Manuscript e150064 Shamsuzzoha 17

Corresponding author. Tel.:+ 966-13-860-7360; E-mail address: [email protected], [email protected]

1

IMC Based Robust PID Controller Tuning for Disturbance Rejection

Mohammad Shamsuzzoha

Department of Chemical Engineering, King Fahd University of Petroleum and Minerals,

Dhahran 31261, Saudi Arabia

Abstract: It is well-known that the IMC-PID controller tuning gives fast and improved

setpoint response but slow disturbance rejection. A modification has been proposed in

IMC-PID tuning rule for the improved disturbance rejection. For the modified IMC-PID

tuning rule, a method has been developed to obtain the IMC-PID setting in closed-loop

mode without acquiring detailed information of the process. The proposed method is

based on the closed-loop step setpoint experiment using a proportional only controller

with gain Kc0. It is the direct approach to find the PID controller setting similar to

classical Ziegler-Nichols closed-loop method. Based on simulations of a wide range of

first-order with delay processes, a simple correlation has been derived to obtain the

modified IMC-PID controller settings from closed-loop experiment. In this method,

controller gain is a function of the overshoot obtained in the closed loop setpoint

experiment. The integral and derivative time is mainly a function of the time to reach the

first peak (overshoot). Simulation study has been conducted for the broad class of

processes and the controllers were tuned to have the same degree of robustness by

measuring the maximum sensitivity, Ms, in order to obtain a reasonable comparison. The

PID controller settings obtained in the proposed tuning method show better performance

and robustness with other two-step tuning methods for the broad class of processes. It

has been also applied to temperature control loop in distillation column model. The

result has been compared to the open loop tuning method where it gives robust and fast

response.

Keywords: PI/PID controller, step test, closed-loop response, IMC-PID, overshoot

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1. Introduction

The proportional-integral-derivative (PID) controllers are the most widely accepted in

industrial applications at the regulatory level. The main reason for this is their

comparatively simple structure, which can be readily understood and which allows them

to be easily implemented in the real world. However, it has been noticed that many

PI/PID controllers are not properly tuned and a lot of effort has been made to

systematically resolve this problem. Therefore, the goal of this research is to develop a

direct approach method of controller tuning from closed loop setpoint data.

There are varieties of PI/PID controller tuning approaches presented in the open

literature and out of that two are extensively used for controller tuning, based either on

open-loop or closed-loop plant tests; the majority of them being of the former type,

employing the process gain (k), time constant () and time delay (). The PID controller

design for the different types of processes based on direct synthesis [1] and Internal

model control (IMC) are among such popular tuning methods [2, 3, 4, 5, 6]. The output

response based on both the approaches has satisfactory performance and robustness.

Recently, Vu and Lee [7], Rao and Chidambaram [8] and Shamsuzzoha et al. [9] have

developed analytical methods for the design of a PID controller cascaded with a second-

order lead-lag filter for various types of time-delay processes for enhanced disturbance

rejection.

Although the PI/PID tuning rule on the basis of IMC and direct synthesis methods gives

excellent performance for setpoint changes, it shows slow output responses to input

(load) disturbances for lag-dominant as well as integrating processes [5, 10, 3].

Skogestad [5] has modified the integral time in SIMC method which is an excellent

remedy for processes with a large time constant to improve load disturbance rejection.

The above two-step approach is based on the open-loop test. It requires first to obtain

process parameters and then calculate PI/PID tuning setting with any other existing

tuning methods. There are two problems associated with this approach. First, to find out

the process model with an open-loop experiment, usually a step test is desirable to get

the process parameters. Sometimes it could be very tedious and may also disturb the

process. The second problem associated with this is the approximation error in getting

the parameters (for example, k, and ) from the open loop step test data.

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It is important to mention at this point that sometimes it is not easy to conduct open-

loop test for the process model identification. There are always chances of the control

variable drifting away from the specified value and eventually leading to products

qualities off-specification. On the other hand, in the closed-loop test, it is easy to control

the process during experiment and thus reduce the effect of disturbances.

The alternative of the open-loop approach is a two-step tuning procedure based on

closed-loop setpoint experiment with a P-controller. It was originally proposed by

Yuwana and Seborg [11], the method is applicable to most of the open-loop stable

systems with dead time. Subsequently, the above method was modified by Jutan and

Rodriguez [12], Lee [13], and Chen [14]. They identified a first-order with delay model

by matching the closed-loop setpoint response with a standard oscillating second-order

step response. Later, for the controller parameters calculation, they mainly utilized the

Ziegler-Nichols [15] tuning rules, which could give tight controller setting but other

tuning rules e.g., IMC-PID by Shamsuzzoha and Lee [3] could also be used. Lee et al.

[16] further reinvestigated the Yuwana and Seborg [11] method by identifying the

processes with a second-order plus dead-time model under closed-loop conditions. They

utilized the Taylor series expansion of the dead-time term with the combination of the

ultimate data matching technique of Chen [14] for second-order plus dead-time model.

However, the resulting Lee et al. [16] method gives relatively better performance and

robustness over the other closed-loop methods [11, 12, 13, 14], albeit at the expense of

increased degree of complexity and computation.

In most of the above mentioned tuning methods based on the closed-loop two-step

technique, at least five measurable quantities were required in the identification test to

obtain the process model. For example, the methods by Yuwana and Seborg [11], Lee

[13], Chen [14] and Lee et al. [16] need to identify first peak of the output response

(cp1), second peak of the output response (cp2), first minimum of the output response

(cm1), half-period of oscillation (t) and the steady-state value (c).

There are few problems associated with the above closed-loop experiment. (i) The

required number of measurable quantities are high i.e., at least five. (ii) For the low

value of overshoot, it is difficult to find the accurate value of the first minimum of the

output response (cm1) and second peak of the output response (cp2) from the step test

experiment. (iii) To obtain the precise value of cm1 and cp2, one has to generate the

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output response of large overshoot with considerable oscillation. Furthermore, there are

several problems associated with the large overshoots. It gives a long settling time and

needs large input changes that are undesirable in chemical process industries.

The other alternative approach to both the above mentioned two step procedures is to

use the one step closed-loop experiments, that directly obtain controller setting without

finding process parameters. A very popular and old method is that of Ziegler-Nichols

(Z-N) [15]. It needs only the ultimate controller gain (Ku) and the period of oscillations

(Pu), which one can obtain directly from an experiment. For a PI-controller the

recommended settings are Kc=0.45, Ku and I=0.83Pu. Ziegler-Nichols [15] closed-loop

tuning method is still very widely used for controller tuning in industrial processes.

However, there are several disadvantages of this method.

The most significant is that in the Z-N method we actually push the process to the limit

of instability as we search for the Ku. Creeping up on the ultimate gain can be very time

consuming, but if we try to save time by making large adjustments in the search for the

Ku, it is very likely that the process will actually become unstable, at least for a brief

period.

The remedy for the above problem is to introduce the relay method of strm and

Hgglund, [17] which requires the feature of switching on/off-control in the system.

One more drawback is that the Z-N [15] tuning setting does not work satisfactorily on

all processes. The prescribed controller settings are somewhat aggressive for lag-

dominant (integrating) processes (Tyreus and Luyben, [18]) and sluggish for dead time

dominant process (Skogestad, [5]). The third disadvantage, of the Ziegler-Nichols [15]

method is that it is not applicable to a simple second-order process.

