PID Control: Theory, Tuning & Applications

• PID control is a feedback strategy that computes control actions based on current, accumulated, and rate-of-change errors for precise regulation.
• It is valued for its straightforward implementation and interpretability, making it a staple in industrial, robotic, and process control applications.
• Modern advancements include digital formulations, adaptive tuning, and nonlinear generalizations that enhance stability, robustness, and performance.

Proportional–integral–derivative (PID) control is a feedback-control strategy in which the control signal is formed from present, accumulated, and rate-of-change information of the regulation error. In continuous time it is commonly written as $$u(t)=K_p e(t)+K_i\int e(t)\,dt+K_d \frac{de(t)}{dt}$$, and in transfer-function form as $C(s)=K_p+\frac{K_i}{s}+K_d s$. Its enduring importance derives from straightforward implementation, interpretability, and an error-driven, often model-free structure that has been used across industrial process control, robotics, optical stabilization, temperature regulation, biochemical reaction networks, and molecular communication systems (Bhandari et al., 2016, Razzaq, 6 Dec 2025).

1. Canonical formulation and control structure

The canonical PID law combines three distinct error channels. The proportional term reacts to the current error and mainly shapes responsiveness; the integral term accumulates past errors and is responsible for offset removal and improved steady-state tracking; and the derivative term anticipates error change, adds damping, and helps reduce oscillations (zhou, 2016). In Laplace form, this structure appears as

$C(s)=K_p+\frac{K_i}{s}+K_d s,$

while in time domain it appears as

$$u(t)=K_p e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d \frac{de(t)}{dt}.$$

This representation remains the common reference point for both classical and modern variants (Bhandari et al., 2016).

In digital control, two discrete-time forms recur. The position form is

$$u(k)=K_p e(k)+K_i\sum_{j=0}^{k} e(j)+K_d\bigl[e(k)-e(k-1)\bigr],$$

whereas the incremental form is

$$u(k)=u(k-1)+K_p\bigl[e(k)-e(k-1)\bigr]+K_i e(k)+K_d\bigl[e(k)-2e(k-1)+e(k-2)\bigr].$$

The incremental form is emphasized in mobile-robot control because it depends only on recent errors and outputs control increments rather than absolute commands, which is described as less disruptive and computationally lighter for digital robot control (zhou, 2016, Yu et al., 2021).

Several works use setpoint-weighted or two-degree-of-freedom formulations. In the “Closed PID-Loop Model” following control formulation, the PID law is expressed with $e_1=br-y$, $e_2=\int_0^\tau (r-y)\,dt$, and $e_3=c\dot r-\dot y$, yielding a compact decomposition $u=T_r r-T_y y$, with

$C(s)=K_p+\frac{K_i}{s}+K_d s$0

This explicitly separates reference weighting from feedback action (Somefun et al., 2020).

In multivariable and joint-space settings, the same structure is retained componentwise. For robot joints, one paper writes

$C(s)=K_p+\frac{K_i}{s}+K_d s$1

with elementwise vectorized gains $C(s)=K_p+\frac{K_i}{s}+K_d s$2 (Wu et al., 9 Nov 2025). This suggests that PID is best understood not as a single equation but as a structured family of error-feedback realizations whose essential semantics remain unchanged across continuous-time, discrete-time, SISO, and MIMO settings.

2. Physical realization, tuning practice, and implementation constraints

A notable feature of PID is the direct interpretability of its coefficients in hardware. In an analog op-amp implementation, the proportional, integral, and derivative actions can be realized by separate inverting blocks with

$C(s)=K_p+\frac{K_i}{s}+K_d s$3

This makes the abstract gains directly programmable as resistor ratios and RC time constants (Bhandari et al., 2016). In the same integrated implementation, the gains are stored in 4-bit registers and realized through switchable resistor and capacitor banks, illustrating how classical PID can be preserved while parameter adaptation is delegated to digital logic (Bhandari et al., 2016).

Practical PID design is shaped as much by sensing and implementation as by the nominal control law. The “split-PID” architecture addresses a common spatially distributed-control problem: when actuator and regulated variable are separated, placing the sensor at the sample gives correct steady-state regulation but low bandwidth, whereas placing it near the actuator improves bandwidth but can introduce offset at the true regulated location. The proposed remedy is to assign the integral term to the far sensor and the proportional/derivative terms to the near sensor, i.e.

