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596

IEEE

TRANSACTIONS

O N MJTOMATIC

CONTROL,

VOL. ac-16

NO.

6, DECEDER 1971

An

Introduction

to Observers

DAVID

G. LUENBERGER, SENOR

MEMBER,

IEEE

Abstract-Observerswhichapproximately reconstruct m i s s i n g

variable information necessary for control are presen ted n an

special topics of the identity observer, a

observer, linea r functional observers , stability prop-

iscussed.

I. INTRODUCTION

T IS

OFTEN

convenient when designing feedback

controI systems to assume nitially th at t.he entire

ate vector of the system to be controlled isavailable

ugh measurement. Thus for th e linear time-invariant

i t )

=

Ax( t )

+

Bu(t)

(1.1)

X

is an

n X

1 state vector, u is an

I’

X

1

input

A

is an n

X

n system matrix, and B is an n

X

T

bution matrix, one might, design a feedback law of

u t )

=

t x t ) , t ) which could be implemented if

mere availa.ble. This is, for example, precisely the

rol lam that resu1t.sfrom solution of a quadratic

optimization problem posed for the syst,em ( l) ,

from numerous other echniques ,hat, nsure

ilit,y and in ome sense improve system performance.

If the entirestate vectorcannot be measured, as is

u t ) = d r x t ) ,

t )

cannot be implemented.

us either nen- approachhat, direct.12: accounts

e nonavailabi1it.y of the entire state vector must be

sed, or a suitable approximation to the sta te vector

t be determined t,hat can be substitut,ed into t>hecon-

law. I n almost every situation the latt.er approach,

at of developing and using an approximate state vector,

vast ly simpler t han a new direct att,ack on the design

Adopting this point of view, t,ha t an approximate s tate

orhe unavailable state,

in the decomposit,ion of a control design problem

o tx o phases. The first phase is design of the control

vector is available. This may

or other design techniques and

a

control law without dynamics. The

is

t.he design of a syst.em th at produces an

pproximation to thestate vector. This syst.em, n-hich

deterministic setting is called an observer, or Luen-

distinguish i t from t,he Iialman filber,

R.

W.

Brockett,, Associate Guest Editor . This esearch

was

supported

Manuscript received July 21, 1971. Paper reconunended by

part by theNational Science Foundation under Gran tGK

16125.

ystems, School of Engineering, Stanford University, Stanford,

Theauthor is with he Depart,ment. of Engineering-Economic

Calif.

9430.5,

current.ly on leave t Office of Science and Technology,

Execut,ive Office of th e President,, Washington, D.C.

ha.s

as

its

inputs the inputs and available outputs of the

system whose stat e is to be approxima.ted and has a state

vector t,hat is linearly related to the desired approxima-

tion. The observer is a dynamic syst,em whose charac-

teristics are somewhat free

t

be determined by the de-

signer, and

it

is through its introduction that dyna.mics

ent.er the overall two-phase design procedure when the

entire state is not available.

The observer wa.s first proposed and developed in [ l ]

andfurther developed in [2]. Since theseearlypapers,

svhich concentrat,ed on observers for purely deterministic

continuous-time linear ime-invariant sy st em , observer

theory has been ext,ended

by

several researchers to include

time-va.rying systems,discrete ystems, and tochastic

systems [3]-[18]. The effect of an observeron ystem

performance (with respect to a quadratic cost functional)

has been examined [ 5 ] , [19]-[22]. New insight,s have been

obtained, and t.he theory has been substantially strea.m-

lined [23]-[25]. At. t.he same t.ime, since 1964, observers

have formed an integral part of numerous control system

designs of which a small percentage have been explicitly

report,ed [26]-[31]. The simplicity of i ts design and its

resolution of the difficulty imposed by missing measure-

mentsmake the observer an attract.ivegeneral design

component

[ F A ]

[32], [33].

In addition

to

their practical utilit,y, observers offer

a

unique heoretical ascination. The associated theory s

intimatelyelated t.0 the fundamentalinearystem

concepts of cont,rollability, observability, ynamic re-

sponse, andst.ability, and provides a simple set.ting in

which all of these concepts interact. This t,heoretical

richness has made t.he observer a n at,tractive area of

re-

search.

This paper discusses the basic elements of observer

design from an elementary ienToint.

For

simplicity

attention isrestricted, as in he earlypapers, to deter-

ministic

continuous-time

linear time-invariant systems.

The approa.ch t.aken in this paper, hon-ever, is influenced

substa,ntially by he simplification and insights derived

from the work of several other a.ut.hors during the past

seven years. In order t.o highlight t,he new t.echniques and

to provide the opport.unit,y for comparison with the

old, many of the example syst,ems presented in t,his paper

are the same as in the arlier papers.

