Echo state network-based adaptive control for nonstrict-feedback nonlinear systems with input dead-zone and external disturbance

1.   Introduction

In recent years, there has been a growing interest in addressing the nonlinear aspects inherent in practical physical systems ^[1,2,3,4,5]. Consequently, there is an increasing demand for control approaches capable of effectively managing complex nonlinear systems. Various control methods have been developed to tackle tracking or stabilization control objectives. One widely adopted approach is the adaptive backstepping technique, recognized as a recursive design procedure suitable for controller design in nonlinear dynamic systems. To address the output maneuvering control problem in nonlinear systems, a study was conducted using the backstepping technique ^[6,7,8]. An adaptive robust backstepping approach was proposed to attain a specified performance level for these nonlinear systems. It is important to note that this approach assumes knowledge or linear parameterization of the structural uncertainties present in the nonlinear systems ^[9,10].

In various practical engineering applications, uncertainties arise not only from nonlinearities but also from parameter variations, contributing to uncertain functions within the systems ^[11]. This complexity hinders the direct application of simple adaptive backstepping control techniques. Researchers have responded by integrating the adaptive backstepping control framework with the approximation capabilities of neural networks (NNs) and fuzzy logic systems (FLSs) to address this challenge ^[12]. For nonlinear systems under feature information, the adaptive control problem has been reported via prescribed performance ^[13]. For strict-feedback nonlinear systems, an adaptive control method has been published via event-triggered mechanism ^[14] For strict-feedback nonlinear systems, an adaptive fuzzy control problem has been published under the impact of control gain unknown functions ^[15]. For high-order nonlinear systems, a multilayer neuro control problem has been reported under the impact of active disturbance rejection ^[16]. For nonlinear mismatched systems, a new integral robust method based on asymptotic tracking has been published ^[17].

Through the incorporation of these techniques, adaptive neural or fuzzy backstepping control strategies have been developed, effectively tackling control issues in nonlinear systems characterized by both uncertain parameters and uncertain functions. These advanced strategies enable the successful management of uncertainties in real-world systems, enhancing their overall control performance.

At this point, radial basis function neural networks (RBFNNs) were employed to tackle time-varying uncertainty. As an alternative approximation technique, RBFNNs were static feed-forward networks with a substantial number of neurons, enhancing the accuracy of the approximation. The authors in ^[18] introduced recurrent neural networks (RNNs) with dynamic feedback structures to model dynamic systems. In addressing the issue of time consumption, echo state network (ESN) was introduced, offering an alternative approach to train RNNs ^[19]. ESN has recently demonstrated its efficacy in improving the approximation performance of uncertain nonlinearities across various nonlinear systems. From the literature, it becomes evident that ESN proves advantageous in effectively approximating unknown functions ^[20].

It is widely recognized that non-strict-feedback nonlinear systems encompass a broader range of system forms compared to systems with strict-feedback nature ^[21]. In real-world applications, various types of nonlinear systems, including the ball and beam system ^[22,23], often exhibit non-strict-feedback structures. A notable characteristic of non-strict-feedback systems is their dependence on all system states when defining the system functions. This poses significant difficulties and challenges when designing control schemes using the backstepping technique ^[]. Consequently, addressing control design for non-strict-feedback systems is an intriguing and valuable topic, both in theoretical research and practical applications, with numerous results documented ^[25,26,27]. For instance, for non-strict-feedback nonlinear systems, an adaptive control method based on fuzzy approximation has been published under function constraints ^[28]. Additionally, a fuzzy adaptive asymptotic method for non-strict-feedback nonlinear systems has been reported ^[29].

