Fuzzy-Adaptive Nonsingular Terminal Sliding Mode Control for the High-Speed Aircraft Actuator Trajectory Tracking

Abstract

High-speed aircraft actuators are critical for precise control of aerodynamic surfaces, demanding fast response, accuracy, and robustness against uncertainties and disturbances. However, the complex nonlinear dynamics of these systems pose significant challenges for conventional control methods. Sliding mode control (SMC) offers robust performance and rapid transient response but is hindered by chattering, which can degrade performance. To address this, this paper proposes an innovative nonlinear control strategy that integrates global nonsingular terminal sliding mode control (NTSMC) for finite-time convergence with fuzzy logic-based adaptive gain tuning to mitigate chattering and suppress oscillations. A prototype actuator and experimental platform were developed to validate the approach. Experimental results demonstrate superior dynamic response and disturbance rejection compared to traditional methods, highlighting the effectiveness of the proposed control strategy.

1. Introduction

High-speed aircraft systems operating at extreme velocities present substantial control challenges in dynamic aerodynamic environments [1,2]. During operational scenarios, guidance systems utilize multidimensional parameters, including velocity differentials and spatial orientation metrics, to generate actuator commands. These inputs are translated into surface deflection controls through signal modulation and power amplification stages. The resultant aerodynamic forces enable precise trajectory regulation by modulating airflow interactions [3,4,5]. Modern aerodynamic platforms exhibit increasingly complex maneuvering capabilities, characterized by rapid energy state transitions and nonlinear response patterns. As the primary role for flight path modulation, the actuation subsystem plays a critical role in determining flight efficiency during high-velocity operations. Its operational effectiveness hinges on achieving millisecond-level synchronization between computational models and physical execution mechanisms [6,7,8].

Electric actuator systems should be designed using advanced multi-objective optimization strategies to balance competing demands across electromagnetic characteristics, mechanical transmission efficiency, and thermal dissipation. Nevertheless, inherent nonlinear dynamics—such as transmission backlash, time-varying friction effects, and aerodynamic hinge moment perturbations—pose significant challenges for precision control, resulting in persistent steady-state oscillations, cumulative positioning inaccuracies, and transient instability during dynamic operations [9,10,11,12,13]. Simultaneously, the Permanent Magnet Synchronous Motor (PMSM) has become the dominant actuator solution for high-performance aerospace applications, owing to its exceptional power density, operational reliability, and compact structural configuration. However, the PMSM exhibits complex multivariable dynamics characterized by strong parameter coupling, intrinsic nonlinearities, and hybrid control challenges associated with electromechanical interactions.

An increasing number of advanced control algorithms are transitioning from theoretical research to practical implementation, finding extensive applications in motor control and continuously driving innovations in PMSM control technologies. Neural network control [14,15] emulates the computational mechanisms of the human brain, learning the mapping between inputs and outputs through training processes. In PMSM applications, neural networks can model the nonlinear relationship between torque and current, thereby enabling precise control. Fuzzy logic control [16,17], based on fuzzy set theory, addresses uncertainties and imprecision between input and output variables. In PMSM control, it achieves system regulation through fuzzification, rule definition, and fuzzy inference. Both neural network and fuzzy control strategies offer self-adaptive capabilities, allowing real-time parameter adjustment and improved disturbance rejection. These features enhance control performance and system adaptability. However, their generalization and robustness may be limited in complex environments due to dependence on training data and inherent model uncertainties. Active disturbance rejection control (ADRC) [18,19] actively estimates and compensates for internal and external disturbances to achieve accurate control. While it performs well in disturbance rejection, its control performance may degrade in the presence of model uncertainties or parameter variations. Model predictive control (MPC) [20,21] relies on a mathematical model to predict future system behavior and optimize control actions accordingly. Although it offers high control accuracy, its computational complexity and dependence on model fidelity limit its real-time applicability. Sliding mode control (SMC) [22,23,24,25] has been widely adopted due to its robustness against uncertainties, insensitivity to bounded disturbances, fast response, favorable transient performance, and computational simplicity.

The design process of SMC can be divided into two distinct phases: the reaching phase and the sliding phase. In the sliding phase of conventional SMC, a linear sliding surface is frequently adopted. The convergence speed of the system can be tuned by adjusting the sliding surface parameters. However, the tracking error cannot converge to zero in finite time under any circumstances. Moreover, a large switching gain is typically required to compensate for disturbances and uncertainties, which consequently exacerbates chattering in the control torque. This chattering not only increases friction between moving mechanical components but also leads to excessive heat generation in the power electronics circuit. To mitigate the chattering phenomenon, several studies [26,27,28] have introduced the concept of a “boundary layer,” which has proven effective in suppressing chattering. However, selecting an appropriate boundary layer thickness involves a trade-off between chattering suppression and control accuracy, making simultaneous optimization of both objectives challenging. Alternatively, researchers in [29,30] have integrated fuzzy control rules into the sliding mode variable structure control, proposing a fuzzy SMC strategy that has successfully achieved chattering attenuation. Furthermore, when the system experiences rapid variations in disturbances, it tends to exhibit poor transient performance.

