6
Design o Con olle s o Time Delay Sys ems:
In eg a ing and Uns able Sys ems
Pe Dos ál, F an išek Gazdoš, and Vladimí Bobál
Facul y o Applied In o ma ics, Tomas Ba a Uni e si y in Zlín
Nad S áněmi 4511, 760 05 Zlín 5,
Czech Republic
1. In oduc ion
The p esence o a ime delay is a common p ope y o many echnological p ocesses. In
addi ion, a pa o ime delay sys ems can be uns able o ha e in eg a ing p ope ies.
Typical examples o such p ocesses a e e.g. pumps, liquid s o ing anks, dis illa ion columns
o some ypes o chemical eac o s.
Plan s wi h a ime delay o en canno be con olled by usual con olle s designed wi hou
conside a ion o he dead- ime. The e a e a ious ways o con ol such sys ems. A numbe
o me hods u ilise PI o PID con olle s in he classical eedback closed-loop s uc u e, e.g.
(Pa k e al., 1998; Zhang and Xu, 1999; Wang and Clue , 1997; Sil a e al., 2005). O he
me hods employ ideas o he IMC (Tan e al., 2003) o obus con ol (P okop and Co iou,
1997). Con ol esul s o a good quali y can be achie ed by modi ied Smi h p edic o
me hods, e.g. (Ås öm e al., 1994; De Pao , 1985; Liu e al., 2005; Majhi and A he on, 1999;
and Ma ausek and Micic, 1996).
P inciples o he me hods used in his wo k and design p ocedu es in he 1DOF and 2DOF
con ol sys em s uc u es can be ound in pape s o au ho s o his a icle (Dos ál e al., 2001;
Dos ál e al., 2002). The con ol sys em s uc u e wi h wo eedback con olle s is conside ed
(Dos ál e al., 2007; Dos ál e al., 2008). The p ocedu e o ob aining con olle s is based on he
ime delay i s o de Padé app oxima ion and on he polynomial app oach (Kuče a, 1993).
Fo uning o he con olle pa ame e s, he pole assignmen me hod exploi ing he LQ
con ol echnique is used (Hun e al., 1993). The esul ing p ope and s able con olle s
ob ained ia polynomial Diophan ine equa ions and spec al ac o iza ion echniques ensu e
asymp o ic acking o s ep e e ences as well as s ep dis u bances a enua ion. S uc u es o
de eloped con olle s oge he wi h analy ically de i ed o mulas o compu a ion o hei
pa ame e s a e p esen ed o i e ypical plan ypes o in eg a ing and uns able ime delay
sys ems: an in eg a ing ime delay sys em (ITDS), an uns able i s o de ime delay sys em
(UFOTDS), an uns able second o de ime delay sys em (USOTDS), a s able i s o de plus
in eg a ing ime delay sys em (SFOPITDS) and an uns able plus in eg a ing ime delay
sys em (UFOPITDS). P esen ed simula ion esul s documen use ulness o he p oposed
me hod p o iding s able con ol esponses o a good quali y also o a highe a io be ween
he ime delay and uns able ime cons an s o he con olled sys em.
