10-IJAEST-Vol-No.4-Issue-No.1-Stability-Enhancement-of-DC-Motor-using-IMC-Tuned-PID-Controller-092-0 by ISERP ISERP

K. Anil Naik et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 1, 092 - 096

Stability Enhancement of DC Motor using IMC Tuned PID Controller K. Anil Naik1 and P. Srikanth1 Department of Electrical Engineering, NIT Hamirpur, Hamirpur, H.P., India email: anilnaik205@gmail.com



Abstract— In this paper PID controller with Internal Model Control (IMC) tuning method for the DC motor is presented for robust operation. The IMC has a single tuning parameter to adjust the performance and robustness of the controller. The proposed tuning method is very efficient in controlling the overshoot, stability and the dynamics of the speed-control system of the DC motor. The results of the IMC tuning method have been compared in the midst of controller with singular frequency (SF) based tuning and Ziegler-Nichols (Z-N) closed loop tuning. A remarkable improvement in stability of the system has been observed with IMC tuning justifying its applicability. Simulated results given in the paper show the feasibility and versatility of the IMC tuning technique in the DC motor. Index Terms—Controller, Internal Model Control, DC motor, Speed, Stability, Tuning.

I. INTRODUCTION

HE DC motors have been popular in the industry control area for a long time, because they have many good characteristics, for example: high start torque characteristic, high response performance, easier to be linear control…etc.[12]. The different control approach depends on the different performance of motors. Because the peripheral control devices are enough, there is the more extensive application in the industry control system. Therefore, the DC motor control is riper than other kinds of motors. Different types of controller have been using in DC motors such as PI and PID controllers. The main disadvantage of PI controller is steady state error is more. And the most popular controllers used in industrial processes is the Proportional Integral Derivative (PID) controller [3]. The real strength of this kind of controller is its simplicity to understand, explain and implement them [4]. The proportional integral derivative (PID) controller is the most common form of feedback in the control systems. PID control is also an important ingredient of a distributed control system and as such these controllers come in different forms [5-7]. And also due to its efficient and robust performance with a simple algorithm, the PID (proportional, integral, and derivative) controllers have been widely accepted in most of the industrial applications [8-12]. Ziegler and Nichols have

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implemented and published their classical methods and also a lot of research is done along the conventional PID controller design [13]. A recent development of modern control system enables us to combine the PID controller with various simple control algorithms in a quick and easy manner to enhance the control performance. However, the classic tuning methods involved in PID controller suffers with a few systematic design problems. Hence, in order to compensate these internal design problems, internal model control (IMC) based tuning approach has been developed. Due to its simplicity, robustness, and successful practical applications it gained a widespread acceptance in designing the PID controller in process industries [14-18]. The analytical method based on IMC principle for the design of PID controller is also developed [19-20]. The resulting structure of the control system is capable of controlling a fast dynamic process by integral control, which results in a striking improvement in performance. Its advantage is even being implemented in many of the industries. However, it has been found from the literature that the IMC-PID controller has not yet been implemented in the DC motor speed control system. Consequently, the present work is a step towards implementing an IMC tuning based PID controller in DC motor. The results with IMC tuned controller have been found to outperform the SF and Z-N tuned PID controllers. The paper is organized as fallows simple model of DC motor is provided in section II. The IMC-PID controller is explained in section III. Results and different plots (frequency and time domain plots) are given in section IV. Finally conclusions are drawn in section V. II. SIMPLE MODEL OF A DC MOTOR A simple model of DC motor as shown in Fig. 1. In this model, the dynamics of the motor itself are idealized; for instance, the magnetic field is assumed to be constant. The resistance of the circuit is denoted by R and the selfinductance of the armature by L. With this simple model and basic laws of physics, it is possible to develop differential equations that describe the behavior of this electromechanical system. The relationships between electric potential and mechanical force are Faraday's law of induction and Ampere’s law for the force on a conductor moving through a magnetic field.

