Intelligent Model-Free Control for Power Line Inspection Robots: Tackling Input Time Delays with Data-Driven Solutions

Abstract

This article presents an innovative approach to model-free adaptive control designed for power line inspection robots facing challenges with input time delays. The strategy begins by employing a compact-form dynamic linearization technique to transform the original system into a data-driven model. Subsequently, utilizing real-time input and output information, the system’s pseudo-partial derivatives are assessed online. Leveraging these assessment parameters, a weighted one-step prediction control mechanism is designed, and a compact-form dynamic linearization model-free adaptive control framework is established. Moreover, the research incorporates compression mapping to thoroughly confirm the convergence of the algorithm, thereby ensuring its stability. Ultimately, the effectiveness and practicality of this control method are substantiated through a series of simulation experiments, demonstrating its robust performance.

1. Introduction

Power line inspection robots (PLIRs) autonomously or remotely carry out routine inspections in power systems, particularly in facilities such as substations and distribution station rooms. These robots conduct routine examinations of electrical infrastructure either autonomously or through remote control, ensuring the reliable functioning of the power grid [1,2]. In power lines, routine inspection often requires professional workers to operate in high-voltage electrified environments, which is not only highly hazardous but also extremely laborious. To address this challenge, considerable research efforts have been made in the last few decades to advance PLIR technology [3,4,5,6,7,8,9].

It is essential to recognize that power line inspection robots (PLIRs) frequently function in challenging and unpredictable environments, including severe weather conditions and areas with substantial electromagnetic interference. These factors can introduce significant disturbances and uncertainties into the system dynamics, thereby compromising the precision and stability of the control systems. Ensuring that PLIRs can maintain accurate tracking and stable performance under such adverse conditions is a critical challenge that must be addressed to guarantee the safety and effectiveness of power line maintenance and inspection operations.

To address these challenges, Lu and his team developed a novel self-detection system for power lines. This system uses drones and real-time GPS data from nearby pylons to navigate efficiently [10]. Similarly, Araar and his team explored two vision-based control schemes, both of which use power lines arranged in parallel as navigational feedback to guide the flight of drones. These approaches offer valuable insights for the autonomous navigation of drones using parallel power lines [11]. In power line-based autonomous navigation methods, it is often required that drones fly above the transmission lines because the parallel features of the lines aid in extracting power line information against complex backgrounds [12]. Building on these principles, Zhou and his team developed a model using prior knowledge of the background, which adapts to environmental changes and addresses the issue of frequently changing tracking parameters [13]. The authors of [14] presented a drone system equipped with advanced embedded processors and a dual-lens stereo vision sensor, capable of generating navigational information for power lines in real time during flight, thus achieving the automated detection of transmission lines. For an underactuated PLIR system with two degrees of freedom and one control input, [15,16] proposed an equilibrium control strategy based on backstepping technology. For a class of underactuated PLIRs, a nonlinear optimal control method was proposed in [17].

The swift progression of information technology is driving transformative changes across numerous industries, including chemical product manufacturing, metal smelting, machinery production, and electronics manufacturing. As production scales expand and both equipment and processes become more intricate, the quality demands for final products are steadily rising. These shifts present significant challenges in constructing precise models for industrial processes using recognition technologies or conventional mechanical principles. In such cases, applying traditional model-based control theory to solve difficult-to-model industrial control problems becomes particularly challenging and sometimes even impossible. However, in modern production processes, vast amounts of data are continually generated and recorded, capturing valuable insights into the operational status of both processes and equipment. In situations where precise process models are elusive, using historical or real-time data to design controllers and manage these intricate production systems shows significant promise and is essential for advancing control theory. Developing and refining methods into data-driven control approaches offers an effective way to tackle the evolving obstacles in contemporary industrial production. Data-driven control has already found practical application in numerous industries and continues to drive technological advancements and improve production efficiency [18,19,20,21,22,23].

Model-free adaptive control represents a strategy that offers substantial benefits by not requiring prior model knowledge of the control system while maintaining stable closed-loop system performance. The core innovation lies in the concept’s introduction of pseudo-gradient vectors and pseudo-partial derivatives. These enable nonlinear systems with discrete-time approximation as a series of linear models that are dynamic in the system’s operational trajectory. By estimating input and output data in real time, the pseudo-partial derivatives and gradient vectors can be determined. This method allows model-free adaptive control to effectively control nonlinear systems without relying on precise models, significantly mitigating the effects of model uncertainties on control performance. Through model-free adaptive control, the controller can self-adjust and optimize, as well as adapt to various environmental changes and inherent uncertainties of the system, thus achieving good control performance without an accurate model. The versatility and practicality of this control technology have led to its wide application in various industries, especially in complex systems where conventional control methods face challenges. Model-free adaptive control has demonstrated its unique value and broad application prospects [24]. Some notable application areas include motion control systems [25,26], industrial automation systems [27,28], and electrical power management systems [29,30]. In these fields, model-free adaptive control technology has demonstrated significant application potential and exceptional performance.

However, the aforementioned research efforts have not fully considered the impact of input time delays on system performance. In practical industrial processes, input time delays are common phenomena, such as signal transmission lags, processing delays, etc., all of which can significantly affect the performance of the control system. Inspired by these studies, this paper focuses on a PLIR system that includes input time delays. To achieve precise tracking of the desired trajectory, this paper aims to develop an advanced control strategy that can effectively deal with input time delays and accurately control system behavior by utilizing available real-time input-output data. The key contributions of this research are outlined as follows:

(1) This study presents an adaptive control technique that operates without a predefined model and relies on a condensed dynamic linearization data model. A key characteristic of this method is that it relies solely on the input and output data of the system, without the need for a mathematical model of the control system. This implies that the method is purely data-driven and is capable of learning and adjusting directly from the data generated by the system’s operation, without the necessity of prior modeling or understanding of the system’s internal structure and parameters. This is particularly important for complex systems, where obtaining an accurate and complete mathematical model is often extremely difficult or even impossible.