While these methods have achieved a certain degree of chattering suppression, none of them fundamentally resolve the issues of chattering and finite-time convergence, which are inherently linked to the switching term in conventional sliding mode control (SMC). The chattering issue can be mitigated by replacing the switching term with a nonlinear function and formulating a terminal sliding mode control (TSMC) strategy. Several studies [31,32] have developed terminal sliding mode surfaces tailored for high-order nonlinear systems. This advancement overcomes the limitation of the discontinuous derivative typically observed in the sliding surfaces of conventional SMC. However, TSMC exhibits slow convergence when the system states are far from the origin, and it is prone to singularities that may lead to unbounded control signals. To overcome the singularity problem, a number of studies [33,34,35,36] have proposed nonsingular terminal sliding mode control (NTSMC) strategies.

During the reaching phase, the system’s dynamic performance becomes highly susceptible to uncertainties and unknown disturbances, often induced by high-frequency switching motions. To eliminate the reaching phase in sliding mode control and enhance robustness throughout the entire response, global sliding mode control (GSMC) techniques have been extensively developed. This is typically accomplished by embedding exponential functions into linear sliding hyperplanes. Researchers have further combined GSMC with terminal sliding mode control (TSMC) to achieve superior tracking performance and improved chattering suppression for nonlinear systems [37,38,39,40,41]. GSMC eliminates the conventional reaching phase by carefully designing the initial conditions, thereby ensuring full-state robustness from the very start of system operation. When integrated with terminal sliding surfaces, this approach theoretically enables finite-time convergence while maintaining global robustness. However, GSMC also presents several limitations. First, calculating the equivalent control requires precise knowledge of the system dynamics, which restricts its practical applicability. Second, GSMC assumes that external disturbances are bounded. Third, chattering remains inevitable in the presence of disturbances and modeling errors, due to the discontinuous nature of the switching control term.

To the best of the authors’ knowledge, the GSMC with NTSMC for robust trajectory tracking of high-speed aircraft actuators has not been investigated in the existing literature. Addressing this gap constitutes the primary motivation of this study. A novel Fuzzy-Adaptive Global Nonsingular Terminal Sliding Mode Control (FAG-NTSMC) strategy is proposed to achieve high-precision and robust control under strongly nonlinear and uncertain conditions. The main innovations and contributions of this paper are summarized as follows:

(i)

Innovative control architecture: To meet the demands for high dynamics and strong robustness in high-speed aircraft actuators, a novel sliding surface is designed by integrating global sliding mode theory with nonsingular terminal attractors. This design eliminates the reaching phase, ensuring that the system state begins on the sliding manifold and converges to the origin in finite time, thereby improving robustness against initial disturbances and avoiding the singularities present in conventional TSMC.

(ii)

Novel fuzzy-adaptive mechanism: A Mamdani-type fuzzy inference system is developed to dynamically tune the switching gain based on the real-time magnitude of the sliding variable. This allows for an intelligent trade-off between fast convergence and chattering suppression, effectively mitigating control-induced oscillations and improving overall tracking smoothness.

(iii)

Enhanced convergence law design: A new reaching law is constructed, incorporating a nonlinear damping term $\left|x_1\right|$·s. This design adaptively accelerates convergence when the error is large and suppresses oscillations near equilibrium. Rigorous Lyapunov-based analysis confirms global boundedness and finite-time stability of the closed-loop system.

(iv)

Comprehensive experimental validation: A custom-built high-speed actuator platform is used to validate the proposed controller under step response, sinusoidal tracking, and parameter perturbation scenarios. Compared to conventional SMC, the proposed FAG-NTSMC achieves a 14.9% reduction in convergence time, 45% improvement in tracking accuracy, and 94.8% suppression of disturbance-induced error growth, demonstrating significant advantages in responsiveness, robustness, and practical applicability.

In summary, this work makes both theoretical and practical contributions by extending classical sliding mode frameworks and validating the approach under realistic operating conditions, thereby offering a promising solution for precision control in aerospace applications. The remainder of this study is structured as follows: Section 2 introduces the nonlinear actuator dynamics and outlines the control challenges. Section 3 presents the design of the fuzzy-adaptive global NTSMC along with stability proofs. Experimental validation and comparative analysis are provided in Section 4, followed by the concluding remarks in Section 5.

