PID and Proportional Controllers Overview

• PID/Proportional Controllers are feedback control algorithms that adjust system inputs based on proportional error, integral, and derivative actions to ensure system stability.
• They are widely used in industrial and scientific applications for their simplicity, ease of implementation, and robust performance in diverse dynamic conditions.
• Tuning techniques vary from classical methods like Ziegler–Nichols to modern optimization and reinforcement learning approaches, balancing performance, noise sensitivity, and robustness.

A proportional-integral-derivative (PID) controller is a canonical feedback mechanism in control engineering consisting of proportional (P), integral (I), and derivative (D) actions on the tracking error to shape closed-loop system dynamics. When only the proportional term is present, the result is a P controller, which generates a control output directly proportional to the instantaneous error. PID and related proportional controllers remain the dominant class of regulators in industrial, embedded, and scientific applications due to their structural simplicity, ease of implementation, and breadth of empirical and theoretical analysis.

1. Mathematical Structure and Variants

Let $e(t) = r(t) - y(t)$ denote the tracking error between reference $r(t)$ and output $y(t)$. The general continuous-time form is

$$u(t) = K_P\,e(t) + K_I\int_0^t e(\tau)\,d\tau + K_D\,\frac{d}{dt}e(t)$$

where $K_P$, $K_I$, and $K_D$ are the proportional, integral, and derivative gains. The P controller omits the $K_I$ and $K_D$ terms. In discrete implementation, difference equations are used, and derivative action is typically filtered to mitigate noise amplification.

Extended PID controllers generalize the basic three-term structure by incorporating higher-order derivatives of the error: $$u(t) = k_0 \int_0^t e(s)ds + k_1 e(t) + k_2 \dot e(t) + \cdots + k_n e^{(n-1)}(t)$$ This allows classical PID intuition and stability results to extend to nonlinear or higher-order plant dynamics (Zhao et al., 2019).

2. Fundamental Properties and Stability Margins

For low-order linear time-invariant (LTI) systems, explicit gain-margin (GM) and phase-margin (PM) formulas can be derived for P, PD, and PID controllers. In first-order minimum-phase plants, all three achieve infinite GM and $r(t)$0 PM; in nonminimum-phase systems, the achievable margins are identical across P, PD, and PID and are given by

$r(t)$1

$r(t)$2

where $r(t)$3 is the plant pole and $r(t)$4 the plant zero (Mao et al., 2023). For second-order plants, PID controllers offer no stability margin advantage over PD. In every maximization of gain or phase margins, the optimal integral gain is driven to zero, leading to the conclusion that integral action does not increase robust stability margins for first- and second-order systems. For process plants with unstable poles and no nonminimum-phase zeros, PID achieves the same margins as any LTI controller.

3. Tuning Methods: Analytical, Optimization, and Data-driven

Classical PID tuning employs rules such as Ziegler–Nichols or Cohen–Coon, based on open- or closed-loop step tests and ultimate gain/period identification. Modern approaches leverage black-box optimization and machine learning, especially for complex or nonlinear systems:

• Evolutionary and Swarm Algorithms: Particle Swarm Optimization (PSO) is used to tune analog PIDs in real time across multi-channel systems, optimizing a weighted sum of steady-state error and overshoot (Bhandari et al., 2016). CMA-ES, with advanced restart and adaptation, efficiently tunes large numbers of coupled PIDs in combustion engine models, significantly outperforming PSO and SHADE in both number of objective evaluations and reliability (Henclova, 2016).
• Reinforcement Learning (RL) and Policy Search: Model-based RL (e.g., PILCO) is used to learn a Gaussian-process dynamics model and optimal policy; the resultant policy is projected into PID form by minimizing the Kullback–Leibler divergence between the RL-optimal and PID controllers, yielding interpretable and robust gains (Jesawada et al., 2022). Entropy-maximizing deep RL schemes (e.g., EMTD3) facilitate automated PID gain discovery via reward signals based on closed-loop performance indices (Chowdhury et al., 2022).
• Fractional and Conformal Mapping Approaches: Fractional-order PID (FOPID/PI$r(t)$5D$r(t)$6) controllers allow additional flexibility in placing open-loop zeros via the order parameters $r(t)$7. Conformal mapping approximates a FOPID by an integer-order PID whose zeros follow a trajectory (“M-curve”) parametrized by the fractional exponents; this reduces control effort compared to standard LQR pole placement (Saha et al., 2012).

4. Robustness, Disturbance Rejection, and Noise Sensitivity

Robust stability and disturbance attenuation remain central design objectives in PID frameworks. The PID–GM approach introduces an explicit nonlinear gain mapping from auxiliary parameters $r(t)$8 to $r(t)$9 such that the closed-loop can be interpreted as a nominal PD control plus an uncertainty/disturbance estimator (UDE). The “virtual singular perturbation parameter” $y(t)$0 governs trade-offs between tracking performance and gain magnitude; as $y(t)$1, all three PID gains increase monotonically, enhancing robustness at the cost of potentially excessive actuator demand (Zhu et al., 21 Apr 2025).