(2) In this study, we pay special attention to the impact of input time delays on the PLIR system’s performance and propose an innovative intelligent control strategy to address this challenge. In practical industrial settings, input time delays caused by factors such as signal transmission lags and system response times can negatively affect the precision and stability of the control system. Therefore, this research is dedicated to developing a control method that can effectively handle this time delay issue, ensuring precise tracking of the predetermined trajectory.

(3) Furthermore, implementing the model-free adaptive control approach within the PLIR system is crucial for enhancing its operational efficiency. By introducing model-free adaptive control, our approach not only has the potential to significantly improve the detection precision and operational efficiency of the PLIR system but also enhances its robustness and reliability in variable environments.

The organization of this article is as follows. Section 2 describes the problem, presenting the structure of the PLIR and the equation in Euler-discretized form. Section 3 focuses on the design of the model-free adaptive control, where a model-free adaptive controller using compact-form dynamic linearization is designed for the PLIR to track the desired trajectory. Section 4 presents the stability analysis and provides a theoretical justification for the stability of the designed control system. Section 5 demonstrates the performance of the control strategy through simulation analysis. Finally, Section 6 enunciates the relevant conclusions drawn from this research.

2. Problem Statement

Consider the PLIR whose mechanical structure is depicted in Figure 1, Figure 2 and Figure 3. Let ${\alpha }_1$ denote the angle of inclination of the robot’s body relative to the $X_1$-axis, and let ${\alpha }_2$ be the angle of the actuator bar from its initial position to the active position. $O_1$ and $O_2$ represent the centers of mass of the robot’s body and the counter-weight box, respectively, which are also active joints. $X_1,X_2,Y_1,Y_2,Z_1,$ and $Z_2$ are specific coordinate axes used to represent the position of a particular point on the robot in the coordinate system. Define $r_1$ as the distance to the robot’s center of gravity and $r_2$ as the distance from the center of gravity to the robot’s core. The mass of the robot and the weight are denoted by $m_a$ and $m_b$, respectively. Let u represent the system control moment and $\tau$ represent the system input delay. The variable w represents the length of the actuator bar, while d denotes the height of the T-shaped base. We can choose parameters such that $m_a{r}_1=m_b{r}_2$ [31].

In light of the influence of unpredictable environmental factors, including signal interruptions and data delays, it is essential for the dynamic model of the system to incorporate these disturbances. The control strategy employs adaptive techniques to counteract the impact of such disruptions, thereby enhancing the system’s stability and reliability across diverse conditions.

According to [31], the state equation is represented by

$$X=\left[\begin{array}{ccc}{x}_1& x_2& \begin{array}{cc}{x}_3& x_4\end{array}\end{array}\right]=\left[\begin{array}{ccc}{\alpha }_1& \begin{array}{cc}{\dot{\alpha }}_1& {\alpha }_2\end{array}& {\dot{\alpha }}_2\end{array}\right].$$

When there is a time delay in the control input, we need to incorporate this delay into our model. Assuming the control input delay is $\tau$, at time j, the control input is actually the control input at time $j-\tau$. Using the first-order Euler method for discretization, we can obtain the discrete model of the PLIR [31]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{x}_1(j+1)=x_1\left(j\right)+T{x}_2\left(j\right)\hfill \\ x_2(j+1)=x_2\left(j\right)+{\displaystyle \frac{T\left[-2m_{b}w(-r_2+wsin{x}_3\left(j\right))cos{x}_3\left(j\right)x_2\left(j\right)x_4\left(j\right)-m_{b}g(dsin{x}_1\left(j\right)+wsin{x}_3\left(j\right)cos{x}_1\left(j\right))+u_1(j-\tau )\right]}{m_a{r_1}^2+m_b\left[d^2+{(-r_2+wsin{x}_3\left(j\right))}^2\right]}}\hfill \\ x_3(j+1)=x_3\left(j\right)+T{x}_4\left(j\right)\hfill \\ x_4(j+1)=x_4\left(j\right)+{\displaystyle \frac{T\left[m_{b}w(-r_2+wsin{x}_3\left(j\right))cos{x}_3\left(j\right){x_2}^2\left(j\right)-m_{b}gwcos{x}_3\left(j\right)sin{x}_1\left(j\right)+u_2(j-\tau )\right]}{m_b{w}^2}}\hfill \end{array}\right.\end{array}$$

(1)

where T is the sampling time.

It is noteworthy to highlight that the control scheme does not utilize system model information, such as system structure or order. The system models discussed previously are solely for generating input-output data and play no role in the design of the controller. The core objective of this research is to achieve precise tracking of the desired time-varying trajectory, a process that is conducted based entirely on input and output data, without relying on the system’s internal model information. Our objective is to engineer a control mechanism that utilizes available input and output data, refining the dynamic system’s response through incremental learning and adjustment processes, with the goal of achieving precise tracking of a specified time-evolving trajectory. This data-driven approach to control holds significant value for complex or unknown system dynamics, as it removes the requirement for an accurate mathematical model of the system, instead extracting the necessary information for control directly from operational data.

The PLIR model described in this section provides a crucial framework for the control strategies developed in the following sections. The next section explains how the control technique is tailored to this specific model.

3. Model-Free Adaptive Control Design

In this section, our model-free adaptive control approach adopts an innovative data-driven method that constructs an approximate model reflecting the current dynamic behavior of the system through real-time analysis of the system’s input and output data. This approximate model is progressively updated and refined with the continuous acquisition of new data, enabling the control strategy to adapt to variations in system parameters.

3.1. Compact-Form Dynamic Linearization

Consider nonlinear systems with discrete time that include input time delays for the PLIR system, which can be described as

$$Y(j+1)=f(Y\left(j\right),\dots ,Y(j-n_Y),U(j-\tau ),\dots ,U(j-\tau -n_U)),$$

(2)

where $U={\left[\begin{array}{cc}{u}_1& u_2\end{array}\right]}^T$ represents the control input, $Y={\left[\begin{array}{cc}{y}_1& y_2\end{array}\right]}^T$ denotes the output vectors of the system, $n_U$ and $n_Y$ denote unknown system orders, and f is an unknown function with nonlinearity.

Before introducing the compact-form dynamic linearization method, we outline the following assumptions about the system.

