Proceedings of the 14th International Middle East Power Systems Conference (MEPCON’10), Cairo University, Egypt, December 19-21, 2010, Paper ID 160.

85

Abstract—This paper presents a gain-scheduling adaptive PI controller scheme for speed control of hybrid stepper motor (HSM) drives. The PI gains are allowed to vary within a pre-determined range and therefore eliminate the problems faced by the conventional PI controller. The performance of the proposed gain scheduled PI controller is simulated and compared with the conventional fixed PI controller under starting, speed reversal, repetitive operation and parameter variations as well as load disturbances. The experimental system of HSM drive is implemented using DSP-DS1102 control board to examine and evaluate the criteria of high performance of the proposed controller under different operating conditions. Simulation and experimental results show a good improvement in transient as well as steady state response of the proposed controller over the conventional fixed PI one.

Index Terms—Gain scheduling adaptive PI control, DSP-Based Control, Hybrid Stepper Motor, Field Oriented Control

I. INTRODUCTION The control of stepper motors has attracted much attention

over the past few years, due to the developments in control theory and the availability of low-cost digital hardware. The hybrid stepper motor (HSM) has been widely known for its higher efficiency and torque capability as compared to other stepper motor types. Attempts have been made to present the dynamic modeling and analysis of hybrid stepper motor using different control laws [2].

HSM are generally operated in open loop due to its special structure. They are mainly used for simple point-to-point positioning tasks in which they were open-loop controlled. In this way, they were driven by a pulse train with a predetermined time interval between successive pulses applied to the power driver, and no information on the motor shaft position or speed was used [3].

In the open loop control, the HSM often use about 50% of its nominal torque since a large torque reserve is required to overcome any load variation. In this classical control scheme there is no feedback of load position to the controller, however the motor must respond to each excitation change. This introduces large overshoot, resonance and torque ripple problems which degrade the operating performances [4]-[5]. Besides, if fast excitation changes are applied, the stepper motor can lose steps and therefore it fails to move the rotor to the new demanded position. This would result a permanent error between the actual load position and the required position and consequently lose its stability and synchronization [4]-[6].

The digital closed-loop principle was introduced to stepper motors in the 1970’s in order to increase positioning accuracy

and reduce their sensitivity to load disturbances [4],[5]. Nowadays, stepper motors are more often closed-loop controlled, in particular, for machine tools and robotic manipulators in which they have to perform high-precision operations in spite of the mechanical configuration changes [6]-[11]. The use of classic closed-loop algorithms such as proportional–integral–derivative (PID) control is inadequate because these algorithms are often sensitive to mechanical configuration changes [6]. Classic methods that make use of linear models for designing controllers are valid only on small variation around an operating point. This problem can be solved by applying advanced closed-loop control techniques such as self-tuning regulation (STR) [7] or nonlinear feedback control [8] where the controller is enforced to adapt itself to the motor operating conditions. Applied to the stepper motor, STR gives better performance than PID regulation because this technique is adaptive to system variations [6]. Nevertheless, this kind of control strategy is difficult to be implemented due to the large amount of floating-point computation, which means an increase in the sampling period. In the adaptive methods, the control laws such as model reference adaptive control and self tuning regulator are nonlinear control laws which are difficult to derive. Furthermore, the complexity grows geometrically with the number of unknown parameters. Moreover, they require tuning of parameters accurately.

Recently, artificial intelligence techniques for closed loop control of stepper motors have been used [6],[13]. Mathematical models of the controlled system are not required. Variations of the parameters and operating conditions of the controlled system do not significantly affect the performance of the controller. Neural network have been applied to the identification and control of nonlinear dynamical systems, but this control approach need state information or plant models. They spent extensive time for online training of neural networks. Also, fuzzy logic control of hybrid stepper motor has been presented in [13]. However, the disadvantages such as designing logic controller need expertise of the human expert and determining parameters of controller by trial and error limits its application.

It is well known that a conventional PI controller is most widely used in industry due to its simple control structure, ease of design and low cost. However, the PI type controller cannot give a good control performance. Moreover, it suffers from disadvantages of slower response, larger overshoots, and oscillation. As the HSM has nonlinear model, the linear PI control is not a good option.

