Designing a cascaded exponential PID controller via starfish optimizer for DC motor and liquid level systems

Izci, Davut; Jabari, Mostafa; Çelik, Emre; Ekinci, Serdar; Bajaj, Mohit; Rubanenko, Olena; Prokop, Lukas

doi:10.1038/s41598-025-28145-9

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Published: 08 December 2025

Designing a cascaded exponential PID controller via starfish optimizer for DC motor and liquid level systems

Scientific Reports volume 15, Article number: 44504 (2025) Cite this article

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Abstract

In this study, a novel cascaded exponential proportional–integral–derivative (exp-PID) controller tuned by the starfish optimization algorithm (SFOA) is proposed for enhancing the transient and steady-state performance of nonlinear dynamic systems. The design objective is to achieve improved adaptability, robustness, and precision under varying operating conditions and external disturbances. The exponential PID structure introduces nonlinear modulation in the proportional and derivative components, enabling smoother control action and superior damping characteristics compared to conventional PID and fractional-order PID designs. The proposed SFOA-based exp-PID controller is validated on two benchmark systems: a DC motor speed control system and a three-tank liquid-level process. Across multiple independent trials, the controller achieved outstanding results, with the DC motor system attaining a rise time of 0.0039 s, settling time of 0.0083 s, and zero overshoot, while the three-tank system reached a rise time of 1.72 s, settling time of 2.47 s, overshoot of 1.5%, and steady-state error of 9.22 × 10⁻⁵%. Comparative analyses with recently developed algorithms (including the flood algorithm, greater cane rat algorithm, mantis search algorithm, and dandelion optimizer) as well as previously reported methods demonstrate the superior convergence behavior, stability, and accuracy of the proposed controller. Statistical evaluations further confirm the method’s robustness and consistent performance across repeated runs.

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Introduction

In modern industrial systems, achieving precise and reliable control over dynamic processes such as direct current (DC) motor speed regulation and liquid-level control remains a significant challenge¹. These systems often exhibit nonlinear behaviors, time-varying dynamics, and sensitivity to external disturbances. While the proportional–integral–derivative (PID) controller has been extensively adopted for its simplicity and satisfactory performance², its fixed gains limit adaptability under changing operating conditions³. Consequently, conventional PID-based systems may experience instability, sluggish transient response, and poor disturbance rejection in dynamic environments⁴. These issues have motivated the development of intelligent and optimization-based tuning strategies capable of improving robustness and adaptability in real-time applications.

Among these, metaheuristic optimization techniques have proven highly effective for controller parameter tuning in nonlinear and uncertain systems. They provide efficient global search capabilities and avoid the limitations of classical gradient-based optimization. Recent studies have introduced several nature-inspired algorithms that emulate biological or physical behaviors to balance global exploration and local exploitation^5,6,7,8,9. Although these algorithms have achieved promising results, their direct implementation in real-time control may be constrained by computational demands. Therefore, integrating metaheuristic optimization with adaptive control structures can enhance both efficiency and robustness in practical systems.

PID controllers remain the cornerstone of industrial control due to their simplicity and reliability^10,11. However, their performance often deteriorates in nonlinear or time-varying systems. To address these limitations, several advanced variants have been proposed. The exponential PID (exp-PID) controller introduces nonlinear exponential terms in the proportional and derivative components, improving transient behavior and disturbance rejection¹². The fractional-order PID (FOPID) controller extends the classical design through fractional calculus, providing more flexible tuning and better handling of system dynamics^13,14,15. The optimal fractional-order time-delayed PID (optimal FOTID) controller further enhances robustness and precision in systems affected by delays and uncertainties¹⁶. These advanced PID designs demonstrate that effective performance relies strongly on intelligent tuning strategies^17,18.

The optimization of control parameters has become essential in modern control engineering, especially in renewable energy and industrial automation systems such as photovoltaic (PV) plants and microgrids^19,20,21. Many studies have demonstrated that metaheuristic-based tuning of advanced PID controllers can significantly improve transient response, steady-state accuracy, and disturbance rejection^22,23,24. Similarly, intelligent and adaptive control schemes such as fuzzy logic control, active disturbance rejection control (ADRC), and model predictive control (MPC) have shown strong potential for regulating nonlinear systems like multi-tank processes^25,26,27,28. Furthermore, hybrid approaches combining global metaheuristic search with local learning or refinement methods have been proposed to accelerate convergence and improve stability in complex systems^29,30,31.

Building upon these research directions, this study proposes a novel control framework that integrates the starfish optimization algorithm (SFOA)⁵ with a cascaded exponential PID (exp-PID) controller. The SFOA is used to optimally tune the exp-PID parameters, including proportional, integral, derivative gains, and exponential modulation factors. This hybrid structure enhances adaptability and robustness, allowing the controller to maintain precise performance across varying operating conditions.

