POSITION CONTROL OF AN INTERIOR PERMANENT MAGNET SYNCHRONOUS MOTOR BY USING ADAPTIVE BACK STEPPING ALGORITHM

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1 Research and Applications (IJERA) ISSN: Vol. 2, Issue 3, May-Jun 212, pp POSITION CONTROL OF AN INTERIOR PERMANENT MAGNET SYNCHRONOUS MOTOR BY USING ADAPTIVE BACK STEPPING ALGORITHM S.S.Sankeswari 1 Chandulal Guguloth 2, Dr.R.H Chile 3 M.Tech (Control systems) M.Tech (Power & Energy Systems) Ph.D (Process Control) M.B.E. Society s, C.O.E.A. M.B.E. Society s, C.O.E.A. S.G.G.S college of Engg., Nanded Abstract: This paper introduces a non linear position controller for an IPMSM (Interior Permanent Magnet Synchronous Motor). The system model equations proide the basis for the controller which is designed using adaptie back stepping technique. By recursie manner, irtual control states of the IPMSM drie hae been identified and stabilizing control laws are deeloped subsequently using Lyapuno stability theory. Various system uncertainties are considered in the design. Using Lyapuno s stability theory, it is approed that the control ariables are asymptotically stable. The complete model is then simulated using MATLAB/Simulink software. Performance of the controller is inestigated at different dynamic operations such as command position change, sudden load change. I. INTRODUCTION The adances in power semiconductor technology, digital electronics, magnetic materials, and control theory hae enabled modern ac motor dries to face challenging high-efficiency and high-performance requirements in the industrial sector. Among AC dries, the Interior Permanent Magnet Synchronous motor (IPMSM) is gaining popularity in industrial fields because of its adantages oer other types of motors such as DC and Induction motors. This is due to its: High power density, higher efficiency, low noise and, Robustness.[1] These features are due to the incorporation of highenergy rare-earth alloys such as neodymium iron boron in its construction. In particular, the interior permanent-magnet synchronous motor (IPMSM) which has magnets buried in the rotor core exhibits certain good properties, such as mechanically robust rotor construction, rotor physical non saliency of the air gap, and small effectie air gap. The rotors of these permanent magnet motors hae complex geometry to ensure optimal use of the expensie permanent magnet material while maintaining a high magnetic field in the air gap. These features allow the IPMSM drie to be operated in high-speed mode by incorporating the fieldweakening technique. Also, the mathematical model of the PMSM is less complex than that of an induction motor. Howeer, the model is still non linear due to the coupling nature of the d-q axis currents (torque and flux components respectiely) making the control task more complex. Usually, high-performance motor dries require fast and accurate response, quick recoery from any disturbances, and insensitiity to parameter ariations. The most effectie method for high performance control of an IPMSM is ector control, or field oriented control[2]. This inoles transforming the machine parameters from the standard abc frame to the synchronously rotating d-q frame using Park s transformation. Using the ector control technique, the torque and flux can be decoupled so that each can be controlled separately. This gies the IPMSM machine the highly desirable dynamic performance capabilities of the separately excited dc machine, while retaining the general adantages of the ac oer dc motors. Originally, ector control was applied to the induction motor and a ast amount of research work has been deoted to this area. The ector control method is releant to the IPMSM drie as the control is completely carried out through the stator, as the rotor excitation control is not possible. Howeer, precise speed control of an IPMSM drie becomes a complex issue owing to nonlinear coupling among its winding currents and the rotor speed as well as the nonlinearity present in the torque equation. The system nonlinearity becomes seere if the

