Control of DC Motors Using Sliding Mode

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Control of DC motors using sliding mode

Conference Paper · January 2012

DOI: 10.1109/IBCAST.2012.6177523

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CONTROL OF DC MOTORS USING SLIDING MODE

Ghulam Murtaza*,A. 1. Bhatti**

Dept. of Electronics Engineering, Mohammad Ali Jinnah University, Islamabad, Pakistan

*[email protected],* *[email protected]

Abstract-DC motors have been extensively used in many

industrial applications for their accurate, simple and continuous control characteristics. This paper addresses the controlling of the speed of DC motor which remains among the vital issues. It presents sliding mode control, integral sliding mode control and dynamic sliding mode control methods to control the speed of DC motors. Also the merits and demerits of each approach are discussed. Sliding mode control (SMC) is robust to the matched uncertainties and the desired performance/speed is attained.

But the main drawback of SMC is chattering, resulting from discontinuous controllers, various techniques are presented to minimize chattering. The performance of the dynamic system with integral and first order dynamic SMC is compared with response of the system with SMC controller. The results show that the integral and specifically dynamic SMC are more robust against matched uncertainties, the desired speed is perfectly tracked and chattering reduction is achieved.

I. INTRODUCTION

Electrical derives involving various types of DC motors turn the wheel of industry. More than 50 percent of the generated electrical energy is consumed in motor derives in the developed countries [8]. DC motors are comprehensively used in various industrial applications such as electrical equipment, computer peripherals, robotic manipulators, actuators, steel rolling mills, electrical vehicles, paper machine etc. Its appli

cations spread from low horse power to the multi-mega watt due to its wide power, torque, speed ranges, high efficiency, fast response, simple and continuous control characteristics[3]

and [8].

Controlling the position/speed of a DC motor is a pivotal issue. Conventionally armature control method has been used for controlling the speed of the dynamic system, but the controllability, being economical and their compatibility with the novel mechanical and electrical equipments like digital systems are the factors which have made its use widespread [2,6].

In the presence of uncertainties and disturbances in a system, an appropriate control needs to be designed to achieve the system's stability and desired performance. Sliding mode control (SMC) is not sensitive to the matched uncertainties and external disturbances. To control a broad class of linear and nonlinear systems, the robustness of SMC has made this technique to be a profound and most appropriate one [2]. A SMC is designed such as to force the system trajectories to

move onto a pre defined surface within finite time and ap

proaches to an equilibrium point all along this surface[ 6]. The closed-loop dynamics of the system is absolutely controlled through the switching function' equations provided the system trajectories remain on this surface. Basically, in sliding mode the system,s order is reduced equal to number of switching surfaces/ that of control inputs, which subsequently reduces the degree of freedom apart from the integral sliding mode control (ISMC)[5] and dynamic sliding mode control (DSMC) or when the system is operating (pre sliding mode) under the designed sliding mode compensator. Robustness and flexibility of designing SMC are its foremost advantages.

In the prior work, the position/speed control problem has been worked out by control schemes as optimal control based feedback linearization [3,11] which is not a robust technique.

Similarly PID the most common control designing technique unable to stabilize the nonlinear plants or in the presence of a bounded but high uncertainties. A SMC , PI/PID SMC and ISMC control designing techniques[2, 1 0] have resolved the subject like robustness, parameter invariance etc. but the core issue with these approaches, the chattering as a result of discontinuous control (the main drawback of SMC) is notably reduced with the dynamic sliding mode control (DSMC) as presented in this work.

The paper has been composed as follows: In Section 2, model of the system is appreciated. In Section 3, open loop stability analysis is discussed. A SMC, Integral SMC and Dynamic SMC controllers are designed in Section 4. In Section 5, simulation results are shown. Finally in Section 6, conclusions are presented.

II. MOTOR DYNAMICS

The structure of the dynamic system is as shown in Figure 1. The control of the motor's speed with great precision is required in most of its applications. A desired speed can be achieved when a desired shaft position is tracked. The reference signal determines the desired speed/position and the control is designed to ensure difference between the reference input and system's output ultimately approaches to its minimum value, preferably zero. In this work voltage is applied as an input to control permanent magnet the DC motor. To achieve the desired performance/speed a constant value of voltage

Proceedings of 2012 9th International Bhurban Conference on Applied Sciences & Technology (IBCAST) 37 Islamabad, Pakistan, 9th - 12th January, 2012

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DC Motor Fig. 1. DC Motor Structure

as a reference signal is applied to the system for simplicity.

