_p\bar q - \Gamma(x)K_p\bar q - K_p\dot q - K_dM_d^{-1}\bar p + z\right]$$

guaranteeing exponential stability provided explicit gain conditions involving inertia, damping, and gyroscopic forces are met. The Lyapunov storage function directly incorporates the integral state and yields explicit tuning rules for convergence rate, overshoot, and input-to-state stability against unmatched disturbances.

For underactuated mechanical systems, PID-PBC constructs two passive outputs (associated with underactuated and actuated coordinates), forms a linear combination, and wraps a PID (including integral) action around the resulting output (Romero et al., 2016). The integral state effectively shapes a closed-loop Lyapunov function with properties analogous to potential energy assignment, thus ensuring global stabilization under broad verifiable conditions.

5. Passivity-Based Integral Action on Non-Euclidean Manifolds and Distributed Systems

Integral action can be intrinsically defined for configuration spaces evolving on Lie groups, where standard integrals of configuration error lack global meaning. In (Zhang et al., 2014), the integral is constructed via parallel transport of the proportional/PD command through the group’s tangent spaces, maintaining a passivity-inspired Lyapunov structure. This architecture guarantees rejection of constant biases (velocity or torque) in attitude or rigid body control, both for SO(3) and SE(3) systems.

For distributed-parameter systems, as in flexible piezoelectric beams (Kosaraju et al., 2019), passivity-based integral controllers leverage the integrability of port-variables to construct PI-like boundary controls. The closed-loop is proven strictly dissipative in terms of a generalized storage, and the integral term ensures zero steady-state error of boundary quantities. The tuning of proportional and integral gains involves trade-offs between vibration damping, rise time, and overshoot.

6. Passivity-Based Integral Action in Data-Driven and Optimization Contexts

Passivity-based integral action is extended to data-driven settings through the parallel design of integral-FIR (iFIR) controllers, where a pure integrator and a finite impulse response filter are combined in discrete time (Wang et al., 2024). Passivity constraints (via KYP LMI, Toeplitz, or PR sampling) ensure that the learned controller maintains closed-loop passivity—regardless of plant uncertainties or nonlinearities—while guaranteeing zero steady-state error.

In convex optimization, primal-dual dynamics can be generalized within a passivity-based framework, constructing controllers with phase-lead modules (stable zeros) in primal, equality, and inequality dual channels (Yamashita et al., 2018). Supplying a stable zero to the integrator is sufficient to recover convergence to KKT points without strict convexity or strong monotonicity, thus extending the classical soft constraint paradigm. The integrated passivity argument yields robustness and improved noise attenuation relative to standard primal-dual flows.

7. Robustness, Adaptation, and Extensions

Passivity-based integral action provides inherent robustness to constant matched disturbances and moderate modeling uncertainties. Extensions to robust integral controllers for pH systems discard the necessity of precise knowledge of dissipation and rely only on conservative upper bounds (Ferguson et al., 2018). Global asymptotic stability can be preserved in the face of model uncertainties or slow actuator dynamics when coupled with invariance principles for (possibly discontinuous) dynamical systems (Gutierrez-Oribio et al., 2022).

Adaptive passivity-based integral control, for example in fuel-cell/boost converter cascades, incorporates online parameter estimators (e.g., hybrid gradient/immersion invariance observers) directly into the passivity-based loop, preserving global convergence and output regulation in the presence of significant parametric uncertainty (Beltran et al., 2024).

---

The passivity-based design of integral action forms a rigorous foundation for high-performance, robust, and globally stable regulation across a broad spectrum of nonlinear, distributed, and high-dimensional control systems, ensuring stability and regulation properties unattainable by direct application of classical integral methods without passivity-theoretic embedding. The integration of energetic and dissipative structures with integral compensation, as formalized in the cited works, represents a unifying principle spanning modern control, optimization, and data-driven controller synthesis.