2. System Description and Problem Formulation

2.1. System Technical Index Requirements

The illustration of the newly designed high-speed aircraft actuator, developed based on the specific application requirements of a particular model, is shown in Figure 1. The primary technical specifications of the actuator are as follows:

Based on the experience gained from model development, the primary technical specifications of the actuator are as follows:

• working angle: −30° to +30°;
• mechanical clearance: ≤0.1°;
• rated torque: 300 N·m;
• rated operational angular speed: ≤$60°/\mathrm{s}$;
• system overshoot at a 30° step: ≤$5\%$.

2.2. Nonlinear Dynamics Analysis of High-Speed Aircraft Actuator

The actuator consists of a PMSM, a reducer, and a controller. The reduction mechanism, comprising a gear reducer and a ball screw pair, converts the high-speed, low-torque output of the motor into the low-speed, high-torque rotation required by the system. The motor’s rotational motion is then converted into linear motion through the ball screw and subsequently transformed back into rotational motion via a rocker arm mechanism.

2.3. Establishment of a Mathematical Model for the Actuator

When analyzing the mathematical model of PMSM, to simplify the analysis, factors that have relatively minor impacts during the experimental process are usually neglected. Consequently, the following assumptions can be made: the electrical conductivity of the rotor’s permanent magnet material is assumed to be zero, and the damping windings are ignored; hysteresis and eddy-current losses are not considered, nor is magnetic circuit saturation; the air-gap magnetic field generated by the permanent magnet and the magnetic field produced by the stator windings exhibit a sinusoidal distribution in space; and the induced electromotive force generated by the phase windings is described in the form of a sine wave [42]. The simplified physical model of the PMSM is illustrated in Figure 2.

Based on the general assumptions for PMSM modeling, this study uses the vector control method with $i_d=0$ in the $d-q$ rotating coordinate system to control the PMSM motor. The stator flux linkage equation can be simplified to

$$\left\{\begin{array}{c}{\psi }_d={\psi }_f\hfill \\ {\psi }_q=L_q{i}_q\hfill \end{array}\right.$$

(1)

where ${\psi }_d$, ${\psi }_q$ is defined as the $d/q$ axis stator flux component, ${\psi }_f$ denotes the permanent magnet flux linkage, $L_q$ signifies the quadrature-axis synchronous inductors, $i_q$ indicates the quadrature-axis current. The stator voltage equation can be simplified to

$$\left\{\begin{array}{c}{u}_d=-{\omega }_e{\psi }_q\hfill \\ u_q=L_q{\displaystyle \frac{d{i}_q}{dt}}+{\omega }_e{\psi }_f+R{i}_q\hfill \end{array}\right.$$

(2)

where $u_d$, $u_q$ is defined as the $d/q$ axis stator voltage component, ${\omega }_e$ denotes the rotor electrical angular velocity, R signifies the stator resistance. The electromagnetic torque equation can be simplified to

$$T_e={\displaystyle \frac{3}{2}}{p}_n{\psi }_f{i}_q$$

(3)

where $T_e$ is defined as the electromagnetic torque, $p_n$ denotes the number of rotor pole pairs. The torque balance equation of the actuator is

$${\ddot{\theta }}_m={\displaystyle \frac{\eta i_s}{J_m}}{\displaystyle \frac{3}{2}}{p}_n{\psi }_f{i}_q+d\left(t\right)$$

(4)

where ${\theta }_m$ is defined as the rudder mechanical angle, $\eta$ signifies the transmission efficiency of the actuator, $i_s$ indicates the transmission ratio of the actuator, $J_m=\left(J_1+{\displaystyle \frac{J_2+J_3}{\eta i_s^2}}\right)$ is defined as the total system inertia, $J_1$ denotes the moment of inertia of the servo motor shaft and pinion, $J_2$ signifies the moment of inertia of the large gear and output shaft, and $J_3$ indicates the moment of inertia of the rudder surface, $d\left(t\right)$ is an uncertain disturbance, and $\left|d\right(t\left)\right|\le D$.

Let $x_1={\theta }_m$, $x_2={\dot{\theta }}_m$, the mathematical model of the actuator can be expressed as

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2={\displaystyle \frac{\eta i_s}{J_m}}{\displaystyle \frac{3}{2}}{p}_n{\psi }_f{i}_q+d\left(t\right)\hfill \end{array}\right.$$

(5)

2.4. Core Challenges

(1)

Transient Performance Limitations

• Finite-Time Convergence vs. Singularity: Traditional TSMC achieves finite-time tracking but suffers from singularities as $e\to 0$, resulting in unbounded control inputs.
• Chattering vs. Robustness: The baseline SMC reduces settling time but amplifies torque ripple due to discontinuous switching via the $\mathrm{sgn}\left(s\right)$ function.

(2)

Robustness–Accuracy Tradeoff in High-Speed Regimes

• Aerodynamic Load Variability: Time-varying load disturbances adversely impact the accuracy and robustness of trajectory tracking.
• Actuator Saturation: Rudder deflection limits and rate constraints require smooth control transitions to avoid mechanical fatigue.

