sliding mode control for second order system (1).pdf

International J. of Research &Innovation in Computer Engineering , Vol 2, Issue 4, ISSN 2249-6580, ( 296-304)

STUDY ON SLIDING MODE CONTROL FOR SECOND

ORDER SYSTEM

1Venkatesha ,

2G G.K.Purushothama,

3N.K.Philip

Abstract

This paper presents a robust control for a

class of systems using Sliding Mode Control

(SMC).The goal is to achieve the system

robustness against disturbances. The

chattering resulting from discontinuous

controller is reduced by using Lyapunov

stabilization technique. The Performance of

the system with external disturbances for both

PID control as well as SMC is compared and

studied.

Keywords: Sliding Mode Control Disturbances, Lyapunov method.

I. Introduction

The sliding mode control (SMC) is a variable

structure control (VSC). The theory of SMC iw

based on the concept of varying the structure of the controller by changing the states of the

system in order to obtain a desired response

[1].Generally, a switching control action is used

to witch between different structures and the

system state is forced to move along the chosen

manifold, called the switching manifold which

determines the closed loop system behavior [2]

[3].

When there are disturbances and uncertainties in

a system, an appropriate control should be

designed so that the system stability and desired system performances are achieved.

1Professor, Dept of E&C Engg., Brindavan

College of Engineering, Bangalore, Karnataka, India.

2Professor, Dept of E&E Engg, Malnad College

of Engg., Hassan, Karnataka, India.

3Scientist / Engineer SG, Control Systems Group, ISRO Satellite Centre, Bangalore,

Karnataka, India.

SMC is insensitive in the presence of external

uncertainties and disturbances. The robustness

properties of SMC have led this approach to be

intensive, popular and suitable method for the

control of wide classes of linear and nonlinear

systems. Various SMC approaches have been

evolved [] during last four decades.

Dynamic systems are controlled in order to

follow prescribed trajectories with certain accuracy. The control is often based on a

mathematical model of the system.

An SMC can be evolved so that the system

trajectories move onto a prescribed surface

(sliding surface) in a finite time and tends to an

equilibrium point along this surface []. The

closed loop dynamics are completely governed

by sliding surface equations as long as the

system trajectories remain on this surface. In

fact, the system in the sliding mode has one

order less than the original system (except when compensator is designed for using SMC

System).There are many advantages for using

SMC, including flexibility of design and

robustness [].

PID controllers have a simple control structure

and systematic tuning methods. Whenever the

system is non-linear or there are bounded

uncertainties in the system, PID controllers are

not perfectly able to stabilize the system.

Particularly, when non linearity is very high or

the bound of uncertainty is large.

In many practical design problems, almost

perfect disturbance rejection or control

performance is required. SMCs may be applied

to the system to obtain these performances. An

SMC enforces the system trajectories to move on

a prescribed surface and remain in it thereafter.

On the other hand, a discontinuous SMC may be

approx- imated by a continuous control. In fact,

the trajectories tend to an equilibrium point

within a boundary of the sliding surface. When the trajectories move on the sliding surface, the

system is internally controlled by a virtual

control, the so called equivalent control. As

already stated, SMCs are insensitive in the

presence of uncertainties and unmodelled

dynamics.


Most of the practical systems can be modeled

with second order systems. Hence the control of

the second order system is usually taken as

baseline.

In this paper, SMC and PID control of second order system are considered and the system

responses are compared when the external bias is

applied as distur-bance to the system. In this

study, PID controllers are selected such that both

the steady state and transient responses are met.

SMC methods yield nonlinear controllers which

are robust against unmodelled dynamics and

against internal and external perturbations.

II. SMC Theory and Analysis

A. SMC Theory: SMC is a particular type of

variable structure control. Variable structure

control systems are characterized by a suite of

feedback control laws and decision rules. The

decision rule, named the switching function

selects a particular feedback control in

accordance with the systems behavior. Note that a bang bang control is a variable structure

control, but is not usually considered as an SMC.

In SMC, VSC systems are designed to drive the

system states to a particular surface in the states

space, named sliding surface. Hence the SMC is

two part controller design. The first part involves

the design of the switching function so that the

sliding motion satisfies design specifications.

The second is concerned with the selection of a

control law that will make the switching surface

attractive to the system state [].

The advantages of SMC is that dynamic

behavior of the system may be designed by the

particular choice of the sliding function.

Secondly, the closed loop response becomes

totally in-sensitive to disturbances. This principle

extends to model parameter uncertainties,

disturbance and nonlinearities that are bounded.

Due to its discontinuous in nature, SMC

introduces chattering in the control loop. This

problem can be overcome by using Lyapunov function in control law.