2 Vol. 2, Issue 3, May-Jun 212, pp IPMSM drie operates in the field weakening region MATLAB/Simulink erify the operation and stability where the direct axis current id. This results in the of the controller and drie system. appearance of a nonlinear term, which would hae anished under the existing ector control scheme with II. MATHEMATICAL MODEL OF id=. IPMSM There hae been significant deelopments in The mathematical model of the IPMSM can be non linear control theory applicable to electric motor gien by the following equations in a synchronously dries. Interestingly, the d q transformation applicable rotating d-q reference frame as [19]. to ac motors can be considered as a feedback linearization transformation. Howeer, with the recent deelopments in nonlinear control theories, a Voltage equations are gien by: modern control engineer has not only found a di systematic approach in dealing with nonlinearities d = Ri d L d Pω d rl q i q 2.1 di but has managed to deelop approaches which had q = Ri q L q Pω q rl d i d Pω r φ m 2.2 not been considered preiously. The surge of such Arranging equations 2.1 and 2.2 in matrix form nonlinear control methods applicable to dld electromechanical systems include ariable-structure d R Pφ r L q i d systems[3], differential geometric approach [4], [5], = q Pφ r R dl q iq Pω r φ m and passiity theory [6]. As electrical drie systems possess welldefined nonlinear model characteristics, they hae 2.3 The deeloped motor torque is being gien by become good candidates for the application of newly T e = 3P [φ 2 m i q L d L q i d i q ] 2.4 deeloped nonlinear control techniques. If the The mechanical Torque equation is gien by knowledge of such nonlinearities can be included in the T e = T L J dω r B design of nonlinear controller, an enhanced dynamic m 2.5 behaior of the motor drie can be accomplished. One ω r = T e T L Bω r 2.6 J ingenious way of designing a nonlinear controller is For conenience, the model is represented as follows. adaptie back stepping (AB) [7] [9]. di d = Ri d Pω r L q i q 1 Recently, the newly deeloped adaptie back L d L d 2.7 d di stepping technique has been used in the design of q = Ri q Pω r L d i d Pω r φ m 1 L speed and position controllers for dc, induction motors q L q 2.8 q dω r and permanent magnet motors [1] [17].This technique = 3P φ 2J m i q L d L q i d i q T L B m ω J J r 2.9 allows the designer to incorporate most system The aboe equations 2.7, 2.8, 2.9 are used to design the nonlinearities and uncertainties in the design of the Simulink model of the machine. controller. In [12], [13], [15] the authors designed a Under balanced steady-state conditions, the electrical nonlinear controller that achiees rotor angular speed angular elocity of rotor ω r is considered constant and and rotor flux amplitude tracking with uncertainties equal to that of the synchronously rotating reference in the rotor resistance and load torque for an frame. In this mode of operation, with the time rate induction motor. Results show that tracking objecties of change of all flux linkages neglected, the steady are achieed with ery little steady state error or state ersions of (2.7), (2.8) and (2.9) become oershoot. Zhou et. al [1] hae deeloped a back d = Ri d ω r L q i q 2.1 stepping based controller for a DC motor and induction motor with uncertainties. In this thesis, a nonlinear back stepping based q = Ri q ω r L d i d φ m 2.11 controller is designed