Haugen [19] has developed the Good Gain Method which is entirely on the basis of the

trial and error approach to find the suitable controller gain and finally the tuning

parameters. Hu and Xiao [20] have developed an analytical PI tuning method which is

similar to the setpoint overshoot method [10]. Skogestad and Grimholt [21, 22] have

claimed that it is hard to obtain a better performance than SIMC, at least for PI control

based on a first order with time delay model. Seki and Shigemasa [23] have proposed

the method to retune the existing controller based on comparing the closed-loop

responses. Veronesi and Visioli [24] have also claimed for retuning of an existing PI

controller for better performance and robustness.

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Recently, Alcantara et al. [25] have addressed the model-based tuning of PI/PID

controller based on the robustness/performance and servo/regulator trade-offs. The

interesting feature of the study has been to show how to shift each compromise based

upon constraint. They have extended the preliminary design concept of balanced

autotuning which was published earlier [26]. K-SIMC method, a modification of SIMC

rule has been proposed recently by Lee et al. [27]. Torrico et al [28] proposed a new and

simple design for the filtered Smith predictor (FSP), which belongs to a class of dead-

time compensators (DTCs) and allows the handling of stable, unstable, and integrating

processes. Recently, several authors [29, 30, 31, 32, 33, 34, 35, 36] have proposed the

modified approach for the enhanced PID controller design based on the open loop

method.

In view of the above discussion about the different types of controller tuning approaches,

it is clear that there is a need for a simple and effective controller tuning method in

closed-loop.

Therefore, the goal of the proposed study is to find a simple and direct controller tuning

technique in closed-loop for the broad class of the processes. No detailed prior

information of the plant process parameters (k, and ) is required to get the modified

IMC-PID controller [4] settings from the closed-loop setpoint experiment. It removes

the shortcomings of the Ziegler-Nichols continuous cycling method and can be an able

substitute for the same. Although the original IMC-PID controller tuning method is

applicable only for the low order processes, the proposed closed loop method is used

even in high order processes without any modification.

To achieve the above mentioned goal the outline of remaining part of this paper is

organized as follows: Section 2 is focused on the modification of the existing IMC-PID

tuning method for the improved disturbance rejection for lag dominant process. In

section 3, the development of the closed-loop setpoint experiment with P-controller for

key information namely overshoot, time tp to reach the (first) overshoot and steady state

value is discussed. Section 4 is dedicated to the development of the correlation between

setpoint response data with modified IMC-PID settings. In section 5, the guidelines for

the selection of P-controller gain (Kc0) are enunciated. Section 6, discusses a simulation

study for the broad class of processes and finally in section 7, the conclusion of the

present study is there.

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2. Modified IMC-PID Controller Tuning Rule

2.1. IMC-PID controller design for first-order with dead time process

The motivation of this section is to review IMC-PID controller tuning proposed by

Rivera et al. [4] for first order process with time delay. In next section, this tuning

method has been utilized as a basis for the development of the proposed closed-loop

method. The first-order time delay process is commonly used as a representation of the

process dynamics for several equipment in chemical industries as:

-

( )1

skeg s

s

(1)

where k, and are the process gain, time constant and time delay, respectively. It is

important to note that the PID controller gives reasonable response in the chemical

industries and the same is given as:

1 1

11

c DI F

c s K ss s

(2)

Kc, I, D and F are the proportional gain, integral time constant, derivative time

constant and lag filter of the PID controller, respectively. The other form of the PID

structure (e.g., series form) can be easily transformed from the ideal form in Eq. (2) by

using a simple calculation [1].

Figure 1 (a) and (b) show the block diagram of the IMC control and equivalent classical

feedback control structures, respectively. In this block diagram, g(s) is the process, g s

process model, q(s) IMC controller and c(s) the classical feedback controller. The

remaining variables are the manipulated variable u, process output variable y, the

setpoint ys, and the input disturbance d at the plant. The relationships in closed-loop

from the setpoint and load disturbance to the output are:

7

q(s) g(s)-

+

g(s)~

+-

yys

d

u

(a) The IMC structure

ys

d

C(s) g(s)yu+

-

(b) Feedback control structure

Figure 1. Block diagram of the IMC and classical feedback control systems.

sgc g

y= y + d1+c g 1+c g

ss s

s s s s (3)

The conventional feedback controller which is equivalent to the IMC controller can be

expressed by the following relationship.

1

q sc s

g s q s

(4)

where g s indicates the process model transfer function, c(s) is conventional and q(s)

is the IMC controller. The standard IMC controller design is divided into two steps as:

Step I: The process model g is decomposed into two parts:

8

M Ag s p s p s (5)

where pM(s) and pA(s) are the portions of the model inverted and not inverted,

respectively, by the controller, pA(s) is typically a non-minimum phase which includes

time delay and right half plane zeros); where pA(0)=1.

Step II: The typical IMC controller is given by

-1Mq s p s f s (6)

where f(s) is the IMC filter and given as 1 ( 1)rcf s s , c is closed loop time constant

which controls the tradeoff between the performance and robustness. The parameter r is

chosen to be a sufficiently big value to form the IMC controller semi-proper. Consider a

first order process with time delay, the IMC controller is given as:

1

1c

sq s

k s

(7)

Therefore, the feedback controller c(s) which is equivalent to the IMC controller is

1

1 sc

sc s

k s e

(8)

Consider approximation of the time delay expression in Eq. (8) using first-order Pade

approximation i.e., - 1- 2 1 2se s s . The resulting controller can be easily

obtained by simple calculation in the form of PID with first order filter (Rivera et al. [4])

as:

2

2c

c

Kk

(9)

2I

(10)

9

2D

(11)

2c

Fc

(12)

The main reason of using 1/1 Pade approximation is to obtain both simple PID control

structure with enhanced performance. It has been found that high order approximation

of the dead time has not any significant advantage in terms of the performance and

stability of the control system. The above PID controller offers fast and smooth set-

point tracking, but has a sluggish disturbance rejection, especially for processes with a

small time-delay/time-constant ratio [3, 1, 10, 5]. To enhance the load disturbance

response, Skogestad [5] suggested the modification in integral time (I) for lag dominant

and integrating process as:

I c =4( +) (13)

Incorporating the above recommended setting for the lag dominant (integrating) process,

the integral time in Eq. (10) has been modified for the improved disturbance rejection

for the small time-delay/time-constant ratio (integrating process) and given as

I c =min , c( +)2

(14)

where c is an arbitrary constant and c=4 has been suggested by Skogestad [5], as we can

see in Eq. (10). This modification of the I has significant advantage for both the lag and

delay dominant processes. The closed loop response has been shown for different value

of c in Figure 2. It is well known that in the majority of process control loops, the

disturbance rejection is the most important task for the controller. To check the faster

disturbance rejection, different values of c (c=4, 3 and 2) have been tested and it was

found that c=3 is the most suitable choice for the modified tuning rule [19, 22]. The

choice of c=3 has impact on the robustness of the system and it will be somewhat lower

than c=4. The other impact should be on the overshoot in the setpoint response and it

will be slightly higher for c=3. In Figure 2, c=4 gives quite sluggish disturbance

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rejection response. In the modified tuning rules, selection of c=0.6 has been the

recommended choice as it gives maximum sensitivity (Ms); approximately 1.73 for the

integrating process, and Ms=1.75 for the delay dominant process. Therefore, the above

tuning method can be simplified for the c= 0.6 and given as:

2

3.2cK

k

(15)

min , 4.82

I

(16)

2D

(17)

0.188F (18)

2.2. Analysis of the effect of integral action

The original IMC-PID rule (Eq. 9-12) gives fast and smooth set-point tracking.

