Optimization of bias current coefficient in the fault-tolerance of active magnetic bearings based on the redundant structure parameters

ABSTRACT To improve the reliability of magnetic bearings, the redundant structures are usually designed to provide the desired bearing characteristics continually by the reconfiguration of the remaining structures when some components fail. Bias current coefficient is one of the key coefficients in the fault-tolerant control (FTC) of magnetically levitated bearings, and its inappropriate value will result in the failure of providing the desired bearing force due to saturation constraints. This paper presents optimization approaches of the bias current coefficient based on the redundant structure parameters. By analysing the range of the bias current coefficient under saturation constraints mathematically, the existence of the optimal solution has been proved, and the model of the electromagnetic force (EMF), load current and the bias current coefficient has been established in this paper. In addition, algorithms to find the optimal solution has been designed for two kinds of optimization objectives, respectively, in FTC of magnetic bearings. Numerical verifications prove the effectiveness and the versatility of the proposed approaches in the different structures.


Introduction
Magnetic bearings, support the rotors by electromagnetic force, with the advantages of no lubrication, no mechanical friction, and controllable support characteristics [1,2], are widely used in high-performance equipment such as aero-engines, energy storage flywheels, nuclear turbine power generation equipment, etc., [1][2][3]. The basic control theory of magnetic bearings is based on the bias current linearization at the equilibrium position, namely, by the displacementforce coefficient and the current-force coefficient, electromagnetic force is linearized [2].
The faults incorporated in magnetic bearings can be divided into 3 types: controller, sensors and actuators. The hot standby configurations of the controller or sensors are the common way to deal with the corresponding faults in the magnetic bearing system [4,5], or more advanced hardware redundancy protocol for achieving high reliability, triple modular redundancy (TMR) [6,7], can be considered in some researches. However, there is special consideration of the faults in the actuators. Because of the requirement of high efficiency in magnetic bearings, it is practically impossible to design some hot standby actuators prepared to replace the failed ones only. Moreover, there is a symmetry constraint on the stator structure [8], the short circuit, break or partial insulation damages in the electromagnetic coils will cause the unexpected EMF, which is defined as the actuator fault [5] and will destroy the original symmetry of the bearing structure, leading to the failure of bearing system and serious impacts [9].
Unlike hot standby configurations, another analytical redundancy design was introduced in [8,10] and can effectively improve the reliability of magnetically levitated bearings. The failed actuators will be isolated and the remaining parts will be continually used and reconfigured to support the magnetically levitated rotor. Similarly, the linearization of EMF generated by redundant supporting structure in magnetic bearings is an important foundation for the realization of FTC, and relevant researches basically follow this idea. Eric et al. [10] proposed the fault-tolerance of magnetic bearings by generalized bias current linearization, in which, the magnetic flux lost due to failed actuator can be compensated by the current distribution, and a linear relationship between the EMF and the controlled current can be derived. Na and Palazzolo [11] optimized the FTC of magnetic bearing and put forward an approach for optimal selection of the current distribution matrix by Lagrange multiplier approach. Moreover, the corresponding experimental verification was performed in their flexible rotor platform [12] to bear the rotor after the failure of an actuator. Subsequently, Ming-Hsiu Li [13] studied the magnetic circuit coupled actuator structure and extended the theory to magnetic bearings of composite structure with radial and axial structure. D. Noh et al. [14] experimentally verified the feasibility of such bias current linearization theory based on the molecular magnetic vacuum pump. In addition, in order to effectively compensate for the model error of the linearization method, Meeker and Maslen [15] estimated factors such as magnetic leakage and edge and eddy current effects and established a more accurate magnetic bearing model. Na and Palazzolo [16] calculated the reluctance of ferromagnetic material path and modelling error due to magnetic leakage. The edge effects were replaced by means of a simple compensation coefficient.
The aforementioned researches have already presented a basic theoretical framework in the faulttolerance of magnetic bearings, however, the following fields need to be concerned. (1) Analysis of the fault-tolerant control system (FTCS) model in magnetic bearings. Arslan A-A. and Khalid M-H. presented a comprehensive state-of-the-art review of FTCS with the latest advances and applications in [17]. Active FTCS (AFTCS) consists of Fault Detection and Isolation (FDI) module [18], a reconfiguration mechanism and a reconfigurable controller [19,20]. Especially, linear regression-based observer model can be used in the fault detection and isolation unit for fault detection, isolation and reconfiguration in the AFTCS [21] to improve the system robustness. Passive FTCS (PFTCS) has no FDI unit and no controller reconfiguration. Rather, the controller works in offline mode in both normal and abnormal conditions with predefined parameters that mask the faulty readings from the components [17]. Hybrid fault-tolerant control system possesses properties of both active fault-tolerant control system and passive fault-tolerant control system. A hybrid fault-tolerant control system was proposed in [22] for air-fuel ratio control of internal combustion gasoline engines based on Kalman filters and triple modular redundancy. (2) Be similar with the common magnetic bearings, the ones with analytical redundancy design have the requirements to linearize the EMF based on a necessary bias design. In order to optimize the EMF output in the FTC of magnetically levitated bearings, Na and Palazzolo [23] designed an optimal selection scheme of current distribution matrix based on the load current limitations and necessary linearization conditions, to lower the load current, compared with the current in [10]. Bias current coefficient is one of the key coefficients in the general current linearization theory [11,14,24,25]. Na et al. [11] gets a certain optimization result by selecting the central intensity of magnetic field as a bias current coefficient. Cheng X. et al. [25] established a FTC model, and through the numerical verification, they pointed out that inappropriate choice of bias current coefficient would cause the failure of generating the desired EMF due to the saturation of the load current or the magnetic field. However, based on what we know, there is no further research about how to find the optimized bias current coefficient in FTCS for magnetic bearings.
Our contribution in this paper is an optimization approach of bias current coefficient based on the redundant supporting structure parameters. We define two kinds of optimization objectives, (1) maximum electromagnetic force under the same structure and saturation constraints and (2) minim intensity of magnetic field under the same structure and EMF output. By analysing the range of the bias current coefficient under saturation constraints mathematically, the existence of the optimal solution has been proved, and the optimization algorithms have been designed to find the optimal solution of bias current coefficient. Numerical verifications prove the efficiency and the versatility of the proposed approaches in different structures. The topic of this paper is not the FTC algorithm but the optimization of bias current coefficient in FTC. The implementation and performance verification of FTC in magnetic bearings can be found in [10][11][12]14,21].
Further contents of the paper are organized as follows: Section 2 describes the structure parameters in fault-tolerance of magnetic bearings. Section 3 provides the mathematical proofs and Section 4 discusses the optimization algorithms of the optimal bias current coefficient. The numerical verification is in Section 5. The conclusion is presented in the last section with future directions. Figure 1 illustrates the redundant structure and magnetic circuit of radial magnetic bearing with n poles [10]. Relying on the coupled magnetic circuit between adjacent magnetic poles by the magnetic yoke, the loss of magnetic flux caused by the failures of some actuators can be compensated to implement the fault-tolerance. The idea relieves the symmetry constraint on stator structure.

