ONTKOL LAP]L1TY EKVAPILIY

ROLLABIL11Y
ONT

peeale
it eale
destd
wtma be
1 ay lal
Slale testsola bM e
Ceat ool sleu

teatrelled u tai Ceatra

u independesl o
da be slal weable vaxiobard

al. f any Gk dali

ig b Leatvl
deatrel
d ipeceibe
ek
ittd
aCsatrellablo
ignal,
syslea
-T1ME
ta LTI
ÞIECRETE

erejore
FOR AN YTE
CONTKOLLABI
LITY
STATE
dsues
P l E T E
tsatrél ysle
d i s e n e l e - t u e

i
asider
x(k+)T:
4x(KT) t HUCKT deupletely s l a l i teatrsliable
tsntrslabMe
slali
deuplenly
sad b be Apizd
auu

l
i toabrel sigoal vkT)
s derdaE
stariu r s u
stariup
frsu
h
a at
t
apieee- &uc
nadS,Suchi a t e n
mads,
bhero exisli
SAplu b ausjened
anule Nser o 0ug be t
A a
NMer slali
sal
xkT)
xíkt)
inslali.
inslali
slali , meat
mect
nn tsamuplap
aaplas
i n t a l

nn a
att
alali X
desired
HUC

xnT)=
Gx(o) + +GHU-T

G Hu-)T|

| H G H ;
GH: Un-2T
x(o)=
x n T G

2
H:GH: GH hee exal
axbeansla
xhT) X
an
 eLOA uAbeuadtd Leslnel
igols 0fo),ufT) un-1)T
al Caw
slal reA e al sli x) r uc
fnal stalk x(nT)

HCAe e NtLa aad

uitust terdilse slal
tsatnellab luly hal

&esBe b baaala d l stal t aa

A b c u sla un at mout n Saip eneds oal
slati
NAde eD(KT) buLNded e mognid e olkT)
ma take
more Mhan
eunk, B
FORM OF THE CONDI110 N FOR COMPLE TE STATE
ALTERNATJVE

CONTROLLABILITY

depaed
xkt)TGx(«T) +HuCkT)
ane dislnet,liun
it u pesei bla
ualuet 6
J iga uLer
at
natrix M,
an alen
hAna
0O 0
MM

Lo o 0

d slal
RtdT à(k) tHVer)
MH
MGM
ad H
ort
M

e o e uat ix H
elkmenli

an

| ro
 al zord hen e ComWpERdag slale uaka le arnal e egalrekad
h aMeanatue tesdilen lor clale coatoltlokiks
an 0;(KT), Heade
edoteauned b Gtberli Tet gues as
ete
C N

L e E R TT E S T

reu MH
glali
csstnslaldle y 0d ealyno
Coupluly
elemenl
u s t be
al
erO

lat, ue natis MGM

has
ordesb
t6m
diagenal

in dislat
t Gare net
Vakue 8 afgnee eigen values,
d~ckat
vauat,
eig.en
eqer haue
doesnt
atrix
1e peesie at
e t

u Sueh
diagenalxalhan Caanenical

Tonda
a
e haabued Lalo
e i
(kT)SH
UKT)
sGS laanBaital fon
Ca +1)T
ordan
is
SGS J
whore

welln u t

F
bloek
leek
Tordan
mdasv
lut e
o
No
No nal Loes perds
Sae ci eigen valuCs e SH
e l e a e n l s
ef a ae
re
all zeao,
and
blcek
Toda
SH
S s a t comeseud
(
L a r w

e 78

eleanerli
ac t
ret
aaW xOre
are
a u e

in
113) esgen
distet

l
MPLE:

oleuag Seas
are Coapletely slale eostroable
H

1 )1 o
0
,kr) 20-2 0 0
)
o -5 xtw

0
slal
tsapletely
not
axe
ens

ellen
Cpntrellable

o
a k t )

0 3
+o ()
-2 , O
-2

akt) 0
-5 oo
0
le
e7-plaae plaae
Csatrstiabi&L
C s a t r a l l a b i l i

S
S lla
allui
Csusicli lily lau b
Coaylcle
glal eCsabrelabi
luew
40% letsslli

Ce aa Cadils a

aafer
a s e
funet
r a e t a .

sul
ulae
toaplli
tsadaus
fer
AOCLsLaRY
tess o aad guhhieet
u e i e l

ele- zee loaellaluan

e l a - z e

slalee
ho
at
i fuactsu.
tantrsllalily

Sali a all
pule
CCus
e
Caaoled m
m ae
dde
t a a e s l e s

directuan
u
Bie
 Coer olieaL pule raerkaciew.
Z+0 2

(zt09)(z+0-2).
otteust bel
Canrellallon U e aeo (Z+0)
AeraA ano dangunalor.

pulse reasesfunckeu
0or slale 2oauSAS,
Alalí taa 0tule

x,(kt

M:GH -0.8 6
0 64

S |HGH 0,

reak o H:GH]#
HEACe Syale L6Ceatve{lalke
n CeatvelLabAe

UT CONTROLLABJLITY:
C O N T R O L L A B J L I T Y :

CoMFLE1E
OUTPU
T

l e a t n e l

saat

MO be &epleta
bde apletoly
-
tene ualeu s a u d o
seele csuust
kacsie o eAlruet
Ad &G esike
A finle
o(KT), dahined
a
coatnellabl,

