A Novel Model-Free Adaptive Proportional–Integral–Derivative Control Method for Speed-Tracking Systems of Electric Balanced Forklifts

Abstract

Similar to many complex systems, the operation process of electric balanced forklifts has characteristics such as time-varying model parameters and nonlinearity. Establishing an accurate mathematical model becomes challenging, making it difficult to apply model-based control methods in engineering practice. Aiming at the longitudinal control system of electric forklifts containing external disturbances, this paper proposes an improved full-format dynamic linearization model-free adaptive PID control (iFFDL-MFA-PID) method. Firstly, the full-format dynamic linearization (FFDL) method is employed to transform the operating system of the electric balanced forklift into a virtual equivalent linear data model. Secondly, the nonlinear residual term and pseudo-gradient (PG) of the data model are estimated using the difference estimation algorithm and the optimal criterion function, respectively. Furthermore, in order to enhance the robustness of the system, the idea of intelligent PID (iPID) is introduced and the principle of equivalent feedback is utilized to derive the iFFDL-MFA-PID control scheme. The design process of this scheme only requires the use of the input and output data of the system, without relying on the mathematical model of the system. Finally, the iFFDL-MFA-PID method proposed in this paper is simulated and tested with the EFG-BC/320 counterbalanced forklift equipped in the Special Equipment Testing Center and compared with the model-free adaptive control method (FFDL-MFAC) and the PID control method. Simulation results show that the speed-tracking error of the electric forklift truck under the action of the iFFDL-MFA-PID algorithm is maintained within ±0.132 m/s throughout the process, achieving higher tracking accuracy and better robustness compared to the MFAC and PID methods.

1. Introduction

With the rapid development of industrial logistics, the role of electric balanced forklifts in the fields of warehousing, loading and unloading, and transportation is becoming increasingly important [1]. These forklifts offer advantages such as zero emissions, ease of operation, and high handling efficiency. However, traditional forklift operations rely heavily on the experience and skills of the driver, which may result in low operational efficiency and safety accidents such as rollovers and rear-end collisions. Therefore, it is of great significance to conduct research on intelligent control technology based on automatic driving for electric balanced forklifts. This research aims to improve forklift operational efficiency and ensure operational safety.

In recent years, autonomous driving technology has seen rapid advancements and demonstrated remarkable achievements in various fields, including smart cars [2,3], unmanned aerial vehicles [4,5], and rail transportation [6,7]. However, when it comes to electric balanced forklifts, the practical application of autonomous driving is still in its early stages. This is primarily due to the complex and ever-changing operating environments of forklifts. These environments involve diverse cargo types, shapes, and weights, as well as unique operational scenarios, which present significant challenges for implementing autonomous driving technology in the forklift industry.

Due to the reasons outlined above, research on automatic driving technology for forklifts has become a hot topic, especially in the areas of visual positioning [8], lidar positioning [9], and adaptive control [10]. For instance, the Vecna company (Cambridge, MA, USA) in the US has launched an automatic navigation forklift that employs multiple sensing technologies, such as lidar, an inertial navigation system, and visual sensors, to achieve accurate positioning and path planning [11]. Similarly, KUKA, a German company (Augsburg, Germany), has developed an automatic forklift based on visual navigation, which captures ground feature points with a camera to enable autonomous navigation and accurate handling of the forklift [12]. In practical research, the tracking of path and time in electric balanced forklifts exhibits a complex nonlinear relationship. On the other hand, the impact of disturbances and load variations makes the vehicle system itself a complex and difficult-to-model nonlinear system. Additionally, traditional forklift operations rely on the driver’s experience and skills, which can lead to low operational efficiency and safety accidents such as tipping over [13]. In order to address these challenges, researchers need to develop innovative control methods to enhance the performance and adaptability of electric balanced forklifts. Currently, most of the research is focused on using two- or four-wheeled mobile robots instead of electric forklifts. Refs. [14,15] applied model predictive control methods based on tracking error models to successfully track the trajectory of mobile robots. Refs. [16,17,18] used neural network control methods to simulate the nonlinear movements of robots and applied them to the trajectory tracking of mobile robots. Ref. [19] applied fuzzy control methods to the trajectory tracking control of mobile robots. Ref. [20] combined reinforcement learning with PID algorithms to apply trajectory tracking control of mobile robots. The aforementioned control methods primarily perform trajectory tracking in the time domain and are based on model-based control. However, they have lower control accuracy and adaptability. Electric forklifts experience significant disturbances due to the increasing load during operation, resulting in a time-varying model. This makes it challenging for the aforementioned control methods to meet the high-precision autonomous driving requirements of time-varying electric forklifts.

According to the aforementioned literature, when analyzing controller design and system stability, it is necessary to obtain the model parameters in advance or linearize the forklift model. These factors lead to mismatches between the controller and the actual model in model-based control methods, resulting in a significant impact on control performance. To address this issue, some scholars have started exploring data-driven control approaches. PID control is one of the earliest control methods and has been widely applied in industrial processes due to its advantages of simplicity and high reliability [21,22]. Moreover, the PID is also a typical data-driven control method. However, due to the nonlinear, time-varying uncertainties; strong disturbances often present in practical systems; and the difficulty in tuning PID controller parameters with unsatisfactory performance, the application of PID control is limited in complex and high-performance systems [23]. To overcome the limitations of traditional PID control, such as poor tracking performance and slow convergence, scholars have actively combined the PID with intelligent algorithms to achieve adaptive changes in PID gains [24]. However, there are still suboptimal phenomena, such as an excessively high sampling frequency. MFAC was proposed by Professor Hou in his doctoral thesis [25]. The basic idea of MFAC is to dynamically linearize a nonlinear system by introducing the concept of pseudo-partial derivatives and establishing an equivalent dynamically linearized data model. Based on this virtual data model, the controller and parameter update laws are designed. This algorithm can achieve both parameter adaptation and structure adaptation and has been applied in various industrial fields after years of development [26,27]. However, traditional MFAC represents the system’s input-output data and uncertainties (primarily including parameter estimation errors and external disturbances) using a dynamic linearized data model. Furthermore, parameter estimation errors and uncertainties are estimated through parameter estimation algorithms. In the case of significant system disturbances, this approach may lead to issues such as over-linearization [28,29]. Additionally, research results indicate [30] that in the absence of external disturbances, conventional MFAC schemes can converge the system output error to zero. However, when measurement disturbances are present, applying conventional MFAC schemes will cause the system’s output error to converge to a non-zero constant. Therefore, the presence of measurement disturbances significantly deteriorates the control performance of the MFAC scheme.

In [31], the advantages of MFAC and iPID were combined to propose an MFAC PID scheme based on compact-form dynamic linearization (CFDL). This approach extends the applicability of iPID to nonlinear and non-affine systems, addressing the issues of poor robustness in the PID and weak disturbance rejection capability in MFAC. However, in the dynamic linearization process in [31], only the output at the next time instant in relation to the input at the previous time instant was considered, resulting in a relatively simplistic design.

Based on the analysis presented above, this paper proposes an iFFDL-MFA-PID scheme by combining the advantages of MFAC and iPID. Firstly, we introduce the gradient estimation algorithm and time difference algorithm based on the general formula for nonlinear systems to effectively estimate time-varying parameters and uncertain nonlinear terms. Moreover, by incorporating additional tracking error information using the iPID algorithm, the iFFDL-MFAC system compensates for the decrease in system information accuracy caused by excessive linearization. Finally, we redefine the learning update law and parameter update law of the system to improve the controller structure. Simulation results demonstrate that the proposed control strategy, iFFDL-MFA-PID, more effectively suppresses the harmful effects of measurement disturbances on system stability, thereby achieving more accurate tracking of the system output. The main contributions of this paper are as follows:

(1)

In the dynamic linearization process, this study fully considers the relationship between the system’s input and output, resulting in a more flexible controller design process. Compared with the CFDL data model previously used, the FFDL method adopted in this study comprehensively considers the relationship between the output change at the next moment and the input and output within a fixed-length sliding time window. The FFDL data model is insensitive to the system’s structure and parameter time-varying characteristics.

(2)

Compared with existing electric forklift operation control algorithms, the iFFDL-MFA-PID scheme does not rely on the system’s dynamic model and only requires its input-output data. It is a data-driven control method that is easy to implement. On the other hand, the FFDL-based MFA-PID control method utilizes more control information from the previous time step compared to the traditional MFAC method, thus achieving better control performance.