for position control of an IPMSM o = Ri = Parks Transformation and Dynamic d q Modeling drie system. According to the proposed control The dynamic d q modeling is used for the study technique, the controller is designed using the motor of motor during transient and steady state. It is done by model equations incorporating arious system conerting the three phase oltages and currents to dqo uncertainties. The design method is similar to the ariables by using Parks transformation [2] one used in [17], howeer, the mechanical parameters Conerting the phase oltages ariables such as motor inertia and load torque are estimated abc to online instead of electrical parameters for better dqo ariables in rotor reference frame the following equations are obtained tracking response. The global system stability is erfied using Lyapuno stability theory. Digital simulations in 271 P a g e

3 d q o = 2 3 S.S.Sankeshwari, Chandulal Guguloth, Dr.R.H Chile / International Journal of Engineering Vol. 2, Issue 3, May-Jun 212, pp d e 1 = θ θ = θ ω r 3.2 q The stabilizing function is determined 2.1 by differentiating o sin θ r sin θ r 12 sin θ r 12 cos θ r cos θ r 12 cos θ r Conert V dqo to V abc a cos θ r sin θ r 1 b = cos θ r 12 sin θ r 1 c cos θ r 12 sin θ r 12 1 III. CONTROLLER DESIGN USING BACK STEPPING ALGORITHM A. Idea of back stepping By using newly deeloped back stepping algorithm we will design a controller. Back stepping is a recursie Lyapuno-based scheme proposed in the beginning of199s. The technique was comprehensiely addressed by Krstic, Kanellakopoulos and Kokotoic in [21]. The idea of back stepping is to design a controller recursiely by considering some of the state ariables as irtual controls and designing for them intermediate control laws. Back stepping achiees the goals of stabilization and tracking. The proof of these properties is a direct consequence of the recursie procedure, because a Lyapuno function is constructed for the entire system including the parameter estimates [21]. B. Controller design The foundation of back stepping is the identification of a irtual control ariable and forcing it to become a stabilizing function. Thus, it generates a corresponding error ariable which can be stabilized by proper selection input ia Lyapuno s stability theory [17]. For position control, we regard the rotor speed and d-q axis currents as the irtual control ariables. The block diagram of adaptie control scheme is shown below. Ѳ*r Ѳr _ S k1 1-k1^2 S^2 ω*r _ ωr C^ k1k2 D^ - - i*d 1/A Voltage controlled Generator V*d V*q the Lyapuno function: V 1 = 1 e to get 3.3 q dv 1 = e d 1 e 1 = e 1 θ ω r o We now choose the first stabilizing function as ω r = k 1 e 1 θ Equation (3.4) indicates the desired elocity for position tracking. The next step is to design a speed controller so that the rotor speed will follow (3.4). Step 2: Now we Define the speed tracking error as e 2 = ω r ω = k 1 e 1 θ ω r 3.5 From equation (4.5), the position error dynamics can be written as e 1 = k 1 e 1 e The speed error dynamics is defined as e 2 = ω r ω r = k 1 2 e 1 k 1 e 2 θ Ai q Bi d i q C Dω r 3.7 Now Define a new Lyapuno function as V 2 = 1 2 e e Differentiate to get dv 2 = e 1 e 1 e 2 e = k 1 e 2 1 e 2 1 k 2 1 e 1 k 1 e 2 θ Ai q Bidiq C Dωr 3.1 Since i d and i q were identified as the irtual control ariables, we Define the reference currents as i q = 1 1 k 2 A 1 e 1 k 1 k 2 e 2 θ C Dω r 3.11 i d = 3.12 Substituting (3.11) and (3.12) back into equation (3.1) would yield V 2 = k 1 e k 2 e Where k 1, k 2 > are design constants. Thus the irtual control is asymptotically stable. Since the parameters C and D are unknown we must use their estimated alues C and D.Parameter A is assumed to be constant. Thus equation (3.11)becomes i q = 1 