Structure parameters in the fault-tolerance of magnetic bearing
The magnetic circuit equation is [10], where, R j and j are reluctance and magnetic flux of the jth pole, respectively; N j and I j are the number of the coil windings and the current of the jth pole, respectively. g(x, y) j , A j and μ 0 are the gap and area of jth pole, and vacuum magnetic permeability, respectively. One can write Then, the equation of coupling magnetic circuit is as following [10] RΦ = NI (7) Note j = B j A j , where B j is the air gap magnetic field intensity of the jth pole. We define A j = A, and B can be described as In the Equation (9), A represents the diagonal matrix of the magnetic pole area. Fx and Fy are the resultant forces of the EMF in the x and y directions, respectively. Then one can write It can be seen that even if an electromagnetic coil fails, the corresponding magneto-motive force is 0, but the Equation (10) can still hold, namely, the reconfiguration of the supporting force is realized by the compensation of magnetic flux. We define The diagonal matrix K is introduced to describe the state of the electromagnetic coil. If a coil fails, the corresponding diagonal element is 0.
Then, Fx and Fy can be described as [10] Current vector is defined as C 0 is the bias current coefficient, i x and i y are the control currents in the x and y directions, respectively. Current distribution matrix W is defined in [10], which satisfies So the Equation (14) can be simplified to Usually, C 0 can be set as a constant to decouple and linearize the relationship of EMF and currents. However, there are constraints in magnetic bearings system, e.g. the magnetic field saturation due to magnetic material or the current saturation of power amplifier. Considering that inappropriate value of C 0 may lead to failure of outputting desired EMF due to the above constraints, how to optimize the bias current coefficient should be taken into consideration. We define two kinds of optimization objectives, (1) maximum electromagnetic force under the same structure and saturation constraints, and (2) minimum intensity of magnetic field under the same structure and EMF output.