In the field of control engineering, non-smooth nonlinearities such as backlash, hysteresis, and dead-zones are commonly observed between actuators and sensors ^[30]. Dead-zones, being non-smooth functions, can significantly degrade system performance and even lead to instability ^[31]. Therefore, it is necessary to consider the presence of dead-zones during the controller design process. In recent years, several research studies have addressed the challenges posed by dead-zones in nonlinear systems ^[32]. For example, a novel adaptive robust backstepping control scheme has been proposed to tackle the dead-zone problem in nonlinear dynamic systems ^[12]. Additionally, an adaptive neural controller utilizing a combination of dynamic surface control (DSC) and neural networks (NNs) has been developed for a specific class of pure-feedback nonlinear systems, which involve unknown control directions and input dead-zones ^[33,34]. Similarly, an adaptive fuzzy control strategy has been proposed for nonlinear switched systems dealing with failures in sensors ^[35]. For stochastic non-strict-feedback systems under dead-zone conditions, an adaptive output feedback method has been reported ^[36]. For fractional-order nonlinear non-strict-feedback large-scale systems with dead-zone input, a decentralized control method has been published ^[37].

The control methods discussed above are limited as they do not effectively handle external disturbances or dead-zones nonlinearities using echo state networks (ESNs). This limitation motivates an investigation into non-strict-feedback nonlinear systems capable of addressing both challenges simultaneously. In many real-world systems, such as mechanical systems with friction or electronic systems with dead-zones, these issues significantly impact control performance. Therefore, there is a pressing need to explore advanced control techniques that can mitigate the effects of external disturbances and dead-zones. Furthermore, ESN emerges as a promising alternative for functional approximation, offering distinct advantages over other techniques like FLSs or RBFNNs. By leveraging ESNs, we aim to develop an effective control framework that can effectively manage external disturbances and dead-zone nonlinearities.

In comparison to existing research, the primary contributions of this work can be summarized as follows:

● Compared with existing results such as ^[6,14,15] that primarily focus on pure-feedback and strict-feedback nonlinear systems, this study addresses adaptive control for nonlinear systems with a non-strict-feedback structure, dealing with challenges posed by dead-zone nonlinearity and external disturbances.

● In comparison to existing results ^[7,19,38], the use of an ESN stands out due to its ability to approximate unknown functions with continuous online updates to the weights. This characteristic is particularly advantageous in dynamic systems where real-time adjustments are crucial for effective control. Additionally, an adaptive controller designed based on backstepping and ESN approximation further enhances the system's adaptability and effectiveness, making it well-suited for handling nonlinearities and uncertainties.

● The proposed controller ensures that all closed-loop system signals exhibit semi-global uniform ultimate boundedness (SGUUB), guaranteeing convergence of the tracking error to a small neighborhood of the origin. The effectiveness of the proposed control method is substantiated through two simulation examples, including a real-world example about the Brusselator model. Furthermore, the utilization of error assessment criteria in the simulation section provides compelling evidence showcasing the superiority and efficacy of the proposed control approach over existing controllers.

The structure of this work is outlined as follows: In Section 2, a comprehensive system description is provided, and preliminary work is discussed. The design procedure for the controller and the stability analysis of the controlled system are presented in Section 3. To showcase the efficacy of the proposed method, Section 4 presents simulation results. Finally, the paper concludes in Section 5.

Notations. $\mathbb{R}^n$ denotes the Euclidean space, and $\bar{x}_i = [x_1, \dots, x_i]^T \in \mathbb{R}^i$ with $i = 1, \dots, n$ and $x = [x_1, \dots, x_n]^T \in \mathbb{R}^n$ represent state vectors. The functions $f_i(\cdot)$ and $g_i(\cdot)$ are nonlinear functions, where $u$ denotes the system input and $y$ represents the system output. The term $d_i(t)$ denotes the external disturbance.

