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IMC PID controller tuning for stable and unstable processes with time delay
Accepted Manuscript Title: IMC PID Controller Tuning for Stable and Unstable Processes with Time Delay Author: Qing Wang Changhou Lu Wei Pan PII: DOI: Reference:
S0263-8762(15)00462-1 http://dx.doi.org/doi:10.1016/j.cherd.2015.11.011 CHERD 2094
To appear in: Received date: Revised date: Accepted date:
26-6-2015 29-8-2015 15-11-2015
Please cite this article as: Wang, Q., Lu, C., Pan, W.,IMC PID Controller Tuning for Stable and Unstable Processes with Time Delay, Chemical Engineering Research and Design (2015), http://dx.doi.org/10.1016/j.cherd.2015.11.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights •The IMC filter for all the stable and unstable processes with time delay which considered in the paper have a unified structure.
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•An imaginary first order filter for the stable and unstable first order processes with time delay
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are proposed.
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•The adjusting parameter can be calculated from the time constant and the time delay of the processes directly.
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•The formulas to calculate the adjusting parameter depend on time delay/time constant ratio
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of the processes.
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Qing Wang, Changhou Lu*, Wei Pan
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IMC PID Controller Tuning for Stable and Unstable Processes with Time Delay
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Key Laboratory of High-efficiency and Clean Mechanical Manufacture, Ministry of Education, School of Mechanical Engineering, Shandong University, Jinan 250061,
M
an
PR China
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Corresponding author * Tel.: +86 531 88392179; fax: +86 531 88395625. Email: [email protected] (C. Lu).
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Abstract: In this paper, a new internal model control (IMC) PID tuning method is
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proposed for stable and unstable processes with time delay. In order to implement pole zero conversion and guarantee the stability of the process, an imaginary
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first-order filter based on pole zero conversion is considered for the stable and
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unstable first order processes with time delay and a first-order lead-lag compensator is serried to PID controller for first order plus integrating and second order unstable
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processes with time delay. Set-point weighting is used to reduce the undesirable overshoot. The adjusting parameter can be calculated by the time constant and time
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delay of the processes directly，and the guidelines to calculate the coefficients have high oneness in form. Simulation works have been performed and compared with
te
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recently reported method, and the proposed tuning method gives consistently better
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performance and robustness for a class of processes with time delay.
KeyWords: pole zero conversion, stable and unstable processes with time delay,
imaginary first-order filter, first-order lead-lag compensator, set-point weighting
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1. INTRODUCTION
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The proportional-integral-derivative controller is widely used in chemical process due to its simple structure. A recent survey shows that the ratio of applications of PID control, conventional advanced control, and model predictive control is about 100:10:1 (Shamsuzzoha, 2013). Conventional PID control could display a good effect in invariant time linear systems without time delays. However, time delay is common in most of the chemical process due to recycle loops and transportation delays. It is difficult for traditional PID controller to guarantee the
ed
stability of time delay processes. Furthermore, it's more difficult to design the PID controller for a process with time delay, which is open-loop
ce pt
unstable. Because of this, the tuning of PID controller is needed. But it is difficult to tune PID parameters properly in industrial processes. It is an important research issue that to design a simple PID controller for process engineers. The direct synthesis method (Chen and Seborg, 2002) and (Lee et al. 1998) and the IMC-PID tuning methods (Skogestad, 2003) and (Shamsuzzoha and Lee, 2007) are two typical tuning
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methods. The direct synthesis method computes the controller which gives the desired closed-loop response to calculate the PID controller parameters. The IMC-PID tuning rule has only one adjustable parameter which can be calculated by the time constant and time delay. Several tuning method have been proposed based on the direct synthesis method and the IMC-PID method for stable and unstable processes with time delay. Wang et al. (2001) proposed the partial internal model control capable of controlling both stable and unstable processes. Oliveira et al.
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(2009) used the Hermite-Biehler theorem designed PID controllers for a class of time delay systems and obtained the set of all stabilizing PID
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controllers. Shamsuzzoha and Lee (2008) design an advanced PID controller, based on IMC method, which cascaded with a lead-lag compensator for second-order processes with time delay. And the adjusting parameter was selected for different robustness levels by evaluating the value of Ms. Rao and Chidambaram (2009) proposed PID controllers in series with a lead-lag compensator for all class of integrating processes with time delay; their method is based on direct synthesis, and using the pole zero cancellation to simplify the controller
ed
and set point weighting is used to reduce the overshoot for dynamics response. The main reason for using the PID controller cascaded with a
ce pt
lead-lag compensator is to provide improved performance without tribulation. And the selection of tuning parameter was to change value of the tuning parameter as equal to half of the time delay of the processes, and to observe whether the selected value achieve a good control performance. If not, the tuning parameter can be increased gradually from this value till good control performances are achieved. The tuning
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parameters are selected by simulation. Jeng et al. (2014) proposed a method for PID controller tuning using the step response data of the process which needn’t to resort the process model. Most of the methods discussed above are either selecting the maximum sensitivity, or carrying out several simulation studies and contrasting the control performances to calculate the adjusting parameters. It's required to design more controllers with more tuning parameters by their
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methods. Their guidelines calculated the adjusting parameters indirectly, therefore it would increase complexity of the calculation. Though
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Jeng’s method calculated the parameters of PID controller directly, it needs to collect required plant data firstly. Therefore, it's desirable to get an easy method of tuning control structure for the operator. In the present work, a very simple and uniform controller design method based on pole zero conversion is proposed for a class of stable and unstable processes with time delay. The adjusting parameter was directly calculated by the time constant and time delay of the processes, and the IMC filter with a unified structure can used for all the stable and unstable processes
ce pt
applied in the system to reduce overshoot.
ed
with time delay. The coefficient get from this method is easy to calculate and have a unified structure in form. And set point weighting is also
2. CONTROLLER DESIGN METHOD
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2.1 Classic IMC-PID controller design
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IMC controller
r
+
Process
+
M an
Set point
Qs
+
y
Ps
+ M s
-
Process model
ed
d Controller
r
+
-
C s
+
Process
y
Ps
ce pt
Set point
+
Figure 1. Block diagram of IMC and classical feedback
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The IMC control diagram is shown in Figure 1, where P s is the process, and M s is the process model, which is used as the internal model, Q s is the IMC controller, and C s is the traditional controller. Utilizing the IMC design procedure, the process model is factorized as: M s M s M s
(1)
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ip t Q s f s / M s
C s
(2)
Q s
1 Q s M s
(3)
ed
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Then the internal model controller Q s can be designed as
Then we can get the controller equation as follows:
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Where M s is the portions of the model inverted and M s is not inverted.
The time delay term is usually approximated by a Pade approximation or Taylor expansion. Although a Pade approximation is more accurate,
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a first order Taylor expansion is appropriate for deriving a simple analytical PID tuning (Skogestad, 2003): e s 1 s
(4)
1 C s K c 1 TD s T s I
(5)
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Writing internal model feedback controller into the structure of PID controller
The designed IMC-PID controller has only one adjustable parameter, which is related to the dynamic performance and robustness of the system (Zhao et al. 2011). 2.2 Proposed method
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M an
For a classic feedback system, we know the closed loop trajectory for set-point is shown as follows: C s Ps y r 1 C s Ps
(6)
f s is a low-pass filter and used in this paper is given as
C s
ed
Then using classic IMC-PID method, we can get:
f s
f s / M s 1 f s e s
1s
1 s
2
s 1 / M s
(7)
(8)
2 s 1 2 s 2
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The main idea of the proposed method is to convert the pole and zero of stable and unstable first-order process. As M s is the process model, it can use pole zero cancellation simplify the controller. Then we make the cancelled pole and zero replace by the designed pole and zero which obtained by an imaginary filter or a lead-lag compensator. The imaginary filter based on the theory of dominant pole is used to complement the
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cancelled pole and zero. For trade-off the dynamic performance and disturbance rejection, we cancel the first-order process which has a lager time constant, and for integrating process, to cancel the approximated process is suggested. Since a first-order filter will influence the stability of the process, the time constant of the imaginary filter and the lead-lag compensator must be designed strictly. The adjusting parameter can be
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calculated from the time constant and the time delay of the processes directly. The formulas to calculate the adjusting parameter depend on time
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delay/time constant ratio of the processes. For a wide range of the ratio, we select the formulas as the case shown below.