Mathematical proofs of the optimal bias current coefficient
Assuming the area of each magnetic pole A, the number of turns of the coil N and the air gaps g 0 are the same, then we introduce them into Equations. (2), (5), (6), (11), can get Equation (16).

W T K T P x KW
The solution of the current-distribution matrix can be divided into two parts, one part is the solution matrix W n×3 , which is only related to the number of magnetic poles and the corresponding angle of each magnetic pole; the other part is a scalar solution which is related to the structure parameters A,N,g 0 and μ 0 in Equation (20). (20) From Equation (19), it can be seen that for a certain redundant structure, its parameters will become a part of the current-distribution matrix W by means of constant. However, Wdetermines the performance of the fault-tolerant control together with the bias current coefficient. Introduce Equation (15) into I = WI C to get Equation (21).
Define the general form of current-distribution matrix as in Equation (22), the current expression as Equation (23) can be derived by Equations (21) and (22).
The ith pole current can be expressed as As shown in Figure 2, we define Then Equation (24) can be expressed as Equation (24) can be simplified as Usually, W is obtained offline and stored in the FTCS, which means that K ij is a constant for the any identified W. Therefore, it can be known from Equation (28) that the value of C 0 directly affects the relationship between the desired EMF and the controlled current of each magnetic pole. the inappropriate value of C 0 will result in the inability to output the desired EMF because of saturation constraint. Moreover, there is an optimal value of C 0 in the any of following four conditions in Figure 3 from Equation (28), and the 4 conditions are from the combinations of K i1 and K i2 in Equation (27). Obviously, the above ranges are symmetrical. In order to simplify the model, we only consider the positive value of C 0 . Therefore the desired EMF can be generated only when C 0 ∈ (C 3 ,

Case A: Maximum electromagnetic force output under the same saturation constraints
Convert Equation (26) to Equation (30), For any identified W and β, when I i takes the maximum value, namely I max , an optimal value of bias current coefficient can be got as in Equation (31), so that   the corresponding structure can provide the maximum EMF.
However, it is necessary to confirm whether the optimal C 0 satisfying Equation (31) exists within the range in Equation (29); if not, namely, when any magnetic pole current reaches the I max , there must be some other magnetic pole current exceeding the I max , as in Figure  4(a). This case means we cannot get the optimal C 0 from Equation (31), and need to consider that two magnetic pole currents reach the I max at the same time, as in Figure 4(b). Assuming the q th and the p th magnetic pole current reach the I max at the same time, the current expression is from Equation (32).
The optimal value of C 0 can be got from Equation (33). Figure 5 illustrates the flow chart of optimization algorithm for case A.

Case B: Minimum intensity of magnetic field under the same electromagnetic force output
From Equations (9), (11) and (21), the following Equation (34) can be obtained.
Since the matrix V is determined by the structure parameters, we define W = V · W, can have Combined with Equation (28), we can have, The relationship between the intensity of magnetic field and C 0 of each magnetic pole can be accurately obtained according to the matrix V and W. Once the desired output EMF is generated, we need to find the optimal C 0 with the minimal value of the maximum intensity of magnetic field of all the poles, compared with other value of C 0 . There are two cases, as shown in Figure 6.
In Figure 6(a), for the lowest point of the kth pole curve, when the magnetic flux intensity of a single magnetic pole can be taken to the minimum value of B k = 2 √ K k1 K k2 F or B k = 0. In Figure 7(b), the lowest point is the intersection of the two curves, therefore the intensity of magnetic field expression corresponding to the sth and rth magnetic poles can be obtained from the Equation   (39).
The solution of Equation (38) is Flow chart of optimization algorithm for case B is in Figure 7, it is obvious that the optimal C 0 can always be got from one of the Equations (31), (33), (37) and (39).

