PI-DÆ: An Adaptive PID Controller Utilizing a New Adaptive Exponent (Æ) Algorithm to Solve Derivative Term Issues

Abstract

This study proposes an enhanced derivative control strategy, named $PI$-$D^{\AE }$, designed to overcome key limitations of the derivative ($D$) term, such as noise amplification, derivative kick $(D$-$k)$, and tuning difficulties. These issues often arise in high-frequency or rapidly changing systems, in which traditional PID controllers struggle. The proposed solution introduces a novel adaptive exponent algorithm (Æ) that dynamically modulates the $D$ term based on the evolving relationship between system output and setpoint. This yields the $PI$-$D^{\AE }$ controller, which adapts in real time to changing conditions. The results show significant performance improvements. Simulation results on two systems demonstrate that $PI$-$D^{\AE }$ achieves a 90% faster response time, a 35% reduction in peak time, and a 100% improvement in settling time compared with conventional PID controllers, all while maintaining a near-zero steady-state error even under external disturbances. Unlike more-complex alternatives such as fuzzy logic, neural networks, or sliding mode control, $PI$-$D^{\AE }$ retains the simplicity and robustness of PID, avoiding high computational costs or intricate setups. This adaptive exponent strategy offers a practical and scalable enhancement to classical PID, improving performance and robustness without added complexity, and thus provides a promising control solution for real-world applications in which simplicity, adaptability, and reliability are essential.

1. Introduction

PID control is widely used in industry to regulate variables such as temperature, speed, and position. Its name derives from its three terms, proportional, integral, and derivative, each with a specific function and a specific gain, $k_P$, $k_I$, and $k_D$. The $\mathrm{P}$ term produces an output proportional to the current error. The $\mathrm{I}$ term eliminates steady-state error, and the $D$ term anticipates future errors using the $D$ of error.

$D$ is considered valuable as it enhances response time and transient behavior and is particularly effective in shortening response time and accelerating recovery from system disturbances [1,2,3]. This improves response speed and reduces overshoot, quickly stabilizing the system. However, $D$ also has drawbacks, such as noise amplification and difficulty in proper tuning. Its sensitivity to noise in measurement signals can lead to unstable responses, and it requires deep knowledge of the system to properly adjust $k_D$.

Additionally, there are phenomena such as the ($Ltd$-$D$) bandwidth-limited derivative control and the “($D$-$kick$) derivative kick”. The $Ltd$-$D$ occurs when a process-variable measurement contains excessive noise due to high-frequency components in the system. $D$ amplifies this noise because it has a high-frequency gain, making it unable to respond effectively to rapid changes and limiting its performance. In practical scenarios, a highly noisy control signal can cause actuator damage, leading to poor actuator performance [4].

$D$-$kick$ occurs with abrupt changes in SP, resulting in a sharp peak in the control signal. This undesired behavior makes $D$ excessively large, potentially causing overshoot or actuator saturation, complicating control and stability.

In summary, $D$ enhances a control system’s performance but must be carefully implemented to avoid undesired effects.

Various strategies have been proposed to mitigate the adverse effects of $D$ [3,4]. Among these, the pure differentiator enhances high-frequency gain and prevents impulses in control signals, often combined with filtering techniques to minimize undesired effects, as demonstrated by Rojas et al. [5]. Another widely adopted method is the linear low-pass filter, proposed by Jin et al. [6], in which a first-order low-pass filter is cascaded with the differentiator. This approach is particularly effective in data acquisition processes, although it tends to slow down transient responses. The speed feedback technique [7], which employs PV instead of the e(t) signal, offers another alternative for refining $D$. Similarly, setpoint filters, as implemented by Arun et al. [8], reduce sensitivity to SP changes and mitigate overshoot. However, tuning these filters for stability and robustness remains a complex task. Pre-filters, analyzed by Singh et al. [9], are often used to achieve smooth control during step changes in SP but fail to address disturbances effectively as they are not integrated into a feedback loop. Finally, nonlinear mean filters, evaluated by Abdel-Razak et al. [10], provide another solution by averaging data points to eliminate noise and disturbances. While these filters effectively smooth control signals and suppress noise, they often increase response times and excessively dampen underdamped processes, limiting their utility in dynamic systems.

These remedies only provide partial solutions. They primarily involve calculating filters to attenuate the controller signal, which cancels $D$ under certain operating conditions.

Besides these approaches, advanced control strategies, such as FL and NN, have been proposed as alternatives to improve the controller’s performance. Fazlollahtabar et al. [11], for instance, demonstrated improved system behavior using FL in a specific plant. While these methods enhance performance, they rely on expert knowledge, demand significant computational resources, and require intricate rule definitions. Similarly, NNs [11] and other adaptive techniques [8]—such as GA, SMCs [11], and fractional-order controllers [12]—offer robustness and adaptability for nonlinear systems but suffer from drawbacks like high computational complexity, convergence issues, chattering effects, and challenging parameter tuning.

The main goal of this paper is to retain the functionality and effectiveness of $D$ without compromising the simplicity of the controller. This study also aims to develop a simple control strategy that avoids the complexity of advanced methods while preserving the $D$ term, unlike filtering approaches that often suppress it.

Furthermore, $D$ continues to present challenges (including Ltd-$D$ and D-kick$)$ that must be overcome. Current solutions primarily focus on attenuating the control signal or applying tuning techniques that fail to address these issues holistically, as previously mentioned. To bridge this gap, this study proposes a novel controller, the $PI$-$D^{\AE }$ controller, which leverages adaptive control theories and self-calculation of the exponential term $D$.

