Adaptive Optics Tip-Tilt Correction Based on Smith Predictor and Filter-Optimized Linear Active Disturbance Rejection Control Method

A tip-tilt mirror (TTM) control method is designed to enhance the control bandwidth and ensure the rejection performance of the adaptive optics (AO) tip-tilt correction system. Optimized with the Smith predictor and filter, linear active disturbance rejection (LADRC) is adopted to achieve the tip-tilt correction. An AO tip-tilt correction experimental platform was built to validate the method. Experimental results show that the proposed method improves the control bandwidth of the system by at least 3.6 times compared with proportional–integral (PI) control. In addition, under the same control bandwidth condition, compared with the Smith predictor and proportional–integral (PI–Smith) control method, the system is more capable of rejecting internal and external disturbances, and its dynamic response performance is improved by more than 29%.


Introduction
AO is the technology for correcting dynamic optical wavefront aberration by detecting and compensating wavefront aberrations in real time [1]. In the field of laser-beam cleaning, AO has the advantage of improving the far-field beam quality of output lasers [2]. In the process of laser-beam cleaning, correcting the tip-tilt aberration is the primary task to stabilize the beam pointing in the detector field of view during the AO closed-loop process [3][4][5]. However, a high-frequency tip-tilt disturbance is introduced because of thermal effects, material defects, and the diversification of AO application scenarios [6]. It affects the laser output beam pointing, making it difficult to stabilize the beam pointing in a stable region. The tip-tilt disturbance leads to severe degradation of the far-field beam quality of the laser, which limits the application of the laser [7].
There are two challenges in AO tip-tilt correction. On the one hand, the time delay caused by the existence of data acquisition and control calculations limits the system control bandwidth [8,9]. On the other hand, the dynamic response performance of a TTM is sensitive to the effects of internal and external disturbances [10][11][12]. The change in the time-delay parameter of the controlled model mainly causes internal disturbances. External disturbances include the vibration of the experimental devices and the instability of the application scenario of the AO system. These two challenges introduce high-frequency tilt disturbances to the system, which exceed the dynamic response capability of the tip-tilt mirror and cause the beam to point away from the detector's field of view [13,14]. Therefore, the impact of the time delay and disturbances on the tip-tilt correction performance of the system needs to be compensated simultaneously to achieve high bandwidth and high dynamic response TTM control in the AO system.
Recently, some control methods were proposed to compensate for the effect of the time delay on the tip-tilt correction performance of AO. Luo et al. [15] embedded the wind speed estimation and prediction control into the AO system and established the control procedure for wind speed estimation and prediction algorithm in the closed loop. They used a combination of geometric interpolation and prediction for compensation correction of the restoration voltage, which reduced the root mean square of the corrected residuals by about 10% and decreased the correction error due to the time delay. However, the method is computationally intensive and requires high computational power on real-time processors. Ruan et al. [16] improved the closed-loop correction performance of the system using a multiple closed-loop control method with limited bandwidth, considering the time-delay limitation. However, the technique failed to improve the system control bandwidth in essence. Based on classical PI control, Li et al. [8] performed Smith prediction to compensate for the effect of time delay on the system correction performance. Smith enhanced the system control bandwidth, but PI control experienced challenges in overcoming the impact on the system correction performance because of the mismatch between the Smith compensation parameter and the system time-delay parameter, which was constrained by internal disturbances. In addition, control methods for disturbance rejection were widely researched. Zhao et al. [17] used a robust control method to design a controller for the AO system. They minimized the H 2 norm under the H Sensors 2023, 23, x FOR PEER REVIEW [13,14]. Therefore, the impact of the time delay an performance of the system needs to be compen bandwidth and high dynamic response TTM con Recently, some control methods were propo time delay on the tip-tilt correction performance o speed estimation and prediction control into the procedure for wind speed estimation and predic used a combination of geometric interpolation and of the restoration voltage, which reduced the roo by about 10% and decreased the correction erro method is computationally intensive and requires processors. Ruan et al. [16] improved the clos system using a multiple closed-loop control meth the time-delay limitation. However, the techniqu bandwidth in essence. Based on classical PI c prediction to compensate for the effect of t performance. Smith enhanced the system control challenges in overcoming the impact on the system mismatch between the Smith compensation p parameter, which was constrained by internal dis for disturbance rejection were widely researched method to design a controller for the AO system. Hꝏ norm constraint, effectively suppressing the the method was computationally extensive, whic delay of the AO system in another way. LQG con effectively identify and reject high-frequency performance depended on the accuracy of the identified the jitter of the beam pointing in real t the accuracy of the calculation. Li et al. [21] im tracking performance of the TTM in optical com output estimation of the system with an improv internal disturbances was not considered in their above improve the tip-tilt correction effect but fail of the time delay and disturbances on the system To provide a simultaneous solution for the e on the correction performance of the system, i proposed a TTM control method with a Smith pre disturbance rejection (FLADRC-Smith) [22]. We control bandwidth of the system and designed a external disturbances. Simulation results showed control bandwidth and dynamic response perform This paper analyzes the influence of FLADR the system bandwidth and noise rejection per mathematical model of the AO tip-tilt correction TTM control method with FLADRC-Smith is veri experimental platform.