3. IMC-PID TUNING RULES
This section proposes the IMC-PID tuning rules for a class of processed with time delay. To calculate the PID coefficient and enhance distance rejection of the stable and unstable first order processes with time delay, an imaginary low pass filter is suggested. And a first order lead-lag
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compensator is serried to PID controller to improve performance for first order plus integrating and second order unstable processes with time
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delay. The resulting of the PID controller tuning rules are listed in Table 1. Case 1 First-order plus time delay (FOPTD)
The model for chemical processes of the FOPDT model is given as follows: Ke s Ts 1
(9)
Ac
Ps
Where K is the gain, is the time delay, and T is the time constant, Assume the first order plus time delay process series an imaginary first-order filter, which isn’t real exist in the close-loop system. To guarantee the stability of the process, the time constant should be chosen cautious, the imaginary filter is obtained as follows:
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A s
1 Bs 1
(10)
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The time constant of the imaginary filter depend on time delay/ time constant ratio of the processes. For a wide range of the ratio, we select the time constant B 2min , T .
Ps
Ke s Ts 1 Bs 1
(11)
s 1Ts 1 Bs 1 s 1Ts 1 Bs 1 2 s 2 s 1 s 1 e K 2 s s 1 2
2 T 2 2T T 2 T
Ac
Make
C s
ce pt
Therefore, the ideal feedback controller is:
ed
Then we can consider the P(s) as follows:
(12)
(13) (14)
For 0 , we know T T T 2 And the tuning parameter is defined as:
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B 2min ,T
(15)
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After simplification, writing internal model feedback controller into the structure of PID controller: C s
s 1 Bs 1 B 1 B s 1 K 2 s K 2 B s B
(16)
The resulting of the PID controller tuning rules are given in Table 1.
ed
Case 2 Second-order plus time delay (SOPTD)
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Consider a second-order plus time delay system as follows, and we just assume that T is larger than L. Ps
Ke s T L Ts 1 Ls 1
(17)
C s
s 1Ts 1 Ls 1 2 s 1 s 1 e s
(18)
The ideal feedback controller is given as follows:
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The formulas to calculate depend on time delay/time constant ratio of the processes. And for a wide range, the tuning parameter is defined as L L2 . The resulting of the PID controller tuning rules are given in Table 1. Case 3 Integrating process plus time delay (IPTD) Consider an Integrating process plus time delay system as follows: 12
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P s
Ke s s
(19)
imaginary filter is obtained as follows:
s Then we can consider the process as P s Ke .
A s
ed
s Bs 1
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Assume the integrating plus time delay process series an imaginary first-order filter, which isn’t exist in the close-loop system. And the
1 , B 0.5 Bs 1
(20)
The integrating process is approximated by a first order process as follows:
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K K s s 1
(21)
where is a large value constant.
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The ideal feedback controller is
C s
s 1 s 1 Bs 1
(22)
2 K 2 s s 1 2
2 2 And just make , and we can get 2 0 . The tuning parameter is defined as 0.5T T T 2 , where T=B. And the
2
PID tuning rules are given in Table 1.
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Case 4 First-order delayed integrating process (FODIP)
The ideal feedback controller is same as IPTD.
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Consider the following FODIP process:
Ps
Ke s s Ts 1
(23)
compensator is given as R s
0.5 s 1 . 0.01s 1
ed
To guarantee the stability and improve performance of the process, we series a lead-lag compensator with the PID controller, and the
ce pt
Then the PID controller in series with a lead-lag compensator as follows: C s
T 1 1 T 1 K 2 T s T
0.5 s 1 s 0.01s 1
(24)
And the tuning parameter is defined as T T T 2 .The PID tuning rules are given in Table 1.
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Case 5 First-order delayed unstable process (FODUP)
Just consider the first-order unstable processes with time delay as follows: Ps
Ke s Ts 1
(25)
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And just like FOPTD, we assume the first order delayed unstable process series an imaginary low filter which is given 1 , B min , T . Bs 1
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as A s
2 2 Then using pole zero conversion, we just make T . We can get 2T T . The tuning parameter is considered
2
as B min , T .
T
To improve the robustness and dynamic performance of the system, derivative coefficient weighting is suggested. And the controller is
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obtained as:
B 1 1 0.5 B s 1 K 2 B s B
(26)
ce pt
C s
The resulting PID tuning rules are listed in Table 1.
If the process is
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Case 6 Second-order delayed unstable process (SODUP)
P s
Ke s Ts 1 Ls 1
(27)
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2 2 T .Then we can get 2T T . And the tuning parameter is defined 2 T
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Using pole zero conversion, we just make
the compensator is given as R s
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as min ,T . Same as FODIP, in order to guarantee the stability of the process, we series a lead-lag compensator with the PID controller, and 0.5 s 1 . The resulting PID tuning rules are given in Table 1. 0.01s 1
Set-point weighting
In order to reduce large overshoot, PID controller is often implemented in the form of
ed
u t K P r y K I e t dt K D
de t dt
(28)
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where e t r y is the tracking error signal between the reference input r and the controlled system output y . K P , K I , and K D are constant of
P, I , D control gains, respectively. And is the set-point weighting coefficient.
4. Stability and robustness
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According to Robust Stability Theorem, the term of the internal model control closed-loop system robust stability is: P jw C jw 1 P jw C jw
1 , lm
(29)
where lm is the bound on the process multiplicative uncertainty (Morari and Zafiriou,1989)
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em
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em is the mismatch level between process and process model.
P s M s , l sup e m m M s
(30)
Considering First-order delayed unstable process, the complementary sensitivity function is: Qs M s M s f s
s 1 e s 2 s 1
(31)
K K e s .
ed
If the gain of the process exists uncertainty P s
Ts 1
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Utilizing the robust stability theorem, and on the basis of the results which list in Table 2. K K
1 P jw C jw P jw C jw
(32)
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Using the same method, we can get the maximum allowable uncertainty in and T . To ensure the robustness of the closed-loop system, the term of disturbance rejection is considered: em j T jw Wm j 1 T jw 1
(33)
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ip t 1 P jw C jw
cr
P jw C jw
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where T jw
, and Wm j is a weight function, which is given by 1 T jw .
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The similar stability and robustness analysis can be done for the remaining processes.
5. Simulation
Stable and unstable processes with time delay are required to realize the simulation. We compare the proposed method with the recently
ed
methods by obtaining their control performances. And we use integral of the absolute error criteria and overshoot to evaluate the performance and robustness of the control system. PID controller parameters and resulting performance indices are listed in Table 2. For fair comparison, the
Example 1
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maximum sensitivity, defined as Ms max 1 / 1 P jw C jw , is used.
Consider a first-order plus time delay process (Lee et al. 2014):
Ac
Ps
e s 0.2s 1
(34)
By using pole zero conversion, the controller parameters of the proposed method are listed in Table 2. The tuning parameter is considered as 0.4 . We define 0.4 as the set-point weighting parameter. According to the parameter of the PID controller, we can calculate the maximum sensitivity Ms=1.6. The proposed method was compared with the method proposed by Lee et al. (2014) and Skogestad (2003), and for 18
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a fair comparison, the same Ms is used to obtain the parameters of their method in the simulation. To calculate the performances of the
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first-order plus time delay process, a unit step input is given in the set point and a negative step disturbance of 0.2 is given at t=25s. Figure 2 shows the responses without mismatches in the model while table 2 shows IAE and overshoot indices. According to the data listed in Table 2, which shows the responses and the value of the performance indices, we can see clearly that the best performance arises from the proposed controller. In the actual situation, there always exist model mismatch. In order to evaluate the robustness of the controller, +20% perturbations in
ed
the process gain and time delay are considered, i.e., P s 1.2 e 1.2 s / 0.2 s 1 as an actual process. The responses when there are 20%
ce pt
mismatches in the plant are shown in Figure 3, IAE and overshoot are given in Table 2. For both no mismatch and mismatch in the plant, the simulation results show that the proposed method give the best control performances both in the set-point tracking and disturbance rejection. The region of stability is evaluated for K, T and individually using the robust stability theorem and the results are shown in Table 4. It is
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found by simulation that the present method is robust for uncertainty in process model parameters.
