The control systems applied on active magnetic bearing are several. A perfect levitation is characterized by maintaining the operating point condition that is characterized by the center of stator coincident with the geometric center of shaft. The first controller implemented for this purpose is PID controller that is characterized by an algorithm that leads the amplifier to produce control current until the operating point condition is not reached, this is obtained by an integration operator. The effect of an integrator is essential but not necessary for a centered levitation for example in the robust control characterized by a dynamic model depended on plant of system so that it depends on angular speed as LQR controller does. In LQR there is not integrator so there is not a perfectly centered section of shaft with center of stator. On contrary PID controller does not depend on angular speed and it can be easily implemented according some simple rules. Predictive control is another interesting controller characterized by a multiple controller operating in different condition in order to get the minimum of cost function, but also in this case the angular speed is introduce for the same reason discussed before.

Active Magnetic Bearings (AMBs) use electromagnets to attract the ferromagnetic cape winding the rotor which is free to rotate with no physical contact with the bearing. This operation, called active magnetic levitation, is unstable unless of a certain control’s algorithm performed respecting the imposed constraining, [

The proposed control algorithms are developed using

The loop-shaping method is commonly used also to obtain tradeoffs of robust stability and robust performance. This technique is a particular optimization problem to guarantee closed loop stability at all frequencies [

The particular configuration shown in this work considers a rotor with four degrees of freedom with eight poles for each active magnetic bearing, having a slope of 45° with regard to horizontal direction so that the force’s resultant supports the rotor along the

Schematic view of 4-axis rotating shaft supported by two radial active magnetic bearings with sensors.

The system is subjected to a state of uncertainty about its mass, cross, and polar moment of inertia dictated by the parameters

The last expression leads to structured uncertainties matrix such as

By introducing a transformation of coordinates (

In this paper, the value of displacement and current gains are taken from a real model of radial active magnetic bearing produced by SKF. The physical meaning on these gain matrices is related to the reaction force produced for the unit of displacement and current, respectively, for displacement and current gains

Due to the transformation coordinates (

In order to provide a stabilizing effect to control the position of the rotor, a suitable control system must be performed because no magnetic levitation can be stabilized without controller [

The weighting function described in (

Block scheme of plant with the introduction of weighting functions as further outputs.

The presence of weighting functions produces an increase of state vector’s variables so that the new plant is P as shown in Figure

Block scheme of plant showing the new plant.

All controllers used in this paper are characterized by a common concept or rather the robustness. The robustness is meant in a double way: robust stability and robust performance. The closed-loop system achieves robust stability if it is internally stable for all possible plant models

Since those weighting functions are introduced in order to provide some characteristic on the system’s output, the robust performance criterion (

Hence, the performance criterion is that the transfer functions from

The simulations are performed by considering the data contained in Table

Data for simulation.

Symbol | Description | S.I. |
---|---|---|

mass of rotor | 2.3 Kg | |

polar moment of inertia | ^{2} | |

transverse moment of inertia | ^{2} | |

distance bearing | 0.241 m | |

distance bearing | 0.139 m | |

distance sensor | 0.241 m | |

distance sensor | 0.119 m | |

pole surface | ^{2} | |

nominal gap | ||

slope of bearings axis | 45° | |

angle of two electromagnet poles | 45° | |

slope of bearings axis | 45° | |

uncertainties percentage | ||

range of uncertainties | ||

displacement gain | 144000 N/m | |

current gain | 38 N/A |

Another set of data are referred to the transfer function introduced in the plant of our system. These transfer functions are essential if a certain performance must be obtained; these performances are usually referred in the frequency domain for example. Some authors usually introduce scalar weighting function in order to describe a certain constant value they want to obtain as a particular output. In this paper, we introduce a transfer function in the output signals or rather the displacement. The introduction of transfer function in Laplace domain “s” means that the displacement must be characterized by a certain dynamic behavior according the frequency variable. This technique is commonly used above all when a flexible structure is taken into account or when some nodes are subjected to vibrations such as in this case. Figures

Sensitivity function describing how the disturbance affects the displacement with LSDP controller.

Sensitivity function describing how the disturbance affects the displacement with

Sensitivity function describing how the disturbance affects the displacement with Sub(

Figures

In a physical sense, the

Figure

Robust stability, nominal and robust performance function described by

Figure ^{0}, 10^{3}] Hz so that the system does not have robust stability in that range. Figure

Robust stability, nominal and robust performance function described by

Robust stability, nominal and robust performance function described by

Obviously the previous results affect the dynamic behavior of the entire system. According to Figure ^{−6} m. All three controllers implemented are capable to support the requirements to reject the disturbance in a different way. The controllers Sub(

Comparison via simulation for disturbance rejection test.

Comparison via simulation for reference tracking test.

The controllers are characterized by state space representation (

This paper shows that a comparison of three different control systems is built for a suspended rotor by active magnetic bearings. The comparison shows that loop-shaping design procedure provides the best performance to eliminate the disturbances and to follow the reference’s tracking when a certain required performances on position and control current are necessary. The present study shows that, once the weighting functions are introduced, only the loop-shaping design procedure is able to lead the system with robust stability and robust performance. The next development is to produce the previous discussion for the flexible rotor under the assumption that sensors are not colocated and with a state of uncertainties on displacement and current control gains.