2. Key Laboratory of Optical Engineering, Chinese Academy of Sciences, Chengdu 610209, China;
3. University of Chinese Academy of Sciences, Beijing 100049, China
Disturbance rejection is one of the hottest topics in the highperformance servoing control areas^{15}. Optical telescopes also face this issue, because the telescopes' imaging quality and their ability to observe faint stars at the diffraction limit is limited by the disturbances induced by wind shaking, equipment vibration and platform vibration^{610}. High control bandwidth is the simplest method to reject vibrations. However, long integration time of the image sensor is always required for a high signal to noise, which restricts the closedloop control bandwidth. Therefore, the vibrations induced by telescope structures could not be fully compensated by classical control loops^{11, 12}. If disturbances are measurable, it is obvious that a disturbance feedforward control strategy is the most effective way to attenuate them^{1315}. Although the structural vibrations can be directly measured by extra inertial sensors, it is difficult to extract useful information to control the TipTilt mirror. Furthermore, the estimation accuracy of disturbance using inertial sensors is affected by random drifting. To overcome these difficulties, one intuitive idea (called basedmodel control) for estimating disturbances through known variables to reject vibration is proposed, such that the disturbances can be compensated by the estimations rather than other sensors. These basedmodel control techniques are attractive in many practical scenarios, such as spacecraft, airplanes, vehicles and other platforms, due to its simplification and effectiveness. Related works are organized as follows. Frequencybased design of ModalControl^{16} is presented to reduce the effects of distortions in groundbased telescopes. Simulations show that the proposed approach is a promising technique to improve the performance of the closedloop control system. An adaptive control algorithm^{17, 18} to reject vibrations is developed and adapted to the complex control architectures. This adaptive vibration cancellation algorithm was integrated into a telescope currently operating at the European Southern Observatory in Chile and verified experimentally. Recently, some popular controllers focusing on Linear Quadratic Gaussian (LQG) control laws or using H_{∞}/H_{2} synthesis methods^{1922} have been successfully implemented. Existing experiments demonstrate that LQG control can achieve better onsky performance than a feedback integrator controller in the condition of the optimally identified process model. As far as these techniques are concerned, the control model and the disturbance model play a crucial role in the TipTilt control system. Large model errors usually deteriorate the closedloop performance, and evenly cause the instability of the control system. Therefore, model identification is the priority when these methods are deployed. Although the control model of the TipTilt mirror exhibits good linearity in the lowfrequency domain, the middlefrequency or highfrequency nonlinearities and dynamics cause difficulties in model identification, resulting in hindering to reject the middlefrequency or highfrequency disturbance.
To relax these conditions, the disturbance observer control^{2327} is reviewed and discussed for TipTilt mirror control system in this paper. Furthermore, the errorbased DOBC controller^{2831} of TipTilt mirror only based on CCD is proposed here. It can be plugged into an existing feedback loop that leads to a generalized version of 1Q (Q is the designed filter) brought in the numerator of the original sensitivity function, resulting in the overall sensitivity function equal to zero in theory at the expected frequencies. To suppress different kinds of vibrations, relevant new Qfilters are also optimized to reject lowfrequency and highfrequency disturbances. In frequency domain, the errorbased DOBC with the improved Qfilters are analyzed in details, and its design procedure and implementation are also introduced. The remainder of this paper is arranged as follows: firstly we present a detailed introduction to conventional proportional integral (PI) control of the TipTilt mirror and give a brief introduction to LQG controller as well as disturbance feedforward (DFF) controller. Secondly, the new errorbased DOBC is proposed, which analyzes its characteristics, gives design process, and also sets up simulations and experiments to testify the errorbased DOBC of TipTilt mirror. And then, a new repetitive control^{3234} based DOBC is proposed. Eventually, we give the conclusions.