(3)

This study designs a time-delay estimator and a gradient-parameter estimator to estimate the nonlinear uncertainties and unknown parameters in the FFDL model instead of using rough parameter estimation algorithms. This approach avoids the problem of over-linearization and further improves the equivalent description of the dynamic linearization within the system. Moreover, the iFFDL-MFA-PID adds compensation for the uncertain system in the control law, enhancing the control system’s robustness.

The remaining structure of this paper is as follows. Section 2 gives the traditional MFAC structure; Section 3 presents the MFAC scheme combined with iPID and proves its stability; Section 4 gives the numerical values of the iFFDL-MFA-PID scheme, as well as the simulation results and an analysis of the electric forklift; and Section 5 provides a summary of this text.

2. Traditional MFAC

2.1. Dynamic Linearization

This study adopts the FFDL method for data linearization, which incorporates the influence of inputs and outputs within a fixed length [32].

Consider the following SISO system

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

(1)

where $u\left(k\right)$ and $I\left(k\right)$ represent the input and output, respectively, and $n_u$ and $n_I$ are the orders.

$\mathbf{H}\left(k\right)$ is defined as follows

$$\mathbf{H}\left(k\right)=[\Delta I\left(k\right),\cdots ,\Delta I(k-L_I+1),\Delta u\left(k\right),\cdots ,\Delta u(k-L_u+1){]}^{\mathrm{T}}$$

(2)

where $L_I$ and $L_u$ are the lengths of the output and input linearization, respectively.

Assumption 1 

([33,34]). The partial derivatives of $f(·)$ for all system variables exist.

Assumption 2 

([33,34]). For any time $k_1\ne k_2\ge 0$ and $\mathbf{H}\left(k_1\right)\ne \mathbf{H}\left(k_2\right)$, one has

$$\left|I(k_1+1)-I(k_2+1)\right|\le p∥\mathbf{H}\left(k_1\right)-\mathbf{H}\left(k_2\right)∥$$

(3)

where p is a constant.

Theorem 1. 

For System (1) that meets Assumptions 1 and 2, there must be a vector called a pseudo-gradient (PG) so that the system can be transformed into the following FFDL data model [32].

$$I(k+1)=I\left(k\right)+{\Lambda }^{\mathrm{T}}\left(k\right)\Delta \mathbf{H}\left(k\right)$$

(4)

where $\Lambda \left(k\right)={[{\Lambda }_1\left(k\right),\cdots ,{\Lambda }_{L_I+L_u}\left(k\right)]}^{\mathrm{T}}$ is bounded.

Proof. 

See [35]. □

2.2. Traditional MFAC Design

Consider the following performance metric functions:

$$J\left(u\left(k\right)\right)=\mathsf{\partial }{\left|u\left(k\right)-u(k-1)\right|}^2+{\left|I_d(k+1)-I(k+1)\right|}^2$$

(5)

where $I_d$ is the reference output and $\mathsf{\partial }>0$ is the weight factor.

By calculating the first-order partial differential with (5) and making it zero, one has:

$$\begin{array}{cc}\hfill u\left(k\right)& =u(k-1)+\frac{\rho {\Lambda }_{L_I+1}\left(k\right)(I_d(k+1)-I\left(k\right))}{\mathsf{\partial }+{\left|{\Lambda }_{L_I+1}\left(k\right)\right|}^2}\hfill \\ & -\frac{{\Lambda }_{L_u+1}\left(k\right)\rho \left({\displaystyle \sum _{i=1}^{L_I}}{\Lambda }_i\left(i\right)\Delta I(k-i+1)+{\displaystyle \sum _{i=L_I+2}^{L_I+L_u}}{\Lambda }_k\left(i\right)\Delta u(k-L_I-i+1)\right)}{\mathsf{\partial }+{\left|{\Lambda }_{L_I+1}\left(k\right)\right|}^2}\hfill \end{array}$$

(6)

where $\rho \in (0,1]$ is the step factor.

Next, $\Lambda \left(k\right)$ needs to be estimated. The following parameter estimation function is given,

$$J\left(\Lambda \left(k\right)\right)={\left|\Delta I\left(k\right)-{\Lambda }^{\mathrm{T}}\left(k\right)\Delta \mathbf{H}(k-1)\right|}^2+\mu {∥\Lambda \left(k\right)-\widehat{\Lambda }(k-1)∥}^2$$

(7)

where $\mu >0$.

According to the minimization function (7), the following estimation algorithm can be obtained

$$\widehat{\Lambda }\left(k\right)=\widehat{\Lambda }(k-1)+\frac{\eta \Delta \mathbf{H}(k-1)\left(\Delta I\left(k\right)-{\widehat{\Lambda }}^{\mathrm{T}}(k-1)\Delta \mathbf{H}(k-1)\right)}{\mu +{∥\Delta \mathbf{H}(k-1)∥}^2}$$

(8)

where $\widehat{\Lambda }\left(k\right)$ is the estimation of $\Lambda \left(k\right)$ and $\eta \in (0,2]$ is a constant.

Therefore, the overall control scheme of FFDL-MFAC is as follows:

$$\left\{\begin{array}{c}u\left(k\right)=u(k-1)+\frac{\rho {\Lambda }_{L_I+1}\left(k\right)(I_d(k+1)-I\left(k\right))}{\mathsf{\partial }+{\left|{\Lambda }_{L_I+1}\left(k\right)\right|}^2}\hfill \\ -\frac{{\Lambda }_{L_u+1}\left(k\right)\rho \left({\displaystyle \sum _{i=1}^{L_I}}{\Lambda }_i\left(i\right)\Delta I(k-i+1)+{\displaystyle \sum _{i=L_I+2}^{L_I+L_u}}{\Lambda }_k\left(i\right)\Delta u(k-L_I-i+1)\right)}{\mathsf{\partial }+{\left|{\Lambda }_{L_I+1}\left(k\right)\right|}^2}\hfill \\ \widehat{\Lambda }\left(k\right)=\widehat{\Lambda }(k-1)+\frac{\eta \Delta \mathbf{H}(k-1)\left(\Delta I\left(k\right)-{\widehat{\Lambda }}^{\mathrm{T}}(k-1)\Delta \mathbf{H}(k-1)\right)}{\mu +{∥\Delta \mathbf{H}(k-1)∥}^2}\hfill \\ \widehat{\Lambda }\left(k\right)=\widehat{\Lambda }\left(1\right)\mathrm{if}∥\widehat{\Lambda }\left(k\right)∥\le \epsilon \mathrm{or}{∥\Delta \mathbf{H}(k-1)∥}^2\le \epsilon \mathrm{or}\mathrm{sign}\left({\widehat{\Lambda }}_{L_I+1}\left(k\right)\right)\ne \mathrm{sign}\left({\widehat{\Lambda }}_{L_I+1}\left(1\right)\right)\hfill \end{array}\right.$$

(9)

3. Improved FFDL Model-Free Adaptive PID Controller

3.1. iFFDL-MFA-PID Design

Consider the following SISO system affected by disturbances

$$I(k+1)=f\left(I\left(k\right),\cdots ,I(k-n_I),u\left(k\right),\cdots ,u(k-n_u)\right)+d\left(k\right)$$

(10)

where $d\left(k\right)$ is a bounded disturbance in the forklift system.

For System (10) to satisfy Assumptions 1 and 2, given $0\le L_I\le n_I$ and $1\le L_u\le n_u$,, there must be a parameter vector $\Lambda \left(k\right)$ so that System (10) can be transformed as

$$\Delta I(k+1)={\Lambda }^{\mathrm{T}}\left(k\right)\Delta \mathbf{H}\left(k\right)+q\left(k\right)$$

(11)

where $\begin{array}{c}q\left(k\right)=f\left(I(k-1),\cdots ,I(k-n_I),\right.\left.u(k-1),\cdots ,u(k-n_u)\right)\\ -f\left(I(k-1),\cdots ,I(k-n_I-1),\right.\left.u(k-1),\cdots ,u(k-n_u-1)\right)+\Delta d\left(k\right)\end{array}$, $q\left(k\right)$ is the sum of the bounded disturbances of the nonlinear remainder.