11. BASIC HEORS

A . Almost any

System

is

an

Observer

Initially, consider t,he problem of observing aree

system

SI,

.e., a syst,em with zero input. If t,he available

output,s of this system are used as inputs

o

drive

anot,her

system Sz, t.he second system will almostalwaysserve

as an observer of the first system In that . its state will


2/7

LUENBERGER: AN INTRODUCTION T O OBSERS’ERS

597

and

Sz,

he observer, is governed by

i t )

= Fz t)

+

G g ( t )

2 .5 )

Fig. 1. A simple observer.

then H = G C. In designing the observer the

m

X n

matrix C is fixed and t.he n

X

m matrix G is arbitrary.

tend o t.ra.ck a linear tra.nsformat.ion of thestate of t,he Thus an ident,itv observer

is

determineduniquelyby

first, system (see Fig.

I ) .

This result forms t,he basis of selection

of

G a,nd takes t,he form

observer theory and explains Ivhy there is a great. deal of

freedom inhe design of an observer.

i t ) = ( A GC‘)z( t )+ G g ( t ) .2 . 6 )

z t ) = T x ( t )

+

eFf z(O) T x ( O ) ] .2 . 1 )

havearbit.rarydynamics if the original system s com-

pletely observable. First, recall t ha t a syst.em ( 2 . 4 ) is

Proof:

We may write immediately

. ,

completely observable if the matrix

Substituting TA FT = H t,his becomes has rank

n.

Generally, if an n

X

n mat.rix A andan

m

X n

i t ) T i t ) = F [ z ( t ) T x ( t ) ]

which has ( 2 . 1 )as a solution.

It should be notjed t,hat t.he two systems S and Sp

need not, have the mnle dimension. Also, it can be shown

[ l ] hat there is aunique solution T t,o theequation

TA FT = H if A and F have no common eigenvalues.

Thus a.ny syst>em

Sz

having different, eigenvalues from A

is an observer for SI n the ense of Theorem 1 .

Next, we n0t.e that he result of Theorem 1 for free

syst.ems can be easily extended to forced systems by

including the input. in t,he observer as well as the original

system. Thus if S is governed by

X t ) = A x ( t )

+

h t ) ( 2 . 2 )

a system Sp governed by

i t )

= Fz( t ) + H x ( ~ ) T B u ( t )2 . 3 )

\\-ill satisfy ( 2 . 1 ) .Therefore, an observer for

a.

system can

be designed by first assuming the system is free and then

incorporating the inputs asn ( 2 . 3 ) .

B .

Identity

Observer

An obviously convenient observer would be one in

which the tra.nsformation T relating t.he st.ate of the

observer to the st,ateof the original system is t,he identity

t.ransformat.ion. This requires t.hat t.he observer S, be of

the same dynamic order as he original syst.em S1 a,nd t.hat

(with

T

=

I )

F

= A H.

Specification of such an ob-

server rests therefore on specificat,ion of t,he matrix

H.

The matrix H is determined part.ly by the fixed out,put

structure of t,he original system and part ly by the input

structure of the observer. If SI, w3h an m-dimensiona,l

output, vectory: is governed by

i t )

=

A x ( t )2 .4 a )

=

C 4 t ) (2.4b)

matrix C satisfy ,his condition we shall say C, ) is

completely observable.

Lenanza

1: Corresponding to t,he real matrices C and A ,

then the set of eigenvalues of A GC can be made to

correspond to the set

of

eigenvalues of any n X n real

matrixbysuitable choice of the real mat,rix C if and

only if ( C , A )s completely observable.

This emma, which is now a cornerstone of linear

syst,em theory, was developed in severa.1 st,epsovera

period of nearlya decade. For the case

112

= 1 , corre-

sponding to single out,put systems, early statements can

be found in Kalman [ 3 4 ] and Luenberger

[ l ] ,

3 5 ] .The

general result s implicit.ly contained in Luenberger [ 2 ] ,

[ 3 6 ] ,and the problem is treabed definitively in Wonham

[ 3 7 ] . A nice proof is given byGopinath [ I . (It was

recent,ly pointed out to me t,hat, Popov [ 3 8 ] published

a

proof of a result of th is type in 1964.) Calcu1at)ion of the

appr0priat.e

G

ma.t,rix .0 achieve given eigenvalue place-

ment for a high-dimensional multivariablesystemcan,

however, be a difficult cornputrational chore.

The result. of this basic lemma translat,es directly into a

result on observers.

Th.eorem 2: An identity observer having arbitrary

dynamics can be designed for ainear t.ime-invariant

system if and only if the syet,em s completely observable.

