Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
6, N
o
. 5
,
O
c
tob
e
r
201
6, p
p
. 2
262
~227
3
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v6
i
5.1
047
0
2
262
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Synthesis of an Optimal Dyna
mic Regulator Based on Linear
Quadrat
i
c Gauss
i
an (L
QG) f
o
r th
e Cont
rol of the Rel
a
tive
Humidity Under Experimental Greenhouse
M. Ou
ta
no
ute
1
, A.
Lac
h
h
a
b
2
, A.
Ed-
d
ahh
a
k
2
, M
.
G
u
erb
aoui
1
, A
.
S
e
lmani
1
, B.
Bouch
i
khi
1
1
Sensors Electr
onic
& Instrumentation
Team, D
e
pa
rtment of Ph
y
s
ics
,
Facu
lty
of
Scien
ces,
Moulay
Isma
ïl
Unive
r
sity
, Me
kne
s
,
Morocc
o
2
Modelling S
y
s
t
ems Control an
d
Telecommunications
Team, Hig
h
School of
Technolog
y
,
Moulay
Isma
ïl
Unive
r
sity
, Me
kne
s
,
Morocc
o
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Mar 12, 2016
Rev
i
sed
Au
g
11
, 20
16
Accepted Aug 25, 2016
This pap
e
r describes one
practical approach
th
at
suggests
a model b
a
sed
techn
i
que to
con
t
rol in r
e
al time
the
relative hum
idity
und
er greenhouse. The
hum
idit
y lev
e
l
is one of
the
m
o
st di
fficult environmental factors to
be
regulated in greenhouse. Moreo
v
er, main
taining
and correcting for more or
less humidity
can be
a
challeng
e for ev
en
the most sophisticated
monitoring
and contro
l eq
uipment. For these ra
isons,
a Linear Quadratic Gaussian
(LQG) controller for relativ
e humidity
regu
latio
n under greenho
use turns out
to be useful. In
deed
a LQG co
ntroller is propo
sed for a relativ
e humid
ity
under a greenho
use control task
. So, the state s
p
ace m
odel
,
which is best
fitting th
e acqu
i
r
e
d data
, was identified using the
Num
e
rical S
ubspace S
t
at
e
S
p
ace S
y
s
t
em
I
D
entifi
cat
ion (N
4S
ID) algorithm
.
Th
e m
a
them
at
ica
l
m
odel
that
is obta
i
ned
will be us
ed for
evalu
a
ting
the p
a
ram
e
ters of
LQ
G strate
g
y
.
The proposed controller is
implemented in two steps,
in one hand, Kalman
filte
r (KF) is used to deve
lop an
observe
r that es
tim
ates the st
ate
of relat
i
ve
humidity
und
er greenhouse. In
the other
hand
, the state f
eedb
a
ck controller
gain is estimated using a lin
ear
quadrat
ic criter
ion
function. Th
e
suggested
optimal implemented cont
roller
using Matlab/Simulink environment is
appli
e
d to an experim
e
nta
l
gree
nhouse. W
e
found, accord
ing to
the results,
that th
e cont
roll
er is abl
e
to le
a
d
the inside r
e
l
a
tive hum
idit
y
to
the desir
e
d
value
with
high
accur
a
c
y
,
r
e
gard
les
s
of th
e
ext
e
rn
al d
i
s
t
urbanc
es
.
Keyword:
Gree
nhouse cli
m
ate control
Kalm
an
filter ob
serv
er
Linear quadratic
ga
ussian
cont
rol
l
e
r
State space m
odel
Subspa
ce ide
n
t
i
fication
Copyright ©
201
6 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
B
e
nachi
r
B
o
uc
hi
k
h
i
,
Sens
ors
,
El
ect
r
oni
c
&
Inst
rum
e
nt
at
i
on
Team
, De
part
m
e
nt
of
Phy
s
i
c
s,
Facu
lty of Scien
ces, Mou
l
ay Ism
a
ïl Un
iv
ersity,
B.P.
1
1201
, Zit
o
un
e, 500
03
,
Mek
n
e
s, M
o
rocco
,
Tel: +21
2
5
355
388
70
; Fax
:
+212
5
355
36
808
.
Em
a
il: b
e
n
ach
i
r
.b
ou
ch
ikh
i
@gmail.co
m
1.
INTRODUCTION
The
gree
n
h
o
u
s
e
en
vi
r
onm
ent
cont
rol
pr
o
b
l
e
m
i
s
consi
s
t
e
d
t
o
creat
e a
fa
v
o
ra
bl
e e
nvi
r
o
n
m
ent
fo
r t
h
e
crop in orde
r to reach prede
t
ermine
d results for hi
gh yield, high quality
and low costs [1],[2]. Real-ti
m
e
m
oni
t
o
ri
ng
o
f
t
h
e
gree
nh
o
u
se
en
vi
r
onm
ent
wi
t
h
se
ns
ors
a
n
d
a
dva
nce
d
s
o
ft
ware
can
g
r
eat
l
y
im
proves
y
i
el
ds
and ec
on
om
i
c
per
f
o
r
m
a
nce by
opt
im
i
z
i
ng pl
ant
gro
w
t
h
[3
]
.
In cont
rol
c
ont
e
x
t
,
i
t
i
s
a
very
di
f
f
i
c
ul
t
t
a
sk t
o
im
pl
em
ent
i
n
pract
i
ce
due t
o
t
h
e c
o
m
p
l
e
xi
ty
of t
h
e
phe
n
o
m
ena i
n
v
o
l
v
e
d
i
n
si
de
g
r
een
h
ous
e d
u
r
i
n
g t
h
e pl
ant
gr
owt
h
p
r
oces
s such as t
h
e
dy
nam
i
cal beh
a
vi
o
r
of
gree
n
h
o
u
se cl
i
m
at
e and co
nt
r
o
l
r
e
qui
rem
e
nt
s,
whi
c
h
prese
n
t
st
ro
n
g
i
n
t
e
ract
i
o
n
s
a
m
ong v
a
ri
abl
e
s,
no
nl
i
n
ea
ri
t
y
and
n
o
n
-st
a
t
i
o
nary
[
4
]
,
[
5
]
.