Unlike traditional approaches, this controller will not rely on training, rule-based systems, complex computations, or filters. Instead, it will employ adaptive control theories to dynamically adjust $D$ through self-tuning mechanisms, including self-calculation of its exponential, effectively enhancing performance without significantly increasing complexity.

The proposed $PI$-$D^{\AE }$ controller aims to enhance the performance of $D$ by effectively reducing noise sensitivity and managing abrupt SP changes. It seeks to prevent system destabilization and output signal saturation, providing a robust and efficient alternative to conventional $PID$ tuning methods. In addition, it is hoped that this controller will become a tool that can be implemented in a wide variety of control systems. Furthermore, this work focuses on the development of an algorithm validated through the simulation and analysis of transient and steady-state responses.

The letter “Æ” is used in this article as a symbolic notation to represent the dynamic combination of proportional and derivative effects in the proposed controller. This choice is inspired by its origin as a ligature of the vowels “a” and “e,” which conceptually parallels the adaptive fusion of two classical control components. Thus, its use not only provides a compact representation of the algorithm but also conveys the integrated nature of these control actions.

The rest of this article is organized as follows. Section 2 presents the $PI$-$D^{\AE }$ controller. Section 3 evaluates its performance on a first-order and inverted pendulum–cart system. Section 4 describes the application of $PI$-$D^{\AE }$ with a detailed analysis of its response in time and under disturbances. Section 5 concludes the study with discussions. Finally, Section 6 proposes the conclusions and opportunities for future improvements.

2. Controller Presentation

$PID$ contains three parts, as follows:

(1)$C_P\left(s\right)=P+I+D$

The parallel structure of $PID$ is selected as shown in Equation (2).

(2)$C_P\left(s\right)=k_P+{\displaystyle \frac{k_I}{s}}+k_D(s)$

Now, a new concept of $\mathrm{\AE }$ is presented:

(3)$C_P\left(t\right)={\displaystyle \frac{U\left(s\right)}{Y\left(s\right)}}=k_{P}e\left(t\right)+{\displaystyle \frac{k_I}{I}}e\left(t\right)+{\left[k_D{\displaystyle \frac{de\left(t\right)}{dt}}\right]}^{\mathrm{\AE }}$

(4)$C_p\left(s\right)=k_P+{\displaystyle \frac{k_I}{s}}+{\left[k_D\left(s\right)\right]}^{\mathrm{\AE }}$

$\mathrm{\AE }$ dynamically adjusts the $D$ system’s behavior by accelerating or mitigating the response, depending on the desired control effect, as shown in Figure 1.

For example, if the exponent is $\mathrm{\AE }=1$, we obtain classic $PID$, and if $\mathrm{\AE }=0$, we obtain $PI$. Extending this concept, $\mathrm{\AE }$ is assumed to take values within the range $(-\mathrm{\infty },\mathrm{\infty })$, including infinitesimally small values approaching zero, offering a continuum of control strategies between $PI$ and $PID$.

This simple and robust idea considers an adaptive exponent ($\AE$) for $D$, proposing that $D$ be enhanced as a dynamic, adaptive parameter instead of a fixed constant.

Based on the classical $PID$ response (the Output signal amplitude system depicted in Figure 2), the proposed concept introduces an analysis of $D$ across its full range of values $(-\mathrm{\infty },\mathrm{\infty })$. This analysis redefines $D$ as a new, independent parameter with a $D^{\mathrm{\AE }}$ function, enabling the transformation of the system’s response. By employing the proposed exponential function, $D^{\mathrm{\AE }}$, the system’s future behavior can be effectively modified, as illustrated in Figure 2 and Figure 3.

According to Liuping Wang [4], maintaining the robustness of a $PID$ design is essential; a positive $D$ ($D$ > 0) can destabilize the system and provide excessive disturbance, while a negative signal ($D$ < 0) is negligible. To prevent instability, $D$ can be deactivated. In the $PI$-$D^{\AE }$ controller, $D^{\mathrm{\AE }}$ remains continuously active, ready to respond to any disturbance or $SP$ change. It dynamically adjusts the behavior of $D$ in real time at any $Y\left(s\right)$, as illustrated in Figure 4 and Figure 5.

$PI$-$D^{\AE }$ employs an ($\mathrm{\AE }$) adaptive exponent within its structure. Unlike other $PID$-based controllers, this approach uses an adaptive exponent with an open-loop adaptive control mechanism, shown in Figure 6, allowing the behavior of $D$ to be modified instantly using only the input-to-output relationship of C_p.

Equation (5) defines $\AE$ as follows:

(5)$\AE =\Lambda \left(\mu -{\displaystyle \frac{\sigma ·R\left(s\right)}{U\left(s\right)+ϵ}}\right)$

where $\mu$ is the maximum limit predefined for $\mathrm{\AE };$ $\sigma$ is a sensitivity factor (100 is the recommended value), which determines how reactive the algorithm will be to changes in $Y\left(s\right);$ $ϵ$ is an indeterminate avoidance value ($ϵ\simeq 0$); and $\Lambda$ is an auxiliary gain (recommended values of 1 or 2) that can be added and is helpful in situations in which the gain must be modified indirectly from $\AE$ without altering the initial tuning.

There are two possible scenarios:

(1). If the input is greater than the output, $U\left(s\right)>Y\left(s\right)$, so the response of exponent $\AE$ is increasing and positive.

For example, if $Y\left(s\right)=0$, $R\left(s\right)=1$, and $\mu =\sigma =1$00, this represents the condition $Y\left(s\right)<R\left(s\right)$, where $e\left(s\right)$ is positive and at its maximum. In this case, $\AE =100$, leading to a rapid response in $D^{\AE }$ as its exponential increases.