The main difficulty in SMC is suppressing chattering while ensuring a rapid response. An inherent trade-off exists between response speed and chattering: faster response times often lead to increased chattering. The stronger robustness of SMC to uncertainties allows for a more conservative controller, which can affect control performance. Trade-offs were also observed between convergence speed, actuator burden, robustness to uncertainties, and control performance. Thus, the next section discusses how to improve control performance while maintaining finite-time convergence and robustness in SMC.

3. Fuzzy-Adaptive Global NTSMC Design and Stability Analysis

3.1. Improved Global NTSMC

Recognizing the challenges posed by the traditional integral term’s inability to be adjusted, this section proposes a new global nonsingular sliding mode controller, specifically designed to meet the actuator’s actual needs. This controller takes advantage of the inherent finite-time convergence mechanism of the terminal sliding mode and the full robustness of the global sliding mode:

$$s\left(t\right)=x_1+{\displaystyle \frac{1}{\beta }}{x}_2^{{\displaystyle \frac{q}{p}}}-f\left(t\right)$$

(6)

where $x_2={\dot{x}}_1, \beta >0$, $q>p$, and q, p being positive odd numbers, and $1<{\displaystyle \frac{q}{p}}<2$. The conditions that $f\left(t\right)$ should satisfy are as follows:

• $f\left(0\right)=x_1\left(0\right)+{\displaystyle \frac{1}{\beta }}{x}_2^{{\displaystyle \frac{q}{p}}}\left(0\right)$;
• as $t\to \infty$, $f\left(0\right)\to 0$;
• $f\left(t\right)$ has a first derivative.

Among them, $f\left(t\right)$ is a nonlinear function and can be designed as

$$f\left(t\right)=f\left(0\right)e^{-at}=\left[x_1\left(0\right)+{\displaystyle \frac{1}{\beta }}{x}_2^{{\displaystyle \frac{q}{p}}}\left(0\right)\right]e^{-at}$$

(7)

where $a>0$.

Substituting (7) into (6), the switching surface function for the GSMC can be obtained as

$$s\left(t\right)=x_1+{\displaystyle \frac{1}{\beta }}{x}_2^{{\displaystyle \frac{q}{p}}}-\left[x_1\left(0\right)+{\displaystyle \frac{1}{\beta }}{x}_2^{{\displaystyle \frac{q}{p}}}\left(0\right)\right]e^{-at}$$

(8)

where $t=0$, $s=0$, indicating that the system state is situated on the sliding mode surface $s=0$ at any initial moment, thereby eliminating the approach mode. Assuming that as $t\to \infty$, $f\left(0\right)\to 0$, this signifies that as time approaches infinity, the nonlinear term approaches zero. Therefore, the global sliding mode switching surface function converges solely to the nonsingular terminal sliding mode surface and passes through the origin.

3.2. Controller Design

When designing SMC laws, commonly used approach laws include the constant velocity approach law, exponential approach law, power approach law, and general approach law. Selecting an appropriate approach law is an effective strategy for mitigating chattering. Experimental research was conducted using a newly identified convergence law:

$$\dot{s}=-k_{1}sign\left(s\right)-k_2\left|x_1\right|s$$

(9)

In the formula, $k_1>0$ and $k_2>0$. The inclusion of the $\left|x_1\right|$ damping term acts as a nonlinear damping mechanism that adaptively modulates the convergence rate of the sliding variable based on the magnitude of the system state $x_1$. Specifically, it accelerates convergence when the state error is large and suppresses oscillations when the error is small. This design not only ensures rapid reaching of the sliding surface but also balances smooth sliding motion and chattering attenuation, thereby enhancing the overall system response quality.

The fuzzy system is employed to dynamically adjust the gain for the exponential term according to the absolute values of the sliding variable s. |s| is first normalized. S stands for small |s|, M stands for middle |s|, and B stands for big |s|. The membership function for the absolute values of the sliding variable s is Gaussian function:

$$\mu \left(\right|s\left|\right)=exp\left(-{\displaystyle \frac{{\left(\right|s|-c)}^2}{2{\sigma }^2}}\right)$$

(10)

where c is the mean and $\sigma$ is the standard deviation for the function. The Gaussian function parameters for different fuzzy sets is shown as follows,

Table 1. Gaussian function parameters for different fuzzy sets.

|s|	S	M	B
c	0	0.5	1
$\sigma$	0.18	0.5	0.18

.