Gain Scheduling Adaptive PI Control of Hybrid Stepper Motor Drives

Mohamed S. Zaky and Ehab M. Ismaeil Electrical Engineering Dept., Faculty of Engineering,

Minoufiya University, Shebin El-Kom (32511), Egypt. Email: ehab_mohammed2008@yahoo.com

86

Gain scheduled scheme for PI controller, as it is simple to implement and provides good performance has been proposed. In the present investigation, PI gains are tuned such that the drive system exhibits satisfactory transient and steady state response under varying operating conditions, such as starting, reversal, repetitive operation, parameter variations and load disturbances. In this paper, a gain scheduling adaptive PI controller for HSM drive is presented. The proposed gain scheduling adaptive PI speed controller is tuning with operating conditions. Their gains are allowed to vary within a pre-determined range and therefore eliminate the problem by the conventional PI controller. Simulation model is established on Matlab/Simulink to test the HSM drive system with the different speed controller under different operating conditions. Experimental system using DSP-DS1102 is built to examine and assess the performance of the proposed controller. The proposed gain scheduling adaptive PI controller is characterized by low computational time, ease of implementation and suitable for practical applications. It has the following features in comparison with conventional fixed PI controller:

1- High performance in terms of trajectory tracking accuracy.

2-Robustness properties against load torque disturbances and parameter variations.

II. FIELD ORIENTED CONTROL OF HSM (A) Motor d-q Model: The HSM can be written in d-q rotor reference frame model for field oriented control as follows [12]:

d dd r q

di R i N idt L L

v= − + ω + (1)

q qmq r d r

di vR Ki N idt L L L

= − − ω − ω + (2)

The developed torque can be expressed as: e m qT K i= (3)

r m v Lq

d K K Tidt J J Jω

= − ω − (4)

ddtθ= ω (5)

where di and qi are direct and quadrature currents.

dv and qv are direct and quadrature voltages.

N is Number of rotor teeth.

dL and qL are direct and quadrature components of

inductance. R is phase resistance

mK is torque constant.

vK is coefficient of viscous friction.

rω is motor speed. θ is electrical position created between d-axis and q-axis

(B)Hysteresis Current Control The current control, which consists of two hysteresis

controllers, is built with Simulink blocks. The motor currents are provided by measurement and compared to the reference currents. The current error is passed through hysteresis controller represented by relay block with band H to produce the inverter gate pulses as shown in Fig. 1

+-

Pulses

ai

ai∗

+-bi

bi∗

Fig. 1 Hysteresis current controllers for producing gate pulses.

(C) Field Oriented Control Priniples In order to simplify the control, direct current id is set to zero. The speed error is passed through speed controller and the result is the torque command component *( )qi . The

synchronous reference frame variable * *( , )q di i are transformed

to stationary reference frame ( , )s sq di i .Then, phase reference

currents ( , )a bi i is obtained. Comparing reference and actual currents and the error pass through hystersis controller to produce the gate pulses. The transformation from synchronous reference frame to stationary reference frame and thus to reference phase currents can be found using (6) and (7), [13]:

cos sin.

sin cos

sd d

sq q

i i

i i

∗

∗

θ θ ⎡ ⎤⎡ ⎤ ⎛ ⎞= ⎢ ⎥⎜ ⎟⎢ ⎥ ⎜ ⎟− θ θ ⎢ ⎥⎢ ⎥ ⎝ ⎠⎣ ⎦ ⎣ ⎦

(6)

1 0.

1 2 3/ 2

sa d

sb q

i ii il

⎛ ⎞ ⎡ ⎤⎡ ⎤⎜ ⎟= ⎢ ⎥⎢ ⎥ ⎜ ⎟−⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎝ ⎠

(7)

According to the mathematical model given above, the block diagram of a speed control for HSM drive system can be represented as shown in Fig. 2.

PWMConverter

HSM

ai

Gain Scheduling Adaptive PI Controller

5 Vdc

Encoder

arefi HysteresisCurrent Control

Driver

−

+

Speed Controller

br efi

bi

d-q to ab

qi∗

0di∗ =

refω

rω

rω

av bv

AxesTransform.

dt∫

rθ

Fig. 2 Block diagram for a field oriented controlled HSM.

III. SPEED CONTROL SCHEMES OF HSM

A. Conventional PI Controller The conventional fixed PI controller can be constructed as follows:

87

( ) ( )c Pc icu K e t K e t dt= + ∫ (8)

The PI controller gains KPc and Kic are tuned at rated conditions and have these values along with the different operating conditions. B. Gain Scheduling Adaptive PI Controller The key feature of the proposed gain scheduling scheme of adaptive control is the reduced amount of computation. An easy and low cost practical implementation of the procedure is possible without employing expensive dedicated computing systems. This scheme is very easy to implement in practice since an existing PI controller is tuned automatically. In this scheme, the gains are allowed to vary over a predetermined range for varying operating conditions. It is well known that the proportional term (KP) is responsible for improving overshoots, rise time response and the integral (KI) term reduces steady state error. When the speed error is large, a large value of proportional gain is necessary for better control effort and similarly when the speed error is small a large value of integral gain is necessary to overcome steady state error. The gain scheduled PI controller output, which is considered as the reference torque of the motor can be described as