In practice, the eigenvalues of tlhe observer a.re selected

t o be negative, so t.hat t.he state of the observer will

converge to t>he tate of the observed system, and t.hey are

chosen t,o be somewhat, more negative t.han the eigen-

values

of the observed system so t.hat convergence is

faster than other system effects. Theoretically, the eigen-

valuescan be moved arbitrarily t,oward minus infinity,

yielding ext.remely rapid convergence. This ends, how-

ever, t,o make the observer ac t like a differentiator and

t,hereby become highly sensitive to noise, and to introduce

other difficukies. The general problem of select,ing good


3/7

59

8 IEEE

T R A N S A C T I O N SN

-4UTOMATIC

C O N T R O L ,

DECEMBER 1971

w+l s+2

Fig. 2 . A second-order syst.em.

U

Fig.

3. Struct.ureof reduced-order observer.

eigenvalues is still not completely resolved but thepractice

of placing t,hem

SO

that he observer is slightly faster hanThe reduced-order observer was first ntroduced in

the rest of t,he (closed-loop) sgrstem seems t.0 be a good one.

[I].

The simple development present,ed in this section is

hasepresentation. We aga.in consider the system

Example:

Consider t,he system shown in Fig. 2. This due to Gopinath [251.

(2.7a)

X t)

= Ax ( t )

+ Bu(t )

y t )

=

CX t)

(3.2a)

(3.2b)

(2.7b)

and assume without loss of generality tha t the m outputs

of the system are linearly independent-or equiva.lentlg

An identity observer isdeterminedby specifying the

observer input vect,or

that the outputdistribution matrix C has rank-m. In this

case it can also be assumed, by possibly introducing a

cha.nge of coordina,tes, that . the matrix

C

akes the form

The resulting observer system matrix is

which

has corresponding characteristic equation

X + (3

+

g1)X + 2 + g1 + g2 = 0. 2.9)

Suppose we decide to make th e observer have two eigen-

valuesequal to -3 . This would give the charact.eristic

equation

(X

+ 3)2

=

X 2 + 6X +

9

=

0.

Matching coeffi-

cientsfrom 2.9)

yields g1 =

3, g2 = 4 he observer is thus

governed by

21 = [ - 54 - 1 1 1 2 1 1

2

+3 1-

111. REDUCED

IMEKSIOK

BSERVER

The ident,ity observer ahhough possessing an ample

measure of simplicity also possesses

a

certain degree of

redundancy. The redundancy &ems from the fact. that

while the observer const,ructs an estimat,e of the entire

st.a.te,part of t,he state asgiven by the system outputs are

already available by direct measurement. This redundancy

can be eliminat.ed and an observer of lower dimension but.

still of arbkrary dynamics can be constructed.

The basic construction of a reduced-order observer s

shown in Fig. 3. If y t ) is of dimension m an observer of

order

n

m is constructed with sta te z t ) t.hat, approxi-

mates

T x ( t )

for some

m

x n mat,rix T , as in Theorem 1 .

Then an estimate i t )

of

x t ) can be determined through

provided tha t he indicat,ed part.itioned mnt,rix is in-

ertible. Thus the

T

associat,ed with the observer must

have

n m

rows that are linearly independent of t.he

rows of

C.

C = [ZiO] .e., C is partitioned into a.n m X

m

identity

matrix and an

m

X

n

mn)

zero matrix.

A n

appropriate

change of coordinates is obtained by selecting an n m)

x n mat.rixD in such a. way that

M

=

[

is nonsingular and using the variables X = M x . ) It is then

convenient to partit.ion t.he state vector as

x = y ]

W

and accordingly writmehe syst.em n the orm

k(t)

=

Al l y ( t )

+

A l d t ) + Blu(t ) (3.3a)

G(t ) =

A a s t ) +

A 2 2 ~ l ( t )+ B 2 ~ ( t ) . (3.3b)

The idea of the construction is thenas ollow.The

vect.or y t) is available for measurement, and

if

n-e dif-

ferentiate it., so is k t ) . Since

u t)

is also measureable

3.3a) provides t.he measurement ~ ( i ) or t,he syst.em

(3.3b) u-hich has state vector w ( t ) and nput.

Aely(t)

+

B2u( t ) .An ident,ity observer of order n m s constructed

for (3.3b) using this measurement. Later t>henecessity to

different,iat,e

is

circumvent,ed.

The justifica6ion of t.he const.ruct.ion

is

based on the

following lemma [%I.

Lemma. 2: If C,

A )

is completely observable, then so is

( A n ,

Am).

The validitmyf t.his stat.ement is, in view of the preceding

discussion, int-uitivelylear.