(2). If the output is greater than the input, $Y\left(s\right)>U\left(s\right)$, so the response of $\AE$ is decreasing and negative.

And, if $Y\left(s\right)=2$, $R\left(s\right)=1$, and $\mu =\sigma =1$00, this represents condition $Y\left(s\right)>R\left(s\right)$, where $e\left(s\right)$ is negative and at its maximum. In this case, $\AE$ is negative, leading to a rapid response in $D^{\AE }$ as its exponential decreases. And $\AE$ results in a maximum negative decrease $\AE =-100$.

$\mu$ and $\sigma$ can be adjusted based on the system’s requirements, such as speed and noise rejection, as suggested by Dastjerdi et al. [13] for $P$. These parameters are associated with a high crossover frequency of $D$, which plays a critical role in achieving stability and optimal control system performance.

The parameters μ, σ, and ϵ directly influence the adaptive response of the exponent Æ and, therefore, the behavior of the $D^{\AE }$ term. Increasing μ allows a wider dynamic range for Æ, making the controller more aggressive and responsive but potentially increasing overshoot. A higher σ makes the controller more sensitive to small changes in output, improving reactivity but also amplifying noise—excessively high σ can destabilize the system in noisy environments. Conversely, a lower σ smooths the response but reduces adaptability. The parameter ϵ ensures numerical stability when the reference R(s) approaches zero. If ϵ is too small, the system may become unstable; if too large, it weakens the effect of Æ. Proper tuning of these parameters enables the controller to balance fast adaptation, robustness, and noise immunity effectively. The values presented in this article were carefully selected as ideal defaults for the initial presentation of the algorithm, providing a robust and balanced behavior across typical operating conditions.

In an additional case, if $Y\left(s\right) \simeq U\left(s\right)$, $\AE$ modifies its response, resulting in a null $D^{\AE }$ response, even if noise variations in $D$ occur.

So, replacing $\AE$ in Equation (4) is achieved as follows:

(6)$C_P\left(s\right)=k_P+{\displaystyle \frac{k_I}{s}}+{\left[k_D\left(s\right)\right]}^{\Lambda \left(\mu -{\displaystyle \frac{\sigma ·Y\left(s\right)}{R\left(s\right)+ϵ}}\right)}$

The proposed adaptive $PI$-$D^{\AE }$ controller is presented in Figure 6. The gain $k_{switch}$ serves as a practical switch that activates or deactivates $D$ in the control system. By assigning a value of 1 to the gain, $D$ is enabled, while a value of 0 disables it.

(7)$C_P\left(s\right)=k_P+{\displaystyle \frac{k_I}{s}}+k_{switch}·{\left[k_D\left(s\right)\right]}^{\Lambda \left(\mu -{\displaystyle \frac{\sigma ·Y\left(s\right)}{R\left(s\right)+ϵ}}\right)}$

(8)$C_P\left(s\right)=k_P+{\displaystyle \frac{k_I}{s}}+k_{switch}·{\left[k_D\left(s\right)\right]}^{\AE }$

(9)$C_P\left(s\right)=P+I+D^{\AE }$

$k_{\Lambda }$ serves as a practical switch that activates or deactivates $D$ in the control system. By assigning a value of 1 to the switch, $D$ is enabled, while a value of 0 effectively disables it, providing a straightforward mechanism for managing its influence on the system’s response.

The total control signal is given as follows:

(10)$U\left(s\right)=U_P\left(s\right)+U_I\left(s\right)+U_D\left(s\right)$

3. Materials and Methods

In this section, two proposed control systems are presented. Simulations and graphs were obtained using MATLAB 2024a and Simulink software.

3.1. Test Plants

3.1.1. First-Order System

A first-order plant as in Equation (11) is considered for testing the controller.

(11)$G\left(s\right)={\displaystyle \frac{K}{1+Ts}}{e}^{-Ls}$

3.1.2. Inverted Pendulum on a Cart IPC

And a second plant [14,15,16] is presented to test the proposed algorithm, an inverted pendulum on a cart, as shown in Figure 7.

Here, M is the mass of the car; p is the pivot point; L is the pendulum; the horizontal direction is $x;$ u is the force; and θ is the angle.

The relevant state-space equations are as follows:

(12)$\dot{x_1}=x_2$

(13)$\dot{x_2}={\displaystyle \frac{\left(M+m\right)gSin{x}_1-m{L{x}_2}^2\left(Sin{x}_{1}Cos{x}_1\right)-uCos{x}_1}{ML+mL{\left(Sin{x}_1\right)}^2}}$

(14)$\dot{x_3}=x_4$

(15)$\dot{x_4}={\displaystyle \frac{u+mL{x_2}^2\left(Sin{x}_1\right)-mgCos{x}_{1}Sin{x}_1}{M+m{\left(Sin{x}_1\right)}^2}}$

where

$x_1=\theta ; the angle of the inverted pendulum.$

$x_2=\dot{\theta }; the angular velocity of the inverted pendulum.$

$x_3=x; the position of the cart.$

$x_4=\dot{x}; the velocity of the cart$.

4. Simulation Results

4.1. First-Order System

The controller was first tested in a first-order plant, where $K=10$, $T=1$ s, and $L=0.1$ were proposed as in reference [17].

(16)$G\left(s\right)={\displaystyle \frac{10}{s+1}}{e}^{-0.1s}$

Figure 8 presents the $PID$ response to emphasize the destabilizing effect of $D$ when different values of ${\tau }_D$ are assigned. These ${\tau }_D$ values are proposed according to the analysis conducted by Ang et al. [17], with an additional value (${\tau }_D=0.4$). The $PID$ and $PI$-$D^{\AE }$ parameters can be found in Table 1.