Based on (11) and the principle of equivalent feedback, we have

$$\begin{array}{cc}\hfill u\left(k\right)& =u(k-1)+\frac{\rho (I_d(k+1)-I\left(k\right)-q\left(k\right))}{\mathsf{\partial }+{\Lambda }_{L_v+1}\left(k\right)}-\frac{\rho {\displaystyle \sum _{i=1}^{L_I}}{\Lambda }_i\left(k\right)\Delta I(k-i+1)}{\mathsf{\partial }+{\Lambda }_{L_I+1}\left(k\right)}\hfill \\ & -\frac{{\displaystyle \sum _{i=L_I+2}^{L_I+L_u}}{\Lambda }_i\left(k\right)\Delta u\left(k-L_I-i+1\right)}{\mathsf{\partial }+{\Lambda }_{L_I+1}\left(k\right)}\hfill \end{array}$$

(12)

To further enhance robustness, by drawing on the ideas of Ref. [31] and incorporating an iPID strategy into the control law (12), it can be concluded that

$$\begin{array}{c}u\left(k\right)=u(k-1)+\frac{\rho (I_d(k+1)-I\left(k\right)-q\left(k\right))}{\mathsf{\partial }+{\Lambda }_{L_I+1}\left(k\right)}-\frac{\rho {\displaystyle \sum _{i=1}^{L_I}}{\Lambda }_i\left(k\right)\Delta I(k-i+1)}{\mathsf{\partial }+{\Lambda }_{L_I+1}\left(k\right)}-\frac{{\displaystyle \sum _{i=L_I+2}^{L_I+L_u}}{\Lambda }_i\left(k\right)\Delta u\left(k-l_I-i+1\right)}{\mathsf{\partial }+{\Lambda }_{L_I+1}\left(k\right)}\\ +k_1(e\left(k\right)-e(k-1))+k_{2}e\left(k\right)+k_3(e\left(k\right)-2e(k-1)+e(k-2))\end{array}$$

(13)

where $e\left(k\right)=I_d\left(k\right)-I\left(k\right)$ is the output error; $k_1,k_2,$ and $k_3$ are the control learning gains; and $q\left(k\right)$ and $\Lambda \left(k\right)$ need to be estimated by the algorithm.

Firstly, we use the difference algorithm in [29] to estimate $q\left(k\right)$

$$\widehat{q}\left(k\right)=\Delta I\left(k\right)-\widehat{\Lambda }{(k-1)}^{\mathrm{T}}\Delta \mathbf{H}(k-1)$$

(14)

Then, we provide the following new function:

$$J\left(\Lambda \left(k\right)\right)=\mu {∥\Lambda \left(k\right)-\widehat{\Lambda }(k-1)∥}^2+{\left|\Delta I\left(k\right)-q(k-1)-\Lambda {\left(k\right)}^T\Delta \mathbf{H}(k-1)\right|}^2$$

(15)

According to the minimization function (15), one has

$$\widehat{\Lambda }\left(k\right)=\widehat{\Lambda }(k-1)+\frac{\eta \Delta \mathbf{H}(k-1)}{\mu +{∥\Delta \mathbf{H}(k-1)∥}^2}\times (\Delta I\left(k\right)-\widehat{q}(k-1)-{\widehat{\Lambda }}^{\mathrm{T}}(k-1)\Delta \mathbf{H}(k-1))$$

(16)

Then, the following reset algorithm is proposed:

$$\widehat{\Lambda }\left(k\right)=\widehat{\Lambda }\left(1\right)\mathrm{if}∥\widehat{\Lambda }\left(k\right)∥\le \epsilon \mathrm{or}{∥\Delta \mathbf{H}(k-1)∥}^2\le \epsilon \mathrm{or}\mathrm{sign}\left({\widehat{\Lambda }}_{L_I+1}\left(k\right)\right)\ne \mathrm{sign}\left({\widehat{\Lambda }}_{L_I+1}\left(1\right)\right)$$

(17)

where $\epsilon >0$ is a sufficiently small positive number.

The control law (13), PG estimation algorithm (16), reset algorithm (17), and difference estimation algorithm (14) together form the iFFDL-MFA-PID scheme, and the control block diagram is shown in Figure 1.

Remark 1. 

The primary distinction between the FFDL data model (11) and the conventional model (4) lies in the expression of the form of the total system uncertainty. Unlike the linear parameter-based approach, the FFDL model considers nonlinear residual uncertainty. Incorporating the estimation of nonlinear residual uncertainty into the control law enables compensation for such uncertainties, thereby enhancing control design flexibility.

3.2. Stability Analysis

The stability analysis includes the boundedness of the parameter vector estimation and the convergence proof for the system error. For ease of description, let $L_I=L_u=1$; other situations follow similarly. Firstly, the following lemmas are introduced.

Lemma 1. 

For any matrix ${\mathbf{B}}_m\in R^{m\times m}$, $b_1,\cdots ,b_m$ are the eigenvalues of ${\mathbf{B}}_m$. Then, the spectral radius of ${\mathbf{B}}_m$ is $s\left({\mathbf{B}}_m\right)=\underset{j}{max}\left|b_j\right|$, $j\in \{1,\cdots ,m\}$.

Lemma 2. 

For any matrix ${\mathbf{B}}_n\in R^{n\times n}$ and a constant $\sigma >0$, there exists a matrix norm $∥·∥$ that satisfies

$$∥{\mathbf{B}}_n∥\le s\left({\mathbf{B}}_n\right)+\sigma$$

(18)

Theorem 2. 

If System (11) adopts the iFFDL-MFA-PID scheme, the system has the following properties:

(1) 

The parameter vector estimation $\tilde{\Lambda }\left(k\right)=\widehat{\Lambda }\left(k\right)-\Lambda \left(k\right)$ is bounded;

(2) 

$e(k+1)$ converges to a small bound if $\left|1-\widehat{\Lambda }\left(k\right)\left(k_1+k_2+k_3+\frac{2\rho }{\lambda +\widehat{\Lambda }\left(k\right)}\right)\right|<1$.

Proof 1. 

Subtracting $\Lambda \left(k\right)$ from both sides of (16) simultaneously yields

$$\begin{array}{cc}\hfill \tilde{\Lambda }\left(k\right)& =\left(1-\frac{\eta \Delta u^2(k-1)}{\mu +|\Delta u\left(k\right){|}^2}\right)\tilde{\Lambda }(k-1)+\frac{\eta \Delta u(k-1)\Delta u(k-2)}{\mu +|\Delta u\left(k\right){|}^2}\tilde{\Lambda }(k-2)\hfill \\ & +\frac{\eta \Delta u(k-1)}{\mu +|\Delta u\left(k\right){|}^2}(q(k-1)-q(k-2))+\Lambda (k-1)-\Lambda \left(k\right)\hfill \end{array}$$

(19)

Define ${\mathbf{M}}_{{\Xi }_1}\left(k\right)=\mathrm{diag}({\Xi }_1\left(k\right),\mathbf{0})$, ${\mathbf{M}}_{{\Xi }_2}\left(k\right)=\mathrm{diag}({\Xi }_2\left(k\right),\mathbf{0})$, and

$${\Xi }_1\left(k\right)=\left(\begin{array}{ccccc}1-\frac{\eta \Delta u^2\left(0\right)}{\mu +|\Delta u\left(0\right){|}^2}& 0& \cdots & 0& 0\\ \frac{\eta \Delta u\left(1\right)\Delta u\left(0\right)}{\mu +|\Delta u\left(1\right){|}^2}& \frac{\eta \Delta u^2\left(1\right)}{\mu +|\Delta u\left(1\right){|}^2}& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & 1-\frac{\eta \Delta u^2(k-2)}{\mu +|\Delta u(k-1){|}^2}& 0\\ 0& 0& \cdots & \frac{\eta \Delta u(k-1)\Delta u(k-2)}{\mu +|\Delta u(k-1){|}^2}& 1-\frac{\eta \Delta u^2(k-1)}{\mu +|\Delta u(k-1){|}^2}\end{array}\right)$$

(20)

$${\Xi }_2\left(k\right)=\mathrm{diag}\left(\frac{\eta \Delta u^2\left(0\right)}{\mu +|\Delta u\left(0\right){|}^2}\frac{\eta \Delta u^2\left(1\right)}{\mu +|\Delta u\left(1\right){|}^2}\cdots \frac{\eta \Delta u^2(k-1)}{\mu +|\Delta u(k-1){|}^2}\right)$$

(21)

Then, (19) can be rewritten as follows

$$\tilde{\Lambda }\left(k\right)={\mathbf{M}}_{{\Xi }_1}\left(k\right)\tilde{\Lambda }(k-1)+{\mathbf{M}}_{\Xi 2}\left(k\right)(\mathbf{q}(k-1)-\mathbf{q}(k-2))+\Lambda (k-1)-\Lambda \left(k\right)$$

(22)

where $\tilde{\Lambda }\left(k\right)={[\tilde{\Lambda }\left(1\right),\cdots ,\tilde{\Lambda }\left(k\right),\mathbf{0}]}^{\mathrm{T}}$, $\Lambda \left(k\right)={[\Lambda \left(1\right),\cdots ,\Lambda \left(k\right),\mathbf{0}]}^{\mathrm{T}}$, $\mathbf{q}\left(k\right)=\left[q\right(1),\cdots ,$${q\left(k\right),\mathbf{0}]}^{\mathrm{T}}$.

