In the fields of earth observation, deep space detection, laser communication, and directional energy weapon, the target needs to be observed and pointed at accurately. Acquisition, tracking, and pointing (ATP) systems are usually designed to stabilize the line of sight (LOS) within submicro radian levels. In the case of an ATP system mounted on a mobile platform, angular disturbances experienced by the mobile platform will seriously affect the LOS. To overcome the problem that the sampling frequency of detectors is usually limited and achieving several hundreds of hertz is difficult, the widebandwidth inertial reference system (WBIRS) and fast steering mirror are usually integrated into ATP systems to mitigate these angular disturbances. To reduce the structural stress, a flexible support providing two rotational degrees of freedom is usually adopted for the system. However, the occurrence of resonant points within the bandwidth will be inevitable. Measurements have to be taken to compensate these lowfrequency resonant points to realize a wide bandwidth and high precision. In this paper, the lowfrequency resonant points of a system were simulated using finite element analysis and tested by a system identification method. The results show that the firstorder resonance happened at 34.5 Hz with a gain of 28 dB. An improved doubleT notch filter was designed and applied in a realtime system to suppress the resonance at 34.5 Hz. The experimental results show that the resonance was significantly suppressed. In particular, the resonance peak was reduced by 79.37%. In addition, the closedloop system settling time was reduced by 36.2%.
HIGHLIGHTS

This study investigates WBIRS, which can stabilize the LOS within submicro radian levels in ATP systems.

An improved doubleT notch filter was designed and implemented in a realtime system to suppress the resonance.

The peak and average resonance attenuation rate reached 63.03% and 79.37%. The settling time was reduced by 36.2%.
1. Introduction
In the fields of earth observation, deep space exploration, longrange laser communication, and directional energy weapons,^{1–5} it is necessary to use acquisition, tracking, and pointing (ATP) systems to stabilize the line of sight (LOS)within submicro radian levels.^{6} Due to the increasing requirements for the maneuverability of ATP systems, ATP systems based on the earth cannot meet practical demands. Moreover, ATP systems set up on mobile platforms such as ships, planes, and satellites were present in historic times. In the case of ATP systems mounted on mobile platforms,^{7} to make the LOS stable, the core problem is to suppress the angular disturbance experienced by the mobile platform.^{8} Integrating the widebandwidth inertial reference system (WBIRS) and fast steering mirror into an ATP system is an effective solution to suppress the influence of angular disturbances on the LOS.^{9}
The basic function of the system is to output a reference beam that is not affected by the vibration of the mobile platform itself.^{10} A group of angular rate sensors is used to measure the angular disturbance. The controller controls the motions of voice coil motors (VCMs) to eliminate the vibration. Then, the light source on the system can output a stable reference beam. The fast steering mirror keeps the LOS consistent with the path of the reference beam, ensuring that the LOS is always stable in the ATP system.^{11}
Research results show that the mobile platforms experience angular disturbances from nearly DC to 200 Hz. Thus, the system is required to suppress the angular disturbances in such a wide bandwidth.^{12} To reduce the structural stress, a flexible support with two degrees of freedom is generally used for the system's support, providing small stiffness in actuating directions and large stiffness in other nonactuating directions.^{13} However, this kind of flexible support will inevitably introduce one to two resonance points within the bandwidth. To ensure the system stability within the designed bandwidth, measurements must be taken to compensate these lowfrequency resonance points.^{14}
Accurate measurement of the resonance point is the premise of suppression. Levy's identification method is very suitable for measuring and analyzing the resonance characteristics of the system. In addition, the whole transfer function of the system can be obtained through this method, which is convenient for the design of the resonance suppression. Presently, numerous methods of resonance suppression exist. They can be divided into two categories: the active suppression method (ASM) and the passive suppression method (PSM). The ASM usually suppresses the resonance by improving the control method or changing the feedback method, whereas the PSM can suppress the resonance by adding special filters into the system without changing the design of the whole control system.
Considering studies on the ASM, in 2015, Lin et al. employed different control methods (incomplete differential method and zeropole cancellation method) to suppress resonance. The study proposed that both methods can suppress resonance to some extent, but both are too difficult for precise parameter determination and circuit implementation.^{15} In addition, Jin et al. adopted different feedback methods to suppress resonance. These methods were designed to match motor inertia and load inertia, according to different ranges of inertia ratio.^{16} In 2017, Du et al. used an adaptive fuzzy logic system to onlineestimate nonlinear parameters in the study of resonance suppression. Then, they designed an adaptive fuzzy controller based on the backward step method. The controller can effectively suppress the influence of resonance in servo systems. However, this study also proved that this method is not suitable for practical engineering.^{17} In 2017, Yang suppressed resonance by compensating the electromagnetic torque and increasing the system damping coefficient. This method was based on accurately extracting the resonancefrequency component in the speed feedback.^{18} The above ASM studies show that the ASM requires changing the control structure or feedback structure. Redesigning the control system and control algorithm may even be needed. They are not suitable for practical engineering. However, the PSMs are simpler and more widely used in engineering. As for studies on the PSM, in 2014, Wang et al., on the study of resonance suppression, proposed a biquadratic resonance suppression digital filter and applied the filter to a practical system. The tracking accuracy and bandwidth were effectively improved.^{19} In 2018, Li et al. proposed an optimization method of notch filtering. This method simplifies the design method of the notch filter that is used in resonance suppression and makes the notch filter easier for application in practical engineering.^{20} These PSM studies show that the PSM is more suitable to the system. Therefore, in the present study, an improved doubleT notch filter was designed and added to the system to suppress the system resonance.
In this paper, the basic working principle of the system is introduced, and then, a thirdorder theoretical model is constructed to describe the system. Finite element analysis and experiments were conducted to obtain a clear review of the system resonance characteristics, and an improved doubleT notch filter was designed and verified in a realtime system.
2. Basic principle of WBIRS
2.1. Working principle
Figs. 1 and 2 show the diagrams of the WBIRS structure and basic working principle, respectively. A twodegreeoffreedom flexure hinge connects the stabilized platform to the base. It provides two rotational freedoms around the xand yaxes. The angular rate sensors are mounted to the stabilized platform. Four VCMs and eddy current displacement sensors (ECDSs) are also mounted between the stabilized platform and the base.^{21} A stable reference light source is mounted on the platform. When the base is experiencing angular disturbances, the inertial sensors send the angular vibration to the controller. The controller controls the VCM to drive the platform moving in the opposite direction to stabilize the platform plane. The RLS outputs a stable reference beam to achieve stability of the reference axis in the ATP system.
The structure of the flexure hinge is shown in Fig. 3. The flexure hinge rotates in the directions of x and yaxes. These two directions are called the main working directions of the flexure hinge. The stiffnesses in the main working directions should be designed to be much smaller than those in the other directions. The openloop characteristics of the system with this structure are illustrated in Fig. 4. It has a typical firstorder resonance and highorder coupling resonance characteristics. The most basic structural resonance is the firstorder resonance f_{1}, which is caused by the design of a flexible support and the whole system structural design. It has the greatest influence on the control stability of the system. This resonance is directly related to the stiffness K_{θ} of the flexible support structure and the rotational inertia J of the whole system, as shown in Eq. (1). The stiffness, moment of inertia, and the theoretical firstorder resonance of the system are presented in Table 1.
Direction .  Rotational inertiaJ(kg ∙ m^{2}) .  StiffnessK_{θ}(N ∙ m/rad) .  Firstorder resonancef_{1}(Hz) . 

