Currently, in the aerospace field, large spacecraft, such as large Earth observation satellites, spacecraft, and space stations, all require large control torques to achieve attitude control. Control torque gyroscopes (MTGs) offer large output torque, stable torque, fast dynamic response, good control linearity, and high efficiency, leading to their rapid development. Based on the frame, MTGs can be divided into single-frame and double-frame MTGs. A single-frame MTG consists of a momentum flywheel or an inner and outer frame supporting the flywheel, rotating at a constant speed. The rotation of the outer frame forces the angular momentum of the momentum flywheel to change direction, thereby adjusting the spacecraft's attitude. Single-frame MTGs have advantages such as simple structure and high control accuracy, making them a commonly used structure in MTG servo systems. However, because these servo systems typically employ traditional PID control methods, they suffer from poor robustness and low accuracy, often losing stability under significant external disturbances. As a highly robust sliding mode variable structure control method, it has the advantage of invariance to external disturbances and parameter perturbations when the matching condition is met. Therefore, it is often used in AC servo control systems [1-2]. However, sliding mode control has a fatal drawback, namely the existence of chattering, which will seriously reduce the control accuracy and stability of the system. Based on this, this paper introduces fuzzy logic into the traditional sliding mode control idea and designs an adaptive fuzzy sliding mode control strategy. The stability of the adaptive sliding mode controller is proved by using the Lyapunov stability theorem, and the simulation model is constructed using actual servo system parameters for experimental verification. [font=黑体] [color=black]SVPWM Control Principle of Control Torque Gyroscope Outer Frame System [/color][/font] When the spacecraft is in steady state, the control torque gyroscope is required to output high-precision and high-stability attitude control torque to ensure attitude control accuracy. For this reason, the outer frame control system is required to have high stability and excellent low-speed performance. The system uses a permanent magnet three-phase brushless DC synchronous motor as the actuator, with two drive schemes: multi-stage transmission and direct transmission. The sine wave driven permanent magnet synchronous motor system offers advantages such as smooth operation, low torque fluctuation, higher dynamic response, and low noise. This paper studies a control torque gyroscope frame control system that uses a sine wave to directly drive the permanent magnet synchronous motor, achieving frame speed control of the single-frame control torque gyroscope to obtain highly stable gyro reaction torque. To improve control system performance and enable the permanent magnet synchronous motor to rotate smoothly at low speeds, thereby outputting high-precision and highly stable attitude control torque, a three-closed-loop control system consisting of a position loop, a speed loop, and a current loop is adopted. The speed mode is implemented through the position loop, which receives speed commands from the controller algorithm and sends them to the speed loop. The speed loop compares the speed command with the actual motor speed fed back; the difference is used by a speed regulator to ensure the actual motor speed matches the command value, eliminating the influence of load torque disturbances and other factors on the stability of the motor speed. The current loop utilizes conventional PID control, comparing the given current with the instantaneous value of the motor current feedback for current setting, obtaining the PWM signal. This signal is then used to control the switching on of power devices via space vector control (SVPWM), providing the motor drive signal. Employing the widely used space vector control, the ultimate goal of motor control is to generate a circular rotating magnetic field, thereby producing constant electromagnetic torque. Based on this objective, the PWM voltage is controlled by tracking the circular rotating magnetic field; this control method is called flux tracking control, which is obtained by adding voltage space vectors, also known as voltage space vector (SVPWM). Space vector pulse width modulation (SPWM) aims to achieve the overall effect of the three-phase output voltage, making the output current waveform as close as possible to an ideal sine wave. A typical three-phase power inverter with load can consist of six MOSFETs, controlled by six trigger signals for switching on and off. When the upper thyristor is turned on, the corresponding lower thyristor is turned off. ua, ub, and uc are the output phase voltages of the inverter. By introducing the Park transformation, the three voltage scalars (three-dimensional) of the three-phase outputs a, b, and c can be transformed into one vector (two-dimensional). [img=159,37]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf41.jpg[/img](1) The generated six PWM pulse signals are input to the six IGBTs of the three-phase bridge arm, so that the switching states of the upper and lower IGBTs of the same bridge arm are opposite, and different conduction modes of the three-phase bridge arm result in different voltage vectors us. According to the voltage space vector defined by equation (1), the inverter can output eight voltage space vectors distributed in space as shown in Figure 1, which correspond to the eight switching modes of the inverter. Figure 1 shows the basic voltage space vector. In the eight switching modes, the inverter output voltage space vector corresponding to states (000) and (111) is zero, called the zero vector, while the other six vectors are called effective vectors. The length of each effective vector is 2ud/3. The symmetrical three-phase sinusoidal variables are synthesized according to equation (1) to obtain a uniformly rotating space vector with a fixed amplitude, and the magnitude of the vector is equal to the peak value of the sinusoidal quantity of each phase. Since the actual vectors that the inverter can generate are