Figure 9, Figure 10 and Figure 11 represent a comparison between $PID$ and $PI$-$D^{\AE }$ when ${\tau }_D=0.03$ and ${\tau }_D=0.1$, which shows its property acceleration response by implementing $PI$-$D^{\AE }$. Additionally, Figure 10 includes the system’s behavior under a disturbance at t = 1.

Furthermore, Figure 12 and Figure 13 present a comparison between $PID$ and $PI$-$D^{\AE }$ when ${\tau }_D=0.2$ and ${\tau }_D=0.4$, respectively. Figure 14 and Figure 15 present a comparison between $PID$ and $PI$-$D^{\AE }$ derivatives when ${\tau }_D=0.2$ and ${\tau }_D=0.4,$ respectively. Table 2 provides a quantitative demonstration of the effectiveness of the $\AE$ algorithm.

Figure 16 shows a comparison between $PID$ at ${\tau }_D=0.03$ and $PI$-$D^{\AE }$ at ${\tau }_D=0.4$, the values at which better responses are observed for each controller. And Figure 17 shows the system’s behavior under a disturbance at t = 1, demonstrating how its signal is recovered faster using $PI$-$D^{\AE }$ than with $PID$.

Table 3 and Table 4 compare $PI$-$D^{\AE }$ vs. $PID$ and the $PI$($D$) vs. the $PI$ controller, named as such because, in this case, $D$ is deactivated (see Table 1). Table 3 provides a response time comparison, and Table 4 provides insight into the activation of $D$. $D$ can be utilized (using the $\AE$ algorithm) to achieve better results, specifically results that would not be attainable with $PID$.

4.2. Inverted Pendulum on a Cart

The $IPC$ model was obtained according to previous studies [14,15,16], $\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} M=2.4 \mathrm{k}\mathrm{g}$, $m=0.23 \mathrm{k}\mathrm{g}$, $L=0.36 \mathrm{m}$, and $g=9.81 \mathrm{m}/{\mathrm{s}}^2$. The state space equations are as follows:

(17)$\dot{x_1}=x_2$

(18)$\dot{x_2}={\displaystyle \frac{\left(2.4+0.23\right)9.81Sin{x}_1-0.23·0.36 {x_2}^2\left(Sin{x}_{1}Cos{x}_1\right)-uCos{x}_1}{2.4 ·0.36+0.23·0.36 {\left(Sin{x}_1\right)}^2}}$

(19)$\dot{x_3}=x_4$

(20)$\dot{x_4}={\displaystyle \frac{u+0.23·0.36 {x_2}^2\left(Sin{x}_1\right)-0.23·9.81Cos{x}_{1}Sin{x}_1}{2.4+0.23{\left(Sin{x}_1\right)}^2}}$

Equilibrium and position control are summed into one control signal; therefore, position control is affected by equilibrium control. To gain a better understanding, let us consider a first $IPC$ (Figure 18), where its equilibrium is controlled by the $PI$-$D^{\AE }$ controller and its position is controlled by the $PID$ controller (affected by the $PI$-$D^{\AE }$ equilibrium controller).

Considering a second $IPC$ (Figure 19), its equilibrium is controlled by a conventional $PID$ controller, and its position is controlled by another $PID$ controller (affected by the $PID$ equilibrium controller).

Furthermore, the traditional derivative term is defined as Equation (21):

(21)$D\left(t\right)=k_D{\displaystyle \frac{de\left(t\right)}{dt}}$

where $k_D$ is a fixed gain, and e(t) is the control error.

In contrast, the proposed term modifies the derivative behavior through an adaptive exponent as Equation (22):

(22)$D^{\AE }\left(t\right)={\left(k_D{\displaystyle \frac{de\left(t\right)}{dt}}\right)}^{\AE }$

where

(23)$\AE \left(t\right)=\mathsf{\Lambda }\left(\mathsf{\mu }-{\displaystyle \frac{\mathsf{\sigma }Y\left(s\right)}{R\left(s\right)+\mathsf{ϵ}}}\right)$

This formulation allows the gain applied to the derivative to change dynamically in real time, based on the system’s input–output ratio $Y\left(s\right)/R\left(s\right)$ Equation (23), modulated by the sensitivity σ, a maximum bound μ, and a robustness constant ϵ.

The behavior of $D^{\AE }$ becomes nonlinear and adaptive:

If Æ = 1, the system behaves identically to the classical PID.

If Æ < 1, the derivative contribution is softened, reducing noise amplification.

If Æ > 1, the derivative contribution is enhanced, accelerating system response.

If Æ → 0, the derivative term vanishes, effectively converting the controller to a PI type in real-time under certain conditions.

To illustrate this, we have also included Figure 3, Figure 5, Figure 18 and Figure 19 comparing the behavior and structure of $PI$-$D^{\AE }$ and $PID$, especially under disturbances, to highlight how the adaptive exponent enhances robustness and mitigates derivative kick and noise amplification.

The following comparisons were employed: the equilibrium control comparison is $PI$-$D^{\AE }$ vs. $PID,$ and the position control comparison is the affected $PID$ (indirectly affected by the $PI$-$D^{\AE }$ equilibrium controller) vs. the affected $PID$ (affected by the $PID$ equilibrium controller). Table 5 and Table 6 show the parameters of the controllers for balance and position, respectively.

The deactivation of $I$ is due to the fact that the authors of the original test [14,15,16] conducted their simulations in this way, and we aimed to respect this condition by offering a comparison with the originally proposed $PID$ controller. This does not preclude the fact that the value of the $\AE$ algorithm can be further explored by comparing it with $PID$ controllers with all three activated terms (as with the first-order system).