$C(s)=K_p+\frac{K_i}{s}+K_d s$4

with $C(s)=K_p+\frac{K_i}{s}+K_d s$5 measured near the heater and $C(s)=K_p+\frac{K_i}{s}+K_d s$6 near the sample (Znaimer et al., 2014). In the reported thermal experiment, heater-probe-only PID led to a sample offset of about $C(s)=K_p+\frac{K_i}{s}+K_d s$7 and drift, sample-probe-only PID removed the offset but responded more slowly, and split-PID preserved sample accuracy while remaining almost as fast as the heater-probe loop (Znaimer et al., 2014).

Derivative implementation is another recurrent practical issue. For time-delay systems, derivative action is not treated as an ideal $C(s)=K_p+\frac{K_i}{s}+K_d s$8 term but as a filtered realization

$C(s)=K_p+\frac{K_i}{s}+K_d s$9

so the derivative channel is $C(s)=K_p+\frac{K_i}{s}+K_d s,$0. One numerical co-design study shows that the filter cannot be appended after tuning an ideal PID; the filter changes the closed-loop characteristic roots and therefore changes stability, decay rate, delay margin, and robustness (Torres-García et al., 17 Apr 2026). A plausible implication is that derivative filtering is not merely an implementation detail but part of the controller definition for delayed systems.

The literature summarized here also repeatedly exposes omissions that matter in deployment. Several adaptive and learning-based PID studies explicitly note that anti-windup, derivative filtering, actuator saturation laws, or detailed noise mitigation are absent from their implementations (Reutov et al., 13 Mar 2025, Razzaq, 6 Dec 2025, Wu et al., 9 Nov 2025). This underscores a persistent distinction between PID as a nominal feedback law and PID as a deployable engineered loop.

3. Structural generalizations of the PID law

A large contemporary literature does not discard PID but alters its internal structure while retaining the error-feedback principle. Representative variants are summarized below.

Variant	Central mechanism	Representative paper
Split-PID	$C(s)=K_p+\frac{K_i}{s}+K_d s,$1 on far sensor, $C(s)=K_p+\frac{K_i}{s}+K_d s,$2 on near sensor	(Znaimer et al., 2014)
Extended PID	Higher-order error derivatives up to $C(s)=K_p+\frac{K_i}{s}+K_d s,$3	(Zhao et al., 2019)
Nonlinear integral PID	Error-dependent integral gain $C(s)=K_p+\frac{K_i}{s}+K_d s,$4	(Ruderman, 2024)
Rolling PID	Repeated finite-horizon gain re-optimization	(zhou, 2016)
PID-GM	Gain mapping from $C(s)=K_p+\frac{K_i}{s}+K_d s,$5 to $C(s)=K_p+\frac{K_i}{s}+K_d s,$6	(Zhu et al., 21 Apr 2025)
Closed PID-loop model following	Gains tied to an implicit second-order CPLM	(Somefun et al., 2020)

The “extended PID” controller generalizes classical PID for high-order affine nonlinear uncertain systems by adding higher-order derivatives of the regulation error,

$C(s)=K_p+\frac{K_i}{s}+K_d s,$7

For systems of relative degree $C(s)=K_p+\frac{K_i}{s}+K_d s,$8, this reduces the closed-loop error dynamics to a perturbed linear chain and yields global or semi-global exponential regulation under explicit gain-manifold conditions (Zhao et al., 2019). In this framework, ordinary PID is recovered when $C(s)=K_p+\frac{K_i}{s}+K_d s,$9.

A different generalization modifies only the integral channel. For the perturbed double integrator $$u(t)=K_p e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d \frac{de(t)}{dt}.$$0, the nonlinear integral extension

$$u(t)=K_p e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d \frac{de(t)}{dt}.$$1

preserves the PD channels and makes the integral gain state dependent (Ruderman, 2024). Because $$u(t)=K_p e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d \frac{de(t)}{dt}.$$2, the effective integral gain is largest near $$u(t)=K_p e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d \frac{de(t)}{dt}.$$3, so the controller behaves like a nominal PID for large errors and a stronger integral controller near the setpoint. The paper proves global asymptotic stability for the unperturbed or constant-disturbance case via the circle criterion and argues, using disturbance-to-output sensitivity, that this nonlinear gain improves residual-error settling under Lipschitz perturbations (Ruderman, 2024).