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K. Anil Naik et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 1, 092 - 096

K   R   b 1  d i  L L i     v K f     L  app ( t ) dt    K m  0    J   J i  y (t )  0 1    [0]vapp (t ) Fig. 1 A simple model of a DC motor driving an inertial load

The torque  seen at the shaft of the motor is proportional to the current i induced by the applied voltage, (1)  (t )  K mi (t ) Where, K m the armature constant, is related to physical properties of the motor, such as magnetic field strength, the number of turns of wire around the conductor coil, and so on. The back (induced) electromotive force, ve.m. f is a voltage proportional to the angular rate

 seen at the shaft,

ve.m. f (t )  Kb (t )

(2)

where , K b the emf constant, also depends on certain physical properties of the motor. The mechanical part of the motor equations is derived using Newton's law, which states that the inertial load J times the derivative of angular rate equals the sum of all the torques about the motor shaft. The result is this equation, d (3) J    i   K f  (t )  K mi (t ) dt Where K f  is a linear approximation for viscous friction. Finally, the electrical part of the motor equations can be described by di (4) vapp (t )  vemf (t )  L  Ri (t ) dt or, solving for the applied voltage and substituting for the back emf, (5) vapp (t ) L di  Ri (t ) Kb (t ) dt This sequence of equations leads to a set of two differential equations that describe the behavior of the motor, the first for the induced current, K di R 1 (6)   i (t )  b  (t )  vapp (t ) dt L L L and the second for the resulting angular rate d 1 1 (7)   K f i (t )  K mi (t ) dt J J State-Space Equations for the DC Motor given the two differential equations derived in the last section, now develop a state-space representation of the DC motor as a dynamic system. The current i and the angular rate  are the two states of the system. The applied voltage, vapp , is the input to the system, and the angular velocity

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(8)

(9)

The transfer function model of the DC motor can be derived from state space model and given in eqn. (10) Km (10) 2 LJs  ( RJ  LK f ) s  ( RK f  K m K b ) III. IMC-PID CONTROLLER Equivalent the typical internal model structure to a single loop PID control structure, as Fig. 2. Where P(s) is the actual process object being controlled, M(s) refer to the model of the process, and G(s) is the IMC primary controller, u refer to output of internal model controller, r, y and d refer to the input, the output and load disturbances, respectively, and Gc(s) is the controller which can get the result of internal model controlling structure after varying equivalently. Designed the IMC controller as eqn. (11), 1

G( s)  M ( s) f ( s)

Where

f (s) 

s  1

Gc ( s ) 

(11)

is the realizable factor. G( s)

1  G( s) M ( s)

 K c (1 

1 Ti

 Td s )

(12)

Fig.2 Equivalent the IMC to general control structure

Consider the plant model M(s) as a DC motor transfer function then Km M (s)  (13) 2 LJs  ( RJ  LK f ) s  ( RK f  K m K b ) Substitute eqn. (13) in eqn. (12) and IMC- PID tuning parameter are given as RJ  LK f RK f  Km Kb LJ KP  , KI  and K D  K m K m K m

 is the output.

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K. Anil Naik et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 1, 092 - 096

IV. RESULTS AND DISCUSSION A standard test model as considered is taken for stability study of DC motor with IMC tuning controller. The test model below shown is completely designed in SISO tool. Fig. 3 shows the block diagram of DC motor driving an inertial load. The DC motor representation includes R= 2.0 Ohms, L= 0.5 Henrys, Km = 0.015 torque constant, Kb = 0.01 emf constant, Kf = 0.2Nms and J= 0.02 kg.m^2.

The step response of the PID controller system with singular frequency tuning in time domain analysis is given in Fig. 5. The whole simulation is done for 50s. The rise time Tr = 9.87s, peak time and settling time Ts =15s for this case are obtained.

Fig. 3 Block diagram of DC motor speed control system

To show the robustness of the DC motor speed controller system with IMC tuning controller, various cases as given below have been considered. The cases considered have been simulated and verified in SISO tool MATLAB/SIMULINK ver 7.6 [21]. Case a: Singular frequency based design tuning Case b: Ziegler-Nichols closed loop design tuning Case c: IMC based design tuning It is mentioned here that the designed values are taken same as have been provided in [21]. A. Singular frequency based design tuning To get the singular frequency based design tuning the Fig. 3 is simulated in SISO tool. The frequency response for such a system is computed using the linear approximation (Bode plot). The magnitude and phase as a function of frequency of such a system are plotted in Fig. 4. From the plotted graph the gain crossover frequency  gc is 0.188rad/sec and phase crossover frequency  pc is 0.988 rad/sec. The gain and phase

Fig. 5 Responses of the system to a step input with Singular frequency based design tuning