limited, it is impossible to output a space vector with continuously changing angles. In order to obtain the rotating voltage space vector, the difference in the action time of each vector can be used to equivalently synthesize the required vector. Taking the third sector as an example, the reference vector vref is synthesized by the two nearest adjacent effective vectors v4, v6 and the zero vector. The equivalent vectors are combined according to the volt-second balance principle, that is: [align=center] v4t4+v6t6=vreft （2） t=t4+t6+t0 （3）[/align][align=left] where t is the PWM period of the system, and t4, t6 and t0 are the action times of u4, u6 and the zero vector, respectively. Simultaneously, the synthesized reference voltage vector vref is decomposed along the α and β axes, i.e.: vref＝uα＋juβ （4） Substituting the expressions for v4 and v6 into equation (2), and comparing equations (2) and (4) above, the real and imaginary parts of the two expressions for vref are made equal, and the required action times of each vector for synthesizing vref are as follows: （5） （6） t0=t-t4-t6 （7） The zero vector's action time t0 only supplements the time outside of t4 and t6, and has no effect on vector synthesis, but t0 > 0 must be satisfied. This ensures that the output waveform is distortion-free and also determines the maximum limit of SVPWM. Similarly, the action time of each vector in other sectors can be calculated. [font=黑体] Fuzzy Sliding Mode Variable Structure Control for External Frame Servo System Sliding Mode Controller Structure Design[/font] The current, speed, and position loops of the position servo system are designed separately. The design principle is: design the inner loop first, then the outer loop; the bandwidth of the inner loop should be higher than that of the outer loop, that is, the frequency response of the inner loop is faster than that of the outer loop; when designing the outer loop, the inner loop is usually equivalent to a first-order inertial element. The current loop and speed loop of the inner loop adopt classic PID control. The position loop adopts the sliding mode variable structure control method. According to the transfer function form of the control system, the output mechanical angular velocity ωn and the output mechanical angle θ of the system are taken as the system state variables. The state space form of the control system can be written as: [img=149,40]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf44.jpg[/img]（8） For the position input signal, the position error is shown in the image [img=58,18]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf45.jpg[/img]. Taking the sliding surface [img=59,17]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf46.jpg[/img], then: [img=182,22]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf47.jpg[/img] From equation (8), we have [img=93,32]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf48.jpg[/img] Substituting this into the above equation, we get [img=121,34]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf49.jpg[/img] Let [img=28,15]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf50.jpg[/img], therefore [img=75,34]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf51.jpg[/img], Then [img=211,33]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf52.jpg[/img]（9） Let [img=76,21]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf53.jpg[/img], where [img=31,17]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf54.jpg[/img], then it is obvious that the sliding surface condition is satisfied, and the design is reasonable and correct. The input of the system after sliding mode control is [img=158,31]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf55.jpg[/img] (10). η should be large enough to compensate for the uncertainty of the system and meet the arrival conditions. However, if it is too large, it will easily cause the controller to saturate and the frequent switching of the control action to damage the actuator. Introducing the concept of boundary layer into variable structure control and replacing the sign function with a saturation function only alleviates the discontinuity of structure switching. As a result, a high gain is generated near the switching surface, which is also accompanied by lag. Therefore, chattering still exists, and the control accuracy of the entire system increases due to the introduction of the boundary layer. How to ensure good transient response and controller robustness in the approaching stage and reduce chattering is the key point to be solved by fuzzy variable structure control method. Here, we can assume that the chattering of the system on the sliding plane during the transition phase is a function of the sliding variables, which is actually obvious. We can further define the following performance index: [align=center] e[img=59,31]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf76.jpg[/img]（11）[/align][align=left] Because chattering can be reflected in the distance between the sliding function and the sliding plane (s=0), and studying the value of this distance at a certain moment is not very meaningful, the author defines this energy index e as the integral of the distance between the sliding function from the initial position to the sliding plane. It indirectly represents the energy accumulation required to reach the sliding plane from the initial position. Considering the relationship between energy and control gain (here referring to the transition process, because the energy function remains unchanged after reaching the sliding plane), we now consider the problem of finding this optimal gain function. The gain function η(s) in this sliding control has uncertainty, and it changes according to the sliding function and its rate of change. Therefore, the approximation capability of fuzzy logic systems can be used for approximation. [font=黑体]Switching control gain acquisition[/font] The design of a fuzzy system mainly includes four parts: fuzzification of input variables, establishment of fuzzy rule base, fuzzy inference mechanism, and defuzzification of output variables. Wang proved that a fuzzy system can approximate any given nonlinear function to any given precision on a closed set [4]. Here, the