To comprehensively evaluate the proposed SFOA-based exp-PID controller, simulations were performed on two representative nonlinear processes: (i) a DC motor speed control system, and (ii) a three-tank liquid-level control system. Each system was tested under identical conditions using multiple objective functions and compared against several recent algorithms. The results demonstrated that the proposed controller achieves superior convergence behavior, minimal overshoot, and faster settling times, consistently outperforming other benchmarked methods. Moreover, statistical consistency across repeated runs confirmed its robustness and stability in handling stochastic search processes.

Although numerous studies have addressed PID optimization through metaheuristic algorithms, most focus primarily on classical or fractional PID structures without incorporating adaptive nonlinear transformations such as exponential or logarithmic functions. The limited exploration of nonlinear cascaded PID forms has restricted performance enhancement in systems exhibiting strong nonlinearity, actuator saturation, or load disturbances. Furthermore, the effectiveness of newly introduced optimizers such as SFOA in tuning multi-parameter nonlinear controllers has not yet been systematically investigated. These gaps highlight the necessity for an optimization framework that combines the nonlinear adaptability of exponential controllers with the global search efficiency of a recent, high-performing metaheuristic algorithm.

The main contributions and novelty of this study can be summarized as follows:

1.
A nonlinear control structure combining exponential error transformation and conventional PID regulation is formulated to achieve fast convergence, reduced overshoot, and high steady-state precision.
2.
The SFOA’s multidirectional arm-extension and adaptive regeneration principles are utilized to optimally tune the controller parameters, ensuring efficient global search and avoidance of local entrapment.
3.
Extensive experiments were performed using several recent optimization algorithms on both electromechanical and process-control systems. The proposed method consistently yielded the best transient and steady-state results, validating its robustness and adaptability.
4.
The controller was evaluated using multiple performance indices to confirm its generalization capability and optimization consistency.
5.
The algorithm’s convergence stability and performance variance were analyzed through multi-run statistical indicators, confirming its reliability in stochastic search environments.

The remainder of the paper is organized as follows. Section 2 summarizes the SFOA, outlining its operators and search mechanism. Section 3 introduces the two case studies (a DC-motor speed regulation system and a three-tank liquid-level system) and presents their mathematical models. Section 4 details the cascaded exp-PID controller and its signal flow. Section 5 formulates the SFOA-based tuning procedure, including the performance index, parameter bounds for both plants, and the optimization workflow. Section 6 reports DC-motor results, first comparing the proposed controller with recent optimizers (FLA, GCRA, MSA, DO) and then with reported works (GWO, ASO, MMPA, LFDNM), together with parameter analyses, time-response plots, and multi-run statistics. Section 7 presents the corresponding three-tank results, with the same two-stage comparison (against recent optimizers and against reported methods (ALC-PSODE, MGO, AOA-HHO, CMA-ES)) and includes quantitative metrics and statistical evaluations. Section 8 concludes the study and outlines directions for future work.

Overview of starfish optimization algorithm

The SFOA is a bio-inspired metaheuristic algorithm that mimics the behaviors of a starfish, including exploration, preying, and regeneration, to solve global optimization problems (as shown in Fig. 1⁵. It is designed to balance exploration and exploitation through a hybrid search pattern. Starfish exhibit three remarkable behaviors that contribute to their survival and efficiency in nature. They explore their environment by extending their five arms, allowing them to navigate and search for resources. When hunting, they rely on their tube feet to capture prey with precision and adaptability. Additionally, their ability to regenerate lost limbs ensures resilience and continuity in their existence. Inspired by these natural strategies, SFOA incorporates these behaviors into an optimization algorithm, promoting thorough exploration and robust convergence.

The SFOA follows a structured mathematical framework composed of three fundamental stages. It begins with an initialization phase, where potential solutions are generated randomly to establish diversity within the search space. In the exploration phase, the algorithm mimics the starfish’s environmental probing, dynamically searching for improved solutions. Finally, the exploitation phase focuses on refining the best-obtained solutions, ensuring convergence toward optimal results with enhanced precision⁵. In the initialization phase, the population of starfish $\:X$ is randomly initialized within the search space:

$$\:X={\left[\begin{array}{ccc}{X}_{11}&\:{X}_{12}\dots\:&\:{X}_{1D}\\\:{X}_{\begin{array}{c}21\\\:.\\\:.\\\:.\\\:.\end{array}}&\:{X}_{\begin{array}{c}22\\\:.\\\:.\\\:.\\\:.\end{array}}\dots\:&\:{X}_{\begin{array}{c}2D\\\:.\\\:.\\\:.\\\:.\end{array}}\\\:{X}_{N1}&\:{X}_{N2}\dots\:&\:{X}_{ND}\end{array}\right]}_{N\times\:D}$$