Conventional control of TipTilt mirrorThe optical configuration of the TipTilt mirror system^{28, 29} in the astronomical telescope is illustrated in Fig. 1(a), mainly including image sensor, splitter mirror, control unit, TipTilt mirror, adaptive mirror, and image processing unit. The image sensor can detect the TipTilt error caused by the telescope's structural vibrations. The classical control structure of the TipTilt mirror is briefly demonstrated in Fig. 1(b).
G(s) is the controlled plant, and C(s) is the PI controller. The time delay
$ {S'_R}(s) = \frac{{E(s)}}{{R(s)}} = \frac{1}{{1 + C(s)G(s){{\rm{e}}^{  \tau s}}}}, $  (1) 
$ {S'_D}(s) = \frac{{Y(s)}}{{D(s)}} = \frac{1}{{1 + C(s)G(s){{\rm{e}}^{  \tau s}}}}, $  (2) 
The TipTilt mirror openloop nominal response can be expressed as follows
$ G(s) = \frac{1}{{\frac{{{s^2}}}{{\omega _n^2}} + 2\frac{\xi }{{{\omega _n}}}s + 1}}\frac{1}{{{T_{\rm{e}}}s + 1}}. $  (3) 
The natural frequency of TipTilt mirror is usually up to kHz, and the damping factor ξ is much smaller than 1, leading to
$ \left\{ \begin{array}{l} \arg [P({\rm{j}}{w_{\rm{c}}})] \ge \frac{{\rm{ \mathsf{ π} }}}{4}\;, \;\;\;\;\;\;\;\;\;\;\;\;\left {P({\rm{j}}{w_{\rm{c}}})} \right = 1\\  20\lg \left {P({\rm{j}}{w_{\rm{g}}})} \right \ge 6\;, \;\;\;\arg [P({\rm{j}}{w_{\rm{g}}})] =  {\rm{ \mathsf{ π} }} \end{array} \right.. $  (4) 
Based on Eq. (4), we
can derive that
$ C(s) = \frac{{\rm{ \mathsf{ π} }}}{{4\tau }}\frac{{{T_{\rm{e}}}s + 1}}{s}. $  (5) 
This controller can stabilize the plant with phase margin no less than 45 degrees and magnitude margin more than 6 dB. Substituting Eq. (5) into Eq. (1), we can have
$ \hat S \approx \frac{1}{{1 + \frac{{\rm{ \mathsf{ π} }}}{{4\tau }}\frac{1}{s}{{\rm{e}}^{  \tau s}}}}. $  (6) 
From Eq. (6),
From above analysis, the closedloop performance is restricted by linear PI controller due to a low bandwidth. For a linear control system driven by additive white Gaussian noise, the linearquadraticGaussian (LQG) control problem^{36, 37} is to determine an optimal control law in the sense of minimizing the expected errors. The basic LQG control structure of TipTilt mirror is depicted in Fig. 2.
The closedloop performance of LQG control is dependent on the optimal controller, which is determined by the estimator. The key process in design LQG control is to choose an optimal metric to evaluate the performance. The accuracy of vibrations model plays an important role in the metric. Therefore, many works about LQG controller concentrate on model identifications in the TipTilt control system. Although LQG controller is implemented in discretetime domain, the closedloop transfer function of TipTilt mirror with the LQG controller can be obtained easily through bilinear transformation when the optimal controller is determined by the estimator.
Disturbance feed forward (DFF) controllerIt is unavoidable for image sensors to induce time delay, resulting in low control bandwidth to limit vibration rejection, so other sensors independent of image sensors are added to measure vibrations directly so that a disturbance feed forward controller is implemented to eliminate vibrations^{37}. Firstly, the vibrations are measured by additional inertial sensors such as accelerometers at the telescope structure and the vibrations are reconstructed by established filter techniques. The reconstructed signals are used feedforward for controlling the TipTilt mirror loop. The basic control structure is shown in Fig. 3.
