An Overview of Model-Free Adaptive Control for the Wheeled Mobile Robot

Abstract

Control technology for wheeled mobile robots is one of the core focuses in the current field of robotics research. Within this domain, model-free adaptive control (MFAC) methods, with their advanced data-driven strategies, have garnered widespread attention. The unique characteristic of these methods is their ability to operate without relying on prior model information of the control system, which showcases their exceptional capability in ensuring closed-loop system stability. This paper extensively details three dynamic linearization techniques of MFAC: compact form dynamic linearization, partial form dynamic linearization and full form dynamic linearization. These techniques lay a solid theoretical foundation for MFAC. Subsequently, the article delves into some advanced MFAC schemes, such as dynamic event-triggered MFAC and iterative learning MFAC. These schemes further enhance the efficiency and intelligence level of control systems. In the concluding section, the paper briefly discusses the future development potential and possible research directions of MFAC, aiming to offer references and insights for future innovations in control technology for wheeled mobile robots.

1. Introduction

Due to the swift advancement of science and technology, wheeled mobile robots have become extensively utilized across various sectors in recent years, including military reconnaissance, industrial automated logistics, space-exploration missions and modern agricultural irrigation [1]. In agricultural scenarios, the effective application of Skid–Steer Mobile Robots necessitates precise trajectory planning and motion-control strategies to handle complex and varied terrain conditions and operational requirements [2]. The hardware components of a wheeled mobile robot primarily consist of a chassis, wheels, drive units and a control system. The chassis serves as the backbone of the structure, supporting all components and bearing the mechanical stresses incurred during motion. The wheels are crucial for locomotion, directly facilitating the robot’s movement across surfaces. Furthermore, drive motors function as the powerhouse that rotates the wheels. The control system acts as its ‘brain’, processing external commands and orchestrating the robot’s mobility accordingly. In these applications, trajectory-tracking control represents a crucial technical challenge. Trajectory tracking is divided into two main forms: path following and real-time trajectory tracking. Path following focuses on ensuring that the robot moves along a predetermined path without strict control over speed. Although this type of tracking is simple, it is often limited in practical scenarios because it cannot precisely control the driving force, which limits its utility. On the other hand, real-time trajectory tracking necessitates the robot to concurrently and precisely adhere to both the position and speed of the designated trajectory, thereby enhancing navigational accuracy and control. Because it can more comprehensively meet the complex requirements of practical applications, real-time trajectory-tracking control is considered to be more effective and practical. Longitudinal slip control adjusts the robot’s forward and backward movements by managing the speed of the drive wheels, preventing slippage caused by excessive acceleration or deceleration. This is crucial for ensuring stable operation on slippery or uneven surfaces. Proper control can prevent skidding, ensuring the robot follows the predetermined path safely. Lateral slip control, on the other hand, corrects deviations by adjusting the steering angle and lateral offset, allowing the robot to move precisely along curved or turning trajectories. Effective lateral control enables the robot to change paths flexibly while maintaining trajectory accuracy during emergency avoidance or complex navigation tasks. Therefore, research and development into real-time trajectory-tracking control technologies for wheeled mobile robots are vital for advancing the performance and reliability of these robots in various demanding environments.

Researchers worldwide employ a variety of approaches to develop and design trajectory-tracking controllers, including sliding mode control [3,4,5,6], backstepping control [7], adaptive control methods [8,9,10], fuzzy logic control [11] and intelligent algorithm-based control technologies [12,13,14,15,16,17,18,19]. In reference [3], a sliding mode trajectory-tracking controller was developed for wheeled mobile robots with nonholonomic constraints. Building on a neurodynamic model, a refined version of the control algorithm was introduced to address the velocity jumps typically seen in conventional sliding mode-tracking controllers. Reference [4] examined the nonholonomic wheeled mobile robot system facing external disturbances and time delay, a sliding mode control strategy was introduced, utilizing a nonlinear disturbance observer to efficiently manage system time delay. Reference [5] discussed the slipping and control incoherence issues encountered by wheeled mobile robots during trajectory tracking and proposed an innovative sliding mode control scheme to address these challenges. Reference [6] presented a sliding mode control method based on a nonsingular recursive structure using reconstruction information obtained from a specially designed disturbance observer, achieving tracking control within a finite time. Additionally, a novel adaptive law was integrated to ensure that the tracking error converges to a small region near the origin within a finite time. A standard integral backstepping controller was implemented to stabilize the robot’s orientation angle. Subsequently, recursive integral backstepping control methods were employed to progressively diminish position errors down to zero in reference [7]. Adaptive controllers were developed in references [8,9,10], employing adaptive neural networks to approximate unknown robot dynamics, proving that the closed-loop system’s tracking error could converge to a small range near zero. Reference [11] proposed and explored an adaptive fractional-order parallel fuzzy proportional-integral-derivative controller for wheeled mobile robots. The controller’s performance was comprehensively assessed through computer simulations under interference from dynamic parameter changes, noise, forced displacement, time delay and posture uncertainty. In references [12,13,14,15,16,17,18,19], controllers were designed by applying intelligent control algorithms, thus simplifying the controller design process. A neural network is a computational system that processes information by emulating the structure and functions of the human brain. This system consists of numerous processing nodes, also referred to as neurons. These interconnected neurons work collaboratively to transmit and process information. In reference [20], a model of the motion equations for a wheeled robot was established, and samples derived from this model were used to train a neural network to ensure precise motion control of the wheeled robot along Euler elastic trajectories. In this approach, the formulation of the motion equations provides the neural network with the necessary input–output relationships, enabling the network to learn and emulate the robot’s responses under various dynamic conditions. In reference [8], an adaptive neural network control scheme was proposed to address the uncertainties associated with wheeled mobile robots under velocity constraints and nonholonomic constraints.

