Super-Twisting-Based Online Learning in High-Order Neural Networks for Robust Backstepping Control of DC Motors Under Uncertainty

Abstract

This paper addresses the speed control problem of a DC motor in the presence of nonlinearities, disturbances, and unmodeled dynamics by proposing a neural backstepping control scheme based on a Recurrent High-Order Neural Network (RHONN). The proposed RHONN serves as an online approximator to compensate for uncertain nonlinear dynamics in a PD-based backstepping controller, enabling the system to handle disturbances, modeling errors, and unmodeled dynamics. Instead of relying on the traditional Extended Kalman Filter (EKF) for RHONN weight adaptation, the neural parameters are updated online using a Super-Twisting Algorithm (STA). As a result, the proposed STA-based learning law provides a simpler and robust covariance-free adaptation mechanism with practical finite-time convergence properties, making it suitable for real-time embedded implementations. The proposed method was evaluated through numerical simulations and implemented on an embedded microcontroller to assess its real-time performance. Simulation results show reductions between 0.04% and 2.04% in steady-state and integral error metrics compared with a tuned PD controller, and improvements up to 25.66% and 23.82% over LQR and MPC in the IMSE index. Experimental results demonstrate good tracking performance, robustness under varying load conditions, and low computational requirements, confirming the practical feasibility.

1. Introduction

In the context of electromobility, the literature identifies Permanent Magnet Synchronous Motors (PMSMs) and Brushless DC (BLDC) motors as two of the most widely used technologies in modern electric propulsion systems. PMSMs are widely adopted in electric vehicles due to their high efficiency, high torque density, and excellent dynamic performance, which make them suitable for high-performance traction applications [1]. Similarly, BLDC motors have gained significant attention due to their relatively simple electronic commutation, reduced maintenance requirements, and high efficiency, making them particularly attractive for applications such as electric bicycles, drones, and compact electric mobility platforms [2]. As a result, both PMSMs and BLDC motors are widely used in contemporary electric mobility systems. Despite the widespread adoption of these technologies, conventional direct-current (DC) motors remain a practical and cost-effective solution for micromobility systems, including electric scooters, small robotic platforms, and educational test benches. Their relatively simple structure, straightforward control strategy, and reduced implementation costs make them attractive for applications where simplicity and affordability are key design considerations [3]. Nevertheless, achieving high-performance control of DC drives remains challenging. The presence of nonlinearities, parameter variations, and external disturbances can degrade control accuracy and may compromise closed-loop stability. Classical control strategies, particularly proportional–integral (PI) and proportional–integral–derivative (PID) controllers, have been widely employed for speed regulation due to their simplicity and ease of implementation. However, their robustness under load variations and sensor noise is often limited [4,5].

To address these limitations, robust control techniques such as Sliding Mode Control (SMC) have been proposed because of their strong disturbance rejection capability and finite-time convergence properties [6]. Nevertheless, the practical implementation of SMC frequently suffers from the well-known chattering phenomenon. To reduce chattering while preserving robustness, higher-order sliding mode control (HOSMC) techniques have been introduced. These approaches significantly reduce the high-frequency switching present in classical SMC while maintaining robustness properties [7]. However, the design of these controllers typically requires accurate knowledge of the system dynamics. To overcome modeling uncertainties, adaptive modeling and system identification techniques have been explored. Artificial intelligence methods such as neural networks and deep learning have demonstrated strong nonlinear approximation capabilities [8,9]. However, deep neural models often require significant computational resources, which limits their applicability in embedded systems with constrained hardware [10].

Given the aforementioned challenges, this work considers a Recurrent High-Order Neural Network (RHONN) as an adaptive model, owing to its strong nonlinear approximation capabilities and relatively low computational complexity compared with deep learning architectures [11]. However, RHONNs are typically trained using the Extended Kalman Filter (EKF), whose implementation requires multiple matrix operations and covariance updates. These computations significantly increase the computational burden, limiting their suitability for real-time applications on resource-constrained embedded platforms [12]. To overcome this limitation, this work proposes an alternative training mechanism for the RHONN based on the Super-Twisting Algorithm (STA). The STA preserves the robustness and fast convergence properties characteristic of sliding-mode-based adaptation while avoiding the high computational complexity associated with EKF-based training [13]. Consequently, the proposed approach enables the development of a robust adaptive control framework suitable for real-time implementation in embedded systems with limited computational resources.

Hence, this paper follows this direction by proposing a control architecture in which an RHONN (defined in [14]) is employed as an adaptive compensator for a PD-based backstepping controller, with online weight adaptation driven by a STA (defined in [15]). The proposed scheme aims to combine the robustness and finite-time properties of sliding-mode learning with the approximation capabilities of RHONNs while maintaining low computational complexity and minimal tuning requirements, making it suitable for real-time embedded implementations. For a clearer visualization of the proposed scheme, Figure 1 presents the overall system block diagram, where the complete algorithm used for the motor speed control can be observed.

2. Literature Review

The evolution of robust and intelligent control methodologies has led to the development of hybrid schemes that integrate high-order sliding-mode control with adaptive neural modeling. Among these approaches, the STA, a second-order sliding-mode technique, has attracted significant attention for its ability to suppress chattering while preserving finite-time convergence and robustness in the presence of parameter variations [16]. Nevertheless, the practical effectiveness of the STA control law is strongly influenced by appropriate gain tuning and the availability of reliable system models, requirements that are often difficult to satisfy in real-world applications [17,18]. To mitigate these limitations, recent research has focused on integrating artificial intelligence techniques within the STA control structure. Neural network-assisted Super-Twisting controllers have been successfully applied to DC motors, wind energy conversion systems, and autonomous vehicles, demonstrating enhanced disturbance rejection, reduced time delay, and smoother control actions [18,19]. In such schemes, system uncertainties are adaptively estimated, and dynamic variations are compensated through shallow and radial-basis neural networks, enabling operation under unknown or time-varying parameters. Moreover, the development of Super-Twisting Zeroing Neural Networks (ST-ZNNs) has further bridged sliding-mode control and neural computation, providing finite-time convergence and global robustness in applications such as robotic manipulators [20]. In parallel with these advances, RHONNs have emerged as an effective framework for identifying nonlinear and time-delayed systems [21]. The inclusion of high-order feedback terms and internal dynamic layers in RHONNs enables the representation of complex temporal dependencies, thereby improving approximation accuracy [22]. Online learning and real-time parameter adaptation in these networks have traditionally been achieved using the EKF [23]. Additionally, discrete-time training strategies based on the Super-Twisting differentiator have been shown to allow neural networks to update their weights without explicit derivative computation, yielding faster convergence and improved numerical stability compared to conventional gradient-based methods [24].

The combination of robust sliding-mode laws with adaptive neural learning mechanisms offers a promising pathway toward intelligent control architectures capable of addressing modeling uncertainties and external disturbances. Neural–robust control schemes have been shown to exhibit Lyapunov-stable behavior and effective uncertainty rejection in applications such as leader–follower systems and vehicular dynamics [12,25]. Concurrently, recent findings on super-convergence phenomena in deep learning suggest that rapid and stable adaptation can be achieved under large learning rates, reinforcing the feasibility of neural-based approaches for real-time embedded implementations [26]. Motivated by these considerations, this work proposes and experimentally validates a continuous hybrid control architecture for DC motor drives. The approach integrates a PD-based backstepping controller with an online-trained RHONN acting as an adaptive compensator, whose weights are updated in real time using a Super-Twisting learning law. The resulting cascaded structure aims to enhance trajectory tracking accuracy, disturbance rejection capability, and control smoothness. To the best of the authors’ knowledge, this is the first reported implementation of a continuous hybrid RHONN-based control scheme in which the STA not only replaces but also outperforms the EKF as the learning mechanism in DC motor applications. In the proposed configuration, the STA serves as an online weight adaptation. This substitution enables finite-time convergence and effective noise attenuation without explicit covariance estimation. The complete system is implemented in real time on an embedded microcontroller, demonstrating the feasibility, robustness, and superior performance of the proposed intelligent high-order control strategy under variable load and uncertainty conditions.

3. Mathematical Background

3.1. RHONN Model

As previously mentioned, several control strategies have explored the use of high-order neural networks as alternatives to conventional linear controllers due to their adaptability and relatively low computational cost compared with traditional deep learning techniques. For example, works such as [23,27] report effective control schemes that employ adaptive approaches based on RHONNs to address nonlinear dynamics and uncertainties in electromechanical systems. These studies show that RHONNs can approximate unknown system dynamics, handle situations in which plant parameters are uncertain or partially known, and compensate for external disturbances. Owing to these characteristics, RHONNs have become an attractive alternative for modeling and control of systems exhibiting nonlinear behavior and uncertainties. Additionally, recent studies have highlighted the relevance of advanced learning mechanisms for improving neural network performance in control and estimation tasks. In particular, Kalman filtering-based training methods have shown significant advantages for neural network learning due to their capability to provide efficient online parameter estimation and fast convergence properties [28].

