. . , Fj„ 9 9 the PI-PD control structure is obtained. The
Settiim the relay height to zero m t^ig- -A
, ^Iiprc nre uiven in (2.22) and (2.23), respectively. At First, the forms of the PI and PD controlleis aic given . v ^
M . ic Hpsiancd based on the stability properties of the process
inner feedback PD controller is designca
.1 , irii-ntillcation procedure given in subsection 2.2.1.6.
model obtained by using the ideiititieatio i
n • . Hnsimied for the stabilized process model Gl{s) that is the Thereafter, the PI controller is dtsigncu
I I rv/iih ihe PD controller,
process model GJs) coupled with tnc
Design of PD controller
mnction model (Fig.2.9) can be written as
The stabilized process translei lunctioi
^:„(.v) = Ojs) ' (l + T.s)c' ±\
(4.21)
. + C,6;,(.v) T.s + KK,i\+
• V ,Kpd for the time delay term m the denominator of
The llrst order Pacle approMi";>P»" '
(4.21 ). L.et us choose
7- =«
■' 1
Then, (4.21) becomes
g:(.v)= Ke
(4.22)
(4.23)
' KK,D
T-
2 ;
sHKK,±\)
I. is „|,p,„cu IVom il"-- 'I---"'"""""" |. of (4 23) that C,,(.v) is capable of stabilizing an
tinstable FOPDT process model and i clocating the pole of a stable process model when
(4.24)
^'7 ^
and K
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4.4 Controller Design lor SISO Processes
K,<^ («5)
KD
Combining (4.24) and (4.25), one obtains
1 < A- < 0.26)
K ' KD
Then, one enn u.se Ihc following proportional gain to design the feedback PD controller
I llf (4.27)
Design of" PI controller
.-...iption becomes C,,C;,(-^) and die basic equations of the Now, the outer loop transiei liinction ue
phase and gain margins criteria can be expicss
Kj</\ =(o/,{KK, ±1).
=| + tun-
^ + taji
where
KK,D^
T- r ^
KK. ±
(4.28) (4.29)
(4.30) (4.31)
1^' i(./%)<^7„(.A'^,.)|-t"
om
^ + arg( { j(o^)C7'„(./n^,«)) ~ + a rg (C. I (ja)^,) G'„, (J (O,,)) = ^
Using (2.6) and (4.23) in (4.28-4.31) g'vcs
(4.32)
W ^o)T,{KK,
(4.33)
(4.34)
(4.35)
(4.36)
88
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4.4 Controller Design for SISO Processes
The .|,p..ox,n.„ion, ta,r'.v = f-i, for i<i, is "secl .o solve (4.34-435) and
44, r, .,or ,or arc assunKcl to be grcalcr lhan one. Now, using .he above appro.xima.ion,
•h / ' /' ' .1,'I'lo cxpres.sions (4.32-4.35) becomes
KK =(0 T -KK, D
,ii,„KK =(t)
KK,D
r -
(f) =7 iti 71
<oJ, co^r
- Dco^
(4.37)
(4.38)
(4.39)
(4.40)
T—V- —
rM_3 7-4.40) gives
" '■"iiilly, the .siimillancoiis soiulion o
K =
r,
V KD j
\( KK,D]
T--
(4.41)
(4.42)
1 +
whcrc
c;
\i^j
O C\ Tf
c. - -A, + and C.' r, ,
u ■ ■ nr d, ), (4.41-4.42), (4.27) and
-1) speciricalioiii. (i'«"^""'' ^ ' ^^1" a given process n.pDconti-ol'ei P'"' paraiiieiei's (K,, T,, K f, 7),).
^'^•22) are used lo cstinniie ibe
^•'<•2.2 Shnulatlon Uesult-S ^,,,,onstraied by several examples. The PI-PD
I ...ni aPpi'O''"^'^ ' x.innd "iveii in subsection 2.2.1,6 is
diis subsection, the coni ^ the mv = , • i , , ,
models obtain eonsidered two typical stable
^"niroller for the procc-ss mo We Haw
,.iliics ol s
, , loi- l^'"^^^esscs aiul two unstable P
, ,oliiCS Ol
. „,ocess" -
ll tlielb
,^,,av The process models and the
"'^Nuned usin.u suitable valdv ,,udy.
^ j|.,, tabulated m Table 4.2.
n m-
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4.4 Controller Design for SISO Processes
'rabic 4.2 Actual process, process nKKici parameters and Pl-PD controller parameters
b.xainplcs Processes
(.v + I)
(.v + l)-'
(5-1)
46^'
(4.V-I)
Process model parameters
K
1.0
2.0
1 .0
4.0 T
1 .0002
4.6127
0.9795
3.9997
D
0.4976
1.7726
0.3998
2.0013
Pl-PD Controller parameters K.-
0.3912
0.2815
0.4460
0.0848
T,
0.1543
0.7070
0.4124
1.8788
Kf
2.0050
1 .1407
2.2136
0.4998
T,
0.2488
0.8863
0.1999
1.0007
^■ample 4.4 ,vample 4.1. The process model parameters and the
msick-,- ,1,0 s,„blo rOPPT process o c.. ^ ^
-PD co„„ollei-paroniccs oblaioed us,.,t i.