In the wake of swift advancements in information technology, numerous sectors such as chemical production, metal refining, mechanical engineering and electronics fabrication have experienced substantial changes. The scale of production continues to expand, and equipment and processes become increasingly complex, while the demand for product quality is also continuously improving. All these factors make it extremely difficult to build models for these industrial processes based on traditional mechanical principles or recognition technologies, and sometimes it may not be possible to implement them at all. However, in the process of industrial production, a large amount of data is continuously generated and stored, which contains important information about the operation of the production process and the status of the equipment. Faced with the challenge of obtaining accurate process models, directly using this real-time or historical data to design controllers to effectively manage these complex processes is not only a practice with broad application prospects but is also crucial for advancing control theory. Refining and elevating this approach to data-driven control theory and methods can help us better meet the various challenges in modern industrial production. Data-driven control technology has been applied in numerous industries [21,22,23,24,25,26,27].

Model-free adaptive control (MFAC) is a highly effective data-driven control method that does not rely on prior model information of the system being controlled, ensuring closed-loop system stability. This control algorithm is praised for its computational efficiency and ease of operation while providing excellent control performance, making it one of the emerging control strategies. At the heart of MFAC is the adoption of newly defined concepts- pseudo-partial derivatives and pseudo-gradient vectors, which allow general discrete-time nonlinear systems to be approximated as a set of dynamic linear models within the local scope of the system’s operating trajectory. The pseudo-partial derivatives and pseudo-gradient vectors of these systems are obtained through online estimation based on input add output data. With this technique, MFAC effectively controls nonlinear systems and eliminates the need for precise system models, thereby reducing the impact of model uncertainties on the control algorithm [28]. References [29,30] presented a robust data-driven control framework based on the Koopman operator for wheeled mobile robots. This framework effectively addressed the modeling error issues encountered when constructing Koopman models. Reference [31] proposed an innovative control strategy for the tracking problem of wheeled mobile robots under velocity saturation. This strategy was a data-driven iterative learning control method with constraints, utilizing a dual-loop control structure. Reference [32] presented a distributed proportional-integral data-driven iterative learning control algorithm for addressing the formation challenges of non-holonomic, velocity-constrained wheeled mobile robots in repeatable operational environments. Reference [33] addressed trajectory-tracking issues for wheeled mobile robots experiencing time delay and bounded disturbances, presenting an enhanced version of the model-free adaptive control approach. The effectiveness and applicability of the model-free adaptive control method have been validated through extensive simulations, experiments and practical applications. Some typical application areas included motion control systems [34,35], industrial control systems [36,37] and power control systems [38,39].

This paper intends to offer a concise overview of MFAC and explore the cutting-edge progress achieved within this domain. The main contributions of this paper are summarized as follows: (1) We have extensively explored MFAC as an efficient control strategy that does not rely on precise system models but only on the system’s input and output signals for implementation. This approach significantly simplifies the design and implementation process of control systems, making it particularly suitable for industrial systems that are difficult to model or whose model parameters frequently change, thereby notably enhancing the flexibility and efficiency of industrial production. (2) The paper elaborately expounds on three distinct dynamic linearization techniques, laying a solid theoretical foundation for MFAC. (3) Furthermore, this paper introduces several advanced MFAC schemes that leverage the latest control theories and technologies, further expanding the application scenarios and capabilities of MFAC. These schemes exhibit outstanding performance in handling complex dynamic systems and uncertain environments, bringing new research and application directions to the field of automatic control. The organization of this paper is as follows: Section 2 presents the model of the wheeled mobile robot. Section 3 introduces the MFAC schemes. Section 4 showcases several sophisticated MFAC strategies that are currently in use. Section 5 is intended to elucidate the unique advantages of MFAC technology through related analysis.