Assumption 1

([24]). The $(n_Y+2)$-th variable’s partial derivative of f is continuously differentiable, with discontinuities occurring only at a limited number of time points.

Assumption 2

([24]). With the exception of a finite number of time points, the function adheres to the generalized Lipschitz condition. This implies that for any $j_1\ne j_2$, $j_1,j_2\ge 0$ and $U(j_1-\tau )\ne U(j_2-\tau )$, the following inequality is valid:

$$∥Y(j_1+1)-Y(j_2+1)∥\le b∥U(j_1-\tau )-U(j_2-\tau )∥,$$

where $Y(j_i+1)=f(Y\left(j_i\right),\dots ,Y(j_i-n_Y),U(j_i-\tau ),\dots ,U(j_i-n_U-\tau ))$, for $i=1, 2$; and $b>0$ is a constant.

In the context of practical applications, the initial presumption is typically a stipulation encountered in the realm of nonlinear system control design. The subsequent presumption dictates an upper boundary for the system’s output alteration rate. When considered from an energy conservation viewpoint, a circumscribed input energy fluctuation should elicit a defined output energy modification within the system.

To simplify the discussion that follows, let $\Delta Y(j+1)=Y(j+1)-Y\left(j\right)$ represent the change in output between two consecutive time instances. Also, $\Delta U(j-\tau )=U(j-\tau )-U(j-\tau -1)$ represents the change in input between two consecutive time instances.

To further investigate the stability of the system, the following lemma is proposed.

Lemma 1.

Considering the nonlinear system (2) that adheres to Assumptions 1 and 2, it is necessary to identify a time-dependent parameter when $∥\Delta U(j-\tau )∥\ne 0$, known as the pseudo-partial derivative, which enables the transformation of system (2) into a concise data model of dynamic linearization.

$$\Delta Y(j+1)=\varphi (j-\tau )\Delta U(j-\tau ),$$

(3)

and further,

$$\Delta Y(j+\tau +1)=\varphi \left(j\right)\Delta U\left(j\right),$$

(4)

where $\varphi \left(j\right)=\left[\begin{array}{cc}{\varphi }_{11}\left(j\right)& {\varphi }_{12}\left(j\right)\\ {\varphi }_{21}\left(j\right)& {\varphi }_{22}\left(j\right)\end{array}\right]$. According to Assumption 2, it can be deduced that $∥\varphi \left(j\right)∥\le b$.

Compared to other linearization techniques for nonlinear functions, compact-form dynamic linearization possesses the following distinctive properties:

(1) The method does not rely on the mathematical model of the control object.

(2) It establishes an exact dynamic linearization data model, not an approximate one.

(3) The dynamic linearization model exhibits a time-varying incremental structure that is simple in construction and has a minimal number of parameters. It is a data model specifically designed for controller design and does not rely on traditional transfer function models. The introduction of the concept of the pseudo-partial derivative eliminates the need for complex high-order controller designs.

3.2. Controller Design

To track the desired trajectory, consider the following cost function:

$$J\left(U\left(j\right)\right)={∥Y_d(j+\tau +1)-Y(j+\tau +1)∥}^2+\lambda {∥U\left(j\right)-U(j-1)∥}^2,$$

(5)

where $Y_d(j+\tau +1)$ is the desired output signal and $\lambda$ represents the weighting factor used to limit the variation in the control input.

Substituting the compact-form dynamic linearized data model (4) into Equation (5), and differentiating Equation (5) with respect to the control input $U\left(j\right)$ and setting it to zero yields the control input algorithm.

$$U\left(j\right)=U(j-1)+{\displaystyle \frac{\rho {\varphi }^T\left(j\right)}{\lambda +{∥\varphi \left(j\right)∥}^2}}(Y_d(j+\tau +1)-Y(j+\tau )),$$

(6)

where $\rho \in \left(0,1\right]$ is the step-size factor, which enhances the versatility of the control scheme.

Since the control input algorithm (6) contains an unknown parameter $\varphi \left(j\right)$, which is time-varying, it is difficult to obtain its exact value directly. Therefore, we need to design an estimation algorithm that only uses the input and output data of the system to estimate the parameter $\varphi \left(j\right)$. To this end, we propose the following estimation criterion function:

$$J\left(\varphi \left(j\right)\right)={∥Y(j+\tau )-Y(j+\tau -1)-\varphi \left(j\right)\Delta U(j-1)∥}^2+\mu {∥\varphi \left(j\right)-\widehat{\varphi }(j-1)∥}^2,$$

(7)

By taking the derivative of Equation (7) with respect to $\varphi \left(j\right)$ and finding the extremum, the estimation algorithm for the pseudo-partial derivative can be derived as

$$\widehat{\varphi }\left(j\right)=\widehat{\varphi }(j-1)+{\displaystyle \frac{\eta \Delta U^T(j-1)}{\mu +{∥\Delta U(j-1)∥}^2}}\left(Y(j+\tau )-Y(j+\tau -1)-\widehat{\varphi }(j-1)\Delta U(j-1)\right),$$

(8)

where $\eta \in \left(0,2\right)$ is a step-size factor, $\mu >0$ is a weighting factor, and $\widehat{\varphi }\left(j\right)$ is the estimation of $\varphi \left(j\right)$.

