Flywheel rotor cross-filter tracking feedback control

In light of the matter that the fixed-parameter cross-filter circuit cannot satisfy the phase compensation requirement in the full speed range, a cross-feedback tracking control method based on cross-filter feedback, in which the high-pass cutoff frequency varies with the nutation frequency, is proposed. The stability analysis of the designed control method is carried out by using the phase angle margin and the root trajectory diagram, and the results show that the cross-feedback tracking control can increase the phase angle margin and significantly enhance the stability of the system. Simulation results of the magnetic levitation rotor system show that the designed control method possesses the benefits of high stability, rapid response time, and strong anti-jamming ability.


Introduction
Active magnetic bearing (AMB) technology has a wide range of applications in aerospace, medical equipment, and energy because of its high accuracy, high efficiency and cleanliness, low wear, ability to withstand large loads, and many other advantages.The flywheel energy storage system has a high rotational speed.The flywheel rotor has a large rotational inertia ratio, a strong gyroscopic effect at high speeds, and significant coupling among the degrees of freedom [1] .Amplifiers, filters, and sensors of the controller generate phase lag, which increases as the speed increases, leading to instability caused by inadequate damping of the rotor's nutation motion modes.
For the problem of suppressing the vortex modes generated by the gyroscopic effect, researchers from both domestic and international academic institutions have conducted extensive studies and proposed many control algorithms.These algorithms are roughly categorized into two groups, one of which is based on modern control theory methods, such as μ integrated control [2] , LQR control [3] , sliding mode control [4] , and robust control [5] .The algorithms are more complex and not easy to realize in engineering.Another type is the cross-feedback control method [6] .The cross-feedback control is widely used for its simplicity, directness, and small computational amount.
To fulfill the requirement of stable control over the full speed range [7] , a tracking high-pass filter can be designed with the cutoff frequency changing with the nutation frequency.This paper adopts the highpass filter whose cutoff frequency varies with the nutation frequency to track and compensate the nutation phase margin, and compares it with the PID control and cross-feedback filtering control with constant cutoff frequency in terms of the control effect and anti-jamming, and finally verifies it by simulation.

Modeling of the dynamics of the AMB-rotor system
The structure of the active magnetic bearing rotor system is shown in Figure 1.It is assumed that the radial four channels of the AMB-rotor control system are completely symmetrical.For the convenience of the study, it is assumed that the displacement sensor and magnetic bearing are located in the same position [8] , i.e., = = , = = , and = = .According to the theory of rotor dynamics, the equations describing the motion of the system in the center-of-mass coordinate system can be expressed as: Where is the rotor mass; and are the mass moments of inertia about the − and the − respectively; is the distance from the magnetic bearing force to the center of mass at the ideal position of the rotor; fxa, fya, fxb, and fyb are the electromagnetic forces in the and directions for magnetic bearing A and magnetic bearing B, respectively; Ω is the rotational speed; = ( + )/2, = ( + )/2, =( − )/(2 ), and =( − )/(2 ) ; the rotor displacement in the xa, y a , x b , and yb direction are xa, y a , x b , and y b ; and are the deflection angles counterclockwise around the − and − , respectively; and are the displacements of the geometric center of the rotor.
Linearizing the electromagnetic force, we can obtain: where is the current stiffness, is the displacement negative stiffness coefficient, and is the radial displacement of the rotor at the sensor position.

Algorithm design
The quantitative relationship between the nutation frequency and the rotational speed under decentralized PID control has been calculated.At the undamped frequency of the control system, the control channel frequency characteristic is a constant value, denoted as [9] . is the nutation frequency, and the nutation vortex direction is the same as the rotational speed, as shown in Equation (3).

Simulation verification
The rotor system is a flywheel rotor under active magnetic bearing control, and Table 1 shows its physical system parameters.
187.5 The parameter selection of the control system is kp=16, 000, kd=160, and ki=32, 000.Simulink simulation is performed on the rotor system, with a speed setting of 60, 000 r/min.After the system is stabilized, the rotor displacement at the stabilization moment corresponding to the three control modes is shown in Figure 5.The rotor radial displacement jumps after the rotor system is stabilized under PID control, and the rotor remains stable after the system is stabilized under crossfiltering control and cross-filter tracking control.Compared with the rotor under cross-filtering control, the rotor stabilized under cross-filter tracking control is more guaranteed to be located in the center of the setup.The vibration is smaller and the system deviation is smaller.Finally, after the AMB-rotor system is stabilized at 0.8 s, a step signal with an amplitude of 0.25 mm is added to the direction.We simulate the anti-disturbance performance of the three control methods.Figure 6 shows the simulation results.The systems with cross-filter control and cross-filter tracking control can return to the stable state faster after being subjected to continuous disturbance, and the systems with cross-filter tracking control are faster and deviate less in the process of restoring stability.

Conclusion
Combined with the solved nutation frequency, high-pass filter cutoff frequency with the nutation frequency change of the cross-tracking filter circuit can be designed.The stability of the cross-tracking filter control is analyzed by using the Bode diagram and the root trajectory diagram, and the stability of the system has been significantly improved.Simulation analyzes the stability and rapidity of the crosstracking filter control.

Figure 1 .
Figure 1.Structural diagram of the AMB-rotor system.It is assumed that the radial four channels of the AMB-rotor control system are completely symmetrical.For the convenience of the study, it is assumed that the displacement sensor and magnetic bearing are located in the same position[8] , i.e., = = , = = , and = = .According to the theory of rotor dynamics, the equations describing the motion of the system in the center-of-mass coordinate system can be expressed as:

Figure 3 .
Figure 3.The phase angles of the three control modes at nutation frequency.Figure 4. The root locus plots for the three control modes.

Figure 4 .
Figure 3.The phase angles of the three control modes at nutation frequency.Figure 4. The root locus plots for the three control modes.

Figure 5 .Figure 6 .
Figure 5. Rotor radial displacement in three control modes.(a) PID; (b) Cross-filtering control and cross-filter tracking control

Table 1 .
Physical parameters of the AMB-rotor system.