2. The Wheeled Mobile Robot

Consider a Wheeled Mobile Robot (WMR). The generalized position and orientation angle of the robot are defined as $q={\left[x,y,\theta \right]}^T$ in the Cartesian coordinate system. Assuming that there is no slippage between the ground and the wheels, it follows that the WMR is subject to non-holonomic constraints. Non-holonomic constraints typically represent restrictions that cannot be directly integrated into finite positions, confining the configuration space of the system. For the WMR, these non-holonomic constraints can be described by the following equation:

$$\dot{x}sin\theta -\dot{y}cos\theta =A{\left(q\right)}^T\dot{q}=0$$

(1)

where the vector $A\left(q\right)={[sin\theta ,-cos\theta ,0]}^T$ represents the constraint force vector. Under the non-holonomic constraints, the kinematic model of the mobile robot can be expressed as:

$$\dot{q}=S\left(q\right)u$$

(2)

here, $u=[v,\omega ]$ represents the velocity of the robot, where v is the linear velocity and $\omega$ is the angular velocity. The transformation matrix function, denoted as $S\left(q\right)$, has the following form:

$$S\left(q\right)=\left[\begin{array}{cc}\cos \theta \hfill & 0\\ \sin \theta \hfill & 0\\ 0\hfill & 1\end{array}\right]$$

(3)

The Euler–Lagrange formulation for the dynamics of the WMR is presented as follows:

$$M\ddot{q}+C\left(\dot{q}\right)\dot{q}+G=E\left(q\right)\tau +A\left(q\right)\gamma$$

(4)

where the control torque applied to the wheel is denoted as $\tau ={[{\tau }_1,{\tau }_2]}^T\in R^2$. The inertia matrix is represented by $M=\left(\begin{array}{ccc}m& 0& 0\\ 0& m& 0\\ 0& 0& J\end{array}\right)\in R^{3\times 3}$, where m and J are the mass and moment of inertia of the WMR, while $C\left(\dot{q}\right)\in R^{3\times 3}$ signifies the centrifugal and Coriolis force matrix. Since the WMR is moving horizontally, the gravitational vector $G\in R^3$ is zero, i.e., $G=0$. The input matrix is given by $E\left(q\right)\in R^{3\times 2}$. The constraint force vector is represented by $\gamma$. The matrix parameters mentioned above have the same precise structure as those referenced in [21].

The system models presented above are used solely to generate the system’s input–output data and do not contribute to the controller design.

3. Model-Free Adaptive Control

MFAC was initially proposed by Hou in 1994 [40]. For the wheeled mobile robot, the essence of motion control lies in the meticulous adjustment of the input signal to compel the robot to move along a predefined trajectory precisely. Consequently, the dynamic system of the WMR can be described as follows:

$$y(k+1)=f(y\left(k\right),\cdots ,y(k-n_y),u\left(k\right),\cdots ,u(k-n_u))$$

(5)

where $y\left(k\right)$ and $u\left(k\right)$ represent the output and input of the WMR at time instant k, respectively, $n_y$ and $n_u$ are the unknown orders of the output and input, and $f(·)$ is an unknown nonlinear function.

3.1. Compact Form Dynamic Linearization

Before introducing the compact form dynamic linearization method, the following assumptions are made for the system.

Assumption 1 

([28]). The partial derivative of f with respect to its $(n_y+2)$-th variable is continuous except at finite points in time.

Assumption 2 

([28]). Except at finite points in time, the system (5) satisfies the generalized Lipschitz condition, meaning that for any $k_1\ne k_2$, $k_1,k_2\ge 0$ and $u\left(k_1\right)\ne u\left(k_2\right)$, it follows that

$$\left|y(k_1+1)-y(k_2+1)\right|\le b\left|u\left(k_1\right)-u\left(k_2\right)\right|$$

where $y(k_i+1)=f(y\left(k_i\right),\cdots ,y(k_i-n_y),u\left(k_i\right),\cdots ,u(k_i-n_u))$, $i=1,2$; $b>0$ is a constant.

For the sake of convenience in the following discussion, let $\Delta y(k+1)=y(k+1)-y\left(k\right)$ denote the output variation between two adjacent time instances, and $\Delta u\left(k\right)=u\left(k\right)-u(k-1)$ denote the input variation between two adjacent time instances.

Lemma 1. 

