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G.L. Raja, A. Ali / ISA Transactions 65 (2016) 394–406
Furthermore, the dynamics of the primary process model is
represented by any one of the following transfer functions:
Gp1 ðsÞ ¼
Gp1 ðsÞ ¼
Gp1 ðsÞ ¼
Gp1 ðsÞ ¼
K 1 ð1 p1 sÞ θ1 s
e
τ1 s þ 1
K1
e θ1 s
τ1 s þ 1
K 1 ð1 p1 sÞ θ1 s
e
τ1 s 1
K1
e θ1 s
τ1 s 1
ð9aÞ
ð9bÞ
ð10aÞ
ð10bÞ
ð11aÞ
Gp1 ðsÞ ¼
K 1 θ1 s
e
s
ð11bÞ
Gp1 ðsÞ ¼
K 1 ð1 p1 sÞ θ1 s
e
sðτ1 s þ 1Þ
ð12aÞ
Gp1 ðsÞ ¼
K1
e θ1 s
sðτ1 s þ 1Þ
ð12bÞ
ð13aÞ
K1
e θ1 s
c2 s2 þ c1 s þ 1
ð13bÞ
Eqs. (10a)-(13a) represent unstable ﬁrst order plus time delay
with positive zero (UFOPTDPZ), integral plus time delay with
positive zero (IPTDPZ), second order integral plus time delay with
positive zero (SOIPTDPZ) and stable second order plus time delay
with positive zero (SOPTDPZ) process models, respectively.
Moreover, (10b)–(13b) represent unstable ﬁrst order plus time
delay (UFOPTD), integral plus time delay (IPTD), second order
integral plus time delay (SOIPTD) and stable second order plus
time delay (SOPTD) process models, respectively.
3. Controller design
The proposed PCCS consists of three controllers (Gc1 , Gcd1 and
Gcd2 ). Controller parameters are derived in terms of known plant
model parameters in this section.
3.1. Secondary disturbance rejection controller Gcd2
IMC approach is used to design the controller Gcd2 . The process
model Gp2 can be factorized into inverting and non-inverting parts
as given below:
ð14Þ
þ
GP2
where
consists of time delays and right half-plane zeros (if
any) whereas GP2
denotes the delay free part of the secondary
process model. The IMC controller is given by
ð15Þ
In the above equation, M is a low-pass IMC ﬁlter whose transfer
function is 1= λ2 s þ1 . Using (8), (14) and (15), Gcd2 is obtained as
Gcd2 ðsÞ ¼
ðτ 2 s þ 1Þ
K 2 λ2 s þ 1
where λ2 represents the adjustable tuning parameter.
ð17Þ
Gp ðsÞ ¼
ð16Þ
K 1 ð1 p1 sÞe θ1 s
ðτ1 þ 0:5K p K 1 p1 θ1 Þs2 þ ð1 0:5K p K 1 θ1 K p K 1 p1 Þs þ K p K 1
ð18Þ
Assuming K p ¼ β =ðK 1 ð0:5θ1 þ p1 ÞÞ, (18) satisﬁes Routh–Hurwitz
stability criterion if 0 o β o1. Based on extensive simulation studies given in Appendix A, the recommended value of β is 0.01.