. ...11,.,. nnrameters by Chidambaian
-■ The set-I
•c nhinincd usn^S
)ntroller paramcteis . , , ,
itrollei" parameters by Chidambaram s method [43] are
he .set-point weighted PID eontio ^ ^ T,= \,T,= 0.25
n r - n 098 with the set-poim
.6, T/ - 0.9, /,/ 1,^ .heir dominant pole placement ir '
n S4. ,-espocP«ly. by
' , ,, nnillt weiem -C- -.-r, 1,
1 T = 0 098 with the sct-p
1 .6, T/ - 0.9, 1.1 I j . tiominant pole placement method and
. u, 0 54 respectively, by
the set-point weigm • ■ loop performances of the proposed
iiinin*'
Jer and Nichols [.^oj ^ set-point change and the step load disturbance are
ChidambaranT.s methods I ^ lejects disturbance rapidly and shows good
I'ol performance in teiins
e.xample considcis «
. c... the idcntincd equivalent FOPDT model
= 30" are used to dcSic,n ^veichted PID eontroiler with the
\AT,\ designed a sei-p
e process. Chidambaram |4N r = ^ 5823, T,/= ' ^ ^ ^ ,^4^ fo,. q,,. higher order process
neters a; - 0.8el, T/
90
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4.4 Controller Design for SISO Processes
using the I^CPD'r model G'„,(.s ) =
2c 15.
. Tlie conlrol peiibrmances oftiie proposed method (3.V + 1)
and C hidambaram's method are shown in Fig. 4.12. The results show that the proposed method can also be extended to design the PI-PD controller for higher order processes.
1 4
1.2
0.8
□ 6
0.4
Proposed
Chidambaram (Z-N)
Chidambaram (dominant pole)
f
ii
5 10
t (Second)
15
l ig. 4. 1 I . Closed loop responses to step input and load disturbance For example 4.4
1 4
Proposed
Chidambaram
1 -
0 8 -
0.6 -
0 4 -
\rV II
0 10 20 30 40 50 60 70 80 90 100
t (Second)
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4.4 Controller Design for SISO Processes
h.uiinp/e 4.6
Conskicr an unslabic FOPDT process = PI-^D controller paranrctcrs arc
-()..) .V
i-'siiniatLxi using ihc gain margin of 4.6 and phase margin of 60". Table 4.2 shows the
, , Lisinc the method discussed in subsection 2.2.1.6 and 'dentilled process model paiamttcis usmg
, ,, Fnr the same process, Padma Sree and Chidambaram
tlie designed controller paramcteis. l oi mc v
,, , ,.,r,:„hied PID controllers using ISE and overshoot
[44] have proposed set-point weiglitea
■
Ti onntml Dcrformances of the proposed method for both the set-
'ninimi/.alion techniques. The conuoi
, A riicinrbance of maanitudc 0.5 are compared with that of
point chaime and the step load distuioanc
^ niPihods [44] as shown in Fig. 4.1 j. The proposed method I'adnia Sree and Chidambaram s mcthoa t j
or- in Iprms of the overshoot, speed oi response and
slutrv., excellent cotitrol pcrlbritiitncct. ft x, ,
r tMQ control method is that the inner leedback PD
^^ottling time. The main advantage of controller manipulates the stabilization p'O
• Proposed
. Padma Sree and Chidambaram (b=0.404) Padma Siee and Chidambaram (b=0.0)
Fiu. 4.13. Clo-scd loop responses tc
5 t (Second)
. , .,nd load disturbance for example 4.6
;p mpm
92
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4.4 Controller Design for SISO i'rocesses
hlxiiniple 4. 7
I^CI Ihc unsuible FOPDT process be G(s) = ■ The eomroller paranrclers obtained by
lire proposed niclhod Ibr Ihc process model osing g. = 4.5 and = 60" arc given in Table 4 2 For .he .same pioeess, lire parame.o,-s of.he PID-P eont,■oiler suggested by Park et
ai. [42] are AV = 0.068. T, = 1.885, T, = 4.296 and A', - 0.35 and the Pi-PD controller
, . I „ Mm nre/f . = 0.131, 7) =2,/w= 0.5 and r</= 1. Fig. 4.14
parameters by Majlii and Atlieilon [ J' '
thp set-Doint input and the static load disturbance of
(a) shows the responses given by the sei p
I vnotimH nroDOsed by Majhi and Athcrton results in better