A novel sliding mode dual-loop control method with fast global convergence

Abstract

This invention belongs to the field of control technology, and particularly relates to a novel sliding mode dual-loop control method with global fast convergence. The method includes: S1: constructing a linear feedback controller to establish a first closed-loop system; S2: constructing a dynamic model of the first closed-loop system; S3: designing an extended state observer; S4: constructing a second closed-loop system using the dynamic model of the first closed-loop system as the controlled object; S5: constructing a sliding mode dual-loop system based on the first and second closed-loop systems, and achieving high-precision control of the angle signal of a piezoelectric fast reflector based on the sliding mode dual-loop system and the estimation results of step S3. This invention effectively avoids singularity problems while ensuring a fast global convergence speed for the system state, effectively improving the dynamic performance and position control accuracy of the system.

Description

Technical Field

[0001] This invention belongs to the field of control technology, and in particular relates to a novel sliding mode dual closed-loop control method with global fast convergence. Background Technology

[0002] With the rapid development of precision optical systems and high-performance control technologies, piezoelectric fast reflectors (PFSMs) are widely used in precision optics and control systems due to their fast dynamic response and high-precision adjustment capabilities. However, the deflection angle of a PFSM exhibits a strong nonlinear hysteresis relationship with the control voltage. This hysteresis effect is one of the main challenges in achieving high-precision and high-stability control. To address this, researchers have proposed various hysteresis modeling methods, including operator-based and differential equation-based methods. However, using these models to characterize the internal mechanisms of the hysteresis effect is very complex, and the model accuracy is often limited by the external environment and parameter identification process. Consequently, the inverse model constructed cannot accurately compensate for the nonlinear dynamics of the system. In terms of closed-loop algorithms, sliding mode control has become a highly regarded control strategy in piezoelectric drive systems due to its strong robustness to system uncertainties and external disturbances. By designing specific sliding surfaces and reaching laws, the controller enables the system state to converge rapidly to a preset sliding surface under the action of the reaching law. Traditional terminal sliding mode improves upon the infinite convergence time problem of linear sliding modes by introducing terminal attractors to construct a nonlinear sliding surface, achieving finite-time convergence of the system state. However, when the error is large, its approach speed decreases significantly, limiting its application to first-order systems. For second-order piezoelectric systems, singularity issues arise. Existing hysteresis models involve internal mechanisms such as optimization algorithm construction, complex parameter identification processes, and mathematical derivation of inverse models. Model accuracy often depends on the external environment, the number of applied model operators, and the accuracy of the identification algorithm. Furthermore, hysteresis models suffer from poor robustness to external disturbances. Summary of the Invention

[0003] In view of this, the present invention aims to provide a novel sliding mode dual-loop control method with global fast convergence to solve the problems of traditional terminal sliding mode control, which achieves finite-time convergence based on nonlinear sliding surfaces. This traditional method is only applicable to first-order systems and introduces singularities in second-order piezoelectric systems. Furthermore, its convergence speed is affected by the terminal attractor, and its approach speed decreases significantly when the error is large. The present invention proposes a novel dual-loop scheme that does not require a hysteresis model or inverse model to compensate for the nonlinearity of the piezoelectric system. Specifically, a linear feedback controller is first constructed to establish the closed-loop system, followed by time-frequency analysis. During the analysis, only the frequency characteristics of the closed-loop system are considered, rather than the nonlinear characteristics of the piezoelectric system, thus avoiding complex hysteresis modeling. To better resolve the contradiction between system approach speed and singularity, the present invention proposes a novel sliding mode control method with global fast convergence, effectively avoiding singularity problems while ensuring a fast convergence speed of the system state globally, effectively improving the dynamic performance and position control accuracy of the system.

[0004] To achieve the above objectives, the technical solution created by this invention is implemented as follows:

[0005] A novel sliding mode dual-loop control method with fast global convergence includes the following steps:

[0006] S1: Construct a linear feedback controller to establish the first closed-loop system;

[0007] S2: Perform time-domain analysis on the first closed-loop system and construct a dynamic model of the first closed-loop system;

[0008] S3: Design an extended state observer and use the extended state observer to estimate the system state of the first closed-loop system and the lumped disturbance of the second closed-loop system.

