a novel method of pid thing for integrating processes
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7/27/2019 A Novel Method of PID Thing for Integrating Processes
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Proceedingsof the 42nd IEEE
Conference on Decision and ControlMaui, Hawaii USA,December 2003
ANovel Method of PID T h in g for Integrating Processes
Jianghua Xu, Huihe Shao
Institute ofAutomation, Shanghai Jiao Tong University, Shanghai 200030, China
Abstract: PID control is widely used to control stable processes,
however, its application to integrating processes is less common.
In this paper, we proposed a new PID controller tuning method
for integrating processes with time delay to meet a new robust
specification. With the proposed PID tuning method, we can
obtain a loop transfer function with the real part close to -0.5.
This guarantees both robustness and performance. Simulation
examples are given to show the performance of the method.
1. Introduction
The proportional-integral-dexivativePID) controllers are still
widely used in the process industries even though more advanced
control techniques have been developed. The main reason is that
the PID controllers have simple structure and are robust to
modeling error and that many advanced control algorithms, such
as model predictive control, are based on the PID control. As
indicated in [I1, more than 95% of the control loops are of PID
type in process control. Over the years, there are many formulas
derived to tune the PID controllers for stable processes, such as
Ziegler-Nichols, Cohen-Coon, internal model control, integral
absolute error optimum @E, IAE, and ITAE), and recently
proposed tuning However, it is difficult to control
integrating processes with time delay. Recently, many tuning
methods for integrating processes have been prop~sed[~~-[~]*['~~.
However, they usually either show poor closed-loop response
such as excessive overshoot and large settling time or have
complicated formulas.
For controller design purposes, many integrating processes
are often approximated by low-order plus time delay model,
which can be identified by P control method [ I 2 ] or relay control
method[']. Because the resulting models are usually imprecise
and the parameters of all physical systems vary with the working
condition and time, robustness is always a primary concern when
analyzing and designing the control system. In this paper, a new
tuning method for the PID controller with setpoint weighting['] is
proposed for integrating processes to meet robustness
specification. The control scheme first presents the internal loop
design strategy. Then, simple and effective PID controller with
setpoint weighting is designed based on robustness specification.
Because of a good loop transfer with the real part closed to -0.5
0-7803-7924-1/03/$17.00 02003 IEEE 139
TuA04-6
in low frequency, the control system guarantees both robustness
and performance. Simulation examples show that the proposed
method achieves better control performance and robustness
compared with other methods.
2. PIDTuning method
For controller design purposes, many of the integrating
processes are adequately described by low-order plus dead-time
transfer function
Ke-'L
s(Ts + 1)G,(s)=-
We first present an inner feedback loop for integrating
processes, the block diagram of the proposed method is shown in
Fig.1. Here, the PD controller (PD) in the inner feedback loop
plays an important role in changing the integrating process to the
stable process.
Fig.1. Block diagram of a two-loop controller for integrating
process
Denote the PID controller transfer function by Gp&) and is
given by
Denote the PD controller transfer function by GpD(s) and is given
by
G (s) =k, + k, s (3)Robustness is always of primary concern for process control
when the control systems are designed and analyzed because the
models used for the design of controllers are usually imprecise
and the parameters of all physical systems vary with the working
condition and time. We introduce a new robustness
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specificationh’ ”,which is defined as
(4)
where the loop transfer function is GXs) = G,(s)G,(s), G,(s) is
controller transfer function and the quantity h is simply the
inverse of maximum of absolute real part of loop transfer
function GIG@, as shown in Fig.2. The new specification is
similar with the gain margin and phase margin specifications.
The following relations are obtained
A,,,> h
(bn,>arccos( +)
i t I m G f ( j @ )*
Re G I ( j @ )
Fig.2. Nyquist curve of the loop transfer function.
It is obvious that the new specification satisfies both the gain
margin and phase margin requirement to a degree. Reasonable
values of h are in the range fiom1.5 to 2.5.
The PD controller transfer function(2) is also written as
GpD(s) k, +k d s= kl(a bs) (7)
We choose
a= 1 , b=T ( 9 )
With the PD controller n the inner feedback loop, the internal
loop transfer function can be obtained as
G l ( s )= GPD( s )Gp ( s )-el K -Ls (IO)
where k is determined based on the new robustness
S
specificationh.
From equation (1 0), we can obtain
(13)
To find the maximum, we note fiom (13 ) that
o+o
Hence,
Considering a internal loop transfer function with a good
shape, i.e. the real part close to -0.5 in low lkequency, we choose
h = 2 . From equation 1 5 ) and h=2, we get
. (16)kl =-2K L
So the PD controller is
(17)1 T
2K L 2KLGpD ( s ) =- -
The PD controller in equation (17) can guarantee both
robustness and performance of the inner feedback loop.