The “rolling PID” approach retains the discrete PID law but periodically re-estimates the gains over a rolling horizon using recent observed data. Its objective functions are horizon-based versions of ISE, ITSE, IAE, and ITAE, and the controller gains are repeatedly updated by solving finite-horizon optimization problems (zhou, 2016). In the reported nonlinear-system simulations, the rolling method with model updating converged to the reference with no overshoot and steadier trajectories than the version without model updating (zhou, 2016).

PID-GM, or PID control with gain mapping, introduces the nonlinear map

$$u(t)=K_p e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d \frac{de(t)}{dt}.$$4

which recasts PID exactly as a nominal PD law plus an uncertainty/disturbance compensator: $$u(t)=K_p e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d \frac{de(t)}{dt}.$$5 The parameter $$u(t)=K_p e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d \frac{de(t)}{dt}.$$6 becomes a virtual singular perturbation parameter: smaller $$u(t)=K_p e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d \frac{de(t)}{dt}.$$7 increases all three PID gains and reduces the ultimate tracking-error bound, but also increases peaking and transient control effort (Zhu et al., 21 Apr 2025).

The CPLMFC framework interprets PID as implicitly shaping a second-order closed-loop model

$$u(t)=K_p e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d \frac{de(t)}{dt}.$$8

which is then matched to a desired second-order form with natural frequency $$u(t)=K_p e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d \frac{de(t)}{dt}.$$9 and damping $$u(k)=K_p e(k)+K_i\sum_{j=0}^{k} e(j)+K_d\bigl[e(k)-e(k-1)\bigr],$$0 (Somefun et al., 2020). This suggests a shift in viewpoint: tuning can proceed from desired closed-loop settling behavior rather than from an explicit plant model.

4. Adaptive, learning-based, and optimization-based PID tuning

A major modern trend is to preserve PID as the inner control law while moving adaptation to an outer supervisory layer. The following directions recur.

Direction	Adaptation mechanism	Representative paper
Mixed-signal self-tuning	PSO updates bit-programmable analog gains	(Bhandari et al., 2016)
RL-assisted threshold or gain tuning	RL/PID hierarchy with PID remaining actuator-facing	(Reutov et al., 13 Mar 2025)
RL gain scheduling for mobile robots	SAC outputs incremental PID gains online	(Yu et al., 2021)
Policy-to-PID distillation	PILCO/MBPO policy converted to PID by KLD fitting	(Jesawada et al., 2022)
Meta-initialized adaptive PID	Meta-learning initializes gains, PPO refines online	(Wu et al., 9 Nov 2025)
Untrained neural adaptation	Error-driven MLP reshapes $$u(k)=K_p e(k)+K_i\sum_{j=0}^{k} e(j)+K_d\bigl[e(k)-e(k-1)\bigr],$$1 online	(Razzaq, 6 Dec 2025)
Event-based self-tuning	Game-theoretic players adapt PI/PID gains on events	(Yuwono et al., 16 Jun 2025)

At the hardware end, a single-chip self-tunable N-input N-output controller combines analog PID loops with a shared PSO tuner. Local analog PID attempts to reject disturbances first; only if the error is not corrected within a stipulated time $$u(k)=K_p e(k)+K_i\sum_{j=0}^{k} e(j)+K_d\bigl[e(k)-e(k-1)\bigr],$$2 is a shared digital PSO block invoked to compute revised $$u(k)=K_p e(k)+K_i\sum_{j=0}^{k} e(j)+K_d\bigl[e(k)-e(k-1)\bigr],$$3 values (Bhandari et al., 2016). The PSO fitness is

$$u(k)=K_p e(k)+K_i\sum_{j=0}^{k} e(j)+K_d\bigl[e(k)-e(k-1)\bigr],$$4

with $$u(k)=K_p e(k)+K_i\sum_{j=0}^{k} e(j)+K_d\bigl[e(k)-e(k-1)\bigr],$$5 and $$u(k)=K_p e(k)+K_i\sum_{j=0}^{k} e(j)+K_d\bigl[e(k)-e(k-1)\bigr],$$6 chosen equal, and the implementation uses 50 particles, 50 iterations, and 4-bit coefficient precision (Bhandari et al., 2016).

