At the present study, two methods would be presented to control active magnetic bearing. Results of the two methods would be compared and would be simulated using MATLAB Simulating software. Finally, one of the two methods would be proposed as the most efficient method. The methods include 1 - adaptive back -stepping control (ABS), which can be applied for controlling linear model of active magnetic bearing (AMB) and 2- pole placement control, which is applicable for controlling system linear model. In both methods, error level around working point can be measured in presence ofimproper external disturbance s. Through control using the two methods, it could be indicate d that magnetic bearing around work point is asymptotically stable. Simulations indicate efficiency of the control methods properly in presence of external disturbance s. Key words: Introduction Over the decades, magnetic bearings have been applied for pu rpose of achieving optimal efficiency, high accuracy and controllingvibrations of rotating machines in the industry. Through eliminating physical contact between the bearing and rotating elements, it would be possible to have access to high rotating speed, which has been previously along with many difficulties using common industrial bearings. Magnetic bearings can be divided to two general categories including active and inactive. Inactive magnetic bearings have been formed of permanent magnets and in this type of bearing, outlet flux can't be controlled; although active magnetic bearings are formed of electromagnetic materials, in which outlet flux can be controlled using changing the current of coils. Therefore, active magnetic bearings are more common th an inactive types because of their controllability. Because of high speed and low space between rotor and stator, high error in performance of the rotor may cause problem in performance of the machine. Hence, controlling status of the rotor is a very impor tant issue; although controlling location of rotor is a hard action to do. PID controller in [3 -5] is very common, which can cause stability of the system. In addition to PID, LQR controller has been also presented for smaller sizes of magnetic bearings in [6]. Displacement of location of rotor needs control, which would be conducted through measurement of bearing current. A new perspective named Active Disturbance Rejection Control (ADRC) developed over the years has been simulated on active magnetic beari ng system in [2 and 12]. ADRC has the capability to reject strong power of external disturbances, when location sensor is not applied and it is just adapted with regulation of bandwidth of the controller and observation. However, inefficiency of ADRC contr oller can be predicted for controlling active magnetic bearing in the steady state for location of rotor [12]. In this study, ABC controlling method has been added to AMB for controlling system linear model. Adaptive Back -stepping Control (ABC) has been de veloped considerably over the decades [13]. ABC control has been composed of three general parts including control feedback, Lyapunov stability and adaptive control (or