Though, it works successfully for any reference signal.

The electrical and mechanical equations of a DC motor's model [3] are given as under:

Va = RaIa + La dIt + Vb

T

= J �� + BdW +

¹¹

T

= KtIa Vb = KbW

where the physical parameters are described as

Va:

Supply voltage,(v)

Vb: Back emf, (v)

Ia:

Armature current, (amp)

La:

Inductance of armature, (h)

Ra:

Armature resistance, (ohm)

Bd:

Damping ratio,(Nms)

Kt:

Torque constant, (NmJamp)

Kb:

Motor constant, (v-s/rad)

J:

Moment of inertia,

(kg.m2/s2)

w: Speed of shaft, (rad/s) 11 : Torque load

Suppose the state vector is defined as:

[Zl, Z2] = [w, Ia]T

Let us define the new parameters as

dl = -Bd/J , d2 = Kt/J, d3 = -Kb/La, d4 = -Ra/La b = 1/ La,

U

= Va

The state space depiction of the model is

il = dlzl + d2z2 - 1)

i2 = d3z1 + d4z2 + bu

y

= Zl

=}

i = Az + Bu +

H and where

A = [dl, d2; d3, d4]

B = [O·..l..]

_{, La '}H

= [_'!l. 0] u

_J"

= V a

Here A is a state matrix, B is the input matrix, and H is a load torque (external disturbance) matrix that produces an undesired effects on the response of the dynamic system.

whereas y is the system's output. To find a straightforward sliding mode system, the system's dynamics is transformed into a controllable canonical form. Evidently, the results may be represented in terms of the actual physical states of the system in state space form [2].

It is assumed that

Xl = Zl

and

Xl = X2

The conversion of system model into canonical form is as below:

Xl = X2

X2 = DIXI + D2X2 + blu ( = Xl

where

DI = d2d3 - dld4, D2 = dl + d4

and

bl = d2b

Later on the system in this form will be used for various controller designing schemes.

III. OPEN Loop STABILIT Y ANALYSIS

Since the eigen values of matrix A are (-1.38 and -86.29) i.e real, distinct and lying in the -ve half plane. So matrix A is Hurwitz and the equilibrium pt (0,0) is a stable node.

However stability of the dynamic system can be analyzed by an alternative way.

A. Lypunov Direct Method

Assume

V(z) = zT pz

is a positive definite Lyapunove candidate function. where

�

is a real positive definite sym

metric matrix. Then the

V(z)

along the system's trajectories will be

V ( z) = ZT Pi + iT pz = _ZTQZ

The derivative of V will be negative definite, if for positive definite symmetric matrix Q, the matrix ^Pobtained by solving the Lyapunove equation

AP+ ATp=-Q

is a real symmetric positive definite. where A is a Jacobian matrix of the system at origin. As the matrix ^Pobtained is

P = [0.389, -0.0029; -0.0029,0.0058]

The matrix ^Pis positive definite because all the principle minors of ^Pare positive, all of its eigen values ( 0.0058,0.3886) are also strictly positive and determinant of ^Pis also +ve. Thus the matrix A is a stability matrix. The Lyapunove function candidate V(z) is a Lyapunove function for the system. Hence by Lyapunov direct method the dynamic system is stable and global asymptotic stability is guaranteed.

IV. CONTROLLER DESIGN FOR DC MOTOR In this section three sliding mode controller designing techniques like Sliding Mode Controller (SMC), Integral Sliding Mode Control (ISMC) and Dynamic Sliding Mode Control (DSMC) are discussed comprehensively to achieve the robustness, desired performance, system stability in the reaching phase as well as in the sliding phase against the parametric variation, un-modeled dynamics, matched uncer

tainties/disturbances and the undesired chattering reduction (which is caused by the switching of the discontinuous control with high frequency and is considered a setback as for the practical implementation of sliding mode methods).