VSC systems are systems where the control law

is deliberately changed during the process to

predefined rules which depend on the state of the

system [ ]. This yields to a switching surface in

the state space in the sliding mode control. In

SMC full state feedback control structure is used

with an addition of a switching term that is

aimed to cancel the effects of uncertainties.

Consider the second order system

is used for illustrating VSC system, where u is

the control input and x is the angular position.

First consider effect of using only a negative

feedback control law (k>0)

By integrating (3), we get

where C is constant of integration.

Equation (4) represents an ellipse in the phase

portrait, is a circle when k=1. Fig1 shows two

ellipses with two different k values k11.

By changing the value of k during the system

process it is possible to move the system states

from an initial point on the phase portrait to the

origin. This is equivalent to decreasing the

distance of an initial point on state space to zero [ Fig2].

Therefore the control law may be changed to

x

x

x x

u(t) = k1 x(t) u(t) = k2 x(t)

Fig 1

Fig 2

x

)3(..................

)2(....).........()(

xxkxx

and

tkxtu

)4(...............22 ckxx

)5......({)(0)(

0)(1

2

xxiftxk

xxiftxktu

)1(.............).........(tux


SMC uses the same principle described above to

take the states to the origin. In the above

described controller approach,

was used for switching controller

input. In SMC, the switching function is chosen

as .

Therefore, SMC would switch with the sign of

, which is called sliding surface [Fig3].

represents a line that

passes through origin in state space. Therefore if

on can find a controller that first reaches

and consequen tly keeps the states so that ,

the system would reach to origin of the state

space.

If a control structure can be established so that

whenever and

and vice versa then it is possible to

keep the states on the sliding surface. Once it is

satisfied then the system will behave as a first

order system, and

slide to the origin on the sliding surface.

However this is only possible if the switching

occurs at zero time, the frequency of the controller is infinity. This is called ideal sliding

motion. This is depicted graphically in Fig 4.

Figure-4

B. SMC Analysis: An nth order uncertain

nonlinear time invariant system with m control

inputs is given by

The function is assumed to be unknown but bounded by some known functions

of the states. Different restrictions can be placed

on this function, which represents parameter

uncertainties or nonlinearities in the system, or

even disturbances.

SMC can be divided into two terms, one being

the linear control and the other being

discontinuous control. The first discussion is on

linear control law. Second, we discuss the

reachability of the sliding surface by using

second order system

B.1 Linear Control Law: The system given by

equation (6) is first assumed to be linear and is

given by

The sliding surface can be represented by

Also,

Once the sliding surface is reached, then

Therefore, from (8), the linear control laws can

be obtained as

which is equivqlent to linear control law

equation , is designed such that

the states would remain on sliding surface. SB

must be non singular. Here S is a design

parameter and B has a rank of m. Because the rank of B is m, the system equations can be

portioned to give equation (7) can be written as,

The compatible transformation of the switching

function would result in

. Therefore during ideal sliding, the

motion

Dividing (11) by S1 and replacing

by M, equation (10-a) can be written as

0, xxs

0, xxs

0, xxs

0,.,.; xmxxxseimxx

0, xxs

0, xxs

xmxxxs ,

0, xmxxxs

)6.().........,,()()()( uxtftButAxtx

),,( uxtf

)7.......().........()()( tButAxtx

nmRSwheretSxtxs ),(),()8........().........()()( tSButSAxtxS

0)( tSx

)9(..........).........()( 1 tSAxSBuL

SASBk 1)(

mmn RxandRxwhere

btuBtxAtxAtx

atxAtxAtx

21

22221212

2121111

)10......()()()()(

)10...()()()(

21 SSS

)11........(..........0)()( 2211 txStxS

1

12 SS

xxxxs ,

xxs ,

Fig 3

0, xxs

0, xxs


Equations (11) & (12) represent ideal sliding

motion. Note that the order of the system is

reduced to m or to equation (10-b) as soon as S is defined so that

B.2 Reachability of the sliding surface: The second order system is given as

Sliding mode control for this system may be

selected as

The linear part of this control input is state

feedback law, which is in agreement with the

structure of equation (5) and the discontinuous

part is to be arranged so that the system is

insensitive to disturbances.

To check the stability of sliding surface, one can

use Lyapunov second method. It states that if the

projection of the system trajectories on the sliding surfaces is stable then the system is

stable. The theorem [4] is reproduced here for

completeness.

Theorem: If there exists a function v(s,x,t)

where s is the distance from the sliding surface

and x is the state variable, which is positive

definite. i.e., it satisfies the following conditions

1. v(s,x,t) > 0 with s 0 and arbitrary x and t.

2. v(s,x,t) is continuous and differentiable. 3. v(0,x,t) = 0 for all x and t and its

derivative ),,( txsv is negative definite every where except the

discontinuity surface, then the system is

stable.