B. Ziegler-Nichols closed loop design tuning To achieve such a system of speed controller PID system, the Fig. 3 is simulated in SISO tool. For this system also the frequency response is computed using the linear approximation (Bode plot). The magnitude and phase as a function of frequency are plotted and is as shown in Fig. 6. From Fig. 6, it is determined that gain crossover frequency  gc is 0.821rad/sec and phase crossover frequency  pc is 7.45rad/sec for this case. The gain and phase margins are Gm = 27.3dB and

m = 74.7deg. Since, gc is less than  pc and

hence in this case also the DC motor speed control system is stable.

margins are Gm = 12.5dB and m = 72.8deg,  gc is less than  pc since  gc should not be greater than  pc for stability of the system. The speed controller of DC motor system with singular frequency based design tuning is stable.

Fig. 6 Frequency response for Ziegler-Nichols closed loop based design tuning

The step response of the Ziegler-Nichols design tuning in time domain analysis is as given in Fig. 7. The whole simulation is done for 50s. Rise time Tr =8.89s, and settling

Fig. 4 Frequency response for Singular frequency based design tuning

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time Ts =16.3s are obtained.

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K. Anil Naik et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 1, 092 - 096

Fig.7 Responses of the system to a step input with Ziegler-Nichols closed loop based design tuning

A. Internal Model Control (IMC) based design tuning This tuning design can be obtained when the Fig. 3 is simulated in SISO tool. The magnitude and phase as a function of frequency for this case are plotted in Fig. 8. It is seen from the figure that gain crossover frequency  gc is 0.224rad/sec and phase crossover frequency  pc is 6.44rad/sec. The gain and phase margins are Gm = 36.2dB and m = 85.5deg. Since  gc is less than  pc (phase crossover frequency) then the magnitude and phase values of the bode plot are more and positive. In this case the gain margin and phase margin are more than SF and Z-N tuning methods and hence the speed controller of DC motor system with IMC tuning is more stable.

Fig. 8 Frequency response for IMC based design tuning

Fig. 9 Responses of the system to a step input with IMC based design tuning

Time and frequency domain responses have been determined to investigate the effectiveness of the PID controller in IMC design tuning. It has been determined that the IMC tuning with controller provides the required stability and performance specifications. The results show that the gain and phase margins are significantly improved with 36.2dB gain margin and 85.5o phase margin. These are obtained from the frequency response of the open-loop system and are as given in Fig.8. It is found from Fig. 8 that the phase margin is significantly improved at the critical frequency of inter-area modes between 0.224rad/sec and 6.44rad/sec. On the other hand, 12.5dB and 27.3dB gain margins for the Singular Frequency and Ziegler-Nichols tuning controllers are obtained which are quite low compared with the IMC tuning controller. Detailed results are as summarized in Table I. The time domain results for closed loop system are presented in Table II. Improved results have been obtained with IMC design tuning controller. Generally, a lower rise time and settling time are preferred for the better performance of the system. From the Table II it has been seen that for SF based tuning rise time is 9.78s and settling time is 15s, for Z-N based tuning rise time is 8.89s and settling time is 16.3s and for IMC tuning rise time is 1.89s and settling time is 3.34s. From the above it is observed that when compare to SF and ZN tuning methods IMC tuning method obtained better settling time and peak rise time, thereby justifying the suitability of the IMC tuning for PID control DC motor speed system. Time domain responses of all types of tuning methods PID controllers with DC motor speed governing system comparison are as shown in Fig. 10 for a step set-point speed signal change.

The step response of the IMC based design tuning in time domain analysis is as shown in Fig. 9. For which the rise time Tr = 1.89s, there is no peak time and peak overshoot %M p = 0 and settling time Ts =3.34s. In this case compare to SF and ZN tuning methods rise time and settling time are improved.