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1.4
1
y, e s n o p s e R
SIMC Lee Proposed
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1.2
0.8
0.6
ed
0.4
0.2
0
5
10
15
20
25 Time/s
30
35
40
45
50
ce pt
0
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Figure 2. Response for perfect model for example 1
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1.4
M an
1.2
1
y, e s n o p s e R
SIMC Lee Proposed
0.8
0.6
ed
0.4
0.2
0
5
ce pt
0
10
15
20
25 Time/s
30
35
40
45
50
Figure 3. Response for a perturbation of +20% in process for example 1
Example 2
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Just consider the following SOPDT process (Lee et al. 2014) Ps
e2 s s 1 0.7s 1
(35)
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The controller parameters obtained for the proposed method are listed in Table 2. The tuning parameter is considered as 3 . And the
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maximum sensitivity, Ms=1.64. The weighting parameter is 0.4 . The proposed controller is compared with Lee et al. (2014) and Skogestad (2003), the same Ms is considered. The responses are shown in Figure 4, and IAE and overshoot are given in Table 2. 1.4
1.2
SIMC Lee Proposed
0.8
0.6
ce pt
y, e s n o p s e R
ed
1
0.4
0.2
Ac
0
0
10
20
30
40
50 Time/s
60
70
80
90
100
Figure 4. Response for perfect model for example 2
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A mismatch of +20% in process gain and time delay is considered. And the process is described as P s 1.2e 2.4 s / s 1 0.7 s 1 . Figure 5
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shows the simulation result of example 2, and IAE and overshoot of the process are given in Table 2. It can be obvious that the IAE and overshoot for the proposed method are the best for disturbance rejection and dynamic performance. 1.4
1.2
SIMC Lee Proposed
ed
1
0.8
0.6
ce pt
y, e s n o p s e R
0.4
0.2
Ac
0
0
10
20
30
40
50 Time/s
60
70
80
90
100
Figure 5. Response for a perturbation of +20% in process for example 2
Example 3
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Consider an integrating process with time delay (Shamsuzzoha and Lee, 2007):
Ps
0.2e7.4 s s
(36)
In the proposed method, we define 100 . The parameter of the filter is 10.1 , and Ms=1.96. A negative step disturbance of 0.1 is given in the load at t=150s. The methods proposed by Shamsuzzoha and Lee (2007) and Chien and Fruehauf (1990) are considered in this paper. The value of for their method is adjusted to obtain the value of Ms=1.96. Fig 6 shows the simulation results when the process model is perfect.
Ac
ce pt
ed
Table 2 shows IAE and overshoot of the process.
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1.4
Shamsuzzoha Chien Proposed
M an
1.2
1
y, e s n o p s e R
0.8
0.6
ed
0.4
0.2
0
ce pt
0
50
100
150 Time/s
200
250
300
Figure 6. Response for perfect model for example 3
Ac
To evaluate the robustness of the process, 20% change in the plant delay and the steady state gain P s 0.24e8.88 s / s , which has a perturbation uncertainty of +20% in gain and time delay of the process. The simulation results are shown in Figure 7. For both no mismatch and mismatch, it is clear that the proposed method gives better control performances.
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1.8
Shamsuzzoha Chien Proposed
M an
1.6 1.4 1.2 y, e s n o p s e R
1 0.8 0.6
ed
0.4 0.2
0
50
100
150 Time/s
200
250
300
ce pt
0
Figure 7. Response for a perturbation of +20% in process for example 3
Example 4
Ac
The FODIP plant model considered in Ali and Majhi (2010): Ps
e4 s s s 1
(37)
And we consider 100 . And the filter parameter is 3.24 , so that Ms=2.3. In order to evaluate the disturbance rejection, a negative step disturbance of 0.1 is given at t=100s. And the methods proposed by Ali and Majhi (2010) and Skogestad (2003) are considered. For a fair 26
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comparison, all compared controllers are tuned to have Ms=2.3 by adjusting . The simulation results are shown in Figure 8. And the
M an
performance indices are given in Table 2. 1.5
1
SIMC Ali Proposed
ed
y, e s n o p s e R
ce pt
0.5
Ac
0
0
50
100
150 Time/s
200
250
300
Figure 8. Response for perfect model for example 4
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us
To evaluate the robustness of the process, 20% change in the plant delay and the steady state gain is considered. P s 1.2 e 4.8 s / s s 1
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Figure 9 shows the responses for a 20% mismatch model. The performance indices are given in Table 2. Improve performance is obtained by the proposed method obviously. 2 1.8 1.6
SIMC Ali Proposed
ed
1.4 1.2
1
0.8
ce pt
y, e s n o p s e R
0.6 0.4 0.2
Example 5
Ac
0
0
50
100
150 Time/s
200
250
300
Figure 9. Response for a perturbation of +20% in process for example 4
The following first-order delayed unstable process is considered (Shamsuzzoha and Lee, 2007):
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P s
e 0.4 s s 1
(38)
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The filter time constant 0.4 is select for the proposed method. Then we can get the Ms=2.75. The weighting parameter is defined as 0.2 . To evaluate disturbance rejection of the process, a negative step disturbance of 1 at t=15s are considered. For comparison, we choose the methods proposed by Shamsuzzoha and Lee (2007) and Lee et al. (2000), is selected by Ms=2.75 to ensure fair comparison. Figure 10 shows the responses for the perfect model. And Figure 11 shows the responses for a 20% mismatch model. P s 1.2e0.48s / 1.2s 1 . The values of
ed
the controller performance indices are listed in Table 3. Clearly, the proposed method performs better than the others. The region of stability is
Ac
ce pt
evaluated for K, T and individually and the results are shown in Table 4.
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2.5
Lee Shamsuzzoha Proposed
M an
2
1.5 y, e s n o p s e R
1
ed
0.5
0
ce pt
0
5
10
15 Time/s
20
25
30
Ac
Figure 10. Response for perfect model for example 5
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2.5
Lee Shamsuzzoha Proposed
M an
2
1.5 y, e s n o p s e R
1
ed
0.5
0
ce pt
0
5
10
15 Time/s
20
25
30
Figure 11. Response for a perturbation of +20% in process for example 5
Example 6
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The unstable process is considered for the simulation as follows (Shamsuzzoha and Lee, 2008) ： P s
e0.939 s 5s 1 2.07s 1
(39)
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The tuning parameter was selected as 0.939 , so that Ms=2.27. And the set-point weighting parameter is 0.2 . A negative step disturbance
M an
of 0.2 are considered. To show the improvement, the method proposed by Shamsuzzoha and Lee, (2007) and Shamsuzzoha and Lee, (2008) are considered. The adjusting parameter are selected to obtain Ms=2.27 for each method. The simulation results are shown in Figure 12.
1.4
ed
1.2
Shamsuzzoha Shamsuzzoha Proposed
1
0.8
ce pt
y, e s n o p s e R
0.6
0.4
0.2
Ac
0
0
10
20
30
40
50 Time/s
60
70
80
90
100
Figure 12. Response for perfect model for example 6
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the
robust
performance
of
us
To
the
process,
a
10%
change
in
all
four
parameters
is
M an
considered. P s 1.1e 1.0329 s / 4.5 s 11.863s 1 The simulation results are shown in Figure 13. The values of the controller parameters are listed in Table 3. 1.4
1.2
Shamsuzzoha Shamsuzzoha Proposed
ed
1
0.8
0.6
ce pt
y, e s n o p s e R
0.4
0.2
Ac
0
0
10
20
30
40
50 Time/s
60
70
80
90
100
Figure 13. Response for a perturbation of +20% in process for example 6
6. Conclusion
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An IMC PID controller based on pole zero conversion design for a class of processes with time delay is proposed. And we provide guidelines
M an
to select the tuning parameters. The adjusting parameter can be calculated from the time constant and the time delay of the processes directly. The IMC filter structure for all the stable and unstable processes with time delay which considered in the paper have a unified structure. And the formulas to calculate the adjusting parameter depend on time delay/time constant ratio of the processes. Several processes are considered in the simulation. The controller proposed in this paper successfully demonstrated better performance than other PID tuning method for a class of
ed
processes with time delay.
ce pt
REFERENCES
Shamsuzzoha, M., 2013. Closed-loop PI/PID Controller Tuning for Stable and Integrating Process with Time Delay. Ind. Eng. Chem. Res. 52, 12973-12992.