There is no specific method to find Lyapunov

function candidate however VI Utkin [5] has discussed the method of using quadratic form to

find the sliding domain.

In this analysis, the sliding surface, can be

written as

by selecting as 1 and using

for m ( one of S1 and S2 is enough to define the

sliding surface). Once sliding surface is reached

the system would behave as the first order

system and is given by

which is insensitive to the disturbances. Then

next step is to reach the sliding surface, which is

possible if

where

Equations (13) and (14) are used to

replace .

The convenient selection of k1 and k2 are

then replacing

results

Since, then replacing

would give

Equation (18) is negative by the definition of the

parameters. Therefore sliding surface can be

reached in finite time.

It can be shown that

if small parameter in equation(18) is neglected.

Therefore sliding mode control input given in

equation (14) takes the system to the sliding

surface in finite time keeps the state on the

sliding surface that decays to the origin provided

that the design parameters are selected as

described above. Here the second order system is

decreased into two first order sub-systems when

the sliding mode control input is applied to the

system. It depends on the selection of poles and m of the two first order systems. First one is responsible to take the system to the sliding

surface and the second one is responsible for

taking the system to the origin by sliding on the

sliding surface. It must be noted that if some part

)12().........()()( 112111 txMAAtx

MAAAs 121111

)13........().........(),()( tutxftx

)14)).......((()()()( 21 tssigntxktxktu

)()( txtmxts

2S

)15...(..........0)()( txtmx

0ss

)16)........(( xxmsss

)17)........()(( 21 fssignxkxkxmsss

),(21 positiveareallmkandmk

swithsignsand

swithxmx

fsssss 2

)( parameterdesignpositivesmallis

f

)18..(..........2 ssss

tests )0()(

)(tx

1

12 SS


of the non linearity or uncertainty term could be

represented in terms of linear combination of the

states then it would be possible to obtain a lower

magnitude of the discontinuous term of the

sliding mode control input given by equation

(14).

State feedback law parameters, k1 and k2 can be

designed by pole placement.

C. PID Controller:

PID Controller is widely used in many control

applications because of its simplicity &

effectiveness. Tuning PID controllers have been

a challenging problem since its appearance

(Ziegler Nichols 1942) in control engineering. Alternative tuning methods have been recently presented including disturbance rejection

magnitude optimum [6], pole placement and

optimization methods [7]. These methods

provide relatively fast and non oscillatory

disturbance rejection responses. In particular,

these methods do not require any additional

tuning parameters. The trade off between

performance, robustness and control activity is

considered while arriving at the PID control

parameters.

The transfer function of a PID controller is given

by

sTd

sTiKpKs

11

III. Design, Simulation and Results

A. SMC Design for Second Order System:

Consider a second order system given by transfer

function

Therefore the control law is given by

The value of can be selected such that the sliding surface is reached in minimum time.

B. Simulation:

The MATLAB Simulink design of the SMC for

second order system is shown in Fig 3.2a. The

SMC for second order system with step bias and

sinusoidal bias as disturbances at input level are

shown in Fig 3.2b and Fig 3.2c respectively.

215 xxxmxs

uxxx

xx

xx

uxxx

sUsssX

sssU

sX

122

21

1

2

2

01.12.0

01.12.0

)()01.12.0)((

01.12.0

1

)(

)(

ux

x

BuAxx

1

0

2.001.1

10

2

1

010

05)(

2

1

x

xtSx

SASBkwheretkxtu 1)()()(

]2.001.1[

2.001.1

50]10[)(

2.001.1

50

2.001.1

10

10

05

]10[)(

1

0

1

0

10

05

1

1

k

SASBk

SA

SB

SB

)5(2.001.1

)(

2121 xxsignxx

ssignkxuL

)19)....(5()(2.0)(01.1)( xxsigntxtxtu


Fig 3.2 a

The simulink design of PID controller for the

second order system is shown in Fig 3.2d and the PID controller for the second order system with

disturbance is shown in Fig 3.2e.

Fig 3.2b

Fig 3.2c

Fig 3.2e

C. Results Discussion:

The results of SMC for second order system

show the stabilized performances and robustness

of SMC.

The phase portait plot of SMC for second order

system for the control law (equation 19) is

depicted in Fig3.3a. This shows that the phase

portrait takes a longer timt to reach origin from

initial conditions for k1 = 1.01 and k2 = 0.2

values. Fig3.3b shows the phase portrait for the

increased values of k where, k1=5.01 and k2=4.8.

It reveals that with increased values of k the

control law can be optimized to reach the

stability at the earliest.

Fig3.3c. and Fig3.3d show the variation of

timewrtxandx .