Rotary in xaxis  3449.423 × 10^{−6}  218.25  40.03 
Rotary in yaxis  3831.379 × 10^{−6}  218.27  37.987 
Torsion in zaxis  3862.081 × 10^{−6}  6.3801 × 10^{4}  646.88 
Pullpressing  1.659  1.43926 × 10^{8}  1482.85 
Direction .  Rotational inertiaJ(kg ∙ m^{2}) .  StiffnessK_{θ}(N ∙ m/rad) .  Firstorder resonancef_{1}(Hz) . 

Rotary in xaxis  3449.423 × 10^{−6}  218.25  40.03 
Rotary in yaxis  3831.379 × 10^{−6}  218.27  37.987 
Torsion in zaxis  3862.081 × 10^{−6}  6.3801 × 10^{4}  646.88 
Pullpressing  1.659  1.43926 × 10^{8}  1482.85 
In the openloop characteristics, highorder resonances f_{2} caused by coupling also exist. Their transfer functions can be approximated as a series of several oscillation elements. Because the highorder resonance occurs in the highfrequency band, which can even be outside the system operating frequency, this resonance has a less influence degree on the control stability of the system than the firstorder resonance. Therefore, this study mainly investigates the suppression of the firstorder resonance. For highorder resonances, theoretical modeling and simulation are performed.
The firstorder resonance influences the control system stability. As illustrated in Fig. 5, after the system openloop transfer function is corrected, the characteristic curve crosses the 0 dB line from the cutoff frequency f_{0}. Due to the existence of a resonance peak, the gain of the frequency band after the cutoff frequency may be greater than 0 dB again. Then, the system will oscillate, failing to meet the stability requirements. The characteristic curve needs to be moved down to make the system stable. However, this causes the cutoff frequency to move leftward, which consequently reduces the system bandwidth. If the resonance suppression of the system is carried out in advance before the correction is carried out, the system stability can be satisfied without sacrificing the bandwidth.
2.2. System modeling
The system dynamic model is shown in Fig. 6.
The system moment balance equation is as follows:
The voltage balance equation of the VCM is as follows:
The definitions and units of specific parameters in the above two equations are presented in Table 2.
Parameter .  Definition .  Parameter .  Definition . 