Therefore, the model-free adaptive control scheme based on compact-form dynamic linearization can be represented by

$$\widehat{\varphi }\left(j\right)=\widehat{\varphi }(j-1)+{\displaystyle \frac{\eta \Delta U^T(j-1)}{\mu +{∥\Delta U(j-1)∥}^2}}\left(Y(j+\tau )-Y(j+\tau -1)-\widehat{\varphi }(j-1)\Delta U(j-1)\right),$$

(9)

$$\begin{array}{c}{\widehat{\varphi }}_{ii}\left(j\right)={\widehat{\varphi }}_{ii}\left(1\right),\begin{array}{cc} & if\begin{array}{cc} & \left|{\widehat{\varphi }}_{ii}\left(j\right)\right|<b_2,\begin{array}{cc} & or\begin{array}{cc} & \left|{\widehat{\varphi }}_{ii}\left(j\right)\right|>\alpha b_2\end{array}\end{array}\end{array}\end{array}\hfill \\ {\widehat{\varphi }}_{ik}\left(j\right)={\widehat{\varphi }}_{ik}\left(1\right),\begin{array}{cc} & if\begin{array}{cc} & \left|{\widehat{\varphi }}_{ik}\left(j\right)\right|>b_1,\end{array}\end{array}\begin{array}{cc} & i\ne k\end{array}\hfill \end{array}$$

(10)

$$U\left(j\right)=U(j-1)+{\displaystyle \frac{\rho {\widehat{\varphi }}^T\left(j\right)}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}(Y_d(j+\tau +1)-Y(j+\tau )),$$

(11)

where ${\varphi }_a$ and ${\varphi }_b$ are positive constants. Similar to [32], a reset rule (10) is introduced in the estimation of the parameter estimation algorithm $\widehat{\varphi }\left(j\right)$, enabling the parameter update law to better track the time-varying parameter. $\widehat{\varphi }\left(1\right)$ is the initial value of $\widehat{\varphi }\left(j\right)$.

Equations (9) through (11) demonstrate that the proposed control scheme relies solely on the input and output data of the closed-loop system, eliminating the need for model information about the controlled system. This characteristic underscores the data-driven nature of the algorithm, which ensures the tracking performance of the system without utilizing pre-existing knowledge of the system’s behavioral principles. As such, the scheme exhibits superior tracking accuracy and robustness, making it particularly suitable for applications where system models are either unknown or too complex to be accurately defined.

4. Stability Analysis

Before verifying the main results, the following assumption and lemma are presented.

Assumption 3

([33]). The pseudo-partial derivative is considered a matrix that is diagonal and dominant such that $\left|{\varphi }_{ik}\right|\le b_1$, $b_2\le \left|{\varphi }_{ii}\left(j\right)\right|\le \alpha b_2$, $i=1,2$, $k=1,2$, $i\ne k$, $\alpha \ge 1$, $b_2>b_1(2\alpha +1)$, and the sign of all the elements remains unchanged.

Lemma 2

([34]). Let $A=\left[a_{ij}\right]\in C^{n\times n}$. For each $1\le i\le n$, the Gerschgorin disk is defined as $D_i=\left\{z|\left|z-a_{ii}\right|\le {\displaystyle \sum _{k\ne i,k=1}^n}\left|a_{ik}\right|\right\}$, $z\in C$. The Gerschgorin domain is defined as a union of all the Gerschgorin disks: $D_A={\displaystyle \bigcup _{i=1}^n}{D}_i$. The eigenvalues of matrix A are located in its Gerschgorin domain $D_A$.

To further validate the stability of the proposed control strategy and the convergence of the tracking error, we present the following theorem, which demonstrates the boundedness of the estimated pseudo-partial derivatives and the error convergence.

Theorem 1.

In the case of the discrete-time nonlinear system (2) that adheres to Assumptions 1–3, selecting the control parameter λ, where $\lambda >{\lambda }_{min}$, and applying a model-free adaptive control approach based on compact-form dynamic linearization (9)–(11) guarantees the following:

(1) The estimated pseudo-partial derivative value is bounded;

(2) The system’s tracking error converges boundedly.

Proof. 

(1) Upon meeting the criteria of the reset algorithm (10), the bounded nature of the estimated pseudo-partial derivative, denoted as $\widehat{\varphi }\left(j\right)$, can be inferred. This inference is based on the premise that its initial estimate, $\widehat{\varphi }\left(1\right)$, is confined within certain limits.

In other cases, let $\widehat{\varphi }\left(j\right)=\left[\begin{array}{c}{\widehat{\varphi }}_1\left(j\right)\\ {\widehat{\varphi }}_2\left(j\right)\end{array}\right]$, ${\widehat{\varphi }}_i\left(j\right)=\left[{\widehat{\varphi }}_{i1}\left(j\right),{\widehat{\varphi }}_{i2}\left(j\right)\right]$, $i=1,2$.

According to the compact-form dynamic linearization data model (4), the parameter estimation algorithm (9) can be written as

$${\widehat{\varphi }}_i\left(j\right)={\widehat{\varphi }}_i(j-1)+{\displaystyle \frac{\eta \Delta U^T(j-1)}{\mu +{∥\Delta U(j-1)∥}^2}}\left(\Delta Y_i(j+\tau )-{\widehat{\varphi }}_i(j-1)\Delta U(j-1)\right),$$

(12)

where $\Delta Y_i(j+\tau )={\varphi }_i(j-1)\Delta U(j-1)$, $i=1,2$.

Subtract ${\varphi }_i\left(j\right)$ from both sides of (12), and let ${\tilde{\varphi }}_i\left(j\right)={\widehat{\varphi }}_i\left(j\right)-{\varphi }_i\left(j\right)$. Then,

$$\begin{array}{c}{\tilde{\varphi }}_i\left(j\right)={\widehat{\varphi }}_i(j-1)-{\varphi }_i(j-1)+{\varphi }_i(j-1)-{\varphi }_i\left(j\right)-{\displaystyle \frac{\eta {\tilde{\varphi }}_i(j-1)\Delta U(j-1)\Delta U^T(j-1)}{\mu +{∥\Delta U(j-1)∥}^2}}\hfill \\ ={\tilde{\varphi }}_i(j-1)+{\varphi }_i(j-1)-{\varphi }_i\left(j\right)-{\displaystyle \frac{\eta {\tilde{\varphi }}_i(j-1)\Delta U(j-1)\Delta U^T(j-1)}{\mu +{∥\Delta U(j-1)∥}^2}},\hfill \end{array}$$