Since IPTDPZ process model is a limiting case of SOIPTDPZ process
model (with τ1 ¼ 0), the above controller settings are also valid for
IPTDPZ process models. Substituting K p ¼ β =ðK 1 ð0:5θ1 þ p1 ÞÞ in
(18), we get the following SOPTDPZ transfer function:
Gp ðsÞ ¼
K 1 ð1 p1 sÞ θ1 s
e
Gp1 ðsÞ ¼
c2 s2 þ c1 s þ 1
1
Gcd2 ðsÞ ¼ MðsÞ
Gp2 ðsÞ
K 1 ð1 p1 sÞe θ1 s
ðsð1 þ τ1 sÞÞ þ K p K 1 ð1 p1 sÞð1 þT d sÞ 1 0:5θ1 s = 1 þ 0:5θ1 s
Assuming T d ¼ 0:5θ1 , the above equation reduces to
K 1 ð1 p1 sÞ θ1 s
e
s
þ
Gp2 ðsÞ ¼ Gp2
ðsÞGp2
ðsÞ
3.2.1. For SOIPTDPZ/IPTDPZ primary process model
Substituting Gcd1 ðsÞ ¼ K p ð1 þ T d sÞ and Gp1 ðsÞ ¼ ðK 1 ð1 p1 sÞe θ1 s Þ
=ðsðτ1 s þ 1ÞÞ in (6) and approximating the time delay term in the
denominator by a ﬁrst order Padé approximation R1;1 ðsÞ [15], we get
Gp ðsÞ ¼
Gp1 ðsÞ ¼
Gp1 ðsÞ ¼
3.2. Stabilizing controller Gcd1
K 1 ð1 p1 sÞe θ1 s
a2 s2 þ a1 s þ a0
ð19Þ
where, a2 ¼ ðτ1 þ 0:5K p K 1 p1 θ1 Þ, a1 ¼ ð1 0:5K p K 1 θ1 K p K 1 p1 Þ and
a0 ¼ K p K 1 . If Gp1 follows IPTD dynamics, then (19) reduces to a
FOPTD transfer function given by ðK 1 e θ1 s Þ=ða1 s þ a0 Þ.
3.2.2. For UFOPTDPZ primary process model
Substituting Gcd1 ðsÞ ¼ K p ð1 þ T d sÞ and (10a) in ð6Þ and approximating the time delay term in the denominator by
1 0:5θ1 s = 1 þ0:5θ1 s , we get
Gp ðsÞ ¼
K 1 ð1 p1 sÞe θ1 s
τ1 s 1 þ K p K 1 ð1 þ T d sÞð1 p1 sÞ 1 0:5θ1 s = 1 þ 0:5θ1 s
ð20Þ
θ1 s
=(ð0:5θ1 p1 K p K 1 Þs2
Assuming T d ¼ 0:5θ1 , we get Gp ðsÞ ¼K 1 e
þ ðτ1 K p K 1 p1 0:5K p K 1 θ1 Þs þ K p K 1 1). Using Routh–Hurwitz
stability criteria, we get the minimum and maximum values of K p
as K pmin ¼1=K 1 and K pmax ¼ τ1 =ðK 1 ðp1 þ0:5θ1 ÞÞ, respectively. In the
present work, K p is assumed as
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
m
τ1
ð21Þ
K p ¼ m K pmin K pmax ¼
K 1 ðp1 þ 0:5θ1 Þ
Assuming m¼ 1 results in large overshoot and settling time for
processes with θ1 =τ1 o 1. Therefore, the following strategy is used
to obtain a suitable value for m: starting from m ¼ 1, m is reduced in
steps of 0.01. For each value of m, b1 ¼ ðτ1 K p K 1 p1 0:5K p K 1 θ1 Þ
and b0 ¼ ðK 1 K p 1Þ are computed until either b1 or b0 becomes
negative. The least value of m for which b1 and b0 remains positive
is used to obtain K p using (21). Substituting (21) and T d ¼ 0:5θ1 in
(20) results in the following SOPTDPZ transfer function:
Gp ðsÞ ¼
K 1 ð1 p1 sÞe θ1 s
ðb2 s2 þ b1 s þ b0 Þ
ð22Þ
where b2 ¼ 0:5θ1 p1 K p K 1 , b1 ¼ ðτ1 K p K 1 p1 0:5K p K 1 θ1 Þ and
b0 ¼ ðK 1 K p 1Þ. Tuning rules that are derived in this subsection for
Gcd1 are summarized in Table 1.
Remark-1. Since the design of Gcd1 involves ﬁrst order Padé
approximation R1;1 ðsÞ [15], the controller settings obtained in
Section 3.2.2 is recommended only for UFOPTDPZ process models
with 0 o ðp1 þ θ1 Þ=τ1 r 1 and UFOPTD process models with
0 o θ1 =τ1 r 1. If the proposed method needs to be applied for
UFOPTD process models with θ1 =τ1 4 1, it is recommended to use
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