[0009] S4: Design a sliding mode controller based on an improved full-order sliding surface and a reaching law, and construct a second closed-loop system using the dynamic model of the first closed-loop system as the controlled object;

[0010] S5: Construct a sliding mode dual closed-loop system based on the first closed-loop system and the second closed-loop system. Based on the sliding mode dual closed-loop system and the estimation results of step S3, achieve high-precision control of the angle signal of the piezoelectric fast reflector.

[0011] Furthermore, in step S1, the expression for the linear feedback controller is:

[0012] ;

[0013] ;

[0014] in, These are all control parameters for a linear feedback controller. For time variables, For the angle output signal of PFSM, This is the output signal of the sliding mode controller; The output signal of the linear feedback controller. This refers to the angular position error of the piezoelectric fast reflector.

[0015] Furthermore, in step S2, the dynamic model of the first closed-loop system is as follows:

[0016] ;

[0017] in, For the angle output signal of the piezoelectric fast reflector, The angular velocity output signal for the piezoelectric fast reflector. These are the equivalent parameters related to the mechanical structure of the first closed-loop system. For time variables, This represents the lumped disturbance experienced by the second closed-loop system. The output signal of the sliding mode controller. Angular velocity, Let ω be the angular acceleration and y be the output signal of the first closed-loop system.

[0018] Furthermore, in step S3, the expression for the extended state observer is:

[0019] ;

[0020] ;

[0021] ;

[0022] ;

[0023] in, State variables The estimated value, The output signal of the first closed-loop system The estimated value, For the i-th state variable Real-time estimates, Here, A, B, and C are the observer gain matrix, and are intermediate parameters with no physical meaning.

[0024] Furthermore, step S4 specifically includes:

[0025] S41: Define the position error function as follows:

[0026] ;

[0027] in, To track errors, The first derivative of the tracking error. The second derivative of the tracking error, For reference position signal, The first derivative of the reference position signal, The second derivative of the reference position signal, For the angle output signal of the piezoelectric fast reflector, The angular velocity output signal for the piezoelectric fast reflector. The first derivative of the angular velocity output signal of the piezoelectric fast reflector;

[0028] S42: Constructing an improved full-order sliding surface:

[0029] ;

[0030] ;

[0031] in, It is a positive number, and , It is a positive number. All are sliding mode variables, and b is the switching function. For sliding mode variables The first derivative, For sliding mode variables The second derivative;

[0032] The approach law is designed as follows:

[0033] ;

[0034] in, c is a positive number;

[0035] S43: Constructing a sliding mode controller based on the position error function, an improved full-order sliding surface, and a reaching law:

[0036] ;

[0037] ;

[0038] ;

[0039] in, These are the equivalent parameters related to the mechanical structure of the first closed-loop system. For the output of the sliding mode controller, These are the equivalent control law and the switching control law, respectively. State variables The estimated value.

[0040] Furthermore, in step S42, when the second closed-loop system state is far from the sliding surface, b>1; when the second closed-loop system state is close to the sliding surface, b<1.

[0041] Furthermore, in step S42, when the state of the second closed-loop system moves away from the sliding surface, the switching item... As the gain increases, when the state of the second closed-loop system approaches the sliding surface, the switching term... The gain decreases.

[0042] Compared with the prior art, the present invention can achieve the following beneficial effects:

[0043] This invention creates a novel sliding mode dual-loop control method with global fast convergence. The improved full-order sliding surface constructed by this invention can not only achieve the finite-time convergence effect of traditional terminal sliding mode, but also improve the system convergence speed problem in terminal sliding mode control. Furthermore, when the system trajectory is in a sliding dynamic state, because the order of the sliding surface is consistent with that of the closed-loop system, the singularity problem caused by differentiation is fundamentally avoided. Attached Figure Description

[0044] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments and descriptions of the invention are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings:

[0045] Figure 1 A flowchart illustrating the novel sliding mode dual-loop control method with global fast convergence described in the embodiments of the present invention;

[0046] Figure 2 A schematic diagram illustrating the principle of the novel sliding mode dual-loop control method with global fast convergence described in the embodiments of the present invention;

[0047] Figure 3 The diagram shows a comparison of the effects of the present invention and conventional control on tracking step positions, as described in the embodiments of the present invention.