2.   Problem formulation and preliminaries

Consider the following non-strict-feedback nonlinear system with input dead-zone and external disturbance as

where $\bar{x}_i = [x_1, \dots, x_i]^T \in \mathbb{R}^i$ with $i = 1, \dots, n$, and $x = [x_1, \dots, x_n]^T \in \mathbb{R}^n$ are state vectors. The functions $f{i}(\cdot)$ and $g_{i}(\cdot)$ denote unknown smooth nonlinear functions with $f_i(0) = 0$, and both $f{i}(\cdot)$ and $g_{i}(\cdot)$ are locally Lipschitz functions. The variables $u$ and $y$ represent the system input and system output, respectively. Additionally, $d_i(t)$ represents external disturbances, satisfying $|d_i(t)| \leq \bar{d}_i$, where $\bar{d}_i$ is a positive constant.

The nonsymmetric dead-zone ^[33] with input $v$ and output $u$ is defined as

where $d_r$ is the right dead-zone break-points and $d_l$ is the left dead-zone break-point, while $c_r$ and $c_l$ represent the slope of the dead-zone.

Controller objectives. The primary objective of this work is to design a controller that ensures the boundedness of all signals in a closed-loop system and guarantees precise tracking of the system output $y$ to the reference signal $y_d(t)$.

2.1.   ESN

ESN has proven effective in approximating nonlinearity owing to its exceptional approximation and learning capabilities ^[19]. The structure comprises three components, illustrated in Figure 1, where $K$ is the number of neurons in the input layer, $N$ is the number of neurons in the hidden layer, and $L$ is the number of neurons in the output layer. The continuous-time state of reservoir neurons is defined as

where $P(Z)$ represents the activation function of the dynamic reservoir, $\tanh(\cdot)$ is the hyperbolic tangent function, $\mu > 0$ represent the leakage rate of each reservoir neuron, and $W_{in} \in \mathbb{R}^{N \times K}$, $W_d \in \mathbb{R}^{N \times N}$, and $W_{fb} \in \mathbb{R}^{N \times L}$ denote the connection weight matrices for input, internal, and feedback connections, respectively. The $u$ is the external input of dimension $K$, and the network's output is defined as

where $W \in\mathbb{R}^{N \times 1}$ is the output weight matrix.

It has been noted in ^[19] that there exists an ESN system $y(Z)$ in the form of (2.4) such that for any given continuous function $f(\cdot): \mathbb{R}^n \rightarrow \mathbb{R}$ over a sufficiently large compact set $\Omega \subset \mathbb{R}$, one has

where $\bar{\epsilon} > 0$ is an arbitrary positive constant. The function $f(Z)$ is approximate as

where $\epsilon$ is the approximation error with $|\epsilon|\leq \bar{\epsilon}$, and $W^*$ represents the ideal weight matrix determined as ^[19]

where $P(Z) = [P_1(Z), \ldots, P_n(Z)]^T$ represents the activation function with $P_j(Z)$ chosen in the Gaussian function form as ^[19]

where $\lambda_i = [\lambda_{j1}, \ldots, \lambda_{iN}]^T$ and $\nu$ represent the center and the width of the Gaussian function.

Remark 2.1. The computational complexity of an ESN is primarily determined by the operations in its hidden layer (reservoir), which have a complexity of $O(N^2)$ due to recurrent connections and nonlinear activations. Additionally, the input layer operation has a complexity of $O(N *K)$, and the output layer operation has a complexity of $O(N*L)$, where $K$ is the number of neurons in the input layer, $N$ is the number of neurons in the hidden layer, and $L$ is the number of neurons in the output layer. Therefore, the overall computational complexity of an ESN is approximately $O(N * K + N^2 + N * L)$, with the reservoir operations typically dominating the complexity. In the special scenario where $L = 1$ and $K = N$, the computational complexity simplifies to $O(N^2)$.

Remark 2.2. ESN stands out as a viable alternative for functional approximation, capable of replacing other techniques like FLSs ^[19], or RBFNNs ^[7], or multidimensional Taylor networks (MTN) ^[38]. Unlike RBFNNs with fixed multilayered feedforward architectures, ESN offers easy and precise training without requiring adjustments to the weights between the input and hidden layers.

