PSO-PID: Particle Swarm Optimized PID

• PSO-PID is a method that applies Particle Swarm Optimization to tune PID controller parameters, supporting designs from classical to fractional and fuzzy adaptations.
• It encodes controller gains as particle positions and iteratively minimizes objective functions like MSE, ITAE, or RMSE through swarm dynamics.
• Empirical results show significant performance gains in diverse applications such as UAV landing, STATCOM regulation, and underwater vehicle depth tracking.

Searching arXiv for recent and foundational PSO-PID papers to ground the article. arxiv_search(query="Particle Swarm Optimized PID controller tuning PSO PID", max_results=10, sort_by="relevance") arxiv_search(query="Nonlinear PID Particle Swarm Optimization latent factorization PID arXiv 2025", max_results=10, sort_by="relevance") arxiv_search(query="fractional order PID particle swarm optimization arXiv", max_results=10, sort_by="relevance") Particle Swarm Optimized PID (PSO-PID) denotes a family of controller-synthesis and parameter-tuning schemes in which Particle Swarm Optimization (PSO) is used to determine the parameters of a proportional–integral–derivative controller or a PID-derived structure. In the surveyed literature, the tuned object may be a classical three-parameter PID, a fractional-order controller $PI^\lambda D^\delta$, a fuzzy-PID with additional scaling factors, a PID-type sliding surface, or a nonlinear PID error-reconstruction law embedded in a learning algorithm rather than a physical feedback loop. Across these variants, the recurring pattern is that PSO treats controller parameters as particle positions, evaluates each candidate by a task-specific fitness function, and iteratively updates the swarm through personal-best and global-best attractors until a stopping criterion is met (Kebbati et al., 23 Sep 2025, 0811.0083, Wang et al., 18 Apr 2025).

1. Formal definition and canonical equations

In its classical control form, PSO-PID combines a standard PID control law with a population-based search over the gains. A representative continuous-time controller used in the literature is

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

where $e(t)$ is the tracking error and $(K_p,K_i,K_d)$ are the proportional, integral, and derivative gains (Kebbati et al., 23 Sep 2025, Paital et al., 2016). In discrete-time robotic implementations, the same structure appears as

$$u(t)=K_P\,e(t)+K_I\sum_{k=0}^{t} e(k)\,T_s + K_D\frac{e(t)-e(t-1)}{T_s},$$

with $T_s=0.05\,\mathrm{s}$ in the reported UAV landing experiments (Wu et al., 2023).

The PSO component is likewise standardized at the level of swarm dynamics. For particle $i$ with position $x_i$ and velocity $v_i$, the update equations recurrently used across the literature are

$$v_i(t+1)=w\,v_i(t)+c_1\,r_1\bigl(p_i-x_i(t)\bigr)+c_2\,r_2\bigl(g-x_i(t)\bigr),$$

$$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$0

where $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$1 is the personal best, $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$2 is the global best, $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$3, and $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$4 are inertia and acceleration coefficients (0811.0083, Ding et al., 13 Feb 2026). Several studies use fixed or conventional coefficients such as $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$5, $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$6, or linearly decreasing inertia from $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$7 to $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$8; others introduce adaptive schedules for faster convergence or to avoid premature stagnation (Wang et al., 18 Apr 2025, Kebbati et al., 23 Sep 2025).

The term therefore does not refer to a single controller equation. In the cited work, it denotes a coupling between a PID-type structure and a PSO-based search mechanism. This includes both plant control and learning-system optimization, such as nonlinear PID-enhanced latent factor models for missing-data recovery (Li et al., 2022, Wang et al., 18 Apr 2025).

2. Optimization workflow and problem formulation

The common workflow begins with a particle encoding of the tunable parameters, followed by repeated evaluation of a fitness function derived from simulation, closed-loop response, or validation error. In the simplest case, each particle carries only three gains, $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$9, as in longitudinal vehicle control, STATCOM voltage regulation, and multivariable PID tuning (Kebbati et al., 23 Sep 2025, Paital et al., 2016, Taeib et al., 2013). More elaborate encodings enlarge the search space. Fractional-order formulations optimize five parameters, $e(t)$0 or $e(t)$1, thereby including the integral and derivative orders (0810.3776, 0811.0083). A constrained fuzzy-PID scheme for underwater vehicles tunes six variables, namely $e(t)$2 (Ding et al., 13 Feb 2026). In adaptive latent-factor models, the PSO particle can represent nonlinear PID gains and auxiliary constants rather than direct controller gains (Li et al., 2022, Wang et al., 18 Apr 2025).