(1)

where $\:{X}_{i,j}$ represents the position of the $\:{i}^{th}$, starfish in the $\:{j}^{th}$ dimension. $\:N$ is the is the population size, and $\:D$ is the number of design variables. In addition, each position is initialized as:

$$\:{X}_{ij}={l}_{j}+\left({u}_{j}-{l}_{j}\right),\:\:\:\:\:\:\:r\in\:\left(\text{1,0}\right)$$

(2)

where $\:{l}_{j},\:$and$\:\:{u}_{j}$ are the lower and upper bounds of the search space, and $\:r$ is a random number.

The exploration phase ensures a diverse search, mimicking how starfish explore their environment. If $\:D>5$, a five-dimensional search pattern is used:

$$\:{Y}_{i,p}^{\left(t\right)}=\left\{\begin{array}{c}{X}_{i,p}^{\left(t\right)}+{a}_{1}\left({X}_{best,p}^{\left(t\right)}-{X}_{i,p}^{\left(t\right)}\right)\times\:{cos}\left(\theta\:\right)\:\:\:\:\:\:\:r\le\:0.5\:\:\:\\\:{X}_{i,p}^{\left(t\right)}-{a}_{1}\left({X}_{best,p}^{\left(t\right)}-{X}_{i,p}^{\left(t\right)}\right)\times\:{sin}\left(\theta\:\right)\:\:\:\:\:\:\:\:\:\:\:r>0.5\end{array}\right.$$

(3)

where $\:{X}_{best,p}^{\left(t\right)}$, $\:t$ and $\:{t}_{max}$ are the best solution in the population, the current iteration, the total number of iterations respectively. In addition, $\:{a}_{1}=(2r-1)\pi\:$, $\:\theta\:=\frac{\pi\:}{2}\left(\frac{t}{{t}_{max}}\right)$. If $\:D\le\:5$, a one-dimensional search pattern is used:

$$\:{Y}_{i,q}^{\left(t\right)}={{E}_{t}X}_{i,p}^{\left(t\right)}+{A}_{1}\left({X}_{{k}_{1},p}^{\left(t\right)}-{X}_{i,p}^{\left(t\right)}\right)+{A}_{2}\left({X}_{{k}_{2},p}^{\left(t\right)}-{X}_{i,p}^{\left(t\right)}\right)$$

(4)

where $\:{X}_{{k}_{1},p}^{\left(t\right)}$, $\:{X}_{{k}_{2},p}^{\left(t\right)}$, $\:{A}_{1}$ and $\:{A}_{2}$ are positions of randomly selected starfish, random numbers between − 1 and 1, respectively. In addition, $\:{E}_{T}=\left(\frac{{t}_{max}-t}{{t}_{max}}\right)\cdot\:cos\theta\:$. After updating the positions, boundary constraints are enforcedç

$$\:{X}_{i,p}^{(t+1)}=\left\{\begin{array}{cc}{Y}_{i,p}^{\left(t\right)}&\:{l}_{p}\le\:{Y}_{i,p}^{\left(t\right)}\le\:{U}_{p}\\\:{X}_{i,p}^{\left(t\right)}&\:otherwise\end{array}\right.$$

(5)

The exploitation phase refines solutions through two key strategies:

Preying Behavior: Moves towards better positions using a two-directional search strategy:

$$\:{Y}_{i}^{\left(t\right)}={X}_{i}^{\left(t\right)}+{r}_{1}{d}_{m1}+{r}_{2}{d}_{m2}$$

(6)

where $\:{d}_{m1}$, $\:{d}_{m\:}$are two randomly selected distances between the best solution and other starfish.

Regeneration Behavior: Ensures diversity by allowing the last starfish $\:(i=N)$, to regenerate:

$$\:{Y}_{i}^{\left(t\right)}=\text{exp}\left(-\frac{t\times\:N}{{t}_{max}}\right)\times\:{X}_{i}^{\left(t\right)}$$

(7)

After updating, boundary constraints are applied:

$$\:{X}_{i}^{(t+1)}=\left\{\begin{array}{cc}{Y}_{i}^{\left(t\right)}&\:\:{l}_{b}\le\:{Y}_{i}^{\left(t\right)}\le\:{u}_{b}\\\:{l}_{b}&\:{Y}_{i}^{\left(t\right)}<{l}_{b}\\\:\:\:\:{u}_{b}&\:\:\:{Y}_{i}^{\left(t\right)}>{u}_{b}\:\end{array}\right.$$

(8)

Case studies

This section examines the application of the SFOA algorithm in optimizing exponential PID controller parameters for two different control systems: a DC motor speed regulation system and a three-tank liquid level control system. These case studies aim to evaluate the adaptability and effectiveness of the proposed optimization method. For the DC motor, the primary objective is to achieve precise and efficient speed regulation, a crucial requirement in numerous industrial applications. Meanwhile, the three-tank system focuses on maintaining stable liquid levels despite complex nonlinear interactions and dynamic disturbances. By developing detailed system models and conducting comparative analyses against well-established optimization techniques, these case studies highlight the SFOA algorithm’s ability to enhance system performance across diverse operating conditions.