Several reinforcement-learning studies keep PID as the low-level controller and let learning update only the gains. In optical beam stabilization for satellite-to-ground communication, a DDPG agent updates $$u(k)=K_p e(k)+K_i\sum_{j=0}^{k} e(j)+K_d\bigl[e(k)-e(k-1)\bigr],$$7 while $$u(k)=K_p e(k)+K_i\sum_{j=0}^{k} e(j)+K_d\bigl[e(k)-e(k-1)\bigr],$$8 remain fixed at baseline values. Relative to manually tuned baseline gains $$u(k)=K_p e(k)+K_i\sum_{j=0}^{k} e(j)+K_d\bigl[e(k)-e(k-1)\bigr],$$9, an RL-derived online-retrained policy reduces the average radial image-plane deviation from $$u(k)=u(k-1)+K_p\bigl[e(k)-e(k-1)\bigr]+K_i e(k)+K_d\bigl[e(k)-2e(k-1)+e(k-2)\bigr].$$0 pixels to $$u(k)=u(k-1)+K_p\bigl[e(k)-e(k-1)\bigr]+K_i e(k)+K_d\bigl[e(k)-2e(k-1)+e(k-2)\bigr].$$1 pixels, described as a $$u(k)=u(k-1)+K_p\bigl[e(k)-e(k-1)\bigr]+K_i e(k)+K_d\bigl[e(k)-2e(k-1)+e(k-2)\bigr].$$2 reduction (Reutov et al., 13 Mar 2025). In line-following mobile robots, a hierarchical SAC-PID controller outputs six gains for two incremental PID loops; on the hardest simulated path, SAC-PID achieves $$u(k)=u(k-1)+K_p\bigl[e(k)-e(k-1)\bigr]+K_i e(k)+K_d\bigl[e(k)-2e(k-1)+e(k-2)\bigr].$$3 success and $$u(k)=u(k-1)+K_p\bigl[e(k)-e(k-1)\bigr]+K_i e(k)+K_d\bigl[e(k)-2e(k-1)+e(k-2)\bigr].$$4, while fuzzy PID achieves $$u(k)=u(k-1)+K_p\bigl[e(k)-e(k-1)\bigr]+K_i e(k)+K_d\bigl[e(k)-2e(k-1)+e(k-2)\bigr].$$5 success and $$u(k)=u(k-1)+K_p\bigl[e(k)-e(k-1)\bigr]+K_i e(k)+K_d\bigl[e(k)-2e(k-1)+e(k-2)\bigr].$$6 (Yu et al., 2021).

Model-based reinforcement learning has also been used to design interpretable PID controllers by first learning a policy with PILCO and then fitting a probabilistic PID controller by minimizing forward Kullback–Leibler divergence between trajectory distributions (Jesawada et al., 2022). In that framework, error, derivative-of-error, and integral-of-error coordinates are extracted from rollouts, and the final control input is represented as a linear combination of proportional, integral, and derivative channels on those coordinates (Jesawada et al., 2022).

Hierarchical meta-learning and RL move further toward cross-platform auto-tuning. For a 9-DOF Franka Panda and a 12-DOF Laikago quadruped, a meta-network predicts PID gains from physical robot features, and PPO then applies multiplicative online updates

$$u(k)=u(k-1)+K_p\bigl[e(k)-e(k-1)\bigr]+K_i e(k)+K_d\bigl[e(k)-2e(k-1)+e(k-2)\bigr].$$7

