Sliding mode control of an unmanned air-vehicle system

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Sliding mode control of an unmanned air-vehicle system Jayakrishna Vanaparthy

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SLIDING MODE CONTROL OF AN UNMANNED AIRVEHICLE SYSTEM By

JAY AKRISHNA V ANAPARTHY

A Thesis Submitted In Partial Fulfillment Of The Requirement For The

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

Approved by:

Agamemnon Crassidis

Dr. Agamemnon Crassidis Department of Mechanical Engineering

(Thesis Advisor)

Dr. Mark Kempski , Department of Mechanical Engineering

Mark Kempski

Dr. Alan Nye Department of Mechanical Engineering

Alan Nye

Dr. Alan Nye Associate Department Head of Mechanical Engineering

Alan Nye

DEPARTMENT OF MECHANICAL ENGINEERING ROCHESTER INSTITUTE OF TECHNOLOGY JULY, 2003

SLIDING MODE CONTROL OF AN UNMANNED AIR-VEHICLE SYSTEM

I, Vanaparthy, Jayakrishna grant permission to Wallace Library of Rochester Institute of Technology to reproduce my thesis in whole or part. Any reproduction will not be for commercial use or profit.

Vanaparthy, Jayakrishna July, 2003

Acknowledgements

I

to thank my advisor, Dr. Agamemnon

wish

Engineering

at

tremendous

the

Rochester Institute

support,

inspiration,

of

patience,

Crassidis, Professor

Technology

at

of

Mechanical

Rochester, NY for his

knowledge, faith

and

friendship

made

completion of my graduate studies possible.

I and

wish

to

express

Dr Alan Nye for

suggestions

I

have

wish

Department

NY for their

being

part of

my

committee

and

for reading my thesis. Their

being greatly appreciated.

and all

help

to my committee members: Dr Mark Kempski

appreciation

to thank Dr. Edward C.

Finally, I sisters and

my

the

office staff at

and support

wish

for

all

of

the Mechanical

Engineering

the Rochester Institute of Technology at

Rochester,

the International Students.

to express my

my brother for their

Hensel, Head

deep

appreciation and gratitude

patience and understanding.

n

to my parents,

ABSTRACT

The

objective of

this study is to design

conditions of system parameters

for

variation

Controller

the

are

from the trim

Unmanned Air Vehicle

an

Controller that is

a

conditions and also robust

System. The PID

(UAV)

for the UAV

control models

stable under

and

for

varying

parametric

Mode

Sliding

that are studied, designed and

system

analyzed.

The system

system

proposed

for

parametric

variation.

and

the modified

ATAN functions

system

with a

control

linear PID

The sliding

for

to

and observed

mode controller was

make

for

to

for the

control

in

the

It

presented

to

in

Velocity

show

this

input. It

gain and nonlinear gain.

chattering

with

effect

was observed

The

that

same model

that the controller is stable but the

was observed

is

tracking

inability

of

the

control a

that the

Single Input Two Output

controller was able

to

stabilize

system robust.

mode and robust asymptotic

variation

negate

effect

the

presence of parameter uncertainties such

on

Switching theory

designed for Unmanned Air Vehicle System

sliding

indicated chattering

order

then extended to

Then, Sliding Mode Controller based was

model

and robustness of

control nonlinear systems.

parametric variation.

the system and

to a nonlinear second order

than the sliding mode controller, this is due to

more

controller

robustness

damping, input

PID Controller

is 10 times

system

The

applied

tested for stability

and

were proposed

demonstrated

inertial mass, stiffness,

error

(SISO))

was

(signum) function. Therefore, in

Saturation

tested

Mode Controller

(Single Input Single Output

switching

as

Sliding

and

Angle

stability in of

Attack

uncertain

method.

uncertainty

UAV

parameters.

the effectiveness of the design

111

under

and

Lyapunov's theory conditions.

systems were

Finally,

Stable

investigated

simulation results are

Table Of Contents

Chapter 1

1.1)

Introduction

Sliding Mode Control

Of Electro-Mechanical Systems

1

2)

Equivalent Control Method

5

3)

Design Concept

6

4)

Cause Of Chattering

6

5)

Boundary Layer Solution.

13

6)

Literature Review

15

Chapter 2

Sliding Mode

2.1)

Introduction To

2.2)

The

Control Method

Sliding Mode

Control Method

2.2.1) Stability Of Sliding Mode 2.2.2) 2.3)

2.4)

2.5)

16

Switching Theory

Development Of A Scalar

19

Control Law

Switching

22

Gain

23

Simulated Nonlinear Second-Order Example

29

2.3.1)

Effect Of Varying Inertial Mass

35

2.3.2)

Effect Of Varying Nonlinear Parameter

40

Effect Of A Thin

Boundary Layer

On

Switching

Controller

43

2.4. 1)

Saturation Function

43

2.4.2)

Atan Function

45

PID Control OfThe Second Order Nonlinear System

Chapter 3

Modeling A Bus

Suspension System

47

Using Sliding Mode

Controller

3.1)

3.2)

50

Bus Model With Two Controller

3.1.1)

Results For Suspension System

53

3.1.2)

Results For Vehicle

54

body

55

Bus Model With Single Controller

3.2.1)

55

Results

IV

Chapter 4

4.1)

Sliding Mode

Control Of Unmanned Air Vehicle System

Aerodynamic Parameters

4.1.1)

Lift Coefficient

4.1.2) Drag 4.1.3)

58 58

Coefficient

59

Coefficient Of Lateral Force

59

4.1.4) Rolling Moment Coefficient

60

4. 1 .5)

Pitching Moment Coefficient

60

4.1 .6)

Yawing Moment Coefficient

60 61

4.1.7) Velocity 4.1.8) Angle 4.1.9)