For the nonlinear system (5) that satisfies Assumptions 1 and 2, when $\left|\Delta u\left(k\right)\right|\ne 0$, there must exist a time-varying parameter ${\varphi }_c\left(k\right)$, referred to as the pseudo-partial derivative, such that the system (5) can be transformed into the following compact form dynamic linearization data model.

$$\Delta y(k+1)={\varphi }_c\left(k\right)\Delta u\left(k\right)$$

(6)

3.2. Partial Form Dynamic Linearization

Lemma 1 demonstrates that the compact form dynamic linearization technique converts a typical discrete-time nonlinear system into a linear, time-varying dynamic data model with a scalar parameter ${\varphi }_c\left(k\right)$. The original system’s potential complex behaviors, including nonlinearity, parameter time-variance and structural time-variance, are condensed and incorporated into the time-varying parameter ${\varphi }_c\left(k\right)$, which may consequently display highly complex dynamics. Looking at it from another perspective, the essence of the compact form dynamic linearization method is to focus on analyzing the dynamic relationship between the output variation at the next moment and the input variation at the current moment. However, the output variation at the next moment may also be influenced by other control input variations from previous moments. Based on this consideration, during the linearization process, we can take into account the influence of all input variations within a fixed-length sliding time window at the current moment on the output variation at the next moment, thus proposing a new partial form dynamic linearization method.

Define vector $U_L\left(k\right)\in R^L$ as the collection of all control input signals within the sliding time window $[k-L+1,k]$ as follows. Assuming the control input signal is a sequence of signals varying with time, then vector $U_L\left(k\right)$ can be represented as:

$$U_L\left(k\right)=[u\left(k\right),\dots ,u(k-L+1)]$$

For the discrete-time nonlinear system (5), two assumptions similar to Assumptions 1 and 2 are proposed as follows.

Assumption 3 

([28]). The function f possesses continuous partial derivatives with respect to the variables from the $(n_y+2)$-th to the $(n_y+L+1)$-th, respectively.

Assumption 4 

([28]). System (5) satisfies the generalized Lipschitz condition, meaning that for any $k_1\ne k_2$, with $k_1,k_2\ge 0$ and for all $U_L\left(k_1\right)\ne U_L\left(k_2\right)$, the following holds:

$$\left|y(k_1+1)-y(k_2+1)\right|\le b∥U_L\left(k_1\right)-U_L\left(k_2\right)∥$$

where $y(k_i+1)=f(y\left(k_i\right),\cdots ,y(k_i-n_y),u\left(k_i\right),\cdots ,u(k_i-n_u))$, $i=1,2$; $b>0$ is a constant.

Let $\Delta U_L\left(k\right)=U_L\left(k\right)-U_L(k-1)$. The subsequent lemma will outline a partial form dynamic linearization technique for system (5).

Lemma 2. 

For the nonlinear system (5) that satisfies Assumptions 3 and 4, given L, when $∥\Delta U_L\left(k\right)∥\ne 0$, there exists a time-varying parameter vector known as the pseudo-gradient ${{\varphi }_L}^T\left(k\right)\in R^L$, such that system (5) can be transformed into the following partial form dynamic linearization data model:

$$\Delta y(k+1)={{\varphi }_L}^T\left(k\right)\Delta U_L\left(k\right)$$

(7)

3.3. Full Form Dynamic Linearization

The traditional partial-form dynamic linearization method primarily focuses on the dynamic relationship between the system’s output variation at the next moment and the input variations within a fixed-length sliding time window at the current moment. However, in reality, the future changes in system output may depend not only on the changes in inputs but also on the system’s own output variations over a certain period in the past. To more comprehensively capture these complex dynamic behaviors, an improved dynamic linearization technique, known as the full form dynamic linearization, can be proposed. This method takes into account not only the variations in control inputs within a certain time window but also the influence of the system’s output changes within this time window on subsequent output variations when constructing models. By integrating this information, full form dynamic linearization is capable of providing a richer set of pseudo-gradient components, which helps to approximate the dynamic characteristics of complex systems more accurately. Compared with the compact form and partial form dynamic linearizations, this method utilizes more historical data in hopes of achieving more precise model predictions and thereby enhancing the performance of control systems.

Define $H_{L_{y,}{L}_u}\left(k\right)\in R^{L_y+L_u}$ as a collection that includes all control input signals within the input-related sliding time window $[k-L_u+1,k]$ and all system output signals within the output-related sliding time window $[k-L_y+1,k]$. In mathematical notation, this can be expressed as:

$$H_{L_{y,}{L}_u}\left(k\right)={\left[y\left(k\right),\cdots y(k-L_y+1),u\left(k\right),\cdots u(k-L_u+1)\right]}^T$$

For the discrete-time nonlinear system of the form (5), two new assumptions similar to Assumptions 1 and 2 can be proposed.

Assumption 5 

([28]). The nonlinear function $f(\cdots )$ has continuous partial derivatives with respect to all its variables.