O(KT) dained e v e
A a l

oulpul g*al
taatos sue a
, Suew nat n T
u t o n s h e u e d

eoda
penods
0O4kT
kT2 4 nT ,

Saapl
umber a rt i a l
dagir valud Me(abasax) n l e ta
vald
startinp ransesned
6 a dasired
ta e goglunp emicde)
meat n
 atix

D:CH
CGH: CGH CG
H m
M mengisral
8ere
ulpul eele

O B S E R V AB J L I T Y :

dicrele-ue Caatnel sean guwe by

Cansider
4x(kr)
X(xt) T
=

e (sapliTol shseouable
Sald ehseavatim

sylem determuined rs e
da e
(0 dhe aysleus
a a i peaiads.
s0plup ensds
slale
numker &4 s
el
eHinbal

fuk
GAAr
a
ehzeuable
Cuey raAnls
MKT)
ådepletely

slal
tnneasunalle s l a vanables
e ca
Tecennuelunp
T e C e u e l a p

toatrel
Leatral
obluw
blzav feaabaek
f cabek
re slate
alu
S l.
Aanhculandy f r l e - iunne
-n e Ssleas
paiculanky
sbicrde

Esesualulely

x+)T Gx(kT).
C o u p e k e

seluben abeue apaton

x(KT xo)

eas
e a s s a t ,
at gwe y(0), 4CT)
gwen
baely
bseuaeky

epleli
detesine X,(o), X,(o). .
.

,x,(o)
lb
poeepoee
ble e sa
e

aT)
k B s s ,

n
deterune
termLAe
ue e
To
Aateninalion ,0),7
,CO).
yT
 For
Coplte obsevakle usleu,
sysleu guen
awen
0) Cx(o),
(T) Cax(o)

n-1)T= e x(o)
X,0
dakermuezlo) *2(0)
,*2(0)
lo),
daterure z,
be able lo necslin
rece&un
n
mut us T
se m-
m -u
eecc
lloes
r,,
a aa l wAvolup
wveluup
u
al (KT) nm
Nm
o
euuaa
liig
euast ,
set
selulusss
aseliueua
Neti eelo set s
L a l a n A

a urtaa nust
u l a e s u s ebla we
eaiuatieng,
xCo).To nm
e eapatien
2qpaiuoes
aC0),« Co) Co) re d
we p
id ee
pn dd
en eenxt
t

xCo), xCo)
KaAVy
naary
n nta l r
ARadly nm xn a t r i x

l e
:
l e
nm

al
al
a l

e
X a t u r s

c at e
dsaAiu.
C a d i u B I N

e
S a u e , Henes
Henee

A a k , s a e ,

e
a
rree meak
metnx
M a t i x

bseahl
t r a a s p e s e

Corlel

madai=
obseait
M a t n

at
k n @ - v

matnx &

Matncas

ai
M a y be slaled

Ma Rask
ALTE
2NATIVE
R N AT I V E LONDIT1ON TOR COMPLE1E OPCERVA PILIT/:

JLBERTS TEST
(k+)T = GxKT)
YCkT= Cx(KT) d a
lsunct, and
a a
cnet,
a u o
values ae dE
ae
watnx
a .

e eigea alóaa
diagoral
diagoaal
ppeee
peet ba
a aa
fse
lona
u us
s
G aló
iPP
ranssoWAie
a tnix
rwet uat
diagoaal
d e a g e a a l

PGP
&

so
al
U n e t

x(T) PR (kT)
w*
w ii
l le
e

Ca
we PGPi(kT)
hea,
P(«T) =
K+)T
ykT)= CPs (kT)

C P > ( ) i t o )

nT)
HeC e r (Far)'o)

[2o)
CP

n disuet Lige
toiert , 2

anes ebseruale aad onl

& toa-plete
ln
he mxn
mXn
Matix ee
Matrix CP
ceULALt
neA ete eleaneal.
Xere
ee
Cenaigl
 betauu, e tou
o demeal eP c
a an slali uaniale
anable Co) wel
2(o)
t aPa e ealpalapalis aud erehoe eaAnst be
Actornud

Aetauned
e
1 natix Uewea muulli pte eigon valu
Go a t `as T i Jonda Caansneal Poru, e i

Csupatel
lose Jorda bletki ase aesoeiale l e
Sae igo valuu

Nw

ah Tovda betk tsalul e all xod elemak

Aret 8
Tes('as)' i(0)
No tsus oesat toovespead b distact ügen
3
Censut a zene elkaua
Valus

EXAMPLE

de Cputeby ebseva
heeieuun u

Z,er)T

x(KtOr 7,(1)
a(K+1)T
Ta T)
K3(kT)
(t)T
4 Kt)T
0 -3
s)T

6 1 1)
yaCer)
x(kT)
|4T
zs()
 deeewa syaltut ant el (aleiy olevob

,(k41)T -1 o17(Kr)
a e +1) T J 2a(e T)J

t)T
0 1
a (ktu)T

-3
3,(er)J
,(kt)T

Z- PLANE
OBSERVABJLIT
IN THE
FOR CoMPLE TE
C O N D I T I O N

tardilieo
tsrdilisr for tsapels
fr tsapNels
't
g ie
eeeaaEt
A C L L a y a a d

CaAeella lsr
taaceilalisa

e l e xeo
pele xe

LOai
at unde 9 Caaeelatou

ebaernabLle

puulse
uw Candele me-de