Mathematical modeling of DC motor system

In this section, explores the implementation of an externally excited DC motor for speed regulation by varying the armature voltage. Through an analysis of the corresponding electrical schematic, a mathematical model is developed to gain a thorough understanding of the motor’s behavior and electrical characteristics, as depicted in Fig. 2.

When the magnetic flux remains constant, the induced voltage $\:{E}_{b}$, is directly proportional to the angular velocity $\:{\omega\:}_{m}$, which represents the rate of change of rotation, $\:\frac{d\theta\:}{dt}$¹⁴.

$$\:{E}_{b}={K}_{b}\frac{d\theta\:}{dt}={K}_{b}{\omega\:}_{m}$$

(9)

The speed of an armature-controlled DC motor is regulated by the armature voltage $\:{E}_{a}$. According to the mathematical model of the DC motor, the circuit can be described as follows.

$$\:{E}_{a}={L}_{a}\frac{d{i}_{a}}{dt}+{R}_{a}{I}_{a}+{E}_{b}$$

(10)

Here, $\:{I}_{a}$, $\:{R}_{a}$, and $\:{L}_{a}$ represent the armature current, armature resistance, and armature inductance of the DC motor, respectively. The torque produced by the armature current results from the combined influence of inertia and friction torques²³.

$$\:T=J\frac{d{\omega\:}_{m}}{dt}+B{\omega\:}_{m}=K{I}_{a}$$

(11)

Here, $\:{\omega\:}_{m}$ denotes the angular speed of the motor shaft, $\:J$ represents the moment of inertia, $\:B$ is the motor friction constant, and $\:K$ is the motor torque constant. By assuming all initial conditions of the system are zero, applying the Laplace transform to Eqs. (9–11) yields the following equation:

$$\:{E}_{b}\left(s\right)={K}_{b}{\omega\:}_{m}\left(s\right)$$

(12)

$$\:\left\{\begin{array}{c}{E}_{a}\left(s\right)=\left({L}_{a}s+{R}_{a}\right){I}_{a}\left(s\right)+{E}_{b}\left(s\right)\\\:T\left(s\right)=\left(Js+B\right){\omega\:}_{m}\left(s\right)=K{I}_{a}\left(s\right)\end{array}\right.$$

(13)

where $\:{K}_{b}$ is electromotive force constant^18,23. Finally, the open loop equation of the system for $\:{T}_{L}=0$ is defined as follows:

$$\:{G}_{DC-motor}\left(s\right)=\frac{K}{\left({L}_{a}s+{R}_{a}\right)\left(Js+B\right)+{K}_{b}K}=\frac{15}{1.08{s}^{2}+6.1s+1.63}$$

(14)

Mathematical modeling of liquid level system

Three-tank liquid level control system

The goal of the three-tank liquid level control system is to regulate and maintain the liquid levels in Tanks 1, 2, and 3, as shown in Fig. 3. To simplify the model, several assumptions are considered. The tanks are open to the atmosphere, ensuring no pressure buildup. The liquid is considered incompressible, maintaining a constant total mass throughout the system. Fluid movement occurs exclusively from higher to lower levels between interconnected tanks, driven by natural gravitational flow. The system is assumed to be entirely leak-free, ensuring no unintended losses. Additionally, the flow rates are governed by the differences in liquid levels, which dictate the system’s dynamic behavior.

These dynamics are captured through the following differential Equations^1,32.

$$\:\frac{{dH}_{1}}{dt}={q}_{in}-{q}_{12}-{q}_{13}$$

(15)

$$\:\frac{{dH}_{2}}{dt}={q}_{12}-{q}_{23}$$

(16)

$$\:\frac{{dH}_{3}}{dt}={q}_{13}+{q}_{23}-{q}_{out}$$

(17)

In this system, $\:{H}_{1}$,$\:{H}_{2}$ and $\:{H}_{3}$ represent the liquid levels in Tanks 1, 2, and 3, respectively. The flow rates $\:{q}_{12}$, $\:{q}_{13}$, and $\:{q}_{23}$ are directly influenced by the differences in liquid levels between the tanks, ensuring that fluid moves from higher to lower levels. This relationship can be expressed mathematically as follows¹:

$$\:{q}_{12}={k}_{12}\left({H}_{1}-{H}_{2}\right)$$

(18)

$$\:{q}_{13}={k}_{13}\left({H}_{1}-{H}_{3}\right)$$

(19)

$$\:{q}_{23}={k}_{23}\left({H}_{2}-{H}_{3}\right)$$

(20)