Remark 2.3. ESN make training easier by assigning random weights to the input and storage layers, leaving only the output weights to be learned. This simplifies the training procedure while ensuring excellent accuracy in capturing complicated temporal dynamics. In contrast to RBFNNs, which handle static data with localized activations, and rule-based fuzzy logic systems, ESNs excel in sequential data tasks.

Algorithm 1 outlines the sequential procedure for the ESN algorithm:

To facilitate the achievement of the control objective, the following preliminary assumptions and lemma are required:

Assumption 2.1. ^[6] The reference signal $y_d(t)$ and its $n$th order derivatives are characterized by continuity and boundedness. Specifically, $\exists$ is a constant $d > 0$ such that the absolute value of $y_d(t)$ is less than or equal to $d$, i.e., $|y_d(t)| \leq d$.

Assumption 2.2. ^[7] The signs of $g_i(\bar{x}_i)$ for $i = 1, \dots, n$ are supposed to be known. Generally, we suppose that $g_i(\xi) \geq b_m > 0$, where $b_m$ is a known constant.

Lemma 2.1. (Young's inequality) ^[1] We have the following inequality for any vectors $x$ and $y$ in $\mathbb{R}^n$

where $q > 0$ and $r > 0$ are constants with $(q-1)(r-1) = 1$.

Assumption 2.3. ^[7] The parameters of the dead-zone $d_r, d_l, c_r, c_l$ are not known, but their sign is known, $d_r\mathrm{\ge }\mathrm{0, \ }d_l\mathrm{\le }\mathrm{0, \ }c_r\mathrm{ > 0, \ }c_l\mathrm{ > 0\ }$, and $\exists$ constants $c_{max}\mathrm{ > 0}$ and $c_{min\mathrm{\ }}\mathrm{ > 0}$ are known such that $c_{max}\mathrm{ = }{\mathrm{max} \{c_r, c_l\}\ }$ and $c_{min}\mathrm{ = }{\mathrm{min} \{c_r, c_l\}\ }$.

By employing the input dead-zone as in ^[33], (2.2) can be represented as

where

Thus, $j\left(t\right)$ can be described as

From (2.10), we have

where $j^* = {\mathrm{max} \{c_ld_l, \ c_rd_r\}\ }$.

The architecture of the presented control system is demonstrated in Figure 2.

3.   Design of adaptive controller and the analysis of stability

In this part, the backstepping technique and ESN are employed to provide an adaptive control method for the non-strict-feedback nonlinear system (2.1). As illustrated below, the recursive backstepping method relies on the introduction of coordinate changes:

where $\eta_0 = y_d$ and $\eta_i$ represents the virtual control law defined as follows:

where $k_i > 0$, $a_i > 0$ are design parameters, $X_i = [\bar{x}_i^T, \bar{\hat{\theta}}_{i-1}^T, \bar{y}^{(i)}_d]^T$ with $\bar{x}_i = [x_1, \cdots, x_i]^T$, and $\hat{\theta}_i$ denotes the estimate of $\theta_i$, which will be provided later.

The control law is chosen as

where $k_n > 0$, $a_n > 0$ are design parameters, $X_n = [\bar{x}_n^T, \bar{\hat{\theta}}_{n-1}^T, \bar{y}^{(n)}_d]^T$ with $\bar{x}_n = [x_1, \ldots, x_n]^T$, and $\hat{\theta}_n$ denotes the estimate of $\theta_n$, which will be provided later.

The adaptive law is selected as

where $r_i > 0$, $\sigma_i > 0$ are design parameters.

Step 1: From (2.1) and (3.1), we have

As $x_2 = e_2 + \eta_1$, (3.6) becomes

The Lyapunov function is chosen as

where $r_1$ represents the design parameter, $b_m$ is a constant defined in Assumption 2.2, and $\tilde{\theta}_1 = \theta_1 - \hat{\theta}_1$ represents the estimation error, where $\hat{\theta}_1$ denotes the estimate of $\theta_1$, which will be provided later.