However, it has a slow disturbance rejection for processes with a small / ratio. The

modified tuning formula given in Eq. (15)-(18) is for the enhanced disturbance rejection.

To show the effect of the integral action, a first-order process with time delay

-( ) 10 1sg s e s has been considered. Figure 2 shows the comparison of the closed-

loop response of the modified IMC-PID controller for four different values of the I

while other tuning parameters (Kc, D and F) are kept constant. The different values of

I obtained by changing the value of c, i.e, c=4, 3 and 2. To test the performance of the

control system, both the load disturbance and setpoint have been added a step change of

magnitude 1 (ys=1 and d=1). The robustness measure, Ms-value is almost the same for

all four closed-loop responses. Although the closed-loop response for the disturbance

rejection of c=2 is better, it gives an unacceptable overshoot in setpoint response. The

output response of the original IMC-PID (I=+/2) gives slow disturbance rejection

while satisfactory setpoint change as shown in figure. The goal of the proposed

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modification in the IMC-PID controller is to obtain fast disturbance rejection while

maintaining the sepoint response.

Therefore, c=3 is a better choice of the integral time and resulting integral time equation

is given as I=3(c+), which is tradeoff between load performance and setpint change.

2.3. Effect of setpoint filter on servo response

The integral action has been increased in the modified IMC-PID tuning rule for the

enhanced disturbance rejection. This modification is applicable to the lag-dominant and

integrating process with time delay. It provides satisfactory improvement in the

disturbance rejection performance while deteriorate setpoint response with large

overshoot. Therefore, leadlag setpoint filter which is the usual practice in industries to

improve the servo response, is recommended to remove the overshoot in setpoint

response. The recommended choice to lead-lag filter is 0.75 1 1r I If s s . To

show the performance improvement a first order process with time delay

-( ) 10 1sg s e s has been considered. The resulting setpoint filter for this case

should be 3.6 1 4.8 1rf s s . Figure 3 shows the closed-loop response of the

modified IMC-PID tuning rule for both with and without setpoint filter. The integral of

the absolute value of the error (IAE)-value is reduced from 3.11 to 2.44 and total

variation (TV) from 14.73 to 11.47, after using the setpoint filter. As expected, the

output response with setpoint filter is fast without any overshoot.

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Figure 2. Closed-loop responses of 10 1

seg s

s

for different value of I (i.e, c=4,

3, 2 and +/2) while other tuning parameters (Kc, D and F) are same for c=0.6. Setpoint change at t=0; load disturbance of magnitude 1 at t=20, (ys=1 and d=1).

0 8 16 24 32 400

0.25

0.5

0.75

1

1.25

Time

Pro

cess

Var

iab

le (

y)

I=3(

c+)=4.8, M

s=1.74

I=4(

c+)=6.4, M

s=1.75

I=(+/2)=10.5, M

s=1.76

I=2(

c+)=3.2, M

s=1.74

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Figure 3. Effect of setpoint filter to remove the overshoot from setpoint response:

Setpoint responses of first-order stable process with time delay 10 1

seg s

s

.

Setpoint change at t=0; load disturbance of magnitude 1 at t=20.

2.4. Effect of the low order lag filter in closed-loop response

The modified IMC-PID tuning has first order lag filter F=0.188. These days most of

the DCS systems usually provide the PID controllers with various equations, lead lag

blocks, filter blocks and pure dead time blocks. Some, however, may allow you to select

a more sophisticated filter. It is straightforward to implement the modified IMC-PID

with lag filter control scheme under the modern DCS system environment. As an

example, a standard block of first-order lag in a well-known DCS system is:

The selection of the right filter parameter always ensures the overall performance

improvement of the control loop. Especially, in the PID controller when the derivative

action is active, if the lag-filter is not used, or if its magnitude is very small, then the

0 8 16 24 32 400

0.25

0.5

0.75

1

1.25

Time

Pro

cess

Var

iab

le (

y)

I=3(

c+)=4.8, M

s=1.74

I=(+/2)=10.5, M

s=1.76

I=3(

c+)=4.8, M

s=1.74 with setpoint lead-lag filter

14

controller will be responding to noise. This may cause the control valve to move

unnecessarily and eventually lead to process upset. On the other hand, if a filter

parameter is large, then it may slow the performance of the controller. Therefore, it is

important to select a proper value of the lag-filter so that the controller responds quickly

to any upsets in the process.

The recommended setting of the filter parameters should not be more than 1/3 of the

process dead time (Buckbee [37]). In the modified IMC-PID, F=0.188 for c=0.6,

which is within the recommended value.

In this paper, the simulation study is based on the ideal form of controller which is

given in Eq. (2). In real practice one has to modify derivative action with the derivative

filter 1D Ds s in the PID control.

A simulation is carried out to show the effect of the lag-filter in the system that

possesses noise in the measurement. A first order process with time delay

-( ) 10 1sg s e s has been considered for the simulation and controller setting is

calculated based on c=0.6. In this comparison, the derivative-filter is used in both the

controllers, with and without lag-filter, with =0.1. Figure 4 shows the comparison of

the closed-loop process response and control variable of the modified IMC-PID

controller. The resulting process variables and the control variables are plotted for the

controller with and without first order lag filter with noise measurement of white noise

of power 2.0E-005.

The derivative filter, which is also included in both the cases, plays an important role in

reducing the measurement noise. It is clear from Figure 4 that the control output is less

noisy for the modified IMC-PID with output-filtered structures. This is because the

proportional action is partially responsible for the amplification of the measurement

noise [38]. Therefore, the control structure with lag-filter which is applied to the whole

control variable is more efficient than that applied to the derivative action only.

15

Figure 4. Load disturbance response (with noise measurement) of first-order with time

delay process 10 1

seg s

s

. The controller tuning parameters are selected for

c=0.6 and the resulting PID setting of the proposed method is

1 0.48 1

6.56 14.8 0.1*0.48 0.188 1

sc s

s s s

with white noise of power 2.0E-005. For the

modified IMC-PID without lag-filter 1 0.48

6.56 14.8 0.1*0.48

sc s

s s

, load disturbance of

magnitude 1 at t=0.

3. Closed-Loop Setpoint Step Test Experiment

This section describes the procedure to find the closed-loop data for the proposed IMC-

PID controller tuning method. The easy and classical approach for closed-loop

experiment is a setpoint step response given in Figure 5. In this experiment one can

keep full control of the process, including the change in the output variable. The time tp

to reach the first overshoot and its magnitude is the simplest to observe in this

experiment.

The closed-loop data extraction procedure is as follows (Shamsuzoha [39]):

0 5 10 15 20 25 30

0

0.05

0.1

0.15

Time

Pro

cess

Var

iab

le (

y)

0 5 10 15 20 25 30-3

-1.5

0

1.5

3

Time

con

tro

l v

aria

ble

(u

)

Modified IMC-PID with derivative filter and first order lag

Modified IMC-PID with only derivative filter

Modified IMC-PID with derivative filter and first order lag

Modified IMC-PID with only derivative filter

16

1. Switch the controller to P-only mode (for example, increase the integral time I to its

maximum value or set the integral gain KI to zero). This kind of controller mode switch

does not upset the industrial process.

2. Make a setpoint change that gives an overshoot between 0.10 (10%) and 0.60 (60%);

about 0.30 (30%) is a good value. Record the controller gain Kc0 used in the experiment.

Most likely, unless the original controller is quite tightly tuned, one will need to

increase the controller gain to get a sufficiently large overshoot.

