Enhanced Sliding-Mode Control for Tracking Control of Uncertain Fractional-Order Nonlinear Systems Based on Fuzzy Logic Systems

Abstract

This study introduces an enhanced Adaptive Fuzzy Sliding-Mode Control (AFSMC) approach based on the fuzzy logic systems (FLSs) to achieve trajectory tracking of multiple-input and multiple-output (MIMO) fractional-order nonlinear systems in the presence of uncertain nonlinear terms and disturbances. An integral SMC approach is proposed for achieving state trajectory tracking control. However, uncertainties in real systems are complex and diverse, not only uncertain bounded disturbances but unknown nonlinear functions. Therefore, in this paper, the FLSs are used not only to approximate unknown functions but also to improve the switching function of the SMC. The stability of the system with designed input control laws is demonstrated through the fractional-order Lyapunov function stability criterion. Subsequently, the simulation results are displayed and serve to validate the efficacy and resilience of the proposed control methodology. These results underscore the ability of the proposed method to perform reliably under various conditions, thereby confirming its robustness as a viable solution.

1. Introduction

Fractional calculus is a mathematical theory that describes the characteristics and applications of fractional-order calculus operators and expands the concept of traditional calculus. The use of fractional operators in control systems has increased due to the quick development of fractional-order theory [1,2,3,4], for example, fractional-order CHEN, whose chaotic and bifurcation characteristics have been widely studied [5,6,7,8]. Fractional-order systems have become increasingly popular in engineering, particularly in the last few years. A novel fractional lumped capacitance model for transient heat conduction was proposed in [9]. Huang and Chen confirmed the widespread existence of fractional order in physics by studying the fractional-order characteristics of the interelectrode capacitance of MOSFETs [10]. It is interesting that researchers have discovered that fractional calculus can better capture the essence of objects and systems than integer calculus. A fractional-order electromagnetic model of PMSM and its identification method were proposed in [11]. Experimental results showed that the fractional-order electromagnetic model had higher accuracy than the integer-order PMSM model.

Traditional control techniques are unable to satisfy the control requirements of complex fractional nonlinear systems in engineering and social production applications. As demonstrated in [12], the UAV model displayed a complex state equation that imposed very high requirements on the robustness and real-time performance of the control algorithm because of the underactuated structure, nonlinear SE(3) dynamics, strong coupling of control allocation, and sensor noise in the real system. The authors of [13] presented a fractional-order strict feedback nonlinear system with fuzzy dead zones. Researchers have proposed a number of fractional intelligent-control methods, including the typical SMC, a type of control method for nonlinear systems, to solve this type of nonlinear system with nonlinear factors and uncertainty factors. Compared with the usual backstepping method, the SMC is more widely used in the lower triangle nonlinear system, and it can solve the exponential explosion in the backstepping control. The traditional SMC is a relatively mature development and is applied in various control technology schemes, such as synchronization control and tracking control. The initial applications of the SMC in [14,15,16,17,18,19,20] showed a strong robustness to interferences and system uncertainties. Aghababa found a fractional-order chaotic system based on the PMSM and proposed the PI-SMC method applied in [14], and the method was improved by Prieto and his partners in [15]. A fractional-order exponential switching function was proposed to enhance the dynamic behavior of the SMC in [17]. The SMC was applied in a single flexible link manipulator and had an advantage compared with the classical PD controller in [20].

Uncertain disturbances or unknown parameters are typically designed as bounded values when creating an adaptive sliding-mode controller. An adaptive control law is then created for the bounded values. However, there are a variety of disturbances and uncertainty factors in the systems application process. Especially, uncertain nonlinear function terms are present in the system as a result of uncertain disturbances that are frequently present in nonlinear functions due to state measurement errors or device losses. In order to solve this problem, the FlSs are introduced to deal with unknown functions. In [21], an AFSMC was designed to control MIMO uncertain subsystem models linearized by the full state input–output feedback linearization control, and the controller can effectively suppress the uncertain bounded external interference. Back in 2013 [22], an integer-order AFSMC approach was proposed that combined the excellencies of the SMC and FLSs, and adaptive fuzzy control was used to calculate the switching function of PD-SMC. As FLSs have had many applications in integer-order systems, we investigate their suitability for fractional-order nonlinear systems. The AFSMC has proved that it can not only handle single or multiple uncertainties but also can be used in the case of fractional or integer-order systems, e.g., in [23]. Sliding-mode control, fuzzy control, network systems, and even infinity control were all combined by the author and compared to sliding-mode control [24]. In [25], the problem of observer-based SMC for fractional-order singular fuzzy systems was studied, and a new fractional-order integral sliding function was constructed.

Thus far, the research of the SMC has integrated fractional calculus, FLS, and adaptive control to deal with uncertainties. Additionally, researchers have made significant progress in choosing sliding surfaces. However, the fundamental problem of the SMC has not been solved. The presence of switching mechanisms will result in chattering. Typical switching strategies include power switching functions that contain both sign functions and sliding-mode surface variables, exponential switching functions that contain both sign functions and sliding-mode surface variables, and sign switching functions that only contain sign functions. These methods have been proved to be effective for system control, but there is chattering. Many studies on how to advance switching technology have been carried out in recent years. In [26], the authors applied the exponential switching function to the fractional-order system and constructed the fractional-order SMC. In fractional-order systems, it was merely a direct application of the exponential switching function. Fractional calculus had a greater impact than integer order because of its memory properties. In [27], the sliding surface in integer-order systems was determined by the system variables. However, the switching mechanism was incorporated with a complex function that combined the sign function and the fractional-order algorithm. The complex switching functions were used to improve the chattering of the SMC [28,29,30]. Even though they could successfully reduce chattering, the controller structure was complicated and more computation was required. In [31], the SMC switching mechanism was modified to reduce chattering by applying the rectifier principle in power systems. In [32], the TS-SMC was used to control single-input and single-output fractional-order systems with lower triangular structure. The FLSs have reached a relatively mature stage of development, their calculation complexity has decreased, and they can be effectively implemented in the control system. It is necessary to enhance the modeling and analysis of uncertainty in fractional-order systems.

In this article, a better performing fractional-order integral AFSMC is proposed to achieve the state trajectory tracking. Additionally, the uncertain nonlinear functions in the system are estimated using FLSs. A review of the SMC literature makes it clear that optimizing SMC requires improving the switching function. Thus, an FLS-based improved switching function is presented. To validate the stability and robustness of the designed controller, disturbances are intentionally added to the simulation process. The primary contribution of this article can be summarized as follows:

• The dynamics of the fractional-order nonlinear PMSM system, which serves as a representative example of fractional order MIMO nonlinear dynamic systems, are demonstrated. Referring to [14,33], the simulation experiments of special PMSM and PMSM with actual parameters are carried out, respectively. FLSs and adaptive control are used effectively to manage unknown nonlinear functions and uncertain disturbances in dynamic systems.
• An enhanced integral AFSMC is created, and its capacity to follow the state trajectory with minimum errors is demonstrated mathematically. Simulation experiments further prove the superiority of the proposed method.
• Compared with the typical switching function, the fuzzy switching function in AFSMC can effectively weaken the chattering and reduce the tracking error. Compared with [32], the system model of this paper is more universal.

2. Theories

Definition 1.

