Speed control of DC motor using sliding modecontrol approach

Prabhudev Ghalimath,Department of Post Graduation,

Mahatma Bashweshwar SocietysCollege of Engineering,Ambajogai, Beed, India

Email: pmgmath@gmail.com

S. S. SankeswariDepartment of Post Graduation,

Mahatma Bashweshwar SocietysCollege of Engineering,Ambajogai, Beed, India

Email: sankeswari@gmail.com

AbstractThis paper presents the sliding mode controllerdesign to regulate the speed control of the direct-current(DC)motor. In this, The sliding surfaces used to derive the differentcontrol schemes presented in this work are based on an inte-graldifferential equation acting on the tracking-error expression.A new approach for the design of sliding mode controllers basedon first-order-plus-dead time model of the system, is developed.This approach results in a fixed structure controller with a set oftuning equations as a function of the characteristic parametersof the model. The controller performance is judged by simulationon model of the DC motor

Index TermsDC motor; Sliding mode control, PID controller,single variable; disturbance; robustness.

I. INTRODUCTIONVariable structure system (VSS) with sliding mode control

was first proposed and elaborated in the early 1950s in theSoviet Union by Emelyanov and several co researchers. Atthe very beginning, VSS is well known as special class ofnonlinear systems for solving several specific control tasks insecond order linear and nonlinear systems. However VSS didnot receive wide acceptance among engineering professionalsuntil 1977, the most interesting fact is that robustness hasbecomes a major requirement in modern control application[1]. The performance of low order control system design, suchas with proportional- integral-derivative (PID) controllers isless effective for electomechanical systems such as DC drives.The parametric uncertainty introduced due to plant-modelmismatch degrades the overall system performance since theseuncertainties are hardly considered in the design of linearcontrollers like PID. During the past few decades, the robustcontrol system design for plant-model mismatch processeshave received considerable attention in control community.Among the established design approaches for robust processcontrol, sliding mode control (SMC) plays an important rolebecause it not only can stabilize certain and uncertain systemsbut also provide the capability of disturbance rejection andinsensitivity to parameter variations [1], [2].

The most distinguishing property of VSS is its ability toresult in very robust control systems. In other words, thesystem is completely insensitive to parametric uncertaintyand external disturbances. Due to its excellent invariance androbustness properties, the VSS concepts have been developed

into practical application mainly in the field of control ofDC servo motors [2][3], robotic manipulators [4][5], PMsynchronous servomotors, induction motors, aircraft control,spacecraft control and flexible space structure control [6].These experiments confirm the theoretical results regardingrobustness of VSS with sliding modes. However, in someof these experimental results [2][4][5], it was found thatthe resulting control is discontinuous and the chattering phe-nomenon which can leads to low accuracy in control system.These problems can be solved by replacing a continuouscontrol [4][5] into the computation of the control input (a signfunction). As a result, the large error behavior of a system isidentical to that with discontinuous control. It can be assumedthat, the behavior of the system in small error region as ahigh gain system and this is similar to that of system withdiscontinuous control. Hence, this high gain effect of slidingmode control based on VSS, suppressed the uncertainties dueto parametric variations, external disturbances and variablepayloads [5]. Besides that the proper selection of the switchingfunctions will avoid chattering problem in the DC drivesystems, hence result in high accuracy control. The choiceof switching functions to control the system states, such thatcurrent, speed or position has been discussed and examinedin detail in literature [7]. In Is], the control of a permanentmagnet synchronous motor under sliding mode controller hasbeen presented which uses a hyperbolic tangent switchingfunction in order to overcome the chattering problem. Underthis control strategy, the dynamic performance of the systemcan be shaped according to the system specification by anappropriate choice of switching function.

It is well known, that the sliding mode control is a popularrobust control method. However it has a reaching phase prob-lem and an input chattering problem (as discussed aboved).These problems cause the sliding mode control (SMC) is veryconservative to be used with other controller design methodsbecause the state trajectory of the sliding mode control systemis determined by sliding mode dynamics, which cannot havethe same order dynamics of the original system. This leads tothe introduction of robust controller design with novel slidingsurface. To overcome the conservatism of the SMC, the novelsliding surface has been used which has the same dynamics of

the nominal original system controlled by a nominal controller.The reaching phase problem, can be eliminated, by using aninitial virtual state that makes the initial sliding function equalto zero. Therefore, it is possible to use the SMC techniquewith various types of controller. The proposed controllerdesign with novel sliding surface is discussed in detail in [9].Besides that, as discussed in [lo], although SMC systems arequalitatively well known for possessing robust performance,the quantitative analysis of the robustness and synthesis of thecontrol system to enhance the robust performance, especiallyagainst a step load disturbance for DC drive is necessary. Thereason why the quantitative analysis is necessary, because ofthe step load application may vary due to certain factors suchas an integral action and smooth control algorithms which areoften incorporated in the practical system. In [10], the analysisin terms of the time domain expressions of the transient speeddeviation and its maximum value due to a step load applicationunder sliding mode control was performed. As a result, the fastresponse speed and robust performances can be achieved.