Mathematical Model of AO Tip-Tilt Correction
The schematic diagram of the AO tip-tilt co far-field CCD detects tip-tilt disturbances and calculates the control voltage and then applies deflection, thus compensating the tip-tilt disturba norm constraint, effectively suppressing the low-frequency disturbances. However, the method was computationally extensive, which again exacerbated the degree of time delay of the AO system in another way. LQG control [18] and hybrid control [19,20] can effectively identify and reject high-frequency tip-tilt disturbances. Still, the control performance depended on the accuracy of the model establishment, and the system identified the jitter of the beam pointing in real time, which required hard real time and the accuracy of the calculation. Li et al. [21] improved the disturbance rejection and tracking performance of the TTM in optical communication systems by optimizing the output estimation of the system with an improved LADRC method. Still, the effect of internal disturbances was not considered in their work. The control methods mentioned above improve the tip-tilt correction effect but fail to simultaneously address the influence of the time delay and disturbances on the system calibration performance.
To provide a simultaneous solution for the effect of the time delay and disturbances on the correction performance of the system, in our previous work, we innovatively proposed a TTM control method with a Smith predictor and filter-optimized linear active disturbance rejection (FLADRC-Smith) [22]. We used a Smith predictor to enhance the control bandwidth of the system and designed a filtering unit to suppress internal and external disturbances. Simulation results showed that the method improved the system's control bandwidth and dynamic response performance.
This paper analyzes the influence of FLADRC-Smith control parameter selection on the system bandwidth and noise rejection performance according to the equivalent mathematical model of the AO tip-tilt correction system. Finally, the effectiveness of the TTM control method with FLADRC-Smith is verified by building an AO tip-tilt correction experimental platform.

Mathematical Model of AO Tip-Tilt Correction
The schematic diagram of the AO tip-tilt correction system is shown in Figure 1. A far-field CCD detects tip-tilt disturbances and feeds back to the controller, which calculates the control voltage and then applies it to the TTM to drive a slight angle deflection, thus compensating the tip-tilt disturbance and stabilizing the far-field spot in the CCD field of view. As an actuator for correction, the TTM produces mechanical resonance phenomena in the high-frequency band because of its inherent elastic structure. However, the first Sensors 2023, 23, 6724 3 of 14 resonance frequency is in the hundreds and thousands of hertz. Therefore, in the frequency range of concern for the system, its ideal transfer function is as follows: the CCD field of view. As an actuator for correction, the TTM produces mechanical resonance phenomena in the high-frequency band because of its inherent elastic structure. However, the first resonance frequency is in the hundreds and thousands of hertz. Therefore, in the frequency range of concern for the system, its ideal transfer function is as follows: ( ) Combined with the τ -second time delay present in the system, the equivalent transfer function of the system can be expressed as follows:  Figure 2 shows the control block diagram of an AO tip-tilt correction system using the FLADRC-Smith control. In Figure 1, r is the set value of the TTM deflection angle, d is the external disturbance signal, u is the voltage control signal of TTM, y is the output deflection angle of TTM, and y has a time delay of τ seconds in the time domain concerning u. The feedback signal comes from the TTM deflection angle measurement sensor, and the error of the deflection angle is sent to the controller.  Combined with the τ-second time delay present in the system, the equivalent transfer function of the system can be expressed as follows: Figure 2 shows the control block diagram of an AO tip-tilt correction system using the FLADRC-Smith control. In Figure 1, r is the set value of the TTM deflection angle, d is the external disturbance signal, u is the voltage control signal of TTM, y is the output deflection angle of TTM, and y has a time delay of τ seconds in the time domain concerning u. The feedback signal comes from the TTM deflection angle measurement sensor, and the error of the deflection angle is sent to the controller. the CCD field of view. As an actuator for correction, the TTM produces mechanical resonance phenomena in the high-frequency band because of its inherent elastic structure. However, the first resonance frequency is in the hundreds and thousands of hertz. Therefore, in the frequency range of concern for the system, its ideal transfer function is as follows:

Design of FLADRC-Smith Control Method
( ) Combined with the τ -second time delay present in the system, the equivalent transfer function of the system can be expressed as follows:  Figure 2 shows the control block diagram of an AO tip-tilt correction system using the FLADRC-Smith control. In Figure 1, r is the set value of the TTM deflection angle, d is the external disturbance signal, u is the voltage control signal of TTM, y is the output deflection angle of TTM, and y has a time delay of τ seconds in the time domain concerning u. The feedback signal comes from the TTM deflection angle measurement sensor, and the error of the deflection angle is sent to the controller.  In Figure 2: where ω 0 is the bandwidth of the extended state observer in rad/s in LADRC, k p and k i are the feedback control law parameters, and ω c is the cutoff frequency of the filtering unit. G d (s) is the Smith predictor model designed according to Equation (2):