Fitness definitions are equally application-dependent. Closed-loop speed control for autonomous vehicles uses mean-square speed-tracking error, $e(t)$3, and the cited work states that in principle the objective could be augmented by overshoot, settling time, or control effort, although $e(t)$4 alone was used there (Kebbati et al., 23 Sep 2025). STATCOM tuning minimizes an ITAE-type criterion involving both voltage deviation and speed deviation, $e(t)$5 (Paital et al., 2016). Underwater fuzzy-PID tuning introduces a scalarized constrained objective with normalized $e(t)$6, control energy, relative overshoot, settling time, saturation occupancy, and an energy penalty that enforces $e(t)$7 (Ding et al., 13 Feb 2026). In recommendation or tensor-recovery settings, the objective is validation RMSE or reconstruction error rather than a time-domain control index (Li et al., 2022, Wang et al., 18 Apr 2025).

A compact cross-section of particle encodings and objective functions is shown below.

Variant	Particle contents	Objective used in the cited work
Classical PID	$e(t)$8	MSE, ITAE-type criteria, or integral error indices
FOPID / $e(t)$9	$(K_p,K_i,K_d)$0 or $(K_p,K_i,K_d)$1	Residual-based pole-placement cost or ITAE
Fuzzy-PID	$(K_p,K_i,K_d)$2	Weighted fitness with ITAE, $(K_p,K_i,K_d)$3, $(K_p,K_i,K_d)$4, $(K_p,K_i,K_d)$5, $(K_p,K_i,K_d)$6, and penalties
Nonlinear PID-enhanced learning	nonlinear PID gains, sometimes plus $(K_p,K_i,K_d)$7	Validation RMSE or reconstruction error
PID-type sliding-mode	$(K_p,K_i,K_d)$8	ISE or weighted tracking-performance index

The implementation details that recur most often are bound clamping on $(K_p,K_i,K_d)$9, clipping or limiting of velocities, random initialization within admissible ranges, and termination either at a maximum iteration count or when the best fitness change falls below a tolerance (0811.0083, 0811.0079, Wang et al., 18 Apr 2025).

3. Principal variants of PSO-PID

Classical three-gain PSO-PID

The most direct form tunes only $$u(t)=K_P\,e(t)+K_I\sum_{k=0}^{t} e(k)\,T_s + K_D\frac{e(t)-e(t-1)}{T_s},$$0 for a fixed PID law. This appears in autonomous longitudinal speed control, where a classical PID regulates nonlinear longitudinal vehicle dynamics and PSO is run offline to minimize speed-tracking MSE (Kebbati et al., 23 Sep 2025). It also appears in power-system applications such as STATCOM voltage regulation in a single-machine infinite-bus system, where the optimized PID modulates the PWM index through the voltage error $$u(t)=K_P\,e(t)+K_I\sum_{k=0}^{t} e(k)\,T_s + K_D\frac{e(t)-e(t-1)}{T_s},$$1 (Paital et al., 2016). In multivariable nonlinear systems represented by a Takagi–Sugeno fuzzy model, PSO is applied to the full set of decentralized MIMO PID loops, with the decision dimension scaling as $$u(t)=K_P\,e(t)+K_I\sum_{k=0}^{t} e(k)\,T_s + K_D\frac{e(t)-e(t-1)}{T_s},$$2 for $$u(t)=K_P\,e(t)+K_I\sum_{k=0}^{t} e(k)\,T_s + K_D\frac{e(t)-e(t-1)}{T_s},$$3 outputs (Taeib et al., 2013).