The organization of the paper is as follow: The systemdescription is given in Section II while Section III describessliding surfaces, design and implementation issues. Simulationresults of speed control of DC motor are included in SectionIV and conclusions are summarized in Section V.

II. MATHEMATICAL MODEL OF A TYPICAL DC MOTORThe BLDC motor provided for this paper is the EC 45

flat 45 mm, brushless, 30 Watt from Maxon motors [12].The parameters used in the modeling are extracted from thedatasheet of this motor with corresponding relevant parametersused. Find below in Table I the major extracted parametersused for the modeling task. The mathematical model of the

TABLE IPARAMETERS OF BLDC MOTOR

Sr. No. Data Unit Value1. Nominal voltage V 122. No load speed rpm 12003. No load current mA 1514. Nominal speed rpm 12005. Torque mNm 596. Nominal current A 2.147. Starting current A 108. Max. efficiency % 779. Stall Torque mNm 25510. Terminal Resistance 1.111. Terminal Inductance mH 0.512. Torque Constant mNm/A 24.513. Speed Constant rpm/V 35.414. Speed/torque gradient rpm/mNm 17.615. Mechanical time constant ms 16.116. Rotor Inertia gcm2 82.517. No. of phase - 3

BLDC motor is modelled on the parameters from table givenabove.

G(s) =1/Kg

mes2 + ms+ 1(1)

where Kg, m and e are the constants and need to calculated.

The term e is calculated using

e =L

3R(2)

e =0.5 103

3 1.10(3)

e = 151.51 106 (4)

The term m is calculated using

m =3RJ

KgKt= 0.0161 (5)

where Ke is

Ke =3RJ

mKt= 0.06902 (6)

Thus the model of the DC motor in the form of transferfunction is

G(s) =14.48

2.44 106s2 + 0.0161s+ 1(7)

III. SLIDING MODE CONTROLThe model of the DC motor, in general form is,

Y (s)

U(s)=

K

mes2 + (m)s+ 1(8)

or

Y (s)[mes2 + (m)s+ 1] = KU(s) (9)

Taking inverse Laplace, the equation in time domain form is

mey(t) + (m)y(t) + y(t) = Ku(t) (10)or

y(t) =1

me[(m)y(t) y(t) +Ku(t)] (11)

The sliding surface used by Camacho [2] is

(t) = 1e(t) + 0

t0

e(t)dt+d

dte(t) (12)

where, 1 = 2 and 0 = 2 and is a tuning parameter,which helps to define the sliding surface. The derivative of 12is

(t) = 1e(t) + 0e(t) + e(t) (13)The error is e(t) = r(t) y(t), Thus above Eq. 13 can bewritten as

(t) = 1[r(t) y(t)] + 0e(t) + r(t) y(t) = 0 (14)We know, from Eq. 11, y(t) = 1

me[(m)y(t) y(t) +

Ku(t)]

1[r(t)y(t)]+0e(t)+r(t)1

me[(m)y(t)y(t)+Ku(t)] = 0.

(15)

The Equivalent controller ueq(t) can be obtained from aboveEq. 15 and given as,meK

[(1

e 1

)y(t) +

y(t)

me+ 0e(t) + r(t) + 1r(t)

]= 0.

(16)The ueq(t) can be simplified by considering

1 =1

e(17)

It has been shown that this choice for 1 is the best for thecontinuous part of the controller [2]. To assure that the slidingsurfaces behave as a critical or over-damped system, 0 shouldbe

0 2

1

4(18)

The switching controller is taken as

usw(t) = Kd(t)

|(t)| + (19)

As we know, derivative of reference signal of setpoint is zero,that is r(t) = r(t) = 0. The Total control law is

u(t) = ueq(t) + usw(t) (20)

u(t) =meK

[y(t)

me+ 0e(t)

]+Kd

(t)

|(t)| + (21)

where,

Kd =0.51

|K|

(

td

)0.76(22)

and = 0.68 + 0.12|K|KD1 (23)

IV. IMPLEMENTATION AND RESULTSA DC motor is used in simulation to obtain the results of

sliding mode control. The results show the effectiveness ofthe proposed controller and this results are compared withconventional PID controller. MathworksTM MATLAB 7.0.1is used for simulation. The systems considered are of higherorder with time delay. The simulink block of equivalent controllaw is shown in Fig. 1 and that of switching control is in Fig.2.