resistant). Moreover, ABC can be applied in helicopters, inverted pendulum, jet enginee ring and induction motor drive systems [13 -20]. It has been depicted in [17] that ABC is more efficient than PI controller in terms of robot. The study has investigated effect of ABC on AMB generally. Controlling overall resistance of the system using dire ct Lyapunov method would be conducted while the process of designing control rules. In addition, proposed ABC in this study is resistant against external disturbances and changes in parameters. However, in [14 -20], change in parameters have been just esti mated and compensated. In this study, in addition to ABC method, another method has been also proposed named Pole Placement for active magnetic bearing control, which is applicable for controlling linear systems. In pole placement method, displacement erro r of rotor would be measured and location of rotor would be controlled. The study has been designed as follows: at the first section, mechanical model of system is constructed and presented. In the next section, ABC controllers are designed and next, resu lts of ABC simulation would be presented. In the section 4, pole placement controller would be designed. Next, results of simulation for pole placement controller would be displayed. At the final step, controlling methods would be compared to each other an d a method would be proposed. System's mechanical modeling Figure 1 illustrates simple model of magnetic actuator. In figure 1, I indicates coil current, g is air gap, N is number of coil laps to core laps, A g indicates cross section. Magnetic field can b e produced by the current and in upward current. Figure 1: magnetic actuator According to ampere in nodes (Kirchhoff) rule, equation 1 would be obtained. In this equation, H is magnetic field, B is flux density, n s is number of loops in p ath 1 when H is constant, n c is number of the other coil. ∑ 𝐻𝑖𝑙𝑖𝑛𝑠 𝑖=1=∑ 𝑁𝑖𝐼𝑖𝑛𝑐 𝑖=1 (1) With the assumption that average permeability of µ is each loop is constant, magnetic flux density would be as follows: 𝐵𝑖=µ𝑖𝐻𝑖 (2) Through combining 1 and 2, the equation would be as follows: ∑𝐵𝑖𝑙𝑖 µ𝑖𝑛𝑠 𝑖=1=∑ 𝑁𝑖𝐼𝑖𝑛𝑐 𝑖=1 (3) For the system in figure 1, two air gaps are existed and air permeability rate ( gµ) is very lower than iron ( Oµ). Therefore: 2𝐵𝑔𝑔 µ𝑔=𝑁𝐼⇒𝐵𝑔=µ𝑔𝑁𝐼 2𝑔 (4) Stored energy (E) in air gap is as follows: 𝐸=1 2𝐵𝑔𝐻𝑔𝐴𝑔2𝑔 (5) Electromagnetic ener gy (f) is equal to energy derivative (E) due to the air gap that can be expressed as follows: 𝑓=𝑑𝐸 𝑑𝑔=𝐵𝑔𝐻𝑔𝐴𝑔=1 µ𝑔𝐵𝑔2𝐴𝑔 (6) According to flux density equation in eq. 4 and other equations, the next equation would be: 𝑓=1 µ𝑔𝐴𝑔(µ𝑔𝑁𝐼 2𝑔)2=µ𝑔𝑁2𝐼2𝐴𝐺 4𝑔2 (7) AMB with a degree of freedom (df) has been illustrated in figure 2. Figure 2: AMB model In figure 2, Fd is force of the disturbances on the rotor and F 1 and F 2 are two electromagnetic forces opposite to each other, which can be measured by equation 7. Through determining input voltages of u 1 and u 2, value of i 1 and i 2 currents can be control led and as a result, output energy can be measured. In figure 2, displacement of rotor from its initial place (x 0) is equal to x. Hence, the equation, according