Int
e
rnal
t
e
m
p
erat
ure
a
n
d
relativ
e hu
m
i
d
i
ty wh
ich
are
very sen
s
itiv
e to th
e ou
tsid
e
weath
e
r, are clo
s
ely lin
k
e
d
tog
e
th
er in
a
greenho
u
s
e,
and
are
t
h
e t
w
o
i
m
port
a
nt
vari
a
b
l
e
s f
o
r
ph
ot
o
s
y
n
t
h
esi
s
an
d
ph
ot
o
m
o
r
p
ho
ge
nesi
s
of t
h
e
pl
ant
[
6
]
-
[
8
]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Synt
hesis
of an O
p
timal
Dy
namic Re
gul
ator Base
d on
Line
ar Quadr
a
tic
Gaussi
an (
L
Q
G
)
.... (
M
.
O
u
t
a
noute)
2
263
Ho
we
ver
,
t
h
e t
uni
ng
of se
ver
a
l
cont
r
o
l
l
e
rs i
n
t
h
e com
p
l
e
x gree
n
h
o
u
se en
vi
r
onm
ent
i
s
a
chal
l
e
ng
e t
o
pr
ocess
en
g
i
n
eers and
o
p
e
rators.
It is
i
m
p
o
r
tan
t
to main
tain
th
e
p
r
op
er
relativ
e humid
ity sin
ce th
e h
u
m
id
ity in
sid
e
th
e
gree
n
h
o
u
se
ha
s a cl
ose
rel
a
t
i
o
n
t
o
c
r
ops
g
r
o
w
t
h
,
beca
use
th
e
h
i
gh
lev
e
l
of hu
m
i
d
ity lea
d
s t
o
create
a su
itab
l
e
envi
ro
nm
ent
t
o
t
h
e em
erge
n
ce an
d de
vel
o
pm
ent
of s
o
m
e
ki
n
d
of
di
seas
es [
9
]
,
[
10]
. T
o
achi
e
ve
t
h
ese
g
o
al
s,
th
e te
m
p
eratu
r
e an
d
relativ
e h
u
m
id
ity
m
u
st
b
e
con
t
ro
lled
op
ti
m
a
lly
b
y
g
i
ven
certain
criteria th
ro
ugh
actu
a
tors
lik
e
h
eater, h
u
mid
i
fier
and
ven
tilato
r [11
]
. In
o
r
d
e
r
to
r
each
a go
od
p
e
rform
a
n
ce
o
f
t
h
e co
n
t
ro
ller,
we n
eed
to
have
a m
a
thematical
m
odel which is capable to desc
ri
be c
o
rrectly as m
u
ch as
possible the
dy
na
m
i
cal
beha
vi
o
u
r
of t
h
e p
r
oces
s
para
m
e
t
e
rs.
C
ont
r
o
l
st
rat
e
g
i
es have be
en
reco
g
n
i
zed as
an effi
ci
e
n
t
an
d co
nsi
s
t
e
nt
w
a
y
t
o
im
prove
gree
n
h
o
u
se
p
r
o
cess au
to
matio
n
.
Th
e d
e
sig
n
o
f
su
ch
syste
m
will en
ab
le u
s
to
m
o
d
i
fy th
e b
e
h
a
v
i
ou
r
of th
e p
l
an
t to
su
it o
u
r
needs in term of specified
stability, perform
a
nce, and
robust
ness obj
e
c
tives [12]. In recent years, large
am
ounts of studies have been conducte
d for
cont
rolling the
clim
atic para
m
e
ters of gree
nhouse system
, where
vari
ous
ba
si
c
and
ad
va
nce
d
cont
rol
st
rat
e
g
i
es, l
i
k
e
pre
d
i
c
t
i
v
e co
nt
r
o
l
,
a
d
apt
i
v
e
c
ont
r
o
l
,
r
o
b
u
st
c
o
nt
r
o
l
an
d
fuzzy
c
o
nt
rol
h
a
ve
been
em
pl
oy
ed a
n
d t
e
st
e
d
i
n
p
r
act
i
ce f
o
r t
h
e
co
nt
r
o
l
l
e
d
g
r
een
h
ouse
[
1
3]
,[
14]
.
In th
is
p
a
p
e
r, t
h
e
p
r
ob
lem
o
f
regu
latin
g th
e
relativ
e hu
m
i
d
ity u
n
d
e
r an exp
i
rem
e
n
t
al g
r
een
hou
se t
o
a
fo
u
v
ar
obl
e l
e
v
e
l
i
s
addre
ssed
.
To ac
hi
eve t
h
is goals,
we
use L
QG tec
h
niqu
e that is one of
th
e m
o
st p
opu
lar
m
odel
based
c
ont
rol
st
rat
e
gi
e
s
i
n
m
oder
n
c
o
nt
r
o
l
t
h
e
o
ri
es a
n
d
i
t
s
ap
pl
i
cat
i
ons
[
1
5]
.
Th
e LQG co
n
t
ro
ller
syn
t
h
e
sis is an appro
a
ch of
d
e
sign
ing
a prop
er regu
lato
r b
y
co
m
b
in
in
g th
e
LQR
a
n
d K
a
lma
n
ob
s
e
rv
er
in
to
an ou
tpu
t
f
e
edb
a
c
k
co
mp
e
n
s
a
to
r.
As all syste
m
sta
t
es usuall
y are
not a
v
ailable, the
syste
m
state
variables are es
tim
a
te
d usi
n
g
a KF,
fo
rm
i
ng t
h
e LQ
G co
nt
rol
l
e
r.
The
pu
r
pos
e o
f
t
h
e st
u
d
y
i
n
g
th
eory o
f
cu
rren
t
con
t
ro
ller
is to
syn
t
h
e
size th
eir co
n
t
ro
l
laws with
specified
prop
rieties wh
ich
p
e
rmit
to
optim
ize a perform
a
nce index and to re
duce
as
wel
l
t
h
e
n
o
i
s
es enc
o
unt
e
r
e
d
sy
st
em
[16]
,
[
17]
.
We
report h
e
rein
, t
h
e
resu
lt
s of t
h
e
d
e
v
e
l
o
pp
ed
m
a
th
e
m
atical
m
o
d
e
lli
n
g
to con
t
ro
l
th
e relative
hum
idity using N4SID al
gorithm
.
For
t
h
at
purpose,
we m
a
ke
use
of the s
t
ate space m
o
del whose
validation
was carri
e
d
o
u
t
,
and t
h
e
n
we h
a
ve el
abo
r
at
ed
an o
p
t
i
m
a
l
cont
rol
base
d o
n
LQG a
p
p
r
oac
h
p
e
rm
it
t
i
ng t
o
assu
r
e
the closed-loop stability and
m
a
intain the variations
of internal relative hum
idity under a
n
experi
m
e
ntal
g
r
eenh
ou
se.
