Maximum sensitivity based analytical tuning rules for PID controllers for unstable dead time processes

chemical engineering research and design 1 0 9 ( 2 0 1 6 ) 593–606

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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd

Maximum sensitivity based analytical tuning rules for PID controllers for unstable dead time processes K. Ghousiya Begum a , A. Seshagiri Rao b,∗ , T.K. Radhakrishnan a a b

Department of Chemical Engineering, National Institute of Technology, Tiruchirappalli 620 015, Tamilnadu, India Department of Chemical Engineering, National Institute of Technology, Warangal 506 004, Telangana State, India

a r t i c l e

i n f o

a b s t r a c t

Article history:

In this paper, maximum sensitivity and internal model control (IMC) based proportional-

Received 6 August 2015

integral-derivative (PID) controllers are designed for unstable first-order plus-dead-time

Received in revised form 20

(UFOPDT) processes. The designed controller parameters are functions of the UFOPDT model

February 2016

parameters and the IMC closed loop tuning parameter. The tuning parameter plays a vital

Accepted 3 March 2016

role and determines the closed loop performance and robustness of the designed con-

Available online 9 March 2016

troller. Systematic guidelines are provided for selection of this tuning parameter based

Keywords:

eters for different time delay to time constant ratios with desired level of robustness.

IMC control

These controller settings allow the operator to deal with the closed-loop control system

on maximum sensitivity. Analytical tuning rules are developed for the controller param-

H2 minimization

performance-robustness trade-off by specifying the robustness level (maximum sensitiv-

PID controller

ity). Simulation studies have been carried out on various UFOPDT processes to explain the

Unstable process

advantages of the proposed analysis.

Maximum sensitivity

© 2016 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

1.

Introduction

Open loop unstable processes arise frequently in the chemical (distillation, polymerization reactors, heat exchangers, etc.) and biological (fermentation) processes and are fundamentally difficult to control than that of the stable processes and the difficulty increases when there exist a time delay. Time delays occur frequently in process control problems, because of the distance velocity lags, recycle loops, and composition analysis loops. The performance specifications that are usually achieved for stable systems are difficult to achieve for unstable systems. The closed loop response for such processes exhibits large overshoots and settling times. As has been widely reported, PID controllers are with no doubt, the controllers most extensively used in the process industry. The PID controller design methods for unstable processes have been addressed by many researchers (Rao and Chidambaram, 2006, 2012). Sree et al. (2004) and Sree and Chidambaram (2006) described occurrence and existence of

∗

unstable systems in engineering processes and provided an excellent overview of controller design techniques for unstable processes. They developed tuning rules based on equating coefficient method for first order unstable processes with time delay. Arrieta et al. (2011) proposed PID tuning based on servo or regulatory operations. Nasution et al. (2011) proposed PID controller design based on IMC method and H2 minimization and Maclaurin series approximation and obtained improved performances over many previous methods. They have developed five different desired closed loop transfer functions and designed the controllers correspondingly based on these five desired closed loop transfer functions and recommended one. Shamsuzzoha and Lee (2008) showed that PID controllers in series with lead/lag compensators provide improved closed loop performances when compared to that of PID controller alone for UFOPDT processes. Further, complex control structures have also been developed such as modified Smith predictor (Rao et al., 2007; Uma et al., 2010; NormeyRico and Camacho, 2008), modified IMC (Tan et al., 2003) and

Corresponding author. Tel.: +91 870 2462633. E-mail address: [email protected] (A.S. Rao). http://dx.doi.org/10.1016/j.cherd.2016.03.003 0263-8762/© 2016 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

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chemical engineering research and design 1 0 9 ( 2 0 1 6 ) 593–606