The transfer function of the Fig. 3 can be derived as follows:
$ Y(s) = \frac{{C(s)G(s){{\rm{e}}^{  \tau s}}}}{{1 + C(s)G(s){{\rm{e}}^{  \tau s}}}}R(s) + \frac{{1  DFF(s)G(s)}}{{1 + C(s)G(s){{\rm{e}}^{  \tau s}}}}D(s). $  (7) 
The vibrations can be totally cancelled under the following condition:
$ DFF(s) = \frac{1}{{G(s)}}. $  (8) 
Obviously, this perfect condition cannot be satisfied due to model errors existed in the TipTilt control system. Although the partial compensation of Eq. (8) is effective to lowfrequency vibrations, the final performance is directly affected by the reconstructed accuracy of vibrations. In fact, DFF is effective only with vibrations generated by the optical platform. However, these disturbances can occur at any point along the optical link due to structural flexibility, and therefore could not be reconstructed precisely to feedforward control of the TipTilt mirror.
Disturbance observer control (DOBC)To avoid complexities of extra load on the TipTilt control system, the DOBC is proposed to improve the closedloop system. In comparisons with LQG controller, this new method can enhance the original PI control system without deteriorating stability. Furthermore, the control system in the DOBC mode can obviously exhibit the openloop and closedloop characteristics in frequency domain. Fig. 4 shows the conventional DOBC structure^{25, 26}. Q(s) is the designed filter. The inverse of the control plant G(s) is described by
The new closedloop sensitivity transfer functions illustrated in Fig. 5 can be expressed as follows:
$ {S_R}(s) = \frac{{1  Q(s)}}{{1 + C(s)G(s){{\rm{e}}^{  \tau s}} + ({{\rm{e}}^{  \tau s}}G(s)G_m^{  1}(s)  1)Q(s)}} $  (9) 
$ {S_D}(s) = \frac{{1  Q(s)}}{{1 + C(s)G(s){{\rm{e}}^{  \tau s}} + ({{\rm{e}}^{  \tau s}}G(s)G_m^{  1}(s)  1)Q(s)}}. $  (10) 
$ \begin{array}{l} W(s) = 1 + {{\rm{e}}^{  \eta s}}C(s)\;G(s) + \left( {G_m^{  1}(s)\;G(s)  1} \right)Q(s)\\ \;\;\;\;\;\;\;\; = \left( {1 + {{\rm{e}}^{  \eta s}}C(s)\;G(s)} \right)\left( {1 + \frac{{\left( {G_m^{  1}(s)\;G(s)  1} \right)Q(s)}}{{1 + {{\rm{e}}^{  \eta s}}C(s)\;G(s)}}} \right), \end{array} $  (11) 
Because the original feedback system is stable, it implies that the stability condition of the closedloop control system has to satisfy the following condition according to Small Gain Theorem:
$ {\left\ {\frac{{\left( {G_m^{  1}(s)\;G(s)  1} \right)\;Q(s)}}{{1 + {{\rm{e}}^{  \eta s}}C(s)\;G(s)}}} \right\_\infty } < 1. $  (12) 
Without doubt, the Qfilter could be designed as a lowpass filter^{26}, because the disturbance to be rejected is usually with low or medium frequency, whereas the sensor noise is with medium or high frequency. Therefore, the errorbased DOBC is able to estimate the disturbance and uncertainty in a low and medium frequency range but filter out the highfrequency measurement noise. The general form of lowpass filters can be described as follows:
$ {Q_{\rm{L}}}(s) = \frac{{\sum\limits_{k = 2}^{m  2} {C_m^k(} \tau s{)^k} + 1}}{{{{(\tau s + 1)}^m}}}. $  (13) 
The general filters of Eq. (13) are subject to the orders of the filter and the time delay. Q_{31}filter^{39, 40} is considered as an optimal lowpass filter for the closedloop performance in terms of bandwidth and robustness.
There is an example of the TipTilt mirror that compares Bode responses of the sensitivity functions in the modes of the PI and the DOBC controller in Fig. 6. The sampling frequency is 2000 Hz, and the time delay approximates with 0.0015. Obviously, Fig. 6 shows that the DOBC controller can achieve an extra improvement below the frequencies of 10 Hz.