Fractional-order PSO-PID

Fractional-order variants extend the search from three to five parameters by introducing integral and derivative orders. The controller takes the form

$$u(t)=K_P\,e(t)+K_I\sum_{k=0}^{t} e(k)\,T_s + K_D\frac{e(t)-e(t-1)}{T_s},$$4

or equivalently $$u(t)=K_P\,e(t)+K_I\sum_{k=0}^{t} e(k)\,T_s + K_D\frac{e(t)-e(t-1)}{T_s},$$5 (0811.0083, Das et al., 2016). In the cited work, fractional orders are used either to satisfy pole-placement specifications derived from peak overshoot and rise time or to minimize ITAE directly (0810.3776, 0811.0079). Practical realization is not treated as automatic; reported implementations use rational approximations such as Oustaloup recursive approximation or Tustin operator-based continued fraction expansion (Das et al., 2016, 0811.0083).

Fuzzy and constrained PSO-PID

A further variant optimizes a fuzzy-PID architecture in which a fuzzy inference block adjusts PID gains online. In the underwater-vehicle depth-control formulation, PSO tunes six offline parameters: the baseline gains $$u(t)=K_P\,e(t)+K_I\sum_{k=0}^{t} e(k)\,T_s + K_D\frac{e(t)-e(t-1)}{T_s},$$6, the input quantization factors $$u(t)=K_P\,e(t)+K_I\sum_{k=0}^{t} e(k)\,T_s + K_D\frac{e(t)-e(t-1)}{T_s},$$7, and the output proportional scaling factor $$u(t)=K_P\,e(t)+K_I\sum_{k=0}^{t} e(k)\,T_s + K_D\frac{e(t)-e(t-1)}{T_s},$$8 applied to the defuzzified corrections $$u(t)=K_P\,e(t)+K_I\sum_{k=0}^{t} e(k)\,T_s + K_D\frac{e(t)-e(t-1)}{T_s},$$9 (Ding et al., 13 Feb 2026). The defining feature of this formulation is not only fuzzy adaptation but explicit incorporation of actuator-oriented constraints through control energy and saturation occupancy.

Nonlinear PID-enhanced learning schemes

In data-driven models, PID action can be applied to learning error rather than a plant output. The NPALF and NPIL models rebuild the instantaneous learning error by proportional, integral, and derivative components with nonlinear gain functions of the current error, and then use the adjusted error in stochastic gradient updates (Li et al., 2022, Wang et al., 18 Apr 2025). In NPIL, the underlying model is a rank-$T_s=0.05\,\mathrm{s}$0 CP latent factorization of a partially observed third-order tensor with Tikhonov regularization and linear biases; the nonlinear PID term replaces the raw stochastic error in updates of $T_s=0.05\,\mathrm{s}$1 (Wang et al., 18 Apr 2025). PSO then adapts the nonlinear gain parameters on a validation subset.

PSO-tuned PID-type sliding surfaces

A hybrid control interpretation appears in PID-type sliding-mode control. There, the sliding surface is

$T_s=0.05\,\mathrm{s}$2

and a modified PSO tunes $T_s=0.05\,\mathrm{s}$3 for an improved power-rate exponential reaching law (Singh et al., 2022). In this construction, PID parameters do not directly define a standalone PID controller; instead they shape the sliding manifold and equivalent dynamics.

4. Objective functions, constraints, and parameter adaptation

The fitness function is the principal mechanism by which PSO-PID formulations differ. Integral error criteria remain common. MIMO nonlinear control experiments report IAE, ISE, and ITSE as discrete sums of per-output tracking errors, and optimize different controller instances against each index (Taeib et al., 2013). Fractional-order design studies alternatively use a pole-residual cost $T_s=0.05\,\mathrm{s}$4 or $T_s=0.05\,\mathrm{s}$5, where $T_s=0.05\,\mathrm{s}$6 are the real and imaginary residuals of the characteristic equation at desired closed-loop poles and $T_s=0.05\,\mathrm{s}$7 or $T_s=0.05\,\mathrm{s}$8 is the phase term (0810.3776, 0811.0079). These objectives differ structurally from time-domain error criteria, even though both are optimized by the same swarm mechanism.