By taking the time derivative of (3.8), we have

where $\bar{f}_1 = g_1e_2-\dot{y}_d+\frac{1}{2}e_1$. As $\bar{f}_1$ contains unknown functions $f_1$ and $g_1$, so an ESN $W^T_1P_1(Z_1)$ is used to approximate it. For $\epsilon_1 > 0$, we have

where $\delta_1(Z_1)$ is the estimation error, and $Z_1 = [x_1, \ldots, x_n, y_d, \dot{y}_d]^T$.

By using Lemma 2.1, one has

where $a_1 > 0$ is design parameter, $b_m$ is defined in Assumption 2, and $\theta_1 = \frac{\|W^{*}_1\|^2}{b_m}$, and $X_1 = [x_1, y_d, \dot{y}_d]^T$.

By using Young's inequality, we have

where $\bar{d}_1 > 0$ is a constant.

Now, by using (3.11), (3.12), virtual control law (3.3), adaptive control law (3.5), and $\tilde{\theta}_1 = \theta_1-\hat{\theta}$, (3.9) becomes

Step $i$ $(2\leq i \leq n-1)$: From (2.1) and (3.2), we have

As $x_{i+1} = e_{i+1}+\eta_i$, (3.14) becomes

The Lyapunov function is described as

where $r_i$ represents the design parameter, and $b_m$ being a constant shown in Assumption 2.2, $\tilde{\theta}_i = \theta_i - \hat{\theta}_i$ represents the estimation error, where $\hat{\theta}_i$ denotes the estimate of $\theta_i$, which will be provided later.

By taking the time derivative of (3.16), we have

where $\bar{f}_i = g_ie_{i+1}-\dot{\eta}_{i-1}+\frac{1}{2}e_i$. Since $\bar{f_1}$ includes functions $f_i$ and $g_i$ that are unknown, an ESN $W^T_iP_i(Z_i)$ is used to approximate it. For $\epsilon_i > 0$, we have

with $\delta_i(Z_i)$ being the estimation error, and $Z_i = [x_1, \ldots x_n, \bar{\hat{\theta}}^T_{i-1}, \bar{y}^{(i)}_d]^T$ with $\bar{\hat{\theta}}^T_{i-1} = [\hat{\theta}_1, \ldots, \hat{\theta}_{i-1}]^T$ and $\bar{y}^{(i)}_d]^T = [y_d, \cdots, y^{(i)}_d]^T$, respectively.

Similarly to (3.11), one has

where $b_m$ is defined in Assumption 2, $a_i >0$ is a design parameter, and $\theta_i = \frac{\|W^{*}_i\|^2}{b_m}$, and $X_i = [\bar{x}_i^T, \bar{\hat{\theta}}_{i-1}^T, \bar{y}^{(i)}_d]^T$ with $\bar{x}_i = [x_1, \cdots, x_i]^T$.

Using Lemma 2.1, one has

where $\bar{d}_i > 0$ is a constant.

Now, by using (3.19), (3.20), virtual control law (3.3), adaptive control law (3.5), and $\tilde{\theta}_i = \theta_i-\hat{\theta}_i$, (3.17) becomes

Step $n$: From (2.1), (2.10), and (3.2), we have

The Lyapunov function candidate is represented as

where $r_i$ denotes the design parameter, and $b_m$ being a constant shown in Assumption 2.2, $\tilde{\theta}_n = \theta_n - \hat{\theta}_n$ represents the estimation error, where $\hat{\theta}_n$ denotes the estimate of $\theta_n$, which will be provided later.