Assumption 6 

([28]). The system (5) satisfies the generalized Lipschitz condition, which means that for any $k_1\ne k_2$, where $k_1,k_2\ge 0$ and $H_{L_{y,}{L}_u}\left(k_1\right)\ne H_{L_{y,}{L}_u}\left(k_2\right)$, the following holds:

$$\left|y(k_1+1)-y(k_2+1)\right|\le b∥H_{L_{y,}{L}_u}\left(k_1\right)-H_{L_{y,}{L}_u}\left(k_2\right)∥$$

where $y(k_i+1)=f(y\left(k_i\right),\cdots ,y(k_i-n_y),u\left(k_i\right),\cdots ,u(k_i-n_u))$, $i=1,2$; $b>0$ is a constant.

Let $\Delta H_{L_y,L_u}\left(k\right)=H_{L_y,L_u}\left(k\right)-H_{L_y,L_u}(k-1)$, the following lemma will outline a full form dynamic linearization method for system (5).

Lemma 3. 

For the nonlinear system (5) that satisfies Assumptions 5 and 6, under the given conditions $0\le L_y\le n_y$ and $1\le L_u\le n_u$, when $∥\Delta H_{L_y,L_u}\left(k\right)∥\ne 0$, there must exist a time-varying parameter vector ${\varphi }_{f,L_{y,}{L}_u}\left(k\right)\in R^{L_y+L_u}$, known as the pseudo-gradient, which allows the system (5) to be transformed into a full-form dynamic linearization data model:

$$\Delta y(k+1)={{\varphi }^T}_{f,L_y,L_u}\left(k\right)\Delta H_{L_y,L_u}\left(k\right)$$

(8)

By applying dynamic linearization techniques, the process of controller design can be greatly simplified. Taking compact form dynamic linearization as an example, after transforming the nonlinear system into a linear time-varying data system, a weighted one-step-ahead cost function can be used to derive the MFAC scheme based on compact form dynamic linearization, as described below.

$${\widehat{\varphi }}_c\left(k\right)={\widehat{\varphi }}_c(k-1)+{\displaystyle \frac{\eta \Delta u(k-1)}{\mu +\Delta u{(k-1)}^2}}\left(\Delta y\left(k\right)-{\widehat{\varphi }}_c(k-1)\Delta u(k-1)\right)$$

(9)

$${\widehat{\varphi }}_c\left(k\right)={\widehat{\varphi }}_c\left(1\right),\begin{array}{cc}& if\left|{\widehat{\varphi }}_c\left(k\right)\right|\end{array}\le \epsilon \begin{array}{cc}& or\end{array}\left|\Delta u(k-1)\right|\le \epsilon \begin{array}{cc}& orsign\end{array}\left({\widehat{\varphi }}_c\left(k\right)\right)\ne sign\left({\widehat{\varphi }}_c\left(1\right)\right)$$

(10)

$$u\left(k\right)=u(k-1)+{\displaystyle \frac{\rho {\widehat{\varphi }}_c\left(k\right)}{\lambda +{\left|{\widehat{\varphi }}_c\left(k\right)\right|}^2}}\left(y^{\ast }(k+1)-y\left(k\right)\right)$$

(11)

where $\lambda >0$, $\mu >0$, $\rho \in \left(0,1\right]$, $\eta \in \left(0,1\right]$, $\epsilon$ is a sufficiently small positive number, ${\widehat{\varphi }}_c\left(1\right)$ is the initial value of ${\widehat{\varphi }}_c\left(k\right)$.

In the aforementioned MFAC scheme based on compact form dynamic linearization, the introduction of reset algorithm (10) is intended to enhance the capability of the pseudo-partial derivative estimation algorithm (9) to track the time-varying parameter.

4. Some Improved MFAC Approaches

In the era of rapid technological advancement, wheeled mobile robots have become a crucial component in the realms of automation and intelligence. These robots are extensively employed across various domains such as logistics, surveillance, exploration and service, where MFAC technologies provide new possibilities for enhancing the autonomy and adaptability of wheeled mobile robots. Here are some advanced model-free adaptive control methods that are propelling the development of wheeled mobile robots:

4.1. Dynamic Event-Triggered MFAC

The WMR often need to navigate and operate efficiently in complex and constantly changing environments while executing tasks. Traditional time-driven control methods may not be the most efficient or suitable choice, as they typically execute control commands according to a fixed schedule without considering the actual state of the system. In contrast, event-triggered control offers a more flexible and responsive control strategy that determines when to execute control actions based on changes in the system’s state, rather than following preset time intervals. Combining the event-triggered mechanism with MFAC brings new possibilities for controlling WMR.