The α-order Caputo fractional derivative of the function is defined as the following form:

$$\begin{array}{c}\hfill {}_{t_0}{}^{C}D_t^{\alpha }f(t)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\mathsf{\Gamma }(m-\alpha )}}{\int }_{t_0}^t{\displaystyle \frac{f^m(\tau )}{{(t-\tau )}^{\alpha -m+1}}}d\tau ,m-1<\alpha <m\hfill \\ {\displaystyle \frac{d^{m}f(t)}{d{t}^m}},\alpha =m\hfill \end{array}\right.\end{array}$$

(1)

m is a positive integer number and represents the integer part of α. Γ is the Gamma function, which satisfies $\mathsf{\Gamma }(1)=1$.

If m is equal to $\alpha$ in the above definition, fractional-order algorithms will be equivalent to integer-order algorithms.

Fractional-order calculus is homogeneous; for any constant, there exists

$$\begin{array}{c}\hfill {}_{t_0}{}^{C}D_t^{\alpha }[af(t)+bg(t)]=a{}_{t_0}{}^{C}D_t^{\alpha }f(t)+b{}_{t_0}{}^{C}D_t^{\alpha }g(t)\end{array}$$

(2)

The relationship between fractional-integral and fractional-differential is

$$\begin{array}{c}\hfill {}_{t_0}{}^{C}D_t^{\alpha }{}_{t_0}{}^{C}I_t^{\alpha }f(t)=f(t)\end{array}$$

(3)

Definition 2.

Global Stability: If there exists a positive defined, continuously differentiable, radially unbounded function $V(x)$ defined in $R^n$, such that $\dot{V}(x)$ is negative definite in $R^n$, then the origin is globally asymptotically stable.

The integer-order differential operation can be regarded as a special case of the fractional-order operation; thus, because of the function $\dot{V}=D_t^{1-\alpha }{D}_t^{\alpha }V$, if there exists a positive define function $V(x)$ and a negative define function ${}_0{}^{C}D_t^{\alpha }V(x)$, the system will be proved to be stable.

Definition 3.

Sliding-mode control: The primary concept of the SMC involves designing the switching sliding-mode surface of the system in accordance with the anticipated dynamic characteristics of the system. The system state is then closed from outside the hyperplane to the switching sliding-mode surface through the use of a sliding-mode controller. Once the system state reaches this surface, the controller ensures that the system reaches its origin along this path. This process offers strong robustness as it depends solely on the characteristics and parameters of the designed switching sliding-mode surface and is unaffected by external interference. To ensure that the system state can be reached and maintained on this surface, it must satisfy a specific formula: $\underset{s\to 0}{lim}s\dot{s}<0$.

Definition 4.

Fuzzy logic systems:The FLSs consist of three main components: the fuzzy rule base, fuzzification, and defuzzification operators. The fuzzy rule base contains inference rules for multiple-input and single-output systems:

$R^l$: If $x_1$ is $A_1^l$ and $x_2$ is $A_2^l$ ⋯ and $x_n$ is $A_n^l$, then y is $B^l$, $l=1,2,\dots ,N.$

By utilizing singleton function, center average defuzzification, and product inference, FLS can be effectively expressed as

$$\begin{array}{c}\hfill \widehat{f}(x)={\displaystyle \frac{{\displaystyle \sum _{l=1}^N}{\overline{y}}_j(t){\displaystyle \prod _{i=1}^n}\mu A_i^l(x_i)}{{\displaystyle \sum _{l=1}^N}({\displaystyle \prod _{i=1}^n}\mu A_i^l(x_i))}}\end{array}$$

(4)

Define the fuzzy basis functions as

$$\begin{array}{c}\hfill {\xi }_l(x)={\displaystyle \frac{{\displaystyle \prod _{i=1}^n}\mu A_i^l(x_i)}{{\displaystyle \sum _{l=1}^N}({\displaystyle \prod _{i=1}^n}\mu A_i^l(x_i))}}\end{array}$$

(5)

where the function $\mu A_i^l(x_i)$ is the membership function. Denote ${\theta }^T=[{\overline{y}}_1,{\overline{y}}_2,\dots ,{\overline{y}}_N]=[{\theta }_1,{\theta }_2,\dots ,{\theta }_N]$ and $\xi (x)=[{\xi }_1(x),{\xi }_2(x),\dots ,{\xi }_N(x)]$, then the fuzzy logic system (4) can be rewritten as

$$\begin{array}{c}\hfill \widehat{f}(x)={\theta }^T\xi (x)\end{array}$$

(6)

Let $f(x)$ be a continuous function defined on a compact set $\mathsf{\Omega }$, then for any constant $\epsilon >0$, there exists an optimal parameter as

$$\begin{array}{c}\hfill {\theta }^{*}=arg\underset{\theta \in \mathsf{\Omega }}{min}[sup|\widehat{f}(x|\theta )-f(x,t)]\end{array}$$

(7)

Thus, the minimum approximation error can be obtained as

$$\begin{array}{c}\hfill \epsilon =f(x,t)-\widehat{f}(x|{\theta }^{*})\end{array}$$

(8)

3. The Design of the Adaptive Fuzzy Sliding-Mode Controller

In this section, a sliding-mode controller will be designed to improve the robustness of MIMO nonlinear fractional-order systems, which contain uncertain disturbances and unknown nonlinear functions. The key to designing this controller is to discover the input control laws $u_i(t),i=1,2,\dots ,n$ and the effective sliding-mode surfaces $s_i(t),i=1,2,\dots ,n$. The system can be written as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}_0{}^{C}D_t^{\alpha }{x}_1=f_1(x)+g_1(x)+d_1(t)+u_1(t)\hfill \\ {}_0{}^{C}D_t^{\alpha }{x}_2=f_2(x)+g_2(x)+d_2(t)+u_2(t)\hfill \\ \begin{array}{cc}& ⋮\end{array}\begin{array}{cccc}& & \begin{array}{ccc}& & \end{array}& ⋮\end{array}\hfill \\ {}_0{}^{C}D_t^{\alpha }{x}_n=f_n(x)+g_n(x)+d_n(t)+u_n(t)\hfill \end{array}\right.\end{array}$$

(9)

where $d_i(t),i=1,2,\dots \dots ,n$ are bounded uncertain disturbances, and $∥d_i(t)∥\le d_i,i=1,2,\cdots \cdots ,$ where $d_i>0.$ $x^T=[x_1,x_2,\cdots ,x_n]\in R^n$ are vector variables of the system. $f_i(x),i=1,2,\dots ,n$ denote known terms of the system, and $g_i(x),i=1,2,\dots ,n$ are the unknown internal structure functions of the system, which are difficult to accurately obtain.

The following flowchart in Figure 1 effectively illustrates the intricate design and implementation of AFMSC. By leveraging the inherent capabilities of the integral SMC, significant enhancements are achieved in improving the switching function, thereby optimizing overall system performance. In scenarios characterized by uncertain interference and unknown nonlinear factors, it is strongly necessary to employ adaptive control strategies and leverage FLS for designing and implementing an equivalent control. These approaches enable robustness against unpredictable disturbances while ensuring precise regulation in dynamic environments. Furthermore, they facilitate adaptation to varying operating conditions, ultimately enhancing system stability and reliability.

Supposing that the reference state signals are continuously differentiable functions, the tracking errors are defined as

$$\begin{array}{c}\hfill e_i=x_i-x_{id}\end{array}$$

(10)

The designed integral sliding-mode surfaces with $e_i$ are defined as follows:

$$\begin{array}{c}\hfill s_i(t)=e_i(t)+m_i{}^{C}I_t^{\alpha }{e}_i(t)-h_i(t),i=1,2,\dots ,n\end{array}$$

(11)

where $m_i>0,i=1,2,\dots ,n$, ${}^{C}I_t^{\alpha }$ is the fractional-integral operator. To make sure the sliding-mode surfaces pass through the coordinate origin, the functions $h_i(t)$ satisfy

• $h_i(0)=e_i(0)+m_i{}^{C}I_t^{\alpha }{e}_i(0)$;
• $h_i(t)\to 0$ as $t\to \infty$;
• $h_i(t)$ have a continuous derivative.

