CETDC superhelix nonsingular terminal sliding mode control method based on disturbance estimation

Abstract

The invention is suitable for the technical field of power electronic converter control, and provides a CETDC superhelix nonsingular terminal sliding mode control method based on disturbance estimation, and the method comprises the steps: building a dynamic mathematical model of CETDC, constructing an extended sliding mode observer, and estimating the lumped disturbance in real time, and a superhelix nonsingular terminal sliding mode controller is designed to be combined with a fast superhelix algorithm reaching law, so that the dynamic response speed and robustness of the system are remarkably improved. Compared with traditional PI control, the method has the advantages that capacitor voltage fluctuation and current buffeting are reduced, high-voltage side power and low-voltage side power are rapidly stabilized, and the method is suitable for a high-voltage large-capacity direct-current power grid scene.

Description

Technical Field

[0001] The present invention belongs to the technical field of power electronic converter control, and in particular relates to a CETDC super-helical non-singular terminal sliding mode control method based on disturbance estimation. Background Art

[0002] With the rapid development of power electronics technology, high-voltage, high-power power electronics have been widely used in fields such as motor drives, reactive power compensation, and distributed generation. In recent years, a large number of high-voltage, high-capacity DC / DC converter topologies based on modular multilevel converters (MMCs) have emerged, significantly promoting the development of flexible direct current transmission (HVDC). Against this backdrop, researchers have proposed a capacitive energy transfer DC / DC converter (CETDC) that combines a series IGBT valve block with a sub-module bridge arm. This converter has the advantages of not requiring a transformer or filter inductor, requiring no AC voltage injection in the bridge arm, and requiring a small number of components. It can meet the low-cost, high-efficiency, and lightweight requirements of high-voltage, high-capacity DC / DC converters in multi-voltage DC grids, and has promising engineering application prospects.

[0003] At present, PI control has become the mainstream control method of CETDC control system due to its advantages such as simple algorithm and easy implementation. However, in actual operation, uncertainties such as slow dynamic response, parameter changes and external disturbances will affect the control performance of the system.

[0004] SMC is a nonlinear control method whose main principle is to change the system output according to the current state during the dynamic process, forcing the system state to move along a predetermined "sliding mode" state trajectory. Due to its robustness to parameter uncertainty and external disturbances, SMC has been widely used in motor drives and power electronic power converters. Therefore, to address the problems of slow dynamic response of bridge arm currents, large capacitor voltage fluctuations, and poor robustness in PI control for CETDC, a super-helical non-singular terminal sliding mode control method for CETDC based on disturbance estimation is proposed. Summary of the Invention

[0005] The purpose of the embodiments of the present invention is to provide a CETDC super-helical non-singular terminal sliding mode control method based on disturbance estimation, aiming to solve the problems raised in the above background technology.

[0006] The embodiment of the present invention is implemented as follows: a CETDC super-helical non-singular terminal sliding mode control method based on disturbance estimation includes the following steps: Step 1: Establish a dynamic mathematical model based on the topology and operation principle of the CETDC converter; Step 2: Based on the mathematical model, an extended sliding mode observer (ESMO) is constructed to estimate the lumped disturbance; Step 3: Based on the mathematical model, a super-helical non-singular terminal sliding mode controller (STNTSMC) is constructed to improve the system control performance.

[0007] Further technical solution, said step 1 comprises the following specific steps: CETDC consists of three identical phases, each of which includes an energy storage bridge arm and two sets of converter valves. The converter valves are composed of IGBT devices connected in series. The energy storage bridge arm is composed of submodules connected in series, and a buffer inductor is connected in series. When the converter valve 1 is turned on and the converter valve 2 is turned off, the energy storage bridge arm is U H For side charging, the state equation is: (1) When the converter valve 1 is closed and the converter valve 2 is open, the energy storage bridge arm is U L Side discharge, the state equation is: (2); in, U ca is the submodule capacitor voltage, n Invest in the number of submodules, N is the total number of submodules, L For the bridge arm buffer inductor, R is the parasitic resistance of the buffer inductor, i pa Indicates the A-phase bridge arm current.

[0008] Combining the above formulas (1) and (2), we get the system dynamic mathematical model: (3); in, S 1. S 2 are the switch states of converter valve 1 and converter valve 2 respectively. , and their values ​​are complementary; Indicates a symbolic function. In the formula, the symbol i pa The size of the change; the number of n As the control input of the controller.