2.
MATE
RIAL
S AND METHODS
2.
1.
Experimental Greenhouse
P
r
ototype
The e
x
peri
m
e
nt
al
gree
nh
o
u
se
use
d
i
n
t
h
ese
expe
ri
m
e
nt
at
i
o
n i
s
e
q
ui
p
p
ed
wi
t
h
a c
o
nt
r
o
l
sy
st
em
t
h
at
allo
ws
bo
th
t
h
e acqu
i
sitio
n an
d au
t
o
m
a
tic
co
n
t
ro
l
of
g
r
een
hou
se clim
a
t
e p
a
ram
e
ters
(Figu
r
e 1)
[1
8]. Th
e
internal
clim
ate is
define
d by
the i
n
ternal te
m
p
erature, t
h
e
internal
relative hu
m
i
d
ity an
d CO
2
con
t
en
t
wh
ich
co
nstitu
tes th
e ou
tpu
t
s of the g
r
eenho
u
s
e,
wh
ile th
e ex
te
rn
al clim
ate is
co
m
p
o
s
ed of
th
e tem
p
eratu
r
e, th
e
relativ
e hu
m
i
d
ity, an
d
th
e
so
l
a
r rad
i
atio
n
t
h
at acts d
i
r
ect
l
y
on t
h
e
ope
rat
i
on
o
f
t
h
e
gree
nh
o
u
se. T
h
e e
x
t
e
r
n
al
clim
ate para
m
e
ters are
unc
ont
rollable i
n
puts
and they a
r
e c
o
nside
r
ed as
dis
t
urbances
.
Fi
gu
re
1.
Ex
pe
ri
m
e
nt
al
green
ho
use
set
-
up
Th
e air tem
p
eratu
r
e,
relativ
e
h
u
m
id
ity, so
lar
ra
di
at
i
o
n
an
d ca
rb
o
n
di
o
x
i
d
e c
o
nce
n
t
r
at
i
o
n
of
g
r
eenh
ou
se ar
e m
easu
r
ed
ev
er
y 5
second
s
by th
eir
r
e
sp
ecti
v
e se
ns
ors
.
T
h
e low-level si
gnals
delivere
d
fro
m
sen
s
o
r
s are con
d
ition
e
d, am
p
lified
an
d
th
en tran
sm
i
tte
d
to
co
m
p
u
t
er syste
m
th
ro
ugh
the PCI-602
4E b
o
a
rd
(f
rom
Nat
i
onal
Inst
rum
e
nt
s®
) as a
dat
a
acq
ui
si
t
i
on
(D
AQ
) de
vi
ce. T
h
e
cont
rol
a
n
d
o
p
e
rat
i
ng
so
ft
wa
r
e
was
d
e
sign
ed
in th
e Matlab
/
Si
m
u
li
n
k
env
i
ron
m
en
t.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 5
,
O
c
tob
e
r
20
16
:
226
2
–
22
73
2
264
2.
2.
Mathematica
l
Mo
delling
wi
th Sta
t
e-Spa
c
e Approa
ch
Thi
s
sect
i
on i
s
dev
o
t
e
d t
o
t
h
e
m
a
t
h
em
ati
c
al
m
odel
l
i
ng of
t
h
e rel
a
t
i
v
e hum
i
d
i
t
y
unde
r gree
n
h
o
u
se
.
Mo
d
e
lling
is an
essen
tial p
r
ecurso
r
in
t
h
e p
a
ram
e
ter esti
m
a
t
i
o
n
pro
c
ess. Th
e i
n
tern
al cli
m
ate
mo
d
e
l
o
f
gree
n
h
o
u
se i
s
essent
i
a
l
fo
r i
m
provi
n
g
e
nvi
ro
nm
ent
a
l
perf
orm
a
nce an
d c
ont
rol
ef
fi
ci
en
cy
. The sy
nt
he
si
s of
reg
u
l
a
t
o
r
s
re
q
u
i
res a
very
prec
i
s
e m
odel
l
i
ng f
o
r
t
h
e
pr
ocess
un
de
r re
g
u
l
a
t
i
o
n.
In the sy
nthesi
s of the L
Q
G cont
roller, a stat
e
space m
odel is identified us
ing the
N4SID. Subs
pace
i
d
ent
i
f
i
cat
i
on
m
e
t
hods
of
fer
an al
t
e
rnat
i
v
e t
o
t
h
e cl
assi
c re
cursi
v
e p
r
edi
c
t
i
on er
r
o
r m
i
nim
i
zat
i
on
m
e
t
h
ods
(e.
g
.
Aut
o
Regresive
m
odel with e
X
ternal
input (ARX)).
The s
u
bsp
ace ide
n
tification m
e
thods
are use
d
to i
d
entify
param
e
ters (m
atrices)
of a
lin
ear tim
e-invariant st
ate sp
ace
m
o
d
e
l fro
m
the in
pu
t/ou
t
pu
t
d
a
ta [19
]
.
A state-s
p
ace
m
odel has bee
n
derive
d t
o
understa
n
d
the
dynamic beha
viour
of t
h
e relat
i
ve hum
i
dity
under greenhouse by
using s
ubspace
N4SID algorithm
.
For
this
we excite the syst
e
m
by sending a voltage
step
to
h
eater
an
d th
en
to
t
h
e hu
m
i
d
i
fier, an
d we co
llect measu
r
em
en
ts o
f
relativ
e
h
u
mid
ity u
n
til a stead
y
state is reache
d
.
Li
near
su
bs
pac
e
i
d
ent
i
f
i
cat
i
o
n
m
e
t
hod
s are
c
once
r
ned
wi
t
h
sy
st
em
s and m
odel
s
o
f
t
h
e
f
o
r
m
[20]
:
(1
)
(
)
(
)
(
)
(
)
()
()
(
)
x
tA
x
t
B
u
t
F
w
t
yt
C
x
t
D
ut
v
t
(1
)
Whe
r
e:
x(t) is the
state vector,
u
(
t) is th
e v
ect
o
r
of inpu
t,
y
(
t
)
i
s
t
h
e
vect
or
o
f
out
put
m
easurem
ent
s
,
w(t) and
v(t
)
are the
process
a
n
d th
e output measurem
ent
noises vectors,
respectively,A,
B, C,
D a
n
d F
are
the system
m
a
trices of a
p
propri
at
e di
m
e
nsi
ons t
o
be est
i
m
at
ed.
