---
title: PID Passivity-Based Control (PID-PBC)
url: https://www.emergentmind.com/topics/pid-passivity-based-control-pid-pbc
type: topic
---

# PID Passivity-Based Control (PID-PBC)

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 [2508.18719]. 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 [1610.06999][2101.05047]. 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 [1907.02314][2005.01671][2303.10981].

## 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 [2303.10981]. 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 [2303.10981].

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
\[
\tilde y=g^\top(\bar x)Q\tilde x,\qquad \tilde x=x-\bar x,\quad \tilde u=u-\bar u,
\]
with incremental storage \(\mathcal H(\tilde x)\) satisfying
\[
\dot{\mathcal H}(\tilde x)= -\tilde x^\top Q\mathcal RQ\tilde x+\tilde y^\top \tilde u,
\]
so that the PID can be interconnected with \(\tilde y\) rather than with a naive tracking error [2101.05047]. 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 [2101.05047].

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
\[
\dot x_c=-g^\top(x_\star)Qx,\qquad
u=-K_Pg^\top(x_\star)Qx+K_Ix_c-K_Dg^\top(x_\star)Q\dot x,
\]
with \(K_P,K_I,K_D\) matrix gains [2101.05047]. In continuous time, the converter review reports that under perfect knowledge of the system parameters the equilibrium is globally exponentially stable for any \(K_P\ge 0\), \(K_D\ge 0\), and \(K_I>0\) [2101.05047]. Closely related robust PI-PBC constructions for partially known nonlinear systems proceed from incremental passivity around \(x^\star\neq 0\), using only actuated-coordinate measurements and the known input matrix structure, again with the storage function shifted to the desired operating point [1503.02935].

## 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 \(G=\begin{bmatrix}0_{s\times m}&I_m\end{bmatrix}\), inertia depending only on unactuated coordinates, constant actuated inertia subblock, separable potential energy, and an integrability condition on the coupling block \(m_{au}(q_u)\) [1610.06999]. Under these assumptions, two passive outputs are constructed:
\[
y_u := -m_{aa}^{-1}m_{au}(q_u)\dot q_u,\qquad
y_a := m_{aa}^{-1}m_{au}(q_u)\dot q_u + \dot q_a,
\]
with corresponding storage functions \(H_u\) and \(H_a\) satisfying \(\dot H_u=u^\top y_u\) and \(\dot H_a=u^\top y_a\) [1610.06999]. The PID is not wrapped around either output individually but around the weighted combination
\[
y_d := k_a y_a + k_u y_u.
\]

The resulting controller is a standard linear PID in \(y_d\),
\[
k_e u = -\left(K_P y_d + K_I z_1 + K_D \dot y_d\right),\qquad \dot z_1 = y_d,
\]
with \(K_P,K_I>0\) and \(K_D\ge 0\) [1610.06999]. The crucial step is that the integrability condition allows \(z_1\) to be expressed as a function of configuration variables, which converts the closed-loop storage into a shaped mechanical energy
\[
H_d(q,\dot q)=\frac12 \dot q^\top M_d(q_u)\dot q + V_d(q).
\]
With suitable gains and integral-state initialization, this \(H_d\) is a Lyapunov function for the desired equilibrium; if \(V_d\) 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 [1610.06999].

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 \(y=G^\top\dot q\), one PI-PBC law is
\[
u=-K_P \dot q_a - K_I\bigl(q_a-q_{a\star}-K_I^{-1}\nabla_{q_a}V(q_{a\star})\bigr),
\]
which produces a closed-loop Hamiltonian
\[
H_d(q,p)=H(q,p)+\frac12\left\|q_a-q_{a\star}-K_I^{-1}\nabla_{q_a}V(q_{a\star})\right\|_{K_I}^2
\]
and damping matrix \(D(q,p)+GK_PG^\top\) [2105.04324]. The same paper derives the no-oscillation tuning condition
\[
\lambda(GK_PG^\top + D_\star)^2 \ge 4\bar\lambda(GK_IG^\top+\nabla^2V_\star)\bar\lambda(M_\star),
\]
and uses energy-based damping identification to estimate the diagonal viscous damping matrix before selecting \(K_P\) and \(K_I\) [2105.04324]. For underactuated or flexible-joint cases, a modified PI-PBC introduces an additional damping term in unactuated velocities,
\[
u=-K_IG_1^\top(q_a-q_{a\star})-K_{Pa}G_1^\top\dot q_a-K_{Pu}\dot q_u,
\]
which preserves port-Hamiltonian structure and is analyzed through exponential-convergence estimates [2105.04324].

## 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,
\[
\dot x=(\mathcal J_0+\sum_{i=1}^m\mathcal J_i u_i-\mathcal R)\nabla \mathcal H(x)+E,\qquad
\mathcal H(x)=\frac12 x^\top Qx,
\]
PID-PBC is built around the shifted passive output \(\tilde y=g^\top(\bar x)Q\tilde x\) [2101.05047]. 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 [2101.05047].