Evidently, the designed MFAC algorithm is premised on the assumption of continuous network communication. However, it is well understood that network resources are finite, and incessant data transmission would deplete these valuable resources. To conserve communication resources effectively, a dynamic event-triggering mechanism has been introduced to facilitate selective data transmission [41]. The set of instances at which events are triggered is defined as $k_i,i=0,1,\cdots$. Based on this, the following dynamic event-triggering function is devised.

$$g\left(k\right)={\displaystyle \frac{1}{\upsilon }}\left|\gamma \left(k\right)\right|+o-\left|m\left(k\right)\right|$$

(12)

where o and $\upsilon$ are chosen to be appropriate positive constants. $m\left(k\right)=y\left(k\right)-y\left(k_i\right)$ representing the measurement error, $y\left(k_i\right)$ denotes the system output at the latest event-triggered instant $k_i$. $\gamma \left(k\right)$ represents the dynamic variable that satisfies

$$\gamma (k+1)=\phi \gamma \left(k\right)+o-\left|m\left(k\right)\right|,\gamma \left(1\right)={\gamma }_0$$

(13)

where the variables $\phi >0$ and ${\gamma }_0$ are referred to as the designed parameter and initial value, respectively.

The controller can receive data packets at the event-triggered sequence $k_i$, which is generated by the following event-triggered conditions:

$$k_{i+1}=inf\{k\in N|k>k_i,g\left(k\right)<0\}$$

(14)

By combining (11) and (14), when $k\in \left[k_i,k_{i+1}\right)$, the control input will be updated according to the following equation:

$$u\left(k\right)=u(k-1)+{\displaystyle \frac{\rho {\widehat{\varphi }}_c\left(k_i\right)}{\lambda +{\left|{\widehat{\varphi }}_c\left(k_i\right)\right|}^2}}\left(y^{\ast }(k+1)-y\left(k_i\right)\right)$$

(15)

The purpose of a DoS attack is to disrupt system performance by preventing data packets from being transmitted between sensors and controllers. The n-th DoS attack interval is defined as $\left[{T^{on}}_n,{T^{off}}_n\right]$, where ${T^{on}}_n\in N$ and ${T^{off}}_n\in N$ are the start and end times of the DoS attack, respectively. The union of the interval $[0,k]$, $k\in N$ for the DoS attack is given by the following expression.

$$E_a(0,k)=\left\{\bigcup _{n\in N}\left[{T^{on}}_n,{T^{off}}_n\right)\right\}\bigcap \left[0,k\right]$$

(16)

Evidently, the time interval without a DoS attack is

$$E_s(0,k)=\left[0,k\right]\backslash E_a(0,k)$$

(17)

Subsequently, in conjunction with the aforementioned DoS attack mechanism, the following development is carried out for the dynamic event-triggered MFAC scheme under non-periodic DoS attacks:

$$\left\{\begin{array}{c}{\widehat{\varphi }}_c\left(k\right)={\widehat{\varphi }}_c(k-1)+{\displaystyle \frac{\eta \Delta u(k-1)}{\mu +\Delta u{(k-1)}^2}}\left(\Delta y\left(k\right)-{\widehat{\varphi }}_c(k-1)\Delta u(k-1)\right)\hfill \\ {\widehat{\varphi }}_c\left(k\right)={\widehat{\varphi }}_c\left(1\right),\begin{array}{cc}& if\left|{\widehat{\varphi }}_c\left(k\right)\right|\end{array}\le \epsilon \begin{array}{cc}& or\end{array}\left|\Delta u(k-1)\right|\le \epsilon \begin{array}{cc}& orsign\end{array}\left({\widehat{\varphi }}_c\left(k\right)\right)\ne sign\left({\widehat{\varphi }}_c\left(1\right)\right)\hfill \end{array}\right.$$

(18)

$$u\left(k\right)=u(k-1)+h\left(k\right){\displaystyle \frac{\rho {\widehat{\varphi }}_c\left(k_i\right)}{\lambda +{\left|{\widehat{\varphi }}_c\left(k_i\right)\right|}^2}}\left(y^{\ast }(k+1)-y\left(k_i\right)\right)$$

(19)

where

$$h\left(k\right)=\left\{\begin{array}{c}\begin{array}{cc}1,\hfill & k\in \left[{T^{off}}_{n-1},{T^{on}}_n\right)\end{array}\hfill \\ \begin{array}{cc}0,\hfill & k\in \left[{T^{on}}_n,{T^{off}}_n\right)\end{array}\hfill \end{array}\right.$$

(20)

and $\epsilon$ is a sufficiently small positive number.

4.2. Model-Free Adaptive Iterative Learning Control

Iterative learning control combined with MFAC is an advanced control method that is particularly well suited for the WMR operating in complex environments with high precision and efficiency. The core of this method lies in the integration of iterative learning and model-free adaptive techniques to enhance the robot’s performance and adaptability.