Figure 20 shows an angle control comparison between $PID$ and $PI$-$D^{\AE }$. Figure 21, Figure 22 and Figure 23 compare $D$ and $D^{\AE }$ behavior. Typically, $D$ is equal to $k_D·de\left(t\right)$. However, a distinction is made. $D$ and $D^{\AE }$ will be considered $D$ control signals. $k_D·de\left(t\right)$ will be considered $D$ behavior; this $D$ behavior will not be the same as $D$ because when $PI$-$D^{\AE }$ is employed, $D$ becomes more complex than $k_D·de\left(t\right)$, so $D\ne D^{\AE }$.

Figure 24 shows a comparison of the position control response between two PID configurations.

Table 3, Table 7 and Table 8 compare the time response improvement of angle and position control, respectively, between $PI$-$D^{\AE }$ and $PID$.

Inverted Pendulum on a Cart with Disturbances

Input disturbances with a randomly assigned amplitude of 0.01 were introduced and uniformly applied to both control systems $\mathrm{P}\mathrm{I}\mathrm{D}$ and $\mathrm{P}\mathrm{I}$-${\mathrm{D}}^{\mathrm{\AE }}$. Similar to the previous system, the simulations considering disturbances are shown in Figure 25, Figure 26, Figure 27 and Figure 28.

Table 9 and Table 10 present a comparison of the analysis of transient and steady-state responses, including an optimization percentage.

5. Discussion

$PI$-$D^{\AE }$ demonstrates significant advantages over the conventional $PID$, particularly in scenarios with varying ${\tau }_D$ values (Table 3 and Table 4). For ${\tau }_D$ = 0.03, $PI$-$D^{\AE }$ shows marginal improvement; however, its rapid response and steady error reduction are evident. As ${\tau }_D$ increases to 0.1, $PI$-$D^{\AE }$ achieves better performance across most criteria, except for an 8% higher overshoot ($M_p$). Crucially, while $PID$ fails to reach a steady state, $PI$-$D^{\AE }$ ensures stability and error convergence to zero.

For higher values of ${\tau }_D$ (0.2 and 0.4), the PID controllers diverge, whereas $PI$-$D^{\AE }$ remains stable and delivers substantial performance improvements. Error accumulation metrics improve by up to 23%, while $t_s$ shows enhancements of up to 61%. Despite a slight increase in $t_r$, the overall performance indicates faster and more-stable setpoint attainment with $PI$-$D^{\AE }$. Additionally, $t_p$ shows minimal improvement at ${\tau }_D$ = 0.2 but is notably reduced by 53% at ${\tau }_D$ = 0.4. Importantly, $M_p$ decreases by up to 11.3% with $PI$-$D^{\AE }$, ensuring better system stability.

$PI$-$D^{\AE }$ dynamically adapts to $D^{\AE }$, effectively addressing issues such as instability and overshoot while accelerating response times. Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 confirm the superior robustness and disturbance rejection of $PI$-$D^{\AE }$ compared with $PID$. Optimizing $D^{\AE }$ mitigates instability, improves response acceleration, and ensures zero steady-state error, achieving enhanced system performance and reliability.

$PI$-$D^{\AE }$ demonstrates significant advantages over $PID$, particularly in equilibrium control. With identical $PID$ tuning, $PI$-$D^{\AE }$ achieves a near-zero steady-state error from the start, while $PID$ requires multiple oscillations before reaching stability. Notably, $PI$-$D^{\AE }$ effectively neutralizes $DK$ observed in $PID$, as shown in Figure 20, maintaining control over the θ angle with minimal oscillations and a stable signal.

Furthermore, $PI$-$D^{\AE }$’s transient response exhibits unique characteristics. While its $t_r$ is faster, it achieves $t_s$ almost immediately, maintaining control near SP without oscillations. This contrasts with $PID$, which takes longer to stabilize. $D^{\AE }$ actively mitigates instability early on, as indicated by its initial high magnitude, compared with negligible $D$ activity in $PID$, which leads to pronounced $DK$.

Regarding $t_p$ and $M_p$, $PI$-$D^{\AE }$ demonstrates a 99% reduction in $M_p$, ensuring minimal overshoot and improved stability. Although occasional oscillations are present, their magnitudes are negligible, contributing to overall system robustness. The error accumulation metrics show significant improvements in three of four criteria, with only ITAE showing a slight increase, which remains inconsequential in absolute terms.

In the overlap analysis of equilibrium and position control signals (Figure 18 and Figure 19), $PI$-$D^{\AE }$ introduces slower transient responses and minor steady-state oscillations below SP. While this behavior slightly impacts position control, the primary focus of this study is equilibrium control. $PI$-$D^{\AE }$ demonstrates superior performance in stabilizing equilibrium by neutralizing $DK$ and enhancing robustness under dynamic conditions. These results confirm its effectiveness as a robust alternative to traditional $PID$ controllers, particularly for applications prioritizing stability and precision.

In IPC systems with disturbances, as shown in Table 9 and Table 10, the $PI$-$D^{\AE }$ controller significantly improved the control signal compared with PID and maintained low $D$ magnitudes throughout the process. Error accumulation across all the criteria improved by over 82%, reaching up to 97% in the best case.

$PI$-$D^{\AE }$ achieved steady-state control from the start, while PID failed to reach equilibrium. Although $t_p$ for $PI$-$D^{\AE }$ was longer, the ($M_p$) overshoot was reduced by 99% relative to $PID$. Sub-impulses in the $PI$-$D^{\AE }$ response were minimal and acceptable when compared with $PID$, and the former consistently rejected oscillations and always maintained equilibrium.

