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故障检测

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1994 On the development of discontinuous observers 59(5).pdf 528KB

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This article was downloaded by: [Tongji University]

On: 15 April 2013, At: 23:46

Publisher: Taylor & Francis

Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,

37-41 Mortimer Street, London W1T 3JH, UK

International Journal of Control

Publication details, including instructions for authors and subscription information:

http://www.tandfonline.com/loi/tcon20

On the development of discontinuous observers

CHRISTOPHER EDWARDS

& SARAH K. SPURGEON

Control Systems Research, Department of Engineering, University of Leicester, Leicester,

LEI 7RH, U.K.

Version of record first published: 12 Mar 2007.

To cite this article: CHRISTOPHER EDWARDS & SARAH K. SPURGEON (1994): On the development of discontinuous observers,

International Journal of Control, 59:5, 1211-1229

To link to this article: http://dx.doi.org/10.1080/00207179408923128

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Any substantial or systematic

reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to

anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contents

will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should

be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims,

proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in

connection with or arising out of the use of this material.

INT.

CONTROL,

1994,

VOL.

59,

No.5,

1211-1229

On the development of discontinuous observers

CHRISTOPHER

EDWARDSt

and

SARAH

SPURGEONt

A framework is proposed for the development and

analysis

of robust observers

for uncertain dynamical

systems.

A variable structure

systems

approach is used.

Emphasis is placed upon the numerical tractability of the associated synthesis

procedure.

A selected numerical example is used to illustrate the proposed

algorithm.

1. Introduction

recent years, the problem of controlling uncertain dynamical systems

subject to external disturbances has

been

a topic of considerable interest. Many

of the proponents of the associated theoretical developments have found it

convenient to assume that the system state vector is available for use by the

control scheme. In practice, it is not always possible to measure the state vector;

in this case, either an alternative design method based solely upon the

output

information is required, or a suitable estimate of the state vector must be

constructed for use in the original control law. This

paper

considers the latter

approach.

From the point of view of robust control, the desirable properties of variable

structure control systems, particularly with a sliding mode,

are

well documented

(Utkin

1992,

DeCarlo

et al. 1988). Such controllers exhibit the property of total

invariance to all matched uncertainty when sliding. Further, in the presence of

only matched uncertainty, the system's dynamic behaviour when in the sliding

mode is wholly described by the surfaces in the state-space which

are

chosen to

specify the discontinuity of the control structure. Despite fruitful research and

development activity in the

area

of variable structure control theory, relatively

few authors have considered the application of the underlying principles to the

problem of estimator design. Utkin

(1981) presents a discontinuous observer

strategy whereby the

error

between the estimated and measured outputs is

forced to exhibit a sliding mode and measurement noise effects

are

reduced.

Dorling and

Zinober

(1983) explore the practical application of this observer to

an uncertain system and report difficulties in the selection of an appropriate

switched gain that will cause the system to exhibit a sliding mode without

excessive chattering.

should be noted that this study did not consider the

application of a continuous approximation to the discontinuous observer action

as traditionally used in the implementation of variable structure control systems.

Walcott

et al. (1987), Walcott and

Zak

(1986, 1988) and Zak and Walcott (1990)

use a Lyapunov-based approach to formulate an observer design which, under

appropriate assumptions, exhibits asymptotic state

error

decay in the presence of

bounded non-linearities/uncertainties. In particular, this method seeks to

render

the observer

error

system totally insensitive to matched uncertainty. This

Received

January

1993.

Control

Systems

Research,

Department of

Engineering,

University

Leicester,

Leicester

LEI 7RH, U.K.

Downloaded by [Tongji University] at 23:46 15 April 2013

1212

C. Edwards and S. K. Spurgeon

framework,

although

intuitively appealing, necessitates

the

use

algebraic

manipulation tools to solve

the

associated

constrained

Lyapunov

problem

effectively for systems

reasonable

order.

This

paper

seeks to build

upon

the

existing

contributions

outlined

above;

§ 2

provides a

detailed

review of

the

design

approaches

Utkin

(1992)

and

Walcott

and

Zak

(1986, 1988). In § 3, an

appropriate

methodology

presented

for

determining

the

magnitude of

the

discontinuous gain

required

the

Utkin

observer

ensure

the

existence

a sliding

mode

despite the

presence

a class

bounded

plant uncertainty. This uses a

Lyapunov

approach;

§ 4

further

develops

the

proposed

framework to

incorporate

the effects

bounded

matched

indeed

other

structured

system uncertainty in

the

design

procedure.

This

effectively considers

the

problem

first posed by Walcott &

Zak

(1986);

however,

here

the

emphasis is placed

upon

the

numerical tractability of

the

solution

method.

A numerical example considering

robust

observer

design for an

aerospace

system is

presented

in § 5.

Current

conceptions of discontinuous observers

Consider

the

uncertain dynamical system described by

x(t)

Ax(t)

Bu(t)

f(x,

u, t) }

yet)

Cx(t)

(1)

where

A E

IR"

X I

lR"

x m

C E

p x lI

where

p""

and

the

unknown

function

1R"

IR+

which

represents

the

system uncertainty, satisfies

Ilf(x, u,

1)11:;;;

p Vx E

u E

(Throughout

this

paper

11·11

refers to

the

euclidean

norm.)