Taking the norm on both sides of (22), we obtain

$$\begin{array}{c}\Vert \tilde{\Lambda }\left(k\right){\Vert }_v\le \Vert {\mathbf{M}}_{{\Xi }_1}\left(k\right){\Vert }_v\Vert \tilde{\Lambda }(k-1){\Vert }_v+{∥{\mathbf{M}}_{{\Xi }_2}\left(k\right)∥}_v{∥\mathbf{q}(k-1)-\mathbf{q}(k-2)∥}_v\hfill \\ +\Vert \Lambda (k-1){\Vert }_v+\Vert \Lambda \left(k\right){\Vert }_v\hfill \end{array}$$

(23)

Since $\left|\Lambda \left(k\right)\right|\le {\overline{b}}_{\Lambda }$, $\left|q\left(k\right)\right|\le {\overline{b}}_o$, so that both $\Lambda \left(k\right)$ and $\mathbf{q}\left(k\right)$ are bounded. Next, we assume that $∥\Lambda \left(k\right)∥\le B_{\Lambda }$ and $∥\mathbf{q}\left(k\right)∥\le B_o$, where $B_{\Lambda }>0$ and $B_o>0$ are two constants.

Since $0<\eta <2$ and $\mu >0$, one has

$$\left|1-\frac{\eta \Delta u^2\left(k\right)}{\mu +|\Delta u\left(k\right){|}^2}\right|<1$$

(24)

According to Lemma 1, one deduces that $s\left({\mathbf{M}}_{{\Xi }_1}\left(k\right)\right)<1$. From Lemma 2, we have

$${∥{\mathbf{M}}_{{\Xi }_1}\left(k\right)∥}_v\le s\left({\mathbf{M}}_{{\Xi }_2}\left(k\right)\right)+{\sigma }_1\le d_1<1$$

(25)

where ${\sigma }_1>0$ and $d_1>0$ are small constants.

Then, according to Lemma 1 and Lemma 2, one has

$${∥{\mathbf{M}}_{{\Xi }_2}\left(k\right)∥}_v\Vert \mathbf{q}(k-1)-\mathbf{q}(k-2){\Vert }_v+\Vert \Lambda (k-1){\Vert }_v+\Vert \Lambda \left(k\right){\Vert }_v\le 2d_2{B}_o+2B_{\Lambda }<{\varsigma }_1$$

(26)

where $0<{\varsigma }_1$ is a constant.

In terms of (22) and (26), we can deduce that

$$\Vert \tilde{\Lambda }\left(k\right){\Vert }_v\le d_1\Vert \tilde{\Lambda }(k-1){\Vert }_v+{\varsigma }_1\le \cdots \le d_1^k\Vert \tilde{\Lambda }\left(0\right){\Vert }_v+\frac{{\varsigma }_1}{1-d_1}$$

(27)

which implies that $\tilde{\Lambda }\left(k\right)$ is bounded. Obviously, $\widehat{\Lambda }\left(k\right)$ is bounded since $\Lambda \left(k\right)$ is also bounded. Then, we assume that $\left|\widehat{\Lambda }\left(k\right)\right|\le {\overline{b}}_{\widehat{\Lambda }}$, where ${\overline{b}}_{\widehat{\Lambda }}>0$ is a constant and $\widehat{\Lambda }\left(k\right)$ are the elements of $\widehat{\Lambda }\left(k\right)$. □

Proof 2. 

According to (13) and (14), we have

$$\begin{array}{cc}\hfill \Delta u\left(k\right)& =\frac{\rho \left(I_d(k+1)-2I\left(k\right)+I(k-1)+\widehat{\Lambda }(k-1)\Delta u(k-1)\right)}{\mathsf{\partial }+\widehat{\Lambda }\left(k\right)}\hfill \\ & +k_1(e\left(k\right)-e(k-1))+k_{2}e\left(k\right)+k_3(e\left(k\right)-2e(k-1)+e(k-2))\hfill \\ & =\left(k_1+k_2+k_3+\frac{2\rho }{\lambda +\widehat{\theta }\left(k\right)}\right)e\left(k\right)+\left(-k_1-2k_3-\frac{\rho }{\mathsf{\partial }+\widehat{\Lambda }\left(k\right)}\right)e(k-1)+k_{3}e(k-2)\hfill \\ & +\frac{\rho \widehat{\Lambda }(k-1)}{\mathsf{\partial }+\widehat{\Lambda }\left(k\right)}\Delta u(k-1)+\frac{\rho }{\mathsf{\partial }+\widehat{\Lambda }\left(k\right)}\left(I_d(k+1)-2I_d\left(k\right)+I_d(k-1)\right)\hfill \end{array}$$

(28)

Combined with the system data model (11), the tracking error can be written as

$$\begin{array}{cc}\hfill e(k+1)& =I_d(k+1)-I(k+1)\hfill \\ \hfill & =e\left(k\right)-\Lambda \left(k\right)\Delta u\left(k\right)+I_d(k+1)-I_d\left(k\right)-q\left(k\right)\hfill \end{array}$$

(29)

Substituting (28) into (29), one has

$$\begin{array}{cc}\hfill e(k+1)& =e\left(k\right)\left(1-\Lambda \left(k\right)\left(k_1+k_2+k_3+\frac{2\rho }{\mathsf{\partial }+\widehat{\Lambda }\left(k\right)}\right)\right)\hfill \\ & -\Lambda \left(k\right)\left(-k_1-2k_3-\frac{\rho }{\mathsf{\partial }+\widehat{\Lambda }\left(k\right)}\right)e(k-1)-\Lambda \left(k\right)k_{3}e(k-2)\hfill \\ & -\frac{\rho \widehat{\Lambda }(k-1)\Lambda \left(k\right)}{\mathsf{\partial }+\widehat{\Lambda }\left(k\right)}\Delta u(k-1)+I_d(k+1)-I_d\left(k\right)-q\left(k\right)\hfill \\ & -\frac{\rho \Lambda \left(k\right)}{\mathsf{\partial }+\widehat{\Lambda }\left(k\right)}\left(I_d(k+1)-2I_d\left(k\right)+I_d(k-1)\right)\hfill \end{array}$$

(30)

Let $\mathbf{e}(k+1)={[e\left(1\right),\cdots ,e\left(k\right),\mathbf{0}]}^{\mathrm{T}}$, ${\mathbf{I}}_d\left(k\right)={[I_d\left(0\right),\cdots ,I_d\left(k\right),\mathbf{0}]}^{\mathrm{T}}$, $\overline{\mathbf{o}}\left(k\right)=\left[o\right(0),\cdots ,$${q\left(k\right),\mathbf{0}]}^{\mathrm{T}}$, $\Xi \left(k\right)=1-\Lambda \left(k\right)\left(k_1+k_2+k_3+\frac{2\rho }{\lambda +\widehat{\Lambda }\left(k\right)}\right)$, and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}w\left(k\right)=\frac{\rho \widehat{\Lambda }(k-1)\Lambda \left(k\right)}{\mathsf{\partial }+\widehat{\Lambda }\left(k\right)}\left(k_1+k_2+k_3+\frac{2\rho }{\mathsf{\partial }+\widehat{\Lambda }(k-1)}\right)-\Lambda \left(k\right)\left(k_1+2k_3+\frac{\rho }{\mathsf{\partial }+\widehat{\Lambda }\left(k\right)}\right)\hfill \\ c\left(k\right)=k_3+\frac{\rho \widehat{\Lambda }\left(k\right)}{\mathsf{\partial }+\widehat{\Lambda }(k+1)}\left(\begin{array}{c}\frac{\rho \widehat{\Lambda }(k-1)}{\mathsf{\partial }+\widehat{\Lambda }\left(k\right)}\left(k_1+k_2+k_3+\frac{2\rho }{\mathsf{\partial }+\widehat{\Lambda }(k-1)}\right)\hfill \\ -\left(k_1+2k_3+\frac{\rho }{\mathsf{\partial }+\widehat{\Lambda }\left(k\right)}\right)\hfill \end{array}\right)\hfill \end{array}\right.\end{array}$$

(31)