Fig 3.3a

XY

simout

To Workspace1

simout

To

Step

-1.2 Slider Gai n

1 Slider Gai n

5 Slider Gai n

-Slider Gai n

-

Slider Gai

Scop

Relay

ss(tf([1],[1 0.2

LTI

du/dt

Derivati v

Add

Add

Add1

Ad

x

x

XY

Graph

simou

t1 To

Workspace1

simo

ut To

Workspace

-

1.2 Slid

er Gain5

1 Slider Gain4 5

Slid

er Gain3

-

4.8 Slid

er Gain1 -5.0

1 Slid

er Gain

Scop

e

Rela

y1

ss(tf([1],[1 0.2

1.01])) LTI

System

du/

dt Derivati

ve

Add

2

Add

1

Ad

d

x x D

x

-80 -60 -40 -20 0 20 40 60 80 -80

-60

-40

-20

0

20

40

60

80

100

120

x

dx

Phase Portrait for gains k1=1.01, k2=0.02


Fig 3.3b

Fig 3.3 c

Fig 3.3d

The phase portrait and the variation of

timewrtxandx of SMC second oder system (Fig 3.2 b) with step bias as disturbances

at input level are shown in Fig3.3e, Fig3.3f and

Fig3.3g respectively.

Also, the phase portrait and the variation

timewrtxandx of SMC second oder system (Fig 3.2c) with sinusoidal bias as

disturbances at input level are shown in shown

in Fig 3.3h, Fig 3.3i and Fig 3.3j respectively

These results reveal that the SMC gives same

result for an undisturbed system as well as for

the disturbed system. This shows the robustness of SMC for the external disturbances.

Figure 3.3e

Fig 3.3f

Fig 3.3g

-70 -60 -50 -40 -30 -20 -10 0 -20

0

20

40

60

80

100

x

dx

Phase Portrait for gains k1=5.01, k2=4.8

0 10 20 30 40 50 60 70 80 90 100 -70

-60

-50

-40

-30

-20

-10

0

time--->

x

x v/s time at gains k1=5.01, k2=4.8

0 10 20 30 40 50 60 70 80 90 100 -20

0

20

40

60

80

100

time--->

dx

dx v/s time at gains k1=5.01, k2=4.8

-70 -60 -50 -40 -30 -20 -10 0 -20

0

20

40

60

80

100

x

dx

0 10 20 30 40 50 60 70 80 90 100 -20

0

20

40

60

80

100

time--->

dx

dx v/s time at k1=5.01, k2=4.8

0 10 20 30 40 50 60 70 80 90 100 -70

-60

-50

-40

-30

-20

-10

0

time--->

x

x v/s time at k1=5.01, k2=4.8

Dx Phase Portrait for gains k1=5.01, k2=4.8


Fig 3.3h

Fig 3.3i

Fig 3.3j

The simulated results for PID controller for a second order system without and with

disturabance at input level is depicted in Fig 3.3k

and Fig 3.3l respectively. The results show that

the PID controller takes longer settling time

compare to that of SMC. The PID controller for

the system with disturbance also introduces

steady state error.

Fig 3.3k

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4Step Response of PID controller with Disturbance

time t----->Am

plitu

de --

-->

Fig 3.3l

IV. Conclusion

In this paper, a robust control system with sliding mode control is presented for a second

order system. According to simulation results,

the SMC controller can provide the properties of

insensitivity and robustness to extenal

disturbances where as the PID controller takes

longer settling time than the SMC. The proposed

SMC controller is a robust controller.

References:

1. http://www.pages.drexel.edu 2. Young K.D., Utkin V.I., Ozguner U., A

control engineers guide to sliding mode control, IEEE Trans. Conttr. Syst., Vol.

7(3), Pg 328 -342, 1999.

3. Utkin V.I., Variable structure systems with sliding modes, IEEE Trans.

Automat. Contr., Vol. AC-22, Pg 212-

222, 1977.

4. J.J.E. Slotine and S. S. Shastry, Tracking and control of Nonlinear

Systems using sliding surfaces with

applications to Robotic Manipulators,

Intl Journal .Contr., Vol 38, No.2,pp 465-492, 1983.

5. M.Fallahi,S.Azadi, Robust control of DC motor using fuzzy sliding mode

10 20 30 40 50 60 70 80 90 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Time---->

Amplitude

0 50 100 150 200 250 -70

-60

-50

-40

-30

-20

-10

0

time--->

x

0 50 100 150 200 250 -20

0

20

40

60

80

100

time--->

dx

dx v/s time at gains k1=5.01, k2=4.5

-70 -60 -50 -40 -30 -20 -10 0 -20

0

20

40

60

80

100

x

dx

phase portrait at gains k1=5.01, k2=4.5


control with PID

compensator,Proceeding of the

international multiconference of

engineering and computer scientists Vol

II, IMECS 2009,March 18 20, 2009, Hong Kong.

sliding mode control for second order system (1).pdf

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