Proceedings of 2012 9th International Bhurban Conference on Applied Sciences & Technology (IBCAST) 38

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A. Sliding Mode Control

In this subsection, the sliding mode controller is designed for the dynamic system with matched uncertainty.

Consider the dynamic system in the controllable canonical form:

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( = Xl

where

DI = d2 d3 - dld4, D2 = dl + d4, bl = d2 b

and

Td = -1)

is supposed/considered as a matched uncer

tainity/external disturbance.

The control system may be summarized with this assump

tion as

Disturbance Phase Plane

0.05 ,---�---, 10,..---,

r·_,:··SV

-0.05 '---' o 0.05 0.1 0.15 0.2

Time(sec) Control Input with SMC

Time(sec)

c o -20

30

�

20 u.

� 10 E

0.5 x1 Switching Function (SMC)

.8

� ⁰

\'--��--��

0.1 0.2 0.3 0.4 Time(sec)

=}

^x

= f(x) + B(x)u + d(t,x)

(2) Fig. 3. Control Law(SMC), Sliding surface, Phase portrait and load torque.

where the function d(t,x) represents the perturbation/external disturbance fulfils the matching condition.

=} d(t,X)E

span

B(x)

and B is full rank.

Suppose the switching function is defined by

s = PXI + X2 ,

s = PX2 + DIXI + D2 X2 + blu + d(t,x)

On the sliding surface

s=O=}s=O

So equivalent control will be

Ueq = -ljbl(DIXI + (p + D2 )X2 + d(t,x))

Suppose control law is defined as

U = Ueq - Msign(s)

with

M

^>

Id(t,x)jbll

which enforces the system's trajectories onto the switching surface within a finite time and make sure that the trajectories remain on this surface subsequently.

The equation of motion of the system under sliding mode will be

Xl = -PXI,

where

p

>

0

is a performance parameter.

which guaranteed the stability as well as insensitivity of the system to matched uncertainties, parametric variations and the external disturbances in the sliding mode.

Speed Responce with SMC

0.8

0.6

0.2

0.2 0.4 0.6 0.8

Time(sec)

Fig. 2. Speed response of the dynamic system with matched uncertainty

Chattering Phenomenon with SMC

0.2 r---�_----�---__,

i

u.

0>

C

�

0.15

0.1

� ^-0.05

-0.1

-0.15

-0.2 '---�'---'---'

0.16 0.165 0.17 0.175

Time(sec)

Fig. 4. Chattering phenomenon with SMC

The system dynamics are governed by the sliding mode (reduced order) equation [2]. There are two shortcoming of SMC technique. First one is the chattering as a result of discontinuous control and second one being un-modeled system dynamics/imperfections may cause instability in the reaching phase (before the occurrence of sliding phase). The simulation result in figure 4. depicts a noticeable chattering of amplitude (0.083). To deal with these issues the two methods integral sliding mode control (ISMC) and dynamic sliding mode control (DSMC) are discussed comprehensively in the subsequent sections.

B. Integral Sliding Mode Control

The robustness characteristics with respect to the variation in the system's parameters and external disturbances can be achieved by using conventional SMC. However as in the reaching phase the system does not possesses the insensitivity property as in the sliding mode. So un-modeled dynamics and disturbance/ parametric variation may cause instability in this phase. Secondly, chattering caused by discontinuous control oftenly exciting the un modeled dynamics and is considered

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as a problem for practical implementation of sliding mode control. The Integral SMC deals with both of the above mentioned issues, the first one by eliminating the reaching phase, unlike the usual SMC approach. The order of system in sliding mode remains same like that of novel system instead of the reduction in its dimension equal to the number of its switching surface/control inputs, which provides more degree of freedom than SMC. So from the very beginning time instant the robustness of the system can be guaranteed throughout the entire system's response [5,10]. Another shortcoming, the chattering is diminished/reduced by passing the discontinuous control through low pass filter (integral term) before the plant to generate the sliding mode.

Let us define control law for the dynamic model of DC motor in controllable canonical form (1) as

0.05

� g-

I-

"

� 0 ...J

-0.05

'5 ⁰

k

�

-100

()

0 -200

Disturbance

0 0.005

Time(sec) Control Law (ISMC)

(

0.005 Time(sec)

Switching Function (ISMC) 100

c 80

�

0 60 u. 0> ₄₀ E c

.8 ₂₀ .� rIJ

0.01 0.02 0.04

Time (sec) Integral Term (ISMC) 100

80 60 M x 40

20

\.