Once t
h
e data have
bee
n
c
o
llected, it can
be
anal
ysed
by assum
i
ng a state space represe
n
tation
with
th
e N4
SID algo
rith
m
an
d fittin
g th
e m
o
d
e
l t
o
th
e pro
cess
data.
The
perce
n
t
a
ge
o
f
t
h
e
o
u
t
p
ut
t
h
at
t
h
e m
odel
r
e
pr
o
duces
(B
es
t
Fi
t
m
e
t
r
i
c
) [2
1]
,
defi
ne
d a
s
:
ˆ
1-
*
100%
yy
Be
s
t
Fi
t
yy
(2
)
In th
is eq
u
a
ti
on
, y is th
e m
easu
r
ed
ou
tpu
t
,
y
i
s
t
h
e si
m
u
l
a
t
e
d m
odel
out
put
,
and
y
is th
e m
e
a
n
o
f
y.
Th
e fi
rst step
fo
r an
adv
a
n
c
ed co
n
t
ro
l
d
e
sign is th
e d
e
velopmen
t o
f
a
d
y
n
a
mic
m
o
d
e
l. Mo
d
e
l
q
u
a
lity
is an
essen
tial asp
ect to
ach
iev
e
satisfactory co
n
t
ro
l p
e
rf
orman
ces. Th
e
main
o
b
j
ectiv
e in
th
is p
a
rt is to
co
m
e
up
wi
t
h
a
val
i
d
m
odel
t
h
at
can
be
use
d
as
a
b
a
si
s f
o
r t
h
e
rel
a
t
i
v
e h
u
m
i
di
t
y
unde
r
gree
n
h
o
u
s
e
co
nt
r
o
l
desi
g
n
.
2.
2.
1.
Relative
humi
d
ity response
to a step of
humidifier
To
d
e
crib
e t
h
e
ev
o
l
u
tio
n
of the relativ
e hu
m
i
d
ity u
n
d
e
r
g
r
een
hou
se,
we excite th
e syste
m
b
y
send
ing
a step input to
the hum
i
di
fier in order to humidify
the air under greenhouse until
reaching a steady st
ate. The
co
llected
d
a
ta
an
d th
e
N4
SID alg
o
rith
m
in
Matlab
are
u
s
ed
to d
e
v
e
lop
t
h
e m
o
d
e
l.
The m
odel
of
t
h
e
hum
i
d
i
f
i
e
r i
s
gi
ven
by
t
h
e
fol
l
o
wi
n
g
st
at
e
-
space
re
pre
s
e
n
t
a
t
i
on:
(1
)
(
)
(
)
(
)
()
()
(
)
()
bb
bb
b
x
tA
x
t
B
u
t
F
w
t
yt
C
x
t
D
u
t
v
t
(3
)
Wh
ere y(t) is t
h
e ou
tpu
t
v
ecto
r
, u(t) is th
e i
n
pu
t v
e
ct
or a
n
d x
(
t
)
i
s
a st
at
e
vect
o
r
o
f
t
h
re
e di
m
e
nsi
onal
.
A
b
, B
b
,
C
b
and F
b
, are t
h
e m
a
trices given
by:
0
.
984
27
0.1
0002
-
0
.027
04
-
0
.0
3097
0.09
140
-
0
.625
12
-
0
.002
44
-
1
.03
5
8
-
0.1
0883
b
A
(4
)
0
.
00
033
0.
00
511
0.
03
481
T
b
B
(5
)
117.43
0.5049
4.4587
b
C
(6
)
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISS
N
:
2088-8708
Synt
hesis
of an Optimal
Dy
namic Re
gul
ator Base
d on
Line
ar Quadr
a
tic
Gaussi
an (
L
QG)
.... (
M
.
Out
a
noute)
2
265
0
b
D
(7
)
0.
003
59
0.
01
092
0
.
013
27
T
b
F
(8
)
And
t
h
e in
itial
state was,
0
0.
35
47
9
0
.
2
0
0
5
6
0.
20
02
T
b
x
(9
)
The i
n
de
x ‘
b
’
of t
h
e m
a
t
r
i
c
es desc
ri
be
d a
b
o
v
e
re
fe
rs to
the h
u
m
i
difier (
b
r
u
m
e
) actuator as t
h
e
con
s
i
d
ere
d
sy
s
t
em
i
nput
. P
o
l
e
s val
u
es
o
f
t
h
e i
d
e
n
t
i
f
i
e
d
m
odel
i
ndi
cat
e ope
n
-
l
o
op st
a
b
l
e
, co
nt
r
o
l
l
a
b
l
e, an
d
obs
er
vabl
e sy
st
em
.
Fi
gu
re 2 s
h
ow
s t
h
e com
p
ari
s
on
of t
h
e o
b
t
a
i
n
ed m
ode
l wit
h
real data,
where it can be
observe
d
that
the s
ubs
pace m
odel ca
ptures t
h
e system
dyna
m
i
cs prope
r
ly.
Fi
gu
re
2.
C
o
m
p
ari
s
on
o
f
sy
st
em
m
odel
f
o
r
h
u
m
i
di
fi
er, si
m
u
l
a
t
e
d st
e
p
-
r
es
po
nse
wi
t
h
t
h
e
expe
ri
m
e
nt
al
measurem
ent
Th
e in
si
d
e
relativ
e h
u
m
id
ity i
s
stab
ilized
at 6
3
.95
% an
d
t
h
e in
itial v
a
lu
e is 3
9
.76
%.
Th
e
m
o
d
e
l b
e
st
fit
m
e
tric at about 90.
5
5 %, a
nd the
n
the sta
t
e space
m
odel describe
s 90.
5
5 % of the
variance in the process
out
put. Results
indicate t
h
at the si
m
u
lation a
n
d expe
rim
e
ntal results
foll
ow
very cl
osely each
othe
r.
2.
2.
2.
Relative
humi
d
ity resp
o
nse
to a step
of
he
ater
In this case,
we excite the syste
m
by
a st
ep inp
u
t
o
f
2.