two-degrees-of-freedom control structures (Liu et al., 2005; Tan, 2010) with more number of controllers and also with more complexity in the design of controllers for UFOPDT processes to improve the closed loop performance. In most of these controllers, there exists performance-robustness trade-off and the tuning parameter should be selected in such a way that the resulting controllers give both nominal as well as robust performance as a function of peak value of the sensitivity function i.e. maximum sensitivity (Ms). Moreover, PID controller cannot provide stabilized responses when the time delay to time constant ratio is greater than 1.2 for unstable systems. It can be observed that complex control schemes such as modified Smith predictor and two degrees of freedom structures where there are more than two controllers involved are not desirable for practical purposes and these modified schemes also are not applicable when the time delay to time constant ratio (To ) exceeds 1.2. Hence, keeping the simplicity into account, properly designed PID controller is better than these modified schemes. However, the designed PID controller should provide good nominal and robust closed loop responses and smooth manipulated variable responses. Alfaro et al. (2010) have developed analytical equations for PI controller for stable systems based on Ms. Arrieta and Vilanova (2012) developed PID tuning rules for stable systems for servo/regulatory problems based on Ms. They formulated an optimization problem with constraints to design the controller. However, such rules do not exist for unstable systems. Based on this motivation, in the present work, an attempt is made to develop analytical tuning rules for PID controllers as a function of Ms for unstable processes using IMC–H2 minimization theory. This approach explicitly considers the control system performance-robustness trade-off aiming to obtain a smooth response to both disturbance and set point step changes and at the same time to guarantee a minimum robustness level. The distinctive feature of the resulting tuning procedure is that the designer may select one of four different robustness levels in the range 2 ≤ Ms ≤ 10.5 for UFOPDT process models with time delay to time constant ratio (To ) in the range 0.10 ≤ To ≤ 0.9. For clear illustration, the paper is organized as follows. Controller design is addressed in Section 2 followed by the proposed Ms based tuning rules in Section 3. Simulation studies are explained in Section 4 followed by discussion in Section 5 and finally conclusions are presented in Section 6.

2.

Controller design

The closed-loop control structure of IMC is shown in Fig. 1, where Gp (s) is the transfer function of the unstable process, Gm (s) is the corresponding transfer function model and QC is the transfer function of the IMC controller. Here, the controller design is addressed for UFOPDT processes. Recently, Vanavil et al. (2014) have developed analytical tuning formulas for PID

d r

+

-

Qc

+

u +

controller with a lead-lag filter for UFOPDT processes. Their method is briefly discussed here. The UFOPDT process transfer function is considered as Gp (s) =

kP e−s P s − 1

(1)

According to IMC principle, the IMC controller QC is equivalent to QC = Q˜ C F

(2)

where F is a filter which is used for altering the robustness of the controller. The filter structure should be selected such that the IMC controller QC is proper and realizable and also the control structure is internally stable. In addition to these requirements, it should be selected such that the resulting controller provides improved closed loop performances. Q˜ C is designed for a specific type of step input disturbance (v) to obtain H2 optimal performance (Morari and Zafiriou, 1989) and is based on the invertible portion of the process model. The process model and the input are divided as Gm = Gm− Gm+

and

v = v− v+

(3)

where the subscript “−” refers to minimum phase part and “+” refers to non-minimum phase part. The Blaschke product of RHP poles of Gm and v are defined as

bm =

k  −s + p i=1

s + p¯ i

i

bv =

and

k˜  −s + p i=1

s + p¯ i

i

(4)

where pi and p¯ i are the ith RHP pole and its conjugate respectively. Based on this, the H2 optimal controller is derived by using the following formula given by Morari and Zafiriou (1989). −1 −1 Q˜ C = bm (Gm− bv v− ) {(bm Gm+ ) bv v− }|∗

(5)

where {. . .} |* is defined as the operator that operates by omitting all terms involving the poles of (Gm+ )−1 after taking the partial fraction expansion. This idea is applied successfully by Nasution et al. (2011) and derived IMC based PID controller. The same derivation for obtaining the IMC controller Q˜ C for UFOPDT processes is given here for clear understanding. Considering perfect model case i.e. Gp = Gm , first, split the process model and input into minimum and non-minimum phase parts as Gm− =

v− =

−kP P (−s + (1/P ))

−kP P (−s + (1/P ))s

and Gm+ = e−s

and v+ = 1

(6)

(7)

Then, the Blaschke product is obtained as

y

Gp

Gm

Fig. 1 – IMC control scheme.

bm = -

(−s + (1/P )) (s + (1/P ))

and

bv =

−s + (1/P ) s + (1/P )

(8)

+

Substituting all expressions in Eq. (5), one will get, (P s − 1) /P Q˜ C = {(e − 1)P s + 1} kP

(9)

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chemical engineering research and design 1 0 9 ( 2 0 1 6 ) 593–606

To get the final form of the IMC controller, here, the filter is selected as

GC =

(˛s + 1)

F=

(s + 1)

(10)

3

Therefore, the IMC controller is obtained as QC =

(P s − 1) /P (˛s + 1) {(e − 1)P s + 1} 3 kP (s + 1)

(11)

According to IMC principle, the closed loop transfer function between output (y) and set point (r) for perfect model conditions (i.e. Gp = Gm ) from Fig. 1 is H=

y = QC Gm r

(12)

{(e/P − 1)P s + 1}(˛s + 1)(P s − 1)



3

kP (s + 1) − {(e/P − 1)P s + 1}(˛s + 1)e−s

Gc (s) =

1 s

 J(0) + J (0)s +

(s + 1)

(13)

3

The first two conditions are satisfied from the above design procedure and third condition can be applied as

 (18)



1 s + d s i

(19)

i = J (0)/J(0) and

d = J (0)/2J (0)

(20)

where

J(0) = 1/pm (0)D(0) J (0) = −[pm (0)D(0) + pm (0)D (0)]/[pm (0)D(0)] J (0) = J (0)

Condition 1: QC must be stable and should cancel the right half plane poles of Gm . Condition 2: QC Gm should be stable. Condition 3: (1 − Gm QC ) at the RHP poles of the process should be zero.