In this paper, let $h_i(t)=h_i(0)e^{-pt},i=1,2,\dots ,n$, where $p>0$. The positive constants $m_i,i=1,2,\dots ,n$ determine the decay rate and velocity reaching the sliding-mode surfaces.

Unlike traditional SMC techniques that use discontinuous switching functions with fixed-form switching functions, like exponential or sign switching functions, this study suggests a novel approach in which FLSs are methodically used to produce adaptive switching signals. The proposed fuzzy switching functions effectively emulate sliding-mode dynamics through real-time fuzzy reasoning algorithms that process multiple-input variables, including sliding surface variables and their derivatives. Specifically, the FLSs establish a nonlinear mapping between the system state errors and the required control compensation through carefully designed membership functions and rule bases derived from Lyapunov stability criteria. The fuzzy switching functions, which are also the switching input control laws, are

$$\begin{array}{c}\hfill {}^{C}D_t^{\alpha }{s}_i=u_{isw}={\widehat{u}}_{isw}(s_i|{\theta }_{iu})\end{array}$$

(12)

As the sliding-mode surface and its differential are approaching zero, a global sliding-mode surface function can be obtained:

$$\begin{array}{c}\hfill {}^{C}D_t^{\alpha }{s}_i={}^{C}D_t^{\alpha }{e}_i+m_i{e}_i-{}^{C}D_t^{\alpha }{h}_i(t)=0\end{array}$$

(13)

Let ${}^{C}D_t^{\alpha }{s}_i=0$, then one can obtain the following functions:

$$\begin{array}{c}\hfill {}^{C}D_t^{\alpha }{e}_i=-m_i{e}_i+{}^{C}D_t^{\alpha }{h}_i(t)\end{array}$$

(14)

Firstly, the system designed must be stable and satisfy the definition of global stability. Let the first Lyapunov function be

$$\begin{array}{c}\hfill V_1(t)=\sum _{i=1}^n{\displaystyle \frac{1}{2}}{e_i}^2\end{array}$$

(15)

From (14), the $\alpha$–th order derivative of $V_1(t)$ is expressed as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^{C}D_t^{\alpha }{V}_1(t)& =\sum _{i=1}^n{e}_i{D}_t^{\alpha }{e}_i\hfill \\ \hfill & =\sum _{i=1}^n{e}_i(-m_i{e}_i+{}^{C}D_t^{\alpha }{h}_i(t))\hfill \\ \hfill & =-\sum _{i=1}^n{m}_i{e_i}^2+\sum _{i=1}^n{e}_i{h}_i(0){}^{\alpha }D_t^C{e}^{-pt}\hfill \end{array}\end{array}$$

(16)

It is obvious that if $t\to \infty$, $e^{-pt}=0$ and ${}^{\alpha }D_t^C{e}^{-pt}=0$; thus, the final formulation can be obtained

$$\begin{array}{c}\hfill {}^{C}D_t^{\alpha }{V}_1(t)\le -\sum _{i=1}^n{m}_i{e_i}^2\le -2m{V}_1(t)\end{array}$$

(17)

where $m=min\left\{m_1,m_2,\cdots ,m_n\right\}$. Based on the fractional Lyapunov stability theorem, the tracking errors of the dynamical system (9) are driven to the origin asymptotically.

Based on AFSMC in this paper, the designed input control laws can be obtained by the following:

$$\begin{array}{c}\begin{array}{c}{u}_i=u_{ieq}-u_{isw};\hfill \\ u_{ieq}=-f_i(x)+{}^{C}D_t^{\alpha }{x}_{id}-m_i{e}_i-{\widehat{g}}_i(x|{\theta }_{ig})+{}^{C}D_t^{\alpha }{h}_i(t)-{\widehat{\delta }}_i;\hfill \\ u_{isw}={\widehat{u}}_{isw}(s_i|{\theta }_{iu}).\hfill \end{array}\end{array}$$

(18)

where ${\widehat{\delta }}_i$ is the excepted value of ${\delta }_i$, which will be mentioned in the following analysis; ${\widehat{g}}_i(x|{\theta }_{ig})={\theta }_{ig}^T\xi (x)$ are the fuzzy of unknown terms $g_i(x,t)$; and ${\widehat{u}}_{isw}(s_i|{\theta }_{iu})={\theta }_{iu}^T\varphi (s_i)$ are the fuzzy of switching functions.

Adaptive control laws are

$$\begin{array}{c}\hfill {}^{C}D_t^{\alpha }{\theta }_{ig}=r_i{s}_i\xi (x)\end{array}$$

(19)

$$\begin{array}{c}\hfill {}^{C}D_t^{\alpha }{\theta }_{iu}=l_i{s}_i\varphi (s_i)\end{array}$$

(20)

$$\begin{array}{c}\hfill {}^{C}D_t^{\alpha }{\widehat{\delta }}_i={\gamma }_i{s}_i\end{array}$$

(21)

According to Definition 4, suppose that the optimal parameters of fuzzy approximation are as follows:

$$\begin{array}{c}\hfill {\theta }_{ig}^{*}=arg\underset{{\theta }_{ig}\in {\mathsf{\Omega }}_{ig}}{min}[sup|{\widehat{g}}_i(x|{\theta }_{ig})-g_i(x)|]\end{array}$$

(22)

$$\begin{array}{c}\hfill {\theta }_{iu}^{*}=arg\underset{{\theta }_{iu}\in {\mathsf{\Omega }}_{isw}}{min}[sup|{\widehat{u}}_{isw}(S_i|{\theta }_{iu})-u_{isw}|]\end{array}$$

(23)

The expected switching functions are sign functions:

$$\begin{array}{c}\hfill {\widehat{u}}_{isw}(s_i|{\theta }_{iu}^{*})={\eta }_i\mathrm{sgn}(s_i)\end{array}$$

(24)

To improve the control performance of FLSs, the minimum approximation error is as follows:

$$\begin{array}{c}\hfill w_i=g_i(x)-{\widehat{g}}_i(x|{\theta }_{ig}^{*})\end{array}$$

(25)

Designing that ${\tilde{\theta }}_{ig}={\theta }_{ig}-{\theta }_{ig}^{*},{\tilde{\theta }}_{iu}={\theta }_{iu}-{\theta }_{iu}^{*}$ and ${\delta }_i=w_i+d_i$, one can obtain the differential forms as

$$\begin{array}{c}\hfill \begin{array}{c}{}^{C}D_t^{\alpha }{\tilde{\theta }}_{ig}={}^{C}D_t^{\alpha }{\theta }_{ig}\hfill \\ {}^{C}D_t^{\alpha }{\tilde{\theta }}_{iu}={}^{C}D_t^{\alpha }{\theta }_{iu}\hfill \\ {}^{C}D_t^{\alpha }{\tilde{\delta }}_i={}^{C}D_t^{\alpha }{\widehat{\delta }}_i\hfill \end{array}\end{array}$$

(26)

Choose the second Lyapunov function:

$$\begin{array}{c}\hfill V_2(t)={\displaystyle \frac{1}{2}}\sum _{i=1}^n(s_i^2(t)+{\displaystyle \frac{1}{r_i}}{{\tilde{\theta }}_{ig}}^2+{\displaystyle \frac{1}{l_i}}{{\tilde{\theta }}_{iu}}^2+{\displaystyle \frac{1}{{\gamma }_i}}{{\tilde{\delta }}_i}^2)\end{array}$$