The constants $\:{k}_{12}$, $\:{k}_{13}$, and $\:{k}_{23}$ are determined by the system’s geometry and the physical properties of the liquid, influencing the rate of flow between tanks. To facilitate analysis, a simplified tank model is adopted, where $\:{q}_{1}$ and $\:{q}_{2}\:$represent the inflow and outflow rates, respectively. The liquid height is denoted by$\:\:\left(h\right)$, while $\:\left(A\right)\:$refers to the cross-sectional area of the tank^1,33. To ensure consistency with previous studies, the system is modeled using a transfer function of the form:

$$\:{G}_{tank}\left(s\right)=\frac{1}{64{s}^{3}+9.6{s}^{2}+0.48s+0.008}$$

(21)

This model allows for the simulation of the system’s dynamic behavior under different flow rates and control strategies. By implementing various control techniques, the flow rates can be adjusted to maintain the desired liquid levels, ensuring the system meets the requirements of specific applications. This approach enables precise regulation, making it adaptable to a wide range of industrial and process control scenarios. You can see brief schematic linking industrial applications with three tanks in Fig. 4.

Concept of exponential PID controller

The cascaded exponential proportional–integral–derivative (exp-PID) controller is designed to enhance the adaptability and precision of control actions in nonlinear and time-varying dynamic systems. In contrast to a conventional PID controller, which applies constant gain values to the error signal and its derivative, the exp-PID incorporates nonlinear exponential transformations to adjust the control effort dynamically¹². This structure provides improved transient response, reduced steady-state error, and enhanced robustness against external disturbances and parameter variations. As illustrated in Fig. 5, the exp-PID controller is composed of two interconnected subsystems: (i) the exponential block, which performs nonlinear modulation of the error dynamics, and (ii) the PID block, which generates the final control action based on the modified signals.

In the exponential block, the instantaneous error $\:e\left(t\right)$ and its time derivative $\:\frac{de\left(t\right)}{dt}$ are first weighted by time-varying scaling coefficients $\:{\tau\:}_{1}$ and $\:{\tau\:}_{2}$, respectively. These scaled quantities are denoted as

$$\:\left\{\begin{array}{c}k\left(t\right)={\tau\:}_{1}e\left(t\right)\\\:l\left(t\right)={\tau\:}_{2}\frac{de}{dt}\end{array}\right.$$

(22)

Both signals are then passed through the nonlinear exponential activation function $\:{f}_{\text{e}\text{x}\text{p}}\left(u\right(t\left)\right)$, mathematically expressed as in (23).

$$\:{f}_{exp}\left(u\left(t\right)\right)=\frac{2}{1+{e}^{-u\left(t\right)}}-1$$

(23)

This function maps the input into the bounded interval $\:[-\text{1,1}]$, thereby suppressing large fluctuations in $\:e\left(t\right)$ and $\:\frac{de\left(t\right)}{dt}$ while amplifying smaller variations near equilibrium. The resulting nonlinear outputs are $\:{f}_{\text{e}\text{x}\text{p}}\left(k\right(t\left)\right)$ and $\:{f}_{\text{e}\text{x}\text{p}}\left(l\right(t\left)\right)$. Each of these outputs is multiplied by corresponding gain coefficients $\:{g}_{1}$and $\:{g}_{2}$. The combined outcome produces the auxiliary control signal $\:\delta\:\left(t\right)$.

$$\:\delta\:\left(t\right)={g}_{1}{f}_{exp}\left(k\left(t\right)\right)+{g}_{2}{f}_{exp}\left(l\left(t\right)\right)$$

(24)

This nonlinear transformation allows the controller to emulate a self-tuning mechanism—strong corrective actions are moderated when the error is large, and fine adjustments are emphasized when the system approaches the desired setpoint. Consequently, the exponential block acts as an adaptive preprocessing unit that enhances system stability and mitigates overshoot in highly dynamic conditions. The output $\:\delta\:\left(t\right)$ from the exponential block serves as the input to a conventional PID framework. The control signal $\:u\left(t\right)$ is generated as:

$$\:u\left(t\right)={K}_{p}\text{\hspace{0.17em}}\delta\:\left(t\right)+{K}_{i}{\int\:}_{0}^{t}\delta\:\left(\tau\:\right)\text{\hspace{0.17em}}d\tau\:+{K}_{d}\frac{d\delta\:\left(t\right)}{dt}$$

(25)

where $\:{K}_{p}$, $\:{K}_{i}$, and $\:{K}_{d}$ denote the proportional, integral, and derivative gains, respectively. The integral term ensures zero steady-state error, while the derivative component enhances damping and anticipates future deviations. By acting on the nonlinear signal $\:\delta\:\left(t\right)$, these classical components inherit the adaptability introduced by the exponential transformation. The overall controller thus realizes a cascaded nonlinear–linear architecture, where the exponential module modifies the input dynamics and the PID module performs fine regulation. This configuration offers both the interpretability of classical PID control and the flexibility of nonlinear modulation.