Of Attack

62

Angle Of Sideslip

63

4. 1.10) Roll Rate

65

4.1.11) Pitch Rate

66

4.1.12) Yaw Rate

66

4. 1.1 3) Pitch Angle

67

4. 1.1 4) Roll Angle

67 72

Mode Control Law

4.2)

Stability Of The Sliding

4.3)

Development Of A Scalar

4.4)

Results

Switching

74

Gain

81

4.5.1) Stability Of The

System Under Change Of Angle Of Attack Input 81

Signal

4.5.1.1)

Velocity=626.81863 Ft/Sec & Angle Of

4.5.1.2)


4.5.2) Stability Of The 4.5.2.1) 4.5.2.2)

Attack=3.0

81

Attack=4.1

83 85

System Under Change Of Velocity Input Signal

Velocity=600 Ft/Sec & Angle Of Attack=3. 6 1029

Velocity=650 Ft/Sec & Angle Of

15

85

Attack=3.6102915

87

4.5.3) Stability Under Change In Input Signals Of Both Velocity And Angle Of 89

Attack Input Signals

4.5.3.1)


4.5.3.2)


Attack=3.0

89

Attack=4.1

91

4.5.4)

Test For Robustness Of The System Under 1 0% Variation In The Coefficients OfThe Parameters Of The System

93

4.5.4.1)


Attack=3.6102915

4.5.4.2)


Attack=3.6102915

4.5.4.3)


Attack=4.1

4.5.4.4)


Attack=3.0

4.5.4.5)


Attack=3.0

4.5.4.6)


Attack=4.1

4.5.4.7)

Test

of

Sensitivity of the

system

93

95 97 99 101

103

to variation of

105

parameters

Conclusion

1 07

Suggestions for future Work

108

References

1 09

Appendix

112

VI

CHAPTER 1 INTRODUCTION

Over the control actions

or

'on-off

past

years, the

has been

intensity

of

highly for

design

of

discontinuous

of systems with

high level. In particular,

maintained at a

regulators ranked

investigation

feedback

at

the

first stage, relay The

systems.

reason was

twofold: ease of implementation and high efficiency of hardware.

In may

arise.

occur

areas

in

The

control action switches at

the system.

from

high-order

The study

of

with

nonlinear

for many

sliding

dynamic

to

have

modes

plants

high

function,

frequency

so-called

sliding

causing the sliding

mode

to

modes

heterogeneous

application aspects.

to be

proven

an efficient

under

operating

processes of modern

state

mode embraces a wide range of

sliding

pure mathematical problems

Systems

problem

discontinuous

systems with control as a

tool to

control complex

uncertainty conditions,

a common

technology.

1.1 Sliding Mode Control OfElectro-Mechanical Systems:

Sliding

modes as a phenomenon

ordinary differential mode control

first

equations

appeared

of the system state switches at

be

enforced

in the

simplest

high

systems.

frequency;

=

With the bounded function f (x), the

(t) (i.e.,

e

=

r(t)

-

x

;

r

in

a

dynamic

right-hand

this

motion

is

called

system with

system governed

sides.

It may happen that the

first-order tracking relay

X(t)

error e

appear

discontinuous

with

in relay

may

The term sliding

control as a

sliding

the state

(t) is the

It may

variable x

(t):

(1.1)

input u(t)

reference

function

mode.

f(x) + u(t)

control

by

as a

input) is

relay function

given

by

of

the

tracking

u(t)

\u0

if

e>0

\-u

if

e

-u0

e, is

equal

to Uq in

an

in the switching device

takes intermediate values for continuous

motion

for t>T is

called

sliding

mode as shown

in the figure 1.1 [1-4].

Fig

Sliding control

but

also

example, the

mode

may

in any dynamic

right

hand

side

is

a

1.1:

Sliding Mode Tracking Control

appear not

system with

only in

a control system with

discontinuities in the

discontinuous function

of

motion equations.

=

u

For

the state in the simple second

order system

x-ax

discontinuous

(a>0)

u

s

consists of two unstable

linear

=

=

-k\x\sign(s)

C0x +

(k>0,c>0)

x

structures as shown

in Figures 1.2

and

(1.3)

1.3.

X(t)

X(t)

Cnx

Fig

1.2 Unstable

structure

+x

(1)

X(t)

Fig

The

system

behavior is

undergoes

discontinuities

constituted

by

at

the

1

.3

analyzed

Unstable

structure

in

space

state

switching line

two families: the first

family

s=0

method.

and

corresponds

(2)

the

The

state

control

u

trajectories

(t) are

to s>0 and u=-A;|x|and the

second corresponds

to

s tl.

trajectories in the switching line is called sliding mode.

represented as a motion

above

trajectories

slope of the

Having

Since, in

switching line s=0, its

equation, i.e.

x+cx

From the

state

leave the switching line i.e. the

the course of sliding mode, the state equation

The

observed

=

0

that the

motion of

on plant parameters and on

the state trajectories in the

any disturbances the

to.

X(t)

Figure: 1.4

Sliding Mode

(NM\

)

plant

may

In

connection

independent

enables order

procedure.

has

input

This

means

interest is

great

s

(x)

reduction,

Furthermore, close

that the

to

zero

which

sliding

leads to

the element

during

modes

implementing

sliding

systems with

decoupling

and

discontinuous

simplification of

discontinuous function

a

its

mode whereas

implements high gain,

element

in

which

output takes

is the

finite

the

(x)

s

values.

conventional tool to

the influence of disturbance and uncertainties in the plant behavior.

suppress

The

high-dimensional plants,

of

Enforcing

partial components.

design an

control

to design methods that allow the overall system motions to be decoupled into

attached

control

the

with

motions

The

in

systems with

sliding

order of the system

is

modes

have

shown

two things:

reduced.