2
V t
o
t
h
e heat
e
r
. Fi
gu
re 3 s
h
ows t
h
e ev
ol
ut
i
o
n
o
f
th
e m
easu
r
ed
an
d sim
u
late
d
relativ
e hu
m
i
d
ity b
y
using
t
h
e
N4SID al
g
o
rith
m
.
Fi
gu
re
3.
C
o
m
p
ari
s
on
o
f
sy
st
em
m
odel
f
o
r
h
eat
er, si
m
u
l
a
t
e
d st
e
p
-
r
esp
o
n
s
e
wi
t
h
t
h
e e
x
pe
ri
m
e
nt
al
measurem
ent
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 5
,
O
c
tob
e
r
20
16
:
226
2
–
22
73
2
266
Th
e resu
lts in
dicate th
at th
e p
r
opo
sed
iden
tificati
on m
e
t
hod
can capt
u
re t
h
e dy
nam
i
c behavi
o
u
r
o
f
th
e ex
p
e
rim
e
n
t
al relativ
e h
u
m
id
ity
su
ccessfu
lly with
a g
o
o
d
accu
r
acy. Th
e b
e
st fit
m
e
t
r
ic ab
ou
t 8
5
.93
% of
th
e ob
tain
ed
m
o
d
e
l, assert th
i
s
ou
tco
m
e.
The
n
, t
h
e st
at
e
-
space
m
odel
i
s
desc
ri
be
d
by
:
0.
98
87
6
0
.
01
18
5
0.0
0
2
0.01
24
0
.
00
19
2
0
.01
2
5
0.
09
06
3
0
.2
10
45
1.
21
75
0
.
61
91
0.
09
38
6
0
.3
7
0
5
0.
16
97
0
.
89
48
5
0
.2
2
46
02
0.
0
9
9
00
7
h
A
(1
0)
0
.
00
169
0
.
0
420
5
0
.
0
7
627
0
.
02
17
T
h
B
(1
1)
1
23.
19
5.
6
062
0.
17
959
0
.
715
94
h
C
(1
2)
0
h
D
(1
3)
0.
00206
0.
00410
0.
00598
0.
00685
T
h
F
(1
4)
And t
h
e initial
state was,
0
0
.
5
303
8
0
.
141
32
0.
014
86
0.
09
06
T
h
x
(1
5)
The in
dex
‘h
’
of the m
a
trices abo
v
e re
fer
s
to th
e heater
actuator as t
h
e
consi
d
ere
d
sy
stem
input.
Poles
values
of t
h
e identified m
odel indicate open
-lo
op sta
b
le, c
o
ntr
o
llable
, an
d
obs
er
vable s
y
stem
.
Accordingly, after ha
ving the syste
m
’s m
odel on st
ate space form
, an optim
al L
QG c
ont
roller will be
im
plem
ented so as
to c
o
ntrol
the relative
h
u
m
idity
unde
r a
n
e
xpe
rim
e
ntal g
r
een
h
ouse
.
2.
3.
D
e
sig
n
of
Linea
r
Quadrat
i
c Gaussian
C
o
nt
ro
ller
The
focus
of this section is t
o
design an opti
m
a
l LQG cont
roller i
n
order t
o
regulate the relative
hum
idity unde
r
gree
nhouse a
t
desire
d state. Indee
d
, the
L
Q
G controller
is the m
odern
state-space c
o
ntrol
technique
for t
h
e design of
optim
a
l dynam
i
c regulators
,
which
requires
a
state-space m
odel
of t
h
e
plant and
com
b
ines m
u
lt
ivariate function such
as
Line
ar
Qua
d
ratic R
e
gulator
(L
QR) a
n
d Kalm
an Filter (K
F)
[
2
2]
.
The
pu
rp
ose
of t
h
e co
nce
p
t of a m
odel
base
d o
p
tim
a
l cont
roller is
to en
ha
nce th
e reg
u
latio
n
per
f
o
r
m
a
nce,
while m
i
nim
i
zing t
h
e c
o
st o
f
co
ntr
o
l e
f
f
o
r
t
as well as r
e
duci
n
g
the
di
stur
bance
ef
fe
ct. To
approach this
problem
,
LQG cont
roller ca
n
b
e
im
plem
ented in tw
o
steps
[
2
3]
:
1.
Desi
gni
ng of a
Kalm
an filter to esti
m
a
te t
h
e
de
sired stat
es that are needed to
be controled;
2. C
a
lculation
of state feed
back c
ontr
o
lle
r gain to
m
i
nim
i
ze
the cost fu
nctio
n base
d
on linear q
u
a
d
ratic
criterion function.
The controllers, which prov
i
d
e inp
u
t signal
s
for the
plant
based
on the
estim
a
ted state
-
vect
or, a
r
e
called com
p
ensators
[24]. KF is
one
of t
h
e state esti
m
a
tion t
h
at ca
n e
s
tim
a
te the state varia
b
le wit
h
the
m
easurem
ent including
noise
.
The co
ntr
o
l m
e
tho
d
LQ
G c
o
nsists of a
n
L
Q
R
with
the
o
b
ser
v
e
r
states of the sy
stem
via the m
e
thod
of t
h
e
KF. In t
h
e case
where
the system
state is a linea
r one or linear aroun
d
an
o
p
e
r
atin
g
poi
nt, the
sy
s
t
em
is
rep
r
ese
n
ted
by
eq
uation
(
1
) i
n
w
h
ic
h w
(
t) a
n
d
v
(
t)
rep
r
ese
n
t w
h
ite ga
uss
i
an n
o
ise
with
zero a
s
m
ean value
,
in
d
e
p
e
nd
en
t, resp
ectiv
ely
[25]-
[
27
].
The estim
ate e
r
ror covaria
n
ce
is de
fine
d as
followi
ng:
ˆˆ
l
i
m
(
()
()
()
(
)
)
T
f
t
P
E
x
t
xt
xt
x
t
(1
6)
The state estim
ator
x
t
is de
rive
d
fr
om
:
ˆˆ
ˆ
()
(
)
e
x
tA
x
B
u
K
y
C
x
D
u
(1
7)
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISS
N
:
2088-8708
Synt
hesis
of an Optimal
Dy
namic Re
gul
ator Base
d on
Line
ar Quadr
a
tic
Gaussi
an (
L
QG)
.... (
M
.
Out
a
noute)
2
267
Th
e estim
atin
g
g
a
in K
e
of
Kal
m
an
is estab
lish
e
d b
y
t
h
e
fo
ll
o
w
i
n
g relation
[28
]
:
1
()
TT
ef
KP
C
C
P
C
R
(1
8)
Wh
ere
P is th
e
so
lu
tion
m
a
trix
of Riccati equatio
n
:
1
.