From Fig. 7, the DOBC with the lowpass filter is verified effectively to mitigate the lowfrequency disturbances.
Bandpass filterHigh rejection bandwidth in the lowpass filter control mode could be limited due to the stability condition of Eq. (8), leading to no mitigation of highfrequency disturbances. To overcome this problem, another way of canceling disturbances^{29} is expressed as follows
$ 1  Q(s)]D(s) = 0. $  (14) 
Suppose that the disturbance D(s) can be defined as:
$ D(s) = {A_i}\sum\limits_{i = 0}^k {\varphi (s, {w_i})} , $  (15) 
where w_{i} is the ith disturbance with the maximum amplitude of A_{i}, and
$ 1  Q(s)]D(s) = \prod\limits_{i = 1}^k {(\frac{{{s^2}}}{{w_i^2}} + 1)} D(s) = 0. $  (16) 
The above condition is impractical, because
$ ESF(s) = 1  Q(s) = \prod\limits_{i = 1}^k {\frac{{\frac{{{s^2}}}{{w_i^2}} + {\zeta _i}\frac{s}{{{w_i}}} + 1}}{{\frac{{{s^2}}}{{w_i^2}} + {\alpha _i}{\zeta _i}\frac{s}{{{w_i}}} + {\beta _i}}}} , $  (17) 
here, the three parameters follow that
$ \begin{array}{l} Q(s) = 1  \prod\limits_{i = 1}^k {\frac{{\frac{{{s^2}}}{{w_i^2}} + {\zeta _i}\frac{s}{{{w_i}}} + 1}}{{\frac{{{s^2}}}{{w_i^2}} + {\alpha _i}{\zeta _i}\frac{s}{{{w_i}}} + 1}}} \\ \;\;\;\;\;\;\; = \frac{{\prod\limits_{i = 1}^k {\left( {\frac{{{s^2}}}{{w_i^2}} + {\alpha _i}{\zeta _i}\frac{s}{{{w_i}}} + 1} \right)}  \prod\limits_{i = 1}^k {\left( {\frac{{{s^2}}}{{w_i^2}} + {\zeta _i}\frac{s}{{{w_i}}} + 1} \right)} }}{{\prod\limits_{i = 1}^k {\left( {\frac{{{s^2}}}{{w_i^2}} + {\alpha _i}{\zeta _i}\frac{s}{{{w_i}}} + 1} \right)} }}. \end{array} $  (18) 
From Eq. (18), there is one differentiator in the numerator of Q(s), and meanwhile Q(s) features a lowpass characteristic since its relative degree is one. Therefore, the Q(s) is a bandpass filter. In a below example, the openloop vibrations are shown in Fig. 8 that include multiple peak areas.
Due to three areas of energetic vibrations existing in the high frequency, ESF(s) can be designed as follows
$ ESF(s) = ES{F_1}(s) \times ES{F_2}(s) \times ES{F_3}(s), $  (19) 
$ {\rm{Where,}}\;ES{F_1}(s) = \frac{{0.000659{s^2} + 2.567{\rm{e}}  5s + 1}}{{0.000659{s^2} + 0.0154s + 1}}, $  (20) 
$ ES{F_2}(s) = \frac{{2.093{\rm{e}}  4{s^2} + 1.447{\rm{e}}  5s + 1}}{{2.093{\rm{e}}  4{s^2} + 0.002894s + 1}}, $  (21) 
$ ES{F_3}(s) = \frac{{5.234{\mathop{\rm e}\nolimits}  5{s^2} + 7.234{\rm{e}}  6s + 1}}{{5.234{\rm{e}}  5{s^2} + 7.234{\rm{e}}  4s + 1}}. $  (22) 
As can be seen in Fig. 9, the 6order notch filter shown in Eq. (19) is designed at the centered frequencies of 6 Hz, 11 Hz, and 21 Hz respectively. As a result, Q(s) is a bandpass filter.