[0048] Figure 4 This is a comparison diagram of the errors of the present invention and conventional control tracking sinusoidal position as described in the embodiments of the present invention. Detailed Implementation

[0049] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not constitute a limitation thereof.

[0050] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0051] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," and "outer," etc., indicating orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation on this invention. Furthermore, the terms "first," "second," etc., are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, features defined with "first," "second," etc., may explicitly or implicitly include one or more of that feature. In the description of this invention, unless otherwise stated, "a plurality of" means two or more.

[0052] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art will understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0053] The invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0054] like Figure 1 As shown, this invention provides a novel sliding mode dual-loop control method with global fast convergence to improve the dynamic performance and position control accuracy of a piezoelectric fast reflector (PFSM). The method includes the following steps: S1: Constructing a linear feedback controller to establish a first closed-loop system; S2: Performing time-domain analysis on the first closed-loop system to construct its dynamic model; S3: Designing an extended state observer and using it to estimate the system state of the first closed-loop system and the lumped disturbance of the second closed-loop system; S4: Designing a sliding mode controller based on an improved full-order sliding surface and a reaching law, and using the dynamic model of the first closed-loop system as the controlled object to construct the second closed-loop system; S5: Constructing a sliding mode dual-loop system based on the first and second closed-loop systems, and using the estimation results from the sliding mode dual-loop system and step S3 to achieve high-precision control of the angle signal of the piezoelectric fast reflector.

[0055] It should be noted that this invention aims to solve the key technical problems of high-precision beam pointing and accurate tracking in piezoelectric driven fast steering mirrors (PFSMs). The strong nonlinear hysteresis relationship between the deflection angle of the PFSM and the control voltage is a major challenge in achieving high-precision and high-stability control. Currently, various models used to characterize the internal mechanism of the hysteresis effect are very complex, making accurate compensation difficult. Model accuracy often depends on the hardware environment and the number of model operators applied; furthermore, hysteresis models suffer from poor robustness to external disturbances. Based on this, this invention proposes a novel dual-closed-loop scheme that eliminates the need for hysteresis or anti-hysteresis models to compensate for the nonlinearity of the piezoelectric system. Specifically, the inner loop first constructs a linear feedback controller to establish the first closed-loop system, followed by time-frequency analysis. During the analysis, only the frequency characteristics of the first closed-loop system are considered, rather than the nonlinear characteristics of the piezoelectric system, thus avoiding complex hysteresis modeling. The outer loop uses a sliding mode controller to track the trajectory. The terminal sliding mode can achieve finite-time convergence of the system state; however, when the error is large, the approach speed will decrease significantly, and singularity problems will occur for second-order piezoelectric systems. To better resolve the contradiction between system approach speed and singularity, this invention proposes a novel globally fast convergence sliding mode control method. This method effectively avoids the singularity problem while ensuring a fast global convergence speed for the system state, effectively improving the system's dynamic performance and position control accuracy.

[0056] In some embodiments, in step S1, the expression for the linear feedback controller is:

[0057] ;

[0058] ;

[0059] in, These are all control parameters for a linear feedback controller. For time variables, For the angle output signal of PFSM, This is the output signal of the sliding mode controller; The output signal of the linear feedback controller. This refers to the angular position error of the piezoelectric fast reflector.

[0060] In some embodiments, the dynamic model of the first closed-loop system in step S2 is as follows:

[0061] ;

[0062] in, For the angle output signal of the piezoelectric fast reflector, The angular velocity output signal for the piezoelectric fast reflector. These are the equivalent parameters related to the mechanical structure of the first closed-loop system. For time variables, This represents the lumped disturbance experienced by the second closed-loop system. The output signal of the sliding mode controller. Angular velocity, Let ω be the angular acceleration and y be the output signal of the first closed-loop system.

[0063] It should be noted that, through time-frequency analysis and using the frequency sweep method (applying a voltage signal with constant amplitude and increasing frequency to the PFSM), the dynamic model of the entire first closed-loop system is established by comparing the amplitude and phase relationship between the input signal and the output angle. This represents the total lumped disturbance experienced by the entire second closed-loop system, including external disturbances and parameter uncertainties. The upper limit of the disturbance can be expressed as a constant. express: .