It

ran be asily proved direct.ly

by applying the definition of comp1et.e observability.

To construct the observer we initially define it in the

form

h t)

= A22

LAn)zir(t)

+

A P I Y ( ~ ) B 2 ~ ( t )

+ L ( m

A d t ) ) LBlU(t) .

(3.4)

In view

of

Lemmas

1

and 2, L can be selected

so

that


4/7

LUENBERGER: AN INTRODUCTION TO OBSERVERS

599

Y Y

U

82 -LE,

W

Fig. 4. Reduced-order observer using derivative.

Y

Fig. 5 . R.educed-order observer.

Azz EAzl has arbi trary eigenvalues. The configuration

of this observer is shown in Fig. 4.

The requireddifferentiation of

y

can be avoided by

modifying the block diagram of Fig. 4 o that of Fig. 5 ,

which is equivalent at th epoint, ~. This yields the desired

final form of t,he observer, which can be written

i ( t ) = ( A z z

LAlz)z( t )

+

(An LAlz)~%(t )

+ (Azl LAll)y(t) + (Bz LBl )u( t ) 3.5)

with

z t )

= G ( t )

5 / t ) . 3.6)

For

this

observer T = [-LIZ]. This const,ruction enables

us

60

st,ate the ollowing theorem.

Theorem 3: Corresponding t.0 an nth-order completely

controllableinearime-invariantystemaving m

linearly ndependent outputs astat.e observer of order

n m ca.n

be

constructed having arbit.rary eigenvalues.

It is important to understand that the explicit. form of

the reduced-order observer given here, obt,ained by par-

tiOioning the system, is o d y one way to find the observer.

In a.ny specific instance, ot,her techniques such as trans-

forming to canonical form or simply hypot,hesizing the

general structure and solving for the unknown parameters

may be algebraically simpler. Theorem

3

guarantees that,

suchmethods will always yield an appropriate result,.

The preceding method used in the derivation is, of course,

often a. convenient one.

Example:

Consider the syst,em shown in Fig.

2

and

t,reated in the exa.mple of Sect,ion

11.

This s a second-

Fig.

6.

Observer for second-order system.

order system with a. single output so a first-order observer

with an arbitmry eigenvalue can be constructed. The C

matrix already has he required form,

C

= 1 0. In this

case A22

GAB

=

- 1

G , which gives t.he eigenvalue

of the observer. Let. us select G

=

2 so tha t the observer

will havets eigenvalue equal t.0 -3 . The resulting

observer a.ttached o the ystem is shown in Fig. 6.

I IT

BSERVING

A

SINGLE

LINEARUNCTIONAL

For some applications an estimate of a single 1inea.r

functional E = a‘x of the st,at,e s a.11 t,hat is equired.

For example, a linear t,ime-invaria,nt, control law for a

single input, syst?em s by definition determined simply by a

linear functional of the system stat.e. The quest,ion arises

then as 60 m-het.her a less complex observer can be con-

structed to yield an estimate of

a

given linear functional

tha.n an observer tha t estimat,es t.he entire st.ate.Of course,

again, it is desired to have freedom in the selection of the

eigenvalues of the observer.

A

major result for bhis problem

[ 2 ]

is that . any given

linear functiona,l of the sta te, say,

E = Q ’ X ,

can be esti-

mat.ed m&h an observer having v

1

arbitrary eigenvalues.

Here v is the

obsemability

index [ 2 ] defined as the least

positive integer for which the matrix

[C’IA’C‘;(Ar)2C’i . . ( A r ) y - l C r ]

has rank n. Since for any completely observa.ble system

v

1

5

n m

and

for

many systems v

1

is far less

than n m, observing a single linear functional of t,he

stat.e may be f a r simpler than observing the entire &ate

vector.

Th e genera.1 form of the observer is exactly analogous

to a

reduced-order observer for the entire state vector.

The estimate of E = a’x is defined by

2 t ) = b’y t) + c’z t ) 4-1)

i t )

= Fz t )

+ H x ( t )

+

T B u ( t )

(4.2)

where

F

H ,

T , B

a.reas in Sect,ion

11-A

and where b and

c

are vectors satisfying

b’C

+ c’T = a’.

Again the mportant, result is that the observer need

only haveorder

v

1.

The precise design technique is

dict,ated by considerations of convenience.

We illust.rate the general result 1vit.h a single example.

The method used in t.his example ca,n, however, be applied

to any multivariable system.


5/7

600

IEEE

TRANS.4CTIONS ON AUTOXL4TIC COSTRO L, DECEM BER 1971

I

Fig.

7.

A fourth-order

system.

XI 2

u

x 3

Fig.