Design of FLADRC-Smith Control Method
where τ 1 is the compensation parameter of the Smith predictor. G 0 and H combine to form the deflection angle measurement sensor, and the controller consists of G 0 , the deflection angle measurement sensor, and G d (s). Accordingly, the closed-loop transfer function M FLADRC−Smith (s) of the control loop can be obtained as: where The characteristic equations of the closed-loop transfer function are processed using the pole placement method. When set up, b 0 ω 3 0 and k i → 0 , B(s) can be factorized into B(s) = s + k i /k p C(s) using the synthetic division method. To ensure that B(s) has strong disturbance rejection in the high-frequency band, and then by finding the rational roots of polynomials using the irreducible fraction, C(s) can be factorized into C(s) = s + 3 2 ω 0 D(s). When D(s) is expanded to D(s) = s + k p ω c E(s), a scaling factor 0 < µ < 1 is introduced to obtain the desired pole as µω 0 , where ω c ≈ µ ω 0 k p . When designing the FLADRC-Smith controller in the AO tilt correction system, the power spectral density of the disturbance signal should be analyzed to determine the distribution band of the disturbance signal to confirm the desired closed-loop pole µω 0 and then adjust k p according to the actual response performance of the system.

Bandwidth Analysis
The error transfer function reflects the steady-state error of the control system and characterizes the system's ability to suppress the error signal. The larger the error attenuation bandwidth, the stronger the rejection capability of the system. Therefore, to analyze the disturbance rejection performance of the control system, it is necessary to analyze the error attenuation bandwidth of the system.
The block diagram of the relationship between the system inputs and the error is shown in Figure 3, which means the error transfer function: where Comparing Equation (7) with Equation (5), it can be observed that the closed-loop transfer function of the AO tip-tilt correction control system has the same pole composition as the error transfer function. When Sensors 2023, 23, 6724 5 of 14 is set up, the error attenuation bandwidth of the control system is µω 0 because of the approximate elimination of the zero value − 7 5 ω 0 and the polar component − 3 2 ω 0 . The rejection performance of the system can be evaluated according to the error attenuation bandwidth.
transfer function of the AO tip-tilt correction control system has the same pole composition as the error transfer function. When is set up, the error attenuation bandwidth of the control system is

Noise Analysis
There are many contradictions in process control, such as the conflict between speed and stability. Similarly, there is a contradiction between control bandwidth and system noise rejection capability. The larger the control bandwidth, the more measurement noise is inevitably introduced while correcting for external disturbances. In Figure 4, the noise n is shown on the system output of the schematic diagram. Accordingly, the noise transfer function can be derived as: where Figure 3. Block diagram of the connection between the input signal and the error signal.

Noise Analysis
There are many contradictions in process control, such as the conflict between speed and stability. Similarly, there is a contradiction between control bandwidth and system noise rejection capability. The larger the control bandwidth, the more measurement noise is inevitably introduced while correcting for external disturbances. In Figure 4, the noise n is shown on the system output of the schematic diagram. Accordingly, the noise transfer function can be derived as: where Considering only the amplitude-frequency characteristics of the noise transfer function, it is noted that ( ) nf M s contains four zeros and five poles. According to the effect of pole-zero on the frequency domain characteristics of the system, it is known that the amplitude-frequency characteristic curve of ( ) nf M s eventually decreases at a rate of −20 dB/dec. Therefore, the system has the ability to suppress noise. Define the frequency corresponding to the amplitude-frequency curve over the −3 dB gain point as the noise attenuation bandwidth. It can be seen that the smaller the value, the stronger the system's ability to suppress noise.
The above analysis shows that the system's control bandwidth and noise suppression capability conflict with each other. To ensure the noise rejection capability of the system, the bandwidth of the system should not be too large. When selecting the control parameters for the FLADRC-Smith controller, the disturbance rejection performance of the system and the control index requirements should be considered comprehensively. Considering only the amplitude-frequency characteristics of the noise transfer function, it is noted that M nf (s) contains four zeros and five poles. According to the effect of pole-zero on the frequency domain characteristics of the system, it is known that the amplitude-frequency characteristic curve of M nf (s) eventually decreases at a rate of −20 dB/dec. Therefore, the system has the ability to suppress noise. Define the frequency corresponding to the amplitude-frequency curve over the −3 dB gain point as the noise attenuation bandwidth. It can be seen that the smaller the value, the stronger the system's ability to suppress noise.

Results
The above analysis shows that the system's control bandwidth and noise suppression capability conflict with each other. To ensure the noise rejection capability of the system, the bandwidth of the system should not be too large. When selecting the control parameters for the FLADRC-Smith controller, the disturbance rejection performance of the system and the control index requirements should be considered comprehensively.