construction method of fuzzy system given in reference [5] is used to design a fuzzy estimator. Therefore, the fuzzy inference form of the gain function η(s) is as follows: [align=center] if s is [img=22,22]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf78.jpg[/img] and [img=12,18]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf79.jpg[/img] is [img=23,23]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf80.jpg[/img], then [img=28,20]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf81.jpg[/img] is [img=25,20]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf82.jpg[/img] （12）[/align][align=left] Where the system state x=（s,[img=12,18]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf79.jpg[/img]）t is the input of the fuzzy estimator,[img=21,23]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf57.jpg[/img] is the fuzzy set defined on the input space; the output of the fuzzy estimator is the unknown function to be estimated; [img=15,20]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf58.jpg[/img] represents the coefficients of the estimation function defined on the output space. Therefore, by using singleton fuzzification, product inference rules, and center average for defuzzification, the estimated expression of the unknown term η(s) can be obtained as follows: [/align][img=179,41]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf59.jpg[/img](13) Further defining the fuzzy basis function to simplify an approximation of η(s), its expression is as follows: [img=182,44]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf61.jpg[/img](14) Then the estimated function of the unknown term η(s) can be further expressed as: [img=124,20]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf62.jpg[/img]（15） Here are, [img=103,27]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf63.jpg[/img], [img=112,28]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf64.jpg[/img]. The optimal estimated parameters are defined as follows: [img=201,34]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf65.jpg[/img]（16） Therefore, the error of the estimated parameters can be expressed as: [img=69,23]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf66.jpg[/img]（17） In this way, we can obtain the fuzzy control expression of the gain function, and further obtain the switching control quantity in equation (10), [img=130,34]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf67.jpg[/img]（18） Here, [img=29,15]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf68.jpg[/img] is a self-designed, arbitrarily small constant, which can be chosen to be small enough that the chattering effect on the sliding surface can be ignored. Theorem 1: For the servo system model shown in equation (8), using the control quantity shown in equation (10), and simultaneously changing the switching control to the form shown in (18), if the gain of the switching control adopts the adaptive rate shown below, [img=76,22]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf69.jpg[/img] (19), then the entire system is stable. Proof: Take the Lyapunov energy function as: [img=130,35]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf70.jpg[/img]（20） Here [img=14,17]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf71.jpg[/img] is a constant greater than zero, and [img=17,23]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf72.jpg[/img] is the optimal value of the gain coefficient estimate. [img=331,207]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf73.jpg[/img]（21） Substituting equation (19) into the equation, we can obtain, [img=62,19]http://www.ca800.com/uploadfile/maga/servo2007-2/___wmf83.jpg[/img]（22） Therefore, the entire sliding mode function is stable, and according to the properties of sliding mode control, the state error of the system converges in finite time. [font=黑体][color=black][b]Simulation and Experimental Results of Servo Control System[/b][/color][/font] A simulation model was built in the Simulink environment. The inner current loop and speed loop adopted PID controllers, and the outer position loop adopted an adaptive sliding mode controller. The entire control torque gyroscope frame system requires the permanent magnet synchronous motor to rotate smoothly at low speeds, thereby outputting high-precision and high-stability attitude control torque. By selecting appropriate inner-loop PID control parameters and outer-loop SMC control parameters, the permanent magnet synchronous motor achieves stable speed and fast response at low speeds. The permanent magnet synchronous motor parameters used in the system are: rated voltage: 48VDC; rated output torque: 10Nm; maximum output peak torque: 15Nm; motor phase resistance: 8.61Ω; phase inductance: 25MH. The simulated servo system model is constructed according to the above parameters. Figures 2(a) and (b) show the system error phase plane simulation curves when the position loop of the control system adopts fuzzy adaptive sliding mode control and ordinary sliding mode variable structure control, respectively. It can be seen from the phase plane diagram that when the sliding mode variable structure control parameters are adjusted using fuzzy rules, the chattering of the sliding mode variable structure control is effectively reduced. Figure 3 shows the system speed response simulation curve when the position loop of the control system adopts fuzzy adaptive sliding mode control, which has good dynamic performance. Figure 2(a) Phase plane curve of fuzzy adaptive sliding mode variable structure control Figure 2(b) Phase plane curve of ordinary sliding mode variable structure control Figure 3 Simulation speed curve of control system [font=黑体][color=black][b]Conclusion[/b][/color][/font] This paper proposes an adaptive sliding mode controller based on fuzzy method for a single-frame controlled torque gyroscope servo system. The control law of the controller is derived based on the Lyapunov stability requirement. Further verification is performed using a parameter-based model of an actual servo system. The results show that the adaptive sliding mode controller effectively reduces the inherent chattering phenomenon of the sliding mode controller while meeting the control requirements of the servo system.