Then, define ${\mathbf{M}}_{{\Xi }_3}\left(k\right)=\mathrm{diag}\left({\Xi }_3\left(k\right)\mathbf{0}\right),{\mathbf{M}}_{{\Xi }_4}\left(k\right)=\mathrm{diag}({\Xi }_4\left(k\right)\mathbf{0})$, and

$$m\left(k\right)={\left[\frac{\rho \Lambda \left(0\right)\widehat{\Lambda }(-1)}{\mathsf{\partial }+\widehat{\Lambda }\left(0\right)},\frac{{\rho }^2\Lambda \left(0\right)\widehat{\Lambda }\left(0\right)}{\mathsf{\partial }+\widehat{\Lambda }\left(1\right)}\frac{\rho \widehat{\Lambda }(-1)}{\mathsf{\partial }+\widehat{\Lambda }\left(0\right)},\dots ,\frac{{\rho }^{k+1}\Lambda \left(k\right){\displaystyle \prod _{m=-1}^{k-1}}\widehat{\Lambda }\left(n\right)}{{\displaystyle \prod _{m=0}^k}(\mathsf{\partial }+\widehat{\Lambda }\left(m\right))},\mathbf{0}\right]}^{\mathrm{T}}$$

(32)

where

$${\Xi }_3\left(k\right)=\left(\begin{array}{ccccc}\Xi \left(0\right)& 0& \cdots & 0& 0\\ -w\left(1\right)& \Xi \left(1\right)& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ -\frac{{\rho }^{k-3}\Lambda (k-1){\displaystyle \prod _{n=2}^{k-2}}\widehat{\Lambda }\left(n\right)}{{\displaystyle \prod _{m=3}^{k-1}}(\mathsf{\partial }+\widehat{\Lambda }\left(m\right))}c\left(1\right)& -\frac{{\rho }^{k-4}\Lambda (k-1){\displaystyle \prod _{n=3}^{k-2}}\widehat{\Lambda }\left(n\right)}{{\displaystyle \prod _{m=4}^{k-1}}(\mathsf{\partial }+\widehat{\Lambda }\left(m\right))}c\left(2\right)& \cdots & \Xi (k-1)& 0\\ -\frac{{\rho }^{k-2}\Lambda (k-1){\displaystyle \prod _{n=2}^{k-1}}\widehat{\Lambda }\left(n\right)}{{\displaystyle \prod _{m=3}^k}(\mathsf{\partial }+\widehat{\Lambda }\left(m\right))}c\left(1\right)& -\frac{{\rho }^{k-3}\Lambda (k-1){\displaystyle \prod _{n=3}^{k-1}}\widehat{\Lambda }\left(n\right)}{{\displaystyle \prod _{m=4}^k}(\mathsf{\partial }+\widehat{\Lambda }\left(m\right))}c\left(21\right)& \cdots & -w\left(k\right)& \Xi \left(k\right)\end{array}\right)$$

(33)

$${\Xi }_4\left(k\right)=\left(\begin{array}{cccc}\frac{\rho \Lambda \left(0\right)}{\mathsf{\partial }+\widehat{\Lambda }\left(0\right)}& 0& \cdots & 0\\ \frac{\rho \Lambda \left(1\right)}{\mathsf{\partial }+\widehat{\Lambda }\left(1\right)}\frac{\rho \Lambda \left(0\right)}{\mathsf{\partial }+\widehat{\Lambda }\left(0\right)}& \frac{\rho \Lambda \left(1\right)}{\mathsf{\partial }+\widehat{\Lambda }\left(1\right)}& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ \frac{{\rho }^{k+1}\Lambda \left(k\right){\displaystyle \prod _{n=0}^{k-1}}\widehat{\Lambda }\left(n\right)}{{\displaystyle \prod _{m=0}^k}(\mathsf{\partial }+\widehat{\Lambda }\left(m\right))}& \frac{{\rho }^k\Lambda \left(k\right){\displaystyle \prod _{n=0}^{k-1}}\widehat{\Lambda }\left(n\right)}{{\displaystyle \prod _{m=1}^k}(\mathsf{\partial }+\widehat{\Lambda }\left(m\right))}& \cdots & \frac{\rho \Lambda \left(k\right)}{\mathsf{\partial }+\widehat{\Lambda }\left(m\right)}\end{array}\right)$$

(34)

Then, (30) can be rewritten as follows

$$\begin{array}{cc}\hfill \mathbf{e}(k+1)& ={\mathbf{M}}_{{\Xi }_3}\left(k\right)\mathbf{e}\left(k\right)-{\mathbf{M}}_{{\Xi }_4}\left(k\right)\left({\mathbf{I}}_d(k+1)-2{\mathbf{I}}_d\left(k\right)+{\mathbf{I}}_d(k-1)\right)\hfill \\ & +{\mathbf{I}}_d(k+1)-{\mathbf{I}}_d\left(k\right)-\overline{\mathbf{o}}\left(k\right)-\mathbf{m}\left(k\right)\Lambda (-1)\Delta u(-1)\hfill \\ & ={\mathbf{M}}_{{\Xi }_3}\left(k\right)\mathbf{e}\left(k\right)-{\mathbf{M}}_{{\Xi }_4}\left(k\right)\left({\mathbf{I}}_d(k+1)-2{\mathbf{I}}_d\left(k\right)+{\mathbf{I}}_d(k-1)\right)\hfill \\ & +{\mathbf{I}}_d(k+1)-{\mathbf{I}}_d\left(k\right)-\overline{\mathbf{o}}\left(k\right)\hfill \end{array}$$

(35)

Define ${∥{\mathbf{I}}_d(k+1)∥}_v\le B_{I_d}$, ${∥\overline{\mathbf{o}}\left(k\right)∥}_v\le B_{\overline{o}}$, where $B_{I_d}$ and $B_{\overline{o}}$ are constants that are more significant than 0.

According to (35), one has

$$\Vert \mathbf{e}(k+1){\Vert }_v\le {∥{\mathbf{M}}_{{\Xi }_3}\left(k\right)∥}_v\Vert \mathbf{e}\left(k\right){\Vert }_v+({∥\mathbf{I}-{\mathbf{M}}_{{\Xi }_4}\left(k\right)∥}_v+{∥2{\mathbf{M}}_{{\Xi }_4}\left(k\right)-I∥}_v+{∥{\mathbf{M}}_{{\Xi }_4}\left(k\right)∥}_v)B_{I_d}+B_{\overline{o}}$$

(36)

Since $\epsilon <\left|\widehat{\Lambda }\left(k\right)\right|$ and $\left|\Lambda \left(k\right)\right|<{\overline{b}}_{\Lambda }$, and according to Lemma 2, we deduce that ${∥{\mathbf{M}}_{{\Xi }_4}\left(k\right)∥}_v$ is bounded.

Further, the parameters are set as follows

$$\left|1-\Lambda \left(k\right)\left(k_1+k_2+k_3+\frac{2\rho }{\mathsf{\partial }+\widehat{\Lambda }\left(k\right)}\right)\right|<1$$

(37)

According to Lemmas 1 and 2, if ${\sigma }_3>0$ exists, then the following inequality holds

$${∥{\mathbf{M}}_{{\Xi }_3}\left(k\right)∥}_v\le s\left({\mathbf{M}}_{{\Xi }_3}\left(k\right)\right)+{\sigma }_3\le d_3<1$$

(38)

where $d_3>0$ is a constant. Therefore, according to (35) and (38), one has

$$\Vert \mathbf{e}(k+1){\Vert }_v\le d_3\Vert \mathbf{e}\left(k\right){\Vert }_v+{\varsigma }_2\le \cdots \le d_3^{k+1}\Vert \mathbf{e}\left(0\right){\Vert }_v+\frac{{\varsigma }_2}{1-d_3}$$

(39)

where $({∥\mathbf{I}-{\mathbf{M}}_{{\Xi }_4}\left(k\right)∥}_v+{∥2{\mathbf{M}}_{{\Xi }_4}\left(k\right)-\mathbf{I}∥}_v+{∥{\mathbf{M}}_{{\Xi }_4}\left(k\right)∥}_v)B_{I_d}+B_{\overline{o}}\le {\varsigma }_2$, ${\varsigma }_2>0$, which means that $\underset{k\to \infty }{lim}\Vert \mathbf{e}(k+1){\Vert }_v=\frac{{\varsigma }_2}{1-d_3}$, and conclusion (b) in Theorem 2 is proved. □

Remark 2. 

It should be noted that the controller parameters of the proposed iFFDL-MFA-PID scheme are generally selected through trial and error. On the other hand, the ranges of these parameters can be determined according to the convergence condition given in the following theorem if the gradient information is available. Further, $k_1$, $k_2$, and $k_3$ can also be selected using the well-known PID control parameter-tuning methods.