The defuzzification for the gain for the exponential term is carried out by the following expression:

$$k_1={\displaystyle \frac{{\sum }_{i=S,M,B}{\mu }_i\left(\right|s\left|\right){\omega }_i}{{\sum }_{i=S,M,B}{\mu }_i\left(\right|s\left|\right)}}$$

(11)

where ${\omega }_i$ is the fuzzy logic output, ${\mu }_i\left(\right|s\left|\right)$ is the membership degree of |s| in fuzz set $i\in \{S,M,B\}$. For a detailed breakdown of how ${\omega }_i$ is computed and its significance within the fuzzy logic framework, please refer to

Table 2. Fuzzy logic used in tuning of the gain for exponential term.

|s|	S	M	B
${\omega }_i$	0	0.5	1

. As shown in Figure 3, the variation of $k_1$ with respect to |s| is presented.

From (8), we can get

$$\dot{s}\left(t\right)={\dot{x}}_1+{\displaystyle \frac{1}{\beta }}{\displaystyle \frac{q}{p}}{x}_2^{{\displaystyle \frac{q}{p}}-1}{\dot{x}}_2+a\left[x_1\left(0\right)+{\displaystyle \frac{1}{\beta }}{x}_2^{{\displaystyle \frac{q}{p}}}\left(0\right)\right]e^{-at}$$

(12)

Combining Formulas (5) and (12)

$$\begin{array}{c}\hfill \dot{s}\left(t\right)={\dot{x}}_1+{\displaystyle \frac{1}{\beta }}{\displaystyle \frac{q}{p}}{x}_2^{{\displaystyle \frac{q}{p}}-1}\left({\displaystyle \frac{\eta i_s}{J_m}}{\displaystyle \frac{3}{2}}{p}_n{\psi }_f{i}_q+d\left(t\right)\right)+a\left[x_1\left(0\right)+{\displaystyle \frac{1}{\beta }}{x}_2^{{\displaystyle \frac{q}{p}}}\left(0\right)\right]e^{-at}\end{array}$$

(13)

From Equations (9) and (13), it can be derived that

$$\dot{s}\left(t\right)={\dot{x}}_1+{\displaystyle \frac{1}{\beta }}{\displaystyle \frac{q}{p}}{x}_2^{{\displaystyle \frac{q}{p}}-1}\left({\displaystyle \frac{\eta i_s}{J_m}}{\displaystyle \frac{3}{2}}{p}_n{\psi }_f{i}_q+d\left(t\right)\right)+a\left[x_1\left(0\right)+{\displaystyle \frac{1}{\beta }}{x}_2^{{\displaystyle \frac{q}{p}}}\left(0\right)\right]e^{-at}=-k_{1}sign\left(s\right)-k_2\left|x_1\right|s$$

(14)

The Lyapunov function is defined as

$$V={\displaystyle \frac{1}{2}}{s}^2$$

(15)

To ensure $\dot{V}\le 0$, the sliding mode controller rate can be designed as

$$i_q=-\beta {\displaystyle \frac{p}{q}}\left({\displaystyle \frac{J_m}{\eta i_s}}{\displaystyle \frac{2}{3}}{\displaystyle \frac{1}{p_n{\psi }_f}}{x}_2^{2-{\displaystyle \frac{q}{p}}}+a\left[x_1\left(0\right)+{\displaystyle \frac{1}{\beta }}{x}_2^{{\displaystyle \frac{q}{p}}}\left(0\right)\right]e^{-at}\right)-{\displaystyle \frac{J_m}{\eta i_s}}{\displaystyle \frac{2}{3}}{\displaystyle \frac{1}{p_n{\psi }_f}}(k_{1}sign\left(s\right)+k_2\left|x_1\right|s)$$

(16)

Combine Equations (13) and (16)

$$\dot{V}=\mathrm{s}\dot{\mathrm{s}}=-{\displaystyle \frac{1}{\beta }}{\displaystyle \frac{q}{p}}{x}_2^{{\displaystyle \frac{q}{p}}-1}(k_2\left|x_1\right|s^2+k_1\left|s\right|-sd\left(t\right))$$

(17)

Let $k_1\ge D$, and D is an upper bound of the disturbance $d\left(t\right)$, it follows that

$$k_1\left|s\right|-sd\left(t\right)\ge 0$$

(18)

Given that $1<{\displaystyle \frac{q}{p}}<2$, it holds that

$$0<{\displaystyle \frac{q}{p}}-1<1$$

(19)

Furthermore, considering that $\beta >0$, and both q and p are positive odd integers, it follows that

$${\displaystyle \frac{1}{\beta }}{\displaystyle \frac{q}{p}}{x}_2^{{\displaystyle \frac{q}{p}}-1}>0$$

(20)

In addition, for any $k_2>0$, we have $k_2\left|x_1\right|s^2>0$. Therefore, it can be concluded that

$$\dot{V}\le 0$$

(21)

This indicates that the sliding mode dynamics are asymptotically stable, thereby confirming the theoretical effectiveness of the proposed sliding mode controller.

