PID Passivity-Based Control (PID-PBC)

• PID-PBC is a nonlinear control strategy that interconnects a PID controller with a passive plant output to establish global closed-loop stability via energy-based arguments.
• It leverages passivity properties and energy shaping techniques in port-Hamiltonian frameworks, offering robust performance in power electronic and mechanical applications.
• The approach encompasses discrete-time implementations, observer-based designs, and safety filters to handle uncertainties and constraints in complex control systems.

PID passivity-based control (PID-PBC) is a nonlinear control methodology in which a PID controller is interconnected with a plant output that is passive, incremental passive, or shifted passive, so that closed-loop stability is established from a storage-function argument rather than from local linearization. In the power-converter literature this principle is stated explicitly as leveraging the passivity property of PIDs for positive gains and wrapping the PID around a passive output to ensure global stability (Moreschini et al., 26 Aug 2025). In mechanical and port-Hamiltonian settings, the same idea appears as energy shaping plus damping injection, often augmented by integral action to regulate nonzero equilibria and to avoid the PDE-based synthesis required by other passivity-based methods (Romero et al., 2016, Zonetti et al., 2021). The resulting body of work includes literal PID laws, PI-PBC and PI-like variants, observer-based implementations, leakage and saturation modifications, extensions to distributed-parameter systems, and passivity-preserving safety filters (Kosaraju et al., 2019, Bobtsov et al., 2020, Califano, 2023).

1. Passivity principle and closed-loop construction

The common analytical core of PID-PBC is the passivity inequality. For an affine nonlinear system $\dot{x}=f(x)+g(x)u$, passivity with storage $S$ and output $y$ is expressed as $\dot S \le y^\top u$; for physical systems, $S$ is energy and $y^\top u$ is power (Califano, 2023). In port-Hamiltonian form,

$$\dot{x}=(J(x)-R(x))\frac{\partial H}{\partial x}+g(x)u,\qquad y=g(x)^\top\frac{\partial H}{\partial x},$$

with $J=-J^\top$ and $R=R^\top\ge 0$, this becomes

$$\dot H=-\frac{\partial H}{\partial x}^\top R(x)\frac{\partial H}{\partial x}+y^\top u \le y^\top u,$$

which is the standard energy balance used by passivity-based design (Califano, 2023).

Within PID-PBC, the regulated variable is usually not the plant state directly, but a passive output defined relative to a desired equilibrium. In power converters this is often an incremental or shifted-passive output. A representative formulation uses

$S$0

with incremental storage $S$1 satisfying

$S$2

so that the PID can be interconnected with $S$3 rather than with a naive tracking error (Zonetti et al., 2021). This is the technical distinction between PID-PBC and classical PID around a linearized model: the controller is built from a passive map of the nonlinear plant, and the stability proof is carried out on the nonlinear closed loop (Zonetti et al., 2021).

The controller channel may be written in a literal PID form or in an equivalent port-Hamiltonian form. For power converters, one standard expression is

$S$4

with $S$5 matrix gains (Zonetti et al., 2021). In continuous time, the converter review reports that under perfect knowledge of the system parameters the equilibrium is globally exponentially stable for any $S$6, $S$7, and $S$8 (Zonetti et al., 2021). Closely related robust PI-PBC constructions for partially known nonlinear systems proceed from incremental passivity around $S$9, using only actuated-coordinate measurements and the known input matrix structure, again with the storage function shifted to the desired operating point (Aranovskiy et al., 2015).

2. Mechanical-system formulations

For underactuated Euler–Lagrange systems, PID-PBC has been developed around passive outputs tailored to the inertial coupling structure. A broad class is characterized by constant collocated actuation $y$0, inertia depending only on unactuated coordinates, constant actuated inertia subblock, separable potential energy, and an integrability condition on the coupling block $y$1 (Romero et al., 2016). Under these assumptions, two passive outputs are constructed: $y$2 with corresponding storage functions $y$3 and $y$4 satisfying $y$5 and $y$6 (Romero et al., 2016). The PID is not wrapped around either output individually but around the weighted combination

$y$7

The resulting controller is a standard linear PID in $y$8,

$y$9

with $\dot S \le y^\top u$0 and $\dot S \le y^\top u$1 (Romero et al., 2016). The crucial step is that the integrability condition allows $\dot S \le y^\top u$2 to be expressed as a function of configuration variables, which converts the closed-loop storage into a shaped mechanical energy

$\dot S \le y^\top u$3

With suitable gains and integral-state initialization, this $\dot S \le y^\top u$4 is a Lyapunov function for the desired equilibrium; if $\dot S \le y^\top u$5 is radially unbounded, the stability result is global, and under stronger coupling and boundedness assumptions LaSalle arguments yield convergence without the full positive-definiteness condition used in the Lyapunov proof (Romero et al., 2016).