Sensitivity with respect to parameter variation and disturbances may be reduced.

The motion

in

equations

switching

to

two

that the state would

a

a

space.

easily feasible in

The sliding

stages.

First,

the

equation of

design the desired dynamics

performance criterion.

As

canonical

are

mode

surface equations and not on control of the system.

should consist of

selected

invariance

order reduction and

of

this

the

the

manifold and such

result, the design is decoupled into two

motion

Hence the design

in

accordance

mode exists

subprograms of

on

mode

with

be found

in this

lower dimensions,

system will possess

the

procedure

sliding

control should

that sliding

finite time interval preceding the sliding motion, the

dynamics depend

manifold with

Then secondly, the discontinuous

reach

a second order system with

is

some such

manifold.

and after

the desired

dynamic behavior.

1

.2

Equivalent Control Method:

Equivalent

intersection vectors

of

control method means replacement of

switching

surfaces

lies in the tangential

by

manifold.

continuous

the discontinuous control in the

control

such

that the state velocity

1.3 Design Concept:

Sliding equations of

modes

discontinuous

two

of

at

the sliding

theory,

dynamic optimization, may be

discontinuous

to

control

enforce

reduced

The first

stages.

surfaces such that

a

stage

discontinuity

such as

Partitioning motion precedes

sliding

overall motion

mode within a

desired

this

stage.

although

The

is usually

into two

since

equal

desired

motions of

may be insensitive to place

of

the

mode

of

properties.

the

The

its dimension is

the

and

equal

to

of control.

lower dimension

the first

-

second motion

is

procedure considerably.

unknown

for any

is to find

the surfaces selected

to dimension

finite time interval

invariance property takes

selection

second stage

may simplify the design

properties

modes

sliding

disturbances,

the

sliding

mode with

In addition,

of

on

stabilization, Eigen structure placement,

in the intersection

mode

which

surfaces,

depending

design is the

of

the first stage. The second problem is of reduced order,

the number of

system

motion would exhibit

applied at

sliding

order

The design methodology for the sliding

surfaces.

methods of conventional control

and

by

governed

discontinuous

the

consists

control

are

plant

parameters

and

system.

1.4 Cause Of Chattering:

1)

Fast switching the

2)

control

of

loop,

the sliding

which were neglected

Digital implementation in

discretization

mode controller often excites

oscillations

exciting

caused

unmodeled

fixed sampling

rates

may lead to

chatter.

oscillations

are

fast dynamics in

system model.

micro controllers with

The term chattering describes the amplitude

in the

the

appearing

by

in

the high

dynamics in the

sensors and actuators neglected

in the

of

phenomenon

sliding

many

frequency closed

finite-frequency

mode

switching

of a

loop. 'Unmodeled

principle

modeling

significantly faster than the main system dynamics.

and

implementations. sliding

mode

finite These

controller

dynamics'

process since

may

they

are

refer

to

generally

A

first

simple

considered and given

order plant with second order unmodeled actuator

dynamics is

by X

=

aX(t)

+

d(X,t) + bw(t) (1.5)

Where a, b and

are unknown parameters within

disturbance d

(t) is

assumed

known

bounds,

w

(t) is

the control variable

to be uniformly bounded for all operating conditions.

w(t)

=

Msign(S)

Where

S(t) is

considered as

assumed

the sliding

=

surface.

Xd(t)-X(t) X

d

(1.6)

is the desired trajectory

with

is

predefined or

to known.

Consider the

parameters as

a

=

0.5, b

Now simulating the

=

follows

\,d(t)

system with

=

0.2sin(100 + 0.3cos(200

the

given

parameters,

,

Time

we observe

constant=0.02 sec

the

following results.

Without Actuator:

Fig: Block diagram representing the

system without a

Actuator

A discontinuous desired trajectory Xd

controller

forces the

as shown

in the

(t)

output

X (t)

of the plant

Fig 1.1. No chattering

to exactly track the

occurs since the control

loop is free of unmodelled dynamics.

Results:

state response

desired

state

1 2

-

^^\

"

^

,

cd

^

x(t)

y en

s ,'

04

\ S

-

-

xd(t) 02

Fig

1 1 Desired Response .

plant input xd-x

mi ii -

u(t)

1

0

-

s(t)

1

o

o

-

J_

0.8

Time

Fig 1.2

Control Input

and

Sliding

Surface

II |

In

|j

I .,

0 9

1

With Actuator:

Subsystem:

CD-*

Block diagram of Control

1

1

0.05S+1

0.05S+1

tracking

w

loop with actuator dynamics neglected in

Since the discontinuous actuator, the

?CD

of the

controller excites the

desired trajectory

by the

ideal

control system

fast switching dynamics

state

the ideal condition, due to chattering in the Control Input.

trajectory is

of

the

worse compared

to

-

-

state response

des red state

1.2

-

-^ z_~ x^-*

\

P

-

\

\

//

O

/'

'

'

V

-

04

xdffl

0.2

Fig 1.3

Desired Response

time

Fig 1.4 Control Input,

Disturbance

10

and

Sliding

Surface

Influence Of Unmodeled Dynamics On The System:

u w

1

>^

? >

l^

0

.0

1

5s+1

1 ^

0.05S+1

"

s

^

Clock

XVsT 1.2

1

1

i

i

i

i

i

0.8 CD

CO

Applying

behavior is

J

J

\

becomes

27

(2.30)

d(t,X,X) switching

as

gain

K>kt)\(i-/3r~l] X(t) +

+

kt)\(l-/Jk-l]\X(t)\

+

d(},X,X%\-p\

+

/3D(t,X,X)

m(t)\(\-fr-1} Xd(t)-AX(t) V

(2.31) The the

control

switching law

discontinuous term in the

most

highly

undesirable

manipulator,

Smoothing smoothing this

control

application, chattering is

since

the

boundary

chattering

essentially

signum

law becomes the

of

i.e.,

to achieve

system

in

a

thin

assigns a

pass

proper performance.

modes,

layer

which can

can eliminate

filter to the

the smoothing

effect.