It is clear, form Fig. 1 that the inputs to the equivalentcontrol law are error signal, reference input signal and outputsignal. The switching control law is taken as

usw(t) = Kd(t)

|(t)| + (24)

with Kd = 0.8312 and = 1.2917. The simulink block ofswitching control law is shown in Fig. 2. The parameters ofthe PID controller are obtained using Zeiglar-Nicholas method.The PID controller is given by

Gc(s) = KP +KIs

+KDs = 11.327 +1381.34

s+ 0.0232s

The performance of the SMC and PID controller in termsof output responses, input responses, error signals is shownin Fig. 3,Fig. 4 and Fig. 5 respectively. The sliding surface is

Fig. 1. Simulink Block of Equivalent Control Law

Fig. 2. Simulink Block of Switching Control Law

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.2

0.4

0.6

0.8

1

1.2

1.4

Time in seconds

Ou

tpu

t sig

na

l

PID ControllerSMC

Fig. 3. Output Responses

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.21

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Time in seconds

Contr

ol S

ignal

PID ControllerSMC

Fig. 4. Input Responses

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20.2

0

0.2

0.4

0.6

0.8

1

1.2

Time in seconds

Contr

ol S

ignal

PID ControllerSMC

Fig. 5. Error Responses

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.210

0

10

20

30

40

50

60

70

80

90

100

Time in seconds

Slid

ing

Su

rfa

ce

Fig. 6. Sliding Surface

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.2

0.4

0.6

0.8

1

1.2

1.4

Time in seconds

Ou

tpu

t sig

na

l

PID ControllerSMC

Fig. 7. Output Responses

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.22

0

2

4

6

8

10

12

14x 1012

Time in seconds

Contr

ol S

ignal

PID ControllerSMC

Fig. 8. Input Responses

shown in Fig. 6. It is clear from the output responses that bothcontroller gives satisfactory resposne while the control effortby PID controller, that is control signal shown in Fig. 4 byPID controller is non-smooth while those by SMC is smooth.

In order to check the controller performance, the outputdisturbance d = 0.5r is added at time t = 0.1sec. Theoutput, input and error signal is shown in Fig. 7-9. It is clearfrom Fig. 8 that the PID controller can not be used as its signalgoes to very high value whenever disturbance occurred.

V. CONCLUSIONSThis paper has presented a sliding surface approach for

the speed control of DC motor. The advantage of slidingmode approach is that it gives smooth controller action. Ithas been shown that the performance of the PID controlleris comparable in terms of output of the system, but whenit comes under the consideration of the controller action,the action given by the PID controller is continuously zig-

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20.2

0

0.2

0.4

0.6

0.8

1

1.2

Time in seconds

Err

or

Sig

nal

PID ControllerSMC

Fig. 9. Error Responses

zag pattern. Again in case of output disturbances, the slidingmode gives the feasible action of the while the PID controllergives very high signal which can not be implementable in realapplications. The SMC gives advantage of robustness againstdisturbances wither it is in input side or output side of thesystem. In case of unmeasurable disturbances, SMC is morecapable and gives satisfied performance. Simulation resultsshow that the proposed sliding mode control strategy producessatisfactory results with less and smooth control efforts and iseffective and promising in the control of uncertain time delaysystems.

REFERENCES

[1] V. I. Utkin, Sliding Modes in Control and Optimization, Springer, Berlin;1992 .

[2] O. Camacho and C. A. Smith, Sliding Mode Control: An Approach toRegulate Chemical Processes, ISA Transactions, vol. 39, 2000 pp 205-218.

[3] Slotine, J.J.E., and Li, W., Applied Nonlinear Control, Prentice-Hall,1991.

[4] Kim, T.; Lee, H.W.; Ehsani, M. Position Sensorless Brushless DCMotor/Generator Drives: Review and Future Trends. IET Electr. PowerAppl. 2007, 1, 557-564.

[5] Weibing Gao, Variable Structure Control of Non-linear Systems: ANew Approach, IEEE transactions on Industrial Electronics, VOL40.No.I, February 1993.

[6] Slotine, J. J., and Sastry, S. S., 1983, Tracking control of non-linearsystems using sliding surfaces with applications to robot manipulators.International Journal of Control, 38, 465 - 492.

[7] John Y Hung, WeibingGao and James C. Hung, Variable StructureControl:A Survey in IEEE transactions on Industrial Electronics, VOL40.No.I, February 1993.

[8] Takahashi, I.; Koganezawa, T.; Su, G.; Ohyama, K. A Super High SpeedPM Motor Drive System by a Quasi-Current Source Inverter. IEEETrans. Ind. Appl. 1994, 30, 683-690.

[9] Jezernik, K., Curk, B., and Harnik, J., 1994 , Discrete-Time ChatteringFree Sliding Mode Control. VSLT94: Proceedings of the Workshopon Robust Control via Variable Structure & Lyapunov Techniques,Benevento, September 1994, pp.319 - 324; 1994

[10] Bianchi, N.; Bolognani, S.; Jang, J.H.; Sul, S.K. Comparison of PMMotor Structures and Sensorless Control Techniques for Zero-SpeedRotor Position Detection. IEEE Trans. Power Electron. 2007, 22, 2466-2475.

[11] EL Sharkawi M and Huang C., Variable structure tracking of DC motorfor high performance applications. IEEE Trans. on Energy Conversion,V 4, 1989, pp. 643-650.

[12] Maxon Ec motor , May 2008 Edition, EC 45 flat 45 mm, brushless, 30watt maxon flat motor.

[13] O. Camacho, R. Rojas, V. G. Gabin, Some long time delay sliding modecontrol approaches, ISA Transactions, vol. 46, 2007 pp 95-101.