A second axis of variation concerns constraints and regularization. Several studies impose only bound constraints on gains and use clamping or clipping to keep particles inside the search region (0811.0083, Wu et al., 2023, Kebbati et al., 23 Sep 2025). Others formulate explicit performance constraints. The underwater-vehicle work introduces an energy-constraint penalty,

$T_s=0.05\,\mathrm{s}$9

along with a soft overshoot tolerance $i$0 (Ding et al., 13 Feb 2026). This formulation is noteworthy because it is designed to suppress “pseudo-improvements achieved solely by increasing control inputs,” a concern that is often only implicit in unconstrained PSO tuning.

A third axis is adaptation of PSO hyperparameters themselves. The literature reports fixed settings such as $i$1 and $i$2, linearly decreased inertia from $i$3 to $i$4, and improved schedules with dynamically adapted $i$5 (Wang et al., 18 Apr 2025, 0811.0083, Kebbati et al., 23 Sep 2025). Modified PSO for sliding-mode tuning uses time-varying acceleration coefficients and decaying inertia to privilege exploration early and local refinement later (Singh et al., 2022). This suggests that “PSO-PID” often includes meta-level design choices beyond the PID structure itself.

5. Representative empirical results

The reported performance of PSO-PID is strongly domain-specific, but several studies document quantitative gains against manually tuned or baseline alternatives.

Domain	Reported result	Source
NILM missing-data recovery	On D1@15%, NPIL achieved RMSE $i$6, MAE $i$7, was $i$8 lower than best competitor M4, converged in $i$9 s versus $x_i$0–$x_i$1 s for others, and reached stable RMSE/MAE within fewer than 10 iterations	(Wang et al., 18 Apr 2025)
Underwater vehicle depth tracking	$x_i$2 decreased from $x_i$3 to $x_i$4, settling time from $x_i$5 s to $x_i$6 s, relative overshoot from $x_i$7 to $x_i$8, control energy from $x_i$9 to $v_i$0, and saturation occupancy from $v_i$1 to $v_i$2	(Ding et al., 13 Feb 2026)
Autonomous longitudinal speed control	PSO-tuned PID achieved MSE $v_i$3 and $v_i$4 in double-lane-change and winding-road cases, tracked speed within $v_i$5 m/s, and showed no observable overshoot or oscillation	(Kebbati et al., 23 Sep 2025)
STATCOM voltage regulation	Under nominal loading, PSO-PID yielded voltage overshoot $v_i$6, voltage settling time $v_i$7 s, speed overshoot $v_i$8, speed settling time $v_i$9 s, and ITAE $$v_i(t+1)=w\,v_i(t)+c_1\,r_1\bigl(p_i-x_i(t)\bigr)+c_2\,r_2\bigl(g-x_i(t)\bigr),$$0, compared with $$v_i(t+1)=w\,v_i(t)+c_1\,r_1\bigl(p_i-x_i(t)\bigr)+c_2\,r_2\bigl(g-x_i(t)\bigr),$$1, $$v_i(t+1)=w\,v_i(t)+c_1\,r_1\bigl(p_i-x_i(t)\bigr)+c_2\,r_2\bigl(g-x_i(t)\bigr),$$2 s, $$v_i(t+1)=w\,v_i(t)+c_1\,r_1\bigl(p_i-x_i(t)\bigr)+c_2\,r_2\bigl(g-x_i(t)\bigr),$$3, $$v_i(t+1)=w\,v_i(t)+c_1\,r_1\bigl(p_i-x_i(t)\bigr)+c_2\,r_2\bigl(g-x_i(t)\bigr),$$4 s, and $$v_i(t+1)=w\,v_i(t)+c_1\,r_1\bigl(p_i-x_i(t)\bigr)+c_2\,r_2\bigl(g-x_i(t)\bigr),$$5 for conventional PID	(Paital et al., 2016)
UAV landing on moving boat	Landing time decreased from $$v_i(t+1)=w\,v_i(t)+c_1\,r_1\bigl(p_i-x_i(t)\bigr)+c_2\,r_2\bigl(g-x_i(t)\bigr),$$6 s to $$v_i(t+1)=w\,v_i(t)+c_1\,r_1\bigl(p_i-x_i(t)\bigr)+c_2\,r_2\bigl(g-x_i(t)\bigr),$$7 s, touchdown vertical velocity changed from $$v_i(t+1)=w\,v_i(t)+c_1\,r_1\bigl(p_i-x_i(t)\bigr)+c_2\,r_2\bigl(g-x_i(t)\bigr),$$8 m/s to $$v_i(t+1)=w\,v_i(t)+c_1\,r_1\bigl(p_i-x_i(t)\bigr)+c_2\,r_2\bigl(g-x_i(t)\bigr),$$9 m/s, planar error was reduced to $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$00 m, and maximum trackable boat speed reached $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$01 m/s, about $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$02 of UAV top speed	(Wu et al., 2023)