4. Implementation and Validation of High-Speed Aircraft System

4.1. Parameter Design of the Actuator

To achieve the optimal design of the actuating mechanism system, a preliminary design of the servo motor must be carried out based on the overall system design requirements and technical specifications. Initially, the power of the servo motor should be estimated to ensure that it can adequately drive the external load torque of the actuating mechanism. The maximum output power of the mechanism, denoted as $P_{\mathrm{jmax}}$, is given by:

$$P_{j max}=M_{j max}\times {\dot{\delta }}_{\max }=942\left(\mathrm{W}\right)$$

(22)

where $M_{\mathrm{jmax}}$ denotes the maximum output torque (N·m), and ${\dot{\delta }}_{max}$ represents the rated operating angular velocity (rad/s).

The maximum output power of the motor, $P_{d max}$, can be calculated based on the maximum output power of the actuating mechanism as follows:

$$P_{d max}\ge P_{\mathrm{jmax}}/\eta$$

(23)

where $\eta$ denotes the efficiency of the actuating mechanism. Assuming $\eta =0.7$, the required motor power should satisfy $P_{d max}\ge {\displaystyle \frac{P_{\mathrm{jmax}}}{\eta }}\approx 1346\mathrm{W}$. Subsequently, the reduction ratio of the system is estimated to ensure that the actuating mechanism achieves a sufficient rated angular speed. The reduction ratio of the actuating mechanism should be designed as follows:

$$\left\{\begin{array}{c}n\times {\displaystyle \frac{1}{i_s}}\times {\displaystyle \frac{360}{60}}\ge 180\\ T\times i_s\times \eta \ge 300\end{array}\right.$$

(24)

When the motor’s no-load speed n is 8000 rpm and the rated torque T is $2.5\mathrm{N}·\mathrm{m}$, it can be concluded that $171\le i_s\le 267$.

In the design of the servo motor, it is essential to define its main structural dimensions within a diameter of $\Phi 80\mathrm{mm}$ to ensure compliance with the spatial layout requirements of the aircraft. Based on engineering experience, the parameters of the PMSM are finally determined as follows,

Table 3. Parameters of the PMSM.

Description	Parameter	Description	Parameter
Back EMF waveform	Sine wave	Rated phase current	11.11 A
Rotor type	Surface mount	Rated speed	8000 rpm
Rotor magnetic flux	0.1299 v.s	Overload torque	13.85 Nm
Stator phase resistance	0.0575 $\mathsf{\Omega }$	Overload speed	5900 rpm
Cross-axis inductance	0.268 mH	Overload current	61.57 A
Direct-axis inductance	0.268 mH	Pole pair number	4
Motor rotational inertia	0.000122 kg·${\mathrm{m}}^2$	Static friction	25 mNm
Peak coefficient of line back EMF	19.2304 Vpeak/krpm	Rated bus voltage	160 VDC
Torque coefficient	0.225 Nm/A	Rated line voltage	113.14 Vrms
Rated torque	2.5 Nm	Back EMF coefficient	0.0136 Vrms/rpm

:

4.2. Design of the Actuating Mechanism

4.2.1. Gear Parameter Design

The gear parameters of the actuating mechanism are listed in

Table 4. Actuator gear parameter design.

Parameter	Driven Wheel	Driving Wheel
Module	0.5	0.5
Number of teeth	59	24
Pitch circle diameter	33.5 mm	10.5 mm
Tooth width	5 mm	5 mm
Tooth angle	20°	20°

.

4.2.2. Design of the Ball Screw Pair

The ball screw pair of the actuating mechanism is designed with the following specifications:

• ball screw pair lead: 4 mm;
• ball screw pair nominal diameter: 12 mm;
• ball screw pair axial clearance: ≤0.01 mm;
• maximum axial load (bidirectional): 6000 N.

4.2.3. Verification of the Reduction Ratio

When selecting reducer stages and distributing reduction ratios, the following general principles should be adhered to: fewer transmission stages are preferred, and a higher reduction ratio closer to the load is more desirable. Considering spatial structural constraints, a two-stage reduction scheme is adopted. Gear reduction is utilized for both the first and second stages, with the second stage integrating a ball screw pair and rudder shaft rocker arm for additional reduction. Whenever feasible, extending the length of the rocker arm can enhance the final-stage reduction ratio, thereby improving the transmission stiffness. Figure 4 illustrates the operational diagram of the actuator.