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Time-Delay Sys ems
114
2. App oxima e ans e unc ions
The ans e unc ions in he sequence ITDS, UFOTDS, USOTDS, SFOPITDS and UFOPITDS
ha e hese o ms:
1() ds
K
Gs e
s
τ
−
= (1)
2() 1ds
K
Gs e
s
τ
τ
−
=− (2)
312
() (1)(1)
ds
K
Gs e
ss
τ
ττ
−
=−+ (3)
4,5() (1)
ds
K
Gs e
ss
τ
τ
−
=±. (4)
Using he i s o de Padé app oxima ion, he ime delay e m in (1) – (4) is app oxima ed by
2
2
dsd
d
s
es
τ
τ
τ
−−
≈+. (5)
Then, he app oxima e ans e unc ions ake o ms
01
121
(2 )
() (2 )
d
A
d
Ksbbs
Gsss
sas
τ
τ
−−
==
++ (6)
whe e 02
d
K
b
τ
= , 1
bK
=
and 12
d
a
τ
= o he ITDS,
01
2210
(2 )
() (1)(2 )
d
A
d
Ks bbs
Gs ss
sasa
τ
ττ
−−
==
−+
+
+ (7)
wi h 02
d
K
b
τ
τ
=,1K
b
τ
=
, 02
d
a
τ
τ
=− , 12d
d
a
τ
τ
ττ
−
= and τd ≠ 2τ o he UFOTDS,
312
(2 )
() (1)(1)(2)
d
A
d
Ks
Gs ss s
τ
ττ τ
−
=−++
01
32
210
bbs
sasasa
−
=
+
+−
(8)
whe e
012
2
d
K
b
τ
ττ
=, 112
K
b
τ
τ
=, 012
2
d
a
τ
ττ
=, 12
112
2( ) d
d
a
τ
ττ
τττ
−−
=,
12 1 2
212
2dd
d
a
τ
τττττ
τττ
+−
= and τd ≠ 2τ1 o he USOTDS, and,
01
4,5 32
21
(2 )
() (1)(2 )
d
A
d
Ks bbs
Gs
ss s sasas
τ
ττ
−−
==
±+ ++
(9)
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Design o Con olle s o Time Delay Sys ems: In eg a ing and Uns able Sys ems
115
whe e 02
d
K
b
τ
τ
=, 1K
b
τ
=
, 12
d
a
τ
τ
=± , 22d
d
a
τ
τ
ττ
±
= and τd ≠ 2τ o he SFOPITDS and
UFOPTDS, espec i ely.
All app oxima e ans e unc ions (6) – (9) a e s ic ly p ope ans e unc ions
()
() ()
A
bs
Gs as
= (10)
whe e b and a a e cop ime polynomials in s ha ul ill he inequali y de
g
de
g
ba<.
The polynomial a(s) in hei denomina o s can be exp essed as a p oduc o he s able and
uns able pa
() () ()as asas
+−
= (11)
so ha o ITDS, UFOTDS, USOTDS and SFOPITDS he equali y
de
g
de
g
1aa
+
=
− (12)
is ul illed.
3. Con ol sys em desc ip ion
The con ol sys em wi h wo eedback con olle s is depic ed in Fig. 1. In he scheme, w is
he e e ence, is he load dis u bance, e is he acking e o , u0 is he con olle ou pu , y is
he con olled ou pu , u is he con ol inpu and GA ep esen s one o he app oxima e
ans e unc ions (6) – (9) in he gene al o m (10).
Rema k: He e, he app oxima e ans e unc ion GA is used only o a con olle de i a ion.
Fo con ol simula ions, he models G1 – G5 a e u ilized.
Bo h w and a e conside ed o be s ep unc ions wi h Laplace ans o ms
0
() w
Ws s
=, 0
()
Vs s
=. (13)
The ans e unc ions o con olle s a e assumed as
()
() ()
q
s
Qs
p
s
=#
#, ()
() ()
s
Rs
p
s
=# (14)
whe e ,andq p
##
a e polynomials in s.
- -
y u u
0
e w R
Q
G
A
Fig. 1. The con ol sys em.
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Time-Delay Sys ems
116
4. Applica ion o he polynomial me hod
The con olle design desc ibed in his sec ion ollows he polynomial app oach. Gene al
equi emen s on he con ol sys em a e o mula ed as i s in e nal p ope ness and s ong
s abili y (in addi ion o he con ol sys em s abili y, also he con olle s abili y is equi ed),
asymp o ic acking o he e e ence and load dis u bance a enua ion. The p ocedu e o
de i e admissible con olle s can be pe o med as ollows:
T ans o ms o basic signals in he closed-loop sys em om Fig.1 ake ollowing o ms ( o
simpli ica ion, he a gumen s is in some equa ions omi ed)
() () ()
b
Ys Ws pVs
d
=+
⎡
⎤
⎣
⎦
# (15)
1
() ( ) () ()Es ap bqWs bpVs
d
=+ −
⎡
⎤
⎣
⎦
## # (16)
() () ()
a
Us Ws pVs
d
=+
⎡
⎤
⎣
⎦
# (17)
whe e
() ()() () () ()ds asps bs s qs=+ +
⎡
⎤
⎣
⎦
##
(18)
is he cha ac e is ic polynomial wi h oo s as poles o he closed-loop.
Es ablishing he polynomial as
() () () s s qs
=
+# (19)
and subs i u ing (19) in o (18), he condi ion o he con ol sys em s abili y is ensu ed when
polynomials
p
# and a e gi en by a solu ion o he polynomial Diophan ine equa ion
()() ()() ()asps bs s ds
+
=
# (20)
wi h a s able polynomial d on he igh side.