SFOA-based exp-PID controller

The SFOA is applied to optimize the parameters of the exp-PID controller for two distinct control systems: DC motor speed regulation and liquid level regulation. The primary objective is to minimize the cost function by fine-tuning the controller parameters, including $\:{K}_{p}$, $\:{K}_{i}$, $\:{K}_{d}$, $\:{\tau\:}_{1}$, $\:{g}_{1}$, $\:{\tau\:}_{2}$ and $\:{g}_{2}$, while ensuring optimal transient and steady-state performance.

Cost function

The cost function ($\:CF$) is designed to evaluate the performance of the exp-PID controller by considering key transient and steady-state characteristics. The cost function is defined as³⁴:

$$\:CF=\left(1-{e}^{-\rho\:}\right)\times\:\left(\frac{{m}_{os}+{e}_{ss}}{100}\right)+{e}^{-\rho\:}\times\:\left({t}_{s}-{t}_{r}\right)$$

(26)

where $\:\rho\:$ is the balancing coefficient set to 1, $\:{m}_{os}$ is the percentage overshoot, $\:{e}_{ss}$ is the percentage steady-state error measured at the end of the simulation time $\:{t}_{sim}$, $\:{t}_{s}$ is the settling time within a ± 2% tolerance band, and $\:{t}_{r}$ is the rise time from 10% to 90% of the final value. The cost function ensures a trade-off between steady-state accuracy and transient response, where the exponential term adjusts the influence of each component. The selection of simulation time varies based on the dynamics of the control system under evaluation. For a DC motor speed control system, the simulation time is set to 2 s to effectively capture the transient response and steady-state performance of the motor. In contrast, for a liquid level control system, a longer simulation time of 250 s is necessary due to the inherently slower response of liquid dynamics. This ensures that the controller’s performance, including settling time, overshoot, and steady-state error, is accurately assessed and optimized for each specific application.

Constraints of optimization problems

The optimization of the exp-PID controller is subject to predefined parameter constraints to ensure stability, robustness, and effective control performance. These constraints define the feasible search space for the optimization algorithm, preventing extreme values that may degrade system performance. For the DC motor speed control system, the parameter bounds are set as shown in Table 1. These bounds ensure the controller operates within a reasonable range for efficient motor speed regulation. For the liquid level control system, a different set of constraints is applied due to the slower response dynamics of fluid systems, as shown in Table 2. These constraints allow the optimization algorithm to tune the controller effectively while ensuring stability and accuracy in maintaining liquid levels.

Table 1 Bounds of exp-PID controller for DC motor speed control system.

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Table 2 Bounds of exp-PID controller for liquid level control system.

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Application stage of SFOA

The SFOA-based tuning of the exp-PID controller follows an iterative process aimed at minimizing the cost function and enhancing system performance. The optimization begins with the initialization of controller parameters within predefined bounds. These parameters are then applied to the exp-PID controller, generating control signals that regulate the system’s response based on the reference input. The controller interacts with the plant model, producing an output that is evaluated using key performance metrics, including overshoot, steady-state error, rise time, and settling time. Once the system response is obtained, the cost function is computed to assess performance, ensuring a balance between transient and steady-state behavior. This cost function value is then processed within the SFOA module, where new controller parameters are generated based on the algorithm’s search and update rules. The process is repeated iteratively, refining the controller parameters with each cycle until an optimal solution is reached. For the DC motor speed regulation system, the optimization process continuously adjusts the controller parameters to enhance speed tracking and minimize errors. The control signal is applied to the plant model, and system response metrics are analyzed to iteratively fine-tune the controller settings, ensuring improved dynamic performance and stability. Similarly, in the liquid level regulation system, the same approach is applied, but with a focus on maintaining stability in slower-response dynamics.

Figure 6 shows the SFOA-based exp-PID design framework for DC motor system. On the other hand, the flowchart representation in Fig. 7 illustrates the structured workflow of SFOA optimization, where the exp-PID controller dynamically adapts to system conditions based on cost function evaluations. The iterative process ensures that the controller minimizes overshoot, reduces steady-state error, and improves settling time. By following this structured optimization framework, the proposed SFOA-based exp-PID controller achieves enhanced adaptability and efficiency, making it well-suited for both fast-response systems like DC motor speed control and slow-response systems like liquid level regulation. The continuous parameter tuning ensures optimal system performance while maintaining stability and robustness in dynamic control applications.