TT
ff
PA
P
P
A
P
C
R
C
P
Q
(19)
In
Eq
uat
i
o
ns
(
1
8
)
a
n
d (
1
9)
, R
f
and Q
f
a
r
e
w
e
i
ght
i
n
g m
a
t
r
i
c
es,
or
desi
gn
p
a
ram
e
t
e
rs of
K
F
.
The sy
nt
hesi
s
of t
h
e
LQR c
o
nt
r
o
l
l
e
r i
s
reve
al
ed t
h
r
o
ug
h fi
ndi
ng a m
a
t
r
i
x
wi
t
h
gai
n
L i
n
w
h
i
c
h t
h
e
opt
i
m
al
feedba
ck c
ont
rol
i
s
gi
ven
by
[
29]
:
.(
t
)
uL
x
(20)
Th
e m
i
n
i
m
i
z
i
n
g
q
u
a
dratic criterion
for
o
b
t
ai
n
i
ng
th
e con
t
rol law
is:
0
()
TT
LQR
Jx
Q
x
u
R
u
d
t
(21)
W
h
ere t
h
e m
a
trix
Q
an
d R
are
p
o
s
itiv
e d
e
fin
ite and
po
si
tiv
e sem
i
d
e
fin
ite m
a
trix
, resp
ectiv
ely. They are
w
e
igh
tin
g p
a
rameters th
at p
e
nalize th
e stat
es
and the c
o
ntrol ef
fort, re
specti
v
ely.
Th
e m
i
n
i
m
a
l
so
lu
tion
of the co
st fu
n
c
tion
g
i
v
e
s th
e st
ate feedb
ack
l
a
w
wh
ich
is i
n
tro
d
u
c
ed
in
equat
i
o
n
(
2
0
)
,
whe
r
e L
i
s
obt
ai
ned
by
s
o
l
v
i
n
g t
h
e
Co
nt
r
o
l
Al
ge
brai
c Ri
cc
at
i
Equat
i
o
n
(C
ARE):
1
-0
TT
SA
A
S
SB
R
B
S
Q
(22)
Th
e
op
ti
m
a
l g
a
in
m
a
trix
L
is then
calcu
lated
by:
1
T
LR
B
S
(23)
Fig
u
re
4
sh
ow
s th
e sch
e
m
a
tic
d
i
agram
o
f
th
e
LQ
G con
t
ro
l tech
n
i
q
u
e
,
w
h
ere th
e term
N
is calcu
lated
in steady state
to elim
inate the sta
tic erro
r by th
e fo
llow
i
ng equ
a
tio
n [30
]
:
1
1
NC
I
A
B
L
B
(24)
An
d w
h
ere
K
e
is th
e
K
F
g
a
in
an
d L is t
h
e LQ
R gain
.
Fi
gu
re 4.
The
LQG
c
ont
r
o
l
l
e
r bl
oc
k di
agra
m
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ISSN
:
2
088
-87
08
IJECE
Vo
l. 6
,
N
o
. 5
,
O
c
tob
e
r
20
16
:
226
2
–
22
73
2
268
The c
ontroller design a
nd s
t
ate estimation are tr
eated s
e
parately in a
ccorda
n
ce wit
h
separation
p
r
i
n
cip
l
e wh
ich
allow
s
to
sep
a
rate the estimatio
n
th
e st
at
e of
t
h
e
ob
ser
v
er a
n
d t
h
e
fe
edbac
k
gai
n
c
ont
rol
l
e
r
to
g
e
th
er i
n
to
t
w
o sep
a
rated prob
lem
s
[3
1
]
.
A
n
estim
a
t
o
r
or ob
server acco
r
d
i
ng
t
o
eq
u
a
t
i
o
n
(1
8) co
m
e
s
above
t
o
t
h
e bl
oc
k d
i
agram
i
n
Fi
gure
4. The
ob
serve
r
gai
n
m
a
t
r
i
x
K
e
d
e
termin
es th
e co
nv
erge
nce spee
d of the
esti
m
a
ted
o
u
t
pu
t
y
t
to
th
e m
eas
u
r
ed
ou
tpu
t
y(t
)
. M
o
reov
er, an
o
p
tim
al feed
b
ack con
t
ro
l law
is
d
e
term
in
ed
b
a
sed
o
n
lin
ear qu
adratic op
timal co
n
t
ro
l
th
eo
ry as expressed
in equ
a
tion
(2
0).
Th
e fu
nd
am
en
tal o
b
j
ectiv
e
of d
e
sign
ing
the K
F
ob
serv
er is to
esti
mate
th
e state v
a
riab
les o
f
t
h
e
gree
n
h
o
u
se i
n
c
l
udi
n
g
rel
a
t
i
v
e
h
u
m
i
di
ty
l
e
vel
.
These
st
at
es can
be
use
d
f
u
rt
her t
o
d
e
si
g
n
L
Q
G
co
nt
r
o
l
l
e
r t
o
dri
v
e the
relative
hum
idity under greenhouse
at a de
sired st
ate.
3.
RESULTS
A
N
D
DI
SC
US
S
I
ON
Th
e exp
e
rim
e
n
t
w
a
s performed
to
ex
am
i
n
e th
e ab
ility o
f
th
e LQG
co
n
t
ro
ller for au
to
m
a
tical
l
y
ad
ju
sting
th
e pro
cess o
f
th
e ou
tpu
t
v
a
riation o
f
relativ
e humid
ity
to
n
e
w
setp
o
i
n
t
s. To
act o
n
th
e g
r
eenh
ou
se
rel
a
t
i
v
e h
u
m
i
dit
y
, we
used
t
h
e
act
uat
o
rs
whi
c
h ar
e c
ont
r
o
l
l
e
d acc
or
di
n
g
t
o
t
h
e si
g
n
of
t
h
e
di
ffe
re
nce
bet
w
e
e
n
th
e setpo
i
n
t
and
th
e m
easu
r
ed relativ
e
h
u
m
id
ity (Fig
ure
5
)
[3
2
]
.
Fi
gu
re
5.
