2012 12th International Conference on Control, Automation and SystemsOct. 17-21, 2012 in ICC, Jeju Island, Korea

PID controller with Feedforward Low Pass Filters for Permanent MagnetStepper Motors

Youngwoo Lee1, Donghoon Shin1 and Chung Choo Chung21Department of Electrical Engineering, Hanyang University, Seoul 133-791, Korea

(Tel: +82-2-2220-4308; E-mail: stork@hanyang.ac.kr (Y. Lee), shin211@hanyang.ac.kr (D. Shin))2Division of Electrical and Biomedical Engineering, Hanyang University, Seoul, Korea

(Tel: +82-2-2220-1724, e-mail: cchung@hanyang.ac.kr)

Abstract: In this paper, PID controller with feedforward low pass filters (FLPFs) is developed to guarantee desiredcurrents for a permanent magnet stepper motor (PMSM). The proposed method consist of position controller and FLPFs.The position controller consists of a PID controller, a commutation scheme, and a Lyapunov based controller. To generatethe desired torque, the PID controller is incorporated. Commutation scheme is developed to generate the desired currents.Lyapunov based controller is used to guarantee the desired currents. The FLPFs are developed to filter out high frequencynoises of measured state and compensate for the phase lag caused by the conventional LPFs. Using the Lyapunov theory,we prove that globally asymptotical stability of currents errors is ensured by the proposed method. Experimental resultsvalidate the effectiveness of the proposed method.

Keywords: PID controller, Commutation scheme, Lyapunov based controller, Low Pass Filter, Microstepping, PositionTracking, phase lag, PMSMs.

1. INTRODUCTIONVarious feedback methods have been studied for

position control of permanent magnet stepper motors(PMSMs) [1], [2]. Microstepping is used to improvethe resolution and motion stability of the PMSM. In mi-crostepping, the desired currents are provided, then thetorque generated by the currents moves the position tothe desired position. Thus, guaranteeing the desiredcurrents are important for position tracking. In indus-trial applications, proportional-integral (PI) controller hasbeen widely used to improve the current tracking perfor-mance [2], [3], [4], [5], [6], [7]. These methods requiredstate feedbacks. Since, in many practical problems wecant measure all state variables due to technical or eco-nomic reasons, a state observer is used to estimate thestates from the output measurement. Recently, a Lya-punov based controller with passive nonlinear observerwas proposed to guarantee the desired currents [8]. How-ever, this method requires the position feedback. Thus,disturbance observer-based controller was proposed toguarantee the desired current using only current feed-back [9]. Recently, full state estimation method wasdeveloped using only position feedback [2]. However,bandwidth of current estimators is determined by resis-tance and inductance so that the bandwidth of estima-tor cannot be arbitrarily chosen. Furthermore, the cur-rent can be easily measured using the motor drivers sincealmost motor driver includes the circuit of the currentsensor. Because of the noises in currents measurement,conventional low pass filters (LPFs) are generally usedto reduce the high frequency noises with current sensorin many applications. But, phase lag in the currents wascaused by the conventional LPFs.

In this paper, we propose PID controller with feedfor-

ward current low pass filters (FLPFs) to improve the po-sition and currents tracking performances. The positioncontroller consists of a PID controller, a commutationscheme, and a Lyapunov-based controller. PID controlleris used to generate desired torque [10]. Commutationscheme is used to transform the desired torque into thedesired currents. Lyapunov-based controller (LBC) is de-signed to guarantee the desired currents [8]. The FLPFsare proposed to reduce measurement noises and com-pensate to the phase lag due to the conventional LPFs.The closed-loop stability is proven using Lyapunov the-ory. Simulations were performed to evaluate the pro-posed method.

2. MATHEMATICAL MODEL OF PMSMAND CONTROLLER DESIGN

2.1 Mathematical modelThe dynamics of PMSM can be represented in the state

space form as follows

=

=1J(Kmia sin(Nr )+Kmib cos(Nr )B)

ia =1L(va Ria +Km sin(Nr ))

ib =1L(vb Rib Km cos(Nr )).