This

such

due to

as

effect

is

flexible

lead to instability.

the chattering. The

To

control signal.

replaced with a saturation

which provides

"chatter"

to

flexible dynamics,

with

boundary

low

system

the sign term in the control equation. In

could excite resonant

function is

layer thickness,

law

undesirable

control

control signal

effect

task, the

for

the closed-loop

will cause

function

accomplish

with

The sliding

a

varying

mode control

following

'S^

i/(0

=

b(t) 1 kt) X(t)

+

k{t)X(t)

+

d(t, X, X)

+

777(0

Xd(t)-AX(t)

+

AXd(t)

Ksat

(2.32) Where

^

is the

boundary

layer thickness,

sat

fs}

is the

saturation

function

of the

sliding

\0; surface

in the thin

boundary

layer.

Smoothing

effect can also

be

accomplished

by

using

(S^ an atan(s

(t))

function instead

function in the

of sat

above equation.

WJ

1

u(t)

=

bit)

A

\

c(t) X(t) +

k(t)X(t) + d(t, X, X) + 777(0 Xd(t)-AX(t) + AXd(t)

28

Katan(s)}

The initial desired

conditions when possible so that the control effort

The

following

constraints

the actual initial states or

states or conditions should match

is

used

for the

is finite

and within reasonable

bounds.

second order system

Xd(0)

=

X(0) (2.33)

Xd(0)

The tracking

error

when the control signal system.

The tracking

function

of time.

the desired

is

X(t)

between the desired

be improved

error can

and actual state

by

changing the

control system yields

bandwidth that is

trajectories is imperfect

discontinuous

to the

smoothed compared

The resulting

control

=

the best

control

law feedback

boundary layer thickness tracking performance,

as

given

available to the system.

2.3 Simulated Nonlinear Second-Order Example:

In the

previous

a more general

section,

sliding

mode control

scalar second-order nonlinear system with parametric uncertainties considered.

damping,

The

control

stiffness, input gain,

function. A stability Lyapunov's direct applying section,

is

law developed

a

of

the

the inertial mass,

The switching system and

law eq

the closed-loop

The tracking uncertainty.

control

results

The

are

damping,

(2.18)

system

compared

simulations

are

control

sliding

a nonlinear second-order system

placed on

assumption

is

is

was

to the desired

repeated

for the

boundary layer.

29

previously

some

was

by

known using

then changed

by

to smooth the control input. In this

stiffness, input gain,

simulated with

by

conducted

considered as an example.

developed in the

was

a

that the inertial mass,

law eq(2.18)

surface

than

be bounded

system

closed-loop

The switching

boundary layer near the

thin

the

and nonlinear parameters can

analysis

method.

with

law was derived for

An

bound

and nonlinear parameters.

previous section

varying

state

assumed

is

applied

to the

parametric uncertainties.

trajectories for each varying

smoothed

control

law

with a

thin

A

second-order

Mechanical

systems are

integrations

of

inertial

damping

transformation,

systems

drag,

such as

about an

displacement

and

entire region.

the

operating

is

model

Consider the

777(0 X(t) +

Where the

parameter m

for

friction

nonlinear and

point and control

second

law

requires

two

systems with a significant

can

poor

provide

following nonlinear

law is

c

is

nonlinear

develop

a suitable

if the

attractive

valid

are

state

for these

throughout the

second-order system as an example

kit) X(t) + c(t)x(t) + fit) Xit) X(t) Xit)

(t), k (t),

to

motion,

systems

performance

control method

control

rotational

nonlinear

are applied

damping.

and

Coulomb friction,

Typically,

linear techniques

law

elasticity

forces in

such as

saturation.

linearized thus the

not

with

centripetal

trajectories are outside the linear region. The sliding systems since

output

systems

include

can

springs,

law. However, the

control

Newton's

systems.

electrical

and

mechanical

second-order since

manipulators,

nonlinear

most

these systems include: liquid-level systems, thermal system,

of

pneumatic,

linearized

describe

the input to obtain the

Nonlinearities for these nonlinear

can

inherently

Examples

effect.

hydraulic,

model

(t), f (t), b (t)

are assumed

to be

=

bit)u(t)

unknown

(2.34)

but bounded

by the following

1

3

trajectories,

o

of

2,

that the track thus

of

the

40% from the

by

varied

value

mass

the

above

controller

the

is

desired

making

the

B

system

parametric

5

10

15

20

25

30

36

50

40

time

Fig

2.24 Desired

Tracking Error

38

robust

under

large

variations.

Initial Conditional riid ctntoPl

nbd cmftcrFl

25 ?rti

;

ri st3

j

artrdrpj

8

6

2

'

1

'

"

"

"

A

I

/

4

:

\

1S

\\

I 1

!

i

i

/

'

Mf\Vj \r

\

\

OS

L1

L

i

4.--.