J 2 s + ··· 2!

the PID controller parameters are obtained as kc = J (0),

This is the desired closed loop trajectory for the filter chosen as per Eq. (10). If the filter structure changes, then the expression for H also changes as per Eq. (12) and hence the final controller. Here,  is the closed loop tuning parameter. The value of ˛ is obtained from the conditions of internal stability for IMC structure. The conditions to be followed for internal stability are

(17)

By considering this as a PID controller in the form



{(e/P − 1)P s + 1}(˛s + 1)e−s



Vanavil et al. (2014) simplified the above form into a PID with lead-lag controller. In this work, it is simplified as a PID controller. To simplify this expression to a PID controller form, Maclaurin series or Laurent series can be used. To do that, let us define J(s) = sGc (s). Expand J(s) using Maclaurin series expansion to obtain the controller Gc as

Gc (s) = kc 1 +

Substituting QC from Eq. (11), one can obtain H=

Substituting all the terms, one can obtain



2

2J (0) pm (0)D(0) + 2pm (0)D (0) + pm (0)D (0) + pm (0)D(0) + pm (0)D (0) J(0)

D(0) = 3 − pA (0);

D (0) = [62 − pA (0)]/2;



D (0) = [63 − p A (0)]/3

pA (0) = ˛ −  + P x pA (0) =  2 − 2˛ + 2P x(˛ − ) pA (0) = − 3 + 3˛ 2 + 3P x 2 − 6˛P x pm (0) = −kP ; pm (0) = −kP (P − ˛ − P x); 2

(1 − QC Gm )|s=1/P = 0

(14)

Substituting QC from Eq. (11) into Eq. (14), the value of ˛ is obtained as 2

˛ = {(/P ) + 3(/P ) + 3}

(15)

Now, Fig. 1 is converted in to a unity feedback control system as shown in Fig. 2 and the corresponding unity feedback controller GC is obtained as GC =

QC 1 − QC Gm

(16)

d r

Gc

+

u

+ +

Gp

-

Fig. 2 – Simple unity feedback control.

y

pm (0) = −kP (2P x(P − ˛) + 2˛P + 2(P − ˛ − P x) ) x = (e(/tp) − 1)

(21) In which ˛ is given in Eq. (15). These are the final expressions for the controller parameters obtained based on IMC method and H2 minimization. If the process model is known, then one has to select proper value of  and then obtain the controller parameters. However, selection of  is very important for unstable processes and there should be systematic guidelines for selection of .

2.1.

Guidelines for selection of 

It is well-known that there is always a trade-off in selecting the desired closed-loop tuning parameter (). For stable processes, fast speed of response and good disturbance rejection are favoured by choosing a small value of  and stability and robustness are favoured by a large value of . However, this statement is not always true for unstable processes. Hence, the choice of  is very important when dealing with unstable processes. To have clear understanding for selection of , in the present work, a systematic analysis is carried out using Ms as the performance index.

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chemical engineering research and design 1 0 9 ( 2 0 1 6 ) 593–606

Ms is the inverse of smallest distance from the Nyquist curve to the critical point (−1,0) in Nyquist plot. Ms value sets a lower bound for the Gain margin (GM) and Phase margin (PM). The lower bound for GM and PM in terms of Ms are related as (Skogestad and Postlethwaite, 2005)

As Ms value increases, the lower bound for GM and PM decreases. In fact, the recommended minimum values of GM and PM are 1.7 and 35 degrees for a typical process. However, for unstable processes, the lower bound for phase margin is lesser than 35. This is the limitation for unstable processes. If the process is delay dominant (ratio of time delay to time constant) is more (>1.2), it is difficult to achieve robust closed loop responses because of the limitations of the controller for controlling a delay dominant unstable process. Hence, one can select  value based on Ms which provides useful information about the lower limits of GM and PM (Anusha and Seshagiri Rao, 2012). More details about Ms can be found in Skogestad and Postlethwaite (2005). In the present work, Ms values are plotted against the tuning parameter , controller parameters and from this understanding, one can select the controller parameters based on the required level of robustness.

