Virtual collocation of sensors and actuators 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 input force, i.e. when all input forces have the distinguishable influence on measurement signals, the system is said to be collocated. When forces act not at locations of sensors, the system is non-collocated. In many real life applications it is not always possible to collocate the vibration control actuator at the location of interest. As an example, an active magnetic bearing (AMB) can be considered, for which eddy current sensors are used to measure shaft displacements. It is obvious that sensors cannot be placed exactly at locations of magnetic coils of the AMB. The shaft segment of a given diameter and some distance between sensors and the rest of the magnetic circuit is required for the proper operation of the AMB. Therefore it is not possible to collocate sensors and actuators in the AMB. Non-collocation complicates the control problem because the dynamics of the structure between the control actuator and the sensor disturbs the performance of the vibration control system. It has been proved, that non-collocated systems have non-minimum-phase zeros at the right half-plane. These zeros decrease the robustness of the control system, increasing its sensitivity to the parameter variation. Furthermore, if these zeros occur within the operating range, the system may become unstable. Non-collocation effects on the stability of the vibration control system can be suppressed by several ways. Phase shifting, time delay or passive vibration absorber methods have been employed in the past. However, none of these methods ensures the required stability margins, wide operational bandwidth, nor good performance. The present paper recommends the virtual (recalculated) collocation. The idea is to calculate displacements at actuator locations given displacements at sensor locations, or to calculate control forces at actuator locations given displacements measured at sensor locations. The paper discusses the former method, as it is a common practice to design local control loops rather than global ones. The approach is illustrated with numerical simulation results of the flexible rotor supported by AMBs and controlled with four PID controllers. The finite element (FE) model of the rotor is developed and transformed to the state-space. The FE model is reduced by the modal truncation technique. Then, an observer for the reduced system is developed. This way shaft displacements at any discretization node along the shaft axis can be simply estimated an introduced to PID controllers. Calculation results demonstrate a good performance of collocated PID control systems, conurming the potential of the method and giving a rationale for its further development.