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K. Anil Naik et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 1, 092 - 096

REFERENCES [1] [2] [3] [4] [5] D. [6] [7] Fig.10 Responses of the system to a step input with all tuning algorithm controllers

[8] [9]

TABLE I FREQUENCY DOMAIN RESULTS Specifications

S F Based Tuning

Z-N Closed loop Tuning

IMC Based Tuning

Gain Margin

12.5

27.3dB

36.2dB

Gain crossover Frequency

0.188r/s

0.821r/s

0.224r/s

Phase margin

72.8o

74.7o

85.5 o

Phase crossover Frequency

0.988r/s

7.45r/s

6.44r/s

[10] [11] [12] [13] [14] [15] [16]

TABLE II TIME DOMAIN RESULTS

[17]

Specifications

S F Based Tuning

Z-N Closed loop Tuning

IMC Based Tuning

Rise time

9.78s

8.89s

1.89s

Settling time (1%)

15s

16.3s

3.34s

V. CONCLUSIONS

[18] [19] [20] [21]

Wang, J. B., Control of Electric Machinery. Gau Lih Book co., Ltd, Taipei Taiwan, 2001. G. Haung and S. Lee, “PC based PID speed control in DC motor,” IEEE Conf. SALIP-2008, pp. 400-408, 2008. Visioli A., "Tuning of PID controllers with fuzzy logic,” in IEE Proceedings - Control Theory and Applications, 2001, Volume 148, Issue 1, pp. 1-8. Zhang J., “Structural research of fuzzy PID controllers,” in Proc. International Conference on Control and Automation, ICCA2005, Northeastern University, Qinhuangdao Hebei China. 2005. Vrancic, B. Kristiansson, S. Strmcnik, and P. M. Oliveira, “Improving performance/activity ratio for PID controllers,” Int. Conf. Control and Automation, 2005, pp. 834-839. P. Kundur, Power system stability and control, McGraw-Hill, 1994. A. Khodabakhshian and M. Edrisi, “A new robust PID load frequency controller,” Control Engg. Practice, vol. 16, pp. 1069-1080, 2008. K. J. Astrom, H. Panagopoulos, and T. Hagglund, “Design of PI controllers based on non-convex optimization” Automatica, vol 34. pp. 585-601, 1998. D. E. Seborg, T. F. Edgar, and D. A. Mellichamp, “Process dynamics and control” John Wiley & Sons, Second edition, New York, 2004. S. Skogestad and I. Postlethwaite, “Multivariable feedback control analysis and design” John Wiley & Sons, New York, 1996. C. L. Smith, A. B. Corripio, and J. J. Martin, “Controller tuning from simple process models” Instrumentation Technological, vol. 22, pp. 3945, 1975. J. G. Ziegler and N. B. Nichols, “Optimum settings for automatic controller, Transactions ASME, vol. 64, pp. 759-766, 1942. O. Montiel, R. Sepúlveda, P. Melin and O. Castillo, “ Performance of a simple tuned Fuzzy controller and a PID controller on a DC motor,” Procee. of IEEE (FOCI 2007), pp. 531-538, 2007. I. L. Chien, and Fruehauf, “Consider IMC tuning to improve controller performance” Chemical Engineering Program, vol. 86, pp. 33-38, 1990. O. Aidan and Dwyer, “Handbook of PI and PID controller tuning rules” Imperial College Press, London, 2003. I. G. Horn, J. R. Arulandu, J. G. Christopher, J.G. VanAntwerp, and R. D. Braatz, “Improved filter design in internal model control” Industrial Engineering Chemical Research, vol. 35, pp. 33-37, 1996. Y. Lee, S. Park, and M. Lee, “Consider the generalized IMC-PID method for PID controller tuning of time-delay processes” Hydrocarbon Processing, 2006, pp. 87-91. Y. Lee, S. Park, M. Lee, and C. Brosilow, “PID controller tuning for desired closed-loop responses for SISO systems” AICHE Journal, vol 44, pp. 106-115, 1998. M. Morari and E.Zafiriou, “Robust Process Control” Prentice Hall, Englewood Cliffs, NJ, 1989. D. E. Rivera, M. Morari, and S. Skogestad, “Internal model control, 4. PID controller design” Industrial Engineering Proceeding Design Deu. vol. 25, pp. 252-258, 1986. MATLAB/SIMULINK ver. 7.6

A new robust IMC tuning based PID controller is proposed for DC motor control system. The proposed tuning method has been found to enhance the stability of the DC motor system. Different cases have been considered and compared to justify the suitability of the IMC tuning PID controller. From Table I it is found that the gain margins IMC tuning controller is 23.7dB higher compared with SF tuning controller and 8.9dB higher when compared with Z-N tuning controller, similarly, rise time and settling time has improved with IMC tuning controller.

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