0

0.01 0.2 0.4

Time (sec)

U = Uo + U1

Fig. 6. Control Law, Switching surface, Integral term and load torque (lSMC) where

Uo,

is continuous control part of the controller and

is designed to achieve the desired performance whereas

U1,

is a discontinuous part to be designed for the elimination of the perturbation/disturbance d(t,x).

Assume the switching function in this case is defined as

s = so(x) + X

, with

so(x), XER2

This switching function is composed of two parts; the

so(x) = eXl + X2

ensures the performance (as in the usual Sliding Mode technique) whereas the term ^Xprovokes the integral term. It can be evaluated as under.

s=s'o+X

s = D1Xl + (e + D2 )X2 + b1UO + b1U1 + d(t, x) + X

Suppose

X = -(DIXI + (e + D2 )X2 + b1UO) withX(O) = -so(x(O))

where

X(O)

is obtained based on the condition s(O)=O (the beginning point for the occurrence of sliding mode ).

=}

s = b1U1 + d(t,x)

So on sliding surface

s = 0 U 1 - - Tel

^b1

U1 = -Msign(s)

with

M

^>

Id(t,x)1

Speed Responce with (ISMC)

0.8

0.6

l

"

� ^0.4

rIJ

0.2

0.1 0.2 0.3

Time (sec)

0.4 0.5

Fig. 5. Speed response with ISMC (matched uncertainty)

Chattering Phenomenon with ISMC

0.2 r--�-�-�-�--�-�-�----'

0.15

0.1

� 0.05

�

rIJ 0>

E C .8 � ^-0.05

-0.1

-0.15

0.245 0.25 0.255 0.26 0.265 0.27 0.275 0.28 Time (sec)

Fig. 7. Chattering phenomenon with ISMC

In the sliding mode the system's equation of motion is i;

= f(x) + b1UO

As in sliding phase the system's equation of motion remains only the linear system and the order of the system in sliding mode is same as that of the novel system. So the system's stability is guaranteed in the reaching phase as well as in the sliding phase.

Since

Uo =

-kx The gains

ki

can be chosen by pole placement

1) Convergence Analysis: By considering

V

Lyapunov candidate function

17= ss

17= s( -M sign(s) + h(t, z)jbd 17<0

if

M

^>

Ih(t, z)jb11

Hence in the integral sliding mode, s = 0 is a stable equilibrium point which ensures the existence of sliding mode in the presence of matched disturbances/uncertainties.

C. Dynamic Sliding Mode Control

The dynamic sliding mode control (DSMC) is also a robust SMC technique to the matched uncertainties/disturbances,

Proceedings of 2012 9th International Bhurban Conference on Applied Sciences & Technology (IBCAST) 40

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parametric variations and un-modeled dynamics. This method retains the main advantages of SMC and ISMC and is more effective than these approaches regarding the chattering min

imization and the performance enhancement and system's stability throughout an entire system's response from initial time instance [1,7]. In this technique the control law consists of a dynamic continuous controller, which ensures the sys

tem's stability and a dynamic discontinuous controller which effectively rejects the uncertainties/external disturbances[I].

For the dynamic model of DC motor in controllable canonical form (1)

�ith

( = Xl

as the output of the system.

( = X2

( = DlXl + D2 X2 + blu

c: = DlXl + D2X2 + bl

^U

c: = DlX2 + D2 (DlXl + D2 X2 + bl u + Td) + h

^U

c: = (Xl,X2 ,U) + blu

where

(Xl, X2 , u) = DlD2 Xl + (Dl + D§)X2 + D2 bu + D2 Td

The system has relative degree one, which shows that the system is input-output linearizable.

Let

(1 = (

, then the transformed system in controllable canonical form will be as

(1 = (2 (2 = (3

(3 = (Xl,X2 ,U) + blu

Suppose the dynamic switching surface is defined as:

S=Pl(+ P2 (+ (

where

PI

and

P2

are the performance parameters.