Analysis in the context of nonlinear stochastic systems reveals that, provided the Lipschitz and partial derivatives of the system nonlinearity and noise intensity are bounded, PID gains can always be scaled sufficiently large (within algebraic design inequalities) to guarantee arbitrarily small steady-state tracking error under bounded reference and disturbance rates (Zhao et al., 2023). Measurement noise is primarily amplified by derivative action; accordingly, practical PID implementations apply filters on the D term, or partially restrict RL/PSO-based exploration of the derivative gain to prevent noise-induced actuator chatter (Reutov et al., 13 Mar 2025Kakani et al., 2024).

5. Application Domains and Performance Comparisons

PIDs are foundational in process control (chemical plants, coupled-tank systems), robotics (manipulator joints), power electronics, optics, and gene regulation:

• In coupled tank level/flow control, PIDs provide $y(t)$2 faster response and lower integral error compared to PI at the expense of overshoot and higher tuning effort; PI is favored for robustness and noise insensitivity (Kakani et al., 2024).
• In 3-DOF robot manipulators, PID control yields lower overshoot and zero steady-state error across all links compared to PD under properly tuned gains; the integral action is particularly important for disturbance rejection and model mismatch (Kabir et al., 2019).
• In laser communication stabilization, RL-assisted PID tuning using DDPG agents offers improved static and adaptive gains (mean error reduction by up to $y(t)$3) over classical Ziegler–Nichols–tuned PIDs, with RL particularly effective in navigating trade-offs between responsiveness and noise amplification (Reutov et al., 13 Mar 2025).
• CRN-based PID architectures implement the full PID structure in synthetic biology, enabling robust regulation of protein expression with modular biochemical reactions, with D block construction achieving improved transient damping and oscillation suppression over pure PI (Whitby et al., 2019).

6. Theoretical Interpretations and Modern Extensions

PID control has been reinterpreted as a combination of homogeneous state feedback (placing poles in the integrator-chain homogeneous system) and a disturbance observer that compensates lumped exogenous and structural uncertainty. For an $y(t)$4-th order chain, a PID structure for $y(t)$5 results from pole placement via

$y(t)$6

where $y(t)$7 parameterize the desired homogeneous closed-loop and $y(t)$8 sets the observer bandwidth (Shi, 2024). This two-parameter tuning, with $y(t)$9 setting closed-loop speed and $$u(t) = K_P\,e(t) + K_I\int_0^t e(\tau)\,d\tau + K_D\,\frac{d}{dt}e(t)$$0 ensuring fast disturbance estimation, reduces PID tuning complexity and links classical controller coefficients to measurable system and observer speeds.

Extensions include observer-based PID, where unavailable derivatives of the output error are reconstructed with high-gain observers, and higher-order PID which accommodates plant models of arbitrary relative degree by feedback on higher-order error derivatives (Zhao et al., 2019). Analysis further unifies classical and modern robust control notions by quantifying stability regions in gain space for SOPTD (second-order-plus-time-delay) processes under Pade delay approximations, and providing polynomial rules for rapid controller synthesis (Das et al., 2018).

7. Practical Implementation and Tuning Trade-offs

Practical realization of PID/Proportional controllers involves analog, digital, and mixed-signal platforms:

• In mixed-signal ASICs for miniature robotics, analog PIDs are equipped with digitally PSO-tuned bit-weighted resistive and capacitive banks, providing sub-millisecond retuning across $$u(t) = K_P\,e(t) + K_I\int_0^t e(\tau)\,d\tau + K_D\,\frac{d}{dt}e(t)$$1-channel systems with minimal power and area footprint (Bhandari et al., 2016).
• In feedforward-enhanced PID architectures (e.g., vapor-compression systems), model-based disturbance cancellation and conditional integration eliminate residual steady-state error with minimal phase lag increase, improving composite performance indices by over $$u(t) = K_P\,e(t) + K_I\int_0^t e(\tau)\,d\tau + K_D\,\frac{d}{dt}e(t)$$2 (Yuan et al., 2018).
• In RL-based tuning, exploration–exploitation scheduling, entropy maximization, and off-policy replay have demonstrated improvements in learning efficiency, convergence to near-optimal gains, and avoidance of poor local minima over traditional deep policy gradient approaches (Chowdhury et al., 2022).
• Constraints in PID tuning include measurement noise, actuator limitations, and plant nonlinearity; practical tuning must balance speed, overshoot, steady-state error, and noise robustness, often via filtered D action, regularization in data-driven optimization, or explicit gain-mapping parameter regimes (Kakani et al., 2024, Zhu et al., 21 Apr 2025).

The coexistence of century-old heuristic rules, modern stochastic optimization, RL-based frameworks, and theoretical reinterpretations illustrates the ongoing synthesis of classical intuition and contemporary algorithmic methodology in PID/Proportional controller design.