The $PI$-$D^{\AE }$ controller stands out for its structural simplicity and computational efficiency, relying on only four parameters (Kp, Ki, Kd, and an adaptive exponent Æ) for implementation. In contrast to more-complex approaches such as FL PID controllers, which require 10 to 30 inference rules along with associated gain parameters, or NN-based PID controllers, which involve more than 20 parameters related to network architecture and training data, the $PI$-$D^{\AE }$ design does not require any prior training. Furthermore, while fuzzy and NN-based controllers entail medium and high computational costs, respectively, the $PI$-$D^{\AE }$ maintains a significantly low computational burden. Regarding performance, the $PI$-$D^{\AE }$ demonstrates high robustness to noise, comparable with that of NN-based controllers, yet with a considerably simpler and more direct implementation. [18,19,20,21,22,23,24,25]. These features make the $PI$-$D^{\AE }$ a highly effective and practical alternative for control systems in which operational efficiency and noise immunity are of critical importance.

6. Conclusions

The $PI$-$D^{\AE }$ algorithm offers substantial improvements over traditional PID controllers by addressing key challenges such as noise sensitivity and abrupt setpoint changes. Its innovative reformulation of D enhances robustness, reduces oscillations, and ensures system stability under dynamic conditions. Additionally, the algorithm’s compatibility with conventional tuning methods broadens its applicability but with considerable advantages over conventional PID, making it a practical solution for modern control systems.

While the initial results demonstrate the $PI$-$D^{\AE }$ algorithm’s superiority in tested systems, further validation across diverse applications and comparison with other advanced control strategies are necessary. The $PI$-$D^{\AE }$ algorithm’s adaptability within the PID structure allows for seamless integration with other control methodologies, presenting a promising avenue for improving performance in a wide range of systems. Future refinements, such as dynamically alternating between D and $D^{\AE }$, could further optimize its effectiveness and expand its potential in control engineering.

Author Contributions

Conceptualization, J.M.B.-F. and J.-C.S.-V.; methodology, J.M.B.-F.; software, J.M.B.-F. and J.P.M.H.; validation, J.M.B.-F., J.P.M.H., and J.-C.S.-V.; investigation, J.M.B.-F., J.P.M.H., and K.M.E.; original draft preparation, J.M.B.-F., J.P.M.H., and K.M.E.; review and editing, J.M.B.-F., J.P.M.H., and J.-C.S.-V.; supervision, J.-C.S.-V. and A.V.-C. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data supporting the findings of this study are available from the corresponding author, J.C.S.V., upon reasonable request following the publication of the paper.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

$PI$-$D^{\mathrm{\AE }}$	Adaptive exponent controller
$D^{\mathrm{\AE }}$	Adaptive exponent derivative term
Æ	Adaptive exponent
$\theta$	Angle
$k_{aux}$	Auxiliary gain
$Ltd$-$D$	Bandwidth-limited derivative control
$C_P\left(s\right)$	Controller system
$t_d$	Delay time
$k_D$	Derivative gain
$D$-$k$	Derivative kick
$D$ u	Derivative response
$D$	Derivative term
${\tau }_D$	Derivative time
$e$ or $e\left(t\right)$ or $E\left(s\right)$	Error
$FL$	Fuzzy logic
$GA$	Genetic algorithm
$R\left(s\right)$	Input system or reference
$IAE$	Integral absolute error
$k_I$	Integral gain
$ISE$	Integral squared error
$I$	Integral term
$ITAE$	Integral time absolute error
$ITSE$	Integral time squared error
${\tau }_I$	Integral time
$IPC$	Inverted pendulum on a cart
$M_p$	Maximum peak
$NN$	Neural network
$Y\left(s\right)$	Output system
$t_p$	Peak time
$x$	Position
$PV$	Process variable
$k_P$	Proportional gain
$P$	Proportional term
$t_r$	Rising time
$SP$	Setpoint
$t_s 2\%$	Settling time at 2%
$t_s 5\%$	Settling time at 5%
$SMC$	Sliding mode

Footnotes

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Figures and Tables

Figure 1 $D^{\AE }$ modification of behavior response. This figure illustrates how the proposed $D^{\mathrm{\AE }}$ term dynamically adjusts the influence of the derivative component. Depending on the value of the exponent Æ, the controller’s behavior can range from PI control (Æ = 0) to classical PID control (Æ = 1) or to even more aggressive responses. This adaptability enhances the system’s transient performance without relying on external filters.

Figure 2 $PID$ vs. PI-$D^{\AE }$ responses. The figure highlights how the PI-$D^{\AE }$ controller modifies the output dynamics compared with a traditional PID. The use of the adaptive exponent Æ results in faster rise times, reduced overshoot, and shorter settling times, demonstrating improved robustness to setpoint changes.

Figure 3 $D$ vs. $D^{\AE }$ responses. This plot shows how $D^{\AE }$ either attenuates or intensifies the derivative action based on the system state. This adaptiveness prevents noise amplification and contributes to improved controller stability under disturbances or time-varying conditions.

Figure 4 $PID$ and $PI-D^{\AE }$ responses with disturbance. The figure demonstrates the superior disturbance rejection capability of the $PI-D^{\AE }$ controller. Compared with the conventional PID, the $PI-D^{\AE }$ recovers equilibrium faster and exhibits reduced oscillatory behavior, ensuring improved robustness in non-ideal environments.

Figure 5 $D$ and $D^{\AE }$ responses with disturbance. This figure presents the response of the derivative terms under perturbations. While the classical $D$ term amplifies noise and generates abrupt control actions, the $D^{\AE }$ term adaptively regulates its magnitude via the Æ exponent, resulting in smoother and safer control signals.