S=Pl(1+ P2 (2 + (�

S = Pl(2 + P2 (3 + (Xl, X2 , u) + hu

For the Dynamic SMC the strong reach ability condition is considered as

S = -k(s + wsign(s))

where

0

^<

wk

^<

1

and k is the upper bound of the magnitude of the perturba

tion/disturbance d(t,x).

Speed Responce (DSMC)

Time (sec)

Fig. 8. Speed response with DSMC (Load torque after (l sec))

Disturbance 0.05,..---,

!g U

!

0 .

-0.05 '---' o 1

Time(s) Control Law

c o

�

u.

'"

50

:E -50 B � ^-100

Switching Function

I

Time(s) Udol Responce 30,---,

=> 20

i

g. ^-0.1

�

¹⁰

8 -0.2 .

�

,---- I .

'----'----

----'

-100

1

Time (s) Time(s)

Fig. 9. Control Law, Switching surface, udot and load torque (DSMC)

Chattering Phenomenon (DSMC)

0.2 ,..---�-�-�-�-�F""'---n

0.15

0.1

�

c _0.05

u.

g>

E B

� ^-0.05

-0.1

-0.15

-0.2 '---�-�---'--�-�---'---'--�---'

1.62 1.622 1.624 1.626 1.628 1.63 1.632 1.634 1.636 Time (sec)

Fig. 10. Chattering phenomenon with DSMC

Its first part ensures the desired performance where as second part plays a vital role in chattering reduction.

u = -1/bl(Pl(2 + P2(3 + (Xl, X2 , u)) - k(s + wsign(s)) 1)

Existence of Sliding Mode: Consider a Lyapunov can

didate function

V = � s2

which satisfies the conditions of Lyapunov candidate function ( i.e. it is a locally positive definite,

V(

O

) =

O,and

V(s)

^>O.

Then the sufficient condition for the existence of sliding mode is

V

<

0

(negative definite).

V = ss

V = s( -k(s + wsign(s)) V

<

0

for positive value of k.

=}

V

is negative definite

Hence the convergence of the dynamic sliding mode control is guaranteed.

V. COMPARISON OF SIMULATION RESULTS To validate the effectiveness of the different controller schemes the simulation results are presented in this section.

The system parameters considered for simulations [3] are as follow:

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Comparison of Chattering Phenomenon

�

i'l 0'2

�

^: ^. ^: ^: SMC ( 0.083)

i

-0^2.235

:

^2.24 ^2.245 ^2.25 ^2.255 ^2.26 ^2.265

! O : �

. . . .

•

. . . .

•

.. . . .

•

. . .

=

_{.. .}C ( 0 034)

:§ : : : :

US -0.2 : ' : :

1.24 1.245 1.25 1.255 1.26 1.265

t::i I ] t I--O�"O�) I

3.235 3.24 3.245 3.25 Time (sec)

3.255 3.26

Fig. 11. Comparison of chattering phenomenon

3.265

Ra

=

2.06, La

=

0.0238, kt

=

0.0235, kb

=

0.02352 Ed

=

12

X

10-4, J

=

1.07

X

10-4, Tl

=

0,11

=

0.03.

Figures 2, 5, and 8 depict the robust speed response of DC motor using SMC, ISMC and DSMC in the presence of load torque (0.03) as a matched uncertainty/external disturbance.

The control laws, switching functions, phase plane, integral term (ISMC), udot (in DSMC) and disturbances for each case are shown in the figures 3, 6 and 9. The convergence of each approach can also be seen from the subplots in each of figure. For the integral sliding mode control (ISMC) and dynamic sliding mode control (DSMC) the dynamic system is stable and robust in the entire state space. The figures 4, 7 and 10 represent the chattering phenomenon respectively whilst the figure 11 corresponds to the comparison of the chattering phenomenon, which shows that the chattering (the shortcoming) of SMC is significantly reduced with dynamic sliding mode controller.

VI. CONCLUSION

In this paper SMC , ISMC and DSMC controller are considered for controlling the speed of DC motor. The rel

ative advantages and limitations of each method are studied.

Robustness, performance and parameter invariance is achieved in the presence of external disturbances, un-modeled dynamics and against the matched uncertainties/disturbances. However, chattering (the main drawback of SMC) is reduced signifi

cantly with the ISMC and DSMC techniques. The Dynamic SMC has shown better results specially in chattering reduction.

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