Taking into account the circumferential thrust and radial load-bearing requirements of the ball screw pair, as well as space limitations, the initial rocker arm length is selected as 48 mm. Assuming the rudder deflection angle is denoted by $\delta$, the reduction ratio from the ball screw pair to the rudder shaft can be expressed as:

$$i_1={\displaystyle \frac{2\pi l}{R_y}}\left(1+{tg}^2\delta \right).$$

(25)

where $R_y$ is defined as the lead of the ball screw pair, l denotes the length of the rocker arm. When the rudder shaft is at the zero position, the reduction ratio is minimized. Conversely, when the rudder deflection angle is ${30}^{\circ }$, the reduction ratio is maximized. The reduction ratio range is given by:

$$75.36\le i_1\le 100.48$$

(26)

Considering the feasibility of the dimensional structure, the parameters of the two-stage gears are adjusted to achieve a reduction ratio from the motor to the screw, as expressed by:

$$i_2={\displaystyle \frac{Z_2}{Z_1}}={\displaystyle \frac{59}{24}}=2.466$$

(27)

where $Z_1$ and $Z_2$ are the tooth numbers of the second-stage reduction gears.

Therefore, the reduction ratio of the transmission mechanism is

$$185.84\le i_s=i_1\times i_2\le 247.78$$

(28)

To meet the design requirements, the reduction ratio of the transmission mechanism must satisfy the condition $171\le i_s\le 267$.

4.3. Prototype Testing and Verification

4.3.1. Experimental Platform Setup

The testing platform of the system is illustrated in Figure 5, comprising the actuator, drive system, PC controller, DC supply, load system, among others. After correctly connecting the hardware circuits of the servo system, the power supply is turned on to energize the system. This chapter relies on the hardware platform of the servo system and integrates existing experimental conditions to conduct thorough experimental validation of the designed control algorithms. Upon completion of the tests, the host computer will be used to read the data and obtain response curves, which will then be compared and analyzed.

4.3.2. Experimental Verification

(1)

Amplitude 30° step response

High dynamic response is a core performance metric for actuator systems in high-speed aircraft. To evaluate the performance differences between the traditional sliding mode controller (SMC) and the improved SMC, and to verify the effectiveness of the proposed enhancements, control parameters for both schemes were configured based on practical engineering experience. The parameter selection followed design guidelines aimed at limiting the maximum allowable overshoot to ≤5% and achieving the shortest possible settling time within a 1% steady-state error band under no-load conditions. The system responses to a 30° step input are depicted in Figure 6.

Experimental results indicate that the basic SMC achieves a settling time of 281 ms, whereas the improved SMC reduces this to 239 ms, corresponding to a 14.9% enhancement in transient convergence. This improvement is primarily attributed to the nonlinear geometric nonsingular terminal sliding mode control surface, which enables finite-time convergence. Additionally, the fuzzy-adaptive gain modulation mitigates excessive control input near the steady-state region, further optimizing performance. The reduced overshoot minimizes the risk of actuator saturation—an essential consideration for high-speed maneuvers within constrained aerodynamic envelopes. Furthermore, the convergence trajectory of the improved SMC exhibits reduced oscillatory behavior, indicating enhanced damping characteristics and overall system stability.

During the step response tracking experiment, once the system reached a stable tracking state, the performance differences between the two control algorithms became apparent through the torque fluctuation data (Figure 7). Although the baseline SMC achieved the tracking objective, it exhibited a torque fluctuation amplitude of up to 1.65 N·m, indicating pronounced control input oscillations during state transitions. In contrast, the improved SMC effectively suppressed high-frequency chattering, reducing the torque fluctuation amplitude to 0.24 N·m—an 85.5% reduction. Furthermore, the proposed fuzzy-adaptive approach dynamically modulates the control gain in real time, enabling smoother transitions and further narrowing the torque fluctuation band to within ±0.24 N·m. The embedded fuzzy logic functions as a dynamic filter, attenuating unnecessary torque variations without compromising tracking accuracy. This significant improvement notably reduces mechanical stress on the actuator, which is of considerable engineering value for prolonging equipment lifespan and enhancing overall system reliability. The experimental results demonstrate that the enhanced SMC not only preserves a high dynamic response but also markedly improves control quality, offering an innovative and effective solution for optimizing dynamic performance in high-precision servo systems.

(2)

Sinusoidal signal tracking

In the sinusoidal trajectory tracking experiment, both the baseline SMC and the improved SMC strategies were implemented on the actuator system. When subjected to a sinusoidal input with a one-second period and a 30° amplitude, the actuator response curves clearly highlighted the dynamic performance disparities between the two control approaches. As shown in Figure 8, the proposed improved controller achieved near-perfect phase synchronization with the reference signal, while the conventional SMC exhibited a noticeable phase lag and amplitude attenuation. Although the basic SMC maintained basic tracking functionality, its root mean square error (RMSE) reached 0.3424°, reflecting significant deviations in both phase and amplitude. In contrast, the improved SMC significantly enhanced the system’s dynamic tracking performance, maintaining higher fidelity to the reference trajectory and demonstrating superior responsiveness and control precision.