to Newton's Law, is as follows: 𝑚𝑥̈=𝐹1+𝐹𝑑−𝐹2 (8) In figure 2, x1 and x 2 are respectively the distance of the rotor from the left and right stator. Through replacing x 1 and x 2 separately instead of g in equation 7, electromagnetic forces F 1 and F 2 can be calculated. Hence, the equation would be as follows: 𝐹1=µ𝑔𝑁2𝑖12𝐴𝑔 4𝑥12 =𝐾 4(𝑖1 𝑥1)2 ,𝐹2=µ𝑔𝑁2𝑖22𝐴𝑔 4𝑥22 =𝐾 4(𝑖2 𝑥2)2 (9) If 𝐾=µ𝑔𝑁2𝐴𝑔, the equation according toKirchhoff's law would be as follows: 𝑢1=𝑅𝑖1+𝐿𝑠𝑑𝑖1 𝑑𝑡+𝐾 2𝑑 𝑑𝑡(𝑖1 𝑥1) , 𝑢2=𝑅𝑖2+𝐿𝑠𝑑𝑖2 𝑑𝑡+𝐾 2𝑑 𝑑𝑡(𝑖2 𝑥2) (10) In eq. 10, R refers to electric resistance of coil, L s is inductance coefficient of coil and 𝐾 2𝑑 𝑑𝑡(𝑖1 𝑥1) and𝐾 2𝑑 𝑑𝑡(𝑖1 𝑥1) are Electromotive force (EMF) of power, which can be created as a result of changing flux density. It is assumed that (𝑖0 .𝑥0 .𝑢0) are nominal values of state variable and x1 ،i1،u1 are respectively equal to place, current and voltage in left side of AMB in figure 2. Hence, the equation would be: 𝑥1=𝑥0−𝑥 .𝑥2=𝑥0+𝑥 𝑖1=𝑖0+𝑖 .𝑖2=𝑖0−𝑖 𝑢1=𝑢0+𝑢.𝑢2=𝑢0−𝑢 Through replacing equations 11 and 13 in eq. 10 and through replacing 9 in 8, system's nonlinear model would be obtained as follows: 𝑥̇=v (11) (12) (13) 33) (14) 𝑣̇=𝐾 4𝑚(𝑖1 𝑥0−𝑥)2 −𝐾 4𝑚(𝑖1 𝑥0+𝑥)2 +𝐹𝑑 𝑚 𝑖̇1=2(𝑥0−𝑥) 2𝐿𝑠(𝑥0−𝑥)+𝐾[−𝑅𝑖1−𝐾 2(𝑥0+𝑥)2𝑣𝑖 0+𝑢1] 𝑖̇2=2(𝑥0−𝑥) 2𝐿𝑠(𝑥0−𝑥)+𝐾[−𝑅𝑖2−𝐾 2(𝑥0+𝑥)2𝑣𝑖1+𝑢2] Using Jacobian transformation, one can obtain nonlinear equations of the system in linear form around work point. Equations of linear state have been presented below. State matrix has been displayed by A. [𝑥̇ 𝑣̇ 𝑖̇]=[0 1 0 2𝑘𝑠 𝑚02𝑘𝑖 𝑚 0−𝑘𝑖 𝐿0+𝐿𝑠−𝑅 𝐿0+𝐿𝑠][𝑥 𝑣 𝑖]+[0 0 1 𝐿0+𝐿𝑠]𝑢+[0 1 𝑚 0]𝑓𝑑 (15) According to equation 9, when i0 is constant, bias voltage can be obtained as 𝑢0=R𝑖0 due to resistance of coil and when i0 is variable, the equa tion between u0 and i0 would be as follows: (16) 𝑑 𝑑𝑡𝑖0=−𝑅 𝐿0+𝐿𝑠𝑖0+1 𝐿0+𝐿𝑠𝑢0 Where; 𝑘𝑠=𝐾 2𝑖02 𝑥03و 𝑘𝑖=𝐾 2𝑖0 𝑥02و 𝐿0=𝐾 2𝑥0 Values of system parameters have been pres ented in table1. Table 1: system's parameters Parameter Symbol Value(unit) Force -Displacement Constant Ks 153740 N/m Force -Current Constant Ki 3500 N/A Coil Self Inductance Ls 130mH Air Gap Inductance L0 95mH Weight of Rotor m 4.6 kg Coil Resistanc e R 9.7 Ω Nominal Air Gap 𝑥0 0.0033 m Bias current 𝑖0 1 A Disturbance Force 𝐹𝑑 6.2 N According to values in table 1 and equation 15, eigenvalues of A matrix can be obtained [207.7905, -207.7669, -0.0667]. Clearly, one of the eigenvalues is positi ve-valued, which indicates that the system is naturally unstable. It has been attempted to design a controller that is able to make the system stable. Designing the controller Until the time that active magnetic bearing is unstable, there is need to get h elp of the controller to control and stabilize location of rotor with presence of external disturbances and uncertain parameters of the system. For this purpose, it is necessary to estimate disturbances exactly, since taking action other than it can make c ontrol ineffective. Firstly, Adaptive Back -Stepping controller has been designed for the system and then, pole placement controller has been designed. Adaptive Back -stepping controller ABC is relied on two main bases: 1 - back -stepping controller and 2 - adaptive law. Back -stepping controller can be applied for stabilizing and controlling rotor's location and adaptive law can be applied for estimating external disturbances. In regard with designing ABC, it is assumed that system is in base state and as a res ult: (17) 𝑥̇=𝑓(𝑥)+𝑔(𝑥)𝜉1 𝜉1̇=𝑓1(𝑥 )+𝑔1(𝑥. 