8. Funct,ional observer.

Example: Consider t,he fourt,h-order systemshown n

Fig. 7. This system ni th available measurements

x1

a.nd

x

has observability index

2.

Thus any linear functional

can be observed T1-it.h a first-order observer. Let us decide

t o

construct an observer with a single eigenvalue equal to

-3 to observe the functionalxz

+

x. .

Initially neglecting thenput

LC

we hypothesize an

observer of t he form

i

= -32 + 91x1+ 93x3.

According to Theorem

1

his has an associated

T

satisfying

r 2 1 01

T I ; ; ;

:]

+ 3T = 9 0

93

p 4.3)

-1

0 0 0

If

T

=

tl

tz t3

t4 ,

we would like

tz

=

1,

t 4

=

1.

Sub-

st,itut.ing hese values n

4.3)

we obtain the equation

unknowns tl,

g1? t3, g3.

tl = ,

t3

= ,

g1

=

-2,

g3

= .

this t,he final observer shonn in Fig.

8

is deduced by

V.

CLOSED-LOOP

ROPERTIES

Once an observerha.sbeen construct.ed for a 1inea.r

m which produces an estimate of t,he state vector

or

f

a

linear transformat.ion of the stat.e vector it is impor-

t, to consider the effect induced by using this estimate

of t.he true value called for by a cont,rol lau-. Of

importance in this respect is the effect, of an

on the closed-loop stability properbies

of

t.he

It

would be undesirable, for example, if an other-

designbeca.me unsta.ble when it was

ntroduction of an observer. Observers,

introduced. In this section we show that if linear

control aw is realized u-ith an observer,

of

the srst,em are t.hnse

of

t h e

observer it,self and hose ,hxt mouldbe obtained if the

control lax- could be directly implemented. Thus an ob-

server does not change the closed-loopeigenvalues

of

a

design but merely adjoins its

own

eigenvalues. Similar

results hold for systems with non1inea.r control

laws

[ a ]

Suppose we have the system

X t)

= Ax( t )

+

Bu( t )

(5.la)

and the control law

If

it were possible t,o realize this control law by use of

availablemeasurements (whichwoulde possible if

K

= RC

for some R ) , then the closed-loop system would

be governed by

X t) =

( A + B K ) x ( t ) ( 5 . 3 )

and hence its eigenvalueswould be the eigenvalues of

A

+

B K .

Now

if the control cannot be realized directJy, an ob-

server of the form

i t )

= Fz( t ) + G p ( t ) + T B u ( t )

(5.4a)

~ t )

Ki t) = Ez t )

+

Dg( t )

(5.4b)

where

TA FT

=

G C

(5.5a)

= E T + D C

(5.5b)

canbeconstructed. From ourprevious theory

( C ,A )

completely observable s sufficient. for t.here to be

G ,

E , D F , T

satisfying

(5.5)

with

F

having arbit,rary eigen-

values. Sett.ing

u t ) = K f t )

eads to the composit,esystem

;]

= [

A

+ BDC

G C + T B D C F

+

T B E z

E

] 1. ( 5 . 6 )

This whole structurecan be simplified by ntroducing

= z T x

and using

x

and as coordinates. Then

(5.6)

becomes, using (5.5)

(5.7)

Thus

the eigenvalues

of

the composite system arc those

of

A

+

BK

and of

F.

me note that in view of Lemma.

1

(applied in its dual

form) if the syst,em (5.1) is completely controllable i t is

possible t,o select.

K t.0

place the closed-loopeigenvalues

arbit,rarilg.

If

this control Ian- is not. realizablebut, the

system scomplet.dyobservable, an observer (of some

order

no

great,er than

n m )

can be const.ruct,ed

so

that

t,hecontrol law can be estimated. Sincc t,heeigenvalues

of the observer are also arbitrary the eigenvalues of the

completecomposite system may beseltct,edarbitrarily.

We thereforet.ate t.he following important result of

linear syst,em theory [I ],

[ 2 ] .