The same review also makes explicit the limits of the nominal design under parameter mismatch. For converters satisfying \(\operatorname{rank} g(x)=n-1\), if the reference is computed from an estimated equilibrium set, the actual closed loop converges not to \(x_\star\) but to a scaled equilibrium \((\gamma x_\star,K_I^{-1}u_\star)\), where
\[
\gamma=\frac{P_{\mathrm{net}}(x_\star)}{P_{\mathrm{loss}}(x_\star)}.
\]
Hence the basic PID-PBC remains stable but can exhibit large steady-state offsets independent of gain tuning [2101.05047]. To address performance and robustness limitations, the paper introduces the leaky variant
\[
\dot x_c=-g^\top(x_\star)Qx-K_LK_I(x_c-x_{c\star}),
\]
which yields a steady-state droop relation
\[
\bar u-u_\star=D(\bar y-y_\star),\qquad D:=K_P+K_L^{-1},
\]
and the monotone-transformation variant \(u=w(v)\), which preserves bounded inputs under saturation [2101.05047].

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
\[
\dot x=\begin{bmatrix}-d_1x_1+1\\-d_2x_2\end{bmatrix}+\begin{bmatrix}-x_2\\x_1\end{bmatrix}u,\qquad y=x_2,
\]
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 \(d_1=0\), or two equilibria when \(d_1>0\), one always unstable and the other only potentially stable at impractically large current and integrator values [2505.23112]. 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
\[
\dot x_c = y_{PI},\qquad u = -K_P y_{PI} - K_I x_c,
\]
with
\[
y_{PI}=x_1^\star x_2-y_\star \hat x_1,\qquad x_1^\star=d_2 y_\star^2,
\]
and the equilibrium \((d_2 y_\star^2,y_\star,0)\) is stated to be asymptotically stable [2505.23112].

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 \(y_m=Cx\) via a GPEBO+DREM observer and attains exact finite-time state reconstruction under an interval-excitation condition,
\[
\int_0^{t_c}\Delta^2(\tau)\,d\tau\ge -\frac{1}{\gamma}\ln(1-\mu),
\]
after which the output-feedback controller becomes the original globally stable full-state PI-PBC [2005.01671]. This implementability theme reappears in the modified boost PID-PBC, where \(\hat x_1\) replaces direct current measurement [2505.23112].

## 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
\[
\rho \ddot w(z,t)=C w_{zz}(z,t)-b\dot w(z,t),\qquad C w_z(1,t)=-\gamma U(t),
\]
the port-Hamiltonian variables are \(e_q=w_z\) and \(e_p=\rho \dot w\), with total energy
\[
\mathcal H=\frac12\int_0^1 \alpha^\top P\alpha\,dz,\qquad P=\mathrm{diag}\left(C,\frac1\rho\right).
\]
Classical passivity holds from \(e_b=-\gamma U(t)\) to \(f_b=\dot w(1,t)\), but the key contribution is a new differential passivity property based on the velocity storage
\[
S=\frac12\int_0^1\left(\frac1C \dot e_q^2+\rho \dot e_p^2\right)dz,
\]
under which the beam is passive from \(\dot U\) to \(-\gamma \dot e_p(1)\) [1907.02314].

That new passivity map yields two constructive PI-like controllers for regulating a nonzero desired tip strain. The first performs output shaping,
\[
\dot\phi = e_p(1,t)-e_p^*(1),\qquad
U(t)=\frac{1}{\gamma}\Big(K_i\phi + K_p\big(e_p(1,t)-e_p^*(1)\big)+U^*\Big),
\]
while the second performs input shaping,
\[
\dot\psi = U-U^*,\qquad
U(t)= -\frac{1}{K_p}\Big(K_i\psi - \gamma\big(e_p(1)-e_p^*\big)+U^*\Big).
\]
For the first controller the shaped storage \(S_d=S+\frac12 K_i(e_p(1)-e_p^*)^2\) satisfies
\[
\frac{d}{dt}S_d = -\int_0^1 b\, e_p^2\,dz - K_p\,\dot e_p^2(1) \le 0,
\]
so \(K_p\) injects damping and \(K_i\) shapes the stored energy through the boundary output error [1907.02314]. The paper emphasizes that this construction avoids Casimir synthesis and that input shaping becomes possible because the new passive port variable \(\dot U\) is itself integrable [1907.02314].

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:
\[
u_{\tt{pidz}}=u_{\tt{dz}}+u_{\tt{pi}},
\]
with
\[
u_{\tt{dz}}=-G^{-1}\left(K_Z\sum_{i=1}^{n}e_i\tanh(\mu_i \tilde q_i)+\beta\right).
\]
The resulting shaped Hamiltonian,
\[
H_d(x)=\frac{1}{2}p^\top M^{-1}(q)p+\frac{1}{2}\tilde q^\top K_I \tilde q +K_Z\sum_{i=1}^{m}\frac{\ln(\cosh(\mu_i\tilde q_i))}{\mu_i},
\]
is used to prove global asymptotic stability of the desired equilibrium [2211.01822]. Near the equilibrium, the paper derives the condition
\[
4\mathcal P M_\star \le \mathcal R^2
\]
under which the local spectrum is real and positive, yielding a non-oscillatory and overshoot-free response; it also warns that overestimating \(k_{zi}\) or choosing \(\mu_i\) too large can violate this condition [2211.01822].