[0064] In some embodiments, in step S3, the expression for the extended state observer is:

[0065] ;

[0066] ;

[0067] ;

[0068] ;

[0069] in, State variables The estimated value, The output signal of the first closed-loop system The estimated value, For the i-th state variable Real-time estimates, Here, A, B, and C are the observer gain matrix, and are intermediate parameters with no physical meaning.

[0070] It should be noted that the extended state observer can estimate the system state and lumped disturbance of the entire first closed-loop system. If we assume the total disturbance of the second closed-loop system... For expansion variables The expression for the extended state observer can be obtained, and the observer gain matrix H can be set as follows: g is a positive constant.

[0071] It is worth noting that traditional sliding mode control assumes an upper bound on the disturbance, but in practice, the upper bound on the disturbance is difficult to determine. In order to suppress the impact of the disturbance on the second closed-loop system, an extended state observer (ESO) is introduced to estimate the total disturbance of the second closed-loop system in real time and feed this result directly back into the control law, so that the control law includes the estimated disturbance term, thereby offsetting the impact of the disturbance on the second closed-loop system.

[0072] In some embodiments, step S4 specifically includes:

[0073] S41: Define the position error function as follows:

[0074] ;

[0075] in, To track errors, The first derivative of the tracking error. The second derivative of the tracking error, For reference position signal, The first derivative of the reference position signal, The second derivative of the reference position signal, For the angle output signal of the piezoelectric fast reflector, The angular velocity output signal for the piezoelectric fast reflector. This is the first derivative of the angular velocity output signal of the piezoelectric fast reflector.

[0076] It should be noted that in practice, traditional terminal sliding surfaces achieve finite-time convergence of the traditional closed-loop system state by introducing non-singular attractors. The traditional terminal sliding surface is as follows:

[0077] ;

[0078] in, The parameter is the controller parameter and is greater than 0. By introducing a non-singular attractor, it is ensured that the conventional closed-loop system converges in finite-time when reaching the sliding surface; however, for second-order conventional closed-loop systems, this approach introduces singularity problems, although later scholars proposed non-singular terminal sliding surfaces. However, its sliding mode remains unchanged. When the state of the traditional closed-loop system moves away from the sliding surface, the convergence speed of the traditional closed-loop system will be severely affected. In order to solve this problem, this invention proposes an improved full-order sliding surface in the following form.

[0079] S42: Constructing an improved full-order sliding surface:

[0080] ;

[0081] ;

[0082] in, It is a positive number, and , It is a positive number. All are sliding mode variables, and b is the switching function. For sliding mode variables The first derivative, For sliding mode variables The second derivative of .

[0083] It should be noted that, The number is positive so that the polynomial It is Hurwitz stable, meaning that all its eigenvalues ​​lie on the left side of the complex plane. Regarding the sliding mode variable, it's worth noting that while increasing the coefficients of a traditional terminal sliding surface can accelerate convergence, excessively increasing the parameters can lead to strong chattering and instability in the traditional closed-loop system. Because the exponential part remains unchanged, the convergence speed decreases as the traditional closed-loop system state deviates from the sliding surface. Therefore, to avoid this problem, this invention proposes an improved full-order sliding surface. Specifically, this is achieved by designing a switching function... When the state of the second closed-loop system moves away from the sliding surface, it causes When the state of the second closed-loop system approaches the sliding surface, it causes To ensure rapid global convergence of the second closed-loop system state, this switching threshold can be determined by parameters. Confirmed. Furthermore, when the trajectory of the second closed-loop system is in a sliding dynamic state, the singularity problem caused by differentiation is fundamentally avoided because the order of the sliding surface is consistent with that of the second closed-loop system.

[0084] To achieve faster convergence and reduce system chattering, the reaching law is designed as follows:

[0085] ;

[0086] in, c is a positive number;

[0087] It should be noted that, based on the traditional reaching law, a power function is introduced into the sliding surface, allowing the reaching law parameters to change with the state of the second closed-loop system. Specifically, when the second closed-loop system moves away from the sliding surface, i.e., when the second closed-loop system moves away from the sliding surface, the switching term... Increasing the gain provides a faster convergence speed; conversely, when the second closed-loop system approaches the sliding surface, the switching term decreases; when the second closed-loop system reaches the sliding surface, the switching term gradually decreases to 0 as the error decreases, which can effectively reduce the chattering of the second closed-loop system.

