Virtual collocation method for a flexible rotor supported by active magnetic bearings

Local vibration control systems of mechanical structures can be collocated or non-collocated. If a sensor is placed at the same location as an actuator the system is said to be collocated. Otherwise, the system is non-collocated. It is not always possible to collocate the vibration control actuator at the location of interest, e.g. in active magnetic bearings (AMBs) eddy current sensors are used to measure displacements of the supported shaft. Due to inevitable electromagnetic interferences the sensors cannot be placed exactly at locations of electromagnetic coils of the AMB. Non-collocation complicates the control problem because the dynamics of the structure between the control actuator and sensor disturbs the performance of the vibration control system. Non-collocated systems have non-minimum-phase zeros at the right half-plane, which may even destabilize the control system. Non-collocation effects were suppressed by several methods including phase shifting, time delay or passive vibration absorber methods. However, none of these methods ensured the required stability, nor a good performance. The present paper recommends the virtual (recalculated) collocation method for local control loops of the AMB. The idea is to calculate displacements at actuators' locations given the displacements at sensors' locations. The approach is illustrated with numerical results of the flexible rotor supported by AMBs and controlled with four proportional integral derivative (PID) controllers. The finite element model of the rotor is developed and transformed to the state-space. The model is reduced by the modal truncation technique. Then, an observer for the reduced system is developed. This way the shaft displacements at any discretization node along the shaft axis are simply estimated and introduced to the inputs of PID controllers. Calculation results demonstrate a good performance of the collocated PID control systems, confirming the potential of the method and giving a rationale for its further development.