* ( ) ( ) ( ) ( )p iT K t e t K t e t dt= + ∫ (9)

Where ( ) ( ) ( ),r re t t t∗= ω − ω ( )pK t is the proportional

gain and ( )iK t is the integral gain. These gains are functions of the speed error ( )e t . The gain ( )pK t is expressed as a

function of speed error as follows [ ( )]

(max) (max) (min)( ) ( ) ke tp p p pK t K K K e −= − − (10)

Where k is a constant which decides the rate at which ( )pK t varies between maximum and minimum values of the

proportional gain. A large proportional gain, (max)pK is used

to speed up the transient response when the speed error ( )e t is large and when the error ( )e t becomes small, a minimum proportional gain (min)pK is used to eliminate overshoots and

oscillations. The integral gain ( )iK t is expressed as a function of speed error signal ( )e t as

-[ ( )](max)( ) ke t

i iK t K e= (11)

Under steady state condition when speed error ( )e t is small, large integral gain is used to overcome the steady state error. When the error is large, a small integral gain is used in order to eliminate the undesirable oscillations and overshoot. In transient condition, a large control signal is used to accelerate or decelerate the motor to the reference value within smallest possible time. During this period, ( )pK t is at its maximum

value and ( )iK t is maintained at its minimum value. Under steady state operating condition, the integral gain ( )iK t is increased to its maximum value. These two gains are varied online as a function of speed error ( )e t .

IV. SIMULATION RESULTS

Simulation model is established in Matlab/Simulink environment based on the introduced mathematical model. The performance of the HSM drive system is tested under different operating conditions. Simulation results include starting operation, step speed command; repetitive operation, speed reversal, parameter variations and load impact are presented. The performance of gain scheduling adaptive PI controller in comparison with conventional PI controller is examined and assessed by computer simulations.

(A) Step Speed Change The HSM drive system is tested under the different speed controllers. Fig. 3 shows the simulated speed responses under conventional fixed PI and gain scheduling adaptive PI controllers. It is obvious that the speed response with fixed PI controller suffers from overshoots and large settling time in comparison to gain scheduling adaptive PI controller. (B) Repetitive Operation The drive system is also tested during repetitive operation with the different controllers. It is clear that the gain scheduling adaptive PI controller exhibits superior performance in comparison with fixed PI controller which suffers from overshoots and large settling time. (C) Speed Reversal The stability and synchronization of the drive system with the different controllers are examined during speed reversal. It is evident that the proposed controller gives a good performance in comparison with the conventional PI controller. (D) Load torque disturbances The superiority of the gain scheduling adaptive PI controller is proved and confirmed under load torque disturbance. Fig. 12 shows the speed response of the HSM drive system with different speed controllers. It is observed that the speed recovers quickly and the speed dip is low with gain scheduling adaptive PI controller which is not the case with fixed PI controller. The variation of ( )pK t and ( )iK t during

transient conditions is shown in Fig. 17. It is obvious that these gains are adjusted to proper values during different operating conditions of the motor to give fast dynamic response without steady state error. (E) Effect of parameter variations The robustness of the proposed gain scheduling adaptive PI controller is also examined under load inertia variations and the results are compared with fixed PI controller is shown in Fig. 16. It is observed that the speed response with the proposed controller is better than the corresponding response with the fixed PI controller in terms of overshoots, rising time and settling time

88

(a) Reference and actual speeds.

(b) Reference and actual rotor position signal.

(c) Developed torque.

Fig. 3 Simulated responses during starting operation with conventional PI controller.

Fig. 5 Simulated responses during repetitive operation with conventional PI controller.

Fig. 7 Simulated responses during speed reversal with conventional PI controller.

Fig. 9 Simulated responses during stop- reverse operation with conventional PI

controller.

(a) Reference and actual speeds.

(b) Reference and actual rotor position signal.

(c) Developed torque.

Fig. 4 Simulated responses during starting operation with gain scheduling adaptive PI controller.

Fig. 6 Simulated responses during repetitive operation with gain scheduling adaptive PI

controller.

Fig. 8 Simulated responses during speed reversal with gain scheduling adaptive PI

controller.

Fig. 10 Simulated responses during stop- reverse operation with gain scheduling

adaptive PI controller

89

(a) Developed torque.

(b) Rotor speed.

(c) Stator currents of phases a and b.

Fig. 11 Simulated responses during sudden load change with conventional PI controller.