[0009] Further technical solution, said step 2 includes the following specific steps: Step 2.1: Construct an extended sliding mode observer to expand the lumped disturbance into state variables for estimation, which serves as feedback compensation for the system. The bridge arm current state equation in formula (3) is modified to: (4); in, and Indicates parameter changes, represents external disturbance or system uncertainty; assuming and are all bounded, formula (4) can be transformed into the following form: (5); in, F is the lumped disturbance, expressed as: (6); because and are all bounded, so F It is also bounded; The system design for Equation (5) is as follows: (7); in, Represents the estimated value of the bridge arm current; represents the lumped disturbance estimate; u smo is the sliding mode control law to be designed; l is the observer gain, and ; The design bridge arm current observation error is , the concentrated perturbation observation error is , thus the error equation is: (8); Select the integral sliding surface as: (9); in, c is the integral gain, and ; The exponential reaching law is selected as: (10); in, k 1 is the switching gain to be designed in the reaching law, is the coefficient of the exponential term to be designed, and 、 ; In order to make and The sliding mode surface converges to 0 in a finite time, and the sliding mode control law is designed as follows: (11); Because in the control process is used as feedback compensation to the system, so ;Will u smo Substituting into formula (7) we can get the expression of ESMO; Step 2.2: Prove the stability of the observer by Lyapunov function to ensure the bridge arm current observation error and concentrated perturbation observation error Converges in finite time.

[0010] Further technical solution, said step 2.2 includes the following specific steps: The Lyapunov function is selected as follows: (12); According to formula (8), V The derivative of 1 is expressed as: ; ; ; ; ; ; ; ; (13); if When ,parameter Should satisfy ;if When ,parameter Should satisfy To ensure ,parameter Should meet: (14); According to the principle of sliding mode control, we know that in order to make the system state reach the sliding surface within a finite time, ; The error equation (8) is simplified to: (15); Solved , where is an arbitrary constant, when the observer gain , which can ensure that the system error converges within a finite time.

[0011] Further technical solution, said step 3 includes the following specific steps: Step 3.1: Construct a superhelical non-singular terminal sliding mode controller; The bridge arm current state equation after ESMO compensation described in step 2 is: (16); definition , , select the non-singular terminal sliding mode surface as: (17); In the formula , p 、 q is a positive odd number, and ; Then taking its derivative we can get: ; (18); Using the improved fast supercoil algorithm, the sign function is replaced by the hyperbolic tangent function Instead, the reaching law of the fast supercoil algorithm is obtained as: (19); Among them, the control parameters and are all greater than zero; Combining Equations (18) and (19), we can obtain the STNTSMC control law of the bridge arm current control loop: ; (20); Step 3.2: Prove the finite-time convergence of the controller. From Equation (17), we can obtain the sufficient condition for the existence of a non-singular terminal sliding surface: (twenty one); Where, k is a positive constant; therefore, the STNTSMC control law designed by formula (20) satisfies this condition, For a limited time, the following conditions will be met The non-singular terminal sliding surface is reached: (twenty two); When the non-singular terminal sliding surface hour, The system dynamics is determined by the following differential equations: (twenty three); from arrive Limited time required It is expressed as follows: (twenty four); Therefore, the non-singular terminal sliding surface makes the tracking error and its derivatives In a limited time Converges to zero internally; Step 3.3: Prove controller stability. In order to verify the stability of the STNTSMC control law, the Lyapunov function is selected V 2: (25); right V 2 Taking the derivative we get: ; ; ; ; (26); if When , the parameters need to be met and are greater than zero; if When , also need to meet the parameters and are all greater than zero; therefore, according to the Lyapunov stability theorem, by selecting the control parameters, the system can be made to converge to the sliding surface, indicating that the constructed controller is asymptotically stable.