Figure 6 Proposed adaptive $PI$-$D^{\AE }$ controller structure. This block diagram presents the architecture of the $PI$-$D^{\AE }$ controller, in which the adaptive exponent Æ modulates the derivative action. The total control signal is composed of independent proportional, integral, and adaptive derivative components. A switching gain is included to enable or disable the derivative term as needed, providing flexibility in controller deployment.

Figure 7 System of IPC.

Figure 8 $\mathrm{P}\mathrm{I}\mathrm{D}$ controller with different ${\tau }_D$ values. (a) This figure presents the response of a classical PID controller using various values of the derivative time constant ${\tau }_D$. (b) The simulation highlights the destabilizing effects of the derivative term as ${\tau }_D$ increases, including excessive overshoot and longer settling times. These results underscore the limitations of fixed-gain derivative action in conventional PID tuning, particularly under dynamic or noisy conditions.

Figure 9 $PID$ and $PI$-$D^{\mathrm{\AE }}$. ${\tau }_D=0.03$.

Figure 10 $PID$ and $PI$-$D^{\AE }$. ${\tau }_D=0.03$ with disturbance.

Figure 11 $PID$ and $PI$-$D^{\mathrm{\AE }}$. ${\tau }_D=0.1$.

Figure 12 $PID$ and $PI$-$D^{\mathrm{\AE }}$. ${\tau }_D=0.2$.

Figure 13 $PID$ and $PI$-$D^{\mathrm{\AE }}$. ${\tau }_D=0.4$.

Figure 14 $PID$ and $PI$-$D^{\mathrm{\AE }}$ derivatives. ${\tau }_D=0.2$.

Figure 15 $PID$ and $PI$-$D^{\AE }$ derivatives. ${\tau }_D=0.4$.

Figure 16 $PID \mathrm{at}$${\tau }_D=0.03$ and $PI$-$D^{\mathrm{\AE }}$ at ${\tau }_D=0.4$.

Figure 17 $PID \mathrm{at}$${\tau }_D=0.03$ and $PI$-$D^{\AE }$ at ${\tau }_D=0.4$ with disturbance.

Figure 18 $IPC\#1$ control. $PI$-$D^{\AE }$ equilibrium control + $PID$ position control.

Figure 19 $IPC\#2$ control. $PID$ equilibrium control + $PID$ position control.

Figure 20 $PID$ and $PI$-$D^{\AE }$ equilibrium control.

Figure 21 $PID$ and $PI$-$D^{\AE }$ equilibrium control of $D$ and $D^{\AE }$ control signals.

Figure 22 Derivative behaviors of $PID$ and $PI$-$D^{\AE }$ equilibrium control.

Figure 23 $PID$ position control derivatives affected by equilibrium controllers.

Figure 24 $PID$ position control affected by equilibrium controllers.

Figure 25 System with disturbance under $PID$ and $PI$-$D^{\AE }$ equilibrium control.

Figure 26 System with disturbance under $PID$ and $PI$-$D^{\AE }$ equilibrium control of $D$ and $D^{\AE }$ signals.

Figure 27 System with disturbance under $PID$ and $PI$-$D^{\AE }$ equilibrium control and derivative behaviors.

Figure 28 System with disturbance and $PID$ position control affected by equilibrium controllers.

Table 1

Controller parameters.

	${\mathit{\tau }}_{\mathit{D}}=0$	${\mathit{\tau }}_{\mathit{D}}=0.03$		${\mathit{\tau }}_{\mathit{D}}=0.1$		${\mathit{\tau }}_{\mathit{D}}=0.2$		${\mathit{\tau }}_{\mathit{D}}=0.4$
	$\mathit{P}\mathit{I}$($\mathit{D}$)	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$
$k_P$	0.644
$k_I$	0.625
${\tau }_I$	1.03
$k_D$	0	0.019		0.064		0.128		0.257
${\tau }_D$	0	0.0303		0.1		0.2		0.4
$\Lambda$			2		2		2		2
$\mu$			100		100		100		100
$\mathsf{ϵ}$			0.01		0.01		0.01		0.01
$k_{\Lambda }$			1		1		1		1

Table 2

Controller comparison.

	${\mathit{\tau }}_{\mathit{D}}=0$	${\mathit{\tau }}_{\mathit{D}}=0.03$		${\mathit{\tau }}_{\mathit{D}}=0.1$		${\mathit{\tau }}_{\mathit{D}}=0.2$		${\mathit{\tau }}_{\mathit{D}}=0.4$
	$\mathit{P}\mathit{I}$($\mathit{D}$)	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$
$t_d$ (s)	0.177
$t_r$ (s)	0.273	0.347	0.273	0.613	0.273	0.392	0.276	0.349	0.300
$M_p$	1.150	1.020	1.149	1.054	1.140		1.111		1.020
$t_p$ (s)	0.372	0.573	0.372	0.897	0.369		0.369		0.570
$t_s 2\%$ (s)	0.745	0.318	0.743		0.723		0.534		0.287
$t_s 5\%$ (s)	0.504	0.292	0.504		0.500		0.486		0.269
$IAE$	0.2105	0.1958	0.2103	0.2457	0.2082	0.5791	0.2017	40.30	0.1904
$ISE$	0.1549	0.1529	0.1549	0.1652	0.1545	0.4547	0.1535	9552	0.1521
$ITAE$	0.0312	0.0261	0.0310	0.0471	0.0300	0.2906	0.0270	35.75	0.0239
$ITSE$	0.0134	0.0125	0.0134	0.0166	0.0132	0.2390	0.0128	9030	0.0123

Table 3

Improvement (%) achieved by $PI$-$D^{\AE }$ for each test.