Wi h ega d o ans o ms (13), he asymp o ic acking and load dis u bance a enua ion a e
p o ided by di isibili y o bo h e ms ap bq
+
##
and
p
# in (16) by s. This condi ion is ul illed
o polynomials
p
#and q
#ha ing o ms
() ()
p
ssps
=
#, () ()qs sqs
=
#. (21)
Subsequen ly, he ans e unc ions (14) ake o ms
()
() ()
q
s
Qs
p
s
=, ()
() ()
s
Rs s
p
s
= (22)
and, a s able polynomial p(s) in hei denomina o s ensu es he s abili y o con olle s ( he
s ong s abili y o he con ol sys em).
The con ol sys em sa is ies he condi ion o in e nal p ope ness when he ans e unc ions
o all i s componen s a e p ope . Consequen ly, he deg ees o polynomials q and mus
ul il hese inequali ies
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Design o Con olle s o Time Delay Sys ems: In eg a ing and Uns able Sys ems
117
de
g
de
g
qp
≤
, de
g
de
g
1 p
≤
+. (23)
Now, he polynomial can be ew i en o he o m
() () () s s sqs=+ . (24)
Taking in o accoun sol abili y o (20) and condi ions (23), he deg ees o polynomials in
(19) and (20) can be easily de i ed as
de
g
de
g
de
g
a==, de
g
de
g
1qa
=
−, de
g
de
g
1pa≥−, de
g
2de
g
da≥. (25)
Deno ing deg a = n, polynomials , and q ha e o ms
0
() ni
i
i
s s
=
=
∑
,
0
() ni
i
i
s s
=
=
∑
, 1
1
() ni
i
i
qs qs
−
=
=∑ (26)
and, ela ions among hei coe icien s a e
00
=
, iii
q
+
= o 1,... ,in
=
(27)
Since by a solu ion o he polynomial equa ion (20) only coe icien s i can be calcula ed,
unknown coe icien s i and qi can be ob ained by a choice o selec able coe icien s
0,1
i
γ
∈ such ha
iii
γ
=
, (1 )
iii
q
γ
=
− o 1,... ,in
=
. (28)
The coe icien s γi di ide a weigh be ween nume a o s o ans e unc ions Q and R.
Rema k: I 1
i
γ
= o all i, he con ol sys em in Fig. 1 educes o he 1DOF con ol
con igu a ion (Q = 0). I 0
i
γ
=
o all i, and, bo h e e ence and load dis u bance a e s ep
unc ions, he con ol sys em co esponds o he 2DOF con ol con igu a ion.
The con olle pa ame e s hen esul om solu ions o he polynomial equa ion (20) and
depend upon coe icien s o he polynomial d. The nex p oblem he e is o ind a s able
polynomial d ha enables o ob ain accep able s abilizing and s able con olle s.
5. Pole assignmen
The polynomial d is conside ed as a p oduc o wo s able polynomials g and m in he o m
() () ()ds gsms= (29)
whe e he polynomial g is a monic o m o he polynomial g
′
ob ained by he spec al
ac o iza ion
() () ()() () ()sa s sa s b s b s
g
s
g
s
ϕ
∗∗∗
′
′
+=
⎡⎤⎡⎤
⎣⎦⎣⎦ (30)
whe e ϕ > 0 is he weigh ing coe icien .
Rema k: In he LQ con ol heo y, he polynomial g
′
esul s om minimiza ion o he
quad a ic cos unc ion
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Time-Delay Sys ems
118
{}
22
0
() ()Je u d
ϕ
∞
=+
∫$ (31)
whe e ()e is he acking e o and ()u
$is he con ol inpu de i a i e.
The second polynomial m ensu ing p ope ness o con olle s is gi en as
2
() ()
d
ms a s s
τ
+
==+ (32)
o bo h ITDS and UFOTDS,
2
21
() ()
d
ms a s s s
τ
τ
+⎛⎞
⎛⎞
==+ +
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎝⎠
⎝⎠ (33)
o he USOTDS, and,
21
()
d
ms s s
τ
τ
⎛⎞
⎛⎞
=+ +
⎜⎟
⎜⎟
⎜⎟
⎝⎠
⎝⎠ . (34)
o bo h UFOPITDS and SFOPITDS.