Iterative learning control methods have been developed for applications that require the execution of the same task within a finite time interval repeatedly to achieve precise tracking performance. The key concept behind iterative learning control is to design a series of suitable control inputs that yield outputs matching the desired trajectory. In reference [42], a modified model-free adaptive linearization method was employed and utilized iteratively. By defining a repetitive nonlinear system, the input and output information for system (5) can be redefined as

$$y_j(k+1)=f(y_j\left(k\right),\cdots ,y_j(k-n_y),u_j\left(k\right),\cdots ,u_j(k-n_u))$$

(21)

where j represents the iteration number. Therefore, the iterative compact form dynamic linearization model can be defined as

$$\Delta y_j(k+1)={\varphi }_j\left(k\right)\Delta u_j\left(k\right)$$

(22)

where $\Delta y_j(k+1)=y_j(k+1)-y_{j-1}(k+1)$, $\Delta u_j\left(k\right)=u_j\left(k\right)-u_{j-1}\left(k\right)$.

A new design approach has been proposed that takes into account the discrete algebraic Riccati equation when defining the weighting matrix for iterative learning control methods in reference [42]. As mentioned in reference [43], the influence of the weighting matrix plays a significant role in affecting the performance of tracking error and input energy in iterative learning control. The design philosophy based on optimal iterative learning control aims to minimize the cost function regarding the ultimate error. We assume there is a target function based on the linear quadratic criterion for each iteration step, as shown below.

$$J\left(u_{j+1}\right)={e^T}_{j+1}{W}_e{e}_{j+1}+{u^T}_{j+1}{W}_u{u}_{j+1}$$

(23)

where $e_j\left(k\right)=y^{\ast }\left(k\right)-y_j\left(k\right)$ is the tracking error, $W_e$ and $W_u$ are designed weighting matrices.

In typical iterative learning control methods, the relationship between the system’s input and output can be described as

$$y_j\left(k\right)=G{u}_j\left(k\right)$$

(24)

here, the estimation of the partial derivative (22) can be utilized to determine the transfer matrix G for the next iteration.

To minimize the objective function (23), the steepest descent method is employed in the iterative learning control strategy. Thus, the update rule for learning is derived by applying the gradient descent method to the performance index $J\left(u_{j+1}\right)$, as shown below.

$$u_{j+1}(k+1)=u_j(k+1)+\sigma G^T{W}_e{e}_j(k+1)-\sigma W_u{u}_j(k+1)$$

(25)

where $\sigma \in R$ represents the learning gain.

5. Related Analysis

This section aims to validate the significant advantages of MFAC methods through an in-depth analysis. Compared with traditional control strategies, MFAC does not rely on precise mathematical models, thereby offering greater flexibility in accommodating the nonlinear characteristics and uncertainties of systems. This approach utilizes real-time data to dynamically adjust controller parameters, enabling effective regulation and optimization of complex systems.

The PID (Proportion Integration Differentiation) control [44] is a feedback control algorithm extensively used in industrial automation and control systems. It operates by calculating the proportional, integral and derivative values of the deviation to adjust the output of the control volume, thereby achieving system stability and performance optimization.

For convenience, we assume that the desired trajectory as

$${\theta }^{\ast }\left(j\right)=\left\{\begin{array}{c}\begin{array}{cc}3& \begin{array}{cc}if& j<400\end{array}\end{array}\hfill \\ \begin{array}{ccc}5& if& 400\le j\le 800\end{array}\hfill \end{array}\right.$$

Figure 1 and Figure 2 represent the outcomes of simulation experiments employing the PID method under identical initial conditions to our study. Upon examination of Figure 1, it becomes evident that the tracking performance exhibited by the WMR is sub-optimal. A comparative analysis between Figure 1 and Figure 3 reveals that the MFAC approach exhibits superior characteristics over the PID method, including accelerated response times, diminished overshoot and reduced settling periods. This attests to the enhanced efficiency and effectiveness of the MFAC scheme in comparison to the traditional PID control methodology.

MFAC method demonstrates significant advantages over traditional control methods, such as PID control. The core strength of MFAC lies in its ability to operate without relying on an exact mathematical model, making it particularly suitable for complex, nonlinear or difficult-to-model systems. In such systems, traditional model-based control methods often struggle to achieve ideal control performance because they require accurate system models to design controllers.

In the real world, external disturbances are inevitable, posing a serious challenge to the stability and performance of control systems. Therefore, it is particularly important to explore in depth the tracking ability of MFAC in an environment with bounded disturbances. To ensure consistency and accuracy in the experiments, we keep the parameter configuration of the WMR unchanged, so as to focus on evaluating the performance of MFAC. We assume that the bounded disturbances is: $d=0.2sin\left({\displaystyle \frac{k}{10}}\right)$. The intended path for the WMR is characterized by the mathematical function $y=sinx$. Furthermore, this system maintains a steady velocity of 1.5 m per second as it traverses along this trajectory. The initial values of the system are set as follows: $\left(x\right(1),y(1\left)\right)=(0,0)$ and $\theta \left(1\right)={\displaystyle \frac{\pi }{4}}$. In the system configuration, we have set the following parameter values: $\eta =1$, $\mu =1$, $\rho =0.6$, $\lambda =1$, $\epsilon ={10}^{-5}$, ${\widehat{\varphi }}_c\left(1\right)=1$. These specific numerical settings for the parameters are aimed at ensuring consistency in system operation, where each parameter is carefully selected to meet the basic requirements and expected operational characteristics of the system.