1 k 2 A 1 e 1 k 1 k 2 e 2 θ C D ω r 3.14 e, Ø, ωr Parameter Estimation 1/s C^ Fig3.1: Block diagram of adaptie control scheme The following are the steps to design an adaptie controller Step 1: Define the position tracking error as e 1 = θ θ 3.1 Where θ is the desired reference trajectory of the rotor angle. The position error dynamics is then D^ id iq Step 3: The goal now is to make i d and i q follow the reference trajectory i d and i q. The final current error signals are defined as i e 3 = q -i q 3.15 e 4 = i d -i q 3.16 Using equations (3.15) and (4.16) the speed error dynamics can be represented by 272 P a g e

4 Vol. 2, Issue 3, May-Jun 212, pp e 2 = e 1 k 2 e 2 Ae 3 Be 4 i q C Dω r 3.17 C. Parameter estimation laws 4.17 Where C = C C, and D = D The update laws are defined as D are the C = n 1 e 2 6 e parameter estimation errors. D = n 2 e 2 ω r 6 e 3 ω r 3.26 Now we define the current error dynamics as Substituting equation ( ) into equation 3.21) e 3 = i q di q = 4 A 5 C 6 Dω r 6 Be 4 i q 6 D A a q 3.19 IV. BLOCK DIAGRAM 4.18 AND DRIVE And SYSTEM e 4 = i d = 2 b d 3.2 A. Block diagram of adaptie control 4.19 scheme Where 4, 5 and 6 are the known signals defined in Vd* Va Ѳr* Vb dq-abc the APPENDIX Vq* Vc PWM inerter IPMSM Ѳr ωr Iq Id Step 4: The final Lyapuno function includes the current errors Ia and parameter estimation errors. Ib V 3 = 1 e e 2 2 e 2 3 e C 2 1 abc-dq D 2 Ic 3.21 d/ 4.2 n 1 n 2 Where n 1,n 2 are adaptie gains. Now differentiate and substitute all error dynamic equations to get would yield V 3 = k 1 e 1 2 k e 2 2 k 3 e 2 3 k 4 e 2 4 Ae 2 e 3 < 2 For sufficiently large k 2 and k 3.Thus it is proen that the complete system is asymptotically stable. Fig.4.1.Block diagram of adaptie control scheme B. Drie system control Based on the control principle in chapter 4 the V 3 = e 1 e 1 e 2 e 2 e 3 e 3 e 4 e 4 1 CC 1 DD 3.22 complete closed loop ector control scheme of the n 1 n 2 IPMSM is shown in Figure 4.1. First the command speed is calculated from the position error and = e 1 k 1 e 1 e 2 e 2 e 1 k 2 e 2 Ae 3 Be 4 i q C Dω r deriatie of the command position using e 3 4 A 5 C 6 Dω r 6 a q e 4 2 b d 1 CC 1 equation3.4 as shown in chapter 3.The speed DD n 1 n 2 controller then generates the command d-q axis currents using equations (3.12) and (3.14) respectiely. Parameters C and D are calculated using equations (3.24) and( 3.25) and then the control oltages q and d are calculated using = k 1 e 2 1 k 2 e 2 2 k 3 e 2 3 k 4 e 2 4 Ae 2 e 3 equations (3.22) and (3.23) respectiely. Then they C e 2 6 e 3 1 C D e n 2 ω r 6 e 3 ω r 1 D are conerted to 3-phase oltages using Park s 1 n 2 Transformation. The PWM signals are generated by comparing the 3-phaseoltages with high e 3 k 3 e 3 4 a q e 4 k 4 e 4 Bi q e 2 6 frequency triangular waeforms. The PWM logic signals operate the inerter switches which run the D motor. The 3-phase currents i abc are conerted to d- 1A 2 bq q currents which are fed back into 4.21 the controller Non linear controller along with the rotor speed and position which completes the closed loop system. The d-q axis reference oltages are chosen to be q = 1 a k 3e Ѳr d = 1 b k 4 e 4 Bi q e 2 6 D 1 Where k 3, k 4 > are design constants. A P a g e

5 Position error e1 Vdq (V) Vdq (V) Idqs (amps) Idqs (A) Position (radians) Position (radians) S.S.Sankeshwari, Chandulal Guguloth, Dr.R.H Chile / International Journal of Engineering Vol. 2, Issue 3, May-Jun 212, pp V. SIMULATION RESULTS AND B. Simulink Block of oerall control system drie with DISCUSSION square wae trajectory A. Simulink Block of oerall control system drie with sine wae trajectory Fig.5.1. Simulink Block of oerall control system drie with sine wae trajectory Fig.5.5.Simulink Block of oerall control system drie with square wae trajectory Theta Reference Theta Actual Theta Reference Theta Actual Fig.5.2. Command position and motor position Fig.5.6 Command position and motor position Ids Iqs Iqs in amps Ids in amps Fig.5.3.d-qaxis currents i ds and i qs Fig.5.7.d-q axis currents i ds and i qs Vq in olts Vd in olts q d Fig.5.4. command oltagesv d and V q Fig.5.8. Command oltagesv d and V q Figure 5.9: Corresponding position error e P a g e