Ac
Chen, D.; Seborg, D.E., 2002. PI/PID controller design based on direct synthesis and disturbance rejection. Ind. Eng. Chem. Res. 41, 4807-4822. Lee, Y.; Park, S.; Lee, M.; Brosilow, C., 1998. PID controller tuning for desired closed-loop responses for SI/SO systems. AIChE J. 44, 106-115.
34
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ip t cr us
Skogestad, S., 2003. Simple analytic rules for model reduction and PID controller tuning. J. Process Control 13, 291-309.
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Shamsuzzoha, M.; Lee, M., 2007. IMC-PID controller design for improved disturbance rejection of time-delayed processes. Ind. Eng. Chen. Res. 46, 2077-2091.
Wang, Q.G.; Bi, Q.; Zhang, Y., 2001. Partial Internal Model Control. IEEE Trans. Indus. Electronics. 48, 976-982.
ed
Oliveira, V.A.; Cossi, L.V.; Teixeira, C.M.; Silva, M.F., 2009. Synthesis of PID controllers for a class of time delay systems. Automatica 45,
ce pt
1778-1782.
Shamsuzzoha, M.; Lee, M., 2008. Design of advanced PID controller for enhanced disturbance rejection of second-order processes with time delay. AIChE J. 54, 1526-1536.
Ac
Rao, A.S.; Chidambaram, M., 2009. Direct synthesis-based controller design for integrating processed with time delay. Journal of the Franklin Institute. 346, 38-56.
35
Page 36 of 42
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Jeng, J.C.; Tseng, W.L.; Chiu, M.S. 2013. A one-step tuning method for PID controllers with robustness specification using plant step-response
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data. Chem. Eng. Res. Des. 92, 545-558.
Zhao, Z.C.; Liu, Z.Y.; Zhang, J.G., 2011. IMC-PID tuning method based on sensitivity specification for process with time-delay. Journal of Central South University of Technology. 18, 1153-1160.
ed
Morari, M.; Zafiriou, E. Robust Process Control. NJ: Prentice Hall Englewood Cliffs, 1989.
ce pt
Lee, J.; Cho, W.; Edgar, T., 2014. Simple Analytic PID Controller Tuning Rules Revisited. Ind. Eng. Chem. Res. 53, 5038-5047. Chien, I. L.; Fruehauf, P. S., 1990. Consider IMC Tuning to Improve Controller Performance. Chem. Eng. Prog. 86, 33-41.
Ac
Ali, A.; Majhi, S., 2010. PID controller tuning for integrating processes. ISA Transactions. 49, 70-78. Lee, Y.; Lee, J.; Park, S., 2000. PID Controller Tuning for Integrating and Unstable Processes with Time Delay. Chem. Eng. Sci. 55, 3481-3493.
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ip t cr us M an
Table Caption Table 1. IMC-PID Controller Tuning Rules
Table 2. PID controller parameters and resulting performance indices for stable examples
ed
Table 3. PID controller parameters and resulting performance indices for unstable examples
Ac
ce pt
Table 4. Stability region for K, T and for the stable and unstable processes
37
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ip t cr us
Table 1 IMC-PID Controller Tuning Rules
2min ,T
2T T 2 T
Ke s Ts 1 Ls 1
L L
2T T 2 T
T T T
Ke s Ts 1 Ls 1
min ,T
min ,T
L K 2
L
L L
2 2
0.5 K 2
0.5
0.5 0.5
2 2
T K 2
T
T T
2 2T T T
K 2
0.5
2 2T T T
L K 2
L
L L
Ac c
Ke s Ts 1
2
d
Ke s s Ts 1
2
ep te
0.5T T T
K 2
an
Ke s Ts 1
Ke s s
TD
M
2
TI
Kc
Process
Imaginary filter
Compensator
1 Bs 1
1 Ts 1
0.5 s 1 0.01s 1 1 Bs 1
0.5 s 1 0.01s 1
Page 39 of 42
ip t cr
Example
Method
Kc
TI
TD
Ms
us
Table 2 PID controller parameters and resulting performance indices for stable examples Perfect model
an
IAE
Example 1
overshoot
e s / 0.2s 1
0.4
0.347
0.567
0.12
SIMC
1.0
0.1
0.2
0
Lee
0.6
0.125
0.2
0.2
1.375
0.435
SIMC
1.8
0.263
Lee
1.7
0.272
Example 3 8.26
0.56
Shamsuzzoha
10.6
Chien
14.1
Example 4
1.2e1.2 s / 0.2s 1
1.00
2.55
1.15
1.6
2.65
1.13
3.76
1.37
1.6
2.61
1.04
3.25
1.22
e2 s / s 1 0.7s 1
1.64
6.07
1.05
7.34
1.23
1
0.70
1.64
6.88
1.19
9.19
1.38
1
0.86
1.64
6.85
1.18
9.13
1.40
0.2e7.4 s / s
0.4
0.24e 8.88 s / s
28.57
3.2
1.96
22.2
1.05
22.0
1.1
0.545
23.4
2.42
1.96
22.7
1.08
24.3
1.2
0.556
34.7
3.1
1.96
29.2
1.30
30.7
1.68
e4 s / s s 1
0.4
1.2e2.4 s s 1 0.7s 1
0.40
Ac c
Proposed
overshoot
1.653
ep te
Proposed
IAE
2.31
d
Example 2
1.6
M
Proposed
20 % mismatch
0.5
1.2e4.8s / s s 1
Proposed
3.24
0.22
11
0.91
2.3
13.36
1.06
16.62
1.28
SIMC
4.15
0.19
33
0.97
2.3
31.94
1.31
38.94
1.84
Ali
4.15
0.19
22.37
2.79
2.3
26.30
1.26
27.00
1.61
0.4
Page 40 of 42
ip t cr
Method
Kc
TI
TD
Ms
Example 5 0.4
2.5
2.67
0.17
2.75
Shamsuzzoha
0.745
2.38
2.37
0.214
2.75
Lee
0.58
2.48
2.58
0.152
6.63
Shamsuzzoha
0.960
6.47
Shamsuzzoha
1.745
3.85
1.2e0.48s / 1.2s 1
2.21
1.01
2.32
1.00
2.62
1.08
2.42
1.02
3.46
2.14
3.44
2.42
Perfect model e
0.939 s
0.2
10 % mismatch
/ 5s 1 2.07 s 1
1.1e
1.0329
/ 4.5s 11.863s 1
5.76
1.326
2.27
4.62
1.01
4.70
1.00
5.65
1.352
2.27
5.17
1.02
5.15
1.01
8.06
1.687
2.27
9.26
1.07
8.76
1.03
ep te
0.939
e0.4 s / s 1
0.2
Ac c
Proposed
20 % mismatch
2.75
d
Example 6
M
Proposed
Perfect model
an
Example
us
Table 3 PID controller parameters and resulting performance indices for unstable examples
Page 41 of 42
Table 4 Stability region for K, T and for the stable and unstable processes K
T
e s / 0.2s 1
Model
Proposed Shamsuzzoha Lee
43% 42% 40%
79% 100% 87%
e0.4 s / s 1
30% 25% 32%
Ac
ce pt
ed
M
an
us
35% 36% 33%
96% 138% 102%
ip t
100% 100% 91%
cr
Proposed SIMC Lee Model
Page 42 of 42
S0263-8762(15)00462-1 http://dx.doi.org/doi:10.1016/j.cherd.2015.11.011 CHERD 2094
To appear in: Received date: Revised date: Accepted date:
26-6-2015 29-8-2015 15-11-2015
Please cite this article as: Wang, Q., Lu, C., Pan, W.,IMC PID Controller Tuning for Stable and Unstable Processes with Time Delay, Chemical Engineering Research and Design (2015), http://dx.doi.org/10.1016/j.cherd.2015.11.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
f s
cr
f s M s s
s
O
2
Ac ce p
te
d
M
D
us
an
Qs
ip t
Page 1 of 42
Highlights •The IMC filter for all the stable and unstable processes with time delay which considered in the paper have a unified structure.
ip t
•An imaginary first order filter for the stable and unstable first order processes with time delay
cr
are proposed.
us
•The adjusting parameter can be calculated from the time constant and the time delay of the processes directly.
an
•The formulas to calculate the adjusting parameter depend on time delay/time constant ratio
Ac ce p
te
d
M
of the processes.