(13)

and from Lemma 1, we know that $∥\varphi \left(j\right)∥\le b$, and then $∥{\varphi }_i(j-1)-{\varphi }_i\left(j\right)∥\le 2b$. Taking the norm on both sides of (13) yields

$$\begin{array}{c}∥{\tilde{\varphi }}_i\left(j\right)∥=∥{\varphi }_i(j-1)-{\varphi }_i\left(j\right)∥+∥{\tilde{\varphi }}_i(j-1)-{\displaystyle \frac{\eta {\tilde{\varphi }}_i(j-1)\Delta U(j-1)\Delta U^T(j-1)}{\mu +{∥\Delta U(j-1)∥}^2}}∥\hfill \\ \le 2b+∥{\tilde{\varphi }}_i(j-1)-{\displaystyle \frac{\eta {\tilde{\varphi }}_i(j-1)\Delta U(j-1)\Delta U^T(j-1)}{\mu +{∥\Delta U(j-1)∥}^2}}∥,\hfill \end{array}$$

(14)

For the second term in (14), squaring it yields

$$\begin{array}{c}{∥{\tilde{\varphi }}_i(j-1)-{\displaystyle \frac{\eta {\tilde{\varphi }}_i(j-1)\Delta U(j-1)\Delta U^T(j-1)}{\mu +{∥\Delta U(j-1)∥}^2}}∥}^2\hfill \\ =\left(-2+{\displaystyle \frac{\eta {∥\Delta U(j-1)∥}^2}{\mu +{∥\Delta U(j-1)∥}^2}}\right)\times {\displaystyle \frac{\eta {∥{\tilde{\varphi }}_i(j-1)\Delta U(j-1)∥}^2}{\mu +{∥\Delta U(j-1)∥}^2}}+{∥{\tilde{\varphi }}_i(j-1)∥}^2,\hfill \end{array}$$

(15)

and there exists $0<\eta <2$, $\mu >0$ such that

$$\left(-2+{\displaystyle \frac{\eta {∥\Delta U(j-1)∥}^2}{\mu +{∥\Delta U(j-1)∥}^2}}\right)<0$$

(16)

Equations (15) and (16) show that there must exist a constant, $0<d_1<1$, that satisfies

$$∥{\tilde{\varphi }}_i(j-1)-{\displaystyle \frac{\eta {\tilde{\varphi }}_i(j-1)\Delta U(j-1)\Delta U^T(j-1)}{\mu +{∥\Delta U(j-1)∥}^2}}∥\le d_1∥{\tilde{\varphi }}_i(j-1)∥,$$

(17)

Substituting (17) into (14) yields

$$∥{\tilde{\varphi }}_i\left(j\right)∥\le d_1∥{\tilde{\varphi }}_i(j-1)∥+2b\le \dots \le {d_1}^{j-1}∥{\tilde{\varphi }}_i\left(1\right)∥+{\displaystyle \frac{2b(1-{d_1}^{j-1})}{1-d_1}}.$$

(18)

Therefore, it can be concluded that ${\tilde{\varphi }}_i\left(j\right)$ is bounded. Given the boundedness of $\varphi \left(j\right)$, it follows that $\widehat{\varphi }\left(j\right)$ is also bounded.

(2) The tracking error of the system is defined as $e(j+\tau )=Y_d(j+\tau )-Y(j+\tau )$. Then, $e(j+\tau +1)$ can be represented by

$$e(j+\tau +1)=Y_d(j+\tau +1)-Y(j+\tau ),$$

(19)

when the sampling time is sufficiently short, assuming $Y_d(j+\tau +1)=Y_d(j+\tau )$. Substituting the compact-form dynamic linearized data model (4) and the control input algorithm (11) into Equation (19), we obtain

$$\begin{array}{c}e(j+\tau +1)=e(j+\tau )-\varphi \left(j\right)\Delta U\left(j\right)\hfill \\ =(I-{\displaystyle \frac{\rho \varphi \left(j\right){\widehat{\varphi }}^T\left(j\right)}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}})e(j+\tau ),\hfill \end{array}$$

(20)

From Lemma 2, we know that

$$D_k=\left\{z\left|\left|z-\left(1-{\displaystyle \frac{\rho {\displaystyle \sum _{i=1}^2}{\varphi }_{ki}{\widehat{\varphi }}_{ki}\left(j\right)}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}\right)\right|\le {\displaystyle \sum _{h=1,h\ne k}^2}{\displaystyle \frac{\rho \left|{\displaystyle \sum _{i=1}^2}{\varphi }_{ki}\left(j\right){\widehat{\varphi }}_{hi}\left(j\right)\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}\right.\right\},$$

(21)

where z represents the eigenvalue of the matrix $(I-{\displaystyle \frac{\rho \varphi \left(j\right){\widehat{\varphi }}^T\left(j\right)}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}})$. $D_k$, $k=1,2$ is the Gerschgorin disk. □

Using the triangle inequality, expression (21) can be written as follows:

$$D_k=\left\{z\left|\left|z\right|\le \left|1-{\displaystyle \frac{\rho {\displaystyle \sum _{i=1}^2}{\varphi }_{ki}\left(j\right){\widehat{\varphi }}_{ki}\left(j\right)}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}\right|+\sum _{h=1,h\ne k}^2{\displaystyle \frac{\rho \left|{\displaystyle \sum _{i=1}^2}{\varphi }_{ki}\left(j\right){\widehat{\varphi }}_{hi}\left(j\right)\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}\right.\right\},$$

(22)

Based on Assumption 3 and the reset algorithm, we know that

$$1-{\displaystyle \frac{\rho {\displaystyle \sum _{i=1}^2}\left|{\varphi }_{ki}\left(j\right)\right|\left|{\widehat{\varphi }}_{ki}\left(j\right)\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}\le 1-{\displaystyle \frac{\rho \left|{\varphi }_{kk}\left(j\right)\right|\left|{\widehat{\varphi }}_{kk}\left(j\right)\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}\le 1-{\displaystyle \frac{\rho {b_2}^2}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}$$