[0012] An embodiment of the present invention provides a CETDC super-helical non-singular terminal sliding mode control method based on disturbance estimation. This method establishes a dynamic mathematical model of the CETDC, constructs an extended sliding mode observer for real-time estimation of lumped disturbances, and designs a super-helical non-singular terminal sliding mode controller in conjunction with a fast super-helical algorithm reaching law. This significantly improves the system's dynamic response speed and robustness. Compared to traditional PI control, this method reduces capacitor voltage fluctuations and current jitter, achieving rapid stabilization of high- and low-voltage side power, making it suitable for high-voltage, high-capacity DC grid scenarios. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] Figure 1 It is a CETDC topology using a series IGBT valve group; Figure 2 This is a schematic diagram of the CETDC converter valve state switching; Figure 3 This is the control block diagram of the bridge arm current loop ESMO+STNTSMC; Figure 4 is the three-phase bridge arm current waveform under PI control; Figure 5 This is the transient process of the bridge arm current under PI control; Figure 6 is the three-phase bridge arm current waveform under ESMO+STNTSMC control; Figure 7 The transient process of the lower arm current is controlled by ESMO+STNTSMC; Figure 8 is the three-phase bridge arm capacitor voltage waveform under PI control; Figure 9 The three-phase bridge arm capacitor voltage waveform under ESMO+STNTSMC control; Figure 10 The low-voltage side current waveform under PI control and ESMO+STNTSMC control; Figure 11 is the power waveform under PI control; Figure 12 This is the power waveform under ESMO+STNTSMC control. DETAILED DESCRIPTION

[0014] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0015] The specific implementation of the present invention is described in detail below with reference to specific embodiments.

[0016] An embodiment of the present invention provides a CETDC super-helical non-singular terminal sliding mode control method based on disturbance estimation, comprising the following steps: Step 1: Establish a dynamic mathematical model based on the topology and operation principle of the CETDC converter; The topology diagram of CETDC is as follows: Figure 1 As shown in the figure, it consists of three identical circuits, each phase including a storage bridge arm and two sets of converter valves. The converter valves are composed of a certain number of semiconductor switching devices connected in series, here using IGBTs; the storage bridge arm is composed of a certain number of sub-modules connected in series, and a small buffer inductor is connected in series. The sub-modules basically adopt a half-bridge or full-bridge structure. The CETDC equivalent circuit is shown in the figure. Figure 2 As shown in the figure, when the converter valve 1 is turned on and the converter valve 2 is turned off, the energy storage bridge arm is U H When the converter valve 1 is closed and the converter valve 2 is open, the energy storage bridge arm is U LSince the commutation switch has the ability to actively cut off the current, the energy storage bridge arm passively switches between the charging and discharging states, realizing the transfer of capacitive energy.

[0017] According to the above topological operation principle and equivalent circuit diagram, its dynamic mathematical model is derived. Because the three phases operate independently, taking phase A as an example (the following observer and controller design process all take phase A as an example, phases B and C are the same, but the phase difference is 120 degrees), when the converter valve 1 is turned on and the converter valve 2 is turned off, the energy storage bridge arm is U H Side charging. Figure 2 The equivalent circuit diagram on the left shows the state equation of the system when charging: (1); in, U ca is the submodule capacitor voltage, n Invest in the number of submodules, N is the total number of submodules, L For the bridge arm buffer inductor, R is the parasitic resistance of the buffer inductor, i pa Indicates the A-phase bridge arm current.

[0018] When the converter valve 1 is closed and the converter valve 2 is open, the energy storage bridge arm is U L Side discharge. Figure 2 The equivalent circuit diagram on the right shows the state equation of the system when discharging: (2); Combining the above formulas (1) and (2) we can get the system dynamic mathematical model: (3); in, S 1. S 2 are the switch states of the converter valves 1 and 2 respectively. , and their values ​​are complementary; Indicates a symbolic function. In the formula, the symbol i pa The size of the change; the number of n As the control input of the controller.

[0019] Step 2: Construct an extended sliding mode observer based on the mathematical model to estimate the lumped disturbance; Step 2.1: Constructing the Extended Sliding Mode Observer In the process of establishing the above mathematical model, there will be unknown disturbances such as parameter changes, which will have a great impact on the control performance of the system. Therefore, this method first establishes ESMO and expands the lumped disturbance into state variables for estimation as feedback compensation of the system.

[0020] The bridge arm current state equation in formula (3) is modified to: (4); in, and Indicates parameter changes, represents external disturbance or system uncertainty. Assume and are all bounded, formula (4) can be transformed into the following form: (5); in, F is the lumped disturbance, which can be expressed as: (6); because and are all bounded, so F It is also bounded.