It is important to note that most of the time it is difficult to extract the required

information accurately from small overshoots (overshoots < 0.10). Therefore, this

experiment does not consider the overshoot less than 0.1. On the other hand, large

overshoots (overshoots > 0.6) give severe oscillations and long settling time and also

require more excessive input changes. For these reasons, it is recommended to use an

intermediate overshoot of about 0.3 (30%) for the closed-loop setpoint experiment.

py

pt

y sy

t0t

uy

sy

py

0y

y

Figure 5. Step test output response in closed-loop with P-only control.

17

3. From the closed-loop setpoint response experiment, obtain the following values (see

Figure 5):

Controller gain used in step test, Kc0

Overshoot = (yp - y) /y

Time from setpoint change to reach first peak output (overshoot), tp

Relative steady state output change, b = y/ys.

The resulting output variables are given as:

Setpoint change : sy = ys y0

Peak output change (at time tp) : py = yp y0

Steady-state output change after setpoint step test: y = y - y0

It is important to note that one can speed up the experiment and there is no need to wait

for the response to settle. The waiting time could be more if the overshoot in the

experiment is somewhat large (overshoot > 0.4). In such circumstances, it is

recommended to finish the experiment once the output process response reaches its first

minimum. In the next step, record the corresponding output, yu and calculate y with

following relationship.

y = 0.45(yp + yu) (19)

The detailed derivation of the relationship in Eq. (19) is available in Shamsuzzoha and

Skogestad [10].

4. Mathematical Correlation between Closed-loop Data and IMC-PID

The main purpose of this research is to find a simple technique to obtain IMC-PID

controller setting in closed-loop mode. Therefore, the aim is to develop a mathematical

correlation between the setpoint response data (Figure 5) and the modified IMC-PID

settings (Eq. (15)-18) with c=0.6. For this reason, 15 first-order with time delay

models g(s)=ke-s/(s+1) that cover a wide range of processes have been considered.

They cover a broad range of processes from dead time dominant to lag-dominant

(integrating) as:

18

/=0.10, 0.20, 0.40, 0.80, 1.0, 1.50, 2.0, 2.50, 3.0, 5.0, 7.50, 10.0, 20.0, 50.0, 100.0

It is possible to scale time with respect to the time delay () and since the closed-loop

response depends on the product of the process and controller gains (kKc) we have

without loss of generality used in all simulations k=1 and =1.

For each of the 15 process models (values of /), we have obtained the modified IMC-

PID settings using Eq. (15)-18) with the choice c=0.6. Furthermore, for each of the 15

processes, we have generated 6 closed-loop step setpoint responses using P-controllers

that give a wide range of fractional overshoots as:

Overshoot= 0.10, 0.20, 0.30, 0.40, 0.50 and 0.60

In total, we have 90 setpoint responses, and for each of these we have recorded data for

four variables as:

The P-controller gain Kc0 used in the experiment, the fractional overshoot, the time to

reach the overshoot (tp), and the relative steady-state change, b = y/ys.

4.1. Selection of Controller Gain (Kc)

The first goal is to find a correlation between the above four data and the corresponding

IMC-PID controller gain Kc. Figure 6 shows the plot between kKc verses kKc0 for 90

setpoint experiments for different values of / ratio. As one can see from the said figure

that the ratio Kc/Kc0 is approximately constant for a fixed value of the overshoot. It is

independent of the value of / ratio and therefore it is given as:

c

c0

K=A

K (20)

where, the ratio A is a function of the overshoot only. In Figure 7, we plot the value of

A, which is obtained as the best fit of the slopes of the lines in Figure 6, as a function of

the overshoot. The equation below (solid line in Figure 7) fits the data in Figure 6,

nicely and it is given as:

A= [1.45(overshoot)2 -2.02 (overshoot)+1.27] (21)

Therefore, the final relationship for the controller gain is given as:

2

c c0 K K 1.45 overshoot 2.02 overshoot 1.27

(22)

19

Figure 6. Plot between experimental P-controller gain kKc0 and corresponding PID

controller gain kKc in Eq. (15).

Figure 7. Plot of variation of A with fractional overshoot using slopes data from Figure

6.

4.2. Selection of Integral Time (I)

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

kKc0

kK

c

0.10 overshoot

kKc=1.0895kK

c0

0.20 overshoot

kKc=0.9094kK

c0

0.30 overshoot

kKc=0.7884kK

c0

0.40 overshoot

kKc=0.6987kK

c0

0.50 overshoot

kKc=0.6282kK

c0

0.60 overshoot

kKc=0.5703kK

c0

0.1 0.2 0.3 0.4 0.5 0.60.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Overshoot(fractional)

A

A = 1.45*(overshoot)2 - 2.02*(overshoot) + 1.27

20

The purpose of this section is to find a simple correlation for the I. The modified IMC-

PID tuning rule in Eq. (16) uses the minimum of two I values. It would be interesting to

search a similar correlation for both the large and small delay in closed-loop method as

well.

(i) Comparatively large time delay process (I1 =+ /2): It is the case of relatively

large delay and the integral time in the IMC-PID tuning rule is I = (+ /2). After

rearrangement of Eq. (15)

I 1.6 ckK (23)

In Eq. (23), there is a requirement of process gain k, and to this effect it may be written

as:

kKc= kKc0.Kc/ Kc0 (24)

The closed loop gain kKc0 for the P-control setpoint experiment can be calculated from

the value of b as:

c0

bkK =

(1-b) (25)

The I relationship can be obtained by substituting kKc from Eq. (24) and Kc/ Kc0=A

into Eq. (23), as:

I

b1.6A

(1-b) (26)

To show the steps in brief, the closed-loop setpoint response is y/ys =

g(s)c(s)/(1+g(s)c(s)). With a P-controller (gain Kc0), the steady-state value is y/ys =

kKc0/(1+kKc0)=b and we derive Eq.(25). The absolute value is included to avoid

problems if b>1, as they may occur sometimes because of imprecise data.

It is feasible to get the value of time delay directly from the closed-loop setpoint

response. Moreover, this is not always straightforward. Shamsuzzoha and Skogestad [10]

have developed a reasonable correlation for the dead time and the setpoint peak time tp

which is direct and easier to observe.

For processes with a relatively large time delay, the ratio /tp varies between 0.27 (for

/= 8 with overshoot=0.1) and 0.5 (for /=0.1 with all overshoots), as evident from

21

Figure 8 for the intermediate overshoot of 0.3, the ratio /tp varies between 0.32 and

0.50. A conservative choice would be to use =0.5tp because a large value increases the

integral time. However, to improve the performance for processes with smaller time

delays, we propose to use =0.43tp, which is only 14% lower than 0.50 (the worst case).

In summary, we have for a process with a relatively large time delay:

0.69(1- )

I p

bA t

b (27)

(ii) Comparatively small time delay process (I2 =4.8). The integral time for a lag-

dominant (integrating) process is given as:

I2=4.8 (28)

For />4.8, it can be seen from Figure 8 that the ratio /tp varies between 0.25 (for

/=100 with overshoot=0.1) and 0.37 (for /=8 with overshoot 0.6). We select the

average value = 0.305tp which is approximately 17% lower than 0.37 (the worst case).

Also note that for the intermediate overshoot of 0.3, the ratio /tp varies between 0.30

and 0.32. In summary, we have for a lag-dominant process:

I2 p =1.46t (29)

Conclusion: The integral time I is obtained in a similar way as in Eq. (16) and it is the

minimum of the above two values as:

I p =min 0.69 , 1.46t(1- )

p

bA t

b

(30)

22

Figure 8. Ratio of process time delay () and setpoint overshoot time (tp) as a function

of overshoot for four first-order with delay processes (solid lines). Dotted lines: Values

of /tp used in final correlations.