(1)

where va, vb and ia, ib are the voltages [V] and currents[A] of phases A and B, respectively. is the rotor (an-gular) position [rad], is the rotor (angular) velocity[rad/s], B is the viscous friction coefficient [Nms/rad], Jis the inertia of the motor [Kgm2], Km is the motor torqueconstant [Nm/A], R is the resistance of the phase wind-ing [], L is the inductance of the phase winding [H], andNr is the number of rotor teeth. Since detent torque in a

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bhlee 978-89-93215-04-5 95560/12/$15 ICROS

PMSM does not significantly affect the torque producedby the motor, it can therefore be ignored. In addition, themagnetic coupling between the phases is also ignored, aswell as the variation in inductance due to magnetic satu-ration. An ideal sinusoidal flux distribution is assumed.

3. CONTROLLER AND PROPOSED FLPFDESIGN

3.1 Lyapunov based controller and FLPF designGenerally, tracking controller is used to guarantee de-

sired currents. We propose the nonlinear tracking con-troller. Lyapunov based controller is needed to improvethe current tracking performance. When the Lyapunovbased controller is used, the current feedback is need.Since measured currents have high frequency noises suchas switching noise of pulse width modulation (PWM)of motor driver and electromagnetic interference noise,the conventional LPFs are used to filter out the high fre-quency noises. However, when the LPFs are used, phaselag in the currents is appeared. To remedy this problem,we propose the current FLPFs as follows

ia = k1(ia ia)+ ida,ib = k1(ib ib)+ idb

(2)

where ia and ib are the filtered currents, k1 is a positivenumber, and ida and idb are desired phase currents. Letus define the current tracking errors, ea and ea, currentsfiltering errors, ia and ib as

ea = ida ia,

eb = idb ib,ia = ia ia,ib = ib ib,

(3)

where ida and idb are desired phase currents. The currentstracking error dynamics by the (3) is represented as

ea =1L[Lida va +Ria Km sin(Nr )],

eb =1L[Lidb vb +Rib +Km cos(Nr )]

(4)

By equation (2), (3), and (4), current tracking and filterederror dynamics are represented as

ea =1L[Ria (ea + ia)] =

ea +(R)iaL

,

eb =1L[Rib (eb + ib)] =

eb +(R)ibL

,

ia =

(k1L+ R)ia +eaL

,

ib =

(k1L+ R)ib +ebL

,

(5)

3.2 PID controller and FLPF designThe mechanical dynamics of PMSM is

=

=1J(Kmia sin(Nr )+Kmib cos(Nr )B)

(6)

where = Kmia sin(Nr ) +Kmib cos(Nr ). The posi-tion and velocity tracking errors are defined as

ez = t

0 dd

t0

d,

e = d ,e =

d

(7)

where d is desired position and d is desired velocity.Therefore, tracking error dynamics by the mechanical er-ror is represented by

ez = d ,e =

d ,

e = d

1J[d e B ]

(8)

where = d e.

Theorem 1: Consider the mechanical error dynam-ics (8) and electrical error dynamics (4). Suppose thatthe PID controller, feedforward controller, and Lyapunovbased controller are designed by

d = KPe +KIez +KDe + Jd +Bd,va = Ria Km sin(Nr )+Lida +(ida ia),vb = Rib +Km cos(Nr )+Lidb +(idb ib)

(9)

where KP, KI , and KD are PID controller gains, is arbi-trary positive constant. To guarantee the desired currents,feedforward controller is used in the mechanical error dy-namics. By the (9), the origins of the mechanical andelectrical error dynamics and filtered signal error dynam-ics are globally exponentially stable.