These approaches enable neural networks to adapt their parameters in real time while maintaining robustness against noise and modeling uncertainties, making them particularly suitable for adaptive control applications. However, although the Extended Kalman Filter (EKF) has shown effective performance for neural network training, its implementation can become computationally demanding for systems with a large number of states or parameters. In particular, EKF-based learning requires matrix inversions and covariance updates whose computational complexity typically scales as $\mathcal{O}\left(n^3\right)$, which can limit its practical implementation in real-time applications on low-cost embedded hardware. For this reason, alternative learning mechanisms with reduced computational burden are desirable. In this context, the Super-Twisting Algorithm (STA), a second-order sliding mode technique, emerges as a viable alternative for neural network training. The STA provides finite-time convergence and strong robustness properties while avoiding the intensive matrix operations required by EKF-based approaches. Consequently, STA-based learning schemes are particularly suitable for adaptive control implementations in low-cost hardware platforms and real-time embedded systems. Based on this motivation, the dynamic model of the RHONN architecture is described next. Figure 2 illustrates a general continuous-time RHONN (inspired by the discrete-time formulation presented in [14]), which can be described by the following:

$${\dot{\widehat{x}}}_i\left(t\right)=-a_i{\widehat{x}}_i\left(t\right)+w_i^T{z}_i(\widehat{x}\left(t\right),u\left(t\right)),i=1,\dots ,n$$

(1)

where ${\dot{\widehat{x}}}_i\left(t\right)$ is the time derivative of the i-th neuron’s state, $a_i>0$ is a positive constant representing the neuron’s decay rate, ${\widehat{x}}_i$ denotes the neuron state, $L_i$ is the number of high-order connections, $I_1,I_2,\dots ,I_{L_i}$ is the family of unordered subsets of $1,2,\dots ,n+m$, n is the state dimension, m is the number of external inputs, $w_i$ is the online-updated weight vector, and $u\left(t\right)$ is the external input vector. On the other hand, the vector $z_i(\widehat{x}\left(t\right),u\left(t\right))$ is computed as follows:

$$z_i(\widehat{x}\left(t\right),u\left(t\right))=\left[\begin{array}{c}{z}_{i_1}\\ z_{i_2}\\ ⋮\\ z_{i_{L_i}}\end{array}\right]=\left[\begin{array}{c}\prod _{j\in I_1}{\xi }_{ij}^{d_{ij}\left(1\right)}\\ \prod _{j\in I_2}{\xi }_{ij}^{d_{ij}\left(2\right)}\\ ⋮\\ \prod _{j\in I_{L_i}}{\xi }_{ij}^{d_{ij}\left(L_i\right)}\end{array}\right]$$

(2)

such that $d_{ij}\left(l\right)$ (for $l=1,\dots ,L_i$) are non-negative integers and ${\xi }_i$ is defined as:

$${\xi }_i=\left[\begin{array}{c}{\xi }_{i_1}\\ ⋮\\ {\xi }_{i_n}\\ {\xi }_{i_{n+1}}\\ ⋮\\ {\xi }_{i_{n+m}}\end{array}\right]=\left[\begin{array}{c}S\left({\widehat{x}}_1\right)\\ ⋮\\ S\left({\widehat{x}}_n\right)\\ u_1\\ ⋮\\ u_m\end{array}\right]$$

(3)

In the previous equation, the activation function $S(·)$ used is the hyperbolic tangent function, defined as:

$$S\left(\zeta \right)=tanh\left(\zeta \right)$$

(4)

where $\zeta$ is any real-valued variable.

3.2. Neural Back-Stepping Controller Design

Adaptive control strategies for estimating uncertain parameters and, in some cases, performing system identification were discussed previously. However, before introducing the adaptation mechanism, it is first necessary to establish a control law capable of guaranteeing accurate speed tracking of the DC motor. For this purpose, the backstepping technique is adopted. Backstepping provides a systematic framework for controller design in nonlinear systems and allows stability to be guaranteed through the construction of suitable Lyapunov functions. In addition, it offers several advantages over other nonlinear control strategies reported in the literature, particularly its recursive design procedure and its ability to handle nonlinearities and parametric uncertainties in a structured manner.

Another important feature of backstepping is that it can be naturally combined with other control techniques in order to enhance robustness or improve disturbance–rejection performance. For instance, neural-based robust control strategies have demonstrated effective performance in nonlinear and uncertain systems such as leader–follower mobile-agent systems [25]. Similarly, the integration of nonlinear control with neural and sliding-mode techniques has shown promising results in power electronic systems for electric vehicle applications [29]. Within this framework, the class of nonlinear systems considered in this work can be represented as follows. Consider the following nonlinear, perturbed, and uncertain plant:

$$\begin{array}{cc}\hfill {\dot{x}}_i\left(t\right)& =f^i\left(x_i\left(t\right)\right)+g^i\left(x_i\left(t\right)\right)x_{i+1}\left(t\right)+d^i\left(t\right),\forall i=1,\dots ,r-1\hfill \\ \hfill {\dot{x}}_r\left(t\right)& =f^r\left(x_r\left(t\right)\right)+g^r\left(x_r\left(t\right)\right)u\left(t\right)+d^r\left(t\right)\hfill \\ \hfill y\left(t\right)& =x_1\left(t\right)\hfill \end{array}$$

(5)

In the strict-feedback structure described in (5), the control input $u\left(t\right)$ influences the output $y\left(t\right)=x_1\left(t\right)$ only after propagating through the internal state dynamics $x_2\left(t\right),\dots ,x_r\left(t\right)$. Consequently, the relative degree of the system is r. Because the input $u\left(t\right)$ does not explicitly appear in ${\dot{x}}_1\left(t\right)$, a direct control design based solely on ${\dot{x}}_1\left(t\right)$ is not possible. Therefore, the output must be differentiated until the control input becomes explicitly visible in the system dynamics. Thus, the first derivative of the output is given by:

$$\dot{y}\left(t\right)={\dot{x}}_1\left(t\right)=f^1\left(x_1\left(t\right)\right)+g^1(x_1,t)x_2\left(t\right)+d^1\left(t\right)$$

(6)

Since $u\left(t\right)$ does not appear in (6), the differentiation process must be continued. By repeatedly applying the chain rule and substituting the system dynamics from (5), it follows that the control input appears only in the r-th derivative of the output:

$$y^{\left(r\right)}\left(t\right)=F\left(x\left(t\right)\right)+G\left(x\left(t\right)\right)u\left(t\right)+\tilde{d}\left(t\right)$$

(7)

where $x\left(t\right)={[x_1\left(t\right),\dots ,x_r\left(t\right)]}^T$ is the state vector. The function $F\left(x\right(t\left)\right)$ represents the nonlinear drift dynamics involving the functions $f^i(·)$, while $\tilde{d}\left(t\right)$ groups disturbances and their derivatives. The term $G\left(x\right(t\left)\right)$ corresponds to the decoupling gain, defined as:

$$G\left(x\left(t\right)\right)=\left(\prod _{i=1}^{r-1}{g}^i\left(x_i\left(t\right)\right)\right)g^r\left(x_r\left(t\right)\right)$$

(8)

The input–output model in (7) explicitly reveals how the control signal affects the output dynamics. The control objective is therefore to design $u\left(t\right)$ such that the system output $y\left(t\right)$ accurately tracks a desired trajectory $y_d\left(t\right)$. To implement the backstepping design, a recursive set of tracking errors is introduced. The first error variable corresponds to the output tracking error:

$$e_1\left(t\right)=y\left(t\right)-y_d\left(t\right)=x_1\left(t\right)-y_d\left(t\right)$$

(9)

Subsequently, a sequence of virtual error variables is defined. Each error $e_i\left(t\right)$ measures the difference between the state $x_i\left(t\right)$ and a stabilizing virtual control input $u_{i-1}^v\left(t\right)$:

$$\begin{array}{cc}\hfill e_2\left(t\right)& =x_2\left(t\right)-u_1^v\left(t\right)\hfill \\ \hfill e_3\left(t\right)& =x_3\left(t\right)-u_2^v\left(t\right)\hfill \\ \hfill & ⋮\hfill \\ \hfill e_r\left(t\right)& =x_r\left(t\right)-u_{r-1}^v\left(t\right)\hfill \end{array}$$

(10)

to illustrate the recursive design procedure, the dynamics of $e_1\left(t\right)$ are computed as:

$${\dot{e}}_1\left(t\right)={\dot{x}}_1\left(t\right)-{\dot{y}}_d\left(t\right)=f^1(x_1,t)+g^1(x_1,t)x_2\left(t\right)+d^1\left(t\right)-{\dot{y}}_d\left(t\right)$$

(11)

by introducing the relation $x_2=e_2+{\alpha }_1$, the expression becomes:

$${\dot{e}}_1=f^1(x_1,t)+g^1(x_1,t)(e_2+{\alpha }_1)+d^1\left(t\right)-{\dot{y}}_d\left(t\right)$$

(12)

Hence, the virtual control $u_1^v\left(t\right)$ is then designed to cancel the nonlinear terms and impose the desired error dynamics ${\dot{e}}_1\left(t\right)=-c_1{e}_1\left(t\right)$ with $c_1>0$. This yields the ideal virtual control

$$u_1^{v*}\left(t\right)={\displaystyle \frac{1}{g^1(x_1,t)}}\left(-c_1{e}_1\left(t\right)-f^1(x_1,t)-d^1\left(t\right)+{\dot{y}}_d\left(t\right)\right)$$

(13)