4.3. Derivative action (D):

In this section, a method has been proposed to obtain the D from the closed-loop step

test data with P-only controller.

Mode I: The process which is close to integrating i.e., >> , integral time is I =4.8 in

IMC-PID tuning formula, and = 0.305tp in the closed-loop. D1 in Eq.(17) can be

approximated as

1

0.3050.15

2 2 2 2

p

D p

tt

(31)

Mode II: For the processes which have , integral time is I=(+0.5) in IMC-PID,

and equivalent to this information in closed-loop, =0.43tp. Assuming =, D2 is

calculated from Eq.(17) as

2 2

2

0.43 0.1433

2 2 3 3 3

p

D p

tt

(32)

0.1 0.3 0.5 0.60

0.1

0.2

0.3

0.4

0.5

Overshoot

/t

p

0.43 (I1

)

0.305 (I2

)

/=0.1

/=8

/=100

/=1

23

Summary: It is clear from the above analysis that D1 and D2 are very close to each

other and the conservative pick of D should be:

0.14D pt (33)

4.4. Low order controller filter from step test data

The modified IMC-PID method has low order lag filter 2F c c , and it

simplifies to F =0.188 for c= 0.6.

The objective of this section is to find the equivalent lag filter F from closed-loop data.

The analytical equation of the first order filter (F=0.188) for the integrating process is

F=0.188=0.1880.305tp=0.057tp.

The lag filter for the relatively large delay is F=0.188=0.1880.435tp= 0.082tp. The

lag filter has significant impact on the processes with relatively small delay (integrating

process). Therefore, the final recommended value for the lag-filter in the modified

tuning method is given as

F=0.057tp

(34)

5. Guidelines for the selection of initial controller gain Kc0

Although the proposed method is valid for the overshoot between 0.10 to 0.60, the

recommended value of overshoot around 0.3 gives almost similar response to IMC-PID.

Therefore, it is important to have guidelines for it.

Initial controller setting Kc01 is applied and resulting overshoot OS1 is achieved, which

is somewhere between 0.1 to 0.60 but not around 0.30. The desired overshoot and P-

controller gain are OS and Kc0 respectively. In this method, the aim is to get the same

performance of the PID tuning rule regardless of the overshoot that resulted in closed-

loop experiment. In theory, calculated Kc for any overshoots from different closed-loop

setpoint tests should be the same and one can write a mathematical relationship as:

2 2

1 1 01 01.45 OS 2.02 OS 1.27 1.45 OS 2.02 OS 1.27 c cK K

(35)

Eq. (35) provides a guideline for the P-controller gain for the subsequent closed-loop

setpoint test. The resulting equation can give initial controller setting for overshoot

around 0.3 as:

24

20 1 1 011.26 1.45 OS 2.02 OS 1.27 c cK K (36)

It is important to note that we are not keen to obtain the exact fractional overshoot of

0.30, so in a few trials one can achieve the desired overshoot (around 0.3) from Eq.(36).

6. Simulation Study

In this section, results of the simulation study have been discussed for the different

types of processes. The investigated models have been studied by other researchers

(Shamsuzzoha and Lee [2, 3, 9]; Skogestad [5]; Shamsuzzoha and Skogestad [10];

Chien and Fruehauf [40]; Luyben [41]; Wang and Cluett [42] and Chen and Seborg

[43]).

In the simulation study, several performance and robustness matrices have been

calculated and compared with other methods. The simulation results of 13 different

processes are listed in Table 1 which clearly shows that the proposed method provides

acceptable controller settings in all the cases. The performance and robustness of the

control system are evaluated by the following indices:

Output performance (y) is quantified by computing the integrated absolute error,

0

IAE= dtsy y

. Manipulated variable usage is quantified by calculating the total

variation (TV) of the input (u), which is the sum of all its moves up and down. If we

discretize the input signal as a sequence [u1,u2,u3.,ui] then i+1 ii=1

TV= u -u

. TV is a

good measure of the smoothness of the signal. To estimate the robustness, we calculate

the maximum closed-loop sensitivity, defined as s M =max 1/[1+g c(j)] . Since Ms is

the inverse of the shortest distance from the Nyquist curve of the loop transfer function

to the critical point (-1, 0), a small Ms-value indicates that the control system has a large

stability margin. The optimistic approach is to have a small value of IAE, TV and Ms at

the same time, but for a well-tuned controller there is a trade-off, which means that a

reduction in IAE implies an increase in TV and Ms and vice versa.

Three different overshoots (approximately 0.1, 0.3 and 0.6) have been considered and

the PI/PID settings obtained, based on step response experiments. The same is

compared with the recently published methods [10] for all process models. Although

25

comparison has been done for three different overshoots, the results have been listed

only for the case of overshoot around 0.3 in Table 1. The closed-loop performance

evaluation has been done by introducing a unit step change in both the set-point and

load disturbance i.e, (ys=1 and d=1).

The results of three methods has been compared and shown in Figure 9-13 for cases 3, 5,

9 and 11. For both the proposed and setpoint overshoot method [10], overshoot around

0.3 is compared with the modified IMC-PID method. The proposed controller setting

response shows smaller overshoot and faster disturbance rejection than the setpoint

overshoot method [10]. The closedloop response for both the setpoint tracking and

disturbance rejection confirms that the proposed method gives better response.

The above presented closed-loop method is on the basis of modified IMC-PID tuning

method. The comparison has been also performed to check the agreement of the

proposed method with the modified IMC-PID tuning method. In all the above cases

(Figure 9-13) response of the modified IMC-PID is also shown which clearly indicates

that the proposed method is perfectly matched with the modified IMC-PID method.

The lower overshoot of around 0.10 usually gives sluggish and more robust PID

controller settings in the proposed method, while a large overshoot, close to 0.6, gives

more aggressive and fast PID-settings.

Figure 9. Closed loop response of

1

1 0.2 1 0.04 1 0.008 1g s

s s s s

(case 3),


0 2.5 5 7.5 100

0.25

0.5

0.75

1

1.25

Time

Pro

cess

Var

iab

le (

y)

Proposed method (overshoot=0.292)

Modified IMC-PID (c=0.0888)

Shamsuzzoha &Skogestad (overshoot=0.292)

26

Figure 10. Closed loop responses of

2

1

1g s

s s

(case 5), Setpoint change at t=0;

load disturbance of magnitude 1 at t=30.

Figure 11. Closed loop responses of 5 1

seg s

s



0 15 30 45 600

1

2

3

4

Time

Pro

cess

Var

iab

le (

y)




0 10 20 30 40 500

0.25

0.5

0.75

1

1.25

Time

Pro

cess

Var

iab

le (

y)




27


20.05 1

seg s

s



Figure 13. Closed loop responses of se

g ss



0 4 8 12 160

0.5

1

1.5

2

Time

Pro

cess

Var

iab

le (

y)



Modified IMC-PI (c=0.615)


0 10 20 30 400

1

2

3

Time

Pro

cess

Var

iab

le (

y)




Corresponding author. Tel.:+ 966-13-860-7360; E-mail address: [email protected], [email protected]

28

Table 1: Comparison of proposed PID controller setting with the setpoint overshoot method (hereafter, SOM method [10]).