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cr
Qing Wang, Changhou Lu*, Wei Pan
ip t
IMC PID Controller Tuning for Stable and Unstable Processes with Time Delay
us
Key Laboratory of High-efficiency and Clean Mechanical Manufacture, Ministry of Education, School of Mechanical Engineering, Shandong University, Jinan 250061,
M
an
PR China
Ac ce p
te
d
Corresponding author * Tel.: +86 531 88392179; fax: +86 531 88395625. Email: [email protected] (C. Lu).
2
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Abstract: In this paper, a new internal model control (IMC) PID tuning method is
ip t
proposed for stable and unstable processes with time delay. In order to implement pole zero conversion and guarantee the stability of the process, an imaginary
cr
first-order filter based on pole zero conversion is considered for the stable and
us
unstable first order processes with time delay and a first-order lead-lag compensator is serried to PID controller for first order plus integrating and second order unstable
an
processes with time delay. Set-point weighting is used to reduce the undesirable overshoot. The adjusting parameter can be calculated by the time constant and time
M
delay of the processes directly，and the guidelines to calculate the coefficients have high oneness in form. Simulation works have been performed and compared with
te
d
recently reported method, and the proposed tuning method gives consistently better
Ac ce p
performance and robustness for a class of processes with time delay.
KeyWords: pole zero conversion, stable and unstable processes with time delay,
imaginary first-order filter, first-order lead-lag compensator, set-point weighting
3
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1. INTRODUCTION
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The proportional-integral-derivative controller is widely used in chemical process due to its simple structure. A recent survey shows that the ratio of applications of PID control, conventional advanced control, and model predictive control is about 100:10:1 (Shamsuzzoha, 2013). Conventional PID control could display a good effect in invariant time linear systems without time delays. However, time delay is common in most of the chemical process due to recycle loops and transportation delays. It is difficult for traditional PID controller to guarantee the
ed
stability of time delay processes. Furthermore, it's more difficult to design the PID controller for a process with time delay, which is open-loop
ce pt
unstable. Because of this, the tuning of PID controller is needed. But it is difficult to tune PID parameters properly in industrial processes. It is an important research issue that to design a simple PID controller for process engineers. The direct synthesis method (Chen and Seborg, 2002) and (Lee et al. 1998) and the IMC-PID tuning methods (Skogestad, 2003) and (Shamsuzzoha and Lee, 2007) are two typical tuning
Ac
methods. The direct synthesis method computes the controller which gives the desired closed-loop response to calculate the PID controller parameters. The IMC-PID tuning rule has only one adjustable parameter which can be calculated by the time constant and time delay. Several tuning method have been proposed based on the direct synthesis method and the IMC-PID method for stable and unstable processes with time delay. Wang et al. (2001) proposed the partial internal model control capable of controlling both stable and unstable processes. Oliveira et al.
4
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(2009) used the Hermite-Biehler theorem designed PID controllers for a class of time delay systems and obtained the set of all stabilizing PID
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controllers. Shamsuzzoha and Lee (2008) design an advanced PID controller, based on IMC method, which cascaded with a lead-lag compensator for second-order processes with time delay. And the adjusting parameter was selected for different robustness levels by evaluating the value of Ms. Rao and Chidambaram (2009) proposed PID controllers in series with a lead-lag compensator for all class of integrating processes with time delay; their method is based on direct synthesis, and using the pole zero cancellation to simplify the controller
ed
and set point weighting is used to reduce the overshoot for dynamics response. The main reason for using the PID controller cascaded with a
ce pt
lead-lag compensator is to provide improved performance without tribulation. And the selection of tuning parameter was to change value of the tuning parameter as equal to half of the time delay of the processes, and to observe whether the selected value achieve a good control performance. If not, the tuning parameter can be increased gradually from this value till good control performances are achieved. The tuning
Ac
parameters are selected by simulation. Jeng et al. (2014) proposed a method for PID controller tuning using the step response data of the process which needn’t to resort the process model. Most of the methods discussed above are either selecting the maximum sensitivity, or carrying out several simulation studies and contrasting the control performances to calculate the adjusting parameters. It's required to design more controllers with more tuning parameters by their
5
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methods. Their guidelines calculated the adjusting parameters indirectly, therefore it would increase complexity of the calculation. Though
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Jeng’s method calculated the parameters of PID controller directly, it needs to collect required plant data firstly. Therefore, it's desirable to get an easy method of tuning control structure for the operator. In the present work, a very simple and uniform controller design method based on pole zero conversion is proposed for a class of stable and unstable processes with time delay. The adjusting parameter was directly calculated by the time constant and time delay of the processes, and the IMC filter with a unified structure can used for all the stable and unstable processes
ce pt
applied in the system to reduce overshoot.
ed
with time delay. The coefficient get from this method is easy to calculate and have a unified structure in form. And set point weighting is also
2. CONTROLLER DESIGN METHOD
Ac
2.1 Classic IMC-PID controller design
6
Page 7 of 42
ip t cr us d
IMC controller
r
+
Process
+
M an
Set point
Qs
+
y
Ps
+ M s
-
Process model
ed
d Controller
r
+
-
C s
+
Process
y
Ps
ce pt
Set point
+
Figure 1. Block diagram of IMC and classical feedback
Ac
The IMC control diagram is shown in Figure 1, where P s is the process, and M s is the process model, which is used as the internal model, Q s is the IMC controller, and C s is the traditional controller. Utilizing the IMC design procedure, the process model is factorized as: M s M s M s
(1)
7
Page 8 of 42
ip t Q s f s / M s
C s
(2)
Q s
1 Q s M s
(3)
ed
M an
Then the internal model controller Q s can be designed as
Then we can get the controller equation as follows:
cr
us
Where M s is the portions of the model inverted and M s is not inverted.
The time delay term is usually approximated by a Pade approximation or Taylor expansion. Although a Pade approximation is more accurate,
ce pt
a first order Taylor expansion is appropriate for deriving a simple analytical PID tuning (Skogestad, 2003): e s 1 s
(4)
1 C s K c 1 TD s T s I
(5)
Ac
Writing internal model feedback controller into the structure of PID controller
The designed IMC-PID controller has only one adjustable parameter, which is related to the dynamic performance and robustness of the system (Zhao et al. 2011). 2.2 Proposed method
8
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M an
For a classic feedback system, we know the closed loop trajectory for set-point is shown as follows: C s Ps y r 1 C s Ps
(6)
f s is a low-pass filter and used in this paper is given as
C s
ed
Then using classic IMC-PID method, we can get:
f s
f s / M s 1 f s e s
1s
1 s
2
s 1 / M s
(7)
(8)
2 s 1 2 s 2
ce pt
The main idea of the proposed method is to convert the pole and zero of stable and unstable first-order process. As M s is the process model, it can use pole zero cancellation simplify the controller. Then we make the cancelled pole and zero replace by the designed pole and zero which obtained by an imaginary filter or a lead-lag compensator. The imaginary filter based on the theory of dominant pole is used to complement the
Ac
cancelled pole and zero. For trade-off the dynamic performance and disturbance rejection, we cancel the first-order process which has a lager time constant, and for integrating process, to cancel the approximated process is suggested. Since a first-order filter will influence the stability of the process, the time constant of the imaginary filter and the lead-lag compensator must be designed strictly. The adjusting parameter can be
9
Page 10 of 42
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calculated from the time constant and the time delay of the processes directly. The formulas to calculate the adjusting parameter depend on time
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delay/time constant ratio of the processes. For a wide range of the ratio, we select the formulas as the case shown below.