:

---I--

:

i

V

:

/""""

u L-----i

0

Fig 2.25

Desired Displacement

i-----i

Trajectory

Fig

2.26

Trajectory initial conditions

0

2

4

6

8

10

12

14

16

lime

Fig 2.27 Desired Tracking

-----I

i

;

1

2468D121415-B20

02468B121416-B2)

Error

39

18

20

Desired

Control

Input

2.3.2 Effect Of Varying Nonlinear Parameter:

Most

practical

include:

of

linear model,

the system. Examples of

a nonlinearties

coulomb rate

amplifier,

high

angle of

friction,

limitation attack,

quantization

system,

dominate the

a

of

many

and

can

(2.34). Figures

f(t)=4.1

and

the

excellent, but shown

in the

display rest of

not

many types

be

bounds

magnitude of

other.

used of

switching

the

of systems.

As

nonlinear

frequently

are

such as

tracking

of

term,

error

for the

for both

encountered

saturation of the of aircraft at

modeling

nonlinear

response of

terms

an explained

listed

the

above

earlier, the sliding

the control of these types of systems.

terms must

be

predetermined

the sliding consider

and

mode control

the

control

system

effort

to ensure

nominal parameters.

sets of values.

40

technique

described

with

the parameters remaining the same. The

control effort

parametric

controller.

the nonlinear

as good as

of

accurately describes the

aircrafts, dead-zones in the

Typically

the effectiveness

the

dynamics,

directly for

the

more

that

form

some

friction, gearing backlash,

complex

and

an example of

in

actuators,

on vehicles or

proper performance of the

variation

of nonlinear

drag

technique

However the form

with

form

response of the

mode control

As

today have

encountered a

which, together

nonlinearties,

dynamics

systems

the

by

with a

equation

nonlinear

tracking

error

is

term again

High chattering activity is

Results: Robustness

of

the system is tested using sliding mode controller, when the

the nonlinear parameter

f(t) of the

system

is

changed

from 5 to 4.1

5H1 15

i

gain of

units

H1

i

EttuJ sblo

ctsrsJsfcte

1

ns

"A"

-i-i-l-

I

I

;

-o

u/

-05

-1

i

-15

02

Fig

46

i

810121416B20

2.28 Desired Displacement

Trajectory

Fig

2.29

Trajectory

_

l\

I 0

2

4

\i

6

i

8

error

/

L

10

tracking

V!

12

14

\1 16

time

Fig 2.30 Desired Tracking Error

41

18

20

Desired

Control

Input

Robustness

of

the system is tested using sliding mode controller, when the gain of

the nonlinear parameter X

f(t) of the system is changed from

1 0

f(t)=6

4

3

5 to 6.4

units

4

5

3

2 5

2

1

5

1

0 5

0

0 5

EHR:zr7J ] i

L\J

:

-

!.\l I \J

-1

:

M \f\ !

!

;

!

U"

C

2.31 Desired

Tracking

Error

f(t)=6 1

10 tim e

2.32 Desired Control Input

When the around

in

system

is tested for

the initial value of

forcing

the

state

5,

variation

the above figures

in the

show

nonlinear parameter

f(t) by

that the controller is still successful

trajectories to track the desired states but at a cost of

chattering.

42

15%

increase in

2.4 Effect OfA Thin Boundary Layer On Switching Controller:

In this section, earlier with a

boundary

thin

The

controller.

switching

control chattering.

usually

As

which can

lead to instabilities

acceptable

e.g.,

such as

In general,

maintaining the

state

placed near

shown

earlier,

high

can excite

not

place of

however, chattering

should

trajectories close to sliding

tracking is

desirable,

frequency

and poor performance.

using chattering in

described of

place

the

smoothed to achieve a more realistic

perfect

chattering is

nominal system

the switching surface in

is

discontinuity

Typically,

Also, chattering

results.

motors.

layer

control

and applicable control effort.

high

for the

simulation results are presented

In

since

high

control effort

unmodeled modes of

certain

dithering be

achieved at a price of

system,

applications, chattering is

to reduce

avoided

stiction effects of

when

possible

while

surface.

2.4.1 Saturation Function:

As

described

an example of

by

equation

how chattering

(2.34). The

can

be eliminated,

controller uses

the

again consider

Saturation Function isat

(s\

)

system

with a

\/

a variation of

*

concluded

,

and

around

Vehicle

that the

Body

controller used

to track the desired displacement

to stability

t/2)

15%

be

of the

it is

within a

time

also observed

the initial

span of

5

sec

in

is

of

successful

1 meter

within

presence of road

that the system is robust even when

value of the parameters of the system.

54

in

3.2 Bus Model With Single Controller:

Now considering the two the

outputs of the system control gain matrix

inversing the B matrix. Now the produce

the

same

following

same vehicle system with a single controller

i.e., Vehicle body

B is

changed

This setup model

results.

is

to

output and

B=[0; 1],

makes

the

and psuedo-inverse

system a

simulated with

Matlab

code and

Suspension System

specific

changes

Simulink file is

3.2.1 Results:

1) Control Input:

3.5 Desired Control Input

55

output.

In this

used

for

above

to

function is

SIMO Model

for lookup.

Fig

to control the

described

presented

in Appendix 2

2) Vehicle Body Output:

i

1

x2

!

j

/

:

/ /

\

V j......

^L...l

3

Fig We

forcing required

observe

the vehicle

4

56

3.6 Desired Response

from the

above

trajectory

settling time 5

!

of the

figure that the

7

Vehicle

8

s

1 c

body

single control

to track the desired displacement

input is of

successful

1 meter

sec.

3) Sliding Surface Output OfA Vehicle System:

S(2)

,

5

4

tim e

Fig

3.7 Desired

Sliding Surface

output of the system of the

56

Vehicle

body

within

in a

4) Sliding Surface

Output

OfA Suspension System:

S(1)

0

^4

/

/ ./.

/ :

.....