The same procedure is recursively applied for $e_2\left(t\right),\dots ,e_r\left(t\right)$. At each step i, a virtual control $u_i^{v*}\left(t\right)$ is designed to stabilize the subsystem $(e_1,\dots ,e_i)$. The final step yields the ideal control input $u\left(t\right)$. However, these ideal control laws can only be implemented if the exact system model $f^i(·)$, $g^i(·)$ and the disturbances $d^i\left(t\right)$ are perfectly known. In practical applications, these quantities are typically unknown or affected by uncertainties [30]. To overcome this limitation, the unknown nonlinear functions involved in the ideal control laws are approximated using Neural Networks (NNs). In particular, the ideal controls $u_i^{v*}\left(t\right)$ and $u^{*}\left(t\right)$ can be interpreted as unknown smooth nonlinear functions ${\varphi }_j\left(t\right)$ of measurable states (e.g., $x_i\left(t\right)$, $y_d\left(t\right)$, ${\dot{y}}_d\left(t\right)$, etc.). These functions can therefore be represented using universal approximators such as Recurrent High-Order Neural Networks (RHONNs), leading to the following representation:

$$\begin{array}{cc}\hfill u_i^{*}\left(X_i\right)& =W_i^{*T}{Z}_i\left(X_i\right)+{ϵ}_i,i=1,\dots ,r-1\hfill \\ \hfill u^{*}\left(X_r\right)& =W_r^{*T}{Z}_r\left(X_r\right)+{ϵ}_r\hfill \end{array}$$

(14)

where $W_j^{*}$ denotes the ideal (unknown) constant weight vector, $Z_j\left(X_j\right)$ represents the vector of neuron activation functions (regressors), and $X_j$ is the input vector to the neural network (e.g., $X_1={[x_1,y_d,{\dot{y}}_d]}^T$, $X_2={[x_1,x_2,y_d,{\dot{y}}_d,{\ddot{y}}_d]}^T$, etc.). The term ${ϵ}_j$ denotes the minimal function approximation error. Since the ideal weights $W_j^{*}$ are unknown, the control laws are implemented using their online estimates ${\widehat{W}}_j\left(t\right)$, which leads to the following neural-based control structure:

$$\begin{array}{cc}\hfill u_i^v\left(t\right)& ={\widehat{W}}_i^T\left(t\right)Z_i\left(X_i\left(t\right)\right),i=1,\dots ,r-1\hfill \\ \hfill u\left(t\right)& ={\widehat{W}}_r^T\left(t\right)Z_r\left(X_r\left(t\right)\right)\hfill \end{array}$$

(15)

3.3. Super-Twisting Learning Algorithm

Traditionally, neural networks used in adaptive control are trained online using gradient-based learning rules or optimization techniques such as the Extended Kalman Filter. Although these approaches have been successfully applied in many applications, they typically guarantee only asymptotic convergence and may suffer from sensitivity to disturbances, modeling uncertainties, and unmodeled dynamics. To address these limitations, the neural network in this work is trained online using a robust adaptation law derived from the Super-Twisting Algorithm (STA). By incorporating second-order sliding-mode principles into the learning process, the proposed STA-based scheme provides practical finite-time convergence of the identification error while enhancing robustness against disturbances and uncertainties.

For each neural network channel i, the objective is to adjust the weight vector ${\widehat{W}}_i\in {\mathbb{R}}^{L_i}$ so that the corresponding identification error is driven to a small neighborhood of zero. To this end, the sliding variable is defined as the instantaneous identification error:

$$s_i\left(t\right)\triangleq e_i\left(t\right)$$

(16)

Based on this definition, the learning law incorporates a corrective term ${\nu }_{st,i}\left(t\right)$ constructed from second-order sliding-mode principles. Its dynamics are given by:

$$\begin{array}{cc}\hfill {\nu }_{st,i}\left(t\right)& =-k_{1,i}{\left|s_i\left(t\right)\right|}^{1/2}\mathrm{sign}\left(s_i\left(t\right)\right)+v_i\left(t\right)\hfill \\ \hfill {\dot{v}}_i\left(t\right)& =-k_{2,i}\mathrm{sign}\left(s_i\left(t\right)\right)\hfill \end{array}$$

(17)

where $k_{1,i}$ and $k_{2,i}$ are positive design gains selected to ensure stable error convergence. In this framework, ${\nu }_{st,i}\left(t\right)$ acts as a robust injection term that corrects the adaptation process and drives the system trajectories toward the sliding manifold associated with the identification error.

3.3.1. Weight Update and Implementation

To prevent parameter drift and maintain bounded adaptation under persistent perturbations, the weight update law combines the STA injection term with a leakage factor, also known as $\sigma$-modification. The resulting continuous-time adaptation law is:

$${\dot{\widehat{W}}}_i\left(t\right)={\mathsf{\Gamma }}_i{Z}_i\left(\chi \left(t\right)\right){\nu }_{st,i}\left(t\right)-{\sigma }_i{\widehat{W}}_i\left(t\right)$$

(18)

where ${\mathsf{\Gamma }}_i={\mathsf{\Gamma }}_i^{\top }>0\in {\mathbb{R}}^{L_i\times L_i}$ is the learning gain matrix that determines the adaptation rate, $Z_i\left(\chi \left(t\right)\right)$ is the regressor vector, and ${\sigma }_i>0$ is a small leakage coefficient introduced to improve robustness and avoid unbounded growth of the estimated parameters.

Remark 1.

If explicit bounds on the weights are required, the leakage term $-{\sigma }_i{\widehat{W}}_i\left(t\right)$ in (18) can be replaced by a projection operator $\mathrm{Proj}({\widehat{W}}_i,·)$ in order to enforce constrained parameter estimates.

For embedded implementation, the continuous-time dynamics in (17) and (18) are discretized using the Euler forward method with sampling period $T_s$. This allows the Super-Twisting auxiliary variable and the neural weights to be updated iteratively at each control instant, making the algorithm suitable for real-time microcontroller-based implementation.

3.3.2. RHONN Predictor and Identification Error Dynamics

To formalize the learning dynamics, let $x\left(t\right)\in {\mathbb{R}}^n$ denote the plant state vector to be identified, either measured or estimated, and let $\widehat{x}\left(t\right)\in {\mathbb{R}}^n$ be its RHONN-based prediction. For each channel $i\in \{1,\dots ,n\}$, the RHONN predictor is written as:

$${\dot{\widehat{x}}}_i\left(t\right)=-a_i{\widehat{x}}_i\left(t\right)+{\widehat{W}}_i^{\top }\left(t\right)Z_i\left(\chi \left(t\right)\right)$$

(19)

where $a_i>0$, ${\widehat{W}}_i\left(t\right)\in {\mathbb{R}}^{L_i}$ is the adaptive weight vector, and $Z_i\left(\chi \left(t\right)\right)\in {\mathbb{R}}^{L_i}$ is the regressor built from knowable signals $\chi \left(t\right)$; for example, $\chi ={[{\widehat{x}}^{\top },u^{\top }]}^{\top }$. The corresponding identification error is defined as:

$$e_i\left(t\right)\triangleq x_i\left(t\right)-{\widehat{x}}_i\left(t\right),e\left(t\right)={[e_1,\dots ,e_n]}^{\top }$$

(20)

The following assumption establishes a standard decomposition of the plant dynamics for each identification channel.

Assumption 1.

For each channel, the plant dynamics can be expressed in the standard “linear-in-parameters + residual” form.

$${\dot{x}}_i\left(t\right)=f_i\left(\chi \left(t\right)\right)+W_i^{*\top }{Z}_i\left(\chi \left(t\right)\right)+{\Delta }_i\left(t\right)$$

(21)

where $f_i\left(\chi \left(t\right)\right)$ is an unknown smooth drift term and ${\Delta }_i\left(t\right)$ groups approximation residuals, external disturbances, and unmodeled dynamics. This representation is less restrictive than assuming a fully known plant structure while still being suitable for robust adaptive identification. Thus, subtracting the predictor model (19) from the plant representation (21) gives the following error dynamics:

$${\dot{e}}_i\left(t\right)=f_i\left(\chi \left(t\right)\right)+a_i{\widehat{x}}_i\left(t\right)-a_i{x}_i\left(t\right)+{\tilde{W}}_i^{\top }\left(t\right)Z_i\left(\chi \left(t\right)\right)+{\Delta }_i\left(t\right)$$

(22)

using the relation $e_i=x_i-{\widehat{x}}_i$, this expression can be rewritten as:

$${\dot{e}}_i\left(t\right)=-a_i{e}_i\left(t\right)+{\tilde{W}}_i^{\top }\left(t\right)Z_i\left(\chi \left(t\right)\right)+\underset{\triangleq {\overline{\Delta }}_i\left(t\right)}{\underbrace{\left(f_i\left(\chi \left(t\right)\right)+{\Delta }_i\left(t\right)\right)}}$$

(23)

where ${\tilde{W}}_i\left(t\right)\triangleq W_i^{*}-{\widehat{W}}_i\left(t\right)$ is the weight estimation error and ${\overline{\Delta }}_i\left(t\right)$ represents the lumped residual perturbation affecting the i-th identification channel. To analyze the convergence properties of the proposed learning scheme and establish the stability of the identification error dynamics, it is necessary to introduce a set of standard assumptions regarding the regressor signals and the residual perturbations. These assumptions are commonly adopted in high-order sliding-mode learning and adaptive identification analysis.

Assumption 2.

The regressor is bounded: $\parallel Z_i\left(\chi \left(t\right)\right)\parallel \le {\overline{Z}}_i$ for all $t\ge 0$.

Assumption 3.

The total residual perturbation ${\overline{\Delta }}_i\left(t\right)$ is bounded and has a bounded derivative:

$$|{\overline{\Delta }}_i\left(t\right)|\le {\overline{\Delta }}_i,|{\dot{\overline{\Delta }}}_i\left(t\right)|\le {\overline{d}}_i,\forall t\ge 0$$

(24)

Remark 2.

Assumption 3 is consistent with STA theory since boundedness of ${\dot{\overline{\Delta }}}_i\left(t\right)$ is required to establish practical finite-time convergence.