Case Process model Methods P-control setpoint experiment Resulting PID-controller settings

Kc0 overshoot tp b Kc I D F Ms Setpoint Load disturbance

IAE (y) TV(u) IAE

(y)

TV(u)

1

1

1 0.2 1s s SOM 15.0 0.322 0.393 0.937 9.031 0.958 - - 1.74 0.30 23.72 0.11 1.81

Proposed 15.0 0.322 0.393 0.937 11.47 0.54 0.052 0.021 1.56 0.32 27.44 0.047 1.79

2

3

0.3 1 0.08 1

2 1 1 0.4 1 0.2 1 0.05 1

s s

s s s s s

SOM 1.5 0.302 4.45 0.60 0.929 3.56 - - 1.56 3.83 1.76 3.83 1.1

Proposed 1.5 0.302 4.45 0.60 1.187 3.67 0.623 0.254 1.48 3.22 2.08 3.10 1.04

3

1

1 0.2 1 0.04 1 0.008 1s s s s

SOM 6.50 0.292 0.615 0.867 4.093 1.50 - - 1.59 0.46 9.13 0.37 1.42

Proposed 6.50 0.292 0.615 0.867 5.228 0.91 0.087 0.035 1.48 0.476 10.89 0.173 1.39

4

2

2

0.17 1

1 0.028 1

s

s s s

SOM 0.80 0.301 4.987 1.0 0.496 12.17 - - 1.77 4.74 1.29 24.51 1.81

Proposed 0.80 0.301 4.987 1.0 0.635 7.30 0.70 1.59 1.59 4.71 1.49 11.50 1.75

5

2

1

1s s

SOM 0.58 0.307 6.19 1.0 0.357 15.10 - - 1.75 6.21 0.90 42.33 1.72

Proposed 0.58 0.307 6.19 1.0 0.456 9.067 0.869 0.354 1.62 6.17 1.07 19.89 1.70

6

20 1 2 1

se

s s

SOM 8.0 0.301 8.425 0.889 4.966 20.56 - - 1.62 5.92 10.99 4.14 1.34

Proposed 8.0 0.301 8.425 0.889 6.348 12.32 1.182 0.481 1.55 7.35 13.67 1.94 1.39

7

2

1

6 1 2 1

ss e

s s

SOM 1.40 0.344 13.67 0.583 0.817 9.602 - - 1.59 11.72 1.60 11.78 1.09

Proposed 1.40 0.344 13.67 0.583 1.046 9.954 1.914 0.779 1.51 10.31 1.88 9.55 1.04

29

Note: only PI controller gives satisfactory response for Case 10 (almost delay process)

Case Process model Methods P-control setpoint experiment Resulting PID-controller settings


IAE(y) TV(u) IAE(y) TV(u)

8

0.36 1 3 1

10 1 8 1 1

ss s e

s s s

SOM 15.0 0.308 0.836 0.94 9.22 2.04 - - 1.75 0.92 21.54 0.23 1.26

Proposed 15.0 0.308 0.836 0.94 11.75 1.23 0.118 0.048 1.92 0.97 28.69 0.11 1.37

9

5 1

se

s

SOM 4.0 0.298 3.049 0.80 2.494 6.538 - - 1.56 2.62 4.96 2.62 1.04

Proposed 4.0 0.298 3.049 0.80 3.187 4.409 0.423 0.172 1.66 2.57 6.61 1.38 1.08

10

2

0.05 1

se

s

SOM 0.30 0.30 2.0 0.23 0.187 0.321 - - 1.61 1.74 1.02 1.74 1.01

Proposed 0.30 0.30 2.0 0.23 0.2384 0.331 - 0.114 2.0 1.93 1.44 1.92 1.44

11 se

s

SOM 0.80 0.302 3.282 1.0 0.496 8.008 - - 1.70 3.94 1.21 16.15 1.55

Proposed 0.80 0.302 3.282 1.0 0.634 4.789 0.459 0.187 1.75 3.84 1.63 7.68 1.60

12

2

2

6

1 36

s

s s s

SOM 0.80 0.304 4.989 1.0 0.495 12.17 - - 1.77 4.76 1.29 24.61 1.81

Proposed 0.80 0.304 4.989 1.0 0.632 7.315 0.701 0.286 1.59 4.73 1.49 11.57 1.75

13

29

1 2 9s s s

SOM 1.25 0.322 1.40 0.56 0.752 0.905 - - 1.72 1.26 1.57 1.23 1.21

Proposed 1.25 0.322 1.40 0.56 0.961 0.943 0.197 0.080 1.70 1.12 1.94 1.0 1.27

30

Table 2: Comparison of performance and robustness of the proposed method with other well-known methods

Case

(Process model)

Methods P-control setpoint experiment PID-controller settings


IAE(y) TV(u) IAE(y) TV(u)

Example 1 7.40.2 se

g ss

SOM 0.74 0.595 21.56 1.0 0.334 52.61 - - 1.71 29.03 0.83 157.37 1.61

Proposed 0.74 0.595 21.56 1.0 0.430 31.48 3.02 1.23 1.76 28.13 1.16 75.82 1.72

Chien & Fruehauf - - - - 0.470 42.60 3.38 - 1.74 25.0 1.27 90.74 1.66

Example 2

0.10.547 0.418 1

1.06 1

ss eg s

s s

SOM 2.1 0.61 3.62 1.0 0.783 11.57 - - 1.75 5.06 2.0 14.80 1.73

Proposed 2.1 0.61 3.62 1.0 1.214 5.27 0.51 0.206 2.01 4.56 3.66 4.43 2.27

Luyben - - - - 1.69 11.50 1.15 - 1.90 3.82 5.40 6.85 2.29

Shamsuzzoha &Lee - - - - 1.867 4.23 0.72 - 1.63 3.24 5.86 2.30 2.38

Example 3

50.005 300 1

20 1

ss eg s

s s

Proposed 3.0 0.31 13.53 1.0 2.348 19.77 1.90 0.772 2.11 15.04 6.33 8.57 1.53

Shamsuzzoha &Lee - - - - 33.87 62.11 14.84 300 1.93 27.04 1.16 14.30 1.60

Rivera et al. - - - - 24.80 58.30 13.10 300 1.55 39.0 1.84 21.44 1.54

Example 4

3

22 1 1 1

seg s

s s

Proposed 1.0 0.60 9.54 0.50 0.58 3.894 1.34 0.54 1.62 8.71 1.40 7.28 1.09

Yuwana and Seborg 1.0 0.60 - - 1.21 7.04 1.76 - 3.70 8.734 5.71 6.17 3.45

Chen 1.0 0.60 - - 1.08 6.45 1.61 - 2.70 7.60 3.69 5.99 2.18

Lee et al. 1.0 0.60 - - 0.855 4.33 1.43 - 1.94 7.41 2.18 5.15 1.28

31

Comparison with Two-Step Open-Loop Method

6.1.1. Example 1: Distillation column model

Distillation is a widely used separation method in the process industries. Its operation is

extremely critical, because of the purity demand of the products. The process model for the

level control in distillation is given by the delay integrating process. The distillation column

model (Shamsuzzoha and Lee [3], Chien and Fruehauf [40] and Chen and Seborg [43] ) was

considered as follows:

7.40.2 se

g ss

(37)

The IMC based two-step method of Chien and Fruehauf [40], closed-loop method by

Shamsuzzoha and Skogestad [10] and the proposed method of the present study were used to

design the PID controller. The performance indices are listed in Table 2, and output response

in Figure 14 for both the unit step change in setpoint and disturbance rejection. It is clear from

Table 2 and Figure 14 that the proposed tuning rule results in the least settling time for

disturbance rejection, followed by that of Chien and Fruehauf [40]. It is important to note that

both the proposed and setpoint overshoot methods [10] are based on the closed-loop test and

they do not require the process model to design PID controller like in the Chien and Fruehauf

[40] method. The above discussion indicates that the suggested method has clear benefit over

the other methods.