In addition

the

matrices

Band

are

assumed to be

full

rank.

The

problem

to be

considered

that

reconstructing

the

state

variables using only

measured

output

information.

2.1. Utkin observer

Consider

initially

the

system described

above

under

the

added

assumptions

that

the

pair

(A,

C) is

observable

and

f(x,

u, r) sa

the

outputs

are to be

considered,

it is logical to effect a

change

coordinates

that

the

outputs

appear

components

the

states.

Without

loss

generality it can be

assumed

that

the

output

distribution matrix can be written as

]

(2)

where

p x

(n-

) ,

Px p

and

det

)

Consequently,

the

transform-

ation

(3)

TB =

[~~J

and

[In-

is non-singular

and

with respect to this new

coordinate

system it

can

seen

that

the

new

output

distribution matrix

CT-

= [0 I

] .

the

other

system

matrices

are

written as

TAT-

[All

Downloaded by [Tongji University] at 23:46 15 April 2013

Development

discontinuous observers

then

the

nominal

system

can

written

XI =

AllXI

+ A

l2y

+ e,« }

y =

A2\xI

+ Azzy + Bzu

where

[~IJ

= Tx

and

XI E

The

observer

proposed

Utkin

(1981) has

the

form

1213

(4)

(5)

j =

Azlxl

AnY

+ Bzu - v

where

(XI, y)

are

the

state

estimates for

(XI'

y),

L E

lR(n-p)xp

is a

constant

feedback

gain matrix

and

the discontinuous

vector

v is

defined

component-wise

Msgn(Yi

- Yi) for

IR+

(6)

where

sgn

represents

the

signum function.

the

error

between

the

estimates

and

the

true

states

are

written

el =XI - XI

and

=Y - Y

then

from (4)

and

(5)

the

following

error

system is

obtained:

Allel

+ A

l2e

} (7)

AZlel

+ Azze

- v

Since

the

pair

(A,

C) is

observable

the

pair

(All,

A2\)

is also

observable.

As a

consequence,

L can be chosen to

make

the

spectrum

All

lie in

the

open

left

half

complex

plane.

Define

further

change

coordinates

T, =

[\-p

and

let

[~iJ

T.[~jJ

Then

the

error

system with

respect

these

new

coordinates

can

written

ei =

Ailei

+ Aizey

Azlei

+ A

- v

(8)

(9)

where

Ail

All

zI,

Aiz

= A

Ai1L

and

A2z = A

AzIL.

can be

shown

using singular

perturbation

theory

that

for large

enough

M a

sliding

motion

can

induced

the

output

error

state in (9).

follows

that,

after

some

finite

time

ts> for all

subsequent

time, e

= 0

and

ey =

Equation

(8)

then

reduces

(10)

which by choice of

represents

a stable system

and

so ei

0 as t

00.

Consequently

and

the

remaining

states

can

constructed

the

original

coordinate

system as

(11)

The

major

practical difficulty in this

approach

the

selection

appropriate

Downloaded by [Tongji University] at 23:46 15 April 2013

1214 C.

Edwards

and

S. K. Spurgeon

gain M to induce a sliding motion in finite time. Dorling

and

Zinober

(1983)

report the need to modify

the

gain

parameter

M during

the

time interval in

order

to reduce excessive chattering; it should be

noted,

however,

that

their

study does

not

attempt

to investigate

the

effect of a continuous approximation to

v. In addition, the above analysis does

not

formally address robustness issues.

2.2. A modification to the Utkin observer

Consider the effect of adding a negative

output

error

feedback

term

to each

equation of the Utkin observer (5). This results in a new

error

system governed

ei =

Ailei

Aiz

Gle

(12)

AZlei

+ A:bey -

Gze

- v

By selecting G

Aiz

and

= A

A~z,

where

is any stable design

matrix of appropriate dimension,

then

(13)

In this form

the

(nominal)

error

system is asymptotically stable for v sa 0 because

the poles of

the

combined system are given by

a(Ail)

a(A~z)

and

so lie in

the

open

left half complex plane. In

the

original Utkin observer, the switching

action

v was potentially required to

make

the

error

system stable.

The

addition

of a

Luenberger

type gain matrix, feeding back

the

output

error,

yields

the

potential to provide robustness against certain classes of uncertainty using the

discontinuous portion

2.3.

Walcott

and

Zak

observer

The

problem considered by Walcott and

Zak

(1986, 1988) involves estimating

the states of a system such as

that

described in (1) so

that

the

error

tends to

zero exponentially despite the presence of matched uncertainty.

is assumed

that

f(x,

u, r) =

B;(x,

where the function

is unknown,

but

bounded,

that

(14)

Wx,

t)11.;;

p "Ix E

is also assumed that

there

exists

aGE

~nxp

such

that

= A -

has

stable eigenvalues and

there

exists a Lyapunov pair

(P,

Q) for A

such

that

the

structural constraint

is satisfied for some

F E

~mxp.

The

proposed observer has the form

.i =

G(Cx

- y) + v

where e = x- x

and

(15)

(16)

Downloaded by [Tongji University] at 23:46 15 April 2013

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