(23)

and

$$\begin{array}{c}\rho {\displaystyle \frac{{\displaystyle \sum _{h=1,h\ne k}^2}\left|{\displaystyle \sum _{i=1}^2}{\varphi }_{ki}\left(j\right){\widehat{\varphi }}_{hi}\left(j\right)\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}\le \rho {\displaystyle \frac{{\displaystyle \sum _{h=1,h\ne k}^2}{\displaystyle \sum _{i=1}^2}\left|{\varphi }_{ki}\left(j\right)\right|\left|{\widehat{\varphi }}_{hi}\left(j\right)\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}\hfill \\ =\rho {\displaystyle \frac{{\displaystyle \sum _{h=1,h\ne k}^2}\left|{\varphi }_{kk}\left(j\right)\right|\left|{\widehat{\varphi }}_{hk}\left(j\right)\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}+\rho {\displaystyle \frac{{\displaystyle \sum _{h=1,h\ne k}^2}{\displaystyle \sum _{i=1,i\ne k}^2}\left|{\varphi }_{ki}\left(j\right)\right|\left|{\widehat{\varphi }}_{hi}\left(j\right)\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}\hfill \\ =\rho {\displaystyle \frac{{\displaystyle \sum _{h=1,h\ne k}^2}\left|{\varphi }_{kk}\left(j\right)\right|\left|{\widehat{\varphi }}_{hk}\left(j\right)\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}+\rho {\displaystyle \frac{{\displaystyle \sum _{h=1,h\ne k}^2}\left|{\varphi }_{kh}\left(j\right)\left|{\widehat{\varphi }}_{hh}\left(j\right)\right|\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}+\rho {\displaystyle \frac{{\displaystyle \sum _{h=1,h\ne k}^2}{\displaystyle \sum _{i=1,i\ne k,h}^2}\left|{\varphi }_{ki}\left(j\right)\right|\left|{\widehat{\varphi }}_{hi}\left(j\right)\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}\hfill \\ \le {\displaystyle \frac{2\alpha b_1{b}_2}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}<1.\hfill \end{array}$$

(24)

By combining (23) and (24), we obtain

$$1-{\displaystyle \frac{\rho {\displaystyle \sum _{i=1}^2}\left|{\varphi }_{ki}\left(j\right)\right|\left|{\widehat{\varphi }}_{ki}\left(j\right)\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}+{\displaystyle \sum _{h=1,h\ne k}^2}{\displaystyle \frac{\rho \left|{\displaystyle \sum _{i=1}^2}{\varphi }_{ki}\left(j\right){\widehat{\varphi }}_{hi}\left(j\right)\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}\le 1-\rho {\displaystyle \frac{{b_2}^2-2\alpha b_1{b}_2}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}=1-\rho {\displaystyle \frac{b_2(b_2-2\alpha b_1)}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}.$$

(25)

Based on Assumption 3, Equation (25) can be simplified as follows:

$$1-{\displaystyle \frac{\rho {\displaystyle \sum _{i=1}^2}\left|{\varphi }_{ki}\left(j\right)\right|\left|{\widehat{\varphi }}_{ki}\left(j\right)\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}+\sum _{h=1,h\ne k}^2{\displaystyle \frac{\rho \left|{\displaystyle \sum _{i=1}^2}{\varphi }_{ki}\left(j\right){\widehat{\varphi }}_{hi}\left(j\right)\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}<1-\rho {\displaystyle \frac{b_2{b}_1}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}.$$

(26)

From Assumption 3 and the reset algorithm, we obtain that ${\varphi }_{ki}\left(j\right){\widehat{\varphi }}_{ki}\left(j\right)>0$ for $i=1,2$ and $k=1,2$. Consequently, if ${\lambda }_{min}>0$ is a positive constant ${\lambda }_{min}>0$, the following inequality is satisfied for $\lambda >{\lambda }_{min}$:

$${\displaystyle \frac{{\displaystyle \sum _{i=1}^2}\left|{\varphi }_{ki}\left(j\right)\right|\left|{\widehat{\varphi }}_{ki}\left(j\right)\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}\le {\displaystyle \frac{{\alpha }^2{b_2}^2}{{\lambda }_{min}+{∥\widehat{\varphi }\left(j\right)∥}^2}}<1.$$

(27)

Choose $0<\rho <1$ and $\lambda >{\lambda }_{min}$ such that

$$0<M_1\le {\displaystyle \frac{b_2{b}_1}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}<1,$$

(28)

By combing (25) and (28), we obtain

$$\left|1-{\displaystyle \frac{\rho {\displaystyle \sum _{i=1}^2}{\varphi }_{ki}\left(j\right){\widehat{\varphi }}_{ki}\left(j\right)}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}\right|+{\displaystyle \sum _{h=1,h\ne k}^2}{\displaystyle \frac{\rho \left|{\displaystyle \sum _{i=1}^2}{\varphi }_{ki}\left(j\right){\widehat{\varphi }}_{hi}\left(j\right)\right|}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}<1-\rho M_1<1.$$

(29)

From (21) and (29), we obtain

$$s\left(I-{\displaystyle \frac{\rho \varphi \left(j\right){\widehat{\varphi }}^T\left(j\right)}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}\right)\le 1-\rho M_1<1,$$

(30)

where $s\left(I-{\displaystyle \frac{\rho \varphi \left(j\right){\widehat{\varphi }}^T\left(j\right)}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}\right)$ is the spectral radius of the matrix $\left(I-{\displaystyle \frac{\rho \varphi \left(j\right){\widehat{\varphi }}^T\left(j\right)}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}\right)$.

Then, there exists $\epsilon >0$ such that

$${∥I-{\displaystyle \frac{\rho \varphi \left(j\right){\widehat{\varphi }}^T\left(j\right)}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}}∥}_v<1-\rho M_1+\epsilon =d_2<1.$$

(31)

Applying the norm to both sides of Equation (20) yields

$${∥e(j+\tau +1)∥}_v={∥(I-{\displaystyle \frac{\rho \varphi \left(j\right){\widehat{\varphi }}^T\left(j\right)}{\lambda +{∥\widehat{\varphi }\left(j\right)∥}^2}})∥}_v{∥e(j+\tau )∥}_v\le d_2{∥e(j+\tau )∥}_v\le \dots \le {d_2}^{j+\tau }{∥e\left(1\right)∥}_v,$$

(32)

Equation (32) reveals that the system’s tracking error is convergent, indicating that the system is capable of tracking the desired trajectory.