(27)

where ${\tilde{\delta }}_i={\delta }_i-{\widehat{\delta }}_i$. Substituting the functions (18), (22), (23) and (25) into the formula, the fractional differential form is deformed as the following formula:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill c\sum _{i=1}^n{}^{C}D_t^{\alpha }{s}_i(t)& =\sum _{i=1}^n{}^{C}D_t^{\alpha }{e}_i+m_i{e}_i-{}^{C}D_t^{\alpha }{h}_i(t)\hfill \\ \hfill & =\sum _{i=1}^n{f}_i(x)+g_i(x)+d_i(t)+u_i(t)-{}^{C}D_t^{\alpha }{x}_{id}+m_i{e}_i-{}^{C}D_t^{\alpha }{h}_i(t)\hfill \\ \hfill & =\sum _{i=1}^n{f}_i(x)+g_i(x)+d_i(t)+(-f_i(x,t)-{\widehat{g}}_i(x|{\theta }_{ig})+{}^{C}D_t^{\alpha }{x}_{id}-m_i{e}_i\hfill \\ \hfill & -{\widehat{u}}_{isw}(s_i|{\theta }_{iu})+{}^{C}D_t^{\alpha }{h}_i(t)-{\widehat{\delta }}_i)-{}^{C}D_t^{\alpha }{x}_{id}+m_i{e}_i-{}^{C}D_t^{\alpha }{h}_i(t)\hfill \\ \hfill & =\sum _{i=1}^n{g}_i(x)-{\widehat{g}}_i(x|{\theta }_{ig})-{\widehat{u}}_{isw}(s_i|{\theta }_{iu})+d_i(t)-{\widehat{\delta }}_i\hfill \\ \hfill & =\sum _{i=1}^n{w}_i+{\widehat{g}}_i(x|{\theta }_{ig}^{*})-{\widehat{g}}_i(x|{\theta }_{ig})-{\widehat{u}}_{isw}(s_i|{\theta }_{iu})+d_i(t)-{\widehat{\delta }}_i\hfill \\ \hfill & \le \sum _{i=1}^n{\tilde{\delta }}_i+{\widehat{g}}_i(x|{\theta }_{ig}^{*})-{\widehat{g}}_i(x|{\theta }_{ig})-{\widehat{u}}_{isw}(S_i|{\theta }_{iu})\hfill \end{array}\end{array}$$

(28)

From Formula (28), the fractional differential form of the Formula (27) is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^{C}D_t^{\alpha }{V}_2(t)& \le \sum _{i=1}^n(s_i(t){}^{C}D_t^{\alpha }{s}_i(t)+{\displaystyle \frac{1}{r_i}}{\tilde{\theta }}_{ig}^T{}^{C}D_t^{\alpha }{\tilde{\theta }}_{ig}+{\displaystyle \frac{1}{l_i}}{\tilde{\varphi }}_{iu}^T{}^{C}D_t^{\alpha }{\tilde{\varphi }}_{iu}+{\displaystyle \frac{1}{{\gamma }_i}}{\tilde{\delta }}_i{}^{C}D_t^{\alpha }{\tilde{\delta }}_i)\hfill \\ \hfill & \le \sum _{i=1}^n(s_i(t)({\tilde{\delta }}_i+{\widehat{g}}_i(x|{\theta }_{ig}^{*})-{\widehat{g}}_i(x|{\theta }_{ig})-{\widehat{u}}_{isw}(s_i|{\theta }_{iu}))+{\displaystyle \frac{1}{r_i}}{\tilde{\theta }}_{ig}^T{}^{C}D_t^{\alpha }{\theta }_{ig}\hfill \\ \hfill & +{\displaystyle \frac{1}{l_i}}{\tilde{\theta }}_{iu}^T{}^{C}D_t^{\alpha }{\theta }_{iu}-{\displaystyle \frac{1}{{\gamma }_i}}{\tilde{\delta }}_i{}^{C}D_t^{\alpha }{\widehat{\delta }}_i)\hfill \\ \hfill & \le \sum _{i=1}^n(s_i({\tilde{\delta }}_i-{\tilde{\theta }}_{ig}^T\xi (x)-{\tilde{\theta }}_{iu}^T\varphi (s_i)-{\widehat{u}}_{isw}(s_i|{\theta }_{iu}^{*}))+{\displaystyle \frac{1}{r_i}}{\tilde{\theta }}_{ig}^T{}^{C}D_t^{\alpha }{\theta }_{ig}\hfill \\ & +{\displaystyle \frac{1}{l_i}}{\tilde{\theta }}_{iu}^T{}^{C}D_t^{\alpha }{\theta }_{iu}-{\displaystyle \frac{1}{{\gamma }_i}}{\tilde{\delta }}_i{}^{C}D_t^{\alpha }{\widehat{\delta }}_i)\hfill \\ \hfill & \le \sum _{i=1}^n(s_i{\tilde{\delta }}_i-s_i{\tilde{\theta }}_{ig}^T\xi (x)-s_i{\tilde{\theta }}_{iu}^T\varphi (s_i)+{\displaystyle \frac{1}{r_i}}{{\tilde{\theta }}_{ig}}^T{}^{C}D_t^{\alpha }{\theta }_{ig}+{\displaystyle \frac{1}{l_i}}{\tilde{\theta }}_{iu}^T{}^{C}D_t^{\alpha }{\theta }_{iu}\hfill \\ \hfill & -s_i{\widehat{u}}_{isw}(s_i|{\theta }_{iu}^{*})-{\displaystyle \frac{1}{{\gamma }_i}}{\tilde{\delta }}_i{}^{C}D_t^{\alpha }{\widehat{\delta }}_i)\hfill \\ \hfill & \le \sum _{i=1}^n{\tilde{\theta }}_{ig}^T(-s_i\xi (x)+{\displaystyle \frac{1}{r_i}}{}^{C}D_t^{\alpha }{\theta }_{ig})+{\tilde{\theta }}_{iu}^T(-s_i\varphi (s_i)+{\displaystyle \frac{1}{l_i}}{}^{C}D_t^{\alpha }{\theta }_{iu})\hfill \\ \hfill & -s_i{\widehat{u}}_{isw}(s_i|{\theta }_{iu}^{*})+{\tilde{\delta }}_i(s_i-{\displaystyle \frac{1}{{\gamma }_i}}{}^{C}D_t^{\alpha }{\widehat{\delta }}_i)\hfill \end{array}\end{array}$$

(29)

According to the adaptive control laws (19)–(21), the fractional differential form of the Formula (29) can be approximated as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^{C}D_t^{\alpha }{V}_2(t)& \le \sum _{i=1}^n-s_i{\widehat{u}}_{isw}(s_i|{\theta }_{iu}^{*})\hfill \\ \hfill & \le \sum _{i=1}^n-{\eta }_i|s_i|\hfill \end{array}\end{array}$$

(30)

To make ${}^{C}D_t^{\alpha }{V}_2(t)<0$, the parameters ${\eta }_i$ must satisfy that ${\eta }_i>0$. From (27) and (30), the AFSMC satisfies the Lyapunov theory, and to verify the correctness and feasibility of the designed method, simulation experiments and the results are given in the next chapter. From the mathematical analysis, the improved AFSMC works well in solving the complex challenges associated with control systems affected by various external factors.