- -

~ ~ ~

~~~ - _ -

----

Theorem 4:

Corresponding tonth order completely


6/7

a.nd hence t,he plant follows the free system. This tracking

property can be used t.o definea closed-loop system for the

plant.

Rather than fix a.t,t,entionon t,he fact that only cert,ain

Fig. 9. A general servodesign. outputs of thelantre available, we concentratmen the

fact. that only certain inputs,as defined by

B ,

are availa.ble.

If we ha.d comp1et.e freedom as to where input.s could be

Fig. 10. Compensator for example.

controllable and completely observable system (5.1)

having m linearly independent out.puts, a dynamic feed-

back syst.em of order n m ca,n be constructed such that,

the 2n eigenvalues of the composite syst,em take ang

preassigned values.

Alt.hough this eigenvalue result for linear t.ime-invariant,

systems is of great theoretical interest, it. should be kept

in mind t,hat. ,he more general key result is that. stabilitg

is not af€ect,edby a (stable) observer. Thus even for non-

linear or t,ime-varying cont,rol laws an observer can supply

a suit,able estimate.

Example: Suppose a feedback cont.rol spst,em

is

t.0 be

designed for the syst.enl shown in Fig. 2 so t.hat its output

closely tracks a dist,urbance input d. The general form of

design is shown in Fig. 9.

For

the particular syst,em shon-n in Fig. 2 let us decide

to design a control law that places the eigenvalues at.

1

i. It is easily found t,hat u

=

x +

x2

will ac-

complish this. If this ~ R Ws implemented with t,he first.-

order observer construct,ed earlier, we obtain the overall

system shown in Fig. 10, which can be verified t,o have

eigenvalues

,

+

i -i.

VI.

DUAL

BSERVERS

The funda.menta1 propert): of one syst.em observing

another can be applied in a reverse direction to obtain a

special kind of controller. Such a c~nt~ro ller ,alled a dual

observer, wa.s introduced by Brasch [33].

Suppose in Fig. 1 he systenl S2 s the given system and

SI is a syst,em tha.t we construct t,o control S2. MTe have

shown that the system SZ .ends to follow SI and hence

SI

ca,n be considered as governing the behavior

of

S,.

To make tshisdiscussion specific suppose t,he plant

X t) = Ax ( t )

+

Bu( t ) (6.la)

y t ) = Cx( t ) (6.lb)

is driven by the free syst,em

i t )

= Fz( t ) (6.2a.)

~ t ) J z ( t ) (6.2b)

where

AP PF = BJ

for some

P .

Then from Theorem 1

we see that in thiscase the vector n = x + P z is governed

by

the equa.tion

n t ) = A n ( t )

supplied, t.he output, imitation would not much matter .

Indeed, if t,he out,put

y t )

=

Cx( t )

could be fed

t.0

t.he

system in the orm

i t ) =

A x @ )+

L y ( t )

03-31

t.hen t,he eigenvalues of the system would be the eigen-

Iralues of A

+

LC.

By

Lemma 1, if the s:stem is observ-

able

L

can be selected to place the eigenvalues arbitrarily.

Thedual observer can be thought of assupplying an

approximation to the desired inputs.

To achieve t,he desired result we construct, thedual

observer in the form

z ( t ) =

Fz( t )

+

M w ( t )

(6.4a.)

u t )

= y t )

+

C P z ( t )

(6.4b)

u t) = J z ( t )

+

N w ( t ) (6.4~)

where

AP PF = BJ (6.h)

L = P M + B N . (6.5b)

Equations (6.5) are dual o (5.5) and will have solution

J , MI N , F wit.h F having arbi t,rary eigenvalues if (6.1)

is conlpletely controllable.

The composite system is

A

+

BNC BJ + BNCP

x .

(6.6)

41

=

[ M C F

+

M C P ] ]

Introducing n = x +

P z

and using

z

and

n

for coordinates

yields the comp0sit.e syst,em

i

[ +

LC

01 ]

MC

F z

which is the dual

of

5.7). The eigenvalues of the com-

posite system a,re thus seen to be the eigenvalues of A

+

LC

and the eigenvalues of

F .

We may therefore st,at.e he

dual of Theorem 4.

Th.eorm

5:

Corresponding

to

an nth-order completely

controllable and complehely observableystem (6.1)

having T linearly ndependent. nputs,adynamic feed-

back system of order n r can be constructed such t,hat

the

2n

eigenvalues of t.he composite system t,alce any

preassigned values.

1711.

CONCLUSIOKS

It

has

been shown t.hat mising &ate-variable infor-

mation, not available for measurement,, ca.n besuit,ably

approximated by an observer. Generally, as more output,

variables aremade available] t.he requiredorder of the