Theorem 1 verifies the stability of the system and ensures that it is capable of tracking the desired trajectory. However, to meet actual engineering needs, such as precision control and rapid industrial processes, the study of stability is not enough; therefore, the system should exhibit faster response times and smoother control inputs to prevent potential damage or defects. Thus, Theorem 2 is presented below to optimize the system’s convergence speed and the smoothness of the control inputs. It employs quantitative methods to meet these advanced performance requirements.

Theorem 2.

For the nonlinear system (2) with discrete time satisfying Assumptions 1–3, if there exists a control parameter $\lambda >{\lambda }_{min}$, $\alpha <1$, $\delta <1$, and the model-free adaptive control Equations (9)–(11) based on compact-form dynamic linearization are utilized, the following results can be obtained:

(1) The system output not only tracks the desired trajectory, but also the convergence speed of the tracking error can be quantified;

(2) The smoothness of the control input can be ensured, which means that the change rate for the control input is limited.

Proof. 

(1) According to Theorem 1, we know that the system tracking error is stable and that

$$e(j+\tau +1)={\displaystyle \frac{(I-\rho \varphi \left(j\right){\widehat{\varphi }}^T\left(j\right))}{(\lambda +|\left|\widehat{\varphi }\left(j\right)\right|{|}^2)}}e(j+\tau ),$$

(33)

To quantify the convergence speed, we consider the spectral radius of the error dynamics, which is the absolute value of the maximum eigenvalue of the matrix ${\displaystyle \frac{I-\rho \psi \left(j\right){\widehat{\psi }}^T\left(j\right)}{\lambda +\left|\right|\widehat{\varphi }\left(j\right){\left|\right|}^2}}$.

Suppose that $\delta$ is the spectral radius; then, $\delta <1$. It seems that the convergence rate of $\left|\right|e(j+\tau )\left|\right|$ is determined by $\delta$. By choosing the appropriate control parameters $\lambda$ and step-size factor $\rho$, we can make $\delta$ as small as possible to accelerate the convergence speed.

(2) Consider the modified control input algorithm

$$U\left(j\right)=U(j-1)+\rho {\widehat{\psi }}^T\left(j\right){\displaystyle \frac{Y_d(j+\tau +1)-Y(j+\tau )}{\lambda +\left|\right|\widehat{\psi }\left(j\right){\left|\right|}^2}}-\gamma \Delta U(j-1),$$

(34)

where $\gamma$ is a positive regulatory factor used to limit the rate of change of the control inputs. We define

$$\Delta U\left(j\right)=U\left(j\right)-U(j-1)=\rho {\widehat{\psi }}^T\left(j\right){\displaystyle \frac{Y_d(j+\tau +1)-Y(j+\tau )}{\lambda +\left|\right|\widehat{\psi }\left(j\right){\left|\right|}^2}}-\gamma \Delta U(j-1).$$

(35)

□

To guarantee the continuity and stability of the control signal, $\Delta U\left(j\right)$ needs to be kept within a reasonable range.

By utilizing the appropriate $\gamma$, we can restrict the growth of $\Delta U\left(j\right)$ so that it avoids drastic changes in the control inputs.

Below we present a stability analysis.

To verify the stability of the system, we consider the following Lyapunov function:

$$V\left(j\right)={\left|\right|e(j+\tau )\left|\right|}^2+{\displaystyle \frac{1}{2\gamma }}{\left|\right|\Delta U\left(j\right)\left|\right|}^2.$$

(36)

Calculate the difference of $V\left(j\right)$:

$$\begin{array}{cc}\hfill \Delta V\left(j\right)& =V(j+1)-V\left(j\right),\hfill \\ \hfill \Delta V\left(j\right)& ={\left|\right|e(j+\tau +1)\left|\right|}^2-{\left|\right|e(j+\tau )\left|\right|}^2+{\displaystyle \frac{1}{2\gamma }}{\left(\right||\Delta U(j+1)||}^2-{\left|\right|\Delta U\left(j\right)\left|\right|}^2).\hfill \end{array}$$

(37)

When $\Delta V\left(j\right)<0$, it means that $V\left(j\right)$ is monotonically decreasing. Then, we need to prove that $\Delta V\left(j\right)<0$.

Assume that

$$\begin{array}{cc}\hfill V\left(j\right)& ={\left|\right|e(j+\tau )\left|\right|}^2+{\displaystyle \frac{1}{2\gamma }}{\left|\right|\Delta U\left(j\right)\left|\right|}^2,\hfill \\ \hfill \Delta V& =V(j+1)-V\left(j\right),\hfill \\ \hfill \Delta V& ={\left|\right|e(j+\tau +1)\left|\right|}^2-{\left|\right|e(j+\tau )\left|\right|}^2+{\displaystyle \frac{1}{2\gamma }}{\left(\right||\Delta V(j+1)||}^2-{\left|\right|\Delta V\left(j\right)\left|\right|}^2),\hfill \end{array}$$

(38)

using the triangular inequality,

$$\left|\right|e(j+1)\left|\right|\le \left|\right|e\left(j\right)\left|\right|+\left|\right|\Delta e\left(j\right)\left|\right|,$$

(39)

where $\Delta e\left(j\right)=e(j+1)-e\left(j\right)$.

Also, due to the system dynamics,

$$\begin{array}{cc}\hfill e(j+1)& =Y_d(j+1)-Y(j+1),\hfill \\ \hfill \Delta e\left(j\right)& =e(j+1)-e\left(j\right)=Y_d(j+1)-Y(j+1)-(Y(j+1)-Y\left(j\right)).\hfill \end{array}$$

(40)

Suppose that the control strategy enables

$$\begin{array}{c}\hfill \left|\right|\Delta e\left(j\right)\left|\right|\le -\alpha \left|\right|e\left(j\right)\left|\right|\end{array},$$

(41)

where $\alpha$ is a positive constant that represents the rate of error reduction.