Numerical verification
By taking an 8-pole symmetrical radial magnetic bearing as an example, its structure is shown in Figure 8 and parameters are presented in Table 1.
Based on Equation (5), we can have Equation (40) for this example.
By introducing the structure parameters into Equation (18), we can have We can find one solution of W in Equation (42) from [10], with identity matrix K. Obviously, the structure parameters of the magnetic bearing are contained in W, as constants.
The computational cost of the proposed algorithm of different cases is shown in Table 2; the values are measured by CCS Profile clock when the proposed algorithm is running in the TMS320F28335 DSP with the core clock of 150 MHz.

Maximum EMF output
Firstly, we consider the condition of no failed actuator. Figure 9(a) illustrates the maximum output EMF under variable I max ; Figure 9(b) describes the relationship between the C 0 and the maximum EMF output when I max is 5 A. we can find the optimal C 0 in Equation (43) based on the proposed approach.

Minimum intensity of magnetic field
When expected EMF is F x F y = 500 N 0 N , we can find the optimal solution C 0 , considering two same failure condition based on the proposed approach, (1) no failed actuator, C 0 = 22.3607, and B max = 1.13 T; (2) the 8th actuator fails, C 0 = 22.3593, and B max = 1.13 T.

Performance simulations of FTC adopting proposed optimization algorithms
The proposed optimization algorithms can help to find the optimal C 0 in the FTC of magnetic bearing. The implementation details of FTC of magnetic bearings can be found in [10][11][12]14,21]. To verify the effectiveness of the proposed optimization approach, the simulations based on Matlab/Simulink was carried on. The structure and parameters of magnetic bearing are in Figure 8 and Table 1, respectively. As in Figure 12(a), curve B demonstrates the trajectory of rotor when the optimized C 0 = 23 is chosen. The failure conditions were designed as follows. (1) at 0.5 s, the rotor was suspended near its equilibrium position (0.001 m); (2) the 8th coil failed at 1 s; (3) then the 6th coil failed at 1.5 s. It is obvious that the can return to the equilibrium position after the failures of coils because the supporting structure was reconfigured under the FTC. Figure 12(b) describes the electromagnetic force in the reconfiguration, it is clear that the output electromagnetic force (curve D) can compensate the disturbances in the reconfiguration to maintain the resultant force (curve C) near the 0 N. The trajectories of rotor (curve A and B) are similar if ignoring the reconfiguration, it means that whether or not the optimal C 0 is chosen, the performance of FTC in this case is similar to effectively deal with the failure of coils. Figure 13 demonstrates different performances if a sinusoidal disturbance with the magnitude of ± 300 N and frequency of 100 rad/s is added into the control loop. The optimized C 0 chosen in FTC can effectively compensate the disturbance(in curve B) compared with the case in curve A, it means that the optimized C 0 can effectively add the EMF output capacity if the extreme cases occur. Moreover, the real-time performance of FTC will have the decisive effects to maintain the stability of rotor in the reconfiguration if some coils fail.

Versatility
To prove the versatility of proposed approach, a different redundant structure in Figure 14 is chosen, while the parameters are in Table 3. A matrix W for this    Figure 14.
It can be seen that the optimal C 0 can be got based on the discussed approach even for this 6-pole structure, which means the good versatility of the discussed approach.

Conclusions
Aiming at the FTC of active magnetic bearings, this paper proposes optimization approaches of bias current Table 3. Parameters of the structure in Figure. 14.

Parameter
Value Unit (1) For a certain structure, there is an optimal C 0 , and only when the bias current coefficient is within a certain range, the desired EMF can be generated. This paper proves the existence of the range. (2) To the two kinds of optimization objectives, (a) maximum EMF under the same structure and  saturation constraints and (b) minim intensity of magnetic field under the same structure and EMF output, we design effective optimization algorithms to find the optimal solution of C 0 . In addition, the versatility of proposed approaches has been proved.
The limitation of the proposed approach is that the basic electromagnetic force linearization theory is based on the assumption, namely, the solution of Equation (14) must consider the constraints in the actual applications, as follows: (1) the saturation of magnetic field; (2) the saturation of current in the power amplifier; (3) the redundancy of the magnetic bearings -the more redundant coils, the more failed ones allowed; (4) the ratio of maxim EMF required and EMF designed; (5) the rotor dynamics required or precision; (6) the failed coils are continuously or discretely arranged.
The future work should focus on the optimization approach of bias current coefficient under certain constraints, and the size of the redundancy required for certain types of faults.