By taking the time derivative of (3.23), we have

where $\bar{f}_n = f_n-\dot{\eta}_{n-1}+\frac{1}{2}e_n$. As $\bar{f}_n$ contains an unknown function $f_n$, an ESN $W^T_nP_n(Z_n)$ is used to approximate it. For $\epsilon_n > 0$, we have

where $\delta_n(Z_n)$ is the estimation error, and $Z_n = [x_1, \ldots, x_n, \bar{\hat{\theta}}^T_{n-1}, \bar{y}^{(n)}_d]^T$ with $\bar{\hat{\theta}}^T_{n-1} = [\hat{\theta}_1, \ldots, \hat{\theta}_{n-1}]^T$ and $\bar{y}^{(n)}_d]^T = [y_d, \ldots, y^{(n)}_d]^T$, respectively.

By using Lemma 2.1, one has

where $\theta_n = \frac{\|W^{*}_i\|^2}{b_m}$, $X_n = [\bar{x}_n^T, \bar{\hat{\theta}}_{n-1}^T, \bar{y}^{(n)}_d]^T$ with $\bar{x}_n = [x_1, \ldots, x_n]^T$, and $a_n > 0$ is the design parameter.

By using Young's inequality, one has

where $\bar{d}_n > 0$ is a constant.

Now, by using (2.13), (3.26), (3.27), real control law (3.4), adaptive control law (3.5), and $\tilde{\theta}_i = \theta_i-\hat{\theta}_i$, (3.24) becomes

Theorem 3.1. Consider the nonlinear system (2.1) with dead-zone nonlinearity (2.2) under Assumptions 2.1–2.3. Assume that ESN can effectively approximate each unknown function $\bar{f}_i$ with $1 \leq i \leq n$, ensuring that the approximation errors $\delta_i$ are bounded. Under the real controller (3.4), virtual control law (3.3) and the adaptive law (3.5) with bounded initial conditions, the proposed controller demonstrates that all signals in the closed-loop system are SGUUB. In addition, the tracking error $e_1$ converges to a confined region around the origin.

Proof. Define $V = \sum_{i = 1}^{n}V_i$. It follows from (3.13), (3.21), and (3.28) that

since

Therefore, (3.29) can be written as

where $a_0 = \min\{2k_ib_m, \sigma_i| 1\leq i \leq n$ and $b_0 = \sum_{i = 1}^{n}\left(\frac{a^2_i}{2}+\frac{\epsilon^2_i}{2}+\frac{\bar{d}^2_i}{2}\right) +\frac{1}{2}j^{*^2}$.

Equation (3.31) implies that in the closed-loop system, all signals are bounded.

Remark 3.1. The performance of the controller is greatly impacted by the design parameters $k_i$, $b_i$, $r_i$, and $\sigma_i$. Tracking errors are decreased by reducing $a_i$ while increasing $k_i$ and $\sigma_i$, but control law complexity is increased. Choosing the right control parameters is crucial for meeting system requirements. Carefully tuning these parameters through the trial and error method involves closely watching how the controller performs. This way, the controller can balance between reducing tracking errors and keeping the control law manageable.

Remark 3.2. In our comparative analysis, it is worth noting that the existing method ^[1] did not account for the influence of nonlinearity and external disturbance in the control approach, which is crucial as nonlinearity can significantly impact system performance. Furthermore, in contrast to our proposed work, ^[6] focuses on pure-feedback nonlinear systems, while our approach addresses the more complex non-strict-feedback nonlinear systems. Additionally, ^[11] deals with saturation nonlinearity, whereas in practical systems, the presence of dead-zones nonlinearity poses a more challenging scenario. These distinctions underscore the unique contributions and practical relevance of our proposed methodology.

Remark 3.3. In ^[39], the authors proposed adaptive control for nonlinear systems with input saturation using FLSs, while in ^[40] NNs were employed for tracking control of nonlinear systems under saturation via command filters. Both approaches primarily address saturation nonlinearity. In contrast, this paper utilizes ESNs to tackle the more complex challenges posed by dead-zone nonlinearity and external disturbances in non-strict-feedback nonlinear systems. By randomly assigning weights in the input and storage layers and training only the output weights, ESNs simplify the training process and achieve high accuracy in capturing complex temporal dynamics.