AO Tip-Tilt Correction Experimental Platform
The AO tip-tilt correction experimental platform is shown in Figure 5. In Figure 5a, the light source wavelength is 635 nm. Before entering the wavefront sensor, the output beam passes through reflector 1 (R1), disturbance TTM1, correction TTM2, and R2. In particular, TTM1 and TTM2 have an aperture of 70 mm, a resonance frequency of 1000 Hz, and a dynamic response range of ±100 µrad. Figure 5b shows the schematic diagram of the wavefront sensor, which contains the Hartmann wavefront sensor, the far-field CCD, and the beam splitter with a sampling frequency of 200 Hz. In this experimental platform, the signal generator controls the Ethernet interface highvoltage amplifier to generate a control voltage with disturbances, which drives TTM1 to produce a slight amplitude deflection and makes the beam point jitter in the y-direction. TTM2 and the deformable mirror are driven by a fiber-optic high-voltage amplifier sending control voltage with a voltage limit of ±2 V. The control algorithm enables TTM2 to reject the disturbance generated by TTM1, and the deformable mirror corrects the residual wavefront aberration.
To measure the time delay of the experimental platform, the static tip-tilt aberration is corrected. As seen from the correction results in Figure 6, the control voltage is applied to TTM2 in the 52nd of the time sequence, but in the 54th, the tip-tilt aberration starts to be compensated. This indicates that there is a time delay τ in the control system, which is two times the sampling period. Accordingly, the time-delay compensation parameter 1 τ in the Smith predictor is set to 0.01 s during the experiment. In Figure 5a, the light source wavelength is 635 nm. Before entering the wavefront sensor, the output beam passes through reflector 1 (R1), disturbance TTM1, correction TTM2, and R2. In particular, TTM1 and TTM2 have an aperture of 70 mm, a resonance frequency of 1000 Hz, and a dynamic response range of ±100 µrad. Figure 5b shows the schematic diagram of the wavefront sensor, which contains the Hartmann wavefront sensor, the far-field CCD, and the beam splitter with a sampling frequency of 200 Hz. In this experimental platform, the signal generator controls the Ethernet interface high-voltage amplifier to generate a control voltage with disturbances, which drives TTM1 to produce a slight amplitude deflection and makes the beam point jitter in the y-direction. TTM2 and the deformable mirror are driven by a fiber-optic high-voltage amplifier sending control voltage with a voltage limit of ±2 V. The control algorithm enables TTM2 to reject the disturbance generated by TTM1, and the deformable mirror corrects the residual wavefront aberration.
To measure the time delay of the experimental platform, the static tip-tilt aberration is corrected. As seen from the correction results in Figure 6, the control voltage is applied to TTM2 in the 52nd of the time sequence, but in the 54th, the tip-tilt aberration starts to be compensated. This indicates that there is a time delay τ in the control system, which is two times the sampling period. Accordingly, the time-delay compensation parameter τ 1 in the Smith predictor is set to 0.01 s during the experiment.

Comparison with PI Control Method
To verify the advantages of the FLADRC-Smith method in enhancing the control bandwidth when applied to the AO tip-tilt correction system, the FLADRC-Smith control method is compared with the PI control method on the experimental platform [23].
First, a sine wave signal with an amplitude of 1 V and a frequency of 1 Hz is generated using a signal generator and applied to the y-direction of TTM1 through an Ethernet interface high-voltage amplifier. At this time, the output beam of the light source produces a sine wave disturbance in the y-direction, as shown by the dotted line in Figure 7a. The power of this disturbance at 1 Hz is 17.5 dB/Hz, as can be seen from the dotted line marked by the circle in Figure 7b. Combined with manual experience and considering the bandwidth of the control system, the dynamic response performance of the TTM, and the need to avoid unstable phenomena such as overshooting in the closed-loop process, the control parameters of the PI controller are set to The residual tip-tilt aberration of TTM2 after correction is shown as dashed lines in Figure 7a, corresponding to the power spectral density curves illustrated with plus dotted lines in Figure 7b. The disturbance power of the tip-tilt aberration signal at 1 Hz is below 0 dB, which tells us that the sine disturbance is effectively corrected. The control parameters of the FLADRC-Smith controller are tuned to guarantee that TTM2 has the same rejection capability for disturbances as when PI control is used, as shown by the solid line in Figure 7a

Comparison with PI Control Method
To verify the advantages of the FLADRC-Smith method in enhancing the control bandwidth when applied to the AO tip-tilt correction system, the FLADRC-Smith control method is compared with the PI control method on the experimental platform [23].
First, a sine wave signal with an amplitude of 1 V and a frequency of 1 Hz is generated using a signal generator and applied to the y-direction of TTM1 through an Ethernet interface high-voltage amplifier. At this time, the output beam of the light source produces a sine wave disturbance in the y-direction, as shown by the dotted line in Figure 7a. The power of this disturbance at 1 Hz is 17.5 dB/Hz, as can be seen from the dotted line marked by the circle in Figure 7b.