𝜉1)𝜉2 𝜉2̇=𝑓2(𝑥 .𝜉 1 𝜉2)+𝑔2(𝑥.𝜉1.….𝜉2)𝜉3 . . . 𝜉𝑘−1̇=𝑓𝑘−1(𝑥 .𝜉 1….𝜉𝑘−1)+𝑔𝑘−1(𝑥 .𝜉1… 𝜉𝑘−1)𝜉𝑘 𝜉𝑘̇=𝑓𝑘(𝑥 .𝜉 1.….𝜉𝑘)+𝑔𝑘(𝑥 .𝜉1 .… 𝜉𝑘)𝑢 In equation 17, x€Rnand ξi(i=1…k) are scholar states. Functions f and g are depended on changes of previous state. When ABC was designed, equations of system state of AMB should be ch anged into eq. 17 and as a result, equation 15 can be rewritten as follows: (18) [𝑥̇ 𝑣̇ 𝑖̇]=[010 𝑎0𝑏 0𝑐𝑑][𝑥 𝑣 𝑖]+[0 0 𝑒]𝑢+[0 𝑓 0]𝐹𝑠 Where; 𝑎=2𝐾𝑠 𝑚 𝑏=2𝐾𝑖 𝑚 ,𝑐=−𝐾𝑖 𝐿𝑠+𝐿0 ,𝑑=−𝑅 𝐿0+𝐿𝑠 ,𝑒=1 𝐿0+𝐿𝑠 ,𝑓=1 𝑚. Equation 18 can be written in form of equations 19, 20 and 21; where, 𝑥3=𝑖𝑥2=1 𝑏𝑣and 𝑥1=1 𝑏x : (19) 𝑥̇1=𝑥2 (20) 𝑥̇2=𝑥3+𝑎 𝑏𝑥1+𝜃=𝑥3+𝜑1(𝜃,𝑥1) (21) 𝑥̇2=𝑢́+𝑐𝑏𝑥2+𝑑𝑥3=𝑢́+𝜑2 In the above equations, disturbance force and control input are respectively equal to θ= (Fs/bm)=0.023 and u’=eu. 1φ and φ 2 are equal to φ1(𝜃,𝑥1)=𝑎 𝑏𝑥1+𝜃and 𝜑2=𝑐𝑏𝑥 2+𝑑𝑥3. Accor ding to the equations and the goal of control that is measuring rotor's location, one can select Lyapunov function and state variables to obtain control function and adaptive law as follows: If assume that: 𝑧1=𝑥1و 𝑧2=𝑥2−𝑎2و 𝑎1=−𝑐1𝑥1 ,𝑧3=𝑥3−𝑎2 Following equations can be obtained: Proposed Lyapunov function equation: 𝑉2=1 2𝑍12+1 2𝑍22+1 2𝑍32(22) If in the eq.22, V2 is differentiated, the equation would be: (23) 𝑉̇2=−𝐶1𝑍12−𝐶2𝑍22+𝑍2𝑍3+𝑍3(𝑢+𝜑2−𝑎̇2) =−𝐶1𝑍12−𝐶2𝑍22+𝑍3(𝑍2+𝑢́+𝜑2−𝑎̇2) When 𝑢’ is selected as 𝑈́=−𝐶3𝑍3−𝑍2−𝜑2+𝑎2̇, differentiated equation of Lyapunov function can be rewritten as follow s: (24) 𝑉̇2=−𝐶1𝑍12−𝐶2𝑍22−𝐶2𝑍32 The equation indicates that derivative of the negative Lyapunov function is semi -certain. As a result, the objective of control is achievable. In this equation, no external disturbance is con sidered. However, if one is existed, adaptive laws should be codified, so that the disturbances can be estimated and eliminated. Estimation of the disturbances would be added to the control laws as feedback at the time of working. Details about estimation of disturbances would be explained here. Here, θ is considered as disturbance and estimated disturbances can be shown by 𝜃̂ . Estimation error can be obtained as 𝜃̃1=θ−𝜃̂1. Quadratic form of the equation would be added to equation 22 and a new form of Lyapunov function would be obtained in e quation 23. (23) 𝑉2=1 2𝑍12+1 2𝑍22+1 2𝛾1𝜃̂12 Derivative of the above Lyapunov function would be obtained as follows: (24) 𝑉̇2=−𝐶1𝑍12−𝐶2𝑍22+𝑍2𝑍3+𝑍2𝜃̃1+1 𝛾1𝜃̂̇ 1𝜃̃1 =−𝐶1𝑍12−𝐶2𝑍22+𝑍3𝑍2+𝜃̃1(𝑍2−1 𝛾1𝜃̂̇ 1) In the equation 24, if adaptive law is selected as 𝜃̂̇ 1=𝛾1𝑧2, negative V 1 would be determined and Z2Z3 can be also omitted from the equation. According to externa l disturbances, a 2 value can be selected as follows: (25) 𝑎1=−𝑍1−𝐶2𝑍2−𝜑1𝑥1−𝜃̂1+𝑎̇=−𝑍1−𝐶2(𝑥2−𝑎1)−𝜑1𝑍1−𝜃̂1−𝐶1𝑥̇1 =−(𝐶1𝐶2+𝜑1+1)𝑍1−(𝐶1+𝐶2)𝑥2−𝜃̂1 Here, 𝑧1=𝑥1و 𝑧2=𝑥2−𝑎1و 𝑎1=−𝑐1𝑥1 has been considered. As a result, a 2 equals: (26) 𝑎̇2=−(𝜕𝑎2 𝜕𝑍1𝑍̇1+𝜕𝑎2 𝜕𝑥2𝑥̇2+𝜕𝑎2 𝜕𝜃̂1𝜃̂̇ 1) =−(𝜕𝑎2 