	${\mathit{\tau }}_{\mathit{D}}=0.03$		${\mathit{\tau }}_{\mathit{D}}=0.1$		${\mathit{\tau }}_{\mathit{D}}=0.2$		${\mathit{\tau }}_{\mathit{D}}=0.4$
	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$
$t_d$ (s)	0.177		0.177		0.177		0.177
$t_r$ (s)	0.347	21.32%	0.613	55.46%	0.392	29.59%	0.349	14.04%
$M_p$	1.020	−12.64%	1.054	−8.15%
$t_p$ (s)	0.573	35.07%	0.897	58.86%
$t_s 2\%$ (s)	0.318	−133.64%
$t_s 5\%$ (s)	0.292	−72.60%
$IAE$	0.1958	−7.40%	0.2457	15.26%	0.5791	65.17%	40.30	99.52%
$ISE$	0.1529	−1.30%	0.1652	6.47%	0.4547	66.24%	9552	99.99%
$ITAE$	0.0261	−18.77%	0.0471	36.30%	0.2906	90.70%	35.75	99.93%
$ITSE$	0.0125	−7.20%	0.0166	20.48%	0.2390	94.64%	9030	99.99%

Table 4

Improvement (%). $PID$ vs. $PI$-$D^{\AE }$.

	${\mathit{\tau }}_{\mathit{D}}=0$	${\mathit{\tau }}_{\mathit{D}}=0.03$	${\mathit{\tau }}_{\mathit{D}}=0.1$	${\mathit{\tau }}_{\mathit{D}}=0.2$	${\mathit{\tau }}_{\mathit{D}}=0.4$
	$\mathit{P}\mathit{I}$($\mathit{D}$)	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$
$t_d$ (s)	0.177
$t_r$ (s)	0.273			−1.09%	−9.89%
$M_p$	1.150	0.08%	0.87%	3.39%	11.30%
$t_p$ (s)	0.372		0.80%	0.80%	−53.22%
$t_s 2\%$ (s)	0.745	0.26%	2.95%	28.32%	61.47%
$t_s 5\%$ (s)	0.504		0.79%	3.57%	46.62%
$IAE$	0.2105	0.09%	1.09%	4.18%	9.54%
$ISE$	0.1549		0.25%	0.90%	1.80%
$ITAE$	0.0312	0.64%	3.84%	13.46%	23.39%
$ITSE$	0.0134		1.49%	4.47%	8.20%

Table 5

Angle control parameters.

	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$
$k_P$	−40
$k_I$	0
${\tau }_I$	0
$k_D$	−8
${\tau }_D$	0.2
$\Lambda$		2
$\mu$		100
$\mathsf{ϵ}$		0.01
$k_{\Lambda }$		1

Table 6

Position control parameters.

	$\mathit{P}\mathit{I}\mathit{D}$
$k_P$	−1.25
$k_I$	0
${\tau }_I$	0
$k_D$	−3.6
${\tau }_D$	2.88

Table 7

Angle control comparison.

	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$	Improvement
$t_d$ (s)
$t_r$ (s)	1.375	24.730	−1698.54%
$M_p$	4.926 × 10−3	3.052 × 10−5	99.38%
$t_p$ (s)	0.606	1.901	−213.69%
$t_s 2\%$ (s) [0.002, −0.002]	2.586		100.00%
$t_s 5\%$ (s) [0.005, −0.005]
$IAE$	0.008693	0.00217	75.03%
$ISE$	2.392 × 10−5	6.572 × 10−7	97.25%
$ITAE$	0.01739	0.0568	−226.62%
$ITSE$	2.874 × 10−5	1.802 × 10−5	37.30%

Table 8

Position control comparison.

	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$	Improvement
$t_d$ (s)	1.938	18.807	−870.43%
$t_r$ (s)	25.269	40.420	−59.95%
$M_p$		1.001 × 10−1
$t_p$ (s)		40.726
$t_s 2\%$ (s)	10.168	31.699	−211.75%
$t_s 5\%$ (s)	7.840	28.030	−257.52%
$IAE$	0.288	1.817	−530.90%
$ISE$	0.0169	0.142	−740.23%
$ITAE$	0.7144	19.08	−2570.77%
$ITSE$	0.02084	1.185	−5586.18%

Table 9

Angle control with disturbance comparison.

	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$	Improvement
$t_d$ (s)
$t_r$ (s)	1.343	25.172	−1774.31%
$M_p$	5.176 × 10−3	3.093 × 10−5	99.40%
$t_p$ (s)	0.618	0.835	−35.11%
$t_s 2\%$ (s) [0.002, −0.002]	2.617		100.00%
$t_s 5\%$ (s) [0.005, −0.005]	0.730		100.00%
$IAE$	0.02108	0.002263	89.26%
$ISE$	3.338 × 10−5	8.439 × 10−7	97.47%
$ITAE$	0.3341	0.05867	82.43%
$ITSE$	0.000161	2.398 × 10−5	85.10%

Table 10

Position control with disturbance comparison.

	$\mathit{P}\mathit{I}\mathit{D}$	$\mathit{P}\mathit{I}$ $\text{-}{\mathit{D}}^{\mathit{\AE }}$	Improvement
$t_d$ (s)	1.876	18.800	−902.132
$t_r$ (s)	15.089
$M_p$	1.019 × 10−1	9.807 × 10−2	3.759
$t_p$ (s)	26.193	34.588	−32.051
$t_s 2\%$ (s)	43.795
$t_s 5\%$ (s)	7.280
$IAE$	0.3216	1.99	−518.781
$ISE$	0.01672	0.1442	−762.440
$ITAE$	1.92	26.58	−1284.375
$ITSE$	0.0225	1.284	−5606.667

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