M (N ∙ m)  Electromagnetic moment of motor  i (A)  Loop current 
J (kg ∙ m^{2})  Moment of inertia of the platform  u (V)  Supply voltage of motor 
θ (rad)  Angle of platform  R (Ω)  Equivalent impedance of the motor 
c (N ∙ m ∙ s/rad)  Damping coefficient  k_{b} (V ∙ s/m)  Counter electromotive force coefficient 
l (m)  Motor mounting radius  v (m/s)  Motor speed 
k (N ∙ m/rad)  Stiffness of hinge  m_{c} (kg)  Mass of electric motor coil 
k_{f} (N ∙ A)  Moment coefficient of motor  x (m)  Motor line displacement 
L_{a} (H)  Equivalent inductive reactance of motor 
Parameter .  Definition .  Parameter .  Definition . 

M (N ∙ m)  Electromagnetic moment of motor  i (A)  Loop current 
J (kg ∙ m^{2})  Moment of inertia of the platform  u (V)  Supply voltage of motor 
θ (rad)  Angle of platform  R (Ω)  Equivalent impedance of the motor 
c (N ∙ m ∙ s/rad)  Damping coefficient  k_{b} (V ∙ s/m)  Counter electromotive force coefficient 
l (m)  Motor mounting radius  v (m/s)  Motor speed 
k (N ∙ m/rad)  Stiffness of hinge  m_{c} (kg)  Mass of electric motor coil 
k_{f} (N ∙ A)  Moment coefficient of motor  x (m)  Motor line displacement 
L_{a} (H)  Equivalent inductive reactance of motor 
Eq. (6) shows that the inertial reference system model has a zerothorder numerator and a thirdorder denominator. According to the polynomial theory, a polynomial with a thirdorder denominator always has a real root. Therefore, the transfer function of the system can be decomposed into a series of an inertial element and a secondorder oscillation element, as shown in Eq. (7).
In addition, due to the limitation of mechanical stiffness, one or more highorder resonance points may exist in the high frequency band of the system. Each highorder resonance point can be expressed by a double secondorder element. The actual model of WBIRS is as follows:
3. Improved doubleT notch filter
A notch filter is a kind of bandstop filter that attenuates the signals at some specific frequency points, and it is similar to a passthrough filter in dealing with signals with other frequencies. Unlike the conventional doubleT notch filter, the improved doubleT notch filter can independently design the notch bandwidth and depth. The transfer function of the improved doubleT notch filter can be expressed as
where k_{1} is the notch depth coefficient, k_{2} is the notch bandwidth coefficient, and ω_{n} is the notch frequency. The amplitude and bandwidth of the improved doubleT notch filter at the notch frequency ω_{n} can be described as A_{ωn} and B_{ωn}, respectively. Their mathematical descriptions are as follows:
The relationships between the frequency response of the improved doubleT notch filter and k_{1}, k_{2} are depicted in Fig. 7. As seen, the notch depth decreases with the increase of k_{1}, and the notch width increases with the increase of k_{2}.
The design process of the improved doubleT notch filter is given as follows:
Measurements need to first be taken to know which frequency should be suppressed.
Supposing that the resonance peak A_{m}needs to be attenuated to the design threshold A_{h}, then
To ensure a certain margin in the design process, the margin parameter α is defined, whose values are between 0.7 and 0.9.
In addition, the notch bandwidth can be expressed as
where ω_{1} and ω_{2} are the intersection frequencies of the openloop characteristic of the system and the notch characteristic, respectively.
Finally, the designed notch filter needs to be discretized. The sampling rate should be the same as that of the controller.
4. Simulation and experiment
4.1. Finite element analysis
Finite element analysis was conducted in Workbench to simulate the system resonance characteristics. The results of modal analysis are shown in Fig. 8 and summarized in Table 3. The first two modal shapes are the rotational motions around the y and xaxes, and the corresponding frequencies are 34.292 Hz and 33.105 Hz, respectively.
Modal order .  Simulation frequency (Hz) .  Theoretic frequency (Hz) .  Modal shapes . 