Comparison with PI Control Method
To verify the advantages of the FLADRC-Smith method in enhancing the control bandwidth when applied to the AO tip-tilt correction system, the FLADRC-Smith control method is compared with the PI control method on the experimental platform [23].
First, a sine wave signal with an amplitude of 1 V and a frequency of 1 Hz is generated using a signal generator and applied to the y-direction of TTM1 through an Ethernet interface high-voltage amplifier. At this time, the output beam of the light source produces a sine wave disturbance in the y-direction, as shown by the dotted line in Figure 7a. The power of this disturbance at 1 Hz is 17.5 dB/Hz, as can be seen from the dotted line marked by the circle in Figure 7b. Combined with manual experience and considering the bandwidth of the control system, the dynamic response performance of the TTM, and the need to avoid unstable phenomena such as overshooting in the closed-loop process, the control parameters of the PI controller are set to The residual tip-tilt aberration of TTM2 after correction is shown as dashed lines in Figure 7a, corresponding to the power spectral density curves illustrated with plus dotted lines in Figure 7b. The disturbance power of the tip-tilt aberration signal at 1 Hz is below 0 dB, which tells us that the sine disturbance is effectively corrected. The control parameters of the FLADRC-Smith controller are tuned to guarantee that TTM2 has the same rejection capability for disturbances as when PI control is used, as shown by the solid line in Figure 7a  Combined with manual experience and considering the bandwidth of the control system, the dynamic response performance of the TTM, and the need to avoid unstable phenomena such as overshooting in the closed-loop process, the control parameters of the PI controller are set to K p = 0, K i = 40. The residual tip-tilt aberration of TTM2 after correction is shown as dashed lines in Figure 7a, corresponding to the power spectral density curves illustrated with plus dotted lines in Figure 7b. The disturbance power of the tip-tilt aberration signal at 1 Hz is below 0 dB, which tells us that the sine disturbance is effectively corrected. The control parameters of the FLADRC-Smith controller are tuned to guarantee that TTM2 has the same rejection capability for disturbances as when PI control is used, as shown by the solid line in Figure 7a and the solid line marked by dots in , where the control parameters are set to ω 0 = 250 rad/s, µ = 0.6, k p = 16. With the parameter settings, the ideal error-attenuation bandwidth of the system is, theoretically, 6.25 Hz maximum for PI control and 23.87 Hz maximum for FLADRC-Smith control.
Next, holding the control parameters of the controller constant, the frequency of the signal generator is increased to 5 Hz, and Figure 8 shows the correction results. In Figure 8a, it can be seen that the amplitude of the tip-tilt aberration becomes larger under PI control compared to the suppression effect on the 1 Hz frequency disturbance signal, while the control of TTM2 using the FLADRC-Smith method still better suppresses the disturbance. As can be seen in Figure 8b, the power of the disturbance signal decreases from 17.5 dB/Hz to below 0 dB at 5 Hz frequency under FLADRC-Smith control. At the same time, it is greater than 10 dB/Hz under PI control, which provides poor rejection of the 5 Hz disturbance signal, indicating that the control bandwidth of the system is below 5 Hz.
Sensors 2023, 23, x FOR PEER REVIEW 9 of 16 p 16 k = . With the parameter settings, the ideal error-attenuation bandwidth of the system is, theoretically, 6.25 Hz maximum for PI control and 23.87 Hz maximum for FLADRC-Smith control.
Next, holding the control parameters of the controller constant, the frequency of the signal generator is increased to 5 Hz, and Figure 8 shows the correction results. In Figure  8a, it can be seen that the amplitude of the tip-tilt aberration becomes larger under PI control compared to the suppression effect on the 1 Hz frequency disturbance signal, while the control of TTM2 using the FLADRC-Smith method still better suppresses the disturbance. As can be seen in Figure 8b, the power of the disturbance signal decreases from 17.5 dB/Hz to below 0 dB at 5 Hz frequency under FLADRC-Smith control. At the same time, it is greater than 10 dB/Hz under PI control, which provides poor rejection of the 5 Hz disturbance signal, indicating that the control bandwidth of the system is below 5 Hz. Finally, to explore the control bandwidth of the system during FLADRC-Smith control, the controller parameters are kept constant, the sine disturbance frequency of the signal generator output is increased, and its ability to suppress high-frequency disturbance signals is analyzed. Figures 9 and 10 show the rejection results for 18 Hz and 23 Hz sine disturbance signals under FLADRC-Smith and PI control, respectively. The comparison shows that TTM2 has difficulty suppressing the 18 Hz sine disturbance under PI control and even indicates the counter effect of amplifying the disturbance for the 23 Hz sine signal. According to Figure 9a, the suppression of the disturbance amplitude by FLADRC-Smith is evident in the time domain, and the power of the disturbance component is suppressed to 6.7 dB/Hz, as shown in Figure 9b. However, the results in Figure 10 show that the FLADRC-Smith control becomes less effective in suppressing the 23 Hz sine disturbance, indicating that the control bandwidth of the system is between 18 Hz and 23 Hz, which is consistent with the results of the theoretical calculations. Finally, to explore the control bandwidth of the system during FLADRC-Smith control, the controller parameters are kept constant, the sine disturbance frequency of the signal generator output is increased, and its ability to suppress high-frequency disturbance signals is analyzed. Figures 9 and 10 show the rejection results for 18 Hz and 23 Hz sine disturbance signals under FLADRC-Smith and PI control, respectively. The comparison shows that TTM2 has difficulty suppressing the 18 Hz sine disturbance under PI control and even indicates the counter effect of amplifying the disturbance for the 23 Hz sine signal. According to Figure 9a, the suppression of the disturbance amplitude by FLADRC-Smith is evident in the time domain, and the power of the disturbance component is suppressed to 6.7 dB/Hz, as shown in Figure 9b. However, the results in Figure 10 show that the FLADRC-Smith control becomes less effective in suppressing the 23 Hz sine disturbance, indicating that the control bandwidth of the system is between 18 Hz and 23 Hz, which is consistent with the results of the theoretical calculations.
By analyzing the suppression effect of different frequency sine disturbance signals above, we can observe that the control bandwidth of the system is between 18 Hz and 23 Hz when the FLADRC-Smith control method is used. In contrast, driving TTM2 to suppress the disturbance signal at 5 Hz under PI control is challenging, and the control bandwidth is less than 5 Hz.
In summary, the FLADRC-Smith control method can reject high-frequency disturbance signals better than the PI control method. It improves the control bandwidth of the system by at least 3.6 times.  By analyzing the suppression effect of different frequency sine disturbance signals above, we can observe that the control bandwidth of the system is between 18 Hz and 23 Hz when the FLADRC-Smith control method is used. In contrast, driving TTM2 to suppress the disturbance signal at 5 Hz under PI control is challenging, and the control bandwidth is less than 5 Hz.
In summary, the FLADRC-Smith control method can reject high-frequency disturbance signals better than the PI control method. It improves the control bandwidth of the system by at least 3.6 times.