St
r
u
c
t
ure
of
t
h
e c
o
nt
rol
sy
st
em
The
LQG
des
i
gn m
e
thod
us
es a state s
p
a
ce m
odel
for
the e
xpe
rim
e
n
t
al gree
nhouse
syste
m
as
d
e
scri
b
e
d
abo
v
e. A
s
long
as it is d
e
sirab
l
e to
h
a
v
e
a K
a
lman
filter th
at re
m
o
v
e
s as m
u
ch
no
ise as possib
l
e.
Th
is
filter is tun
e
d b
y
adju
stin
g th
e d
e
si
g
n
p
a
ram
e
ters, wh
ich are Q
f
and R
f
m
a
t
r
i
ces,
fo
r
bot
h sy
st
e
m
i
nput
s
whic
h a
r
e the
heater and t
h
e
hum
i
difier. After s
o
m
e
tr
ials an
d
erro
rs, th
e tun
i
ng
m
a
trices w
e
re set to
:
-4
=
1
.
039
3*
10
fc
Q
(25)
5
8
.
06
02
*
1
0
fc
R
(26)
-4
3.
08
49
*
1
0
fb
Q
(27)
= -207.
2780
b
N
(28)
Th
e co
m
p
u
t
ed
K
a
lm
an
g
a
in
matrix
is g
i
v
e
n b
e
low
:
0.
0731
0
.
7264
-
0
.
8274
T
eb
K
(29)
0.
08
61
1
.
0
9
9
7
-
0
.
3
90
4 0.
795
2
T
ec
K
(30)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN:
208
8-8
7
0
8
Synt
hesis
of an O
p
timal
Dy
namic Re
gul
ator Base
d on
Line
ar Quadr
a
tic
Gaussi
an (
L
Q
G
)
.... (
M
.
O
u
t
a
noute)
2
269
Likewise, and
after som
e
trials and errors, the optim
al gain m
a
trix is calculated in M
a
t
l
ab by
usi
ng
the selected m
a
trices Q a
n
d
R whic
h a
r
e the
desi
gn
pa
ram
e
ters o
f
t
h
e L
Q
R
co
ntroller:
1379
-
5
.9
52.4
-
5
.9
0
-
0
.2
52.4
-
0
.2
2.0
b
Q
(3
1)
1
b
R
(3
2)
1
517
.6 -
6
9
.
1 2
.
2 -
8
.8
-
69.
1
3.1
-
0
.1
0.4
=
2.
2
-
0
.1
0
0
-
8
.8
0.4
0
0
.
1
c
Q
(3
3)
1
c
R
(3
4)
After calculating the
gains
of
the LQR cont
roller,
we
got the followi
n
g results:
=
8
458
.
3
1
64.
3 -20
7
.
4
b
L
(3
5)
=
-207.
2780
b
N
(3
6)
4
=10
*
3.
0929
-
1
.
2730
1.
2267 -
1
.
6
2
c
L
(3
7)
3.
00
71
c
N
(3
8)
Whe
n
c
o
m
b
ining t
h
e LQR
re
gulato
r-la
w
de
sign
with
the
Kalm
an estim
a
tor de
sign,
we
can get the
LQG
c
o
m
p
ensator t
h
at we
wi
ll use f
o
r c
ontr
o
lling t
h
e
relative
hum
idity
un
der
an
ex
perim
e
ntal g
r
een
h
o
u
s
e.
Figu
re 6
(a) a
nd
(b
) desc
rib
e
the evol
utio
n o
f
external te
m
p
erature a
nd the
relative hum
idity,
respectively
,
i
n
an inter
v
al of
20
ho
ur
s o
f
ex
perim
e
ntation with testing the L
Q
G cont
roller in
or
der t
o
reg
u
late the
rel
a
tive h
u
m
i
dity
un
de
r a
n
e
xpe
r
i
m
e
ntal green
h
ous
e.
Figu
re
6.
C
o
ntr
o
l test o
n
t
h
e
g
r
een
h
ouse:
(a)
M
easure
d
e
x
te
rnal tem
p
erature and
(b)
Relative hum
i
dity in an
interval tim
e of
20 hours
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
No. 5
,
Octob
e
r
20
16
:
226
2
–
22
73
2
270
In
fact,
fo
r sa
m
e
ty
pes of
pl
ants, the
ideal
com
f
ort relativ
e h
u
m
i
dity
level is taken
fr
o
m
40 % to 6
0
% in winter m
ont
hs. Based on this r
eality, and as shown
in Figure
7, the
setpoints of the
internal relative
hum
idity
are chan
ge
d res
p
ect
ively
at 20 ho
urs
of rec
o
rd
by increasing and
decreasi
n
g the step of referenc
e
traj
ectory in order to test t
h
e
pe
rform
a
nce of the L
Q
G c
ontroller.
Figu
re
7.
Ex
pe
rim
e
ntal results f
o
r
rela
tive h
u
m
i
dity
reg
u
lating f
o
r seve
ral steps
The
results are obtain
e
d
withi
n
t
h
e m
odel-
ba
sed
LQ
G c
o
ntr
o
ller strate
gy
t
h
at is im
plem
ented
o
n
t
h
e
M
a
tlab/Sim
u
link e
nvi
ro
nm
ents. Fr
om
this Figure
,
it can
be observed that the inte
rnal relative h
u
m
i
dity
reaches its set
poi
nts in s
p
ite of
dist
urba
nces
that are the e
x
ternal m
e
teor
ological c
o
nditions whic
h are a
c
ting
on
gree
n
h
o
u
se
. Ho
we
ver
,
th
e contr
o
ller
maintains the
m
easured i
n
tern
al relative
hum
idity with s
m
all
deviatio
ns a
r
ou
nd
the
setp
oint.
In order to get m
o
re visibilit
y of the evol
ution
of
the internal relative humidity and different desired
values
of the
set point, w
e
prese
n
t the be
havi
ou
r o
f
internal relative
hum
idity
for 3 h
o
u
r
s as sh
ow
n in
Figu
re
8.
O
bvi
ously
, t
h
ese
re
sults in
dicate that m
o
re tim
e
is required
for the relativ
e hum
i
dit
y
to attain the
setpoi
nt when
it decreases from
an uppe
r level to lower
one. This m
eans that
the heater
m
ode is solici
t
ed to
decrease the level of relativ
e
hum
idity which takes m
o
re time in com
p
arison the acting
of t
h
e hum
i
difier to
increase
due t
o
the e
x
ternal fa
ctors.
Figu
re
8.