With the PD controller, the internal closed-loop ransfer
function is given by
G’ s G,(s) - Ke-L p (18)) = 1+ G p D G p ( s ) Ts2+ s +k,K(Ts + l)e-”
Considering Taylor series expansion, the time delay term in
the denominator of equation (18) can be approximated by
e-” E I - L S (19)
Then equation (18) is given by
G ; s )IG, , (s )= Ke-” (20)
(1 - k,KL)Ts* + (I - ,K L t ,KT)s+ k,K
Here, Gm(s) denotes the second-order plus time-delay model
obtained lkom the Taylor series expansion method.
Because the characteristic equation of G,(s) should have
negative poles to be stable, the following condition must be
satisfied fiom the Routh-Hurwitz stability criterion:
(21)kl C-KL
The equation ( 1 6 ) satisfies the stability criterion (21) .
For convenience of the outer loop controller design, the equat ion
(20) is rewritten as
The PID controller ransfer function (2) is also written as
K K .
k k kWhere A = % , B = P and C=-.Thecontroller
zeros are chosen to be equal to the poles of model Gm(s), that is
T 7 B = - ~ K L k ~ K T- IKLA = - and c = k, .Hence
K K ’
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where k is determined based on new robustness specificationh.
From equation (24), we can obtain
(25)
IReG,(M)G,O’dl= 13To find the maximum, we note from (25) that
Hence,
(27)=U
A
Considering a outer loop transfer hc t i o n with a good shape,
i.e. the real part close to -0.5 in low frequency, we choose k=2.
From equation (27) and h=2, we get
0.5k = -
L
Hence PID settings are given as
0.5(1- klKL + kl K T )
KLOSk,
0.5(1- klKL)T
KL
From equation ( 1 6) and (29), we get
Let’s reconsider the block diagram of Fig.1, where
e(s) = r(s)- ( s ) The process input u(s) can be written easily
as the following equation.
, K b = K , + k , , K i = K i ,
(31)r(s)- ( s ) ) - (k, + kds)y(s)
KIntroducing b - P
KP -t kP
c=- Kd and K i = Kd +kd ,we have
Kd -I-d
The expression (32) is the same as PID controller with setpoint
weighting[’], thus the block diagram of Fig. 1 can be changed into
a PID control loop without an inner feedback loop, whereK‘p,Ki,
Kd and setpoint weighting b, c are new PID settings. The new
PID setting for integrating processes are given by
* 0 . 2 5 ( 3 L + T )K, =
KL2
0.75TKL
K : , =-
(33)
(34)
( 35 )
1
3c = - (37)
3. Simulation examples
The following will give the comparisons between the proposed
PID tuning method and other method.
Example I Consider an integrating process with small dead time.
e-o.2s
G p ( s )= ~(0.8s 1)
The proposed method yields the PID control settings as
K’,=8.75, Kic6.25, K>=3.0, b4.714 and c4.33. The control
performance of the proposed method is compared with
Astrom-Hagglund PID tuning method [2] for integrating
processes. The comparison of performance is shown in Fig.3
(proposed method, solid line; Mom-Hagglund method, dashed
line), where a step load disturbance is added at t-20s. The
proposed methods shows better control performance for both
setpoint change and load disturbance.
1 5
I
0 5 1 0 1 5 m x ~ 3 5
Fig.3. Comparisonof process responses
3
Example 2 Consideran ntegrating process with large dead time
e-2’
G , ( s ) = -s(s +1)
The proposed method yields the PID control settings as
Kb4.4375, K’i=0.0625, Ki4.315, b4.4286 and ~ 4 . 3 3 3 . he
control performance of the proposed method is compared with
Tan’s PID tuning methods [8] for integrating processes. Fig.4
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Fig.4. Comparison of process response
Because we introduced the Taylor series expansion method in
designing PID controllers, this produced model error especially
in high frequency. However, we consider desired robust
specification and a loop transfer function with the real part close
to -0 .5 in low frequency['],which guarantees both robustness and
performance.
4. Conclusions
In this paper, we proposed a novel PID tuning formulas for the
integrating processes with time delay. We adopted two-loop
design technique and obtained a tuning method for PID
controllers with setpoint weighting. The proposed tuning method
is very simple and shows better performance in controlling
integrating processes compared with other method. With the
proposed tuning method, we can obtain a loop transfer function
with the real part close to -0 .5 in low frequency, which guarantee
both robustness and performance. Simulation results have been
given to show the performance of the method.
Literature Cited
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