Comparison with PI-Smith Control Method
The Smith predictor is sensitive to the change in the time-delay parameter during the system correction process. Therefore, to further verify the system's disturbance rejection performance and dynamic response performance when the FLADRC-Smith control method is used, the FLADRC-Smith control method is compared with the PI-Smith control method in the experiment.
During the experiments, the control parameters of the FLADRC-Smith controller are  Figure 11 and Figure 12. Figure 11 and Figure 12 illustrate that the system has the same  By analyzing the suppression effect of different frequency sine disturbance signals above, we can observe that the control bandwidth of the system is between 18 Hz and 23 Hz when the FLADRC-Smith control method is used. In contrast, driving TTM2 to suppress the disturbance signal at 5 Hz under PI control is challenging, and the control bandwidth is less than 5 Hz.
In summary, the FLADRC-Smith control method can reject high-frequency disturbance signals better than the PI control method. It improves the control bandwidth of the system by at least 3.6 times.

Comparison with PI-Smith Control Method
The Smith predictor is sensitive to the change in the time-delay parameter during the system correction process. Therefore, to further verify the system's disturbance rejection performance and dynamic response performance when the FLADRC-Smith control method is used, the FLADRC-Smith control method is compared with the PI-Smith control method in the experiment.
During the experiments, the control parameters of the FLADRC-Smith controller are  Figure 11 and Figure 12. Figure 11 and Figure 12 illustrate that the system has the same