observer is decreased.


7/7

IEEE

TRANSACTIONS ON

AUTOMATIC CONTROL, DECEMBER

1971

Although the introductory reat,ment given in his [281

P. V.

Nadezhdin, The optimal control law

in

problanswith

arbi trary nit ial conditions,’’

Eng. Cybern.

USSR),pp.

170

174. 1968.

a.per is restricted o ime-invariantdeterministic

con-

syst,ems, much of the t,heory can be [291

E E.

‘‘APPlicat,ion of observers andopthum

filters to

t.0

more general sit.uations. The references cited

inertial systems,” presented at. the I FAC Symp. Multivariable

Cont,rol SF tems , Dusseldorf, Germany, 1968.

this paper should be consulted for these extensions. [30] D.Q : Xayne and,:. Murdock,“Modalcontrol of linear time

mvarlant systems,

Int. J . Contr.,

vol.

11,

no. 2 pp. 223-227,

1970.

REFERENCES

111 D.

G.

Luenberger, “Observing the st.ate of a inear system,”

IE EE Trans.

X i l ,

Electron., vol. MIL-8, pp. ?&SO Apr. 1964.

[2] ---,“Observers formultivariableystems, IEEE Trans.

[3]

M.

Aoki and J. R. Huddle, “Est,imation of stat e vector of a

dubma t. Contr., vol.

AC-11,

pp. 190-197, Apr. 1966.

linear stochastic syst.em wit.h a constrained estimator,” IEEE

Trans. Automat. Contr.

(Short. Papers), vol. AC-12, p p. 432-

433.

A I I P .

1967.

231

W. A. Wolovich, “On state estimation of observable systems,”

in 1968 Joint Automatic Control Conf., Preprints, pp. 210-220.

J. J.

Bongiorno and D. C. Youla, “On observers in multi-

variablecontrol ystems,”

Int .J. Contr.,

vol.

8,

no. 3, pp.

H. F. Williams: “A solution of the multivariable observer for

linearime arying discrete syst.ems,” R e . 2nd Asilomar

Conf.Circuits and Systems,

pp . 124-129, 1968.

F.

Dellon and P.

E.

Sarachik,“Optimal control of unstable

linear plants wit.h inaccwible state,, ”

IEEE Tran.s. Automat.

Contr., vol. AC-13, pp. 491-495, Oct. 1968.

a unified theory

of minimum order observers, 3rd Hawaii

R. H. Ash and I.

Lee, “State

est.imation in linyr systems-

Sys. Conf.,

Jan. 1970.

G. W Johnson, “A deterministic heory of estimation and

control,” in 1969

JointAutonatic Control

Conf.,

Preprints,

pp . 155-160; also in IEEE Trans. Automat. Contr. (Short

Papers), vol. AC-14, pp. 380-384, Aug. 1969.

E.

Tseand 33. Athans, Optimal minimal-order observer-

estimators ordiscrete inear the -va ryi ng systems,” IEEE

K. G. Brammer, “Lower order opt,imal linear filtering of non-

Trans. Automat. Contr., vol. AC-1.5, pp. 416426, -4ug. 1970.

statlonary ra.ndom sequences,” IEEE Trans. Automat. Contr.

S D. G. Cumming, “Design of observers of reduced dynamics,”

(Corresp.), vol. AC-13, pp. 198-199, Apr. 1968.

Electron. Left., vol. 5, pp . 213-214, 1969.

multivariable cont.rol systems,” presented at the Nat. lectron.

R . T. K. Chen, “On the construction of st ate observers n

Conf., Dec. 8-10, 1969.

L. Novak, The design of anoptimal observer or linear

discrete-t,ime dynamical systems,” n

Rec. 4th dsilomar Conf.

Circuits and Systems, 1970.

Y.

0.

Yiiksel and

J. J.

Bongiorno, “Observers for linear multi-

variable systems Kith applications,” t,hk issue, pp. 603-613.

B. E Bona, “Designing observers for time-varying st.ate

systems,“ n Rec.4thAsi1oma.r Conf. Circuits a.nd Systems,

1970.

A. K. Newman, “Observing nonlinear he-varying systems,”

IE EE Trans. Automat. Contr.,

to be published.

M .

31.Kewmann,

“A

continuous-t.ime reduced-order filter for

Int. J . Contr.,

vol. 11, no. 2, pp . 229-239, 1970.

estimating he tate vector

of

a inear stochast.ic system,”

V.

V.

S. Sarma nd B. L. Deekshatulu, Optimal cont.rol

when some of the s tate variables are not measurable,” Int.

J .

B. Porterand

>,I.

A. Woodhead, “Performance of opt imal

Contr., vol.

7,

no. 3,pp . 251-256, 1968.

measurable,” Int. J . Contr., vol. 8, no. 2, pp. 191-195, 1968.

control systems when some of the tate variables are not

AI. 11.Newman, “Optimal and sub-optimal control using an

observer when some of the statevariables are not measurable,”

I. G .

Sarnla and C. Jayaraj, “On the use of observers in finite-

time opt imal regulator problems,” Int. J . Contr., vol. 11, no. 3,

~I ---- -

221-243, 1968.

h t . J.CWtr., VOl. 9, pp . 281-290, 1969.

DD. 489-497. 1970.

IEEE Trans. Autontat. Conk. (Corresp.), vol. 4C-15, pp.

W .

&I.