7

\

\

:

I

\

:

;

^

8

9

1

/ j

0 )

2

1

Fig

J

3.9 Desired

5) Suspension System

Sliding

in 5 seconds,

7

6

)

Surface Output

of the

Suspension System

Output:

Fig Above figures

1

show

and also

the

3.10 Desired Suspension Output

that the

vehicle

body

single controller

under parametric variation.

57

is

reaches

the desired displacement of lm

successful

in making

the system robust

CHAPTER 4

Sliding Mode In the control

law

3,

chapter

was

control method

was an effective

meet

control

on

first

a

order nonlinear

the stability methods

is designed to

Now considering the

the system

by

a single

Aerodynamic Parameters

4.1.1

Lift Coefficient: Coefficient

Aircraft

matrix model

described in the

control

is

input using sliding

considered and a control

previous sections.

these outputs

with

mode

of Lift

is

given

The

law is

system consists of six

four inputs.

aerodynamic equation of the aircraft model

4.1

of

of

the required specifications. It was shown, that the Lyapunov's direct method

outputs and a controller

Equation

two outputs

suspension system was considered, and a

tool to ensure stability [9].

In this section,

derived based

bus

a second order nonlinear

designed to

to

Control Of Unmanned Air Vehicle System

[13]

by

(4.1)

CT L

a +

+CT

=CT

L ^a

L o

CLqa+CLm

2V

a

+

CT

de

+

Lde

a

Where

Coefficient

CL

,

C, Ja

,

Ct

Lq

,

of Attack

a

Angle

c

Reference Chord

Vt

Velocity Pitch Rate

q

CT

of Lift

5e

Elevator Deflection

5f

Flap Deflection

CjL*

,

cT

L8e

and

Constant

C,

'%

a

58

CT

Ldf

df

4.1.2

Drag Coefficient:

Equation

Coefficient

of

r

of

is

+r

=r

D

Drag

Do

represented

a +

Da

2Vt

by

CD

+

a

q+CD ^

q

Cn D

de

+

de

Cn D

)

a

df df

(4.2)

Where C

CDQ>

CDa

'

C Dq>

Coefficient

D

CD.

CDde'

a

of

Drag

Constants

CDdf

Coefficient OfLateral Force:

4.1.3

Equation

of Lateral

C

Force is

given

+C

=C

J3

by

CY

+

p+CYr

+

CV

da

+

YBa

Cv dr Ydr

Where

Cy

Coefficient

P

Angle

Yfi

,CY

P

sideslip

r

Yaw Rate

5a

Aileron Deflection

or

Rudder Deflection

,CY r

Constants

andC

,C

,C

lateral force

Roll Rate

P

CY Yo

of

of

rda

*dr

59

(4.3)

4.1.4

Rolling Moment Coefficient:

Equation

of

Coefficient

C 1

Rolling Moment is given by

of

=C

+C}

lo

lp

P+

Cj

-X-

2Vt

\.

C, 7

+

r

p+C} r

P

_

da + C,

dr

ldr

da

(4.4)

Where

C, C,

,C,

lo

lB

4.1.5

,Cj

and

,C,

,Cj

lr

1P

Rolling Moment Coefficient

Ida da

C,

Constants

7, ldr

Pitching Moment Coefficient:

Equation

of

Pitching Moment Coefficient is represented by

C

+C

=C

m

m0

a +

ma

CmqP+Cm,

2Vt \

+

a

C

de m

de

J

a

+

Cv df xdf

(4.5)

Where C

Coefficient

of

m

C

,C

mo

,

ma

Cm

,

mq

Cm m*

C

,

and

Pitching Moment

Constant

Cv

Ydf

mde

a

4.1.6

Yawing Moment Coefficient:

Equation

of

Coefficient

C n

of

Yawing Moment is

J3

+C

=C

nQ

np

C

+

2V{

by

given

p+C

r

nr j

np

+

da + C

C

nda

dr

ndr

(4.6) Where Coefficient

C

C

,C

nQ

,C

np

np

,C

nr

X

Constants

andC

nda

dr

60

of

Yawing Moment

4.1.7

Velocity:

Nonlinear Differential Equation

qsCD

V,=

of

Velocity is represented by

qsCy

-Tcosa

cos/? +

m

sin

B+

m

g[(cos 9 cos
0

4

m

j cos(P)PFr2^CD(S(e))5(g) 2

w

1

1

cos(P)pR

2 1

^CD(S(/))8(/)

2

cos(P)rcos(a)

p F?2ssin(p)C }'o w

/w

m

^p ^ssin(P) bCYpp

^

2P^25sin(P)CY(p)P

+

+

+

^

p Vt

s

sin^) b CYrr m

m

m

1pFr25sin(P)CY(5(a))5(a)

\

Vt2

p

s

sin(P)

Cy(6(r)) 5(r )

+ W

7W

+

g cos( P ) cos( 9 ) cos( (|)

)

sin( a )

-

g cos( P ) sin( 0 ) cos( a )

+ gcos(9) sin((j))sin(P)

(4.8)

61

4.1.8

Angle OfAttack:


alphadot

=

gR

+

q

-

of Attack of

Attack is

tan( P ) (/? cos( a )

+

by

qbarsRCL

sin( a ) )

+ r

(cos(8)cos((|))cos(a)

represented

Vt cos( P )

m

rSin(a)i? mVt

cos( P )

sin(0)sin(a))

Frcos(p)

(4.9)

Substituting the

equation

(4.1) in the

above equation and

4

wcos(P) cos(oc)

~

4 4

m

m

cos(p) +

cos(

~

p)

4mcosi$)

4

mgR

tan (

psRc

p)

r

4

CLalpkadM

m

cos(P) + p

iAmcos(:)

(4

+ psRcCLalphadot)Vt

4

+

+

psRcCLalphadol

cos(P)

+

sin(0) sin(a) p

sR c

CLalphadol)

Vt

m

cos(P) + p

s

R

c

CLautdot

2psRCH5{e))5je)Vt

psRcC^q

4mcosi^)

m

wgi?