Comparison with PI-Smith Control Method
The Smith predictor is sensitive to the change in the time-delay parameter during the system correction process. Therefore, to further verify the system's disturbance rejection performance and dynamic response performance when the FLADRC-Smith control method is used, the FLADRC-Smith control method is compared with the PI-Smith control method in the experiment.
During the experiments, the control parameters of the FLADRC-Smith controller are followed, that is, ω 0 = 250 rad/s, µ = 0.6, and k p = 16. Then, adjust the control parameters of PI-Smith to K p = 0, K i = 84 so that the system control bandwidth is the same under both control methods. With this parameter setting, the suppression effects of the two control methods for high-frequency disturbances at 18 Hz and 23 Hz are shown in Figures 11 and 12. Figures 11 and 12 illustrate that the system has the same control bandwidth and rejection capability for high-frequency disturbances under FLADRC-Smith and PI-Smith control.
To more visually compare the dynamic response capabilities of FLADRC-Smith and PI-Smith when suppressing disturbances, a signal generator is used to generate squarewave signals with frequencies of 1 Hz and 5 Hz, respectively, which are applied to TTM1 through an Ethernet interface high-voltage amplifier, where the duty cycle of the squarewave signal is 50%, as shown in Figure 13. In the experiment, the setting time required to recover ±5% of the steady-state value of the response of TTM2 is used as the dynamic response performance evaluation index. control bandwidth and rejection capability for high-frequency disturbances under FLADRC-Smith and PI-Smith control.  To more visually compare the dynamic response capabilities of FLADRC-Smith and PI-Smith when suppressing disturbances, a signal generator is used to generate squarewave signals with frequencies of 1 Hz and 5 Hz, respectively, which are applied to TTM1 through an Ethernet interface high-voltage amplifier, where the duty cycle of the squarewave signal is 50%, as shown in Figure 13. In the experiment, the setting time required to recover ±5% of the steady-state value of the response of TTM2 is used as the dynamic response performance evaluation index. Figure 14 compares the results of TTM2 for suppressing 1 Hz and 5 Hz square-wave disturbance signals driven by FLADRC-Smith and PI-Smith. It can be seen that the system produces a transient response with both control methods at the rising and falling edges of the square-wave signal, and the tip-tilt disturbance can be quickly suppressed. When suppressing the 1 Hz square-wave signal, the setting time for PI-Smith control of TTM2 to resume stable tracking is 0.1153 s. In contrast, the setting time for FLADRC-Smith control is 0.0551 s, which improves the dynamic response performance by 52.2% compared to PI-Smith, as shown in Figure 14a. The setting time for FLADRC-Smith control when suppressing a 5 Hz square-wave signal is 0.0557 s, which improves the dynamic response performance by 45.1% compared to PI-Smith, as shown in Figure 14b.   To more visually compare the dynamic response capabilities of FLADRC-Smith and PI-Smith when suppressing disturbances, a signal generator is used to generate squarewave signals with frequencies of 1 Hz and 5 Hz, respectively, which are applied to TTM1 through an Ethernet interface high-voltage amplifier, where the duty cycle of the squarewave signal is 50%, as shown in Figure 13. In the experiment, the setting time required to recover ±5% of the steady-state value of the response of TTM2 is used as the dynamic response performance evaluation index. Figure 14 compares the results of TTM2 for suppressing 1 Hz and 5 Hz square-wave disturbance signals driven by FLADRC-Smith and PI-Smith. It can be seen that the system produces a transient response with both control methods at the rising and falling edges of the square-wave signal, and the tip-tilt disturbance can be quickly suppressed. When suppressing the 1 Hz square-wave signal, the setting time for PI-Smith control of TTM2 to resume stable tracking is 0.1153 s. In contrast, the setting time for FLADRC-Smith control is 0.0551 s, which improves the dynamic response performance by 52.2% compared to PI-Smith, as shown in Figure 14a. The setting time for FLADRC-Smith control when suppressing a 5 Hz square-wave signal is 0.0557 s, which improves the dynamic response performance by 45.1% compared to PI-Smith, as shown in Figure 14b.   Figure 14 compares the results of TTM2 for suppressing 1 Hz and 5 Hz square-wave disturbance signals driven by FLADRC-Smith and PI-Smith. It can be seen that the system produces a transient response with both control methods at the rising and falling edges of the square-wave signal, and the tip-tilt disturbance can be quickly suppressed. When suppressing the 1 Hz square-wave signal, the setting time for PI-Smith control of TTM2 to resume stable tracking is 0.1153 s. In contrast, the setting time for FLADRC-Smith control is 0.0551 s, which improves the dynamic response performance by 52.2% compared to PI-Smith, as shown in Figure 14a. The setting time for FLADRC-Smith control when suppressing a 5 Hz square-wave signal is 0.0557 s, which improves the dynamic response performance by 45.1% compared to PI-Smith, as shown in Figure 14b.  Further analysis of the voltages of the two control methods when driving TTM2 to suppress 1 Hz and 5 Hz square-wave signals, as shown in Figure 15, indicates that the voltage response of TTM2 is faster using FLADRC-Smith when the rising and falling edges of the square-wave movement come in. Theoretically, according to the setting of Smith predictor parameters, the maximum time-domain boundary of the internal disturbance that the Smith predictor can stably suppress is four times the sampling period. The control parameters are kept constant to compare the two controllers' robustness. That is, 1 τ is 0.01 s for both control methods. Further analysis of the voltages of the two control methods when driving TTM2 to suppress 1 Hz and 5 Hz square-wave signals, as shown in Figure 15, indicates that the voltage response of TTM2 is faster using FLADRC-Smith when the rising and falling edges of the square-wave movement come in. Theoretically, according to the setting of Smith predictor parameters, the maximum time-domain boundary of the internal disturbance that the Smith predictor can stably suppress is four times the sampling period. The control parameters are kept constant to compare the two controllers' robustness. That is, τ 1 is 0.01 s for both control methods. Increase τ to four times the sampling period, that is, 0.02 s, at which time the closed-loop state of the AO tip-tilt correction experimental platform is shown in Figure 16. With a delay time of four times the sampling period, PI-Smith and FLADRC-Smith are used to drive TTM2 to suppress the square-wave disturbance signal, as shown in Figure 13, and Figure 17 shows the correction results.