The correction results are given in Fig. 10. The closedloop errors are 2.32 μrad RMS and 0.63 μrad RMS respectively. In the DOBC mode, the RMS errors are less than 30% of that in the original loop. Peak disturbances still appear at 6 Hz, 11Hz and 21 Hz in the PI control mode, while they disappear in the DOBC mode.
Repetitive control of TipTilt mirrorIn this chapter, a learningtype control strategy called repetitive control is applied to a new Qfilter for coping with unknown disturbances. An improved Qfilter^{41} based on moving average filter is proposed to reduce the waterbed effect, which implies that additional gain amplifications can be mitigated between both periodic frequencies in comparison with conventional repetitive controller.
Qfilter design and performance analysisThe classical repetitive controller (CRC) is expressed as follows:
$ {Q_{{\rm{CRC}}}}({{\rm{e}}^{  sT}}) = {{\rm{e}}^{  sNT}}q({{\rm{e}}^{  sT}}, l). $  (23) 
A lowpass filter
$ \begin{array}{l} q({{\rm{e}}^{  sT}}, l) = {a_l}{{\rm{e}}^{slT}} + {a_{l  1}}{{\rm{e}}^{s(l  1)T}} + \cdots {a_0} + \cdots \\ \;\;\;\;\;\;\;\;\;\;\;\; + {a_{l  1}}{e^{  s(l  1)T}} + {a_l}{e^{  slT}}, \end{array} $  (24) 
here l is a positive integer, and the coefficients meet
$ {E_{{\rm{CRC}}}}({{\rm{e}}^{  sT}}) = 1  {{\rm{e}}^{  sNT}}q({{\rm{e}}^{  sT}}, l). $  (25) 
The maximum value of Eq. (25) is equal to 2. The magnitude response of Eq. (25) is shown in Fig. 11, showing waterbed effect at nonperiodic frequencies.
For relaxing disturbance amplification at the nonrepetitive frequencies, we proposed an improved repetitive controller (IRC), and the new extra sensitivity function is given below
$ {E_{{\rm{IRC}}}}({{\rm{e}}^{  sT}}) = \frac{{1  {{\rm{e}}^{  sNT}}q({{\rm{e}}^{  sT}}, l)}}{{1  \alpha {{\rm{e}}^{  sNT}}q({{\rm{e}}^{  sT}}, l)}}\;, \;\;\;\alpha \in [0, 1]. $  (26) 
Due to
$ {\left {{E_{{\rm{IRC}}}}({{\rm{e}}^{  sT}})} \right^2} = \frac{{2  2\cos (w/N)}}{{1 + {\alpha ^2}  2\alpha \cos (w/N)}}. $  (27) 
The maximum value of Eq. (27) at the frequencies of
The Qfilter of the IRC is derived below
$ \begin{array}{l} {Q_{{\rm{IRC}}}}({{\rm{e}}^{  sT}}) = 1  {E_{{\rm{IRC}}}}({{\rm{e}}^{  sT}})\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \frac{{(1  \alpha ){{\rm{e}}^{  sNT}}q({{\rm{e}}^{  sT}}, l)}}{{1  \alpha {{\rm{e}}^{  sNT}}q({{\rm{e}}^{  sT}}, l)}}. \end{array} $  (28) 
In this section, a design example of the TipTilt mirror under the condition of structural vibrations is exhibited. The sensitivity transfer functions of the TipTilt mirror with IRC, CRC and I (integrator) are shown in Fig. 13. The benchmark line in blue is the closedloop rejection with I controller, of which the rejection bandwidth is about 80 Hz, and no more than 20 dB suppression at the basic frequency of repetitive disturbance. IRC rejects the vibration peak with slightly magnifications of other frequencies, leading to an efficient improvement.
