To provide precise tip-tilt correction, the tip-tilt mirror (TTM) control with high bandwidth and high dynamic response is crucial for adaptive optics (AO) systems. The time delay of the system, which is a significant factor restricting the error attenuation bandwidth of the system, is 2-3 times the sampling period. Additionally, the dynamic response performance of TTM will be decreased by the vibration of the experimental device, the unstable operating environment, and other factors brought on by external system disturbances, as well as the time-delay parameters of the system model brought on by internal troubles. While strategies such as the linear quadratic Gaussian (LQG) control method and hybrid control can suppress the high frequency jitter of the beam, and the Smith predictor control method can compensate for the time delay in the system. Nevertheless, the common problem with these methods is that the control effectiveness depends on the accuracy of the model, which results in the inability to suppress disturbances. In order to achieve high bandwidth and high dynamic response TTM control in AO systems, a control method utilizing the Smith predictor and filter-based linear active disturbance rejection (FLADRC-Smith) is proposed to synchronously compensate for the effects of time delay and internal and external disturbances on the tip-tilt correction performance.
In order to increase error attenuation bandwidth, the FLADRC-Smith control method employs a Smith predictor to modify the linear active disturbance rejection control (LADRC) method. The AO system is considered a pure time-delay system. Therefore, a filter is designed to modify the control amount of TTM in order to achieve LADRC that is resistant to high frequency disturbance. In this paper, characteristics of the error attenuation transfer function of the control system are analyzed from a frequency domain perspective, and the control system is optimized and made simpler. Meanwhile, the stability of the control system when the time-delay parameters of the system model are varied is ensured by parameter constraints, and the connection between the error attenuation bandwidth and the system performance in suppressing internal and external disturbances is analyzed. The formula for the error attenuation bandwidth is supplied in the analysis, along with a straightforward method for tuning the parameters.
MATLAB/Simulink is used to establish the simulation model of AO tip-tilt correction to verify the effectiveness of the FLADRC-Smith control method. Firstly, it is confirmed that FLADRC-Smith can increase the error attenuation bandwidth of the system. The error attenuation bandwidth is 25 Hz with proportional-integral (PI) control and 114 Hz with FLADRC-Smith control, showing a 4.56 times improvement with FLADRC-Smith control, according to simulation results after properly setting the controller parameters (Fig. 6). Then, to verify the disturbance rejection ability of FLADRC-Smith control method, it is compared with the PI-Smith control method. The controller parameters are set properly so that both control methods result in the same system error attenuation bandwidth. When the Smith parameter and time-delay parameter of the system model are both set to 0.0025 s, both control strategies can quickly and stably track a constant value signal. However, when the time-delay parameter of the system model is mismatched to 0.008 s, the dynamic performance index is the transient time required to achieve and maintain the response of TTM at ±5% of the input signal amplitude. FLADRC-Smith control, compared with PI-Smith control, improves the dynamic response of TTM by 23.9% (Fig. 8). Finally, the FLADRC-Smith control method is compared with the PI-Smith control method in a second-order oscillatory time-delay system. The controller parameters are reasonably tuned so that the system error attenuation bandwidth is the same for both control methods (Fig. 9). Both control methods can track the constant value signal quickly and steadily when the Smith parameter and time-delay parameter of the system model are set to 0.0025 s. But when the time-delay parameter of the system model is mismatched to 0.008 s, the response transient time of TTM with PI-Smith control is 0.0564 s, while the FLADRC-Smith overshoot does not exceed ±5% of the input signal amplitude. It indicates that the required transient time is 0 when using the FLADRC-Smith control method, which significantly improves the dynamic response performance of TTM (Fig. 10).
The FLADRC-Smith control method can improve the bandwidth and the dynamic response performance of TTM effectively. In a pure time-delay system, this method improves the error attenuation bandwidth by 4.56 times compared with PI control. Under the same error attenuation bandwidth condition, the method improves the dynamic response performance of TTM by more than 20% with stronger suppression of internal and external disturbances compared with the PI-Smith control method in a pure time-delay system and a second-order oscillatory time-delay system.