On Franka, the combined method improves MAE from $$u(k)=u(k-1)+K_p\bigl[e(k)-e(k-1)\bigr]+K_i e(k)+K_d\bigl[e(k)-2e(k-1)+e(k-2)\bigr].$$8 to $$u(k)=u(k-1)+K_p\bigl[e(k)-e(k-1)\bigr]+K_i e(k)+K_d\bigl[e(k)-2e(k-1)+e(k-2)\bigr].$$9, a $e_1=br-y$0 improvement, with Joint 2 improving from $e_1=br-y$1 to $e_1=br-y$2, an $e_1=br-y$3 improvement (Wu et al., 9 Nov 2025). On Laikago, however, MAE remains $e_1=br-y$4 before and after RL, leading the authors to introduce the “optimization ceiling effect”: RL helps when baseline PID has localized high-error joints, but not when the baseline is already uniformly strong (Wu et al., 9 Nov 2025). This suggests that online gain adaptation is conditional rather than universally beneficial.

Two recent alternatives avoid conventional training overhead. EC-PIDUNN replaces fixed gains with online, error-driven gain adaptation produced by an untrained MLP. The architecture lifts the scalar error into

$e_1=br-y$5

concatenates it with $e_1=br-y$6, and updates gains using

$e_1=br-y$7

inside the modified law

$e_1=br-y$8

On Ackermann-steered vehicle control, classical PID gives steering overshoot $e_1=br-y$9 and steering settling time $e_2=\int_0^\tau (r-y)\,dt$0, whereas EC-PIDUNN gives $e_2=\int_0^\tau (r-y)\,dt$1 overshoot and steering settling time $e_2=\int_0^\tau (r-y)\,dt$2; speed overshoot drops from $e_2=\int_0^\tau (r-y)\,dt$3 to $e_2=\int_0^\tau (r-y)\,dt$4 (Razzaq, 6 Dec 2025). By contrast, an event-based game-theoretic self-tuner treats $e_2=\int_0^\tau (r-y)\,dt$5 as players in a state-based potential game and updates gains only when thresholded performance degrades. In a printing-press temperature loop, gradient-based learning reduced average settling time from $e_2=\int_0^\tau (r-y)\,dt$6 s to $e_2=\int_0^\tau (r-y)\,dt$7 s relative to best-response learning under static load, and from $e_2=\int_0^\tau (r-y)\,dt$8 s to $e_2=\int_0^\tau (r-y)\,dt$9 s under variable load and disturbances (Yuwono et al., 16 Jun 2025).

5. Stability, robustness, and what PID can theoretically guarantee

Theoretical treatments of PID in the cited literature range from classical margin analysis to nonlinear stochastic tracking. For low-order unstable SISO LTI plants, explicit formulas for maximal gain and phase margins achievable by P, PI, PD, and PID have been derived. For first-order plants

$e_3=c\dot r-\dot y$0

the maximal margins achievable by PI are identical to those of P, and for first-order nonminimum-phase plants $e_3=c\dot r-\dot y$1 the relation to arbitrary LTI control is

$e_3=c\dot r-\dot y$2

For second-order nonminimum-phase plants, the maximizing PID controller is effectively PD, and the paper concludes that integral action does not improve robust stabilization margins beyond what PD already achieves (Mao et al., 2023). This directly addresses a common misconception: integral action is crucial for offset removal and tracking, but not automatically for classical gain- and phase-margin maximization.

At the nonlinear end, extended PID control provides global or semi-global exponential regulation for a class of high-order affine nonlinear uncertain systems. Under uniform relative degree $e_3=c\dot r-\dot y$3, a known-sign input gain bounded away from zero, and coordinate-growth conditions, the gains can be chosen from an open unbounded parameter manifold so that $e_3=c\dot r-\dot y$4 exponentially fast (Zhao et al., 2019). An observer-based implementation replaces unavailable derivatives with a high-gain differential observer and preserves the same qualitative conclusion (Zhao et al., 2019).

For stochastic nonlinear systems with time-varying references, a classical PID controller

$e_3=c\dot r-\dot y$5

has been shown to yield global mean-square stabilization and bounded tracking under bounded partial derivatives of the drift and diffusion, bounded $e_3=c\dot r-\dot y$6, bounded $e_3=c\dot r-\dot y$7, and bounded diffusion along the reference trajectory (Zhao et al., 2023). The steady-state tracking error is shown to depend on three factors: the change rates of the reference and disturbance, the intensity of the random noise, and the selected PID gains. Under the scaled-gain family

$e_3=c\dot r-\dot y$8

the paper proves practical tracking with arbitrarily small mean-square error as $e_3=c\dot r-\dot y$9, while also noting that control effort grows with decreasing $u=T_r r-T_y y$0 (Zhao et al., 2023).