Figure 4 vividly demonstrate the excellent tracking performance of the WMR under the influence of disturbances, facilitated by MFAC technology. These illustrations clearly capture how the WMR, in the face of external disturbances, achieves close adherence to a predetermined trajectory through MFAC. Even with the impact of disturbances, the trajectory tracking of the WMR remains almost unaffected, showcasing the significant potential of MFAC in enhancing system disturbance rejection capabilities and stability. Furthermore, the adaptive nature of MFAC means it can adjust control parameters in real-time in response to changes in system dynamic characteristics, thereby enhancing the flexibility and robustness of the control system. This allows MFAC to maintain high control performance in the face of system parameter changes or external disturbances, which is crucial in practical applications, especially in fields like aerospace, robotics and precision manufacturing where system dynamics can change rapidly due to various factors.

Traditional control strategies rely on precise models and struggle to handle model uncertainties and system nonlinearities. MFAC designs controllers directly using input–output data, eliminating the need for precise models, and thus can better adapt to model uncertainties. It naturally addresses nonlinear characteristics, adjusts control strategies based on real-time data and responds flexibly to parameter changes and external disturbances, maintaining stability and performance.

In contrast, traditional control methods like PID control, despite excelling in many standard application scenarios, may require frequent manual adjustments to parameters when dealing with systems characterized by high uncertainty and complexity. The reliance on models can also limit their flexibility in responding to rapidly changing system characteristics. Optimizing a PID controller is a crucial step in ensuring the success of a control system design. This process involves the careful adjustment of three parameters: proportional, integral and derivative, aiming to optimize both the dynamic response and steady-state performance of the system. When the system’s deviation is significant, the integral action should be deactivated; however, as the controlled variable approaches the setpoint and the error decreases, the integral action should be re-engaged. Furthermore, the rate of integral action can be modulated based on the magnitude of the deviation: it should be slowed down when the deviation is large and sped up when the deviation is small. This flexible adjustment mechanism facilitates more precise and stable control outcomes. Therefore, MFAC offers a more modern and automated solution, opening new possibilities for the design and implementation of advanced control systems.

6. Conclusions

This paper outlines and discusses three dynamic linearization methods for MFAC, along with the latest advancements in these techniques. It explores the fundamental concepts of dynamic linearization, its modes of expression, characteristics and the corresponding MFAC technologies. Additionally, it reviews a range of advanced MFAC strategies. Dynamic linearization is dependent on the closed-loop input and output data of the controlled system. Consequently, there are no modeling errors or unmodeled dynamics during the dynamic linearization process. The data models derived from dynamic linearization are equivalent transformations of the original nonlinear system. The goal of the MFAC method is to directly design the controller using process data or knowledge acquired from the data to solve system control problems.

Although a variety of MFAC methods have been proposed in the literature, the theory of MFAC is still in its infancy. The following is a brief discussion on the perspectives of MFAC theory and some promising research topics.

(1) Leveraging efficient data processing methods and their applications within the MFAC framework showcase significant promise for advancements in research and practical implementations. Current hardware technology is capable of supporting the computational requirements for online implementation of these offline algorithms. Given that both online and offline data contain a wealth of valuable information about system operation and behavior, it is particularly important to find ways to use this information and patterns to design robust model-free adaptive controllers. Applying the insights extracted from offline and online data to controller design presents a significant challenge.

(2) In the realm of MFAC theory, robustness is a critical issue. In conventional model-based control theories, robustness pertains to a system’s capability to manage uncertainties or unmodeled dynamic behaviors. However, in MFAC methods, given the absence of unmodeled dynamics, a new definition for robustness must be established. In practical applications, data can be compromised by external disturbances or lost due to sensor, actuator or network faults. Hence, we propose that research on the robustness of MFAC should primarily address the effects of data noise and data loss.

(3) In the area of optimization theory for MFAC, the work done so far is relatively limited. This topic deserves more attention and in-depth research.

(4) To transcend the current limitations of MFAC, a novel strategy can be explored in the future: utilizing output error and output error rate as critical performance metrics, and through precise weight allocation to these parameters, a new control law can be derived. This innovative approach allows for a more adaptive and responsive MFAC system, capable of rapidly tracking desired trajectories while enhancing control accuracy and efficiency.