Firstorder (a)  33.105  37.987  Rotary in yaxis 
Secondorder (b)  34.292  40.03  Rotary in xaxis 
Thirdorder (c)  603.39  646.88  Torsion in zaxis 
Fourthorder (d)  1301.7  1482.85  Pullpressing in zaxis 
Modal order .  Simulation frequency (Hz) .  Theoretic frequency (Hz) .  Modal shapes . 

Firstorder (a)  33.105  37.987  Rotary in yaxis 
Secondorder (b)  34.292  40.03  Rotary in xaxis 
Thirdorder (c)  603.39  646.88  Torsion in zaxis 
Fourthorder (d)  1301.7  1482.85  Pullpressing in zaxis 
4.2. System identification
Levy's method was adopted to identify the system model. The main idea of Levy's method is to minimize the error between the measured and estimated outputs according to the generalized error criterion.^{22}
The block diagram of the experimental system is shown in Fig. 9. A multiharmonic sinusoidal signal with frequency ranging from 1 to 600 Hz and an interval of 0.1 Hz is used as the analog input signal of the driver. The driver converts these analog signals into a pulsewidthmodulation wave to control voltagecontrolled amplifiers. Eddy current displacement sensors measure the rotation angle of the platform. The DAQ device simultaneously collects the excitation signal and the eddy current sensor output signal and transfers these signals to the host computer via serial ports.
The identification results are presented in Table 4. The fitting degrees (Rs) of the identification exceed 90%, indicating that the results are reliable. Bode diagrams of identifications are shown in Fig. 10. The lowfrequency resonance is shown to be between 34 Hz and 35 Hz, which differs slightly from the numerical calculation result (33.105 Hz). This difference is mainly caused by machining errors and the material parameter error. Different from the active resonance suppression method, the notch compensator has a degree of robustness. Therefore, the resonance frequency can be selected as 34.5 Hz.
Times .  Excitation signal (mV) .  Identification result .  Resonance frequency .  Fitting degree Rs . 

3  500  $ G ( s ) = 1 . 37606 \xd7 10 6 ( s + 112 . 747 ) ( s 2 + 4 . 89015 s + 37474 . 8 ) $  34.5  0.936 
3  1000  $ G ( s ) = 1 . 33891 \xd7 10 6 ( s + 114 . 207 ) ( s 2 + 4 . 78291 s + 37475 . 5 ) $  34.5  0.988 
3  1500  $ G ( s ) = 1 . 33831 \xd7 10 6 ( s + 116 . 088 ) ( s 2 + 4 . 79000 s + 37436 . 7 ) $  35  0.905 
Times .  Excitation signal (mV) .  Identification result .  Resonance frequency .  Fitting degree Rs . 

3  500  $ G ( s ) = 1 . 37606 \xd7 10 6 ( s + 112 . 747 ) ( s 2 + 4 . 89015 s + 37474 . 8 ) $  34.5  0.936 
3  1000  $ G ( s ) = 1 . 33891 \xd7 10 6 ( s + 114 . 207 ) ( s 2 + 4 . 78291 s + 37475 . 5 ) $  34.5  0.988 
3  1500  $ G ( s ) = 1 . 33831 \xd7 10 6 ( s + 116 . 088 ) ( s 2 + 4 . 79000 s + 37436 . 7 ) $  35  0.905 
4.3. Design and simulation of doubleT notch filter
According to the system identification results in Section 3, the doubleT notch filter can be designed as.
ω_{n} = ω_{0} = 2π ∙ 34.5 = 216.7 rad/s,A_{m} = 28 dB.
ω_{1} = 32 rad/s,ω_{2} = 316.02 rad/s
Then, the notch filter model can be obtained:
The transfer function is discretized by bilinear transformation as (the sampling frequency is 50 kHz)
The improved doubleT notch filter is verified in simulation. The first simulation corresponds to the openloop frequency response of the system. The frequency responses of the identified model and the model after the notch filter is applied to the system are compared in Fig. 11. The results show that the resonance of the system around 34.5 Hz is significantly suppressed. The second simulation shows the ability of the designed notch filter to process different frequency signals. The result of mixing signals (10 Hz, 34.5 Hz, and 50 Hz) processed by the designed notch filter is shown in Fig. 12. The third simulation corresponds to the closedloop step response of the system. A closedloop model simulation system was built based on the Simlink module of Matlab. Before and after the designed notch filter was added to the system, step response curves of the system were recorded (Fig. 13). The oscillation of the system was significantly suppressed using the notch filter. The system eventually became stable.
4.4. Experimental verification
In this section, the verification of the designed improved doubleT notch filter in a realtime system is discussed. The designed improved doubleT notch was discretized and physically realized in the NI7845Rfieldprogrammable gate array module. The openloop experiment and closedloop experiment were performed. Figs. 14 and 15 show the physical system of the experiment.
The identified result of the openloop system after the notch filter was applied to the system is compared with the original openloop system in Fig. 16. It is easy to see that the 28 dB resonance peak of the system at 34.5 Hz was well suppressed. In addition, the openloop bandwidth of the system was improved by about 125.6 rad/s.
The experimental structure of the closedloop experiment is shown in Fig. 14. The voltage command signal is given by the upper computer software to make the system work in resonant condition. The system resonances before and after the notch filter was added are compared. The closedloop tracking performances of the system before and after the notch filter was added are also compared.
As shown in Fig. 17(a) and (b), the resonance phenomenon was significantly suppressed when the system operated in the resonant state after the action of the improved doubleT notch. As presented in Table 5, the resonance peak attenuation rate was 63.03%, and the average resonance attenuation rate was 79.37%.
Item .  Command signal amplitude .  Peak oscillation amplitude .  Average oscillation amplitude . 