4. Simulation and Results

In this section, an example is presented to show the effectiveness of the proposed method using the algorithmic solution of Caputo fractional-order differential equations. PMSM plays an important part in the electric system, and it is of urgency for mechanical and electrical engineers to research and manufacture the PMSM servosystems of high performance.

Because of the stator voltages and the mechanical angular velocity, the three-dimensional fractional-order state function of the PMSM is given as follows:

$$\begin{array}{c}\left\{\begin{array}{c}{}_C{}^{0}D_t^{\alpha }{i}_d=-{\displaystyle \frac{R_s}{L_d}}{i}_d+{\displaystyle \frac{L_q}{L_d}}{\omega }_e{i}_q+{\displaystyle \frac{1}{L_d}}{u}_d\hfill \\ {}_C{}^{0}D_t^{\alpha }{i}_q=-{\displaystyle \frac{R_s}{L_q}}{i}_q-{\displaystyle \frac{L_d}{L_q}}{\omega }_e{i}_d+{\displaystyle \frac{{\psi }_f}{L_q}}{\omega }_e+{\displaystyle \frac{1}{L_q}}{u}_q\hfill \\ {}_C{}^{0}D_t^{\alpha }{\omega }_e={\displaystyle \frac{n_p}{J}}\left[{\displaystyle \frac{3}{2}}{n}_p\left({\psi }_f{i}_q+\left(L_d-L_q\right)i_d{i}_q\right)\right]-{\displaystyle \frac{n_p}{J}}T-{\displaystyle \frac{B}{J}}{\omega }_e\hfill \end{array}\right.\end{array}$$

(31)

where $i_d$, $i_q$, and ${\omega }_e$ are the state variables of the system, representing the stator current and rotor angular velocity of the d and q axes, respectively; $u_d$, $u_q$, $L_d$, and $L_q$ represent the d-axis voltage, q-axis voltage, and stator inductance, respectively; ${\psi }_f$ and $T_l$ are permanent magnetic flux and external mechanical torque, respectively; $n_p$, $R_s$, J, and B are the number of magnetic pole pairs, stator resistance, rotational inertia, and viscous damping coefficient, respectively.

Considering [14], the parameters of this system can be designed as $B=7.2$, ${\psi }_f=2.667$, $J=4$, $u_d=-12.7$, $n_p=4$, $u_q=2.34$, $R_s=2.857$, $L_d=6.45$, $L_q=7.125$, and $T=0.525$. Let the initial values of the variables be 0.1, and the reference signals be $x_{1d}=x_{2d}=x_{3d}=sin(t)$. At this time, the MIMO system is constructed by the state equation structure of PMSM, and the model does not represent the actual model of PMSM.

Incorporate the above parameters into (31), where the fractional-order operator $\alpha$ is chosen as 0.95, and the external disturbances are selected as $d_1=sin(t)$, $d_2=2$, and $d_3=cos(t)$ after ten seconds of the control process. Especially in the first ten seconds, the disturbances are zero. In this simulation model, there are three state variables, and the fuzzy of the nonlinear functions has three inputs and a single output. Let $r_1=r_2=r_3=1$, $l_1=500$, $l_2=600$, $l_3=800$, $r_1=2$, $r_2=6$, and $r_3=4$. Taking the first nonlinear function $g_1(x)$ as an example, the FLSs of $g_1(x)$ have five rules for each state variable, which can be described as $A_1^l=exp(-{((x+1.2-0.6(l-1))/0.3)}^2),l=1\cdots 5$ and 125 fuzzy basis functions. The five rules of FLSs of $s_1$ are $A_1^l=exp(-{((x+0.5-0.25(l-1))/0.125)}^2),l=1\cdots 5$.

The sliding-mode surfaces are as follows:

$$\begin{array}{c}\hfill s_i(t)=e_i(t)+10I_t^{0.95}{e}_i(t)-(e_i(0)+10I_t^{0.95}{e}_i(0))e^{-4t},i=1,2,3\end{array}$$

(32)

Design the fuzzy adaptive laws as follows:

$$\begin{array}{c}\hfill {}^{C}D_t^{0.95}{\theta }_{ig}=s_i\xi (x),i=1,2,3\end{array}$$

(33)

$$\begin{array}{c}\hfill {}^{C}D_t^{0.95}{\theta }_{1u}=500s_1\varphi (s_1)\end{array}$$

(34)

$$\begin{array}{c}\hfill {}^{C}D_t^{0.95}{\theta }_{2u}=600s_2\varphi (s_2)\end{array}$$

(35)

$$\begin{array}{c}\hfill {}^{C}D_t^{0.95}{\theta }_{3u}=800s_3\varphi (s_3)\end{array}$$

(36)

$$\begin{array}{c}\hfill {}^{C}D_t^{0.95}{\widehat{\delta }}_1=2s_1\end{array}$$

(37)

$$\begin{array}{c}\hfill {}^{C}D_t^{0.95}{\widehat{\delta }}_2=6s_2\end{array}$$

(38)

$$\begin{array}{c}\hfill {}^{C}D_t^{0.95}{\widehat{\delta }}_3=2s_3\end{array}$$

(39)

4.1. Normal External Disturbances

Under the assumption of the aforementioned parameters and external disturbances, there are no other external disturbances.

The fractional-order PMSM is obviously a nonlinear chaotic system. In Figure 2, the trajectory in the diagram is densely entangled in the three-dimensional space ($x1,x2,x3$) and staggered without repeated routes. There is no periodic ring structure, the state diffuses in the phase space after adding external interference after 10 s, and the position of the limit cycle is shifted. It can be seen that the system is sensitive to disturbance, which will cause the trajectory to deviate from the original path quickly during operation. Therefore, Figure 2 depicts the phase trajectory of the nonlinear power of the system at the tenth second when a disturbance occurs, with an initial state of zero. The results demonstrate that the fractional-order PMSM exhibits aperiodic, irregular, and nonlinear oscillations, leading to an instability that significantly impacts unit operation safety and stability. The three-dimensional phase trajectory coordinates exhibit typical chaotic characteristics. Figure 3 displays the fuzzy of uncertain nonlinear functions $g_i(x)$. Figure 4 shows that the sliding-mode surfaces $s_i$ of each state function approach zero. Control inputs based on SMC with different switching functions are shown in Figure 5, from which we can discover that control inputs with sign switching function have dramatic positive and negative changes. At the same time, control inputs with exponential switching function have smoother changes after adding disturbances. There is a smooth and continuous control curve even after the addition of perturbations based on the fuzzy switching function in this paper. The control inputs in this paper are the mildest that best perform the tracking control and have minimum damage to the device.

As shown in Figure 6d, all three states can follow the desired trajectory $sin(t)$, while the tracking effects of the three methods are different in detail from the tracking error analysis of the three states in Figure 6a–c. In one second, all three ways can achieve good tracking effect, and the adjust time of the above three methods is less than 0.2 s, which is in a normal range. Although each error with fuzzy switching functions in this paper has a large adjust time compared with the other two, the error with fuzzy switching function has the minimum value, and after adding the perturbation, the error does not change. However, after adding the disturbances, errors with exponential switching function have a great influence, and the jitter is increased slightly. Generally speaking, the SMC in this paper gives the best tracking results.

4.2. The High-Frequency External Disturbances Within 1

In this section, disturbances are added to the state functions, which have a high frequency but an amplitude of less than 1. Specifically, add $cos(10t)$ to $d_1(t)$, add a random number with frequency 0.1 to $d_2(x,t)$, and add $sin(10t)$ to $d_3(x,t)$.