Figure 14 shows the correction results in three different kinds of controllers. The resulting TipTilt errors are 1.28, 0.91 and 0.66 μrad RMS respectively with I, CRC and IRC. The CRC has reduced TipTilt error at the steady state, less than 72% of the original error, while in IRC mode the error is about 52% of the original error. The spectra are shown in Fig. 15 when the repetitive controllers are employed. The peak vibrations at the basic frequency of 6.6 Hz (such as 6.6 Hz, 13.2 Hz, 19.8 Hz and 26.4 Hz and so on) are significantly reduced.
Conclusion
In this paper, the control methodologies of TipTilt mirror to reject structural vibrations in optical telescopes are reviewed and discussed. We focus on the survey of an errorbased disturbance observer controller of the TipTilt mirror. In this mode, a generalized version of 1Q is cascaded into the original sensitivity function, and therefore the design of the Qfilter plays a vital role in the closedloop control system of the TipTilt mirror. The lowpass filter, bandpass filter and repetitive filter of the Qfilter are proposed to cope with vibrations in different frequency modes. The implementation of the proposed DOBC structure, the optimization of the control parameters and the analysis of the closeloop characteristics from the viewpoint of its practical implementation are provided in this paper. The key problem of designing the improved Qfilter is to determinate disturbance frequencies. A scanning method^{42} can make sure the accuracy in detection of the interesting vibrations. The control bandwidth of the TipTilt mirror is not improved with this proposed controller, but disturbance attenuation is greatly enhanced at the disturbance frequencies. Furthermore, this improved control mode cannot cause big magnifications at nondisturbance frequencies, and therefore an effective improvement of the closedloop performance is obtained. In comparison with existed methods, this improved technique can be carried out in real time if disturbance frequencies are given. Disturbance suppression, especially beyond the Nyquist frequency may be a continuing and attractive research in the mechanical control area. Nonlinear techniques^{4346} such as sliding mode control, repetitive control, backstepping control and active disturbance rejection control are promising for further rejection performance in the high precision control system.
AcknowledgementsThis work was in part supported by Youth Innovation Promotion Association, Chinese Academy of Sciences
Competing interestsThe authors declare no competing financial interests.
1. 
Chen X, Tomizuka M. Overview and new results in disturbance observer based adaptive vibration rejection with application to advanced manufacturing. Int J Adaptive Control Signal Process
29, 14591474 (2015) [Crossref] 
2. 
Shtessel Y, Edwards C, Fridman L, Levant A. Sliding Mode Control and Observation
(New York: Springer, 2014).

3. 
Tomizuka M. Control methodologies for manufacturing applications. Manuf Lett
1, 4648 (2013) [Crossref] 
4. 
Gao Z Q. Active disturbance rejection control: a paradigm shift in feedback control system design. In Proceedings of 2006 American Control Conference (IEEE, 2006); http://doi.org/10.1109/ACC.2006.1656579.

5. 
Han J Q. From PID to active disturbance rejection control. IEEE Trans Ind Electron
56, 900906 (2009) [Crossref] 
6. 
Deng C, Tang T, Mao Y et al. Enhanced Disturbance Observer Based on Acceleration Measurement for Fast Steering Mirror Systems. IEEE Photonics Journal
9, 111 (2017) [Crossref] 
7. 
Lozi J, Guyon O, Jovanovic N, Singh G, Goebel S et al. Characterizing and mitigating vibrations for SCExAO. Proc SPIE
9909, 99090J (2016) [Crossref] 
8. 
Rao C H, Gu N T, Zhu L, Huang J L, Li C et al. 1.8m solar telescope in China: Chinese large solar telescope. J Astron Telescopes Instrum Syst
1, 024001 (2015) [Crossref] 
9. 
MacMartin D G. Control challenges for extremely large telescopes. Proc SPIE
5054, 275286 (2003) [Crossref] 
10. 
Gawronski W. Control and pointing challenges of large antennas and telescopes. IEEE Trans Control Syst Technol
15, 276289 (2007) [Crossref] 
11. 
Petit C, Sauvage J F, Fusco T, Sevin A, Suarez M et al. SPHERE extreme AO control scheme: final performance assessment and on sky validation of the first autotuned LQG based operational system. Proc SPIE
9148, 91480O (2014) [Crossref] 
12. 