3. IMC-PID TUNING RULES
This section proposes the IMC-PID tuning rules for a class of processed with time delay. To calculate the PID coefficient and enhance distance rejection of the stable and unstable first order processes with time delay, an imaginary low pass filter is suggested. And a first order lead-lag
ed
compensator is serried to PID controller to improve performance for first order plus integrating and second order unstable processes with time
ce pt
delay. The resulting of the PID controller tuning rules are listed in Table 1. Case 1 First-order plus time delay (FOPTD)
The model for chemical processes of the FOPDT model is given as follows: Ke s Ts 1
(9)
Ac
Ps
Where K is the gain, is the time delay, and T is the time constant, Assume the first order plus time delay process series an imaginary first-order filter, which isn’t real exist in the close-loop system. To guarantee the stability of the process, the time constant should be chosen cautious, the imaginary filter is obtained as follows:
10
Page 11 of 42
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A s
1 Bs 1
(10)
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The time constant of the imaginary filter depend on time delay/ time constant ratio of the processes. For a wide range of the ratio, we select the time constant B 2min , T .
Ps
Ke s Ts 1 Bs 1
(11)
s 1Ts 1 Bs 1 s 1Ts 1 Bs 1 2 s 2 s 1 s 1 e K 2 s s 1 2
2 T 2 2T T 2 T
Ac
Make
C s
ce pt
Therefore, the ideal feedback controller is:
ed
Then we can consider the P(s) as follows:
(12)
(13) (14)
For 0 , we know T T T 2 And the tuning parameter is defined as:
11
Page 12 of 42
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B 2min ,T
(15)
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After simplification, writing internal model feedback controller into the structure of PID controller: C s
s 1 Bs 1 B 1 B s 1 K 2 s K 2 B s B
(16)
The resulting of the PID controller tuning rules are given in Table 1.
ed
Case 2 Second-order plus time delay (SOPTD)
ce pt
Consider a second-order plus time delay system as follows, and we just assume that T is larger than L. Ps
Ke s T L Ts 1 Ls 1
(17)
C s
s 1Ts 1 Ls 1 2 s 1 s 1 e s
(18)
The ideal feedback controller is given as follows:
Ac
The formulas to calculate depend on time delay/time constant ratio of the processes. And for a wide range, the tuning parameter is defined as L L2 . The resulting of the PID controller tuning rules are given in Table 1. Case 3 Integrating process plus time delay (IPTD) Consider an Integrating process plus time delay system as follows: 12
Page 13 of 42
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P s
Ke s s
(19)
imaginary filter is obtained as follows:
s Then we can consider the process as P s Ke .
A s
ed
s Bs 1
M an
Assume the integrating plus time delay process series an imaginary first-order filter, which isn’t exist in the close-loop system. And the
1 , B 0.5 Bs 1
(20)
The integrating process is approximated by a first order process as follows:
ce pt
K K s s 1
(21)
where is a large value constant.
Ac
The ideal feedback controller is
C s
s 1 s 1 Bs 1
(22)
2 K 2 s s 1 2
2 2 And just make , and we can get 2 0 . The tuning parameter is defined as 0.5T T T 2 , where T=B. And the
2
PID tuning rules are given in Table 1.
13
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Case 4 First-order delayed integrating process (FODIP)
The ideal feedback controller is same as IPTD.
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Consider the following FODIP process:
Ps
Ke s s Ts 1
(23)
compensator is given as R s
0.5 s 1 . 0.01s 1
ed
To guarantee the stability and improve performance of the process, we series a lead-lag compensator with the PID controller, and the
ce pt
Then the PID controller in series with a lead-lag compensator as follows: C s
T 1 1 T 1 K 2 T s T
0.5 s 1 s 0.01s 1
(24)
And the tuning parameter is defined as T T T 2 .The PID tuning rules are given in Table 1.
Ac
Case 5 First-order delayed unstable process (FODUP)
Just consider the first-order unstable processes with time delay as follows: Ps
Ke s Ts 1
(25)
14
Page 15 of 42
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And just like FOPTD, we assume the first order delayed unstable process series an imaginary low filter which is given 1 , B min , T . Bs 1
M an
as A s
2 2 Then using pole zero conversion, we just make T . We can get 2T T . The tuning parameter is considered
2
as B min , T .
T
To improve the robustness and dynamic performance of the system, derivative coefficient weighting is suggested. And the controller is
ed
obtained as:
B 1 1 0.5 B s 1 K 2 B s B
(26)
ce pt
C s
The resulting PID tuning rules are listed in Table 1.
If the process is
Ac
Case 6 Second-order delayed unstable process (SODUP)
P s
Ke s Ts 1 Ls 1
(27)
15
Page 16 of 42
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2 2 T .Then we can get 2T T . And the tuning parameter is defined 2 T
us
Using pole zero conversion, we just make
the compensator is given as R s
M an
as min ,T . Same as FODIP, in order to guarantee the stability of the process, we series a lead-lag compensator with the PID controller, and 0.5 s 1 . The resulting PID tuning rules are given in Table 1. 0.01s 1
Set-point weighting
In order to reduce large overshoot, PID controller is often implemented in the form of
ed
u t K P r y K I e t dt K D
de t dt
(28)
ce pt
where e t r y is the tracking error signal between the reference input r and the controlled system output y . K P , K I , and K D are constant of
P, I , D control gains, respectively. And is the set-point weighting coefficient.
4. Stability and robustness
Ac
According to Robust Stability Theorem, the term of the internal model control closed-loop system robust stability is: P jw C jw 1 P jw C jw
1 , lm
(29)
where lm is the bound on the process multiplicative uncertainty (Morari and Zafiriou,1989)
16
Page 17 of 42
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em
M an
em is the mismatch level between process and process model.
P s M s , l sup e m m M s
(30)
Considering First-order delayed unstable process, the complementary sensitivity function is: Qs M s M s f s
s 1 e s 2 s 1
(31)
K K e s .
ed
If the gain of the process exists uncertainty P s
Ts 1
ce pt
Utilizing the robust stability theorem, and on the basis of the results which list in Table 2. K K
1 P jw C jw P jw C jw
(32)
Ac
Using the same method, we can get the maximum allowable uncertainty in and T . To ensure the robustness of the closed-loop system, the term of disturbance rejection is considered: em j T jw Wm j 1 T jw 1
(33)
17
Page 18 of 42
ip t 1 P jw C jw
cr
P jw C jw
us
where T jw
, and Wm j is a weight function, which is given by 1 T jw .
M an
The similar stability and robustness analysis can be done for the remaining processes.
5. Simulation
Stable and unstable processes with time delay are required to realize the simulation. We compare the proposed method with the recently
ed
methods by obtaining their control performances. And we use integral of the absolute error criteria and overshoot to evaluate the performance and robustness of the control system. PID controller parameters and resulting performance indices are listed in Table 2. For fair comparison, the
Example 1
ce pt
maximum sensitivity, defined as Ms max 1 / 1 P jw C jw , is used.
Consider a first-order plus time delay process (Lee et al. 2014):
Ac
Ps
e s 0.2s 1
(34)
By using pole zero conversion, the controller parameters of the proposed method are listed in Table 2. The tuning parameter is considered as 0.4 . We define 0.4 as the set-point weighting parameter. According to the parameter of the PID controller, we can calculate the maximum sensitivity Ms=1.6. The proposed method was compared with the method proposed by Lee et al. (2014) and Skogestad (2003), and for 18
Page 19 of 42
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a fair comparison, the same Ms is used to obtain the parameters of their method in the simulation. To calculate the performances of the
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first-order plus time delay process, a unit step input is given in the set point and a negative step disturbance of 0.2 is given at t=25s. Figure 2 shows the responses without mismatches in the model while table 2 shows IAE and overshoot indices. According to the data listed in Table 2, which shows the responses and the value of the performance indices, we can see clearly that the best performance arises from the proposed controller. In the actual situation, there always exist model mismatch. In order to evaluate the robustness of the controller, +20% perturbations in
ed
the process gain and time delay are considered, i.e., P s 1.2 e 1.2 s / 0.2 s 1 as an actual process. The responses when there are 20%
ce pt
mismatches in the plant are shown in Figure 3, IAE and overshoot are given in Table 2. For both no mismatch and mismatch in the plant, the simulation results show that the proposed method give the best control performances both in the set-point tracking and disturbance rejection. The region of stability is evaluated for K, T and individually using the robust stability theorem and the results are shown in Table 4. It is
Ac
found by simulation that the present method is robust for uncertainty in process model parameters.