The simulation results are basically consistent with Section 4.1. From Figure 7d, all three states can follow the desired trajectory $sin(t)$. Firstly, compared with Figure 7a–c, the chattering phenomenon is the most obvious in Figure 7b, especially the error based on the SMC with exponential switching function, which starts to display obvious chattering. But the error based on the SMC with fuzzy switching function always displays almost no chattering. Secondly, when we compare Figure 7 to Figure 6, the value of errors is almost uniform, while the chattering becomes worse, except for the error based on the SMC with fuzzy switching function. The comparative analysis of tracking performance under small-amplitude reference signals reveals that both the conventional SMC employing exponential switching function and the proposed fuzzy-adaptive SMC satisfy the prescribed accuracy thresholds, with errors maintained below 0.005. However, the experimental validation demonstrates that the fuzzy switching function exhibits superior chattering suppression characteristics.

Compared with Figure 6d, Figure 7d shows different adjust time. From

Table 1. The tracking errors of $x_i-x_{id}$ based on SMC with different switching functions.

The Switching Function of the SMC	Errors in 10 s (× 10−3)	Errors after 10 s (× 10−3)	Adjust Time in 10 s (s)
sign switching function	1.5	2	0.015
	2	3	0.03
	2.5	3	0.03
exponential switching function	1.2	5	0.016
	1	0.8	0.04
	2	3	0.06
fuzzy switching function	0.08	0.1	0.15
	0.05	0.05	0.3
	0.02	0.03	0.25

, we can obtain state tracking errors when the system reaches steady state. The system errors are less than 0.5%, which satisfies the tracking accuracy, as the amplitude of the desired state trajectory is 1. It is intuitive to see that the tracking accuracy of the SMC with the fuzzy switching function is the highest. However, the adjustment time is also the longest. In order to discuss the reason for this result, the attenuation oscillation curve $sin(t)/t$ is selected as the input to simulate the function of each switching function in the SMC. As shown in Figure 8, as the input decreases, the amplitude of each switching function decreases. The amplitude of the fuzzy switching function decreases significantly with the decrease in the input, and the switching is smoothest when the positive and negative signs of the input change. Therefore, chattering can be well weakened in the SMC with fuzzy switching function. Similarly, the amplitude of each switching function is set to be the same at $t_1$, and then the amplitude of the fuzzy switching function is the largest at $t_0$, and $t_0<t_1$. This is the reason why the SMC with fuzzy switching function has a large adjust time and overshoot.

4.3. The High-Frequency and High-Amplitude External Disturbances

The additional external disturbances, which have high frequency and high amplitude, are supplemented in this section. In particular, we add $10sin(10t)$ to $d_3(x,t)$, $10cos(10t)$ to $d_1(x,t)$, and a random number with frequency 0.1 and amplitude 10 to $d_2(x,t)$.

By analyzing Figure 9, after adding the internal disturbance at 10s, the state based on the SMC with sign switching function can no longer follow the predetermined trajectory. Then, according to Figure 9a–c, although the method with exponential switching function can follow the predetermined trajectory, its following error and chattering are more serious than that of the method in this paper, especially in the case of adding random disturbances.

Finally, by comparison, the methods with exponential switching function and with fuzzy switching function can effectively track in the three preset cases, but the method with sign switching function will lose effect for large amplitude interference. In particular, when the added disturbance is an elementary function and has deformation, the method with exponential switching function can also track the predetermined trajectory. The chattering is weak, but when the disturbance is a random value, the chattering is increased, while the method in this paper can not only track the predetermined trajectory but also has the weakest chattering and tracking error in the face of any disturbance. Therefore, this paper well verifies that the proposed SMC with fuzzy switching function can improve the chattering problem of synovial control.

5. The Fuzzy Sliding-Mode Method for Fractional-Order PMSM Vector Control

The actual model of fractional-order PMSM is given according to [33], and the vector control state-space representation of PMSM can be derived under parameter uncertainties and mismatches in (40).

$$\begin{array}{c}\left\{\begin{array}{c}{}^{C}D^{\alpha }{i}_d={\displaystyle \frac{1}{L_d}}{u}_d-{\displaystyle \frac{R}{L_d}}{i}_d+{\omega }_e{\displaystyle \frac{L_q}{L_d}}{i}_q+g_d\hfill \\ {}^{C}D^{\alpha }{i}_q={\displaystyle \frac{1}{L_q}}{u}_q-{\displaystyle \frac{R}{L_q}}{i}_q+{\omega }_e{\displaystyle \frac{L_d}{L_q}}{i}_d-{\displaystyle \frac{1}{L_q}}{\omega }_e{\psi }_f+g_q\hfill \\ {}^{C}D^{\alpha }{\omega }_e={\displaystyle \frac{3}{2}}{\displaystyle \frac{n_p^2}{J}}{i}_q[(L_d-L_q)i_d+{\psi }_f]-{\displaystyle \frac{n_p}{J}}{T}_L-{\displaystyle \frac{B}{J}}{\omega }_e+g_e\hfill \end{array}\right.\end{array}$$

(40)

where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{g}_d=-{\displaystyle \frac{\mathsf{\Delta }{L}_d}{L_d(L_d+\mathsf{\Delta }{L}_d)}}(u_d-R{i}_d-\mathsf{\Delta }R{i}_d+L_q{\omega }_e{i}_q+\mathsf{\Delta }{L}_q{\omega }_e{i}_q+{\omega }_e\mathsf{\Delta }{\psi }_f)\hfill \\ g_q=-{\displaystyle \frac{\mathsf{\Delta }{L}_q}{L_q(L_q+\mathsf{\Delta }{L}_q)}}(u_q-R{i}_q-\mathsf{\Delta }R{i}_q-L_d{\omega }_e{i}_d-\mathsf{\Delta }{L}_q{\omega }_e{i}_d-{\omega }_e\mathsf{\Delta }{\psi }_f-{\omega }_e{\psi }_f)\hfill \\ g_e={\displaystyle \frac{3}{2}}{\displaystyle \frac{n_p^2}{J}}{i}_q[(\mathsf{\Delta }{L}_d-\mathsf{\Delta }{L}_q)i_d+\mathsf{\Delta }{\psi }_f]\hfill \end{array}\right.\end{array}$$

(41)

The control model of fractional-order PMSM includes the d–q axis current dynamic equation and the speed dynamic equation. In the current state equation, the disturbances $d_d$ and $d_q$ may come from inverter nonlinearity (such as dead time effect), current sensor noise, resistance or inductance parameter perturbation, or harmonic interference, and they belong to the mechanical property disturbances. As for $d_e$, the disturbance that belongs to the mechanical properties may come from the load mutations such as mechanical vibration, friction disturbance, moment of inertia uncertainty, and so on.