[0021] For the system represented by formula (5), the following ESMO can be designed: (7); in, Represents the estimated value of the bridge arm current; represents the lumped disturbance estimate; u smo is the sliding mode control law to be designed; l is the observer gain, and The design bridge arm current observation error is , the concentrated perturbation observation error is , from which we can get the error equation: (8); Select the integral sliding surface as: (9); in, c is the integral gain, and .

[0022] The exponential reaching law is selected as: (10); in, k 1 is the switching gain to be designed in the reaching law, is the coefficient of the exponential term to be designed, and 、 .

[0023] In order to make and The sliding mode surface converges to 0 in a finite time, and the sliding mode control law is designed as follows: (11); Because in the control process is used as feedback compensation to the system, so .Will u smo Substituting into formula (7) we can get the expression of ESMO.

[0024] Step 2.2: Prove observer stability Theorem 1: ESMO chooses appropriate parameters and , making and , which can make the observer's state converge to zero in a finite time.

[0025] Proof: The Lyapunov function is selected as follows: (12); According to formula (8), V The derivative of 1 can be expressed as: ; ; ; ; ; ; ; ; (13); if When ,parameter Should satisfy ;if When ,parameter Should satisfy Therefore, to ensure ,parameter Should meet: (14); It can be seen that choosing the appropriate exponential gain , which can make the observer asymptotically stable. According to the principle of sliding mode control, we know that in order to make the system state reach the sliding surface in a finite time, The error equation (8) can be simplified as: (15); Solved , where is an arbitrary constant, when the observer gain , which can ensure that the system error converges within a finite time.

[0026] Step 3: Based on the mathematical model, a superhelical non-singular terminal sliding mode controller is constructed to improve the system control performance.

[0027] Step 3.1: Construct a superhelical non-singular terminal sliding mode controller; To address the slow response speed and poor robustness of the traditional PI control in the CETDC bridge arm current control loop, this method designs a super-helical non-singular terminal sliding mode controller. The bridge arm current state equation after ESMO compensation described in step 2 is: (16); definition , , select the non-singular terminal sliding mode surface as: (17); In the formula is a positive odd number, and . Then taking its derivative we can get: ; (18); In order to solve the problem of chattering caused by the reaching law of traditional sliding mode control, this invention uses an improved fast super spiral algorithm (FSTA). In order to further reduce chattering, the sign function is replaced by the hyperbolic tangent function. Instead, the reaching law of the fast supercoil algorithm is obtained as: (19); Among them, the control parameters and are greater than zero. Compared with the standard STA approach law, FSTA has more and The two terms act on proportional and integral control respectively. Therefore, FSTA can be regarded as a combination of STA and PI control, which is beneficial to improving the control performance of the system.

[0028] Combining Equations (18) and (19), we can obtain the STNTSMC control law of the bridge arm current control loop: ; (20); The control block diagram of the super-helical non-singular terminal sliding mode control system based on disturbance estimation proposed by this method is as follows: Figure 3 The control strategy is mainly applied to the bridge arm current control loop. The bridge arm current reference value is jointly generated by the direct power control of the previous loop and the energy balance control of the energy storage bridge arm. The non-singular terminal sliding surface is selected based on the error between the actual bridge arm current value and the reference value to generate the FSTA reaching law. The integral sliding surface is selected based on the error between the actual bridge arm current value and the estimated value. The feedback estimate is obtained through ESMO, and finally the STNTSMC control law is obtained.

[0029] Step 3.2: Prove the finite-time convergence of the controller. From formula (17), we can get the sufficient condition for the existence of non-singular terminal sliding surface: (twenty one); Where, k is a positive constant. Therefore, the STNTSMC control law designed by formula (20) satisfies this condition. For a limited time, the following conditions will be met The non-singular terminal sliding surface is reached: (twenty two); When the non-singular terminal sliding surface hour, The system dynamics can be determined by the following differential equations: (twenty three); from arrive Limited time required It can be expressed as follows: (twenty four); Therefore, the non-singular terminal sliding surface can make the tracking error and its derivatives In a limited time Converges to zero internally.