Simulation results on DC motor speed control system

Comparison with more recent algorithms

To assess the performance and robustness of the proposed SFOA-based exp-PID controller in regulating DC motor speed, a detailed comparative analysis was conducted with several recent metaheuristic algorithms. The competing optimizers included the flood algorithm (FLA)⁶, greater cane rat algorithm (GCRA)⁷, mantis search algorithm (MSA)⁸ and dandelion optimizer (DO)⁹. Each algorithm was implemented under identical simulation conditions, with a uniform population size and iteration limit, and each trial was repeated thirty times to ensure statistical consistency.

Statistical performance evaluation

The quantitative performance of each optimizer was evaluated in terms of the best, worst, mean, and standard deviation (SD) values of the defined cost function. The corresponding results are summarized in Table 3. Among all methods, the SFOA-based exp-PID achieved the lowest mean cost value of 1.6378 × 10⁻³, demonstrating both high accuracy and stability across independent runs. In contrast, the DO-based exp-PID exhibited the highest mean cost value (2.1472 × 10⁻³), indicating relatively weaker convergence and higher variance. The narrow SD of the proposed method (3.48 × 10⁻⁵) further confirms its reliable convergence characteristics and lower sensitivity to stochastic variations within the search process.

The convergence behavior of the cost function for all algorithms is illustrated in Fig. 8. The curve corresponding to the proposed SFOA-based exp-PID shows a markedly faster decline during the early iterations and stabilizes at a lower final value, signifying efficient exploration followed by refined local exploitation. By contrast, the DO-based controller displays sluggish convergence and stabilizes at a substantially higher cost level, while the remaining algorithms (FLA, GCRA, MSA) reach intermediate performance.

Table 3 Statistical performance of different algorithms.

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Optimized controller parameters

The optimized controller parameters derived from each algorithm are listed in Table 4. The SFOA-based exp-PID attained comparatively higher proportional ($\:{K}_{p}$) and integral ($\:{K}_{i}$) gains, as well as elevated exponential shaping coefficients ($\:{\tau\:}_{1}$, $\:{\tau\:}_{2}$), which collectively enhanced its transient and steady-state characteristics. These parameter trends indicate that the proposed optimization strategy effectively identifies gain combinations that accelerate response without introducing instability. The superior parameter balance also ensures smoother actuator behavior and reduced control effort.

Table 4 Optimized controller parameters for proposed and different optimization methods.

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Dynamic response analysis

The dynamic performance of each controller was investigated under a step-input reference signal. The transient responses shown in Fig. 9 reveal that the SFOA-based exp-PID achieves the fastest rise and settling times while entirely eliminating overshoot. The FLA- and MSA-based designs follow closely but show slightly slower convergence. In contrast, the DO-based exp-PID exhibits the slowest transient response, confirming its reduced adaptability in dynamic operating conditions.

A further assessment of steady-state behavior is presented in Fig. 10, where the motor speed tracking accuracy of each controller is compared. The SFOA-based and FLA-based controllers maintain the closest adherence to the reference speed, producing nearly zero steady-state error. Meanwhile, the DO- and GCRA-based designs display minor but noticeable deviations, reflecting their less effective fine-tuning of the integral and exponential parameters.

Quantitative performance indicators

Key performance metrics (including rise time, settling time, percentage overshoot, and steady-state error) are summarized in Table 5. The proposed SFOA-based exp-PID achieved a rise time of 0.0039 s, a settling time of 0.0083 s, zero overshoot, and an extremely small steady-state error of 2.53 × 10⁻⁴ %, confirming its outstanding dynamic precision. These improvements correspond to reductions of approximately 18–25% in transient durations compared with the next-best alternatives. Conversely, the DO-based exp-PID recorded the poorest results, with both the highest rise and settling times (0.0054 s and 0.0108 s) and a larger residual error (0.0067%).

Table 5 Comparative performance metrics for proposed and other optimization methods.

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Discussion

The overall comparison demonstrates that the starfish-driven tuning mechanism substantially enhances both convergence quality and control precision relative to other contemporary optimizers. By balancing its regenerative exploration and directional exploitation phases, SFOA avoids premature convergence and effectively searches the multimodal parameter landscape of the cascaded exp-PID controller. Consequently, it delivers superior transient behavior, minimal overshoot, and high steady-state accuracy in the DC-motor speed regulation task. The consistency between the statistical evidence (Table 3), parameter analysis (Table 4), convergence curve (Fig. 8) and temporal responses (Figs. 9 and 10) confirms that the SFOA-based exp-PID provides the most reliable and efficient control strategy among all tested algorithms.