0 50 100 150 200 2500

1

2

3

4

Time

Pro

cess

Var

iab

le (

y)


Chien &Fruehauf


32

Figure 14. Closed loop responses of distillation column model 7.40.2 se

g ss

(Example 1),


6.1.2. Example 2: Boiler steam drum

The process of boiler steam drum is an example of an integrating process with inverse

response which has the following process transfer function (Luyben [41]).

0.10.547 0.418 1

1.06 1

ss eg s

s s

(38)

The PID controllers were designed using the proposed method and the setpoint overshoot

method [10] based on the closed-loop test for an overshoot of around 0.61. The other two

well-known model based methods [2, 41], are also tested and compared with the proposed

method. Figure 15 shows the closed-loop output responses for a unit step change introduced

in both the setpoint and load disturbance. The controller setting parameters including the

performance indices are listed in Table 2. Figure 15 shows that the proposed method gives

better response than that obtained from the method of Shamsuzzoha and Skogestad [10].

Although Luyben's [41] method gives a smaller peak, it has slow response and takes long time

to settle. It is important to note that the process model is required to obtain the controller

settings for both the model based methods [2, 41].

Figure 15. Closed loop responses of boiler steam drum model

0.10.547 0.418 1

1.06 1

ss eg s

s s

(Example 2), Setpoint change at t=0; load disturbance of magnitude 1 at t=20.

0 10 20 30 40

0

0.5

1

1.5

2

2.5

Time

Pro

cess

Var

iab

le (

y)



Shamsuzzoha &Lee

Luyben

33

6.1.3. Example 3: Paper machine dryer cans model

Consider the following process of paper machine dryer cans [2, 42].

50.005 300 1

20 1

ss eg s

s s

(39)

The PID controller parameter settings for the proposed method based on the closed-loop test

for overshoot 0.31 and those of Shamsuzzoha & Lee [2] and Rivera et al. [4] are presented in

Table 2. The PID controller settings for the latter two methods were taken from Shamsuzzoha

& Lee [2]. Figure 16 shows the closed-loop output responses for a unit step change introduced

in both the setpoint and load disturbance for these three design methods.

Shamsuzzoha & Lee [2] previously demonstrated the superiority of their method over that of

Rivera et al. [4] and Wang and Cluetts [42]. Figure 16 clearly shows that Rivera et al.s

method has a large overshoot and long settling time. The proposed method shows a clear

advantage over the others and exhibits a lower IAE value and fast settling time with small

overshoot in disturbance rejection.

On the basis of the above discussion it is clear that the controller settings of the proposed

method provide satisfactory performance and robustness for regulatory problems for a broad

class of processes.

0 100 200 3000

0.5

1

1.5

Time

Pro

cess

Var

iab

le (

y)


Shamsuzzoha &Lee

Rivera et al.

34

Figure 16. Closed loop responses of Paper Machine Dryer Cans model

50.005 300 1

20 1

ss eg s

s s

(Example 3), Setpoint change at t=0; load disturbance of magnitude

1 at t=150.

6.2. Comparison with two-step closed-loop method

The proposed method is also compared with the two-step procedure based on closed-loop

setpoint experiment with a proportional controller (Kc0). In most of the two-step tuning

procedures, first they identify a first-order with time delay model by equating the closed-loop

setpoint response with a standard oscillating second-order step response. Once the model

parameters are obtained, one can use any well-known tuning method e.g., IMC-PID tuning

rule [2]. The proposed method is a direct approach for the controller setting parameters and

identification of few parameters is required. Probably the simplest to observe in the closed-

loop experiment are the time tp to reach the (first) overshoot and its magnitude.

To compare the results of both the direct proposed method and the two-step method based on

closed-loop test, a high order process with significant time delay has been considered below

as:

Example 4:

3

22 1 1 1

seg s

s s

(40)

The PID controller setting data of the Yuwana and Seborg [11], Chen [14] and Lee et al. [16]

were taken from Lee et al. [16] for the initial controller setting Kc0=1. In the proposed method

Kc0=1 is also selected to obtain the PID setting. The performance and robustness matrix is

listed in Table 2. The performance of the all four methods is compared and shown in Figure

17. The figure shows that the proposed tuning method gives acceptable performance with

high robustness. For the same value of Kc0, the proposed method gives significantly robust

(Ms=1.62) closed-loop response with very low value of TV compared to the other methods.

The same observations have been found for the several other processes, though they are not

shown.

Figure 18 shows the manipulated variable (MV) response for example 4 as the representative

case. The response shows that the proposed method has smooth controller output with less

effort in comparison with the other methods. The value of TV is significantly less for the

proposed method among all the others.

On the basis of the above discussion it is again clear that the proposed method scores over the

other two-step closed-loop tuning methods for a broad class of the processes.

35


3

22 1 1 1

seg s

s s

(Example 4), Setpoint change at

t=0; load disturbance of magnitude 1 at t=50.

Figure 18. MV plots of

3

22 1 1 1

seg s

s s

(Example 4), Setpoint change at t=0; load

disturbance of magnitude 1 at t=50.

6.3. Robustness Test

0 20 40 60 80 1000

0.5

1

1.5

2

Time

Pro

cess

Var

iab

le (

y)

Proposed method Kc0

=1, Ms=1.62

Lee, Cho and Edgar Kc0

=1, Ms=1.94

Chen Kc0

=1, Ms=2.70

Yuwana and Seborg Kc0

=1, Ms=3.70

0 20 40 60 80 100-0.25

0

0.25

0.5

0.75

1

1.25

1.5

1.75

Time

MV

(u)

Proposed method Kc0

=1, setpoint TV=1.40, disturbance TV=1.0


=1, TV=2.18, disturbance TV=1.28

Chen Kc0


Yuwana and Seborg Kc0


36

It is important to perform a comparison on fair basis for all the tuning methods. It can be

achieved only if the performance comparison is done for the same level of robustness, e.g.

same Ms-value. The other approaches for the investigation of the robustness of all compared

methods are to check the closed-loop response with uncertainty in different process

parameters. Therefore, the robustness of the different controllers are evaluated by inserting a

perturbation uncertainty in all the three parameters (k, and ). To show the closed-loop

response of the model mismatch, a high order process with time delay (example 4) has been

considered. A case has been selected for 50% in the dead time uncertainty and 25% in both

the gain and time constant simultaneously towards the worst case model mismatch, as follows

24.51.25 1.5 1 0.75 1sg s e s s . The simulation results for the plant-model mismatch

are given in Figure 19 for both the servo and regulatory problems. It should be mentioned that

the controller settings used in simulation are those calculated for the process with nominal

process parameters. It is clear from Figure 19 that the proposed controller tuning method has

an excellent setpoint and load response for model mismatch. Although the closed-loop

response for both the Yuwana and Seborg [11] and Chen [14] two-step methods is not shown

in the said figure, it give an unstable oscillatory response. The better closed-loop response for

the nominal case by Lee et al. [16] two-step controller tuning method is achieved by

sacrificing the robustness of the closed-loop system.

Figure 19. Effect of parameters uncertainties in both the proposed and Lee et al. method.