[0030] Step 3.3: Prove controller stability. In order to verify the stability of the STNTSMC control law, the Lyapunov function is selected V 2: (25); rightV 2 Taking the derivative we get: ; ; ; ; (26); if When , the parameters need to be met and are greater than zero; if When , also need to meet the parameters and Therefore, the Lyapunov stability theorem shows that by selecting appropriate control parameters, the system can converge to the sliding surface, indicating that the constructed controller is asymptotically stable.

[0031] In order to demonstrate the superiority of the performance of this method, a simulation experiment is conducted to compare it with PI control: Table 1 shows the nominal parameters used in the CETDC simulation.

[0032] Table 1 CETDC simulation parameters ; First, the three-phase bridge arm current waveforms under the PI control and the ESMO+STNTSMC control strategy proposed in this invention are simulated and compared. Figure 4-Figure 7 As shown. Figure 4-Figure 7 It can be seen that the speed at which the high and low voltage side bridge arm currents reach steady state under the ESMO+STNTSMC method is significantly faster than that under the PI control. Figure 5 and Figure 7 As can be seen from the results, during transient conditions, the ESMO+STNTSMC method achieves significantly smaller fluctuations in the bridge arm current on the high and low voltage sides than the PI control method, and the waveform quality is better than that of the PI control method. Clearly, the proposed method has better steady-state and transient performance for the bridge arm current than the PI control method.

[0033] Figure 8 and Figure 9 The figure shows a simulation comparison of the three-phase bridge arm capacitor voltage. As can be seen, the amplitude of the three-phase capacitor voltage fluctuations under the ESMO+STNTSMC method during transient conditions is generally smaller than that under PI control (especially for phases B and C), and the three-phase waveforms stabilize more quickly. Therefore, this method reduces the fluctuations in the bridge arm capacitor voltage.

[0034] Figure 10The figure shows a simulation comparison of the low-voltage side current. As can be seen from the figure, the overshoot of the low-voltage side current under the ESMO+STNTSMC method during transient conditions is significantly smaller than that under PI control. The maximum current under PI control exceeds 1200A, while the maximum current under the ESMO+STNTSMC method is only 1140A. Furthermore, the current stabilizes faster under the ESMO+STNTSMC method than under PI control, and the jitter amplitude of the stable current is also smaller. Therefore, this method accelerates the response speed of the low-voltage side current and has better steady-state and transient performance.

[0035] Finally, the high and low voltage side powers are simulated and compared, such as Figure 11 and 12 As shown in the figure, the ESMO+STNTSMC method reaches a stable power value significantly faster (approximately 0.007s) than the PI control method (approximately 0.045s). Furthermore, during transient conditions, the ESMO+STNTSMC method also produces significantly less power overshoot than the PI control method. Therefore, this method accelerates the power response on the high and low voltage sides of the system, resulting in better steady-state and transient performance.

[0036] In summary, the ESMO+STNTSMC control strategy proposed in the present invention accelerates the dynamic response speed of CETDC, reduces the fluctuation of capacitor voltage, and improves the estimation accuracy of disturbances and the robustness of the system, thereby achieving more rational control performance.

[0037] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A CETDC super-helical non-singular terminal sliding mode control method based on disturbance estimation, characterized in that: The following steps are involved: Step 1: Establish a dynamic mathematical model based on the topology and operation principle of the CETDC converter; Step 2: Construct ESMO to estimate the lumped disturbance based on the mathematical model; Step 3: Construct STNTSMC based on the mathematical model to improve the system control performance.

2. The CETDC super-helical non-singular terminal sliding mode control method based on disturbance estimation according to claim 1 is characterized in that: The step 1 includes the following specific steps: CETDC consists of three identical phases, each of which includes an energy storage bridge arm and two sets of converter valves. The converter valves are composed of IGBT devices connected in series. The energy storage bridge arm is composed of submodules connected in series, and a buffer inductor is connected in series. When the converter valve 1 is turned on and the converter valve 2 is turned off, the energy storage bridge arm is U H For side charging, the state equation is: （1）； When the converter valve 1 is closed and the converter valve 2 is open, the energy storage bridge arm is U L Side discharge, the state equation is: （2）； in, U ca is the submodule capacitor voltage, n Invest in the number of submodules, N is the total number of submodules, L For the bridge arm buffer inductor, R is the parasitic resistance of the buffer inductor, i pa Indicates the A-phase bridge arm current; Combining the above formulas (1) and (2), we get the system dynamic mathematical model: （3）； in, S 1. S 2 are the switch states of converter valve 1 and converter valve 2 respectively. , and their values ​​are complementary; Indicates a symbolic function. In the formula, the symbol i pa The size of the change; the number of n As the control input of the controller.