4. Simulation Experiment

The experiments are divided into two groups. The first involves a typical numerical simulation, primarily aimed at comparing the effectiveness of the proposed iFFDL-MFA-PID method to that of the FFDL-MFAC method. The second employs the proposed method to control an electric balanced forklift to validate its efficacy. Moreover, the FFDL-MFAC and traditional PID methods are included in the comparison analysis using indicators such as the MSE, MAE, settling time, and rise time. These comparisons aim to demonstrate the superiority of the proposed method.

4.1. Experiment 1

To enhance the applicability of the proposed method, we extended iFFDL-MFA-PID to a MIMO system and verified its effectiveness by simulating a typical controlled object. We compared and simulated the iFFDL-MFA-PID and FFDL-MFAC algorithms using time-varying signals. The first group of simulation objects consisted of a system with three inputs and three outputs, as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{I}_1(k+1)=1.5\times I_1\left(k\right)+0.12\times {I_1}^2(k-1)+0.7\times u_1\left(k\right)\\ +0.3\times u_2\left(k\right)+0.3\times u_3\left(k\right)+0.5\times d\left(k\right)\\ I_2(k+1)=1.6\times I_2\left(k\right)+0.14\times I_{{}_2}^2(k-1)+0.4\times u_1\left(k\right)\\ +0.6\times u_2\left(k\right)+0.3\times u_3\left(k\right)+0.5\times d\left(k\right)\\ I_3(k+1)=1.6\times I_3\left(k\right)+0.15\times {I_3}^2(k-1)+0.1\times u_1\left(k\right)\\ +0.7\times u_2\left(k\right)+0.6\times u_3\left(k\right)+0.5\times d\left(k\right)\end{array}\right.\end{array}$$

(40)

where $d\left(k\right)$ represents a white-noise signal with a mean of zero and a variance of 0.05. The predetermined trajectory settings were as follows:

$$\begin{array}{c}\hfill I_d=\left\{\begin{array}{c}\mathrm{lins}(0,50,100)\times 5/9,0<k\le 100\\ \mathrm{lins}(250,250,50)/9,100<k\le 150\\ \mathrm{lins}(250,290,50)/9,150<k\le 200\\ \mathrm{lins}(290,290,600)/9,200<k\le 800\\ \mathrm{lins}(290,0,200)/9,801<k\le 1000\end{array}\right.\end{array}$$

(41)

The controller parameters were set to $\widehat{\Lambda }\left(1\right)=\widehat{\Lambda }\left(2\right)=\mathrm{diag}\{0.55,0.6,0.55\}$, $\eta =0.75$, $\mu =1$, $\rho =0.15$, $\mathsf{\partial }=0.2$, $k_1=0.45$, $k_2=0.7$, $k_3=0.05$, $L_I=1$, and $L_u=1$.

Figure 2 depicts the system output under the influence of the iFFDL-MFA-PID and FFDL-MFAC methods, whereas Figure 3 shows the corresponding output error graphs.

Table 1. Performance indexes.

Method	FFDL-MFAC	iFFDL-MFA-PID
Rise time (s)	5	3
Adjust time (s)	27	15
MAE	$3.41\times {10}^{-2}$	$1.15\times {10}^{-2}$
MSE	$7.29\times {10}^{-4}$	$2.32\times {10}^{-4}$

presents the corresponding control performance indicators. By zooming in on the details in the magnified graphs in Figure 2, it can be observed that at the 100th and 200th sampling points, the FFDL-MFAC method experienced phase delay and poor disturbance rejection capabilities, resulting in delayed tracking phenomena and significant oscillations. This significantly diminished the rapid response characteristics of the model-free adaptive control algorithm. On the other hand, the iFFDL-MFA-PID method exhibited no phase delays, promptly providing feedback on the output signal. It maintained the fast response characteristic of the model-free adaptive control algorithm while eliminating disturbances. At the 200th sampling point, even when the reference signal changed, the iFFDL-MFA-PID method continued to maintain its convergence, further demonstrating the adaptability of the proposed algorithm.

Additionally, by examining the local magnified graphs in the output curve in Figure 2 and the time indicators in Table 1, it can be observed that the iFFDL-MFA-PID method achieved stability within 15 s under varying reference outputs, whereas MFAC required more than 27 s. Furthermore, the iFFDL-MFA-PID method exhibited approximately half the rise time of the MFAC. Moreover, from the error change curve, it is evident that the iFFDL-MFA-PID method not only possessed fast tracking characteristics but also effectively suppressed the influence of external disturbances, consistently outperforming traditional MFAC throughout the tracking process. The error-related indicators shown in Figure 3 and Table 1 indicate that regardless of whether the reference signal was constant or changing, the tracking performance of the proposed iFFDL-MFA-PID method was superior to that of traditional MFAC. The MAE and MSE values of the iFFDL-MFA-PID method were only $1.15\times {10}^{-2}$ and $2.32\times {10}^{-4}$, respectively, whereas the MAE and MSE values of traditional MFAC were $3.41\times {10}^{-2}$ and $7.29\times {10}^{-4}$, respectively. In conclusion, the iFFDL-MFA-PID method achieves nonlinear compensation for unknown disturbances such as measurement noise without relying on a model. It further introduces the iPID method to optimize the adaptive controller, resulting in a low-order controller with superior simulation results, robustness, and control performance.

Figure 4 illustrates the control input curves corresponding to each control scheme. In the iFFDL-MFA-PID scheme, all the control inputs changed smoothly in each stage. Even when the predetermined signal changed, the control input adjusted at a lower rate, effectively alleviating the chattering phenomenon of the system. On the other hand, the FFDL-MFAC scheme exhibited significant control input changes during the predetermined signal-change stage. The amplitudes of these changes were large, leading to a more pronounced chattering phenomenon in the system.

(1)

MAE

$$\mathrm{MAE}=\frac{1}{nT}\sum _{s=1}^n\sum _{i=1}^T\left|I_s\left(i\right)-I_d\left(i\right)\right|$$

(42)

(2)

MSE

$$\mathrm{MSE}=\frac{1}{nT}\sum _{s=1}^n\sum _{i=1}^T{\left|I_s\left(i\right)-I_d\left(i\right)\right|}^2$$

(43)

In summary, the iFFDL-MFA-PID method not only maintains the fast response characteristic of the MFAC method but also benefits from the strong adaptive ability of intelligent PID parameters, achieving an optimal control effect for systems affected by noise disturbances. Furthermore, the parameter adjustment process based on the iFFDL-MFA-PID method is simpler and less challenging.

4.2. Experiment 2: Simulation of Longitudinal Dynamic Control of Electric Forklift

The electric forklift, with its advantages of environmental protection, zero emissions, simple operation, and high efficiency, finds wide applications in various industries, including logistics warehousing, manufacturing, port terminals, supermarkets, and logistics parks. The working site of an electric forklift is illustrated in Figure 5. By utilizing electric forklifts, logistics efficiency can be improved, labor intensity reduced, and the goals of energy saving and emission reduction achieved [36].

(1)

Structure and composition: A general electric forklift comprises a chassis, cab, fork assembly, power system, and control system. The chassis is the infrastructure of the forklift, which supports and carries the components of the whole forklift. The cab is where the operator works, including the seat, joystick, instrument panel, etc. Fork assemblies are used to load, unload, and carry goods.

(2)

Power type: Electric forklifts use two main power types: battery and power line supply. Battery-powered forklifts are powered by a battery pack, which has the characteristics of high flexibility and wide adaptability, making them suitable for indoor and outdoor operations. Power line-powered forklifts obtain power by connecting to a fixed power line, making them suitable for continuous logistics operations.

(3)

Load capacity: The load capacity of electric forklifts is usually between 1 and 10 tons. According to different work needs and scenarios, a forklift with an appropriate load capacity can be selected to meet the handling needs.

(4)

Operation mode: The operation of the electric forklift is usually completed by the operator through control devices such as joysticks, buttons, or touch screens. The operator can control the driving, steering, lifting fork, tilting fork, and other actions of the forklift.

(5)

Safety and intelligence: Modern electric forklifts are equipped with various safety equipment and auxiliary systems, such as safety warning lights, reverse radar, anti-collision devices, driver status monitoring, etc., to improve operational safety. At the same time, some electric forklifts also have intelligent functions, such as navigation systems, automatic storage devices, etc., which improve work efficiency and accuracy.

The drive system is a crucial component of an electric forklift and typically consists of a traction motor, control system (including motor driver, controller, and various sensors), mechanical deceleration and transmission, wheels, and other related components. Currently, there are two common types of speed regulation systems for electric forklifts: DC and AC drives. The operational control of an electric forklift mainly involves the following aspects:

(1)

Start and stop: An electric forklift is usually started and stopped by pedals or buttons. When starting, the operator needs to ensure the safety of the surrounding environment and press the start switch or pedal to start the electric forklift. When stopping, the operator needs to release the start switch or pedal, and the electric forklift will stop running.