3

Proof: For stability analysis, a composite Lyapunovcandidate function Vcl is defined as

Vcl =d2(e2z + e

2 + e

2 +(ez+ e + Je)2)

+1 d

2L(e2a + e2b + i2a + i2b).

(10)

Derivative of V1 with respect to time is given by

Vcl = d[Kie2z +(1Kp)e2 +(1 2Kd)e2 + 2K1eze+ 2K2eze + 2K3e e +(K4ea +K5eb)(ez + e + e)]+ (1 d)[1(e2a + e2b)+2(i2a + i2b)+R(eai2a + ebi2b)]

(11)

where K1 =1KpKi

2 , K2 =12KiKd

2 , K3 =12KpKd

2 ,

K4 = Km sin(Nr)2 , K5 = Km cos(Nr)

2 , 1 = , 2 =

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dai

d

bi

ai

bi

av

bv

ai

bi

d

dt

d

dt

d

bi

d

ai

dd

d

+

++

Fig. 1 Block diagram of the proposed method

k1L+RL .

The vector are defined as followse1 = [ez,e ,e ], e2 = [ea,eb, ia, ia], ex = [e1,e2].

Vcl =eTx Qex. (12)

where Q =[Q1 Q2

Q2 Q3

]

Q1 = dKi K1 K2K1 1Kp K3

K2 K3 1 2Kd

,

Q2 = dK4 K5 0 0K4 K5 0 0

K4 K5 0 0

,

Q3 = (1 d)

1 0 R2 0

0 1 0 R2R2 0 2 00 R2 0 2

.

By Schur complement, we see that Q > 0 is equivalent toQ3 Q2Q11 Q2 > 0.Since there always exist d > 0 and K1,K2,K3,K4,K5,1,2to satisfy the stability conditions Vcl < 0, the origin of exis globally exponentially stable.

In the mechanical dynamics (6), torque is generated bythe current. But, actual input of PMSM is not torque, butthe currents ia, ib. Therefore, we propose a commutationscheme defined as

ida =

Kmsin(Nr )

idb =

Kmcos(Nr ).

(13)

4. SIMULATION RESULTSWe performed the simulations to evaluate the po-

sition and currents tracking performance of the pro-posed method. Simulator was implemented by Mat-lab/simulink. To evaluate the performance of the pro-posed method, simulations were executed for two cases. Case 1: Conventional LPFs method. Case 2: Proposed FLPFs method.Random noises were injected as measurement noises inthe each phase currents. Random noises in the currents

Table 1 PMSM and controller parametersParameter Value Parameter Value

J 4.8 105 B 5.0 103Km 1.25 Nr 50R 4.5 L 14.8 50 KP 50KI 0.1 KD 0.001

0 0.5 1 1.5 20

2

4

6

8

10

12

14

16

Time [s]

Posi

tion

prof

ile [ra

d]

Fig. 2 Position profile

and velocity was about 5 % of the rated currents. ThePMSM and controller parameters listed in Table 1 wereused. A position profile is shown in Fig. 1. The cur-rent tracking performances in the each phase are shownin Fig. 2. The current tracking error in the case 1 was rela-tively larger than error amplitude in the case 2. Phase lagsin the current are shown in Fig. 3. In case I, the relativelylarge phase lag appeared due to the use of LPF. On theother hands, the phase lag was reduced by the proposedFLPF. Note that the amplitudes of the currents were de-creased due to the nonlinearities of the PWM drivers. Po-sition tracking errors of both cases are shown in Fig. 4.We observed that the current tracking error in the firstcase was relatively larger than that in the Case II. There-fore, the position tracking error of both methods are re-lated with a reduction of the current tracking errors.