A related mechanical line of work emphasizes that transient oscillations depend strongly on the natural damping and on how the passivity-based gains are tuned. For port-Hamiltonian mechanical systems with passive output $\dot S \le y^\top u$6, one PI-PBC law is

$\dot S \le y^\top u$7

which produces a closed-loop Hamiltonian

$\dot S \le y^\top u$8

and damping matrix $\dot S \le y^\top u$9 (Chan-Zheng et al., 2021). The same paper derives the no-oscillation tuning condition

$S$0

and uses energy-based damping identification to estimate the diagonal viscous damping matrix before selecting $S$1 and $S$2 (Chan-Zheng et al., 2021). For underactuated or flexible-joint cases, a modified PI-PBC introduces an additional damping term in unactuated velocities,

$S$3

which preserves port-Hamiltonian structure and is analyzed through exponential-convergence estimates (Chan-Zheng et al., 2021).

3. Power-electronic realizations

Power converters are one of the most developed application domains for PID-PBC because averaged converter models are naturally port-Hamiltonian and the desired operating points are typically nonzero. For converters without switching sources,

$S$4

PID-PBC is built around the shifted passive output $S$5 (Zonetti et al., 2021). In the nominal case, the review on converter PID-PBC states that the closed-loop equilibrium is globally exponentially stable for any positive gains, with Lyapunov function combining plant energy, derivative damping, and integral energy (Zonetti et al., 2021).

The same review also makes explicit the limits of the nominal design under parameter mismatch. For converters satisfying $S$6, if the reference is computed from an estimated equilibrium set, the actual closed loop converges not to $S$7 but to a scaled equilibrium $S$8, where

$S$9

Hence the basic PID-PBC remains stable but can exhibit large steady-state offsets independent of gain tuning (Zonetti et al., 2021). To address performance and robustness limitations, the paper introduces the leaky variant

$y^\top u$0

which yields a steady-state droop relation

$y^\top u$1

and the monotone-transformation variant $y^\top u$2, which preserves bounded inputs under saturation (Zonetti et al., 2021).

Boost-converter voltage regulation is a particularly informative case because it exposes the difference between classical PI and PID-PBC. For the scaled boost model

$y^\top u$3

the paper comparing classical PI and nonlinear passivity-based control proves that direct voltage PI control can generate either a unique always-unstable equilibrium when $y^\top u$4, or two equilibria when $y^\top u$5, one always unstable and the other only potentially stable at impractically large current and integrator values (Fang et al., 29 May 2025). The same paper recalls three nonlinear passivity-based voltage-feedback controllers, including a PID-PBC with observer-based current reconstruction. Its certainty-equivalent form is

$y^\top u$6

with

$y^\top u$7

and the equilibrium $y^\top u$8 is stated to be asymptotically stable (Fang et al., 29 May 2025).

Practical implementation without full state measurement is a recurring issue. For a broad class of switched power converters, an observer-based PI-PBC reconstructs the state from standard measurements $y^\top u$9 via a GPEBO+DREM observer and attains exact finite-time state reconstruction under an interval-excitation condition,

$$\dot{x}=(J(x)-R(x))\frac{\partial H}{\partial x}+g(x)u,\qquad y=g(x)^\top\frac{\partial H}{\partial x},$$0

after which the output-feedback controller becomes the original globally stable full-state PI-PBC (Bobtsov et al., 2020). This implementability theme reappears in the modified boost PID-PBC, where $$\dot{x}=(J(x)-R(x))\frac{\partial H}{\partial x}+g(x)u,\qquad y=g(x)^\top\frac{\partial H}{\partial x},$$1 replaces direct current measurement (Fang et al., 29 May 2025).