3.4. Finite-Time Property of the STA Subsystem

Lemma 1

(Practical finite-time convergence of the STA subsystem). Consider the perturbed super-twisting subsystem:

$$\begin{array}{cc}\hfill {\dot{s}}_i\left(t\right)& =-k_{1,i}{\left|s_i\left(t\right)\right|}^{1/2}\mathrm{sign}\left(s_i\left(t\right)\right)+v_i\left(t\right)+{\Delta }_i\left(t\right)\hfill \\ \hfill {\dot{v}}_i\left(t\right)& =-k_{2,i}\mathrm{sign}\left(s_i\left(t\right)\right)\hfill \end{array}$$

(25)

where the perturbation satisfies, for all $t\ge 0$,

$$|{\Delta }_i\left(t\right)|\le {\overline{\Delta }}_i,|{\dot{\Delta }}_i\left(t\right)|\le {\overline{d}}_i$$

(26)

Then, there exist gains $k_{1,i},k_{2,i}>0$ such that the trajectory $(s_i\left(t\right),v_i\left(t\right))$ reaches in finite time a compact set:

$${\mathcal{S}}_i\triangleq \left\{(s_i,v_i):|s_i|\le {\rho }_i,\left|v_i\right|\le {\eta }_i\right\}$$

with ${\rho }_i,{\eta }_i>0$ depending on $({\overline{\Delta }}_i,{\overline{d}}_i)$ and on $(k_{1,i},k_{2,i})$. In particular, $s_i\left(t\right)$ reaches in finite time the neighborhood:

$${\mathsf{\Omega }}_i\triangleq \{s_i:|s_i|\le {\rho }_i\}$$

Moreover, in the ideal case ${\Delta }_i\left(t\right)\equiv 0$, the origin $(s_i,v_i)=(0,0)$ is reached in finite time.

Proof. 

Fix i and omit the subscript to simplify notation. Define the auxiliary variable:

$$x_1\triangleq s,x_2\triangleq v+\Delta \left(t\right)$$

using (25), the dynamics become:

$$\begin{array}{cc}\hfill {\dot{x}}_1& =x_2-k_1{\left|x_1\right|}^{1/2}\mathrm{sign}\left(x_1\right)\hfill \\ \hfill {\dot{x}}_2& =-k_2\mathrm{sign}\left(x_1\right)+\dot{\Delta }\left(t\right)\hfill \end{array}$$

(27)

by (26), the additive term $\dot{\Delta }\left(t\right)$ is bounded by $\overline{d}$.

• Step 1: Unperturbed finite-time convergence. Consider first the ideal case $\dot{\Delta }\left(t\right)\equiv 0$. The system (27) is homogeneous with respect to the dilation:

  $$(x_1,x_2)\mapsto ({\lambda }^2{x}_1,\lambda x_2)$$

  of negative degree. Therefore, the origin is finite-time stable. In particular, there exists a continuous, positive definite Lyapunov function $V(x_1,x_2)$ and constants $c_1,c_2>0$ such that, along solutions of (27) with $\dot{\Delta }\equiv 0$:

  $$\dot{V}\le -c_1{V}^{1/2}$$

  (28)

  which implies finite-time convergence of $(x_1,x_2)$ to the origin; hence, $(s,v)\to (0,0)$ in finite time.
• Step 2: Practical finite-time convergence under bounded $\dot{\Delta }$. When $\dot{\Delta }\left(t\right)$ is nonzero but bounded, the same Lyapunov function yields an estimate of the following form:

  $$\dot{V}\le -c_1{V}^{1/2}+c_2\left|\dot{\Delta }\left(t\right)\right|\le -c_1{V}^{1/2}+c_2\overline{d}$$

  (29)
  Hence, for all $V>{\left({\displaystyle \frac{c_2}{c_1}}\overline{d}\right)}^2$, we have $\dot{V}<0$. Therefore, all trajectories enter the compact set in finite time:

  $$\mathcal{V}\triangleq \left\{(x_1,x_2):V(x_1,x_2)\le {\left({\displaystyle \frac{c_2}{c_1}}\overline{d}\right)}^2\right\}$$
  Since V is positive definite and radially unbounded, there exist $\rho ,\eta >0$ (depending on $\overline{d},k_1,k_2$) such that:

  $$\mathcal{V}\subseteq \{(x_1,x_2):|x_1|\le \rho ,|x_2|\le \eta \}$$
  Returning to original variables, $\left|s\right|=|x_1|$ and $\left|v\right|\le |x_2|+|\Delta |\le \eta +\overline{\Delta }$, which proves that $(s,v)$ reaches in finite time a compact set of the claimed form.
• Step 3: Gain selection. The constants $c_1,c_2$ depend on $k_1,k_2$. Increasing $k_1,k_2$ increases the decay rate and reduces the steady neighborhood size; thus, one can choose $k_1,k_2$ such that the resulting $\rho ,\eta$ meet a prescribed practical bound.

□

3.5. Boundedness and Practical Convergence of the Learning Loop

Proposition 1

(Bounded weights and practical convergence of the identification error). Under Assumption 2 and Assumption 3, the learning dynamics given by (17), (18), and (23) satisfy:

1.

The adaptive weights ${\widehat{W}}_i\left(t\right)$ are uniformly bounded for all $t\ge 0$.

2.

The identification error $e_i\left(t\right)$ is ultimately bounded and reaches a compact set

$${\mathsf{\Omega }}_i\triangleq \{e_i:|e_i|\le {\rho }_i\}$$

(30)

where ${\rho }_i$ depends on $({\overline{\Delta }}_i,{\overline{d}}_i)$, the gains $(k_{1,i},k_{2,i})$, and ${\sigma }_i$.

Proof. 

Let $s_i=e_i$. From (23),

$${\dot{s}}_i=-a_i{s}_i+{\tilde{W}}_i^{\top }{Z}_i\left(\chi \right)+{\overline{\Delta }}_i\left(t\right)$$

(31)

Consider the Lyapunov function candidate

$$V_i\left(t\right)={\displaystyle \frac{1}{2}}{\tilde{W}}_i^{\top }{\mathsf{\Gamma }}_i^{-1}{\tilde{W}}_i+c_i\left|s_i\right|$$

(32)

where $c_i\in {\mathbb{R}}_{+}$ to satisfy the conditions of the Lyapunov candidate function. Using ${\dot{\tilde{W}}}_i=-{\dot{\widehat{W}}}_i$ and (18),

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{1}{2}}{\tilde{W}}_i^{\top }{\mathsf{\Gamma }}_i^{-1}{\tilde{W}}_i\right)=-{\tilde{W}}_i^{\top }{Z}_i\left(\chi \right){\nu }_{st,i}\left(t\right)+{\sigma }_i{\tilde{W}}_i^{\top }{\mathsf{\Gamma }}_i^{-1}{\widehat{W}}_i$$

(33)

moreover, for $s_i\ne 0$, ${\displaystyle \frac{d}{dt}}\left|s_i\right|=\mathrm{sign}\left(s_i\right){\dot{s}}_i$. Therefore,

$${\dot{V}}_i=-{\tilde{W}}_i^{\top }{Z}_i{\nu }_{st,i}+c_i\mathrm{sign}\left(s_i\right)\left(-a_i{s}_i+{\tilde{W}}_i^{\top }{Z}_i+{\overline{\Delta }}_i\right)+{\sigma }_i{\tilde{W}}_i^{\top }{\mathsf{\Gamma }}_i^{-1}{\widehat{W}}_i$$

(34)

The leakage cross-term is bounded via Cauchy–Schwarz and Young’s inequality as ${\sigma }_i{\tilde{W}}_i^{\top }{\mathsf{\Gamma }}_i^{-1}{\widehat{W}}_i\le {\displaystyle \frac{{\sigma }_i}{2}}\parallel {\mathsf{\Gamma }}_i^{-1/2}{\tilde{W}}_i{\parallel }^2+{\displaystyle \frac{{\sigma }_i}{2}}{\parallel {\mathsf{\Gamma }}_i^{-1/2}{\widehat{W}}_i\parallel }^2$, which prevents weight drift and yields uniform ultimate boundedness. Finally, the lemma provides a practical finite-time attenuation mechanism for the STA subsystem, which, together with the leakage term, implies boundedness of ${\widehat{W}}_i\left(t\right)$ and the convergence of $s_i\left(t\right)$ to ${\mathsf{\Omega }}_i$. □

Remark 3.

The result is practical: exact convergence $e_i\left(t\right)\to 0$ can only be claimed in the ideal case ${\overline{\Delta }}_i\left(t\right)\equiv 0$ with negligible approximation residuals.

3.6. Discrete-Time Implementation (Embedded Real-Time)

For real-time implementation with sampling period $T_s$, (17) and (18) are discretized using the forward Euler:

$$\begin{array}{cc}\hfill v_i[k+1]& =v_i\left[k\right]-T_s{k}_{2,i}\psi \left(s_i\left[k\right]\right)\hfill \\ \hfill {\nu }_{st,i}\left[k\right]& =-k_{1,i}{\left|s_i\left[k\right]\right|}^{1/2}\psi \left(s_i\left[k\right]\right)+v_i\left[k\right]\hfill \\ \hfill {\widehat{W}}_i[k+1]& =(1-{\sigma }_i{T}_s){\widehat{W}}_i\left[k\right]+T_s{\mathsf{\Gamma }}_i{Z}_i\left(\chi \left[k\right]\right){\nu }_{st,i}\left[k\right]\hfill \end{array}$$

(35)

where $\psi (·)$ is a smooth sign substitute. A typical choice is $\psi \left(s\right)=\mathrm{sat}(s/\varphi )$ with $\varphi >0$, satisfying $\left|\psi \right(s\left)\right|\le 1$ and $\psi (-s)=-\psi \left(s\right)$.