(2)

Direction control: Electric forklifts usually use a steering wheel to control the driving direction. The steering wheel is pushed forward to drive the electric forklift forward and pulled backward to drive the electric forklift backward. When turning, the operator can turn to the left or right to select the lever or steering wheel. When making turns, the directional lever or steering wheel is turned to the left or right.

(3)

Speed control: The speed of the electric forklift can usually be adjusted by the accelerator pedal or the acceleration and deceleration button.

(4)

Lifting control: Electric forklifts are usually equipped with forklifting functions, which can be used for loading and unloading goods. The lifting and lowering of the fork can be controlled by lifting the lever or buttons. Usually, the lifting lever is pushed upward and the fork will rise; the lifting lever is pulled down to lower the fork.

(5)

Auxiliary function control: Some electric forklifts also have auxiliary functions, such as tilt-angle adjustment, a side-shift function, rotary seat control, etc. These functions can be operated through the corresponding control devices to meet different work requirements.

This paper focuses exclusively on the longitudinal speed control of electric forklifts, which encompasses drive control, brake control, and relevant mode switching. We assume that the forklift is operating on an inclined road and consider the forklift itself as rigid. The force distribution acting on the forklift is depicted in Figure 6. The tires on the vehicle axle experience longitudinal and normal forces, whereas other external forces acting on the forklift include air resistance, rolling resistance, and gravity. The force equilibrium equation in the forward direction of the forklift can be expressed as [37]

$$m{a}_{\mathrm{h}}=\frac{F_{\mathrm{t}}-F_{\mathrm{f}}-F_{\mathrm{w}}-F_{\mathrm{b}}-F_g}{\delta }$$

(44)

where $F_{\mathrm{t}}=\frac{T_{\mathrm{e}}{i}_{\mathrm{g}}{i}_0{\eta }_{\mathrm{T}}}{r};F_{\mathrm{f}}=mgfcos\theta ;F_{\mathrm{w}}=\frac{1}{2}{C}_{\mathrm{D}}A\rho u_{\mathrm{r}}^2;F_{\mathrm{b}}=k_{\mathrm{b}}{p}_{\mathrm{b}};$ and $F_{\mathrm{g}}=mgsin\theta$. $F_{\mathrm{t}}$ is the driving force; $F_{\mathrm{f}}$ is the rolling resistance; $F_{\mathrm{w}}$ is the air resistance; $F_{\mathrm{b}}$ is the wheel braking force; $T_{\mathrm{e}}$ is the motor torque; $i_{\mathrm{g}}$ is the motor transmission ratio of the electric balanced forklift; $i_0$ is the transmission ratio of the reducer; ${\eta }_{\mathrm{T}}$ is the transmission efficiency; f is the rolling resistance coefficient; $C_{\mathrm{D}}$ is the air resistance coefficient; A is the windward area; $\rho$ is the air density; $u_{\mathrm{r}}$ is the speed of the forklift on the ramp; $k_{\mathrm{b}}$ is the braking-force proportional coefficient; $p_{\mathrm{b}}$ is the braking force; m is the total mass of the forklift; g is the gravity acceleration; $\theta$ is the ramp angle; and $\delta$ is the rotation mass conversion coefficient. From (44), the desired motor torque $T_{\mathrm{des}}$ can be obtained by the expected acceleration $a_{\mathrm{des}}$:

$$T_{\mathrm{des}}=\frac{\left(mgfcos\theta +mgsin\theta +\frac{1}{2}{C}_{\mathrm{D}}A\rho u_{\mathrm{r}}^2+k_{\mathrm{b}}{p}_{\mathrm{b}}+\delta m{a}_{\mathrm{h}}\right)r}{i_{\mathrm{g}}{i}_0{\eta }_{\mathrm{T}}}$$

(45)

Under the driving conditions, there is a certain relationship between the steady-state output torque $T_{\mathrm{e}}$ and the speed $n_{\mathrm{e}}$ of the motor, in which any parameter can be obtained from the other two parameters.

For the braking conditions, the relationship between $p_{\mathrm{b}}$ and $F_{\mathrm{b}}$ applied to the wheels can be regarded as linear. Therefore, under the braking condition, we can obtain the desired braking force $p_{\mathrm{des}}$ as

$$p_{\mathrm{des}}=\left(\frac{\delta m{a}_{\mathrm{h}}+\frac{T_{\mathrm{e}}{i}_{\mathrm{g}}{i}_0{\eta }_{\mathrm{T}}}{r}+mgfcos\theta +mgsin\theta +\frac{1}{2}{C}_{\mathrm{D}}A\rho u_{\mathrm{r}}^2}{k_{\mathrm{b}}}\right)$$

(46)

Electric forklifts are susceptible to changes in the environment and road conditions during operation, which can impact model parameters such as the quality parameters, resistance coefficient, and motor torque, making it challenging to obtain accurate dynamic models. To address these issues, the iFFDL-MFA-PID method is an ideal choice for achieving longitudinal autonomous driving control of electric forklifts.

This paper collected real-time control force and speed data from electric forklifts and employed the recursive least-squares method to identify the online longitudinal operational model parameters of the forklifts, allowing for real-time corrections. Equations (44)–(46) present the dynamic equations for controlling the force and speed of electric forklifts.

$$A\left(z^{-1}\right)I\left(k\right)=B\left(z^{-1}\right)u(k-1)+\frac{C\left(z^{-1}\right)V\left(k\right)}{1-z^{-1}}$$

(47)

where $I\left(k\right)$ and $u\left(k\right)$ are the speed and control force of the electric forklift, respectively, and the corresponding system parameters can be defined as

$$\left\{\begin{array}{c}A\left(z^{-1}\right)=1+a_0{z}^{-1}+\cdots +a_{n_a}{z}^{-n_a}\hfill \\ B\left(z^{-1}\right)=b_0+b_1{z}^{-1}+\cdots +b_{n_b}{z}^{-n_b}\hfill \\ C\left(z^{-1}\right)=1+c_1{z}^{-1}+\cdots +c_{n_c}{z}^{-n_c}\hfill \end{array}\right.$$

(48)

We rewrite (47) in the following form

$$\begin{array}{c}\Delta I\left(k\right)=[1-A\left(z^{-1}\right)]\Delta I\left(k\right)+B\left(z^{-1}\right)\Delta u(k-1)+V\left(k\right)\\ ={\mathbf{M}}^{\mathrm{T}}\left(k\right)\mathbf{Y}\left(k\right)+V\left(k\right)\end{array}$$

(49)

where $\mathbf{M}\left(k\right)$ and $\mathbf{Y}\left(k\right)$ are the data vectors and model parameters, respectively, defined as

$$\left\{\begin{array}{c}\mathbf{M}\left(k\right)={[-\Delta I(k-1),\cdots ,-\Delta I(k-n_a),\Delta u(k-1),\cdots ,\Delta u(k-n_b-1)]}^{\mathrm{T}}\hfill \\ \mathbf{Y}\left(k\right)={[a_1,\cdots ,a_{n_a},b_0,\cdots ,b_{n_b}]}^{\mathrm{T}}\hfill \end{array}\right.$$

(50)

The parameter vector $\mathbf{Y}\left(k\right)$ to be estimated is calculated using the following recursive least-squares method with a forgetting factor [38].

$$\left\{\begin{array}{c}\widehat{\mathbf{Y}}\left(k\right)=\widehat{\mathbf{Y}}(k-1)+\mathbf{K}\left(k\right)[\Delta I\left(k\right)-{\mathbf{M}}^{\mathrm{T}}\left(k\right)\widehat{\mathbf{Y}}(k-1)]\hfill \\ \mathbf{K}\left(k\right)=\frac{\mathbf{P}(k-1)\mathbf{M}(k-1)}{\lambda +{\mathbf{M}}^{\mathrm{T}}\left(k\right)\mathbf{P}(k-1)\mathbf{M}\left(k\right)}\hfill \\ \mathbf{P}\left(k\right)=\frac{1}{\lambda }\left[\mathbf{I}-K\left(k\right){\mathbf{M}}^{\mathrm{T}}\left(k\right)\right]\mathbf{P}(k-1)\hfill \end{array}\right.$$

(51)

where $\widehat{\mathbf{Y}}\left(k\right)$ is the estimated value of the model parameters; $\mathbf{K}\left(k\right)$ and $\mathbf{P}\left(k\right)$ are the parameter matrices; $\mathbf{P}\left(0\right)=({10}^4-{10}^{10})\mathbf{I}$; and $\lambda$ is the forgetting factor, with values ranging from 0.9 to 1.