The coe icien s o he polynomial d include only a single selec able pa ame e ϕ and all
o he coe icien s a e gi en by pa ame e s o polynomials b and a. Consequen ly, he closed
loop poles loca ion can be a ec ed by a single selec able pa ame e . As known, he closed
loop poles loca ion de e mines bo h s ep e e ence and s ep load dis u bance esponses.
Howe e , wi h espec o he ans o m (13), i may be expec ed ha weigh ing coe icien s γ
in luence only s ep e e ence esponses.
Then, he monic polynomial g and de i ed o mulas o hei pa ame e s ha e o ms
32
210
()
g
ss
g
s
g
s
g
=
+++
(35)
o bo h ITDS and UFOTDS, whe e
2
01221
2
21 14 2 4
,,
dd d
KK
gggKgg
τϕ ϕτ
ϕ
τ
⎛⎞
==+=+
⎜⎟
⎜⎟
⎝⎠ (36)
o he ITDS, and,
22
01 2
22 2 2
21
211 1 1
,4 1,
11
24
dd
dd
dd
d
K
ggKgK
gg
ττ τ
τ
τττϕ ϕ
ϕ
ττ τ τ
ττ ϕ
⎛⎞
== ++
⎜⎟
⎝⎠
=++
(37)
o he UFOTDS, and,
432
3210
()gs s gs g s gs g
=
++++ (38)
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Design o Con olle s o Time Delay Sys ems: In eg a ing and Uns able Sys ems
119
o USOTDS, SFOPITDS and UFOPITDS, whe e
2
012
22 222
12 12 12 12
22 2
12
213 3 2
222 2 2 2
12 12 1 2
21 4 1 4
,
4( )
241 2411
,
dd d
d
ddd
KKK
ggg
K
ggg g g
τττ τττ ϕ
ϕτττττ
ττ τ
ϕτττϕ
τ
ττ ϕ τ τ τ
⎛⎞
==++
⎜⎟
⎜⎟
⎝⎠
++
=− + =+++
(39)
o he USOTDS, and,
2
012
2
213 3 2
22
24
11
,,
41 1 2 41
2,
dd
dd d
KK
K
ggg
ggg K g g
ττ ϕ ττ
ϕτ
ττ ττ
ϕ
ϕττ
⎛⎞
==+
⎜⎟
⎜⎟
⎝⎠
⎛⎞
=+ − =++
⎜⎟
⎜⎟
⎝⎠
(40)
o bo h SFOPITDS ans UFOPITDS.
The ans e unc ions o con olle s a e
21
0
() qs q
Qs sp
+
=+, 2
210
0
() ()
s s
Rs ss p
+
+
=+ (41)
o bo h ITDS and UFOTDS, and,
232
321 3210
22
10 10
() , () ()
qs qs q s s s
Qs Rs
spsp sspsp
+
++++
==
++ ++ (42)
o he USOTDS, SFOPITDS and UFOPITDS.
6. Con olle pa ame e s
Fo he sake o limi ed space, o mulas de i ed om (20) o all conside ed sys ems oge he
wi h condi ions o he con olle s’ s abili y a e in oduced in he o m o ables. Pa ame e s i
and qi in (41) and (42) can hen be calcula ed om i acco ding o (28).