6 Vol. 2, Issue 3, May-Jun 212, pp COMMENTS: In order to erify the effectieness of the proposed adaptie scheme, digital simulations hae been performed using MATLAB/Simulink software. The oerall control bock diagram is shown in Figure 4.1 and a detailed block diagram of the adaptie controller is shown in Figure 3.1. Simulation work is done and some sample results are shown here. Case1: Sine wae trajectory Figure 5.1 shows the rotor position following the command position for a sinusoidal reference trajectory θ r = 1 sin 2t. The rotor position follows the reference trajectory with ery little error and no oershoot as shown in fig.5.2. Figure 5.3 shows the d-q axis currents and figure 5.4 shows the command oltages V d andv q. Case2: Square wae trajectory Figure 5.5 shows the rotor position for a square wae trajectory. The actual position conerges to the reference position in a short time with no oershoot and no steady state erroras shown in fig5.6. Fig.5.7 and 5.8 shows the corresponding currents and oltages. Fig 5.9 is the corresponding position error. The error between actual and command position is conerge to zero. Since the control oltages guarantee asymptotic stability, the error alues conerge to zero. By obsering aboe case1& and2, we can say that using adaptie controller we can reduce the error between command and actual signals, and a good tracking can be achieed. So by using the adaptie control technique good tracking can be performed and accuracy of system will achieed. V. CONCLUSION Successful implementation of adaptie back stepping control for the position tracking of the IPMSM drie has been illustrated in this thesis. It has been shown that the IPMSM drie belongs to a class of nonlinear system for which adaptie back stepping technique can be effectiely used. By recursie manner, irtual control states of the IPMSM drie hae been identified and stabilizing control laws are deeloped subsequently using Lyapuno stability theory. The detailed deriation for the control laws has been proided. The controller was designed using adaptie back stepping with uncertainties of inertia, load torque and friction. As shown by the simulation results, position tracking was achieed with ery little steady state error and oershoot. Global stability of the complete drie system has been erfied by Lyapuno stability theory. The alidity of the proposed control technique has been established in simulation at different input conditions. A. FUTURE SCOPE In this thesis the control drie is implemented with adaptie controller for position tracking only. In future it can be implemented by considering speed also, and the hard ware implementation can be seen. APPENDIX 3 = 1 k = Ri q Pω r L d i d Pω r φ m L q 2 = Ri d Pω r L q i q L d e 1 k 1 k 2 e 2 θ C Dω r 4 = 1 A 1 k 1 2 k 1 e 1 e 2 k 1 k 2 e 1 k 2 e 2 θ D References: D 2 ω r C CD k 1 k 2 e 3 Di q 1 5 = 1 A 3 k 1 k 2 6 = 1 A k 1 k 2 [1].Lau, J; Uddin, M.N.; Nonlinear adaptie position control of an Interior Permanent magnet Synchronous Motor. IEEE International Conference on Industry applications: 25, page(s): [2]. F. Blaschke, The principle of field orientation as applied to the new transector closed-loop control system for rotating-field machines, Siemens Re., May 1972 ol. 34, pp [3]. V. Utkin, Sliding Modes in Control Optimization : Springer