5. CONCLUSIONSIn this paper, we developed the PID controller with

proposed FLPFs to improve the position tracking perfor-

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0 0.5 1 1.5 2

0.40.2

00.20.40.6

Current tracking error

Time [s]

Curre

nt tr

ackin

g er

ror[A

]

Phase APhase B

(a)Case I

0 0.5 1 1.5 2

0.40.2

00.20.40.6

Current tracking error

Time [s]

Curre

nt tr

ackin

g er

ror[A

]

Phase APhase B

(b)Case IIFig. 3 Current tracking performances of both methods

1.1 1.12 1.14 1.16 1.18 1.22

1

0

1

2Phase A current

Time [s]

Phas

e A

curre

nts[A

]

Desired currentActual current

(a)Case I

1.1 1.12 1.14 1.16 1.18 1.22

1

0

1

2Phase A current

Time [s]

Phas

e A

curre

nts[A

]

Desired currentActual current

(b)Case IIFig. 4 Phase A current of both methods in constant

velocity period

0 0.5 1 1.5 2

0

2

4

6x 103 Position tracking error

Time [s]

Posi

tion

track

ing

erro

r[rad

]

(a)Case I

0 0.5 1 1.5 2

0

2

4

6x 103 Position tracking error

Time [s]

Posi

tion

track

ing

erro

r[rad

]

(b)Case IIFig. 5 Position tracking performances of both methods

mance in the PMSM. The previous approaches are basedon the conventional LPFs. However, the conventionalLPFs cause the phase lag in the current tracking. Tocompensate the phase lag and reduce the high frequencynoises in the current measurement, proposed FLPFs wereused. Desired torque generated by PID and feedforwardcontroller was used. Commutation scheme was used togenerate the desired currents. To guarantee the desiredcurrents, Lyapunov based controllers were used as a cur-rent tracking controller. Simulation results validate theeffectiveness of the proposed method.

REFERENCES[1] M. Bodson, N. Chiasson, T. Nonotnak, and B.

Rekowski High-performance nonlinear feedbackcontrol of a permanent magnet stepper motor,IEEE Trans. Contr. Syst. Techn., vol. 1, no. 1, Mar.,1993.

[2] W. Kim, C. Yang, and C. Chung, Design andimplementation of simple field oriented controlfor permanent magnet stepper motors without DQtransformation, IEEE Trans. on Magnetics, Vol.47, No. 10, pp. 4231-4234, Oct., 2011.

[3] G. Baluta, Microstepping mode for stepper motorcontrol, in Proceedings of the IEEE InternationalSymposium on Signals, Circuits and Systems Vol. 1,pp. 1-4, Jul. 2007.

[4] A. Bellini, C. Concari, G. Franceschini, and A.Toscani, Mixed-mode PWM for high-performancestepping motors, IEEE Trans. on Industrial Elec-tronics, Vol. 2, pp. 1212-1217, Nov. 2004.

[5] Core technologies, Motors and control sys-tems for precise motion control [online], Avail-able: http://www.1-core.com/library/auto.motion-control/

[6] S. Manea, Stepper motor control with dsPIC DSCsmicrochip application note, pp. 1-26, 2009.

[7] W. Kim, I. Choi, and C. Chung, Microstep-ping with PI feedback and feedforward for per-manent magnet stepper motors, in Proceedings ofICCAS-SICE, Fukuoka International Congress Cen-ter, Fukuoka, Japan, pp. 603-607, Aug. 18-21, 2009.

[8] W. Kim, I. Choi, and C. Chung, Lyapunov-basedcontrol in microstepping with a nonlinear observerfor permanent magnet stepper motors, in Proceed-ings of American Control Conference, pp. 4313-4316, 2010.

[9] W. Kim, D. Shin, and C. Chung, Observer-basedvariable structure control in microstepping for per-manent magnet stepper motors in Proceedings ofIEEE/ASME International Conference on AdvancedIntelligent Mechatronics, pp. 1053-1057, 2010.

[10] W. Kim, D. Shin, and C. Chung, Microsteppingwith nonlinear torque modulation for position track-ing control in permanent magnet stepper motors, inProc. Conf. Dec. Contr., 2011, pp. 915-921.

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