𝜕𝑍1𝑍̇1+𝜕𝑎2 𝜕𝑥2(𝑥3+𝜑1𝑥1+𝜃)+𝜕𝑎2 𝜕𝜃̂1𝜃̂̇ 1) It is assumed that 𝜃̂2 is an estimation of θ1 and estimation error is equal to 𝜃̃2=θ1−𝜃̂2. Consequently, Lyapunov function can be rewritten as follows: (27) 𝑉2=1 2𝑍12+1 2𝑍22+1 2𝑍32+1 2𝛾1𝜃̂12+1 2𝛾2𝜃̂22 If 𝑧3=𝑥3−𝑎2, the control law, which was differentiated, would be expressed as follows: (28) 𝑢́=−𝐶3𝑍3−𝑍2−𝜑2+𝑎̇2 Through replacing 26 in 28, the equat ion would be as follows: (29) 𝑢́=−(𝐶3+𝑑)𝑥3−[𝐶3(𝐶1𝐶2+𝑎 𝑏+1)+𝐶1]𝑥1−[𝐶3(𝐶1+𝐶2)+1+𝑐𝑏]𝑥2 −𝐶3𝜃̂1+𝑎̇2 Finally, through differentiation of Lyapunov function in equation 27, the obtained equation w ould be: (30) 𝑉̇2=−𝐶1𝑍12−𝐶2𝑍22−𝐶3𝑍32−1 𝛾2𝜃̂̇ 2𝜃̃2−𝑍3𝜕𝑎2 𝜕𝑥2𝜃̃2 To have semi -certain negative value for equation 30, there is need to eliminate existed error of 𝜃̃2in the equation. If the adaptive law is selected as 𝜃̂̇ 2=−𝑧2𝛾2𝜕𝑎2 𝜕𝑥2, derivative of V2 would be: (31) 𝑉̇2=−𝐶1𝑍12−𝐶2𝑍22−𝐶3𝑍32 Now that derivative of Lyapunov function has gotten semi -certain negative value, system has been stabilized properly around its work point. Adaptive back -stepping control law can be obtained finally as follows: (32) 𝑢́=−(𝑎 𝑏(𝐶1+𝐶2+𝐶3)+2𝐶1+𝐶3+𝐶1𝐶2𝐶3)𝑥1 −(𝐶1𝐶2+𝐶2𝐶3+𝐶1𝐶3+𝑐𝑏+𝜑1+𝐶1+𝐶2+3)𝑥2−(𝐶1+𝐶2 +𝐶3+𝑑)𝑥3 The equation 32 displays final ABC controller, which includes 3 feedback states. According to Lyapunov function, AMB system has been controlled by ABC and has achieved to steady state.Closed loop circuit of system has been illustrated in figure 3. Simulation results of controlled system by ABC Simulation results have been presented below. An important issue is that amount of estimated disturbances has been considered to 0.023 (Θ=0.023). Figure 4 illustrates state variables with presence of estimated disturbance. Here, adaptive Figure 3: Closed lo op circuit along with controller coefficients have been considered constant 𝛾1=1 𝑎𝑛𝑑 𝛾2=1; although back Lyapunov coefficients of c1, c2 and c3 have been changed. It indicates that maximum value of rotor displacement is equal to 0.5mm, which can guarantee that rotor would never touch the stator. Through change in the parameters and coefficients, it could be found that the system still is stable and this indicates that controller system is resista nt and effective against changes. Control function output is shown in figure 4. Figure 4: system output, along with ABC controller a) rotor displacement with different Lyapunov coefficients b) controller's output (control signal) Controlling by pole placement method (a) (b) In this method, one can transfer system's poles to the desired places. Hence, name of the method is pole placement control method. First, equations of the system should be transferred from state space to transfer function. State space equat ions of the system are according to equation 18. Through transferring the equations to conversion function, the system would be changed into equation 33. 