Several papers also clarify what is not yet guaranteed. EC-PIDUNN argues for stability architecturally and empirically, but explicitly provides no Lyapunov analysis, closed-loop boundedness proof, or formal convergence guarantee (Razzaq, 6 Dec 2025). The game-theoretic event-based tuner proves convergence to Nash equilibrium and closed-loop stability only under the assumptions that the action set contains stabilizing gains and the barrier slope dominates the utility gradient (Yuwono et al., 16 Jun 2025). The RL-assisted optical-stabilization study keeps PID in the loop precisely because it provides a safe baseline, yet it offers no formal closed-loop stability proof for the RL-tuned system (Reutov et al., 13 Mar 2025). Collectively, these results indicate that PID theory is strongest where structure is explicit and weakest where adaptation is delegated to learned or heuristic outer loops.

6. Applications, contemporary significance, and open questions

PID remains prominent because it continues to adapt to new domains without losing its basic structure. In robotics, it underlies self-tunable analog chips for miniature robots (Bhandari et al., 2016), RL-supervised line following and quadruped/manipulator control (Yu et al., 2021, Wu et al., 9 Nov 2025), and untrained-neural gain adaptation for Ackermann vehicles and pan-tilt systems (Razzaq, 6 Dec 2025). In optical systems, it serves as the embedded servo in MEMS-mirror beam stabilization for satellite-to-ground communication and QKD terminals (Reutov et al., 13 Mar 2025). In process industries, it remains the basis of temperature-control architectures, including event-based self-tuning on printing-press thermal loops (Yuwono et al., 16 Jun 2025). In molecular communication, ART-Rx treats adaptive threshold detection as a PID problem: $u=T_r r-T_y y$1 and reports average BEP reductions such as $u=T_r r-T_y y$2 under varying interferer counts relative to a static statistically optimal threshold (Ni et al., 2024). In synthetic biology, dual-rail biochemical reaction networks now support proportional, integral, and derivative blocks, enabling a full CRN PID controller that reduces oscillations in a microRNA-regulated gene-expression model relative to PI (Whitby et al., 2019).

Several recurring issues define the current research frontier. One is filtering and noise. Derivative action is repeatedly identified as noise-sensitive, and one time-delay study argues that the derivative low-pass filter fundamentally changes spectral properties and must be co-designed with the gains (Torres-García et al., 17 Apr 2026). A second is hyperparameter and initialization sensitivity. RL from scratch fails to converge within $u=T_r r-T_y y$3M steps in the meta-learning-plus-RL study (Wu et al., 9 Nov 2025), and untrained-neural gain shaping is reported to be sensitive to $u=T_r r-T_y y$4 and neural hyperparameters (Razzaq, 6 Dec 2025). A third is reproducibility. Some works omit explicit network sizes, baseline gains, or stabilizing parameters, which limits direct replication (Razzaq, 6 Dec 2025). A fourth is objective dependence: the best PID structure for robust stabilization margins need not be the best for disturbance rejection, settling, or offset elimination. The conclusion that integral action adds no robust-stability margin in certain low-order LTI problems (Mao et al., 2023) coexists with multiple studies in which integral action or nonlinear integral modulation is central to steady-state accuracy and late-phase convergence (Ruderman, 2024, Znaimer et al., 2014).

Taken together, these developments present PID control less as a fixed historical algorithm than as a durable architectural template. Its core remains the same—feedback organized around present, accumulated, and predictive error information—but its realizations now include analog RC hardware, sensor-split structures, gain mappings, singular-perturbation reinterpretations, rolling optimization, policy-distilled gains, event-driven games, untrained neural adaptation, and biochemical or molecular embodiments. This suggests that PID control endures not because its classical form is universally sufficient, but because its structure remains flexible enough to absorb new sensing, computation, and robustness mechanisms while preserving a compact and interpretable feedback law.