MacMartin D G, Thompson H A. Vibration budget for observatory equipment. J Astron Telesc Instrum Syst
1, 034005 (2015) [Crossref] 
13. 
Böhm M, Pott J U, Kürster M, Sawodny O, Defrère D et al. Delay compensation for real time disturbance estimation at extremely large telescopes. IEEE Trans Control Syst Technol
25, 13841393 (2017) [Crossref] 
14. 
Glück M, Pott J U, Sawodny O. Piezoactuated vibration disturbance mirror for investigating accelerometerbased tiptilt reconstruction in large telescopes. IFACPapersOnLine
49, 361366 (2016) [Crossref] 
15. 
Ken F, Susumu Y, Nobutaka B, Shinichiro S, Atsuo T et al. Accelerometer assisted high bandwidth control of tiptilt mirror for precision pointing stability. In Proceedings of IEEE on Aerospace Conference (IEEE, 2011); http://doi.org/10.1109/AERO.2011.5747383.

16. 
Agapito G, Battistelli G, Mari D, Selvi D, Tesi A et al. Frequency based design of modal controllers for adaptive optics systems. Opt Express
20, 2710827122 (2012) [Crossref] 
17. 
Muradore R, Pettazzi L, Fedrigo E. Adaptive vibration cancellation in adaptive optics: An experimental validation. In Proceedings of 2014 European Control Conference 24182423 (IEEE, 2014); http://doi.org/10.1109/ECC.2014.6862434.

18. 
Pettazzi L, Fedrigo E, Muradore R, Haguenauer P, Pallanca L. Improving the accuracy of interferometric measurements through adaptive vibration cancellation. In Proceedings of 2015 IEEE Conference on Control Applications 95100 (IEEE, 2015); http://doi.org/10.1109/CCA.2015.7320616.

19. 
Yang K J, Yang P, Chen S Q et al. Vibration identification based on LevenbergMarquardt optimization for mitigation in adaptive optics systems. Appl Opt
57, 28202826 (2018) [Crossref] 
20. 
Böhm M, Pott J U, Kürster M, Sawodny O. Modeling and identification of the optical path at ELTs a case study at the LBT. IFAC Proc Volumes
46, 249255 (2013) [Crossref] 
21. 
Castro M, Escárate P, Zuñiga S, Garcés J, Guesalaga A. Closed loop for tiptilt vibration mitigation. In Applications of Lasers for Sensing and Free Space Communications 2015 (OSA, 2015); https://doi.org/10.1364/AOMS.2015.JT5A.28.

22. 
Petit C, Conan J M, Kulcsár C, Raynaud H F. Linear quadratic Gaussian control for adaptive optics and multiconjugate adaptive optics: experimental and numerical analysis. J Opt Soc Am A
26, 13071325 (2009) [Crossref] 
23. 
Radke A, Gao Z Q. A survey of state and disturbance observers for practitioners. In Proceedings of 2006 American Control Conference (IEEE, 2006); http://doi.org/10.1109/ACC.2006.1657545.

24. 
Sariyildiz E, Ohnishi K. Stability and robustness of disturbance observerbased motion control systems. IEEE Trans Ind Electron
62, 414422 (2015) [Crossref] 
25. 
Chen W H, Yang J, Guo L, Li S H. Disturbanceobserverbased control and related methodsan overview. IEEE Trans Ind Electron
63, 10831095 (2016) [Crossref] 
26. 
Kim J S, Back J, Park G. Design of Qfilters for disturbance observers via BMI approach. In Proceedings of the 14th International Conference on Control, Automation and Systems 11971200 (IEEE, 2014); http://doi.org/10.1109/ICCAS.2014.6987741.

27. 
Zheng M H, Zhou S Y, Tomizuka M. A design methodology for disturbance observer with application to precision motion control: an Hinfinity based approach. In Proceedings of 2017 American Control Conference 35243529 (IEEE, 2017); http://doi.org/10.23919/ACC.2017.7963492.