Fig. 13 Simulated responses during sudden change of speed with conventional PI

controller.

J = 2Jo J = Jo J = 0.5Jo

Fig. 15 simulated speed responses under load inertia variations with fixed PI

controller

(a) Developed torque.

(b) Rotor speed.

(c) Stator currents of phases a and b.

Fig. 12 Simulated responses during sudden load change with gain scheduling adaptive PI controller

Fig. 14 Simulated responses during sudden change of speed with gain scheduling

adaptive PI controller.

J = 2Jo J = Jo J = 0.5Jo

Fig. 16 simulated speed responses under load inertia variations with gain

scheduling adaptive PI controller.

Fig.17 Variations of Kp(t) and Ki(t) for starting and load disturbance.

90

V. SYSTEM IMPLEMENTATION

The basic configuration of the experimental system is shown in Fig. 18. It consists of a HSM interfaced with a digital control board DS1102 based on a Texas Instruments TMS320C31 Digital Signal Processor for real-time control. Rating and parameters of the HSM are given in the appendix. Stator currents are measured and filtered using analogue circuitry. Hall-effect sensors are used for this purpose. The measured current signals are acquired by the A/D input ports of the DSP control board. This board is hosted by a personal computer on which mathematical algorithms are programmed and downloaded to the board for real-time control.

The motor phases are fed by H-bridge MOSFET converters connected to a 5 V DC voltage source. The motor phase currents are independently controlled by two hysteresis-based controllers which generate the MOSFET drive signals by comparing the measured currents with their references. The output switching commands of the DSP control board are obtained via its digital ports and interfaced with the converter through opto-isolated gate drive circuits.

VI. EXPERIMENTAL RESULTS

The experimental system of Fig. 19 is built in the laboratory to test the performance of the HSM drive system under different operation conditions. Fig. 20 show the experimental phase current Ia, Ib at speed reference 314 rpm. This includes step change of reference speed 314rpm, respectively. Fig. 21 shows the reference and measured speeds under speed command 314 rpm, Fig. 22 shows the reference and measured speeds under step speed change. It is obvious that the measured speed and consequently the rotor position reaches the steady state value smoothly without overshoot or undershoot. Moreover, the drive system has a fast dynamic response and takes a minimum rise time to reach the steady state value. However, the measured speed contains ripples which increase at low speeds. Experimental results are presented also during speed reversal, Fig.23 It is clear that the measured speed follows the reference speed smoothly. Moreover, experimental results are presented with repetitive operation to test the precise operation of the drive system and fast dynamic response, Fig. 24 It is observed that the drive system preserves its stability and synchronization with fast start and stop operations. This proves the supremacy of the proposed closed loop control of HSM drive system.

VII. CONCLUSION

In this paper, a gain scheduling adaptive PI controller for a HSM has been presented. The HSM drive system has been examined experimentally and by computer simulations with the proposed controller and the conventional fixed PI one. The gains of the proposed controller has been varied and tuned such that the drive system exhibits satisfactory transient and steady state response under varying operating conditions. Experimental and simulation results show the effectiveness of this approach, and demonstrate the usefulness of proposed controller in high performance drives. The proposed controller method has shown a good performance in comparison with

the conventional fixed PI controller in terms of trajectory tracking, load inertia variations and load disturbances. Furthermore, the proposed method has characterized by simplicity, low computation time and ease of implementation. A comparison between the conventional fixed PI controller and a gain scheduling adaptive PI one has shown the advantages of the proposed scheme.

HSMDC Gen.

PWM Inverter

DC Power Supply

Current Sensors

Isolation and Gate

Driver

Encoder

Digital I/O

Ports

A/D

Encoder Interface

Gain schedulingωref

dq→ab

HysteresisController

iq id=0

iabrefiab ia

ib

MatlabSimulink

ISA Bus

DSP-DS1102 Control Board

OscilloscopeD/A

θ

PWM Signals

va vb

Fig 18 Block diagram of DSP-based real- time implementation of gain

scheduling adaptive PI controller for HSM drive system.

Fig. 19 A picture of the overall experimental system of HSM using DSP.

Time [sec]

Phas

es c

urre

nt [A

] Ia

Ib

Fig. 20 Experimental Phases current Ia, Ib at speed reference at 314 rpm.

Time [sec]

Spee

ds [r

pm] Reference

Measured

91

Fig. 21 Experimental results during speed command reference at 314 rpm using gain scheduling adaptive PI control.

Time [sec]

Spee

ds [r

pm] Reference

Measured

Fig. 22 Experimental results during step change of speed reference at 314 rpm

using gain scheduling adaptive PI control.