2psRCUa)aVt

2p sRCLoVt 4mcosi^)

+ pSRcCLalphadot)Vt

4

cos(0) cos(

)

by

r sin(

-

)

(4.19) 4.1.14 Roll Angle:


of Roll

phidot

:=p

+

Angle is

represented

tan( 0 )

cos( )

r

+

by

tan( 0 ) q sin(

)

(4.20) Where

m

mass

P

rolling

q

pitch rate

r

yaw rate

q

Dynamic

rate

pressure

R

conversion

s

reference area

a

Angle

of attack

P

Angle

of

?

Roll

9

Pitch

Consider the

sideslip

angle

angle

equations

nonlinear matrix

factor

(4.8), (4.10), (4.12), (4.14), (4.16)

[14]

67

and

(4.18)

and solve

in the form

of

fej-

i

1

I 0 o

8

S _g

bo

CO

-%.

+ CO

8

O O

-H

co CO

O

co

O o

O co

o CO

o

o

o

U

OS

o

OS

Cl

O O

CL

o

o x\

05

O

t*

5

-x

ai

.1

^^t-

? bo +

I

^ c -H

cl

co

(N

+

QS 8

CO

QS

O O

^x +

co

O

St-

o

ca,

t-x ca.

o

CO

O O

bo

CO >t-

O O

+

CO

O o -In

+

I

OS 03

O O

ca.

03

^

CO

bo +

Qi

u OS

CN

On

CN

B x]

05

1.

0

:x

0=L 'co

0 O

s

u

OS

OS

os

-0

S

03

-Ci

0

05

05

ex

03

-x

Conclusion:

and

The

simulation

perfect

>

tracking

Controller Attack trim

>

of

the

the aircraft

under parametric

was able to achieve

from its trim

their initial condition

Keeping

Angle

Velocity from 650(ft/sec)

>

an

and

Angle

of

their

other parameters at

condition of

(4.24)

50

of

stabilizing the

also

(4.9)

seconds

Attack,

of

respectively

the

when

(4.1)

to

within

other parameters

from

(4.8), keeping Velocity

to

trim a

final

(4.16)

time,

condition and

value of

shows

which

even when

condition at

robust even when

the

changing the

600(ft/sec)

is the desired

both

and

that the transients

demonstrate that the transients

50 seconds,

from their trim

of

of

condition.

the parameters

Velocity

and

Angle

of

the same time.

(4.48)

that the

Sliding

Mode Controller

coefficients of system parameter are varied

nominal values.

Mode Controller is

Angle

of

4.1

and

deviation

626.818 to

from the figures (4.25) to

the system

Sliding

to

3.0

in the figures from

results within

are changed

10% from their

and

Velocity

their trim conditions.

initial

(4.17)

observed

makes

minimal

of attack and other parameters at

their desired

Attack

by

keeping

3.6103 to

respectively, the figures from

Figures from

It is

for

results

variation,

condition of

parameters are reached within

reach

>

variations.

the desired transients for the Angle

as shown

and other parameters at

>

desired

achieve

(initial) conditions.

was varied

the

different parametric

achieve the

the desired 50 seconds time with

>

Mode Controller's ability to

Sliding

and robustness under

designed to

was

Controller it

demonstrates

Attack

when

system and

successful

they

in tracking the desired

are varied

making the

for

Velocity

from their initial conditions,

and also

results

system robust under parametric variation.

107

Suggestions for Future Work:

> Test for stability

and robustness of

under variation of pitch rate

> Test

of

stability

parameters yaw rate

and robustness

i.e., for

for the

input

variation

in

the sliding

signal.

for

variation

angle

of

in lateral

sideslip,

unmanned air vehicle system

108

mode controller

is

moment

roll rate

and

recommended.

REFERENCES

Gene

[1] Control

of Dynamic

F.Franklin, J.David Powell Systems"

4

Edition,

Prentice

and

Abbas Emami-Naeini, "Feedback

Hall, Upper Saddle River, New Jersey.

[2]

MATLAB User's Guide, 1997.

[3]

Kuo, B.C., 1991, Automation Control Systems, 6th Edition, Prentice Hall,

Englewood Cliff, New Jersey.

Romane M. DeSantis "PID/

[4] speed

drives

Sliding

Mode Controller for the high accuracy

and position servos".

[5]

Nagy

N. Benjiamin "the

Sliding

mode

Control

of

a

two

link Robot

Manipulator".

Cem Unsal

[6] Antilock

Braking

and

System"

Pushkin

kachroo, "Sliding Mode Measurement Control for

IEEE Transactions

Control Systems Technology, March

on

1999.

Philips, CL.

[7] Hall,

Englewood

Using Sliding Control, Vol.

Estimation

H.T., Digital Control Systems, 2ndEdition, Prentice

Cliffs, New Jersey.

Slotine, J. J., E.

[8]

[9]

andNagle,

Surfaces

with

and sastry.

1983, "Tracking

Application to Robot

control of

Nonlinear Systems

Manipulator", International Journal

of

38,No.2,pp.465-492.

Slotine, J. J., E., Hedrick, J.K.

Using Sliding Observers",

Control, Athens, Greece,

and

Misawa, E.A., 1986, "Nonlinear State

Proceedings

pp.332-339.

109

25th

of

Conference

on

Decision

and

[10]

Slotine, J.J.,

E

and

Li, W., 1991, Applied Nonlinear Control, Prentice Hall,

Englewood Cliffs, New Jersey.