[0088] S43: Constructing a sliding mode controller based on the position error function, an improved full-order sliding surface, and a reaching law:

[0089] ;

[0090] ;

[0091] ;

[0092] in, These are the equivalent parameters related to the mechanical structure of the first closed-loop system. For the output of the sliding mode controller, These are the equivalent control law and the switching control law, respectively. State variables The estimated value.

[0093] In some embodiments, in step S42, when the second closed-loop system state is far from the sliding surface, b>1; when the second closed-loop system state is close to the sliding surface, b<1.

[0094] In some embodiments, in step S42, when the second closed-loop system state moves away from the sliding surface, the switching item... As the gain increases, when the state of the second closed-loop system approaches the sliding surface, the switching term... The gain decreases.

[0095] The entire system block diagram of the sliding mode controller consists of Figure 2 As stated above.

[0096] The closed-loop stability analysis is as follows:

[0097] Consider the Lyapunov function of the entire second closed-loop system. ,right Differentiating, we get:

[0098] ;

[0099] It is obvious that when hour, The second closed-loop system is asymptotically stable, and the above equation can be written as follows: ,in, Based on known mathematical theory, it can be guaranteed that the sliding variable will be reduced within a finite time. It converges to zero, and its convergence time is ,in, This is the initial state of the sliding mode controller.

[0100] When the state of the second closed-loop system reaches the sliding surface, that is... The designed sliding surface ensures that the error function of the second closed-loop system also converges to 0 in finite time. The proof is as follows:

[0101] when When considering Lyapunov functions ,right Differentiating, we get:

[0102] ;

[0103] It is obvious that The sliding surface is asymptotically stable, which indicates the state of the second closed-loop system. Within a limited time ( Converging to .

[0104] when At that time, the sliding surface is: , determined earlier Values ​​to guarantee polynomials Satisfying the Hurwitz conditions, the state of the second closed-loop system can start from any initial conditions. Departure, within a limited time ( Inner edge sliding surface Converging to its equilibrium point At this time At that time, the second closed-loop system error .

[0105] To verify the effectiveness of the improved full-order sliding mode controller method proposed in this invention, i.e., its stronger tracking capability, experiments were conducted under the same conditions using PI control, traditional sliding mode control, and the control proposed in this invention.

[0106] To verify the effectiveness of the algorithm, Figure 3 The tracking performance of different algorithms on step signals was compared, such as... Figure 3 As shown, by using this invention, the overshoot is greatly reduced. The overshoot of the PI control algorithm is about 12.5%, while the overshoot of this invention is 0.3%. In terms of steady-state time, the steady-state time of the PI control algorithm is about 0.01s, while the steady-state time of this invention is about 0.007s. Figure 4 The tracking performance of different algorithms on sinusoidal signals was compared. As can be seen from the error graph, the PI control algorithm has the largest tracking error, which is about 0.01 mrad. The traditional sliding mode error is about 0.003 mrad. The error of the present invention is the smallest, which is about 0.002 mrad, showing better tracking capability.

[0107] A novel dual-loop sliding mode control strategy with global fast convergence is invented, which includes an improved full-order sliding surface and a reaching law, and a sliding mode controller constructed therefrom. This controller, combined with an extended state observer, estimates disturbances and the state of the first closed-loop system in real time, minimizing the impact of disturbances and improving tracking performance. Figure 3 This demonstrates that, in step experiments, the proposed controller can provide a faster convergence speed than the traditional sliding mode controller. Figure 3 In the synovial dual-closed-loop system, the reference input is used as the reference input. As the starting point, the input is sent to the sliding mode controller, and the control signal output by the sliding mode controller... It is fed into the first closed-loop system as a reference input for the first closed-loop system. After processing by the PID controller, the output control signal is... It directly affects PFSM. The actual output of the first closed-loop system is directly measured by the sensor and compared with the reference input. Forming a second closed-loop system, with This forms the first closed-loop system. The control system also contains an ESO (Environmental Safety Organization), which... and Used as input, it is used to estimate the state and disturbance of the first closed-loop system, and its output disturbance is... Feedback is sent to the sliding mode controller to compensate for disturbances in the control system and enhance control robustness.

[0108] Figure 4 This indicates that the proposed controller exhibits better tracking performance, with the tracked trajectory converging to the reference trajectory more effectively, demonstrating stronger robustness.