Comparison with reported works

A comparative investigation was conducted between the proposed SFOA-based exponential PID (exp-PID) controller and several control methods previously reported in the literature to further validate its efficiency and robustness for DC motor speed regulation. The benchmarked methods included the grey wolf optimization (GWO)-based PID¹⁵, atom search optimization (ASO)-based PID¹⁴, marine predator algorithm (MMPA)-based PID³⁰, hybrid Lévy flight distribution and Nelder–Mead algorithm (LFDNM)-based PID¹⁷. Each controller was tested under identical simulation conditions and reference inputs to ensure an equitable comparison of transient and steady-state behaviors.

Transient response comparison

The transient characteristics of the DC motor speed control system are illustrated in Fig. 11, where the speed responses obtained using the five control methods are compared. The proposed SFOA-based exp-PID exhibits the fastest rise and settling times, reaching the desired speed in a substantially shorter period than the alternative approaches. This improvement results from the nonlinear exponential preprocessing within the cascaded structure, which accelerates corrective actions during the initial response phase. Unlike the GWO- and ASO-based controllers, which display moderate oscillations before stabilization, the SFOA-based exp-PID demonstrates zero overshoot, ensuring a smooth approach to the reference value. The MMPA-based PID, on the other hand, exhibits pronounced overshoot and a longer settling period, suggesting an excessive exploratory search pattern that slows convergence in fine-tuning the control parameters. The LFDNM-based PID achieves acceptable transient performance but remains inferior in both speed and damping compared with the proposed approach. The smooth, monotonic transition of the SFOA-based controller emphasizes its superior damping characteristics and adaptability to dynamic changes in the armature current and torque response.

Steady-state response comparison

The steady-state behavior for the same set of controllers is presented in Fig. 12. It is evident that the SFOA-based exp-PID maintains the closest tracking to the reference speed, with an almost negligible steady-state deviation. In contrast, the GWO- and ASO-based controllers exhibit small but visible steady-state offsets, while the LFDNM-based design shows slight ripple around the target speed. The MMPA-based controller performs the worst in this aspect, as it fails to sustain stable operation and produces residual oscillations even after the system reaches steady state. The results confirm that the exponential term incorporated into the SFOA-tuned controller enhances integral compensation without inducing overshoot, allowing it to maintain steady operation and high accuracy despite disturbances or model nonlinearities.

Quantitative performance evaluation

A comprehensive comparison of quantitative performance metrics (including rise time, settling time, percentage overshoot, and steady-state error) is presented in Table 6. The proposed controller achieves a rise time of 0.0039 s and a settling time of 0.0083 s, outperforming all benchmarked techniques by a significant margin. Both times are an order of magnitude faster than those of the GWO- and ASO-based controllers (0.1388 s and 0.0692 s, respectively). The zero-overshoot characteristic of the SFOA-based exp-PID eliminates transient instability, which is particularly beneficial for precision electromechanical systems. The steady-state error achieved by the proposed controller is 2.53 × 10⁻⁴ %, which is nearly two orders of magnitude smaller than the smallest error among the reference methods. In contrast, the MMPA-based PID suffers from a large overshoot (7.01%) and elevated steady-state error (0.4358%), while the LFDNM-based PID, though relatively stable, records higher residual error (0.0051%) and slower convergence. These observations collectively demonstrate that the proposed optimization framework achieves both rapid transient convergence and precise steady-state accuracy.

Table 6 Comparative performance metrics for proposed and other optimization method.

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Discussion

The superior performance of the SFOA-based exp-PID can be attributed to its balanced exploration–exploitation mechanism, which enables efficient global search during the early optimization stages and refined local adjustment near the optimum. The regenerative and multi-directional behaviors of the starfish algorithm prevent premature convergence while ensuring smooth parameter evolution, leading to stable and robust control actions. In contrast, the MMPA tends to over-explore the solution space, reducing tuning precision for low-dimensional control problems, whereas the ASO and GWO methods often stagnate in local minima due to their limited exploitation capability. The hybrid LFDNM approach improves convergence but lacks the nonlinear adaptation provided by the exponential transformation. Consequently, the SFOA-based exp-PID achieves the most favorable trade-off between fast response, minimal overshoot, and near-zero error, as corroborated by the results in Table 6; Figs. 11 and 12. Furthermore, the study adopts a linearized DC-motor model around its nominal operating point to maintain consistency with previously reported analyses. Although this simplification facilitates comparison, future investigations may extend the analysis to include the motor’s nonlinear electromagnetic effects and load disturbances to verify the controller’s performance under real-world conditions.