According to Figure 17a, the rising and falling edge moments of the 1 Hz square-wave disturbance signal's peak error generated by TTM2 during PI-Smith control is 1.2138 µrad (the hollow circle mark). The setting time to recover the stable correction is 0.2356 s (the solid circle mark). At the same time, FLADRC-Smith drives TTM2, producing a peak error of 0.5579 µrad (the hollow square marker). It can recover the stable correction in 0.1654 s (the solid square marker), improving the dynamic response performance of the system by 29.8%. Once the square-wave signal frequency is raised to 5 Hz, PI-Smith cannot suppress the tip-tilt disturbance to 0 µrad at the duty cycle set by the square wave when the rising and falling edges come. Still, the system can be restored to a stable tracking control state within 0.0911 s using FLADRC-Smith, which significantly improves the dynamic response performance of the system. The voltage applied to TTM2 under the two control methods is then analyzed as shown in Figure 18. Increase τ to four times the sampling period, that is, 0.02 s, at which time the closedloop state of the AO tip-tilt correction experimental platform is shown in Figure 16. With a delay time of four times the sampling period, PI-Smith and FLADRC-Smith are used to drive TTM2 to suppress the square-wave disturbance signal, as shown in Figure 13, and Figure 17 shows the correction results.  According to Figure 17a, the rising and falling edge moments of the 1 Hz squarewave disturbance signal's peak error generated by TTM2 during PI-Smith control is 1.2138 µrad (the hollow circle mark). The setting time to recover the stable correction is 0.2356 s (the solid circle mark). At the same time, FLADRC-Smith drives TTM2, producing a peak error of 0.5579 µrad (the hollow square marker). It can recover the stable correction in 0.1654 s (the solid square marker), improving the dynamic response performance of the system by 29.8%. Once the square-wave signal frequency is raised to 5 Hz, PI-Smith cannot suppress the tip-tilt disturbance to 0 µrad at the duty cycle set by the square wave when the rising and falling edges come. Still, the system can be restored to a stable tracking control state within 0.0911 s using FLADRC-Smith, which significantly improves the dynamic response performance of the system. The voltage applied to TTM2 under the two control methods is then analyzed as shown in Figure 18.  Increase τ to four times the sampling period, that is, 0.02 s, at which time the closedloop state of the AO tip-tilt correction experimental platform is shown in Figure 16. With a delay time of four times the sampling period, PI-Smith and FLADRC-Smith are used to drive TTM2 to suppress the square-wave disturbance signal, as shown in Figure 13, and Figure 17 shows the correction results.  According to Figure 17a, the rising and falling edge moments of the 1 Hz squarewave disturbance signal's peak error generated by TTM2 during PI-Smith control is 1.2138 µrad (the hollow circle mark). The setting time to recover the stable correction is 0.2356 s (the solid circle mark). At the same time, FLADRC-Smith drives TTM2, producing a peak error of 0.5579 µrad (the hollow square marker). It can recover the stable correction in 0.1654 s (the solid square marker), improving the dynamic response performance of the system by 29.8%. Once the square-wave signal frequency is raised to 5 Hz, PI-Smith cannot suppress the tip-tilt disturbance to 0 µrad at the duty cycle set by the square wave when the rising and falling edges come. Still, the system can be restored to a stable tracking control state within 0.0911 s using FLADRC-Smith, which significantly improves the dynamic response performance of the system. The voltage applied to TTM2 under the two control methods is then analyzed as shown in Figure 18. As can be observed in the dashed line in Figure 18a, the PI-Smith control has a large overshoot at the rising and falling edges, with a peak overshoot of 11.16%. The FLADRC-Smith control method shown in the solid line has a peak overshoot of only 2.22%, reducing the shock to TTM2 due to voltage overshoot during the tip-tilt correction process. Additionally, as can be seen in Figure 18b, the control voltage calculated with PI-Smith is small compared to that calculated with FLADRC-Smith for TTM2, making correcting the As can be observed in the dashed line in Figure 18a, the PI-Smith control has a large overshoot at the rising and falling edges, with a peak overshoot of 11.16%. The FLADRC-Smith control method shown in the solid line has a peak overshoot of only 2.22%, reducing the shock to TTM2 due to voltage overshoot during the tip-tilt correction process. Additionally, as can be seen in Figure 18b, the control voltage calculated with PI-Smith is small compared to that calculated with FLADRC-Smith for TTM2, making correcting the 5 Hz square-wave disturbances difficult. Figures 14 and 17 indicate that in the absence of internal perturbations, TTM2 generates less overshoot, suppresses more quickly, and restores the stable correction state when TTM2 is controlled using FLADRC-Smith compared to PI-Smith, improving the dynamic response performance of the system. When the internal disturbance reaches the maximum boundary in the theoretical time domain, the PI-Smith becomes less able to suppress the disturbance because of the difficulty of the PI in compensating for the need of the Smith predictor for the accuracy of the time-delay parameter. At the same time, the FLADRC-Smith can guarantee the complete suppression of the disturbance.
In addition, according to the comparison of the control voltages of TTM2 calculated using the two control methods in Figures 15 and 18, when FLADRC-Smith is used, the overshoot of the transition process of the control voltage is slight in the moments of the rising and falling edges of the square-wave signal, which reduces the transient shock to TTM2 due to the sudden change in voltage during the tip-tilt correction process.

Conclusions
This paper adopts the FLADRC-Smith control method for the AO tip-tilt correction experimental platform using the mathematical model of the AO system. A detailed performance analysis was performed, and comparisons were conducted with the PI and PI-Smith control methods. The experimental results show that the FLADRC-Smith method enhances the control bandwidth of the system by at least 3.6 times compared with PI control, which improves the dynamic response performance of the control system compared with the PI-Smith control method. The FLADRC-Smith method achieves tip-tilt correction control with high bandwidth and high dynamic response performance, which is advantageous for applications in laser-beam cleaning, astronomical observation, etc.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author. The data are not publicly available for privacy reasons.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: AO adaptive optics TTM tip-tilt mirror PI proportional-integral PI-Smith Smith predictor and proportional-integral LADRC linear active disturbance rejection FLADRC-Smith Smith predictor and filter-optimized linear active disturbance rejection