Before add notch  1 V  0.211 V  0.063 V 
After add notch  1 V  0.078 V  0.013 V 
Rate of oscillation decay  –  63.03%  79.37% 
Item .  Command signal amplitude .  Peak oscillation amplitude .  Average oscillation amplitude . 

Before add notch  1 V  0.211 V  0.063 V 
After add notch  1 V  0.078 V  0.013 V 
Rate of oscillation decay  –  63.03%  79.37% 
As shown in Fig. 17(c) and (d), after the improvement of the doubleT notch, the settling time of the system was significantly reduced under the same control parameters, and the tracking performance was significantly improved. As shown in Table 6, the settling time was reduced by 36.2%. According to the highorder system empirical Eq. (17), the settling time t_{s} is inversely proportional to the cutoff frequency ω_{c}, so that the system bandwidth couldbe increased by 36.2%.
5. Conclusion
In this study, the singleaxis openloop transfer function model of an ATP system was constructed. The resonance characteristic of the system was analyzed, and the influence of firstorder resonance on the system performance as well as the existence of highorder resonance were determined. According to the system design parameters, the theoretical firstorder resonance is 37.987 Hz. Finite element analysis was conducted to simulate the system resonance characteristics. The simulation value of firstorder resonance was 33.105 Hz. The simulation values of highorder resonances were 603.39 Hz and 1301.7 Hz. The test value of firstorder resonance obtained by the identification experiment was 34.5 Hz. In addition, the actual accurate model of the system was obtained. Finally, an improved doubleT notch filter was designed and utilized to suppress the system resonance. Simulation and experimental results show that the designed doubleT notch filter could effectively suppress the resonance points in the system bandwidth. The openloop experiment showed that the peak resonance attenuation rate reached 63.03% and the average resonance attenuation rate reached 79.37%. For the closedloop experiment, oscillations in the system response were suppressed. Moreover, the settling time of the system was reduced by 36.2%, and the closedloop bandwidth was expanded, providing a guarantee for the research and development of highperformance WBIRSs.
Acknowledgments
This work was supported by the Research on Key Problems of Wideband Inertial Reference Based on Magnetohydrodynamics (Grant number 61733012); National Natural Science Foundation of China Youth Project (Grant number 61703303); Tianjin Natural Science Foundation Youth Project (Grant number No. 17JCQNJCo4100); State Key Laboratory of Precision Testing Technology and Instruments Open Project (Grant number No. PILAB1705); and 2017 Tianjin Education Commission Research Project (Grant number 2017KJ086)
References
Dong Li is a research engineer on the study of key problems of wideband inertial reference which based on magnetohydrodynamics. He obtained a bachelor's degree in engineering from Harbin Engineering University in 2017 and obtained a master's degree in engineering from Tianjin University in 2019. He is mainly engaged in the design and implementation of automatic control systems, system identification and resonance suppression.
Tengfei Wu obtained Ph.D. in engineering from Tianjin University in 2013 and worked as an assistant researcher in State Key Laboratory of Precision Measurement Technology and Instruments. He mainly designed and developed dynamic tuning gyroscopes and magnetic fluid dynamic gyroscopes. His main research direction is sensing and information processing technology, precision measurement and automated measurement equipment.