The four coefficients in (50) calculated using the recursive least-squares method were input into (47) to obtain the control model.

To validate the effectiveness of the proposed iFFDL-MFA-PID algorithm, the operating speed, position, and other relevant information of the electric forklift were recorded. The parameters of the EFG-BC/320 electric balanced forklift, as well as the control parameters of the iFFDL-MFA-PID algorithm, are shown in

Table 2. EFG-BC/320 electric balanced forklift (Jungheinrich Company, Shanghai, China) and iFFDL-MFA-PID parameters.

Parameter Name	-	Parameter Name	-
Forklift model	EFG-BC/320	Control method	iFFDL-MFA-PID
Maximum speed	18 km/h	Load capacity	2000 kg
Overall length	3096 mm	Power system	Electric drive (48 V)
System parameter	$\begin{array}{c}\hfill i_{\mathrm{g}}=3.95,C_{\mathrm{D}}=0.75,\hfill \\ \hfill k_{\mathrm{b}}=1.23,A=1.85\hfill \\ \hfill f=0.05,m=3820\mathrm{kg}\hfill \end{array}$	Control parameter	$\begin{array}{c}\hfill I\left(1\right)=I_d\left(1\right),\eta =0.85\hfill \\ \hfill \mu =0.9,\rho =0.1\hfill \\ \hfill \mathsf{\partial }=0.25,k_1=0.35\hfill \\ \hfill k_2=0.2,k_3=0.01\hfill \\ \hfill L_I=1,L_u=1\hfill \end{array}$
Input and output data	Speed, position, and driving-force sensors	Control accuracy	±0.132 m/s (velocity tracking error)

. Simulation verification was conducted using the iFFDL-MFA-PID, FFDL-MFAC, and PID methods, respectively. The tracking error, control force, and acceleration were compared and analyzed to demonstrate the superiority of the iFFDL-MFA-PID method. The parameters of the other contrasting control methods were set as follows:

(1)

FFDL-MFAC method: $I\left(1\right)=I_d\left(1\right)$, $\eta =0.85$, $\mu =0.9$, $\rho =0.1$, $\mathsf{\partial }=0.25$, $L_I=1$, and $L_u=1$.

(2)

PID method [39]: $I\left(1\right)=I_d\left(1\right)$, $K_1=0.65$, $K_2=0.7$, and $K_3=0.06$.

Figure 7 shows the speed-tracking curves of the electric forklift with the iFFDL-MFA-PID, MFAC, and PID methods, and Figure 8 shows their error curves. In Figure 7 and Figure 8, it can be seen that the speed-tracking error range of the electric forklift under the action of MFAC is in the range of [−0.128 km·h⁻¹, 0.194 km·h⁻¹], whereas that of the PID is in the range of [−0.228 km·h⁻¹, 0.313 km·h⁻¹]. For iFFDL-MFA-PID, firstly, the MFAC law was obtained by using the principle of equivalent feedback. Secondly, to enhance the system’s robustness, the idea of an intelligent PID was introduced, and the delay estimation method was used to estimate the disturbance, parameter error, and other uncertainties of the data model. Further ensuring convergence, the speed-tracking error of the electric forklift remained stable throughout the entire process in the range of [−0.078 km·h⁻¹, 0.124 km·h⁻¹], with minimal disturbance, thereby meeting the speed-tracking accuracy requirements. This method has higher tracking accuracy and better robustness compared to the FFDL-MFAC method.

(1)

MAE

$$\mathrm{MAE}=\frac{1}{T}\sum _{k=1}^T\left|I_d-I\left(k\right)\right|$$

(52)

(2)

MSE

$$\mathrm{MSE}=\frac{1}{T}\sum _{k=1}^T{\left|I_d-I\left(k\right)\right|}^2$$

(53)

Furthermore, the following performance indicators were introduced to evaluate the iFFDL-MFA-PID, FFDL-MFAC, and PID methods.

Table 3. Control schemes’ performance indexes.

Method	PID	FFDL-MFAC	iFFDL-MFA-PID
Adjust time (s)	35	20	7
MAE	$5.67\times {10}^{-2}$	$3.26\times {10}^{-2}$	$1.14\times {10}^{-2}$
MSE	$2.15\times {10}^{-3}$	$8.62\times {10}^{-4}$	$1.48\times {10}^{-4}$

presents the control performance of these three control methods. The calculation method for the performance indicators is shown in (52) and (53), where k and T represent the current sampling time and total time, respectively. The settling time is an important indicator of a control system’s performance. In this study, we observed the response curve of the system when the reference signal changed at the 50th second to identify the time point at which the output first reached and remained within a certain range of stability under the influence of the input. From Table 3, it is evident that the settling time of the iFFDL-MFA-PID method was approximately 7 seconds, which was less than half of the settling time of the MFAC method and significantly lower than that of the PID method. The iFFDL-MFA-PID method achieves an MAE of 1.14 × 10${}^{-2}$ and an MSE of 1.48 × 10${}^{-4}$, whereas the MFAC method yielded an MAE of 3.26 × 10${}^{-2}$ and an MSE of 8.62 × 10${}^{-4}$. These values are much smaller than those of the PID method, indicating higher precision and better stability. The results demonstrate that the control strategy proposed in this paper can more effectively suppress the harm caused by measurement disturbances to the system’s stability, thereby achieving more accurate tracking of the system output.

Figure 9 shows the control force (driving force/braking force) variation curves of the electric forklift under the action of the iFFDL-MFA-PID, FFDL-MFAC, and PID methods. As shown in the figure, the control forces of the electric forklift under the FFDL-MFAC and PID methods changed frequently and were greatly affected by interference. The control force change using the iFFDL-MFA-PID method was relatively stable, and the amplitude was smaller than that of the FFDL-MFAC method. This ensures the operational safety of the electric forklift and that no rollovers or other phenomena occur during the handling process.

Figure 10 shows the acceleration variation curves of electric forklifts using iFFDL-MFA-PID, FFDL-MFAC, and PID methods. As shown in the figure, the acceleration changes using the FFDL-MFAC and PID methods were too fast. In contrast to the iFFDL-MFA-PID method, the acceleration of the electric forklift changed slowly, which ensures the safe operation of the electric forklift and that there is no rollover in the handling process.

These results indicate that the control strategy proposed in this paper can effectively mitigate the adverse effects caused by measurement disturbances on system stability, leading to more accurate tracking of the system’s output.

5. Conclusions

This paper introduces the iFFDL-MFA-PID algorithm to address the challenges posed by sensor measurement noise and other disturbances in electric cargo forklift systems. The algorithm is specifically designed to handle uncertain disturbed systems. The effectiveness of the controller is verified through both theoretical analysis and simulation experiments. Firstly, the gradient estimation algorithm and time difference algorithm are employed based on the nonlinear general equation of the electric cargo forklift system. These algorithms effectively estimate time-varying parameters and uncertain nonlinear terms, enabling accurate estimation of changing parameters and nonlinear terms in the system. Furthermore, the proposed control strategy leverages the iPID algorithm and utilizes additional tracking error information to compensate for the decrease in system information accuracy caused by excessive linearization in the iFFDL-MFAC approach. This ensures better system output tracking by taking advantage of the benefits offered by the iPID algorithm. Additionally, the learning update law and parameter update law of the system are restructured to enhance the controller’s structure, resulting in improved overall performance and stability. The simulation results demonstrate that the proposed control strategy effectively suppresses the harmful effects of disturbances on system stability, achieving more accurate tracking of the system output. This ensures the safe operation of electric forklifts and helps prevent accidents such as rollovers and rear-end collisions during operation. In addition, in future work, we plan to extend the theory of autonomous driving for cars to forklift systems. By integrating image recognition technology, we can make the forklift autonomous driving system more intelligent, safe, and efficient, providing drivers with a safer driving experience.

It should be noted that adaptively adjusting the controller parameters without relying on model information is crucial. For instance, PID control parameter-tuning methods, such as iterative feedback tuning and virtual reference feedback tuning, are data-driven methods. Therefore, in future research, we plan to develop parameter-tuning mechanisms that reference these methods to enhance control performance. While the adaptive adjustment of controller parameters without relying on model information is crucial, we recognize that in specific engineering applications, reliance on model information remains vital. Therefore, the purpose of this study is not to completely negate the importance of model information but to emphasize the advantages and practicality of adaptively adjusting controller parameters in the absence of detailed model information.