Supporting by active magnetic bearings (AMBs), the AMBs-flexible rotor system works near and above the first-order bending critical speed. Because of the deformation of the shaft, the displacement of other positions on the shaft cannot be inferred from the displacement of the two AMBs through simple mathematical calculations. Therefore, when controlling other positions is needed, an estimator construct by the observer is the key to tracking and obtaining the real-time displacement.
As the classical observer in control theory, the Luenberger observer can obtain the state of system by output. However, the design of the Luenberger observer needs to consider mathematical model error and continuous disturbance, which leads to less application in industry and lagging tracking. The Unknown Input Observer (UIO) is designed to keep track of unknown inputs in the system. Since the unbalance force in AMBs-flexible rotor system can be regarded as unknown input. It can be estimated by UIO to realize fast tracking and estimation.
There are two preconditions in the design of UIO. The first one is that the invariant zeros of the system (A, Cm, Bu) are all in the left half plane. This condition is equivalent to the disturbance input matrix Bu being full of rank and the system matrix A being stable. Since the AMB under the PID controller can provide positive stiffness and damping similar to mechanical bearings, the system matrix A is considered stable. The second condition is the observer matching condition, requiring rank(CmBu)=rank(Bu). While meeting these two conditions, the form of UIO can be designed as follows.
However, the observer matching condition is only satisfied in some systems. In the AMBs-flexible rotor system, the output matrix Cm is the position of the displacement sensors, and the unbalanced force acts on the acceleration of the system, resulting in rank (CmBu)=0. In order to achieve the matching of observers, this paper proposes an implementation method for constructing auxiliary outputs. By taking the second derivative of the output equation, the new output includes the original output y and auxiliary output $ \ddot{\boldsymbol{y}}-\boldsymbol{C}_{\mathrm{m}} A \boldsymbol{B} \boldsymbol{u}-\boldsymbol{C}_{\mathrm{m}} \boldsymbol{B} \dot{\boldsymbol{u}}$. The new input matrix CmABu is full of rank, satisfying the matching condition.
The state observability needs to be solved before observing the AMBs-flexible rotor system. The first is the observable condition of the disturbance system, which requires that the number of disturbances cannot exceed the number of sensors. The second is the observable condition of the flexible rotor system, which requires at least n displacement sensors to estimate n nodes, including rigid modes. When observing the first-order bending mode, six displacement sensors are required. In addition, the observability of the system is calculated using the Gramian factor, and the positioning node of the sensor cannot be located near the mode shape node.
The tracking performance of the observers is studied by calculating the frequency response in the SISIO case. Compared with the Luenberger observer, the LESO with an extended disturbance state performs better but still suffers from phase lag when the input frequency exceeds the observer bandwidth. The UIO with auxiliary output can track the physics system accurately over a large frequency area.
Simulations and experiments are carried out to verify the proposed method. Tracking performance refers to the ability of the observer to track the existing output. Estimation performance refers to the ability to obtain the target position. In this paper, the existing outputs are 6 sensors of AMBs and disk A. The position to be estimated is the node of disk B. Simulations verify the estimating and tracking performance of UIO and the observability conditions. In the experiment, the second derivative in the auxiliary output is replaced by the square of the frequency multiplied by-1. A synchronous resonator is used to suppress the noise effect. The experimental result proves that the proposed method can track all system outputs with high precision in the region of 0~6 000 r/min and accurately estimate the displacement of the target position.
李翁衡, 祝长生. 主动电磁轴承-柔性转子系统振动位移的高精度跟踪和估计方法[J]. 电工技术学报, 2023, 38(12): 3151-3164.
Li Wengheng, Zhu Changsheng. High Precision Tracking and Estimation Method for Vibration Displacement of Active Magnetic Bearings-Flexible Rotor System. Transactions of China Electrotechnical Society, 2023, 38(12): 3151-3164.
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2023, Vol. 38 