19
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1.4
1
y, e s n o p s e R
SIMC Lee Proposed
M an
1.2
0.8
0.6
ed
0.4
0.2
0
5
10
15
20
25 Time/s
30
35
40
45
50
ce pt
0
Ac
Figure 2. Response for perfect model for example 1
20
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1.4
M an
1.2
1
y, e s n o p s e R
SIMC Lee Proposed
0.8
0.6
ed
0.4
0.2
0
5
ce pt
0
10
15
20
25 Time/s
30
35
40
45
50
Figure 3. Response for a perturbation of +20% in process for example 1
Example 2
Ac
Just consider the following SOPDT process (Lee et al. 2014) Ps
e2 s s 1 0.7s 1
(35)
21
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ip t cr us
The controller parameters obtained for the proposed method are listed in Table 2. The tuning parameter is considered as 3 . And the
M an
maximum sensitivity, Ms=1.64. The weighting parameter is 0.4 . The proposed controller is compared with Lee et al. (2014) and Skogestad (2003), the same Ms is considered. The responses are shown in Figure 4, and IAE and overshoot are given in Table 2. 1.4
1.2
SIMC Lee Proposed
0.8
0.6
ce pt
y, e s n o p s e R
ed
1
0.4
0.2
Ac
0
0
10
20
30
40
50 Time/s
60
70
80
90
100
Figure 4. Response for perfect model for example 2
22
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us
A mismatch of +20% in process gain and time delay is considered. And the process is described as P s 1.2e 2.4 s / s 1 0.7 s 1 . Figure 5
M an
shows the simulation result of example 2, and IAE and overshoot of the process are given in Table 2. It can be obvious that the IAE and overshoot for the proposed method are the best for disturbance rejection and dynamic performance. 1.4
1.2
SIMC Lee Proposed
ed
1
0.8
0.6
ce pt
y, e s n o p s e R
0.4
0.2
Ac
0
0
10
20
30
40
50 Time/s
60
70
80
90
100
Figure 5. Response for a perturbation of +20% in process for example 2
Example 3
23
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M an
Consider an integrating process with time delay (Shamsuzzoha and Lee, 2007):
Ps
0.2e7.4 s s
(36)
In the proposed method, we define 100 . The parameter of the filter is 10.1 , and Ms=1.96. A negative step disturbance of 0.1 is given in the load at t=150s. The methods proposed by Shamsuzzoha and Lee (2007) and Chien and Fruehauf (1990) are considered in this paper. The value of for their method is adjusted to obtain the value of Ms=1.96. Fig 6 shows the simulation results when the process model is perfect.
Ac
ce pt
ed
Table 2 shows IAE and overshoot of the process.
24
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ip t cr us
1.4
Shamsuzzoha Chien Proposed
M an
1.2
1
y, e s n o p s e R
0.8
0.6
ed
0.4
0.2
0
ce pt
0
50
100
150 Time/s
200
250
300
Figure 6. Response for perfect model for example 3
Ac
To evaluate the robustness of the process, 20% change in the plant delay and the steady state gain P s 0.24e8.88 s / s , which has a perturbation uncertainty of +20% in gain and time delay of the process. The simulation results are shown in Figure 7. For both no mismatch and mismatch, it is clear that the proposed method gives better control performances.
25
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ip t cr us
1.8
Shamsuzzoha Chien Proposed
M an
1.6 1.4 1.2 y, e s n o p s e R
1 0.8 0.6
ed
0.4 0.2
0
50
100
150 Time/s
200
250
300
ce pt
0
Figure 7. Response for a perturbation of +20% in process for example 3
Example 4
Ac
The FODIP plant model considered in Ali and Majhi (2010): Ps
e4 s s s 1
(37)
And we consider 100 . And the filter parameter is 3.24 , so that Ms=2.3. In order to evaluate the disturbance rejection, a negative step disturbance of 0.1 is given at t=100s. And the methods proposed by Ali and Majhi (2010) and Skogestad (2003) are considered. For a fair 26
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ip t cr us
comparison, all compared controllers are tuned to have Ms=2.3 by adjusting . The simulation results are shown in Figure 8. And the
M an
performance indices are given in Table 2. 1.5
1
SIMC Ali Proposed
ed
y, e s n o p s e R
ce pt
0.5
Ac
0
0
50
100
150 Time/s
200
250
300
Figure 8. Response for perfect model for example 4
27
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us
To evaluate the robustness of the process, 20% change in the plant delay and the steady state gain is considered. P s 1.2 e 4.8 s / s s 1
M an
Figure 9 shows the responses for a 20% mismatch model. The performance indices are given in Table 2. Improve performance is obtained by the proposed method obviously. 2 1.8 1.6
SIMC Ali Proposed
ed
1.4 1.2
1
0.8
ce pt
y, e s n o p s e R
0.6 0.4 0.2
Example 5
Ac
0
0
50
100
150 Time/s
200
250
300
Figure 9. Response for a perturbation of +20% in process for example 4
The following first-order delayed unstable process is considered (Shamsuzzoha and Lee, 2007):
28
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ip t cr us
P s
e 0.4 s s 1
(38)
M an
The filter time constant 0.4 is select for the proposed method. Then we can get the Ms=2.75. The weighting parameter is defined as 0.2 . To evaluate disturbance rejection of the process, a negative step disturbance of 1 at t=15s are considered. For comparison, we choose the methods proposed by Shamsuzzoha and Lee (2007) and Lee et al. (2000), is selected by Ms=2.75 to ensure fair comparison. Figure 10 shows the responses for the perfect model. And Figure 11 shows the responses for a 20% mismatch model. P s 1.2e0.48s / 1.2s 1 . The values of
ed
the controller performance indices are listed in Table 3. Clearly, the proposed method performs better than the others. The region of stability is
Ac
ce pt
evaluated for K, T and individually and the results are shown in Table 4.
29
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2.5
Lee Shamsuzzoha Proposed
M an
2
1.5 y, e s n o p s e R
1
ed
0.5
0
ce pt
0
5
10
15 Time/s
20
25
30
Ac
Figure 10. Response for perfect model for example 5
30
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2.5
Lee Shamsuzzoha Proposed
M an
2
1.5 y, e s n o p s e R
1
ed
0.5
0
ce pt
0
5
10
15 Time/s
20
25
30
Figure 11. Response for a perturbation of +20% in process for example 5
Example 6
Ac
The unstable process is considered for the simulation as follows (Shamsuzzoha and Lee, 2008) ： P s
e0.939 s 5s 1 2.07s 1
(39)
31
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ip t cr us
The tuning parameter was selected as 0.939 , so that Ms=2.27. And the set-point weighting parameter is 0.2 . A negative step disturbance
M an
of 0.2 are considered. To show the improvement, the method proposed by Shamsuzzoha and Lee, (2007) and Shamsuzzoha and Lee, (2008) are considered. The adjusting parameter are selected to obtain Ms=2.27 for each method. The simulation results are shown in Figure 12.