[11]

Zribi, M., Ahmad, S.

and

Sira-Ramirez, H, 1 994, "Dynamical Sliding Mode

Control Approach for Vertical Flight Regulation in Helicopters", IEEE Proceedings, Control

Theory

[12]

Applications, Vol.141,

and

Roberts

L.Woods, Kent

pp.

19-24.

L. Lawrence 1997,

"Modeling

and

Simulation Of

Dynamics Systems", Prentice Hall, Upper Saddle River, New Jersey.

[13]

R.D.

Young

and

U.Ozguner, "Variable Structure Systems, Sliding Mode

and

Nonlinear Control", Springer.

[14] John

Etkin

and

Reid, "Dynamics

of

Flight-Stability

Control"

3rd

and

Edition,

Wiley & Sons, Inc., New York, New York.

[15]

Roger W. Pratt, "Flight Control Systems-Practical Issues in Design

and

Implementation.

[16]

Elbrous M. Jafarov

and

Ramazan Tasaltin, "Robust

Sliding

for the Uncertain MIMO Aircraft F-18", IEEE Transactions On Aerospace

Systems, Vol. 36, No.4 October 2000, ppl 127-1 141

[17] Motor Drive on

Alenka without

Speed

Electronics

.

Sensor"

Proceeding

of the

1998 IEEE International Conference

Triestle, Italy 1-4 September 1998.

Xiao-Yun

Nonlinear Systems:

and

Control

Hren, Karel Jezernik, "Robust Sliding Mode Control Of Induction

Control Application

[18]

-Mode

Lu, Sarah K. Spurgeon, "Dynamic Sliding Mode Control for

Stability Analysis.

110

Jian-Xin Xu, Ya-Jun Pan

[19]

"

Tong-Heng Lee,

and

On the

Sliding

Mode

Control For DC Servo Mechanism In the Presence Of Unmodeled Dynamics", Triennial World

[20] Control

Workshop

Congress, Barcelona, Spain.

Spurgeon, S.K., Edward, E.,

Using on

15th

Foeter, N.P., "Robust Model Reference

and

A sliding Mode Controller/ Observer

Variable Structure Systems, VSS'96;

Scheme", 1996 IEEE

proceedings

December

International

5-6, 1996, Vol.

19,pp.36-41.

Yuri

[21]

B.

Shtessel

Manifolds", Vol.332B, No.6,

Using Sliding

Surfaces" ,

Y Liu

[23]

Electro hydraulic Servo

H.

.S.S.,

"Tracking

of

Methods"

Sliding

and

Control

Sliding

of

proceeding

Nonlinear Systems pp.465-492.

Mode Control to

an

on

Mode With Model

Sliding

pp.1221-1235.

pp.850-858.

Slotine, "Sliding Controller Design For Non-Linear

Controller

of the

of

Gopalswamy, "Nonlinear Flight Control

Vol.13, Issue: 5, 1990,

Makota Yokoyama, Karl

Dampers"

Vol.57, 1993,

Swaminathan

Jean Jacques And E.

Following Sliding Mode

Sliding

Flexible Mechanical Load".

International Journal Of Control,

[27]

Dynamic

via

Control, Vol. 38, 2 1983,

Handroos, "Applications

system with

J. Karl Hedrick

[26] Systems"

Sastry

International Journal Of Control,

[25] Design Via

Tracking

AJ.Fossard, "Helicopter Control Law Based

[24] Following"

and

International Journal

and

Output

pp.735-748.

Slotine J.J-. E

[22]

"Nonlinear

Vol.40, 2, 1984,

Hedrick

for

and

Semi-Active

Ill

421-434.

Shiegehiro Suspension

American Control conference,

pg 2652-2657.

pp.

Toyama, "A Model Systems

with

MR

Arlington, VAJune25-27, 2001

APPENDIX

112

E:

\tnesisl\independent_study\crassidis_datafile.m

July 31,

1

Page

2003

9:21:51

PM

APPENDIX-1

CODE

%

FOR

SECOND ORDER NONLINEAR

SYSTEM

DATAFILE

Inertial

m=2;

damping

c=0.5,

k=l

.

3

mass coefficient

,

stiffness

f=5.0,

nonlinear

b=2;

input

mmin=l;

minimum

deviation

of

mass

mmax=3;

maximum

deviation

of

mass

4 ;

cmin=0

.

cmax=0

.75;

parameter

gain

deviation

mininmum

of

damping coefficient damping coefficient

maximun

deviation

of

kmin=l;

minimum

deviation

of

stiffness

kmax=l

maximun

deviation

of

stiffness

minimum

deviation

of

f (t)

5;

maximun

deviation

of

f (t)

95;

minimum

deviation

of

input

gain

maximum

deviation

of

input

gain

.75;

fmin=4 ;

fmax=6 bmin=l bmax=2

.

.

.

2;

(mmin*mmax) A0

.

ccap=

(cmin*cmax) A0

.

kcap=

(kmin*kmax) A0

.

bcap=(bmin*bmax) A0

.

mcap=

mue=

(mmax/mmin) A0

gamma=

.

5;

estimated

inertial

5;

estimated

damping

5;

estimated

stiffness

5;

estimated

input

5;

(cmax/cmin) A0

.

5;

kbar=

(kmax/kmin) A0

beta=

(bmax/bmin) A0. 5;

dcap=

(fmax+fmin) /2

.

;

5;

(mmax/mmin) A0

.

(cmax/cmin) A0

.

kbar=roman

lamda=100;

time

D=abs (dcap-f ) ;

actual

n=0.10;

constant

coefficient

gain

5 5

k

(bmax/bmin) A0 nonlinear

mass

.

5

gain

constant systems

nonlinear

positive

value

gain

behaviour

rr LU _l

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o o HI

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