Time [sec]

Spee

ds [r

pm]

Reference

Measured

Fig. 23 Experimental results during speed reversal at 314 rpm using gain

scheduling adaptive PI control..

Time [sec]

Spee

ds [r

pm]

Reference Measured

Fig. 24 Experimental results during repetitive operation at 314 rpm using gain scheduling adaptive PI control.

APPENDIX Table I Parameters of HSM

Rated voltage (volt) 24 Coefficient of viscous friction [N.m.s/rad] 0.00047

Phase Resistance (Ohm) 0.37 Torque constant 0.153

Phase Inductance (mH) 0.9 Rotor Inertia (Kg.m2) 15.62e-5

Number of rotor teeth 50 Table 2 Gains of conventional PI speed controller

Proportional gain 0.15 Integral gain 1

Table 3 Gains of gain scheduling adaptive PI speed controller

Proportional gain max 0.8 Proportional gain min 0.6 Proportional gain 1

REFERENCES

[1] P. Acarnley, "Stepping motors-a guide to theory and practice," 4th Edition, the Institution of Electrical Engineering, London, 2002.

[2] D. Xu and Y. Jiang, "A method and implementation of fully digitized continuous microstep for step motor", IEEE IEMDC97, pp. TC2/9.1- TC2/9.3, May 18-21, 1997.

[3] T. R. Fredriksen, "Application of the closed-loop stepping motor," IEEE Trans. on Automatic Control, vol. AC-13, pp. 464–474, Oct. 1968.

[4] B. C. Kuo, "Closed-loop and speed control of step motors," The 3rd Annual Symposium Incremental Motion Control Systems and Devices, Urbana-Champaign, IL, May 6–8, 1974.

[5] F. Betin, M. Deloizy, and C. Goeldel, "Closed loop control of stepping motor drive: comparison between PID control, self tuning regulation and fuzzy logic control," European Power Electronics Journal, vol. 8, no. 1–2, pp. 33–39, June 1999.

[6] C. Rusu, I. Birou and E. Szoke, "Model Based Design Controller for the Stepper Motor," IEEE International Conference on Automation, Quality and Testing, Robotics (AQTR), vol. 2, pp. 175–179, May 22-25, 2008.

[7] M. Bodson, J. N. Chiasson, R. T. Novotnak, and R. B. Rekowski, "High Performance NonLinear feedback control of permanent-magnet stepper motor", IEEE Transactions on Control Systems Technology, vol. 1, Issue 1, pp. 5–14, March 1993.

[8] J. B. Grimbleby, "Simple algorithm for closed-loop control of stepping motors," IEE Proc.-Electric Power Appl., vol. 142, no. I, pp. 5-13, Jan. 1995.

[9] P. Crnosija, B. Kuzmanovic, and S. Ajdukovic, "Microcomputer Implementation of Optimal Algorithms for Closed-Loop Control of Hybrid Stepper Motor Drives," IEEE Trans. on Ind. Electron., vol. 47, no. 6, pp. 1319-1325, Dec. 2000.

[10] A. Rubaai, M. J. Castro-Sitiriche, M. Garuba, and L. Burge, "Implementation of artificial neural network-based tracking controller for high-performance stepper motor drives," IEEE Trans. on Ind. Electron., vol. 54, no. 1, pp. 218-227, Feb. 2007.

[11] Persson J, Perriard Y: " Steady state Kalman filtering for sensorless control of hybrid stepper motors",IEMDC03, IEEE International Electric Machines and Drives Conference, vol. 2, no. 4, pp. 1174-1177, June 2003.

[12] Kellermann. H, Hildinger. P, Brandenburg. G, Heinzl. J, "Field oriented position control of hybrid stepper motor conrol", Proceeding of the EPE Conference, vol. 3, pp. 908-913, September 1995.

[13] A. Rubaai, and R. Kotaru, "Adaptation Learning Control Scheme for a High-Performance Permanent-Magnet Stepper Motor Using Online Random Training of Neural Networks," IEEE Transactions on Industry Applications, vol. 37, no. 2, pp. 495-502, March/April 2001.

[14] M. Defoort, F. Nollet, T. Floquet, and W. Perruquetti, "A Third-Order sliding-Mode Controller for a Stepper Motor," IEEE Transactions on Industrial Electronics, vol. 56, no. 9, pp. 3337-3346, September 2009.

[15] K. W. Tsui, N. C. Cheung, and K. C. Yuen. "Novel Modeling and Damping Technique for Hybrid Stepper Motor," IEEE Transactions on Industrial Electronics, vol. 56, no. 1, pp. 202-211, January 2009