1.4
ed
1.2
Shamsuzzoha Shamsuzzoha Proposed
1
0.8
ce pt
y, e s n o p s e R
0.6
0.4
0.2
Ac
0
0
10
20
30
40
50 Time/s
60
70
80
90
100
Figure 12. Response for perfect model for example 6
32
Page 33 of 42
ip t cr show
the
robust
performance
of
us
To
the
process,
a
10%
change
in
all
four
parameters
is
M an
considered. P s 1.1e 1.0329 s / 4.5 s 11.863s 1 The simulation results are shown in Figure 13. The values of the controller parameters are listed in Table 3. 1.4
1.2
Shamsuzzoha Shamsuzzoha Proposed
ed
1
0.8
0.6
ce pt
y, e s n o p s e R
0.4
0.2
Ac
0
0
10
20
30
40
50 Time/s
60
70
80
90
100
Figure 13. Response for a perturbation of +20% in process for example 6
6. Conclusion
33
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ip t cr us
An IMC PID controller based on pole zero conversion design for a class of processes with time delay is proposed. And we provide guidelines
M an
to select the tuning parameters. The adjusting parameter can be calculated from the time constant and the time delay of the processes directly. The IMC filter structure for all the stable and unstable processes with time delay which considered in the paper have a unified structure. And the formulas to calculate the adjusting parameter depend on time delay/time constant ratio of the processes. Several processes are considered in the simulation. The controller proposed in this paper successfully demonstrated better performance than other PID tuning method for a class of
ed
processes with time delay.
ce pt
REFERENCES
Shamsuzzoha, M., 2013. Closed-loop PI/PID Controller Tuning for Stable and Integrating Process with Time Delay. Ind. Eng. Chem. Res. 52, 12973-12992.
Ac
Chen, D.; Seborg, D.E., 2002. PI/PID controller design based on direct synthesis and disturbance rejection. Ind. Eng. Chem. Res. 41, 4807-4822. Lee, Y.; Park, S.; Lee, M.; Brosilow, C., 1998. PID controller tuning for desired closed-loop responses for SI/SO systems. AIChE J. 44, 106-115.
34
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ip t cr us
Skogestad, S., 2003. Simple analytic rules for model reduction and PID controller tuning. J. Process Control 13, 291-309.
M an
Shamsuzzoha, M.; Lee, M., 2007. IMC-PID controller design for improved disturbance rejection of time-delayed processes. Ind. Eng. Chen. Res. 46, 2077-2091.
Wang, Q.G.; Bi, Q.; Zhang, Y., 2001. Partial Internal Model Control. IEEE Trans. Indus. Electronics. 48, 976-982.
ed
Oliveira, V.A.; Cossi, L.V.; Teixeira, C.M.; Silva, M.F., 2009. Synthesis of PID controllers for a class of time delay systems. Automatica 45,
ce pt
1778-1782.
Shamsuzzoha, M.; Lee, M., 2008. Design of advanced PID controller for enhanced disturbance rejection of second-order processes with time delay. AIChE J. 54, 1526-1536.
Ac
Rao, A.S.; Chidambaram, M., 2009. Direct synthesis-based controller design for integrating processed with time delay. Journal of the Franklin Institute. 346, 38-56.
35
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Jeng, J.C.; Tseng, W.L.; Chiu, M.S. 2013. A one-step tuning method for PID controllers with robustness specification using plant step-response
M an
data. Chem. Eng. Res. Des. 92, 545-558.
Zhao, Z.C.; Liu, Z.Y.; Zhang, J.G., 2011. IMC-PID tuning method based on sensitivity specification for process with time-delay. Journal of Central South University of Technology. 18, 1153-1160.
ed
Morari, M.; Zafiriou, E. Robust Process Control. NJ: Prentice Hall Englewood Cliffs, 1989.
ce pt
Lee, J.; Cho, W.; Edgar, T., 2014. Simple Analytic PID Controller Tuning Rules Revisited. Ind. Eng. Chem. Res. 53, 5038-5047. Chien, I. L.; Fruehauf, P. S., 1990. Consider IMC Tuning to Improve Controller Performance. Chem. Eng. Prog. 86, 33-41.
Ac
Ali, A.; Majhi, S., 2010. PID controller tuning for integrating processes. ISA Transactions. 49, 70-78. Lee, Y.; Lee, J.; Park, S., 2000. PID Controller Tuning for Integrating and Unstable Processes with Time Delay. Chem. Eng. Sci. 55, 3481-3493.
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Table Caption Table 1. IMC-PID Controller Tuning Rules
Table 2. PID controller parameters and resulting performance indices for stable examples
ed
Table 3. PID controller parameters and resulting performance indices for unstable examples
Ac
ce pt
Table 4. Stability region for K, T and for the stable and unstable processes
37
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Table 1 IMC-PID Controller Tuning Rules
2min ,T
2T T 2 T
Ke s Ts 1 Ls 1
L L
2T T 2 T
T T T
Ke s Ts 1 Ls 1
min ,T
min ,T
L K 2
L
L L
2 2
0.5 K 2
0.5
0.5 0.5
2 2
T K 2
T
T T
2 2T T T
K 2
0.5
2 2T T T
L K 2
L
L L
Ac c
Ke s Ts 1
2
d
Ke s s Ts 1
2
ep te
0.5T T T
K 2
an
Ke s Ts 1
Ke s s
TD
M
2
TI
Kc
Process
Imaginary filter
Compensator
1 Bs 1
1 Ts 1
0.5 s 1 0.01s 1 1 Bs 1
0.5 s 1 0.01s 1
Page 39 of 42
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Example
Method
Kc
TI
TD
Ms
us
Table 2 PID controller parameters and resulting performance indices for stable examples Perfect model
an
IAE
Example 1
overshoot
e s / 0.2s 1
0.4
0.347
0.567
0.12
SIMC
1.0
0.1
0.2
0
Lee
0.6
0.125
0.2
0.2
1.375
0.435
SIMC
1.8
0.263
Lee
1.7
0.272
Example 3 8.26
0.56
Shamsuzzoha
10.6
Chien
14.1
Example 4
1.2e1.2 s / 0.2s 1
1.00
2.55
1.15
1.6
2.65
1.13
3.76
1.37
1.6
2.61
1.04
3.25
1.22
e2 s / s 1 0.7s 1
1.64
6.07
1.05
7.34
1.23
1
0.70
1.64
6.88
1.19
9.19
1.38
1
0.86
1.64
6.85
1.18
9.13
1.40
0.2e7.4 s / s
0.4
0.24e 8.88 s / s
28.57
3.2
1.96
22.2
1.05
22.0
1.1
0.545
23.4
2.42
1.96
22.7
1.08
24.3
1.2
0.556
34.7
3.1
1.96
29.2
1.30
30.7
1.68
e4 s / s s 1
0.4
1.2e2.4 s s 1 0.7s 1
0.40
Ac c
Proposed
overshoot
1.653
ep te
Proposed
IAE
2.31
d
Example 2
1.6
M
Proposed
20 % mismatch
0.5
1.2e4.8s / s s 1
Proposed
3.24
0.22
11
0.91
2.3
13.36
1.06
16.62
1.28
SIMC
4.15
0.19
33
0.97
2.3
31.94
1.31
38.94
1.84
Ali
4.15
0.19
22.37
2.79
2.3
26.30
1.26
27.00
1.61
0.4
Page 40 of 42
ip t cr
Method
Kc
TI
TD
Ms
Example 5 0.4
2.5
2.67
0.17
2.75
Shamsuzzoha
0.745
2.38
2.37
0.214
2.75
Lee
0.58
2.48
2.58
0.152
6.63
Shamsuzzoha
0.960
6.47
Shamsuzzoha
1.745
3.85
1.2e0.48s / 1.2s 1
2.21
1.01
2.32
1.00
2.62
1.08
2.42
1.02
3.46
2.14
3.44
2.42
Perfect model e
0.939 s
0.2
10 % mismatch
/ 5s 1 2.07 s 1
1.1e
1.0329
/ 4.5s 11.863s 1
5.76
1.326
2.27
4.62
1.01
4.70
1.00
5.65
1.352
2.27
5.17
1.02
5.15
1.01
8.06
1.687
2.27
9.26
1.07
8.76
1.03
ep te
0.939
e0.4 s / s 1
0.2
Ac c
Proposed
20 % mismatch
2.75
d
Example 6
M
Proposed
Perfect model
an
Example
us
Table 3 PID controller parameters and resulting performance indices for unstable examples
Page 41 of 42
Table 4 Stability region for K, T and for the stable and unstable processes K
T
e s / 0.2s 1
Model
Proposed Shamsuzzoha Lee
43% 42% 40%
79% 100% 87%
e0.4 s / s 1
30% 25% 32%
Ac
ce pt
ed
M
an
us
35% 36% 33%
96% 138% 102%
ip t
100% 100% 91%
cr
Proposed SIMC Lee Model
Page 42 of 42