Further evidence comes from nonlinear MIMO control and anaesthetic drug infusion. For a two-output Takagi–Sugeno fuzzy model, PSO-PID reduced overshoot, rise time, and settling time relative to Ziegler–Nichols tuning; for output $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$03, overshoot decreased from $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$04 to $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$05 under the ISE-optimized design, with rise time and settling time reduced from $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$06 s and $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$07 s to $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$08 s and $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$09 s, respectively (Taeib et al., 2013). In the anaesthetic-control study, the best fractional-order controller (FOPID1) reached $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$10 versus $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$11 for PID and converged in approximately 120 iterations to $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$12, whereas PID plateaued at $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$13 (Das et al., 2016).

These results are not directly interchangeable because each study defines a different plant, search space, and fitness function. A plausible implication is that PSO-PID performance must be interpreted relative to the encoded objective rather than as a universally transferable gain in “PID quality.”

6. Implementation issues, misconceptions, and research directions

A common misconception is that PSO-PID is a single standardized tuning recipe. The literature instead shows a heterogeneous design space. The particle may encode three gains, five fractional-order parameters, six fuzzy-PID quantities, nonlinear PID error-law coefficients, or a PID-type sliding surface plus reaching-law parameters (0810.3776, Ding et al., 13 Feb 2026, Li et al., 2022, Singh et al., 2022). Likewise, the objective may be MSE, ITAE, IAE, ISE, ITSE, pole-residual error, reconstruction RMSE, or a constrained composite criterion. The shared element is the PSO search mechanism, not a unique controller architecture.

Another recurring issue is the distinction between offline optimization and online deployment. Several studies explicitly perform PSO offline in MATLAB or simulation, after which the optimized PID runs online with negligible computational cost (0810.3776, Kebbati et al., 23 Sep 2025). Other work inserts an additional approximation layer between offline optimization and runtime use. In the UAV landing system, a neural network is trained on a $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$14 grid of PSO-optimized gains over altitude and boat speed, and the deployed controller retrieves real-time gains as $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$15 (Wu et al., 2023). This indicates that PSO-PID may function as a calibration stage rather than a runtime search procedure.

For fractional-order controllers, digital realization is itself a nontrivial design problem. The cited implementations use Oustaloup recursive approximation over a specified frequency range or Tustin operator-based continued fraction expansion of order $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$16 to approximate $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$17 and $$u(t)=K_p\,e(t)+K_i\int_0^t e(\tau)\,d\tau + K_d\frac{d\,e(t)}{dt},$$18 (Das et al., 2016, 0811.0083). In such cases, reported controller superiority depends not only on PSO tuning but also on the approximation method used to realize fractional operators.

Recent extensions point toward broader hybridization. Reported directions include improved PSO with adaptive inertia and acceleration coefficients (Kebbati et al., 23 Sep 2025), joint tuning of nominal PID and fuzzy scaling parameters under energy constraints (Ding et al., 13 Feb 2026), PSO combined with neural interpolation for gain scheduling (Wu et al., 2023), and possible online PSO, hybrid PSO-GA, or multi-objective Pareto analyses for changing operating conditions (Kebbati et al., 23 Sep 2025, Ding et al., 13 Feb 2026). In data-centric applications, PSO-PID has also moved beyond physical feedback control into optimization of nonlinear PID-enhanced learning updates for latent-factor and tensor models (Li et al., 2022, Wang et al., 18 Apr 2025).

Taken together, the literature presents PSO-PID not as a single algorithmic artifact but as a broad metaheuristic design pattern: PSO supplies a derivative-free search over PID-type parameterizations, while the control or learning performance is determined by the chosen architecture, objective, constraints, and realization method.