4.   Simulation results

Example 4.1. Consider the following non-strict-feedback nonlinear system as

where $f_1 = x_1x^2_2+x^2_1\sin(x_2)$, $f_2 = x^2_1x_2e^{x_2}+x_1\cos(x_1x_2)$, $g_1 = 1.5+0.5\sin(x_1)$, $g_2 = 1.5+\sin(x_1x_2)$, $d_1(t) = 0.5\cos(t)$, $d_2(t) = \sin(0.5t)$. The reference signal is given as $y_d = 0.5(\sin(t)+\sin(0.5t))$.

The virtual control law $\eta_1$, actual control law $v$, and adaptive laws are chosen as

The parameters are selected as $k_1 = 6, k_2 = 10$, $a_1 = 1, a_2 = 1$, $r_1 = 3, r_2 = 3$, $\sigma_1 = 0.05, \sigma_2 = 0.05$. The initial conditions are chosen as $x_1(0) = 0.5, x_2(0) = 0$, $\hat{\theta}_1(0) = 0, \hat{\theta}_2(0) = 0$. The dead-zone parameters are chosen as $d_r = 0.5, d_l = -0.6, c_r = 1, c_l = 1.2$. The center vectors and widths chosen for the Gaussian function are represented as $\lambda_i = [-1, -0.5, 0, 0.5, 1]$ and $\nu = 2$. The simulation results are illustrated in Figures 3–7, providing a detailed analysis of the system's performance. Figure 3 presents the system output $y = x_1$ alongside the reference signal $y_d$, demonstrating the effective tracking performance achieved under the proposed control method. Figure 4 visually represents the tracking error $e_1$, showing its convergence around a small region near the origin. Furthermore, Figure 5 offers insights into the state variable $x_2$, indicating its bounded behavior. Both the control signal $v$ and system input $u$ are illustrated as bounded in Figure 6. Additionally, Figure 7 showcases the boundedness of the adaptive laws. The simulation results confirm that the designed controller is suitable for achieving precise tracking performance and ensuring the boundedness of all closed-loop signals in nonlinear systems, particularly in the presence of external disturbances and input dead-zone.

Moreover, to validate the superiority of the proposed control schemes over the existing RBFNN controller ^[7], MTN controller ^[38], and fuzzy controller ^[19] for this particular system, a comparative analysis is conducted using assessment error criteria. For a given pair of data points $(y_i(t), y_{id}(t))$ in the interval $t\in [1, P]$, Table 1 displays the assessment error criteria.

Sum of squared error (SSE): $\text{J}_{\text{SSE}}$ = $\sum_{t = 1}^{P}(y_i(t) - y_{id}(t))^2$; maximum absolute error (MAE): $\text{J}_{\text{MAE}}$ = $\max_{1 \leq t \leq P}|y_i(t) - y_{id}(t)|$; normalized mean squared error (NMSE): $\text{J}_{\text{NMSE}}$ = $\frac{\sum_{t = 1}^{P}(y_i(t) - y_{id}(t))^2}{\sum_{t = 1}^{P}(y_i(t)-\bar{y_{id}})^2}$; best fit rate (BFR): $\text{J}_{\text{BFR}}$ = $1 - \frac{\sum_{t = 1}^{P}(y_i(t) - y_{id}(t))^2}{\sum_{t = 1}^{P}(y_i(t)-\bar{y_{id}})^2} \times 100\%$; mean squared error (MSE): $\text{J}_{\text{MSE}}$ = $\frac{\sum_{t = 1}^{P}(y_i(t)-y_{id})^2}{P}$; root mean squared error (RMSE): $\text{J}_{\text{RMSE}}$ = $\sqrt{\frac{1}{P}\sum_{t = 1}^{P}(y_i(t) - y_{id}(t))^2}$ where $\bar{y_{id}}$ represents the mean of $y_{id}(t)$.