[0109] It should be understood that the various forms of processes shown above can be used to reorder, add, or delete steps. For example, the steps described in this invention disclosure can be executed in parallel, sequentially, or in different orders, as long as the desired result of the technical solution disclosed in this invention can be achieved, and this is not limited herein.

[0110] The specific embodiments described above do not constitute a limitation on the scope of protection of this invention. Those skilled in the art should understand that various modifications, combinations, sub-combinations, and substitutions can be made according to design requirements and other factors. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of this invention should be included within the scope of protection of this invention.

Claims

1. A novel sliding mode dual closed loop control method with global fast convergence, characterized in that: Specifically comprising the following steps: S1: constructing a linear feedback controller to establish a first closed-loop system; S2: performing time domain analysis on the first closed-loop system to construct a dynamic model of the first closed-loop system; S3: designing an extended state observer and using the extended state observer to estimate the system state of the first closed-loop system and the lumped disturbance of a second closed-loop system; S4: designing a sliding mode controller based on an improved full-order sliding surface and a reaching law, and constructing the second closed-loop system with the dynamic model of the first closed-loop system as a controlled object; Step S4 specifically comprises: S41: defining a position error function as: ； wherein is a tracking error, is a first derivative of the tracking error, is a second derivative of the tracking error, is a reference position signal, is a first derivative of the reference position signal, is a second derivative of the reference position signal, is an angle output signal of the piezoelectric fast steering mirror, is an angular velocity output signal of the piezoelectric fast steering mirror, is a first derivative of the angular velocity output signal of the piezoelectric fast steering mirror; S42: constructing an improved full-order sliding surface: ； ； wherein is a positive constant, and , is a positive constant, are sliding mode variables, and b is a switching function, is a first derivative of the sliding mode variable , is a second derivative of the sliding mode variable . The reaching law is designed as: ； wherein is a positive number, c is a positive number; S43: constructing a sliding mode controller based on the position error function, the improved full-order sliding surface and the reaching law: ； ； ； wherein, is an equivalent parameter related to the first closed-loop system mechanical structure, is an output of the sliding mode controller, are an equivalent control law and a switching control law, respectively, is a state variable is an estimate of the state variable ; S5: constructing a sliding mode double closed-loop system based on the first closed-loop system and the second closed-loop system, and realizing high-precision control of the angle signal of the piezoelectric fast steering mirror based on the sliding mode double closed-loop system and the estimation results of step S3.

2. The novel sliding mode dual closed loop control method with global fast convergence according to claim 1, characterized in that: In step S1, the expression of the linear feedback controller is: ； ； wherein are control parameters of the linear feedback controller, is a time variable, is an angular output signal of the PFSM, is an output signal of the sliding mode controller; is an output signal of the linear feedback controller, is an angular position error of the piezoelectric fast steering mirror.

3. The novel globally fast converging sliding mode dual closed loop control method according to claim 1, wherein: In step S2, the dynamic model of the first closed-loop system is: ； wherein is the angle output signal of the piezoelectric fast steering mirror, is the angular velocity output signal of the piezoelectric fast steering mirror, is the equivalent parameter related to the mechanical structure of the first closed loop system, is the time variable, denotes the lumped disturbance to which the second closed loop system is subjected, is the output signal of the sliding mode controller, is the angular velocity, is the angular acceleration, y is the output signal of the first closed loop system.

4. The novel globally fast converging sliding mode dual closed loop control method according to claim 1, wherein: In step S3, the expression of the extended state observer is: ； ； ； ； wherein is an estimate of the state variable , , is an estimate of the output signal of the first closed loop system, is a real-time estimate of the i-th state variable , is an observer gain matrix, A, B and C are intermediate quantities without physical meaning.

5. The novel fast converging global sliding mode dual closed loop control method according to claim 1, wherein: In step S42, when the state of the second closed-loop system is far away from the sliding surface, b>1; when the state of the second closed-loop system is close to the sliding surface, b<1.

6. The novel globally fast converging sliding mode dual closed loop control method according to claim 1, wherein: In step S42, when the second closed-loop system state is far from the sliding surface, the switching term increases in gain, when the second closed-loop system state is close to the sliding surface, the switching term decreases in gain.

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