Remark 4.

The use of $\psi (·)$ is recommended to mitigate quantization effects and measurement noise, improving robustness in fixed-point embedded implementations.

3.7. Standing Assumptions (Outer-Loop Analysis)

Before presenting the stability analysis of the outer-loop controller, a set of standard standing assumptions is introduced. These assumptions establish the regularity conditions of the strict-feedback system, the smoothness of the reference trajectory, and the boundedness properties required for the neural learning layer. Under these conditions, the closed-loop tracking error dynamics can be properly analyzed using Lyapunov arguments.

Assumption 4

(Strict-feedback regularity and non-singularity). Consider the strict-feedback plant described by Equation (5). The functions $f^i(·)$ and $g^i(·)$ exhibit local behavior The Lipschitz condition is satisfied on an open domain $\mathcal{D}\subset {\mathbb{R}}^r$, which encompasses the closed-loop trajectories. Furthermore, there are established constants ${\underline{g}}_i,{\overline{g}}_i>0$ such that:

$$0<{\underline{g}}_i\le \left|g^i\left(x_i\right)\right|\le {\overline{g}}_i,\forall x\in \mathcal{D},i=1,\dots ,r$$

Assumption 5

(Bounded reference and derivatives). The reference $y_d\left(t\right)$ is r times differentiable and $y_d^{\left(k\right)}\left(t\right)$ is bounded for $k=0,1,\dots ,r$.

Assumption 6

(Learning-layer finite-time bounded residual (used, not re-proved)). Let the RHONN approximators generate ${\widehat{u}}_i^{*}\left(X_i\right)={\widehat{W}}_i^{\top }\left(t\right)Z_i\left(X_i\right)$ as estimates of the ideal (backstepping) functions $u_i^{*}\left(X_i\right)=W_i^{*\top }{Z}_i\left(X_i\right)+{ϵ}_i\left(X_i\right)$, $i=1,\dots ,r$. Assume the previously analyzed ST learning law guarantees the existence of a finite time $T_L>0$ and known constants ${\overline{\vartheta }}_i>0$ such that, for all $t\ge T_L$:

$$|{\vartheta }_i\left(t\right)|\le {\overline{\vartheta }}_i,i=1,\dots ,r$$

where ${\vartheta }_i\left(t\right)$ aggregates NN approximation errors ${ϵ}_i(·)$, disturbances/unmodeled dynamics, and the residual identification/weight mismatch after the learning transient.

Remark 5.

Assumption 6 is the only point where ST finite-time (practical) learning is used in the outer-loop analysis. No additional learning proof is required in this section.

3.8. Outer-Loop Controller and Error Dynamics

The backstepping error coordinates are defined as follows (as in (9) and (10)):

$$e_1\left(t\right)=x_1\left(t\right)-y_d,\left(t\right)e_i\left(t\right)=x_i\left(t\right)-u_{i-1}^v\left(t\right),i=2,\dots ,r$$

where $u_{i-1}^v$ denotes the implemented virtual control. The practical neural backstepping laws are selected as

$$u_i^v\left(t\right)={\widehat{u}}_i^{*}\left(X_i\left(t\right)\right)-c_i{e}_i\left(t\right),i=1,\dots ,r-1$$

(36)

and the actual input as

$$u\left(t\right)={\widehat{u}}_r^{*}\left(X_r\left(t\right)\right)-c_r{e}_r\left(t\right)$$

(37)

with $c_i>0$. Based on standard backstepping algebra (nominal couplings canceled by design), the tracking-error dynamics admit the nearest-neighbor triangular form

$$\begin{array}{cc}\hfill {\dot{e}}_1\left(t\right)& =-c_1{e}_1\left(t\right)+{\beta }_1\left(t\right)e_2+{\vartheta }_1\left(t\right)\hfill \\ \hfill {\dot{e}}_i\left(t\right)& =-c_i{e}_i\left(t\right)+{\beta }_i\left(t\right)e_{i+1}-{\beta }_{i-1}\left(t\right)e_{i-1}+{\vartheta }_i\left(t\right),i=2,\dots ,r-1\hfill \\ \hfill {\dot{e}}_r\left(t\right)& =-c_r{e}_r\left(t\right)-{\beta }_{r-1}\left(t\right)e_{r-1}+{\vartheta }_r\left(t\right)\hfill \end{array}$$

(38)

where ${\beta }_i\left(t\right)$ is induced by strict-feedback gains and coordinate transformations. Based on Assumption 4, there exist known constants ${\overline{\beta }}_i>0$ such that

$$|{\beta }_i\left(t\right)|\le {\overline{\beta }}_i,\forall t\ge 0,i=1,\dots ,r-1$$

(39)

Remark 6.

The learning layer affects (38) only through the additive residual $\vartheta \left(t\right)$, which is uniformly bounded for $t\ge T_L$ by Assumption 6.

3.8.1. Auxiliary Coercivity Lemma

Lemma 2

(Uniform positive definiteness of the backstepping interconnection form). Let $Q\left(t\right)\in {\mathbb{R}}^{r\times r}$ be the symmetric tridiagonal matrix with diagonal entries $c_i$ and off-diagonal entries $-{\beta }_i\left(t\right)/2$, $i=1,\dots ,r-1$. If there exists $\delta >0$ such that

$$c_1\ge {\displaystyle \frac{1}{2}}{\overline{\beta }}_1+\delta ,c_i\ge {\displaystyle \frac{1}{2}}({\overline{\beta }}_{i-1}+{\overline{\beta }}_i)+\delta ,i=2,\dots ,r-1,c_r\ge {\displaystyle \frac{1}{2}}{\overline{\beta }}_{r-1}+\delta$$

(40)

then $Q\left(t\right)⪰\delta I$ uniformly for all $t\ge 0$.

Proof. 

For each row i, the sum of absolute off-diagonal terms is bounded by ${\displaystyle \frac{1}{2}}({\overline{\beta }}_{i-1}+{\overline{\beta }}_i)$ (with ${\overline{\beta }}_0={\overline{\beta }}_r=0$). Condition (40) implies strict diagonal dominance of $Q\left(t\right)-\delta I$. Since $Q\left(t\right)$ is symmetric, Gershgorin’s theorem yields ${\lambda }_{min}\left(Q\left(t\right)\right)\ge \delta$ uniformly in t. □

3.8.2. ISS/UUB of the Tracking Errors

Theorem 1

(Main theoretical result: ISS and UUB of the outer-loop tracking errors). Suppose Assumptions 4–6 hold. Choose $c_i>0$ such that (40) holds for some $\delta >0$. Define $\overline{\vartheta }\triangleq \sqrt{{\sum }_{i=1}^r{\overline{\vartheta }}_i^2}$. Then, for all $t\ge T_L$, the solution of (38) satisfies the ISS estimate

$$\parallel e\left(t\right)\parallel \le e^{-\delta (t-T_L)}\parallel e\left(T_L\right)\parallel +\left(1-e^{-\delta (t-T_L)}\right){\displaystyle \frac{\overline{\vartheta }}{\delta }}$$

(41)

Consequently, $e\left(t\right)$ is uniformly ultimately bounded and enters the ball

$${\mathsf{\Omega }}_e\triangleq \left\{e\in {\mathbb{R}}^r:\parallel e\parallel \le {\displaystyle \frac{\overline{\vartheta }}{\delta }}\right\}$$

in finite time. In the ideal case $\vartheta \left(t\right)\equiv 0$, the origin is exponentially stable and $e\left(t\right)\to 0$ exponentially.

Proof. 

Consider $V\left(e\right)={\displaystyle \frac{1}{2}}{e}^{\top }e$. For $t\ge T_L$, differentiating V along (38) gives

$$\dot{V}=-\sum _{i=1}^r{c}_i{e}_i^2+\sum _{i=1}^{r-1}{\beta }_i\left(t\right)e_i{e}_{i+1}+e^{\top }\vartheta \left(t\right)$$

Introduce

$$Q\left(t\right)=\left[\begin{array}{ccccc}{c}_1& -{\beta }_1\left(t\right)/2& 0& \dots & 0\\ -{\beta }_1\left(t\right)/2& c_2& -{\beta }_2\left(t\right)/2& \ddots & ⋮\\ 0& -{\beta }_2\left(t\right)/2& \ddots & \ddots & 0\\ ⋮& \ddots & \ddots & c_{r-1}& -{\beta }_{r-1}\left(t\right)/2\\ 0& \dots & 0& -{\beta }_{r-1}\left(t\right)/2& c_r\end{array}\right]$$

so that

$$-\sum _{i=1}^r{c}_i{e}_i^2+\sum _{i=1}^{r-1}{\beta }_i\left(t\right)e_i{e}_{i+1}=-e^{\top }Q\left(t\right)e$$

Hence,

$$\dot{V}\le -e^{\top }Q\left(t\right)e+\parallel e\parallel \parallel \vartheta \left(t\right)\parallel$$

Based on Lemma 2, $Q\left(t\right)⪰\delta I$, so $e^{\top }Q\left(t\right)e\ge \delta {\parallel e\parallel }^2=2\delta V$. Moreover, based on Assumption 6, $\parallel \vartheta \left(t\right)\parallel \le \overline{\vartheta }$ for $t\ge T_L$. Thus,

$$\dot{V}\le -2\delta V+\sqrt{2V}\overline{\vartheta }$$

let $w\left(t\right)\triangleq \sqrt{2V\left(t\right)}=\parallel e\left(t\right)\parallel$. For $w\left(t\right)>0$,

$$\dot{w}={\displaystyle \frac{\dot{V}}{w}}\le -\delta w+\overline{\vartheta },t\ge T_L$$

Solving the scalar comparison inequality yields (41). The UUB claim follows by letting $t\to \infty$. If $\overline{\vartheta }=0$, then $\dot{w}\le -\delta w$, and exponential convergence follows. □

Remark 7

(Strict Two-Layer Interpretation). Following the finite transient period $T_L$, the ST learning loop ensures that the residual $\vartheta \left(t\right)$ remains uniformly bounded. The outer-loop backstepping error dynamics exhibit Input-to-State Stability (ISS) with respect to this residual, yielding a tunable ultimate bound of $\parallel e\parallel \le \overline{\vartheta }/\delta$. Consequently, a reduced learning mismatch (smaller $\overline{\vartheta }$) and/or an increased stability margin δ (achieved through larger $c_i$) results in a more constrained tracking neighborhood.