Experimental data show that the RMSE of the improved algorithm was reduced to 0.1881°, corresponding to a 45% decrease in tracking error and a significant enhancement in control accuracy. As illustrated in Figure 9, the phase lag inherent in the baseline SMC results in consistently elevated tracking errors across the entire trajectory cycle. The superior performance of the proposed controller stems from the integration of an exponential nonlinear reaching law and real-time gain adaptation. These design features enable more aggressive corrective actions in high-error regions while ensuring smoother control near zero-crossings—precisely where chattering typically becomes prominent. This balance contributes to both improved tracking fidelity and reduced mechanical stress on the actuator.

Torque fluctuation data provide direct evidence of the performance disparity between the two control algorithms. As shown in Figure 10, the control torque profile under the baseline SMC resembles a square-wave pattern, characterized by abrupt directional changes, particularly near the zero-crossing points of the trajectory. This behavior is a classical manifestation of chattering. In contrast, the improved SMC produces a smooth torque profile that closely follows the sinusoidal reference, effectively minimizing high-frequency excitation of the mechanical system. This smoother control behavior translates into reduced energy dissipation, lower acoustic noise, and prolonged actuator lifespan—factors that are especially critical in airborne applications where reliability and durability are paramount.

(3)

The ability to resist disturbances caused by changes in system parameters

To evaluate the adaptability of each control strategy to variations in system parameters, practical degradation factors such as elevated temperatures from prolonged motor operation and mechanical wear over extended use were considered. In this study, a 30% increase in both motor and stator resistance was assumed, along with a reduction in motor inductance of up to 20%, and a 100% increase in the system’s moment of inertia—conditions representative of thermal effects, aging, and load changes.

The sine wave tracking error curves presented in Figure 11 clearly illustrate the impact of these parameter perturbations on control performance. Under nominal conditions, the baseline sliding mode controller (SMC) achieved a root mean square error (RMSE) of 0.3424°. However, following the introduction of parameter variations, the RMSE increased to 0.4748°, reflecting a 38.7% degradation in tracking accuracy. In contrast, the improved SMC exhibited remarkable robustness, with the RMSE increasing only marginally from 0.1881° to 0.1917°, corresponding to a mere 2% rise—representing a 94.8% reduction in error growth compared to the baseline controller.

Figure 11 further evaluates the robustness of both control schemes under parameter uncertainties simulating aging, thermal drift, and manufacturing variability. The significant RMSE increase observed in the baseline controller underscores its reliance on fixed gain parameters, which become suboptimal under altered dynamics. Conversely, the improved SMC leverages a fuzzy-adaptive mechanism that dynamically tunes control gains in response to evolving error characteristics, thereby compensating for plant degradation in real time. This robustness is particularly critical for long-duration missions and applications where hardware performance may deteriorate over time.

5. Conclusions

In this study, a novel fuzzy-adaptive global nonsingular terminal sliding mode control strategy was proposed and successfully applied to the control of a high-speed aircraft actuator system. Comprehensive experimental validation has demonstrated the remarkable advantages of the proposed method over conventional sliding mode control (SMC) approaches. The main achievements are summarized as follows:

• Improved dynamic response: In a 30° step response test, the system’s adjustment time was reduced from 281 ms to 239 ms, achieving a 14.9% improvement in transient convergence speed, which significantly enhances trajectory responsiveness. Electromagnetic torque fluctuation amplitude was reduced from 1.65 N·m to 0.24 N·m, achieving an 85.5% reduction in torque ripple, which effectively lowers mechanical stress and improves actuator longevity.
• Enhanced tracking accuracy: Under a sinusoidal input with an amplitude of 30°, the root mean square error (RMSE) decreased from 0.3424° to 0.1881°, reflecting a 45.0% reduction in steady-state error, and demonstrating superior tracking precision.
• Robustness against parameter uncertainties: When system parameters such as resistance, inductance, and inertia were perturbed (up to 100%), the RMSE increased by only 2% with the proposed method, compared to 38.7% with traditional SMC—equating to a 94.8% reduction in error growth rate, validating the controller’s high robustness.

Moving forward, several avenues should be considered for further exploration and expansion in this research:

• Engineering Application Verification: Apply the proposed nonlinear dynamic control method to an actual high-speed aircraft system for system validation, thoroughly assessing the method’s effectiveness and feasibility in practical engineering scenarios.
• System Parameter Adaptation: Integrate the sliding mode observer to achieve precise estimation of the system state. Introduce an adaptive control strategy to enhance the system’s adaptability to various flight requirements under different working conditions, thereby improving overall system robustness.
• Multi-objective Optimization: Consider the optimization of multiple performance indicators, such as dynamic response speed and robustness. Implement a comprehensive approach in the actuator design and control methods, aiming to achieve more balanced and improved overall system performance.

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