A well-conceived design of the outer speed ring can mitigate the impact of disturbances on the system, minimize speed fluctuations, and enable the system to operate in a stable state. $i_d=0$ is a vector control method that is widely used. In this control method, $i_d=0$ is maintained and the motor torque is controlled by controlling $i_q$. Given condition $i_d=0$, the system states (40) are shown as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{C}D^{\alpha }{i}_q={\displaystyle \frac{1}{L_q}}{u}_q-{\displaystyle \frac{R}{L_q}}{i}_q-{\displaystyle \frac{1}{L_q}}{\omega }_e{\psi }_f+g_q^{\prime }+d_q(t)\hfill \\ {}^{C}D^{\alpha }{\omega }_e={\displaystyle \frac{3}{2}}{\displaystyle \frac{n_p^2}{J}}{i}_q{\psi }_f-{\displaystyle \frac{n_p}{J}}{T}_L-{\displaystyle \frac{B}{J}}{\omega }_e+g_e^{\prime }+d_e(t)\hfill \end{array}\right.\end{array}$$

(42)

and the internal interferences can be rewritten as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{g}_q^{\prime }=-{\displaystyle \frac{\mathsf{\Delta }{L}_q}{L_q(L_q+\mathsf{\Delta }{L}_q)}}(u_q-R{i}_q-\mathsf{\Delta }R{i}_q-{\omega }_e\mathsf{\Delta }{\psi }_f-{\omega }_e{\psi }_f)\hfill \\ g_e^{\prime }={\displaystyle \frac{3}{2}}{\displaystyle \frac{n_p^2}{J}}{i}_q\mathsf{\Delta }{\psi }_f\hfill \end{array}\right.\end{array}$$

(43)

There is an expected value of the velocity ${\omega }_e^{*}$, and we take the space error as the system state:

$$\begin{array}{c}\hfill x_1={\omega }_e^{*}-{\omega }_e\end{array}$$

(44)

$$\begin{array}{c}\hfill {}^{C}D_t^{\alpha }{x}_1=-{}^{C}D_C^{\alpha }{\omega }_e\end{array}$$

(45)

In order to make the actual electrical angular velocity of the motor track the reference value ${\omega }_e^{*}$, it is necessary to change the electromagnetic torque output of the motor by adjusting $i_q$, thus affecting the speed and electrical angular velocity of the motor. In this case, $i_q$ is used as a control variable. By adjusting the value of $i_q$ reasonably, the running state of the motor can be changed, and, finally, the error will be gradually reduced until a satisfactory control accuracy is achieved. The state function can be described as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{C}D^{\alpha }{x}_1=-{\displaystyle \frac{3}{2}}{\displaystyle \frac{n_p^2}{J}}{\psi }_f{i}_q+{\displaystyle \frac{n_p}{J}}{T}_L+{\displaystyle \frac{B}{J}}{\omega }_e-g_e^{\prime }-d_e(t)\hfill \\ y=i_q\hfill \end{array}\right.\end{array}$$

(46)

Design a simple sliding-mode surface:

$$\begin{array}{c}\hfill s_e=a{x}_1\end{array}$$

(47)

where $a>0$.

Design $i_q^{*}$, that is, the control rate u is

$$\begin{array}{c}\hfill i_q^{*}={\displaystyle \frac{2J}{3n_p^2{\psi }_f}}({\displaystyle \frac{n_p}{J}}{T}_L+{\displaystyle \frac{B}{J}}{\omega }_e-g_e^{\prime }-d_e(t)+{\displaystyle \frac{\lambda }{a}}sgn(s_e))\end{array}$$

(48)

where $d_e(t)$ is uncertain and $g_e^{\prime }$ is the unknown internal nonlinear disturbance. In order to ensure the smooth progress of the control, adaptive control and FLSs must be used.

Assume that $d_e(t)$, $d_q(t)$, and $d_d(t)$ are bounded and $∥d_e(t)∥\le d_e$, $∥d_q(t)∥\le d_q$, and $∥d_d(t)∥\le d_d$. $d_e^{*}$, $d_q^{*}$, and $d_d^{*}$ are the expected values of uncertain external disturbances. $\widetilde{d_e}$, $\widetilde{d_q}$, and $\widetilde{d_d}$ are the estimation errors of uncertain external disturbances.

$$\begin{array}{c}\left\{\begin{array}{c}\widetilde{d_e}=d_e^{*}-d_e\hfill \\ \widetilde{d_q}=d_q^{*}-d_q\hfill \\ \widetilde{d_d}=d_d^{*}-d_d\hfill \end{array}\right.\end{array}$$

(49)

Assume that $\widehat{g_e^{\prime }}$ is the fuzzy approximation form of nonlinear function $g_e^{\prime }$.

$$\begin{array}{c}\hfill \widehat{g_e^{\prime }}={\theta }_e^T\xi (x)=g_e(x|{\theta }_e)\end{array}$$

(50)

where $x={\left[\begin{array}{ccc}{\omega }_e& i_q& i_d\end{array}\right]}^T$. It is also necessary to take adaptive control due to unknown function $g_e^{\prime }$. We design ${\widehat{g_e^{\prime }}}^{*}$ as the optimal function of $\widehat{g_e^{\prime }}$.

$$\begin{array}{c}\hfill {\widehat{g_e^{\prime }}}^{*}=g_e(x|{\theta }_e^{*}),\end{array}$$

(51)

$$\begin{array}{c}\hfill \delta ={g_e}^{\prime }-{\widehat{g_e^{\prime }}}^{*}={g_e}^{\prime }-g_e(x|{\theta }_e^{*});\end{array}$$

(52)

thus, $\widetilde{{\theta }_e}={\theta }_e^{*}-{\theta }_e$ is the error between the fuzzy approximation and optimal fuzzy approximation. According to the above two estimates, the function (48) can be improved as

$$\begin{array}{c}\hfill i_q^{*}={\displaystyle \frac{2J}{3n_p^2{\psi }_f}}({\displaystyle \frac{n_p}{J}}{T}_L+{\displaystyle \frac{B}{J}}{\omega }_e-g_e(x|{\theta }_e)-d_e^{*}+{\displaystyle \frac{\lambda }{a}}sign(s_e))\end{array}$$

(53)

And the adaptive control laws are designed as

$$\begin{array}{c}\hfill {}^{C}D_t^{\alpha }{d}_e^{*}=-a{\alpha }_2{s}_e\end{array}$$

(54)

$$\begin{array}{c}\hfill {}^{C}D_t^{\alpha }{\theta }_e=-a{\alpha }_1{s}_e\xi (x)\end{array}$$

(55)

It is clear that the system (46) is stable under the control rate. The Lyapunov function is selected as $V_1={\displaystyle \frac{1}{2}}{s}_e^2+{\displaystyle \frac{1}{2{\alpha }_1}}{\widetilde{{\theta }_e}}^2+{\displaystyle \frac{1}{2{\alpha }_2}}{\widetilde{d_e}}^2$. The fractional differential form is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^{C}D_t^{\alpha }{V}_1& =s_e{}^{C}D_t^{\alpha }{s}_e+{\displaystyle \frac{1}{{\alpha }_1}}{\widetilde{{\theta }_e}}^T{}^{C}D_t^{\alpha }\widetilde{{\theta }_e}+{\displaystyle \frac{1}{{\alpha }_2}}\widetilde{d_e}{}^{C}D_t^{\alpha }\widetilde{d_e}\hfill \\ \hfill & =s_e{}^{C}D_t^{\alpha }{s}_e-{\displaystyle \frac{1}{{\alpha }_1}}{\widetilde{{\theta }_e}}^T{}^{C}D_t^{\alpha }{\theta }_e+{\displaystyle \frac{1}{{\alpha }_2}}\widetilde{d_e}{}^{C}D_t^{\alpha }{d}_e^{*}\hfill \\ \hfill & =a{s}_e{}^{C}D_t^{\alpha }{x}_1-{\displaystyle \frac{1}{{\alpha }_1}}{\widetilde{{\theta }_e}}^T{}^{C}D_t^{\alpha }{\theta }_e+{\displaystyle \frac{1}{{\alpha }_2}}\widetilde{d_e}{}^{C}D_t^{\alpha }{d}_e^{*}\hfill \end{array}\end{array}$$

(56)