Modified process with 50% high , 25% high k and 25% low from original value of

0 20 40 60 80 1000

0.5

1

1.5

2

Time

Pro

cess

Var

iable

(y)

Proposed method Kc0

=1


=1

37

Example 4, 24.51.25 1.5 1 0.75 1sg s e s s : Setpoint change at t=0; load disturbance of

magnitude 1 at t=50.

6.4. Application to the Distillation column

The case study demonstrates the application of the proposed tuning method in the distillation

column temperature control loop. The dynamic model of the distillation column in Aspen-

Hysys is selected from Luyben [44] to show the simplicity and effectiveness of the

proposed method.

The depropanizer column considered in this case study produces a distillate product that is 98

mole% propane. At 110F, the vapor pressure of propane is slightly higher than 200psia.

Therefore, an operating pressure of 200 psia is kept in the condenser. The boiler pressure is

estimated by assuming a pressure drop over each tray of 5 inches of liquid in this high-

pressure column. The liquid density of this hydrocarbon system is about 30 lb/ft3. The column

has 30 trays and is fed on tray 15, and the pressure in the reboiler is 202.6 psia.

The column feed is 100 lb-mol/hr of a mixture of propane (30 mol%), isobutene (40 mol%)

and n-butane (30 mol%) at 90F. The specified purity of distillate is 98 mol% propane. The

specified impurity of propane in the bottoms is 1.0 mol%. The design reflux ratio is 3.22 and

the design reboiler heat input is 1.02106 Btu/hr.

Luyben [44] suggested Reflux-Vapor Boilup (RV) control structure of the depropanizer and is

shown in Figure 20. The suggested tuning parameters of the different loops are kept

unchanged except the temperature loop. The flow controller has Kc=0.5, I=0.3 minutes, and

two level controllers Kc=2.0 each. The pressure controller is tuned using normal slow setting

with Kc=1.0 and the integral time is I=20.0 minutes. For the temperature loop, Luyben [44]

applied relay-feedback test and found ultimate gain (Ku=32) and the ultimate period (Pu=7.3

minutes). Finally he obtained the PI setting using the TL [18] method as Kc=10.0 and I=16.0

minutes.

In the proposed method, overshoot around 0.30 gives satisfactory performance and

robustness. Start the test in closed-loop using a P-controller with gain Kc0. The magnitude of

the gain Kc0 should be selected such that it gives overshoot around 0.30 for a setpoint change

of magnitude ysp. From the setpoint experiment, read off the maximum response, yp, the

steady state response y, and the time to reach the first peak (tp). It is assumed that the process

output has value y0 before the setpoint change occurs. Step test in temperature loop is shown

in Figure 21.

38

Figure 20. Depropanizer column flowsheet with controllers installed, pressure controller is

not shown in main flowsheet, and it is installed in sub-flowsheet.

Figure 21. The closed-loop responses with a P-controller (controller gain Kc0 = 8.0) of a

depropanizer temperature loop.

39

Figure 22. The closed-loop setpoint responses of the depropanizer temperature loop with a

PID-controller, setpoint change of magnitude +5F at t=100 minutes; reverse setpoint change

of magnitude -5F at t=150 minutes.

Figure 23. Closed-loop response for step changes in feed flow rate as a disturbance at t=15

minutes from 100 to 120 lb-mol/hr, at 120 minutes from 120 to 80 lb-mol/hr.

125.2

125.5

125.8

126.1

126.4

126.7

0 50 100 150 200 250

Main-

Stag

e Tem

pera

ture

(25_

Main T

S)F

Time (Minutes)

Modified IMC-PID open-loop method

Proposed closed-loop method

40

Process output before the setpoint change (y0) = 125.7F, and manipulated variable (OP) =

50.60%, a step test is conducted for setpoint change ( sy )= ys y0=130.7-125.7=5.0, with the

P-controller of Kc0= 8.

Note: It is important to eliminate the impact of the integral action in the step test and for that

substitute I =1000 (sufficiently large value).

Based on the closed-loop setpoint response to a step change of amplitude ys =5oF as shown

in Figure 21, the overshoot and other parameters are calculated as:

p

0

( y y ) 132.37 130.7Overshoot 0.334

y 130.7 125.7

py yOS

y y

The relative steady-state change of the process output is:

0

s s 0

y 130.7 125.7b 1.0

y y y 130.7 125.7

y y

It shows that the process is almost integrating and the value of peak time tp=107.83-

100.0=7.83 minutes. The PID parameter settings can be calculated as

A=1.45(OS)2 2.02(OS)+1.27= 1.45(0.334)2 -2.02(0.334)+ 1.27=0.757

c c0K =K A=8.0*0.757=6.056

For the integral time, I

I p pb

=min 0.69A t , 1.46t1-b

I1.0

=min 0.69*0.757* *7.83, 1.46*7.831.0 1.0

I=11.43 minutes

D=0.14*tp=0.14*7.83=1.10 minutes

The effectiveness of the proposed method has been checked for the setpoint change in the

temperature loop and closed-loop response is shown in Figure 22. The response is

significantly fast and smooth without any oscillation.

The proposed closed-loop method has been compared with the modified IMC-PID controller

for disturbance rejection. The results for the two disturbances in feed flowrate are shown in

Figure 23. At 15 minutes the feed is increased from 100 to 120 lb-mol/hr and at 120 minutes a

large change in the feed flow rate is made, and is finally dropped to 80 lb-mol/hr. Figure 23

clearly shows the advantage of the proposed method in disturbance rejection. Although the

41

proposed method is based on the modified IMC-PID tuning rule, it gives better and more

robust closed-loop response. It seems that the proposed method is less sensitive with

approximation error in different parameters during step test, whereas modified IMC-PID is

very sensitive with the time delay measurement.

7. Conclusion

This study presented a unified PID controller with lag filter design based on the closed loop

approach for several types of processes and thereby key points are summarized below:

1. The integral time has been modified for the classical IMC-PID controller design and it is

recommended to use min , 4.82

I

for Ms=1.74.

2. A closed-loop tuning method has been developed for the IMC-PID controller setting

using step test in setpoint change. The experiment is conducted in closed-loop using a P-

controller with gain Kc0. The PID-controller settings are then obtained directly from the

following data from the setpoint experiment:

a. Overshoot, (yp - y) /y

b. Time to reach first peak, tp

c. Relative steady state output change, (b) = y/ys.

d. The steady state value can be calculated by y = 0.45(yp + yu) for fast

completion of the experiment.

3. The proposed PID tuning with lag filter is:

a. c c0K =K A

b. I p =min 0.69 , 1.46t

(1- )p

bA t

b

c. 0.14D pt

d. F=0.057tp

where A=[1.45(overshoot)2 -2.02 (overshoot)+1.27]

4. The method is valid with satisfactory results for overshoot around 0.1 to 0.6, an

overshoot of around 0.3 is recommended for the best performance and robustness.

42

5. The initial controller gain which provides overshoot around 0.3 in closed-loop test can

be calculated from the following equation as:

20 1 1 011.26 1.45 OS 2.02 OS 1.27 c cK K

6. The simulation results illustrate the better performance and robustness of the proposed

method for different classes of processes. A simple closed-loop step test is required to

obtain the IMC-PID controller setting which gives the appropriate controller settings for

acceptable performance and robustness for a broad range of process models.

Acknowledgement: The authors would like to acknowledge the support provided by King

Abdulaziz City for Science and Technology (KACST) through the Science & Technology

Unit at King Fahd University of Petroleum & Minerals (KFUPM) for funding this work

through project No. 11-ENE1643-04 as part of the National Science Technology and

Innovation Plan.

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