Verlag, [4]. R. Marino, S.Peresada and P. Valigri, Adaptie input-output linearizing control of induction motors, IEEE Trans. Automat. Contr.,Feb. 1993, ol. 38, pp [5].M. Ilic-Spong, R. Marino, S. Peresada, and D. Taylor, Feedback linearizing control of switched reluctance motors, IEEE Trans. Automat.Contr., May 1987, ol. 32, pp [6]. R. Ortega, P. J. Nicklasson, and G. Espinosa, Passiity-based control of the general rotating electrical machines, in Proc. IEEE Conf. Decision and Control, 1994, pp [7]. M. Krstic, I. Kanellakopoulos, and P. Kokotoic, Nonlinear and Adaptie Control Design. New York: Wiley, [8]. J. J. Carroll and D. M. Dawson, Semi global position tracking control of brushless DC motors using output feedback, in Proc. IEEE Conf. Decision and Control, 1994, pp [9]. D. M. Dawson, J. J. Carroll, and M. Schneider, Integrator back stepping control of a brush DC motor turning a robotic load, IEEE Trans. Contr. Syst. Technol., Sept. 1994,ol. 2, pp P a g e

7 Vol. 2, Issue 3, May-Jun 212, pp [1]. J. Zhou, Y. Wang, and R. Zhou, Adaptie back stepping control of separately excited dc motor with uncertainties, in Conf. Rec. International Conf. on Power System Technology, ol. 1, 2, pp [11]. H. Tan and J. Chang, Adaptie back stepping control of induction motor with uncertainties, in Conf. Rec. American Control Conference, ol. 1,1999, pp [12]. H. Tan, Field orientation and adaptie back stepping for induction motor control, in Conf. Rec. IEEE-IAS Annual Meeting, ol. 4, 1999, pp [13]. H. Tan and J. Chang, Adaptie position control of induction motor systems under mechanical uncertainties, in Conf. Rec. IEEE International Conf. on Power Electronics and Drie Systems, ol. 2, 1999, pp [14]. C. I. Huang, K. L. Chen, H. T. Lee, and L. C. Fu, Nonlinear adaptie back stepping motion control of linear induction motor, in Conf. Rec. American Control Conference, ol. 4, 22, pp [15]. D. F. Chen, T. H. Liu, and C. K. Hung, Nonlinear adaptie-back stepping controller design for a matrix-conerter based pmsm control system, in Conf. Rec. IEEE-IES Annual Meeting, ol. 1, 23, pp [16]. J. Zhou and Y. Wang, Adaptie back stepping speed controller design for a permanent magnet synchronous motor, in IEE Proceedings in Electric Power Applications, ol. 149, 22, pp [17]. M. Vilathgamuwa, M. A. Rahman, K. Tseng and M. N. Uddin, Nonlinear control of interior permanent magnet synchronous motor, IEEE Transactions on Industry Applications, ol. 39, no. 2, pp , Mar./Apr. 23. [18]. Enrique L. Carrillo Arroyo, modeling and simulation of permanent magnet synchronous motor drie system, Master degree dissertation, uniersity of Puerto Rico Mayaguez campus 26. [19]. M. N. Uddin, Intelligent control of an interior permanent magnet synchronous motor, Ph.D. dissertation, Memorial Uniersity of New-found land, St. John, Oct. 2. [2]. B. K. Bose, Modern power electronics and AC dries: Prentice Hall, 22. [21]. Jing Zhou, Changyun Wen, Adaptie Back stepping Control of Uncertain Systems. Auther s Profile Mr.S.S Sankeshwari receied his M.tech degree from Walchand College of Engineering. He had 12 years teaching experience, presently working as a Asst. professor and HOD (EEP), M.B.E.S.College of Engineering Ambajogai. He is member of arious professional societies like MISTE. Mr.Chandulal Guguloth receied his M.Tech degree from National Institute of Technology Karnataka(NITK). Presently he is working as a Lecturer in M.B.E.S.College of Engineering Ambajogai,Maharastra. Dr. R.H Chile receied his Ph.D degree from I.I.T Roorkee. He had 22 years teaching experience, presently working as a Asso.Professor in Instrumentation Department S.G.G.S college of Engineering, Nanded.. He is member of arious professional societies. He guided many M.Tech and PhD Students. 276 P a g e