3. The CETDC super-helical non-singular terminal sliding mode control method based on disturbance estimation according to claim 2 is characterized in that: The step 2 includes the following specific steps: Step 2.1: Construct an extended sliding mode observer to expand the lumped disturbance into state variables for estimation, which serves as feedback compensation for the system. The bridge arm current state equation in formula (3) is modified to: （4）； in, and Indicates parameter changes, represents external disturbance or system uncertainty; assuming and are all bounded, formula (4) can be transformed into the following form: （5）； in, F is the lumped disturbance, expressed as: （6）； because and are all bounded, so F It is also bounded; The system design for Equation (5) is as follows: （7）； in, Represents the estimated value of the bridge arm current; represents the lumped disturbance estimate; u smo is the sliding mode control law to be designed; l is the observer gain, and ; The design bridge arm current observation error is , the concentrated perturbation observation error is , thus the error equation is: （8）； Select the integral sliding surface as: （9）； in, c is the integral gain, and ; The exponential reaching law is selected as: （10）； in, k 1 is the switching gain to be designed in the reaching law, is the coefficient of the exponential term to be designed, and 、 ; In order to make e i and The sliding mode surface converges to 0 in a finite time, and the sliding mode control law is designed as follows: （11）； Because in the control process is used as feedback compensation to the system, so ;Will u smo Substituting into formula (7) we can get the expression of ESMO; Step 2.2: Prove the stability of the observer by Lyapunov function to ensure the bridge arm current observation error and concentrated perturbation observation error Converges in finite time.

4. The CETDC super-helical non-singular terminal sliding mode control method based on disturbance estimation according to claim 3 is characterized in that: The step 2.2 includes the following specific steps: The Lyapunov function is selected as follows: （12）； According to formula (8), V The derivative of 1 is expressed as: ； ； ； ； ； ； ； ； （13）； if When ,parameter Should satisfy ;if When ,parameter Should satisfy To ensure ,parameter Should meet: （14）； According to the principle of sliding mode control, we know that in order to make the system state reach the sliding surface within a finite time, ; The error equation (8) is simplified to: （15）； Solved , where is an arbitrary constant, when the observer gain , which can ensure that the system error converges within a finite time.

5. The CETDC super-helical non-singular terminal sliding mode control method based on disturbance estimation according to claim 4 is characterized in that: The step 3 includes the following specific steps: Step 3.1: Construct a superhelical non-singular terminal sliding mode controller; The bridge arm current state equation after ESMO compensation described in step 2 is: （16）； definition , , select the non-singular terminal sliding mode surface as: （17）； In the formula , p 、 q is a positive odd number, and ; Then taking its derivative we can get: ； （18）； Using the improved fast supercoil algorithm, the sign function is replaced by the hyperbolic tangent function Instead, the reaching law of the fast supercoil algorithm is obtained as: （19）； Among them, the control parameters and are all greater than zero; Combining Equations (18) and (19), we can obtain the STNTSMC control law of the bridge arm current control loop: ； （20）； Step 3.2: Prove the finite-time convergence of the controller. From Equation (17), we can obtain the sufficient condition for the existence of a non-singular terminal sliding surface: （21）； Where, k is a positive constant; therefore, the STNTSMC control law designed by formula (20) satisfies this condition, For a limited time, the following conditions will be met The non-singular terminal sliding surface is reached: （22）； When the non-singular terminal sliding surface hour, The system dynamics is determined by the following differential equations: （23）； from arrive Limited time required It is expressed as follows: （24）； Therefore, the non-singular terminal sliding surface makes the tracking error and its derivatives In a limited time Converges to zero internally; Step 3.3: Prove controller stability. In order to verify the stability of the STNTSMC control law, the Lyapunov function is selected V 2: （25）； right V 2 Taking the derivative we get: ； ； ； ； （26）； if When , the parameters need to be met and are greater than zero; if When , also need to meet the parameters and are all greater than zero; therefore, according to the Lyapunov stability theorem, by selecting the control parameters, the system can be made to converge to the sliding surface, indicating that the constructed controller is asymptotically stable.

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