Wonham, “Dynamic observers-geometric theory,”

238-259. Am. 1970.

regulator logic using state-variable concepts,’’

Proc. IEEE,

4. . Brygon,-Jr., and D. G . Luenberger, “The synt.hesis of

vol. 58, pp. 1SO3-1811, Nov. 1970.

B.

Gopinat.h, “On the control of linear multiple inpubo utput .

systems,’’

Bell Syst. Tech. J . ,

Mar. 1971.

1). K. Frederick, and G. F. Franklin, ‘‘Design of piecewise-

linear swit.ching functions or elay control systems,”

IEEETrans. Automat. Contr.,

vol. AC-12, pp. 380-387, Bug. 1967.

Inform. Contr.,

vol.

13, pp. 316-353,

1968.

J. D. Simon and S K. AIitter, “A theory of modal control,”

[31] A. M. Foster and

P.

A. Orner, “A design procedure for a

class

of

distributedparameter control syst.ems,” ASMEPaper

70-

WA/Aut-6.

[32]

D. &I.

Wiberg,

Schau.m’s Outline S e ~ e s : tate Space and Linear

Systems. New York: McGraw-Hill, 1971, ch. 8

[33]

F.

M.Brasch, Jr., “Compensators, observers and controllers,”

to

be published.

[34] R.

E.

Kalman, “Mat.hematica1 descript.ion of linear dynamical

systems,” SIAX SOC. nd.Appl.Nath.) J. A p p l . Math.,

ser. A, vol. 1,pp . 15 192, 1963.

[35] D.

G.

Luenberger, “Invert ible solutions to the operator equa-

tion

T A

BT = C,” in Proc. Am. Math. Soc., vol. 16, no. 6,

[36] --,“Canonical forms for linearmultivariable system,”

IEEE

Trans.

Automat. Contr.

(Short Papers), vol. AC-12,

pp . 290-293, June 196i.

[37]

R . X.

Wonham, “On pole assignment

in

mult.i-input control-

lable linearsystem,’’

IEEE Tra.ns.Automat. Contr.,

vol. AC-12,

[38] 5’. hI. Popov, Hyperst.abilityandoptimality of automatic

pp. 660-665, Dec. 1967.

systems wit,h several control functions,”

Rev.

Roum.

Sci.

Tech.,

Ser . Electrotechn. Energ.,

vol. 9, pp. 629-690, 1964.

pp . 1226 1229, 1965.

David G.

Luenberger (S’57-kI’@-S-MJ7l) was

born in Los Angeles, Cali. , on September 16,

1937. He received the

B.S.

degree from the

California Inst,it.ute of Technology, Pas-

adena, in 1959 andhe

M.S.

and PbD.

degrees from Stanfordniversity, Palo

Alto, Calif., in 1961 and 1963, respectively,

all in electricalengineering.

Since 1963 he has been on t.he faculty of

Stanford Universit,y, where presently he is a

Professor of Engineering-Economic Systems

and of Electrical Engineering. He

is

also d i a t e d with the Depart-

ment of Operations Research.

His

activit.ies have been centered pri-

marily in t,he graduate program, where he has t,aught ourses in opti-

mization, control, mathematical programming, and information

theory. His research areas have included observability of linear sys-

tems, the applicat.ion of functional analysis t.o engineering problems,

optimal cont.ro1, mathematical programming, and optimal planning.

His experience includes summer employment a t Hughes Aircraft

Company and Westinghouse ResearchLaboratories from 1959 to

1963, and service as a consultant

to

West.inghouse, StanfordRe-

search Institute, Wolf Management Semices, and Inta sa, Inc.

This

experience has included work on the control of a large power generat-

ing plant, he numerical solution of part id differential equations,

optimization of trajectories,optimal planning problems, and nu-

merous additionalproblems of optimization and control. He has

helped formdate and solve problems of control, optimization, and

general analysis relating t.0 electric power, defense, water resource

management., telecommunications, air traffic control, education, and

economic planning. He is the author of Optimization

b y

Vector Space

Xethods (Wiley, 1969) and has authored or co-authored over 35

technical papers. Current lyhe

is

on eave from Stanford

at

the

office of Science and Technology, Executive Office of the President,

Washingt.on, D.C.He assist,s th e Science Adviser -with program

planning, review and evaluation, and with formulation of science

policy in areas of civilian technology.

Dr. Luenberger s

a

member of Sigma Xi, TauBetaPi, he

Society for Industrialand Applied Mathematics, he Operations

Research Society of America, and heManagement Science In-

stitute. He served

as

Chairman (Associate Editor) of the Linear

from 1969 to 1971.

Systems Comnlit.tee of the

IEEE

Group on Automatic Cont,rol

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