To provide a clearer graphical representation of the proposed control strategy, the flowchart shown in Figure 3 illustrates the complete algorithm employed for the DC motor speed regulation. This diagram summarizes the sequence of operations required for the implementation of the control scheme, including the computation of the control input, the neural adaptation mechanism, and the update of the system states.

The flowchart is constructed based on the theoretical developments presented throughout this section, particularly the stability analysis derived from the Lyapunov framework. In this sense, each step of the algorithm directly follows from the mathematical formulation and guarantees that the adaptive learning law and control input preserve the stability properties demonstrated in the Lyapunov-based proof. Therefore, the diagram provides an intuitive and structured overview of the implementation procedure, allowing readers to easily translate the theoretical results into a practical algorithm suitable for real-time applications. Furthermore, for reproducibility and implementation purposes, readers are encouraged to consult the Data Availability Statement section at the end of this article, where a link to the resources and code used in this work is provided.

3.9. Model of Permanent Magnet Direct Current Motor

To validate our RHONN backstepping technique, we apply it to a DC drive. To derive the dynamic model from the equivalent circuit illustrated in Figure 4, Kirchhoff’s voltage law is utilized for the electrical subsystem, while the mechanical dynamics are established through the application of Newton’s law of motion. The resulting continuous-time state-space representation is given in (42), where descriptions of each parameter are summarized in

Table 1. Parameters of the DC drive model.

Symbol	Description	Unit
R	Armature resistance	$\mathsf{\Omega }$
L	Armature inductance	H
$K_t$	Torque constant	N·m/A
$K_b$	Back EMF constant	V·s/rad
J	Rotor inertia	kg·m2
B	Viscous friction coefficient	N·m·s/rad
$i\left(t\right)$	Armature current	A
$\omega \left(t\right)$	Angular speed	rad/s

. This formulation is consistent with commonly adopted DC motor models reported in the literature, such as the one presented in [31,32].

$$\left[\begin{array}{c}\dot{i}\left(t\right)\\ \dot{\omega }\left(t\right)\end{array}\right]=\left[\begin{array}{cc}-{\displaystyle \frac{R}{L}}& -{\displaystyle \frac{K_e}{L}}\\ {\displaystyle \frac{K_t}{J}}& -{\displaystyle \frac{B}{J}}\end{array}\right]\left[\begin{array}{c}i\left(t\right)\\ \omega \left(t\right)\end{array}\right]+\left[\begin{array}{c}{\displaystyle \frac{1}{L}}\\ 0\end{array}\right]u\left(t\right)$$

(42)

3.10. DC Drive Controller Design

The state-space defined in (42) must be rewritten to a set of differential equations, considering unknown perturbations $d^i\left(t\right)$ (such as parametric disturbances, torque inputs, etc.) to apply the controller shown in Section 3.2:

$$\dot{\omega }={\displaystyle \frac{K_t}{J}}i\left(t\right)-{\displaystyle \frac{B}{J}}\omega \left(t\right)+d^1\left(t\right)$$

(43)

$$\dot{i}=-{\displaystyle \frac{R}{L}}i\left(t\right)-{\displaystyle \frac{K_e}{L}}\omega \left(t\right)+{\displaystyle \frac{1}{L}}u\left(t\right)+d^2\left(t\right)$$

(44)

all terms are rearranged into packing functions of the following states:

$$\dot{\omega }=f_1(i,\omega ,d^1,t)+b_{1}i$$

(45)

$$\dot{i}=f_2(\omega ,i,d^2,t)+b_{2}u\left(t\right)$$

(46)

The set of functions and terms $f_1,f_2,b_1,b_2$ are approximated by two RHONNs. Then, starting from the tracking error:

$$e_1\left(t\right)={\omega }_d\left(t\right)-\omega \left(t\right)$$

(47)

after the error derivative is obtained, the following result is achieved:

$$\dot{e_1}\left(t\right)=\dot{{\omega }_d}\left(t\right)-\dot{\omega }\left(t\right)$$

(48)

The following stable error dynamic is proposed to ensure convergence of the tracking error:

$$\dot{e_1}\left(t\right)=-\alpha e_1\left(t\right)$$

(49)

Equating (48) and (49) and substituting the identified dynamics with the RHONN model in (45) yields:

$$-\alpha e_1\left(t\right)=\dot{{\omega }_d}\left(t\right)-f_1(i,\omega ,d^1,t)+b_1\left(t\right)i\left(t\right)$$

(50)

From this, the first virtual control $u_1^{*}\left(t\right)$ is obtained. This virtual control can also be interpreted as the current that forces the DC drive to follow a mechanical speed reference; therefore, we redefine it as $i_d$:

$$i_d\left(t\right)={\displaystyle \frac{f_1(i,\omega ,d^1,t)-\dot{\omega }\left(t\right)-\alpha e_1\left(t\right)}{b_1\left(t\right)}}$$

(51)

Then, a robust PD controller acting on $i_d$ and canceling the dynamics of (46) is used with the following structure:

$$u\left(t\right)={\displaystyle \frac{K_p{e}_2\left(t\right)+K_d{e}_2\left(t\right)-f_2(\omega ,i,d^2,t)}{b_2\left(t\right)}}$$

(52)

where:

$$e_2\left(t\right)=i_d\left(t\right)-i\left(t\right)$$

(53)

Based on the RHONN structure defined in (1) and considering the system dynamics given in (45), the following neuron structure is proposed to approximate the angular speed dynamics:

$${\dot{\omega }}^{*}=w_{11}tanh\left(\omega \right)+w_{12}{tanh}^2\left(\omega \right)+w_{13}{tanh}^3\left(\omega \right)+w_{14}{tanh}^4\left(\omega \right)+w_{15}i$$

(54)

where

• $f_1=w_{11}tanh\left(\omega \right)+w_{12}{tanh}^2\left(\omega \right)+w_{13}{tanh}^3\left(\omega \right)+w_{14}{tanh}^4\left(\omega \right)$,
• $b_1=w_{15}$,
• $w_{1,n}$ denotes the synaptic weights of the first neuron.

For the current dynamics described in (46), the following RHONN representation is used:

$$\begin{array}{cc}\hfill {\dot{i}}^{*}& =w_{21}tanh\left(\omega \right)+w_{22}tanh\left(i\right)+w_{23}{tanh}^2\left(\omega \right)+w_{24}{tanh}^2\left(i\right)\hfill \\ \hfill & +w_{25}tanh\left(\omega \right)tanh\left(i\right)+w_{26}{tanh}^2\left(\omega \right){tanh}^2\left(i\right)+w_{27}v\left(t\right)\hfill \end{array}$$

(55)

where

• $f_2=w_{21}tanh\left(\omega \right)+w_{22}tanh\left(i\right)+w_{23}{tanh}^2\left(\omega \right)+w_{24}{tanh}^2\left(i\right)+w_{25}tanh\left(\omega \right)tanh\left(i\right)+w_{26}{tanh}^2\left(\omega \right){tanh}^2\left(i\right)$,
• $b_2=w_{27}$,
• $w_{2,m}$ denotes the synaptic weights of the second neuron.

4. Materials and Methods

4.1. Simulation Environment

The simulation results were obtained using the MATLAB R2021A/Simulink environment. The control strategy was implemented using standard Simulink blocks and MATLAB Function blocks for custom algorithms. The DC motor plant was modeled using the DC Motor block from the Simscape Electrical library, configured with the electromechanical parameters detailed in

Table 2. Parameters of the DC drive Simscape block.

Symbol	Description	Value
$R_a$	Armature resistance	$6.2\mathsf{\Omega }$
$L_a$	Armature inductance	$0.5\mathrm{mH}$
$K_b$	Back-EMF constant	$0.04943\mathrm{V}\mathrm{s}{\mathrm{rad}}^{-1}$
J	Rotor inertia	$1\times {10}^{-4}\mathrm{kg}{\mathrm{m}}^2$
$I_{nom}$	Rated armature current	$0.8\mathrm{A}$

. The parameters used in the simulation correspond to the real DC motor plant presented in

Table 3. Parameters of the non-linear friction model.

Symbol	Description	Value
$B_v$	Viscous friction coefficient	$5\times {10}^{-5}\mathrm{N}\mathrm{m}\mathrm{s}{\mathrm{rad}}^{-1}$
$T_c$	Coulomb friction torque	$5\times {10}^{-4}\mathrm{N}\mathrm{m}$
$T_{nl}$	Non-linear friction torque	$2.5\times {10}^{-4}\mathrm{N}\mathrm{m}$
k	Smoothing gain	$100\mathrm{s}{\mathrm{rad}}^{-1}$

. The armature resistance and inductance were measured experimentally using laboratory instrumentation; the nominal current was taken from the datasheet; and the inertia and back-EMF constants were estimated based on the motor dimensions.