4. Distributed-parameter systems and nonlinear actuator effects

PID-PBC ideas have also been extended beyond lumped finite-dimensional models. For a viscously damped Euler–Bernoulli piezoelectric beam with boundary actuation

$$\dot{x}=(J(x)-R(x))\frac{\partial H}{\partial x}+g(x)u,\qquad y=g(x)^\top\frac{\partial H}{\partial x},$$2

the port-Hamiltonian variables are $$\dot{x}=(J(x)-R(x))\frac{\partial H}{\partial x}+g(x)u,\qquad y=g(x)^\top\frac{\partial H}{\partial x},$$3 and $$\dot{x}=(J(x)-R(x))\frac{\partial H}{\partial x}+g(x)u,\qquad y=g(x)^\top\frac{\partial H}{\partial x},$$4, with total energy

$$\dot{x}=(J(x)-R(x))\frac{\partial H}{\partial x}+g(x)u,\qquad y=g(x)^\top\frac{\partial H}{\partial x},$$5

Classical passivity holds from $$\dot{x}=(J(x)-R(x))\frac{\partial H}{\partial x}+g(x)u,\qquad y=g(x)^\top\frac{\partial H}{\partial x},$$6 to $$\dot{x}=(J(x)-R(x))\frac{\partial H}{\partial x}+g(x)u,\qquad y=g(x)^\top\frac{\partial H}{\partial x},$$7, but the key contribution is a new differential passivity property based on the velocity storage

$$\dot{x}=(J(x)-R(x))\frac{\partial H}{\partial x}+g(x)u,\qquad y=g(x)^\top\frac{\partial H}{\partial x},$$8

under which the beam is passive from $$\dot{x}=(J(x)-R(x))\frac{\partial H}{\partial x}+g(x)u,\qquad y=g(x)^\top\frac{\partial H}{\partial x},$$9 to $J=-J^\top$0 (Kosaraju et al., 2019).

That new passivity map yields two constructive PI-like controllers for regulating a nonzero desired tip strain. The first performs output shaping,

$J=-J^\top$1

while the second performs input shaping,

$J=-J^\top$2

For the first controller the shaped storage $J=-J^\top$3 satisfies

$J=-J^\top$4

so $J=-J^\top$5 injects damping and $J=-J^\top$6 shapes the stored energy through the boundary output error (Kosaraju et al., 2019). The paper emphasizes that this construction avoids Casimir synthesis and that input shaping becomes possible because the new passive port variable $J=-J^\top$7 is itself integrable (Kosaraju et al., 2019).

Nonlinear actuator effects have been incorporated into the same passivity-based architecture. For fully actuated mechanical systems with symmetric or asymmetric dead-zones, a smooth inverse approximation of the dead-zone is embedded into a PI-PBC baseline: $J=-J^\top$8 with

$J=-J^\top$9

The resulting shaped Hamiltonian,

$R=R^\top\ge 0$0

is used to prove global asymptotic stability of the desired equilibrium (Chan-Zheng et al., 2022). Near the equilibrium, the paper derives the condition

$R=R^\top\ge 0$1

under which the local spectrum is real and positive, yielding a non-oscillatory and overshoot-free response; it also warns that overestimating $R=R^\top\ge 0$2 or choosing $R=R^\top\ge 0$3 too large can violate this condition (Chan-Zheng et al., 2022).

5. Sampled-data, uncertainty, and digital realizations

A major practical issue for PID-PBC is that passivity is not preserved automatically under discretization. The discrete-time converter paper states this explicitly and redefines both controller and plant output so that the sampled closed loop remains passive (Moreschini et al., 26 Aug 2025). For bilinear port-Hamiltonian power converters with assignable equilibrium $R=R^\top\ge 0$4, midpoint discretization gives

$R=R^\top\ge 0$5

and the discrete-time output is not taken at $R=R^\top\ge 0$6 but at the midpoint,

$R=R^\top\ge 0$7

With incremental storage $R=R^\top\ge 0$8, this yields

$R=R^\top\ge 0$9

which is the discrete-time shifted-passivity relation (Moreschini et al., 26 Aug 2025).