28. 
Tang T, Qi B, Yang T. YoulaKucera parameterizationbased optimally closedloop control for tipTilt compensation. IEEE Sens J
18, 61546160 (2018) [Crossref] 
29. 
Tang T, Yang T, Qi B, Cao L, Ren G et al. Errorbased plugin controller of tiptilt mirror to reject telescope's structural vibrations. J Astron Telesc Instrum Syst
4, 049004 (2018) [Crossref] 
30. 
Chen X, Jiang T Y, Tomizuka M. Pseudo YoulaKucera parameterization with control of the waterbed effect for local loop shaping. Automatica
62, 177183 (2015) [Crossref] 
31. 
Jiang T Y, Chen X. Transmission of signal nonsmoothness and transient improvement in addon servo control. IEEE Trans Control Syst Technol
26, 486496 (2017) [Crossref] 
32. 
Zhou K L, Wang D W, Zhang B, Wang Y G. Plugin dualmodestructure repetitive controller for CVCF PWM inverters. IEEE Trans Ind Electron
56, 784791 (2009) [Crossref] 
33. 
Cho Y, Lai J S. Digital plugin repetitive controller for singlephase bridgeless PFC converters. IEEE Trans Power Electron
28, 165175 (2013) [Crossref] 
34. 
Chen X, Tomizuka M. New repetitive control with improved steadystate performance and accelerated transient. IEEE Trans Control Syst Technol
22, 664675 (2014) [Crossref] 
35. 
Mahani N K Z, Sedigh A K, Bayat F M. Performance evaluation of nonminimum phase linear control systems with fractional order partial polezero cancellation. In Proceedings of the 9th Asian Control Conference (IEEE, 2013); http://doi.org/10.1109/ASCC.2013.6606329.

36. 
Stengel R F. Optimal Control and Estimation
(New York: Dover Publications, 1994).

37. 
Siouris G M. Errata to an engineering approach to optimal control and estimation theory. IEEE Aero Electron Syst Mag
12, 37 (1997) [Crossref] 
38. 
Glück M, Pott J U, Sawodny O. Investigations of an accelerometerbased disturbance feedforward control for vibration suppression in adaptive optics of large telescopes. Pub Astron Soc Pacific
129, 065001 (2017) [Crossref] 
39. 
Kempf C J, Kobayashi S. Disturbance observer and feedforward design for a highspeed directdrive positioning table. IEEE Trans Control Syst Technol
7, 513526 (1999) [Crossref] 
40. 
Kim B K, Chung W K. Advanced disturbance observer design for mechanical positioning systems. IEEE Trans Ind Electron
50, 12071216 (2003) [Crossref] 
41. 
Tang T, Xu N S, Yang T, Qi B, Bao Q L. Vibration rejection of TipTilt mirror using improved repetitive control. Mech Syst Signal Process
116, 432442 (2019) [Crossref] 
42. 
Guesalaga A, Neichel B, O'Neal J, Guzman D. Mitigation of vibrations in adaptive optics by minimization of closedloop residuals. Opt Express
21, 1067610696 (2013) [Crossref] 
43. 
Huang Y, Xue W C. Active disturbance rejection control: Methodology and theoretical analysis. ISA Trans
53, 963976 (2014) [Crossref] 
44. 
Chen W H. Disturbance observer based control for nonlinear systems. IEEE/ASME Trans Mech
9, 706710 (2004) [Crossref] 
45. 
Won D, Kim W, Shin D, Chung C C. Highgain disturbance observerbased backstepping control with output tracking error constraint for electrohydraulic systems. IEEE Trans Control Syst Technol
23, 787795 (2015) [Crossref] 
46. 
Liu L P, Fu Z M, Song X N. Sliding mode control with disturbance observer for a class of nonlinear systems. Int J Autom Comput
9, 487491 (2012) [Crossref] 