The Proportional–Derivative (PD) controllers were tuned based on the nominal plant model utilizing the Simulink PID Tuner app to achieve a robust response. The resulting tuning parameters were determined as $K_p\approx 7.4373$, $K_d\approx 0.0016$, with a derivative filter coefficient of $N\approx 709.51$. To evaluate the controller’s performance under realistic conditions, a non-linear friction model was introduced into the mechanical system. This disturbance was applied to the Simscape mechanical rotational port via a MATLAB Function block. The total friction torque, ${\tau }_f$, incorporates viscous friction, Coulomb friction, and a non-linear term to approximate Stribeck effects, governed by:

$${\tau }_f=B_v\omega +T_c\mathrm{sign}\left(\omega \right)+T_{nl}tanh\left(k\omega \right)$$

(56)

where $\omega$ represents the angular velocity.

4.2. Real-Time Implementation Setup

The experimental validation was conducted using a low-cost hardware-in-the-loop (HIL) configuration. The control algorithm was directly deployed from Simulink to the microcontroller using the Simulink Support Package for Arduino Hardware, ensuring a seamless transition from simulation to real-time execution. The specific hardware components utilized in the experimental setup are summarized in

Table 4. Experimental hardware platform.

Component	Description	Details (Manufacturer/Version/Reference)
Microcontroller	Arduino Uno (ATmega328P, 16 MHz)	Arduino SA, Ivrea, Italy
Motor Driver	L298N dual H-bridge driver	STMicroelectronics, Geneva, Switzerland
DC Motor	12 V brushed DC motor, 0.8 A nominal current	Generic manufacturer
Speed Sensor	Incremental encoder mounted on motor shaft	Generic manufacturer
Control Signal	PWM generated by Arduino Uno	Firmware via Arduino IDE (2.3.7)
Communication	Serial communication between Arduino and PC	USB serial protocol
Data Processing Software	MATLAB for signal acquisition and visualization	MathWorks, Natick, MA, USA; Version R2021A

.

5. Results

The results are organized into two main experimental stages. The first stage focuses on simulation-based validation and comparative analysis among several control strategies, including neural back-stepping, a tuned Proportional–Derivative (PD) controller, Linear Quadratic Regulator (LQR), Multi-Layer Perceptron (MLP), Model Predictive Controller (MPC), and the proposed RHONN controller. The second stage evaluates the real-time implementation of the RHONN controller on the physical system.

5.1. Simulation

During the simulation stage, all controllers were evaluated under identical operating conditions to ensure a fair comparison (controller parameters are described in

Table 5. Summary of implemented controllers.

Controller	Description	Value
MLP	A one-hidden-layer feedforward neural network with online adaptation (backpropagation-like update using the sign of the plant gradient and normalization for stability). The network output was scaled and saturated within the range $[0,12]$. Additionally, the training process was carried out in MATLAB using its optimization algorithms to facilitate parameter adjustment of the model.	Hidden neurons: $N=10$ Activation: $tanh(·)$ Learning rate: $\eta =0.1$ Output scaling: $u=6(tanh\left(z_2\right)+1)$, saturates to $[0,12]$ Weight init: $W_1,W_2\sim 0.1·\mathcal{N}(0,1)$ Normalization factor in updates: $1+{\mathbf{a}}_1^{\top }{\mathbf{a}}_1$
MPC	A finite-horizon predictive controller based on an online-identified scalar model (Self-Tuning Regulator) was employed. The parameters a and b were estimated using the Recursive Least Squares (RLS) algorithm, while a Riccati recursion was used to compute the control gain over the prediction horizon. Additionally, the parameter identification and training procedures were carried out in Matlab using its optimization algorithms to facilitate parameter adjustment.	Prediction horizon: $N=10$ Cost weights: $Q=1.0$, $R=0.01$ RLS forgetting factor: $\lambda =0.97$ Control limits: $u_{min}=-0.1$, $u_{max}=0.9$
LQR	Adaptive discrete LQR (Self-Tuning Regulator). The system dynamics were identified online using the Recursive Least Squares (RLS) algorithm through the scalar model $e(k+1)=\widehat{a}e\left(k\right)+\widehat{b}u\left(k\right)$. Based on these estimated parameters, an iterative solution of the algebraic Riccati equation was employed to compute the LQR gain, approximating the infinite-horizon optimal control law. Additionally, the parameter identification and training procedures were carried out in Matlab using its optimization algorithms to facilitate parameter adjustment.	Model form: $e_{k+1}=a^e_k+b^u_k$ Cost weights: $Q=1.0$, $R=0.01$ RLS forgetting factor: $\lambda =0.97$ Riccati iterations: $N_{ricc}=20$
PD	The control action was computed using proportional and derivative terms to regulate the system’s response and reduce the tracking error. Additionally, the controller parameters were tuned and validated using Matlab and its optimization tools.	Proportional gain: $K_p=7.4373$ Derivative gain: $K_d=0.0016$

) with the main objective of achieving accurate reference tracking while minimizing the tracking error and improving overall dynamic performance. As shown in Figure 5, all evaluated control strategies are capable of following the desired reference; however, clear differences can be observed in terms of transient response, overshoot, and steady-state accuracy. In particular, the RHONN controller exhibits a smoother behavior and consistently reduced tracking error throughout the entire simulation interval. To better illustrate the transient behavior, a 1-s zoom of the reference tracking is shown in Figure 6, where the superior dynamic response of the RHONN controller becomes more evident, achieving faster convergence and reduced oscillations compared to conventional controllers. In contrast, the PD and LQR controllers exhibit larger deviations during rapid reference changes, whereas the MLP and MPC controllers exhibit slower adaptation to the reference signal. A quantitative performance comparison is provided in Figure 7, where standard error-based performance indices are evaluated, including the Integral of Time-weighted Absolute Error (ITAE) [33], Integral of Time-weighted Squared Error (ITSE) [34], Integral of Absolute Error (IAE) [33], Integral of Squared Error (ISE) [34], and Integral of Mean Squared Error (IMSE), which are commonly used to assess both transient and steady-state behavior in control systems. As illustrated in Figure 7, the RHONN controller consistently achieves the lowest values across all evaluated indices, indicating a significant reduction in both error magnitude and error duration compared to the PD controller and the remaining strategies, with particularly noticeable improvements in time-weighted metrics such as ITAE and ITSE. Overall, these simulation results demonstrate that the proposed RHONN-based control strategy outperforms both classical and intelligent controllers in terms of reference tracking accuracy, transient response, and robustness, thereby validating its effectiveness and motivating its implementation and evaluation under real-time experimental conditions, as discussed in the following subsection.

Table 6. Performance metrics and relative reduction for each controller.

Controller	ITAE		ITSE		IAE		ISE		IMSE
	Val	Red.	Val	Red.	Val	Red.	Val	Red.	Val	Red.
RHONN	136.60	–	2615	–	10.29	–	188.20	–	12.55	–
PD	138.20	−1.17%	2616	−0.04%	10.50	−2.04%	188.40	−0.11%	12.56	−0.08%
LQR	152.50	−11.64%	3273	−25.16%	11.36	−10.40%	236.50	−25.66%	15.76	−25.58%
MPC	152.60	−11.71%	3225	−23.33%	11.47	−11.47%	233.10	−23.86%	15.54	−23.82%
MLP	174.60	−27.82%	2695	−3.06%	15.68	−52.38%	195.90	−4.09%	13.06	−4.06%

summarizes the quantitative performance metrics obtained for each evaluated controller, including ITAE, ITSE, IAE, ISE, and IMSE, together with their corresponding relative reductions with respect to the proposed RHONN controller. The table provides a comprehensive numerical comparison that complements the graphical results presented in Figure 7. As observed, the RHONN controller achieves the lowest absolute values for all considered performance indices, confirming its superior tracking accuracy and dynamic response. Classical controllers such as PD exhibit only marginal degradation relative to RHONN, whereas optimal and predictive approaches, namely LQR and MPC, show a noticeable increase in error-related metrics, particularly in time-weighted and squared-error indices. The MLP-based controller shows the largest performance degradation, especially in the IAE metric, indicating reduced robustness and poorer steady-state behavior. Overall, the relative reduction values reported in Table 6 clearly demonstrate that the RHONN controller consistently outperforms the remaining control strategies across all evaluation criteria.

5.2. Real Time

During the real-time experimental stage, the proposed RHONN controller was implemented on a physical test platform in order to evaluate its performance under practical operating conditions. The complete proposed algorithm was implemented on an Arduino Uno microcontroller, which performed real-time computation of the control law and online adaptation of the RHONN parameters. The experimental system consisted of a 12 V brushed DC motor rated at 0.8 A, equipped with an incremental encoder to measure the shaft speed. The encoder signal was acquired by the Arduino to estimate the motor speed in real time, which was then used as feedback for the controller. The control signal generated by the proposed algorithm was produced as a PWM signal from the Arduino and subsequently amplified using an L298N motor driver module. This driver stage allowed the low-power PWM signal generated by the microcontroller to provide sufficient voltage and current to drive the motor, acting as the power interface between the controller and the plant. The hardware platform and main components of the experimental setup are summarized in Table 4, and an image of the setup is shown in Figure 8.