The corresponding midpoint-discretized PID is

$$\dot H=-\frac{\partial H}{\partial x}^\top R(x)\frac{\partial H}{\partial x}+y^\top u \le y^\top u,$$0

with controller storage

$$\dot H=-\frac{\partial H}{\partial x}^\top R(x)\frac{\partial H}{\partial x}+y^\top u \le y^\top u,$$1

The paper proves

$$\dot H=-\frac{\partial H}{\partial x}^\top R(x)\frac{\partial H}{\partial x}+y^\top u \le y^\top u,$$2

so the discrete PID defines an output-strictly passive map, and the interconnection with the shifted-passive sampled plant is globally stable for all $$\dot H=-\frac{\partial H}{\partial x}^\top R(x)\frac{\partial H}{\partial x}+y^\top u \le y^\top u,$$3, $$\dot H=-\frac{\partial H}{\partial x}^\top R(x)\frac{\partial H}{\partial x}+y^\top u \le y^\top u,$$4, $$\dot H=-\frac{\partial H}{\partial x}^\top R(x)\frac{\partial H}{\partial x}+y^\top u \le y^\top u,$$5 (Moreschini et al., 26 Aug 2025). The structural role of the implicit midpoint rule is central: it is second-order, symplectic, and preserves geometric structure more effectively than naive schemes, which is why Euler discretization exhibits poorer performance and can lose stability (Moreschini et al., 26 Aug 2025).

Uncertainty is handled in several ways across the literature. Robust PI-PBC for partially known nonlinear plants uses only the input matrix $$\dot H=-\frac{\partial H}{\partial x}^\top R(x)\frac{\partial H}{\partial x}+y^\top u \le y^\top u,$$6 and the measured actuated coordinates, exploiting a Lyapunov structure $$\dot H=-\frac{\partial H}{\partial x}^\top R(x)\frac{\partial H}{\partial x}+y^\top u \le y^\top u,$$7 with unknown positive coefficients in the actuated component (Aranovskiy et al., 2015). The resulting implementable controller,

$$\dot H=-\frac{\partial H}{\partial x}^\top R(x)\frac{\partial H}{\partial x}+y^\top u \le y^\top u,$$8

preserves the passivity proof despite unknown scaling, and global Lyapunov stability follows for all diagonal positive definite $$\dot H=-\frac{\partial H}{\partial x}^\top R(x)\frac{\partial H}{\partial x}+y^\top u \le y^\top u,$$9, with asymptotic stability under detectability (Aranovskiy et al., 2015). In fuel-cell/boost converter regulation, the same PI-PBC logic is coupled with an immersion-and-invariance estimator for unknown $S$00 and $S$01, yielding practical stability and recovery of voltage regulation after load changes (Cisneros et al., 2023).

6. Safety, constraints, and the scope of the term

Passivity-based controllers are often interconnected with additional supervisory layers, but such layers do not preserve passivity automatically. For affine nonlinear systems with a passive nominal controller $S$02, the control-barrier-function safety filter preserves passivity if and only if, when the barrier constraint is active,

$S$03

where $S$04 is the dissipation margin of the passive closed loop (Califano, 2023). In port-Hamiltonian mechanical systems, generalized energy-based barrier functions of the form

$S$05

always satisfy this passivity-preservation condition, so the safety filter acts like a state-dependent damper rather than a fixed damping injection (Califano, 2023).

An analogous constraint-handling result appears in task-space manipulator control. Standard task-space passivity-based control uses the storage

$S$06

and nominal force law with $S$07 (Kurtz et al., 2021). Near kinematic singularities, however, $S$08 becomes ill-conditioned, and standard constrained formulations lose passivity when the constraint is active. The proposed remedy is a quadratic program that optimizes both torque and reference-system input while enforcing the passivity constraint $S$09 and an ECBF condition based on the manipulability index $S$10 (Kurtz et al., 2021). The result is simultaneous singularity avoidance, passivity, and feasibility.

Taken together, these works indicate that the label PID-PBC covers a family of designs rather than a single algebraic template. Some papers use a literal PID with proportional, integral, and derivative terms around a passive output (Romero et al., 2016, Zonetti et al., 2021, Moreschini et al., 26 Aug 2025). Others use PI-PBC or PI-like constructions that preserve the same passivity logic, such as nonlinear voltage regulation for fuel-cell/boost systems (Cisneros et al., 2023) and output- or input-shaping controllers for piezoelectric beams (Kosaraju et al., 2019). This suggests two recurrent misconceptions should be avoided. First, PID-PBC is not merely classical PID applied to a nonlinear plant; in the converter literature it is explicitly constructed from port-Hamiltonian passivity and analyzed on the full nonlinear model (Zonetti et al., 2021). Second, neither digital implementation nor constraint handling preserves passivity by default; both require dedicated constructions such as midpoint discretization or passivity-preserving CBF filtering (Moreschini et al., 26 Aug 2025, Califano, 2023).