𝐺(𝑠)=0.06012 𝑠3+0.04211 𝑠2−3.889 𝑒4𝑠−1637 (33) Through calculating roots of conversion function, it could be found that system poles are equal to [207.7905, -207.7669, -and 0.0667]. Positive pole indicates instability of the system. In order to make system stable, the pole should be destroyed. In pole placement method, new places would be determined, to which previous poles of the system are being transferred. For example, it is desired to consider new poles equal to [ -0.5, -1, -1.2]. Through selecting these poles, one can stabilize the system, since all roots of conversion function are plac ed in left hand of the coordinate axis. This type of controller can conduct estimation due to the system's poles and desired places, to which previous poles should be transferred. Through multiplying it in input signal, control signal would be produced and through applying it in system, new poles would be created. Figure 5: system's loop, along with pole placement controller In this study, it has been assumed that source input is equal to 0. Simulation results of controlled system by pole plac ement controller As it is assumed that new places of system's poles are equal to [ -0.5, -1, -1.2], gains of the controller would be equal to [1.7466, 0.6469 , 0.0005]e+6. Through multiplying the values in input signal, control signal would be produced and f inally, the system would be stabilized through applying control signal. Output of the system has been displayed in figure 6 after adding pole placement controller. (a) Figure 6: system output a) rotor displacement diagram, b) controller's output (cont rol signal) It could be observed that in this method, the system has been appeared properly in work point. Maximum displacement amount for rotor is also equal to 0.012mm, which indicates that rotor would never touch the stator. Conclusion At the pre sent study, Active Magnetic Bearing system, which is naturally a nonlinear system, has become linear around working point using Lagrange. Then, two methods of adaptive back - stepping control and pole placement control have been applied to control rotor's st atus and location (b) in ABC system. The main aim by conducting this is stabilizing the closed loop system and controlling rotor's displacement in presence of disturbance and uncertain parameters of the system. Here, performance of the two controllers is comp ared. As it is obvious from simulation results (figures 4 -6), ABC controller is resistant against disturbances and performs properly through changing parameters of the system. The time for achieving response of steady state using ABC is shorter than the ti me of using pole -placement controller. In addition, control signal at the time of using ABS has more stability and has less fluctuations and changes than produced control signal by pole -placement controller. In addition, the distance of rotor and stator wh ile using ABC is higher than pole placement and this can provide better safety margin for preventing contact of rotor and stator. All mentioned issues indicate that performance of ABC controller is better than pole-placement controller. In future, it will be studied in this field that how Lyapun ov coefficients can be selected in a manner that system can have better performance? Also, the aim is to implement ABC controller practically for controlling magnetic bearing system, since it was proved that the sys tem has better action and performance than pole -placement controller system.