02 1 0
(2 )
4
dd
p
ggg
ττ
=+ + , 00
1
g
K
=
110
1()
d
gg
K
τ
=+
, 210
(2 )
4
dd
gg
K
ττ
=+
p0 > 0 o all τd
Table 1. Con olle pa ame e s o he ITDS
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Time-Delay Sys ems
120
210
0
2( )2
2
2
d
d
d
ggg
p
τ
ττ
ττ
⎡⎤
+
++
⎢⎥
⎣⎦
=−
00
g
K
τ
=, 1 010
1()
d
pgg
K
ττ
=++
⎡
⎤
⎣
⎦, 202
1()1 pg
K
τ
=
−−
⎡
⎤
⎣
⎦
p0 > 0 o τd < 2τ
Table 2. Con olle pa ame e s o he UFOTDS
31 2 1 0 1
01
2
22( )
2
2
d
d
d
gggg
p
τ
ττ
τ
ττ
⎡⎤
++++
⎢⎥
⎣⎦
=−, 131
1
pg
τ
=+
1
00
g
K
τ
=,
()
101120
1()
d
pg g
K
τττ
⎡
⎤
=+++
⎣
⎦
12 2
2120312121
1
14 4 1
1
dd
pggg
K
ττ τ
ττ τ ττ
τττ
⎡
⎤
⎡
⎤
⎛⎞⎛⎞
⎛⎞
⎢
⎥
=+−−+++−
⎢
⎥
⎜⎟⎜⎟
⎜⎟
⎜⎟
⎜⎟⎜⎟
⎢
⎥
⎢
⎥
⎝⎠
⎝⎠⎝⎠
⎣
⎦
⎣
⎦
2
31023
1
1
() pgg
K
ττ
τ
⎡
⎤
=−−−
⎢
⎥
⎣
⎦
p1 > 0 o all τd , p0 > 0 o τd < 2τ1
Table 3. Con olle pa ame e s o he USOTDS
02 1 0
(2 )
4
dd
pg g g
ττ
=+ + , 13
1
pg
τ
=
+
00
1
g
K
=, 11 0
1()
d
g g
K
ττ
=++
⎡
⎤
⎣
⎦, 2100
1(2 )(2 ) 2
4ddd
gg g
K
ττ τ ττ
=+++
⎡
⎤
⎣
⎦
310
2
4
dd
gg
K
ττ τ
=+
⎡
⎤
⎣
⎦
p1, p0> 0 o all τd
Table 4. Con olle pa ame e s o he SFOPITDS
2
3210
0
4
4(2 ) 24
2
dd
d
d
gggg
p
ττ
ττ
τ
ττ
⎛⎞
+
++++
⎜⎟
⎜⎟
⎝⎠
=−, 13
2
pg
τ
=
+
00
1
g
K
=, 11 0
1()
d
g g
K
ττ
=++
⎡
⎤
⎣
⎦,
20321
44
18 8
11
ddd d
pggg
K
ττ
τ
τ
ττ ττ
⎡⎤
⎛⎞ ⎛⎞
=−−−+−−
⎢⎜⎟ ⎜⎟
⎥
⎜⎟ ⎜⎟
⎢⎝⎠ ⎝⎠ ⎦
⎣, 3023
12
()2 pgg
K
τ
τ
⎡
⎤
=−−−
⎢
⎥
⎣
⎦
p1 > 0 o all τd , p0 > 0 o τd < 2τ
Table 5. Con olle pa ame e s o he UFOPITDS
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Design o Con olle s o Time Delay Sys ems: In eg a ing and Uns able Sys ems
121
7. Simula ion esul s
The simula ions we e pe o med by MATLAB-Simulink ools. Fo all simula ions, he uni
s ep e e ence w was in oduced a he ime = 0 and he s ep load dis u bance a e
se ling o he s ep e e ence esponses.
7.1 ITDS
In he ans e unc ion (1), le K = 1. The esponses in Fig. 2 o τd = 5 show he e ec o ϕ
upon he con ol quali y. An inc easing alue ϕ imp o es con ol s abili y, and, by choosing
i s alue highe , ape iodic esponses can be ob ained. Simula ion esul s shown in Fig. 3
demons a e he in luence o pa ame e s γ on he con ol esponses. Thei smalle alues
accele a e s ep e e ence esponses bu hey do no a ec load dis u bance esponses. Highe
alues o γ can lead o o e shoo s and oscilla ions. The e ec o pa ame e s γ on he con ol
0 20 40 60 80 100 120 140 160 180
0.0
0.2
0.4
0.6
0.8
1.0
1.2
y( )
Time
ϕ = 100
ϕ = 400
ϕ = 900
w
Fig. 2. ITDS: con olled ou pu esponses (τd = 5, = - 0.1, γ1 = γ2 = 0)
0 20 40 60 80 100
0.0
0.2
0.4
0.6
0.8
1.0
y( )
Time
γ1 = γ2 = 0
γ1 = γ2 = 0.25
γ1 = γ2 = 0.4
w
Fig. 3. ITDS: con olled ou pu esponse (τd = 5, = - 0.1, ϕ = 900).
0 50 100 150 200
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
y( ), u( )
Time
γ1 = γ2 = 0
γ1 = γ2 = 0.4
y( )
5xu( )w
Fig. 4. ITDS: Con ol inpu and con olled ou pu esponses (τd = 5, ϕ = 900)
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