Because of the control rate (53), we can obtain ${}^{C}D_t^{\alpha }{x}_1$:

$$\begin{array}{c}\begin{array}{cc}\hfill {}^{C}D_t^{\alpha }{x}_1& \le -{\displaystyle \frac{3}{2}}{\displaystyle \frac{n_p^2}{J}}{\psi }_f{i}_q^{*}+{\displaystyle \frac{n_p}{J}}{T}_L+{\displaystyle \frac{B}{J}}{\omega }_e-g_e^{\prime }-d_e\hfill \\ \hfill & =g_e(x|{\theta }_e)-(g_e(x|{\theta }_e^{*})+\delta )+(d_e^{*}-d_e)-{\displaystyle \frac{\lambda }{a}}sign(s_e)\hfill \\ \hfill & ={\theta }_e^T\xi (x)-{\theta }_e^{*T}\xi (x)-\delta +\widetilde{d_e}-{\displaystyle \frac{\lambda }{a}}sign(s_e)\hfill \\ \hfill & =-{\widetilde{{\theta }_e}}^T\xi (x)-\delta +\widetilde{d_e}-{\displaystyle \frac{\lambda }{a}}sign(s_e)\hfill \end{array}\end{array}$$

(57)

Thus, ${}^{C}D_t^{\alpha }{V}_1$ is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^{C}D_t^{\alpha }{V}_1& \le a{s}_e(-{\widetilde{{\theta }_e}}^T\xi (x)-\delta +\widetilde{d_e}-{\displaystyle \frac{\lambda }{a}}sign{s}_e)-{\displaystyle \frac{1}{{\alpha }_1}}{\widetilde{{\theta }_e}}^T{}^{C}D_t^{\alpha }{\theta }_e+{\displaystyle \frac{1}{{\alpha }_2}}\widetilde{d_e}{}^{C}D_t^{\alpha }{d}_e^{*}\hfill \\ \hfill & ={\widetilde{{\theta }_e}}^T(-a{s}_e\xi (x)-{\displaystyle \frac{1}{{\alpha }_1}}{}^{C}D_t^{\alpha }{\theta }_e)+\widetilde{d_e}(a{s}_e+{\displaystyle \frac{1}{{\alpha }_2}}{}^{C}D_t^{\alpha }{d}_e^{*})-a{s}_e\delta \hfill \\ \hfill & =-\lambda s_{e}sign(s_e)-a{s}_e\delta \hfill \\ \hfill & =-\lambda \left|s_e\right|-a{s}_e\delta \hfill \end{array}\end{array}$$

(58)

where $\delta$ is the minimum approximation error between $g_e^{\prime }$ and ${\widehat{g_e^{\prime }}}^{*}$. Design $\lambda >a\delta$, then ${}^{C}D_t^{\alpha }{V}_1$ can be guaranteed to be negative. According to Lyapunov’s second method, $V_1(x)$ is positive definite and ${}^{C}D_t^{\alpha }{V}_1(x)<0$, and then the system is asymptotically stable.

In the process of simulation, we keep parameters of fractional-order PMSM in [33] and $a=2$, ${\alpha }_1={\alpha }_2=2$, $\lambda =8$, $\delta =3$. The selection rule of fuzzy rules is to keep the error 0 at the zero of the rule, and then the width of the membership function is reduced in order to improve the accuracy. As for the uncertain nonlinear function $g_e$, we hope that this uncertainty term does not exist, so we keep the expected value 0 at the zero of the rule, too.

$$\begin{array}{c}\hfill {}^{C}D_t^{0.95}{\theta }_e=-4s_e\xi (x)\end{array}$$

(59)

$$\begin{array}{c}\hfill {}^{C}D_t^{0.95}{d}_e^{*}=-4s_e\end{array}$$

(60)

From the above analysis, we obtain the expected value $i_q^{*}$, and next consider the system as a second-order system with states $i_q$ and $i_d$.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{C}D^{\alpha }{i}_d={\displaystyle \frac{1}{L_d}}{u}_d-{\displaystyle \frac{R}{L_d}}{i}_d+{\omega }_e{\displaystyle \frac{L_q}{L_d}}{i}_q+g_d\hfill \\ {}^{C}D^{\alpha }{i}_q={\displaystyle \frac{1}{L_q}}{u}_q-{\displaystyle \frac{R}{L_q}}{i}_q+{\omega }_e{\displaystyle \frac{L_d}{L_q}}{i}_d-{\displaystyle \frac{1}{L_q}}{\omega }_e{\psi }_f+g_q\hfill \end{array}\right.\end{array}$$

(61)

We can obtain through function (32) that the sliding-mode surfaces are

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{s}_q(t)=e_q(t)+10I_t^{0.95}{e}_q(t)-(e_q(0)+10I_t^{0.95}{e}_q(0))e^{-4t}\hfill \\ s_d(t)=e_d(t)+10I_t^{0.95}{e}_d(t)-(e_d(0)+10I_t^{0.95}{e}_d(0))e^{-4t}\hfill \end{array}\right.\end{array}$$

(62)

Design the fuzzy adaptive laws as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{C}D_t^{0.95}{\theta }_{g_q}=s_q\xi (x)\hfill \\ {}^{C}D_t^{0.95}{\theta }_{g_d}=s_d\xi (x)\hfill \end{array}\right.\end{array}$$

(63)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{C}D_t^{0.95}{\theta }_{u_q}=200s_q\varphi (s_q)\hfill \\ {}^{C}D_t^{0.95}{\theta }_{u_d}=300s_d\varphi (s_d)\hfill \end{array}\right.\end{array}$$

(64)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{C}D_t^{0.95}{\widehat{\delta }}_q=4s_q\hfill \\ {}^{C}D_t^{0.95}{\widehat{\delta }}_d=3s_d\hfill \end{array}\right.\end{array}$$

(65)

In

Table 2. The tracking errors of ${\omega }_e$, $i_q$, and $i_d$.

Control Mathod	Errors in 10 s (× 10−3)	Errors After 10 s (× 10−3)	Adjust Time in 10 s (s)
PI	2	2.5	0.1
	1.5	1	0.1
	1.5	2	0.1
SMC with sign switching function	2	2.5	0.1
	1	0.8	0.1
	1	0.8	0.1
SMC with fuzzy switching function	1	1	0.3
	0.8	0.8	0.3
	0.8	0.5	0.35

, it is easy to conclude that the errors of AFSMC are the smallest and the adjustment time is the largest, which is basically consistent with the previous analysis results. From Figure 10, it is clear that all three methods can track the desired rotational speed very well, even with a step here. As we use SMC with sign switching function in the design of the speed ring, the advantage of the SMC with fuzzy switching function is not well shown in Figure 11. For the expected value with an amplitude of 1000, the average error is 2 in this paper, which is within its accuracy requirement. The advantages of the SMC with fuzzy switching function in reducing chattering and error can be well verified in Figure 12. In particular, both error $e_q$ and error $e_d$ have small chattering with respect to SMC in this paper after a step in the speed occurs; at least macroscopically, the chattering frequency is reduced by half. Through the simulation of the actual system, the method proposed in this paper is effective in reducing the chattering and errors in theory.

6. Conclusions

To sum up, the study described in this paper combines three different but related theoretical frameworks: FLSs, adaptive control, and SMC. An inventive control approach designed for fractional-order MIMO systems with uncertain nonlinear dynamics and external disturbances was developed. The incorporation of adaptive control and FLSs not only improves flexibility but also fosters adaptability, establishing a robust foundation for optimal performance across a wide range of engineering applications. In the future, the research on the switching function still needs to be continued, and it is expected to solve the problem of large overshoot and adjustment time. For example, intelligent methods such as fuzzy rule bases and even neural networks can be combined with SMC.