In the experiments, the Arduino executed the complete control algorithm in real time, including the RHONN-based plant identification and the backstepping controller described in the previous sections. The microcontroller processed encoder measurements to estimate motor speed, updated the RHONN parameters online to identify plant dynamics, and computed the control action applied to the motor via the PWM signal. The control algorithm was implemented using a sampling time of $T_s=0.0005$ s, which corresponds to the minimum sampling period allowed by the Arduino–Simulink communication library used for the embedded implementation. For monitoring purposes, the Arduino continuously transmitted the measured signals, including the motor speed and control input, to a host computer through serial communication. These signals were used only for visualization and analysis and were displayed in MATLAB, which served as a graphical interface for observing the experimental responses.

The experimental tests were conducted to evaluate the real-time performance of the proposed control scheme under practical operating conditions and to analyze its behavior when implemented on embedded hardware. Figure 9 shows the reference tracking performance obtained in the real-time experiment. It can be observed that the RHONN-based plant identification, combined with the backstepping controller, accurately tracks the desired reference while maintaining stable system operation. Compared with the simulation results, a slight degradation in tracking accuracy was observed; however, the overall dynamic behavior remained consistent. This difference is mainly attributed to practical factors present in the physical system, such as sensor noise, nonlinearities introduced by the motor driver, and computational delays associated with the embedded implementation.

A one-second zoom of the tracking response, shown in Figure 10, provides a closer view of the transient behavior. In this interval, a slight increase in tracking error and small oscillations are observed during rapid reference variations. Nevertheless, the controller maintains fast convergence and satisfactory transient performance, indicating adequate robustness under practical operating conditions. As previously observed, the proposed control scheme achieves satisfactory reference tracking and stable operation in real-time conditions. However, to provide a more objective comparison between the real-time implementation and simulation results, a quantitative evaluation is conducted using commonly adopted performance metrics reported in the literature. A quantitative comparison between the simulation and real-time execution is shown in Figure 11, where several standard performance indices are evaluated, including ITAE, ITSE, IAE, ISE, and IMSE. As expected, these error metrics showed slightly higher values in the real-time experiment than in the simulation, confirming minor performance degradation. Nevertheless, the increase remained moderate, indicating that the proposed control scheme preserves most of its performance when transferred from simulation to the physical system.

Additionally, the identification capability of the RHONN is analyzed in Figure 12 and Figure 13, which illustrate the convergence of the estimated angular-speed and armature-current dynamics, respectively. In both cases, the RHONN can identify the system behavior in real time, producing dynamics that closely match those observed in the simulation. Overall, the real-time experimental results confirm the practical feasibility of the proposed RHONN controller. Despite real-world disturbances and hardware constraints, the controller maintains stable reference tracking and reliable dynamic identification, demonstrating its suitability for real-time motor control applications.

6. Discussion

The results presented in this paper validate the effectiveness of the proposed continuous hybrid control scheme, which integrates an RHONN with a STA learning law. The primary objective was to address the limitations of conventional control strategies in DC drives, specifically regarding robustness against nonlinear friction and parameter uncertainty, while maintaining a computational footprint low enough for embedded implementation.

6.1. Performance Analysis vs. Conventional and Optimal Strategies

The simulation results (Table 5) clearly demonstrate that the RHONN-based controller outperforms classical linear controllers (PD) and optimal strategies (LQR, MPC) across all evaluated metrics. The linear controllers exhibited performance degradation under the non-linear Stribeck friction model introduced in the plant. This is an expected limitation, as fixed-gain linear controllers cannot naturally compensate for state-dependent nonlinearities without high-gain feedback, which often amplifies noise. Furthermore, while the Model Predictive Controller (MPC) provided competitive results [35], it relies heavily on an accurate mathematical model of the system. In the presence of unmodeled dynamics—such as the complex friction added in the simulation—the MPC’s predictive capability diminishes. In contrast, the RHONN controller demonstrated superior adaptability. The structural plasticity of the High-Order Neural Network allowed it to capture these unmodeled dynamics online, effectively acting as a dynamic compensator. This is evidenced by the $23.82\%$ reduction in the Integral of Mean Squared Error (IMSE) compared to the MPC. Additionally, the RHONN outperformed the static Multi-Layer Perceptron (MLP), confirming that recurrent connections and high-order terms are essential for capturing temporal dependencies and the dynamic behavior of electromechanical systems.

6.2. The Role of Super-Twisting Learning

A key contribution of this work is the replacement of the conventional Extended Kalman Filter (EKF) with a Super-Twisting Algorithm (STA) for neural network training. Although the EKF is a powerful estimator, its implementation requires covariance matrix updates, resulting in a computational complexity on the order of $\mathcal{O}\left(L^2\right)$ or $\mathcal{O}\left(L^3\right)$, where L denotes the number of neural weights. Such computational requirements often limit its applicability in high-frequency learning scenarios, particularly when using low-cost microcontrollers or embedded platforms.

In contrast, the proposed STA-based learning law significantly reduces the computational burden to $\mathcal{O}\left(L\right)$ while preserving strong robustness properties. The theoretical analysis presented in Section 3.7 demonstrates that the proposed learning scheme guarantees finite-time convergence of the identification error to a bounded compact set. This theoretical result is supported by the experimental findings. In particular, the identification results for angular speed ($\omega$) and armature current (i), shown in Figure 12 and Figure 13, exhibit fast convergence and stable behavior, avoiding the numerical instabilities often encountered in gradient-based methods under noisy conditions. Furthermore, the STA operates as a robust differentiator, attenuating measurement noise while driving the neural weights toward the optimal manifold defined by the system error dynamics.

6.3. Real-Time Feasibility and Hardware Constraints

The successful deployment of the algorithm on an Arduino Uno (ATmega328P) serves as a strong proof of concept for the “Green AI” and efficient computing paradigm. Most modern neural control strategies require DSPs or FPGAs. By demonstrating that a robust adaptive neural controller can run on an 8-bit, 16 MHz microcontroller with limited RAM (2 KB), this work opens pathways for deploying intelligent control in cost-sensitive industrial applications. Comparing the simulation and real-time results (Figure 10), a slight increase in error metrics (ITAE, ITSE) was observed in the physical implementation. This degradation is attributed to physical phenomena not fully captured in the simulation, such as sensor quantization noise, discrete-time sampling delays ($T_s$), and H-bridge dead-time effects. However, the system maintained stability and high tracking accuracy, validating the robustness of the STA-driven weight adaptation against real-world uncertainties.

Although the proposed algorithm demonstrates significant improvements in adaptability and performance compared to conventional approaches and several modern learning-based techniques, certain practical considerations must still be acknowledged for specific applications. For instance, advanced drive functionalities such as four-quadrant operation may require additional implementation considerations, particularly when the control algorithm is deployed on hardware platforms with limited computational resources.

In contrast, traditional controllers such as PD or PID remain attractive in some industrial environments due to their extremely low computational cost and minimal programming complexity, often requiring only a few lines of code. Nevertheless, the proposed approach remains a practical alternative, especially given that the processing capabilities of modern embedded systems have significantly improved while hardware costs have continued to decrease. These technological advances make it increasingly feasible to implement more sophisticated adaptive and intelligent control strategies in real-world applications. Furthermore, the results presented in this work demonstrate that the proposed RHONN–STA framework can already operate successfully on low-cost embedded hardware. Future work will therefore focus on extending the proposed framework to include more advanced motor-drive features, including four-quadrant operation, improved low-speed performance, and the integration of protection mechanisms, such as overcurrent management, to enhance industrial applicability.

6.4. Limitations

Despite promising results, the proposed scheme relies on properly tuned Super-Twisting gains ($k_1,k_2$). While the STA is robust, improper gain selection can lead to chattering in the weight updates, which may propagate to the control signal. Although the $\sigma$-modification in the update law helps mitigate parameter drift, an adaptive gain-tuning mechanism could further improve ease of deployment.

7. Conclusions and Future Work

This paper presented a novel robust control scheme for DC motor drives, integrating a PD-based backstepping controller with an online-trained RHONN. A primary contribution of this research was the implementation of a STA for real-time weight adaptation in the neural network, effectively replacing the computationally intensive EKF typically used in these applications.

Simulation results demonstrated that the proposed RHONN-STA architecture successfully serves as a dynamic compensator, accurately identifying and mitigating nonlinearities and unmodeled dynamics, including complex Stribeck friction. The proposed method consistently outperformed classical linear controllers (PD), optimal and predictive strategies (LQR, MPC), and shallow static neural networks (MLP) across standard error metrics. Notably, it achieved significant reductions in the Integral of Mean Squared Error (IMSE) and time-weighted indices, demonstrating superior transient and steady-state performance.

Furthermore, the practical feasibility of the proposed approach was validated through real-time experimental deployment on a low-cost, resource-constrained 8-bit microcontroller. Despite physical hardware limitations, sensor quantization noise, and discrete-time sampling delays, the STA-driven learning law maintained high tracking accuracy, fast convergence, and robust stability. This experimental validation confirms that advanced, high-order intelligent control algorithms can be efficiently executed in real-time without relying on expensive, high-end DSPs. Future research will focus on extending this adaptive control architecture to multivariable, heavily cross-coupled systems, such as Permanent Magnet Synchronous Motors (PMSMs) and Induction Motors. Additionally, the integration of variable-gain Super-Twisting algorithms will be investigated to further optimize the trade-off between convergence speed and chattering mitigation for high-precision electromechanical applications.

Future research will focus on two main directions: first, extending this architecture to multivariable systems such as Permanent Magnet Synchronous Motors (PMSMs) and Induction Motors, where cross-coupling effects are more significant; and second, investigating the implementation of variable-gain Super-Twisting algorithms to eliminate the trade-off between convergence speed and chattering reduction, further optimizing the controller for high-precision robotic applications.