C
o
ntr
o
l test o
n
t
h
e
g
r
een
h
ouse: c
o
n
t
rolled
relative hum
idity
with the
p
r
op
ose
d
L
Q
G
co
ntr
o
ller i
n
[5]
,
[8]
h
The
LQG
re
gulator is a
powerful
m
e
thod for the c
o
ntrol
of li
n
ear s
y
ste
m
s in the
state-space
rep
r
ese
n
tation
due
to t
h
e L
Q
R
techni
que
w
h
ich
ge
nerates
controllers with
gua
r
antee
d
closed
loop
stability
r
obu
stn
e
ss prop
er
ty.
Figure
9
(a) and (b) illustrate
cl
early the com
port
m
ent of t
h
e two
comm
a
n
d variables associated t
o
the heater and
the hum
idifier actuator,
respectively, in order to
m
a
intain the internal rel
a
tive hum
idity at i
t
s
desire
d set
poi
n
t
towa
rds
this
r
e
gulatio
n.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN:
208
8-8
7
0
8
Synt
hesis
of an O
p
timal
Dy
namic Re
gul
ator Base
d on
Line
ar Quadr
a
tic
Gaussi
an (
L
Q
G
)
.... (
M
.
O
u
t
a
noute)
2
271
Figu
re
9.
C
o
ntr
o
l test o
n
t
h
e
g
r
een
h
ouse: c
o
n
t
rol
sig
n
als
cal
culated by
the
pr
o
pose
d
LQ
G
co
ntr
o
ller
associated t
o
:
(a)
The
heate
r
and (b) T
h
e
hum
i
di
f
i
er
actu
a
to
r in
an
in
terv
al ti
m
e
o
f
[6
],
[7] h
The LQG controller turns on the h
eater
when the
relative hum
idity gets ove
r the
refe
rence and it
turns the hum
i
difier i
f
relative hum
i
dit
y
gets under t
h
e
reference.
Results of t
h
e test indicated that
the
tim
e
constant
of t
h
e
heater syste
m
was
rathe
r
large
in the
pu
r
pose
of
re
achin
g the de
sired relative
hum
idity
leve
l
.
Th
ese r
e
su
lts wer
e
pro
b
a
bly d
u
e
to
lo
ng
ti
m
e
con
s
tants o
f
co
ntr
o
ller sy
stem
includi
ng se
ns
ors a
n
d the
dis
t
urbances
, like
external tem
p
erature a
n
d rel
a
tive
hum
idity, which affect the int
e
rnal
para
m
e
ters. B
a
se
d
on
th
e o
b
ser
v
atio
n, t
h
e e
vol
ution
o
f
the inter
n
al
relative
hum
idity was adjusted.
The
use
d
c
ont
roller is
suitab
l
e but at t
h
e e
xpa
nse
of
actuators fre
q
uent ac
tivity. This
com
p
ensator
perm
its a goo
d
perf
orm
a
nce whe
r
e we kee
p
the relative
hu
m
i
dity unde
r g
r
een
h
ouse
fo
r a long
perio
d
o
f
tim
e
and
with a m
i
nim
u
m
powe
r
whic
h is gene
r
a
ted to the
hea
t
er and
hum
idifier actuato
rs.
The ad
va
ntage
of an
observer-based controller is
the
possibility to optimise th
e state feedback
gain m
a
trix taking m
easure
m
ent
noise and actuator sat
u
ration into acc
ount.
In addition, the LQR-based
c
ont
rollers provide
reliable closed-
loo
p
sy
stem
perf
orm
a
nce de
spite o
f
st
och
a
stic plan
t
dis
t
ur
bance
.
T
h
e
s
e ex
perim
e
ntal results s
h
o
w
the
efficiency
of
the
pr
op
ose
d
st
rategy
to c
o
ntr
o
l the
re
lative
hum
idity
unde
r an
ex
pe
rim
e
n
t
al gree
nh
o
u
se
sy
stem
rega
rdless
the
pos
sible m
i
sm
atch bet
w
ee
n t
h
e
real p
r
oc
ess
an
d its ide
n
tifi
e
d m
odel.
4.
CO
NCL
USI
O
N
The a
u
tom
a
tion a
n
d
hi
gh
ef
ficiency
o
n
gree
nh
o
u
se e
nvi
ro
nm
ent
m
onitor
i
ng a
n
d c
ontr
o
l are cr
ucial
fo
r agric
u
ltura
l pro
ducti
on
. This pa
per de
s
c
ribes the p
r
ac
tical application of a
n
optim
al dy
nam
i
c regulato
r
usin
g m
odel b
a
sed Li
near
Q
u
ad
ratic Ga
uss
i
an (L
Q
G
) m
e
tho
d
of
the
relative h
u
m
i
dity
unde
r
gree
n
h
o
u
s
e. T
h
e
Num
e
rical Subspace State Space System
I
d
entification (N4S
ID) is use
d
to ide
n
tify the basic m
ode
l of the
LQG
regulator. The obtained
m
odel wa
s va
lidated with th
e expe
rim
e
ntal data. This LQ
G re
gulato
r
co
nsists
of an
optim
al
state-feedbac
k
LQR co
ntrolle
r and
Kalm
an
filter. In this
c
a
se, the separa
tion pri
n
ciple
allows
desig
n
in
g a dy
nam
i
c regulato
r
base
d on
Li
near Qua
d
ratic Gaus
sian strate
gy
, where a pe
rform
a
nce criterion is
mini
mized in
order to
regul
a
te th
e internal relative humi
dity
of
the
g
r
eenh
o
u
se. A
n
obs
er
ver base
d on
Kalm
an filter is used to estimate the re
lative hum
i
dit
y
as a state variable
i
n
greenhouse pro
cess. Then, t
h
e
use
o
f
LQG r
e
g
u
l
ato
r
d
e
m
o
n
s
tr
ates th
e sig
n
i
f
i
can
t ad
v
a
n
t
ag
es fo
r tr
ack
i
ng
d
e
sir
e
d
setp
o
i
n
t
s
wh
ich
is
a good
solutio
n
fo
r t
h
e
relative
hum
idity
unde
r
gree
n
h
o
u
se
co
ntr
o
l.
REFERE
NC
ES
[1]
N. Bennis,
et
al., “Greenhouse clim
ate modellin
g and robust co
ntrol,”
Computers and Electronics in Agricultur
e
,
vol. 61
, pp
. 96-1
07, 2008
.
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