When satisfying the triangular inequality

$$\begin{array}{cc}\hfill {\left|\right|e(j+1)\left|\right|}^2& \le \left(\right|\left|e\left(j\right)\right||+{\left|\right|\Delta e\left(j\right)\left|\right|)}^2={\left|\right|e\left(j\right)\left|\right|}^2+2\left|\right|e\left(j\right)\left|\right|.\left|\right|\Delta e\left(j\right)\left|\right|+{\left|\right|\Delta e\left(j\right)\left|\right|}^2\hfill \\ \le {\left|\right|e\left(j\right)\left|\right|}^2+2\left|\right|e\left(j\right)\left|\right|.\left|\right|\Delta e\left(j\right)\left|\right|+{\left|\right|\Delta e\left(j\right)\left|\right|}^2\hfill \\ \hfill & \le {\left|\right|e\left(j\right)\left|\right|}^2-{2\alpha \left|\right|e\left(j\right)\left|\right|}^2+({\alpha }^2-1)\left|\right|e\left(j\right){\left|\right|}^2,\hfill \end{array}$$

(42)

choose ${\alpha }^2<1$, so ${\left|\right|e(j+1)\left|\right|}^2{<\left|\right|e\left(j\right)\left|\right|}^2$. Then, we can obtain $\Delta V\left(j\right)={\left|\right|e(j+1)\left|\right|}^2-{\left|\right|e\left(j\right)\left|\right|}^2<0$ such that $\Delta V\left(j\right)<0$. Therefore, the proof of Theorem 2 is complete.

5. Simulation

In this section, we simulate the tracking capability of the PLIR system to demonstrate the effectiveness of the proposed model-free adaptive control method based on compact-form dynamic linearization. In the subsequent simulation experiments, all mathematical models serve solely as the controlled system input and output data generators, and no information from these models is incorporated into the design of the model-free adaptive controller.

The parameters for the PLIR are $m_a=63$ kg, $m_b=27$ kg, $r_1=0.18$ m, $r_2=0.42$ m, $w=0.5$ m, and $d=0.5$ m. The parameters for the model-free adaptive controller based on compact-form dynamic linearization are $\eta =1$, $\rho =1$, $\lambda =0.5$, and $\mu =1$, ${\widehat{\varphi }}_1={\widehat{\varphi }}_2=\left[\begin{array}{cc}0.5& 0\\ 0& 0.5\end{array}\right]$. The initial values of the system are $x_1\left(1\right)=x_1\left(2\right)=x_3\left(1\right)=x_3\left(2\right)=0.5$, $x_2\left(1\right)=x_2\left(2\right)=x_4\left(1\right)=x_4\left(2\right)=0$, $u_1=u_2={\left[0,0\right]}^T$, the time delay $\tau =0.01s$, and $T=0.02s$. The desired trajectories are

$$\left\{\begin{array}{c}{y}_{1d}\left(j\right)=0.5+0.25cos\left({\displaystyle \frac{0.25\pi j}{100}}\right)+0.25sin\left({\displaystyle \frac{0.5\pi j}{100}}\right)\hfill \\ y_{2d}\left(j\right)=0.5+0.25sin\left({\displaystyle \frac{0.25\pi j}{100}}\right)+0.25sin\left({\displaystyle \frac{0.5\pi j}{100}}\right)\hfill \end{array}\right.$$

The results of the simulation experiments are presented in Figure 4, Figure 5, Figure 6 and Figure 7. A detailed examination of Figure 4 and Figure 5 clearly indicates that by implementing the model-free adaptive control method based on compact-form dynamic linearization introduced in this research, the PLIR system can successfully track the desired trajectory. This finding demonstrates the effectiveness of the proposed control method. Furthermore, Figure 6 illustrates the situation of the control inputs for the PLIR system. The variations in the control inputs can be observed in the figure, reflecting the activity of the controller in adjusting the inputs to achieve the desired trajectory tracking. Lastly, Figure 7 reveals a critical characteristic, which is the boundedness of the pseudo-partial derivative matrix. This property is essential for ensuring the stability and reliability of the control system. In summary, through this series of analyses, we can conclude that the model-free adaptive control strategy based on compact-form dynamic linearization presented in this article not only enables the PLIR to accurately track the expected trajectory but also exhibits good stability and reliability during the control process. These simulation results provide strong support for the practical application of this control method.

To provide a thorough demonstration of the effectiveness of the proposed model-free adaptive control approach using conventional model-based adaptive control methods, we present the results of the simulation experiments under identical input conditions in Figure 8, Figure 9, Figure 10 and Figure 11. Figure 8 and Figure 9 show the tracking performance of $y_1\left(j\right)$ and $y_2\left(j\right)$, respectively. Figure 10 illustrates the control input, and Figure 11 depicts the estimation of the pseudo-partial-derivative matrix. Additionally, by conducting a performance comparison of different control methods, the advantage of the model-free adaptive control approach is demonstrated, as shown in

Table 1. Performance comparison of different control methods.

Control Method	Output Tracking Error	Control Input Variation	Response Time
Model-free adaptive control	0.15	±2.5	2.1 s
Conventional model-based adaptive control	0.08	±1.9	1.5 s

. In the future, the method proposed in this paper will continue to improve in terms of tracking accuracy, control input, response speed, etc.

6. Conclusions

This study introduces a novel model-free adaptive control method based on compact-form dynamic linearization to address the challenge of control input time delays in PLIR systems. To begin with, the core advantage of this method lies in its complete reliance on real-time input and output data throughout the controller design and stability analysis process, thereby eliminating the need for an accurate system model. This is significant for practical applications where obtaining a precise model of the system can be challenging or even impossible in many instances. Moreover, the design of this model-free adaptive control method is more straightforward. Unlike traditional adaptive controls, it does not require complex system identification and control law design procedures, making it easier to implement and apply in engineering practice. Finally, through rigorous theoretical analysis, the convergence of the error in this method is assured. This means that regardless of the changes the system faces, the control scheme can ensure that the system output tracking error tends to a minimum in a stable state, which is key to ensuring the precision and reliability of the control system.

Future research will aim to combine data-driven approaches, such as machine learning, reinforcement learning, and deep learning, with model-free adaptive control techniques to improve control performance in challenging environmental conditions. Moreover, conducting field tests in realistic settings will be crucial to confirming the robustness of the proposed control strategy in the face of environmental disturbances.