The results presented in Table 1 reveal a marginal improvement in the proposed control scheme compared to the RBFNN controller ^[7], MTN controller ^[38], and fuzzy controller ^[19] for this particular system. This improvement serves as evidence for the effectiveness of our proposed control method.

Example 4.2. In this example, a real-world application of the Brusselator model is used to show the efficacy of the proposed strategy. A simplified Brusselator model describes a specific group of chemical reactions ^[41]

where $f_1 = a-(b+1)x_1$, $f_2 = bx_1-x^2_1x_2$, $g_1 = x^2_1$, $g_2 = 2+\cos x_1$, $d_1(t) = 0.5\cos(t)$, $d_2(t) = \sin(0.5t)$. The reference signal is given as $y_d = 0.5(\sin(t)$.

The virtual control law $\eta_1$, actual control law $v$, and adaptive laws are chosen as

The parameters are $a = 1$, $b = 3$, $k_1 = 10, k_2 = 10$, $a_1 = 1, a_2 = 1$, $r_1 = 3, r_2 = 3$, and $\sigma_1 = 0.05, \sigma_2 = 0.05$. The initial conditions are set as $x_1(0) = 0.4, x_2(0) = 0.4$, and $\hat{\theta}_1(0) = 0, \hat{\theta}_2(0) = 0$. The dead-zone parameters are set at $d_r = 0.5, d_l = -0.6, c_r = 1, c_l = 1.2$. The center vectors and widths chosen for the Gaussian function are represented as $\lambda_i = [-1, -0.5, 0, 0.5, 1]$ and $\nu = 2$. The simulation results are depicted in Figures 8–12, offering an intricate analysis of the system performance. Figure 8 showcases the system output $y = x_1$ alongside the reference signal $y_d$, highlighting the effective tracking performance achieved with the proposed control method. Moving on, Figure 9 visually represents the tracking error $e_1$, depicting its convergence around a small region near the origin. Furthermore, Figure 10 provides insights into the behavior of the state variable $x_2$, indicating its bounded nature. Both the control signal $v$ and system input $u$ are shown to be bounded in Figure 11. Additionally, Figure 12 showcases the boundedness of the adaptive laws. These simulation results affirm that the designed controller is adept at achieving precise tracking performance and ensuring the boundedness of all closed-loop signals in nonlinear systems, especially when confronted with external disturbances and input dead-zone.

To validate the superiority of the proposed control scheme over the existing RBFNN controller ^[7], MTN controller ^[38], and fuzzy controller ^[19] for this specific system, a comparative analysis is conducted using assessment error criteria defined in Example 1. Table 2 shows that the suggested control strategy performs marginally better than the RBFNN controller ^[7], MTN controller ^[38], and fuzzy controller ^[19] for this system. This improvement supports the effectiveness of our proposed control mechanism.

5.   Conclusions

This study addresses adaptive control for non-strict-feedback nonlinear systems with dead-zone and external disturbances. It uses an ESN to approximate the unknown nonlinear function and designs an adaptive controller using the backstepping method and Lyapunov analysis with ESN approximation. The proposed controller ensures SGUUB of all signals in the closed-loop system, as shown through a thorough Lyapunov stability analysis. Two examples and error assessment criteria validate the method's effectiveness. However, it's important to note that the tuning of control parameters through the trial and error method can be time-consuming, which serves as a limitation of the approach. In future research work, our focus will extend to stochastic nonlinear systems incorporating input quantization.

Author Contributions

Hadil Alhazmi: Writing–original draft, Resources, Methodology; Mohamed Kharrat: Conceptualization, Methodology, Writing–review & editing. All authors have reviewed the results and approved the final version of the manuscript.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in creating this article.

Acknowledgments

This work was funded by the Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R528), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of interest

The authors confirm no conflicts of interest.