## 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 [2508.18719]. For bilinear port-Hamiltonian power converters with assignable equilibrium \(x^\star\), midpoint discretization gives
\[
x_{k+1}=x_k+\delta f(z_k)+\delta g(z_k)u_k,\qquad z_k=\frac12(x_{k+1}+x_k),
\]
and the discrete-time output is not taken at \(x_k\) but at the midpoint,
\[
y_k=(g^\star)^\top Q z_k,\qquad y^\star=(g^\star)^\top Q x^\star.
\]
With incremental storage \(H(\tilde x_k)=\frac12 \tilde x_k^\top Q\tilde x_k\), this yields
\[
\frac{1}{\delta}\Delta H(\tilde x_k)= -\tilde z_k^\top QRQ\tilde z_k + \tilde y_k^\top \tilde u_k \le \tilde y_k^\top \tilde u_k,
\]
which is the discrete-time shifted-passivity relation [2508.18719].

The corresponding midpoint-discretized PID is
\[
u_k = -K_P \tilde y_k - \frac12 K_I(\xi_{k+1}+\xi_k) - \frac{1}{\delta}K_D \mathcal C\,\Delta x_k,
\qquad \mathcal C=(g^\star)^\top Q,
\]
with controller storage
\[
H_c(\tilde \xi_k,\tilde x_k)=\frac12 \tilde \xi_k^\top K_I \tilde \xi_k + \frac12 \tilde x_k^\top \mathcal C^\top K_D \mathcal C\,\tilde x_k.
\]
The paper proves
\[
\frac{1}{\delta}\Delta H_c = -\tilde y_k^\top K_P \tilde y_k -\tilde y_k^\top \tilde u_k,
\]
so the discrete PID defines an output-strictly passive map, and the interconnection with the shifted-passive sampled plant is globally stable for all \(K_P>0\), \(K_I>0\), \(K_D\ge 0\) [2508.18719]. 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 [2508.18719].

Uncertainty is handled in several ways across the literature. Robust PI-PBC for partially known nonlinear plants uses only the input matrix \(G=[0\;G_2]^\top\) and the measured actuated coordinates, exploiting a Lyapunov structure \(H(x)=H_u(x_u)+H_a(x_a)\) with unknown positive coefficients in the actuated component [1503.02935]. The resulting implementable controller,
\[
u=-K_P\widetilde\Phi(x_a)+z,\qquad \dot z=-K_I\widetilde\Phi(x_a),
\]
preserves the passivity proof despite unknown scaling, and global Lyapunov stability follows for all diagonal positive definite \(\Gamma_P,\Gamma_I\), with asymptotic stability under detectability [1503.02935]. In fuel-cell/boost converter regulation, the same PI-PBC logic is coupled with an immersion-and-invariance estimator for unknown \(R_P\) and \(1/R_L\), yielding practical stability and recovery of voltage regulation after load changes [2302.08697].

## 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 \(u=\beta(x)+\nu\), the control-barrier-function safety filter preserves passivity if and only if, when the barrier constraint is active,
\[
-\frac{L_g S_{\mathrm{cl}}(x)\,L_g h(x)^\top}{L_g h(x)L_g h(x)^\top}\,\Psi(x;\beta)\le d_p(x),
\]
where \(d_p(x):=-L_{f_{\mathrm{cl}}}S_{\mathrm{cl}}(x)\ge 0\) is the dissipation margin of the passive closed loop [2303.10981]. In port-Hamiltonian mechanical systems, generalized energy-based barrier functions of the form
\[
h(q,p)=-K_e(q,p)+\alpha_E \bar h(q)+\bar E
\]
always satisfy this passivity-preservation condition, so the safety filter acts like a state-dependent damper rather than a fixed damping injection [2303.10981].

An analogous constraint-handling result appears in task-space manipulator control. Standard task-space passivity-based control uses the storage
\[
V(q,\dot q)=\frac{1}{2}\dot e^T \Lambda \dot e + \frac{1}{2}e^T K_P e,
\qquad \Lambda=(JM^{-1}J^T)^{-1},
\]
and nominal force law with \(\dot V=-\dot e^T K_D \dot e\le 0\) [2109.13349]. Near kinematic singularities, however, \(\Lambda\) 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 \(\dot V\le 0\) and an ECBF condition based on the manipulability index \(\mu(q)=\sqrt{\det(JJ^T)}\) [2109.13349]. 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 [1610.06999][2101.05047][2508.18719]. Others use PI-PBC or PI-like constructions that preserve the same passivity logic, such as nonlinear voltage regulation for fuel-cell/boost systems [2302.08697] and output- or input-shaping controllers for piezoelectric beams [1907.02314]. 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 [2101.05047]. Second, neither digital implementation nor constraint handling preserves passivity by default; both require dedicated constructions such as midpoint discretization or passivity-preserving CBF filtering [2508.18719][2303.10981].

Source: https://www.emergentmind.com/topics/pid-passivity-based-control-pid-pbc