3.

To=0.1

10

c

PM ≥ 2 sin−1 (1/2Ms)

12

To=0.2 To=0.3

8 6 4 2

2

2.5

3

3.5

Maximum Sensivity

4

4.5

(a) 3.5

To=0.4 To=0.5 To=0.6

3

c

GM ≥ Ms/(Ms − 1);

14

2.5

Tuning rules based on Ms 2

3.1.

Tuning equations

3

3.5

4

4.5

5

5.5

6

6.5

Maximum Sensivity (b) 2.1 2 To=0.7 To=0.8 To=0.9

1.9 1.8

c

In the present work, Ms values are plotted against the controller parameters and from this plot one can select the Ms value and obtain the controller parameters based on the required level of robustness. The designer may resolve the performance-robustness trade off by selecting the appropriate robustness allowed for the control system according to the expected variation of the controlled process parameters. This will give the designer a closed loop control system behaviour with the highest speed that can be obtained for the specified minimum robustness. Three models were considered for carrying out this analysis such as (1) Model 1: kP = 1,  = 0.4,  P = 1; (2) Model 2: kP = 4,  = 2,  P = 4; (3) Model 3: kP = 1,  = 0.5,  P = 5; which are extensively published models in the literature (Shamsuzzoha and Lee, 2008; Tan et al., 2003; Liu et al., 2005; Park et al., 1998).

1.7 1.6 1.5 1.4

The controller parameters (kc ,  i ,  d ) equations obtained in Eq. (20) are functions of To (= / p ) and / p . Based on the simplified expressions obtained and using the process model gain kP and time constant  P and the normalized dead time To , the controller normalized parameters as considered as kc = kc ∗ kP

(22)

i =

i P

(23)

d =

d P

(24)

These normalized controller parameters values are plotted against the Ms values for three ranges of normalized dead times 0.1–0.3, 0.4–0.6, 0.7–0.9. Initially a graph between Ms and the tuning parameter  is drawn. The maximum and minimum values of Ms are selected in order to analyze the performance and robustness trade off. For that particular range of Ms, the controller parameter’s behaviour is analyzed. Fig. 3

1.3

5

6

7

8

9

10

Maximum Sensivity (c)

Fig. 3 – Normalized gain variation with Ms. (a) normalized dead times 0.1–0.3; (b) normalized dead times 0.4–0.6; (c) normalized dead times 0.7–0.9. shows the graph between Ms and normalized proportional gain kc . As the normalized dead time To increases the effect of Ms in terms of the tuning parameter , the controller gain kc decreases. Accordingly, as the robustness degree is increased the normalized controller gain kc decreases. This can be visualized from Fig. 3a–c drawn for the normalized dead times in the range of 0.1–0.9. Fig. 4 shows the graph between Ms and normalized integral time i . However, the behaviour of the integral time is entirely different when compared to normalized controller gain kc . As the normalized dead time To increases, the effect of Ms in

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chemical engineering research and design 1 0 9 ( 2 0 1 6 ) 593–606

0.13

4.5 To=0.1 To=0.2 To=0.3

4 3.5

0.12 0.11 0.1

3

d

0.09 0.08

2

0.07

1.5

0.06

i

2.5

0.05

1

0.04

0.5 0

0.03

2

2.5

3

3.5

4

2

2.5

3

3.5

4

4.5

Maximum Sensivity

4.5

Maximum Sensivity

(a)

(a)

0.28

9 0.26

To=0.4 To=0.5 To=0.6

8 7

0.22

i

d

6

0.2

5

0.18

4

0.16

3 2 1

To=0.4 To=0.5 To=0.6

0.24

3

3.5

4

4.5

5

5.5

6

6.5

Maximum Sensivity 3

3.5

4

4.5

5

5.5

6

6.5

(b)

Maximum Sensivity (b)

0.44 0.42

22 To=0.7 To=0.8 To=0.9

20

0.38

d

18

To=0.7 To=0.8 To=0.9

0.4

16

0.36

14

i

0.34

12 0.32

10

0.3

8

6

7

8

9

10

Maximum Sensivity

6 4

5

(c)

5

6

7

8

Maximum Sensivity

9

10

(c) Fig. 4 – Normalized integral time variation with Ms. (a) normalized dead times 0.1–0.3; (b) normalized dead times 0.4–0.6; (c) normalized dead times 0.7–0.9.

terms of the tuning parameter , the integral time i decreases. Similarly, as the robustness degree is increased, the integral time i increases. This can be observed from Fig. 4a–c which are drawn for the normalized dead times in the range 0.1–0.9. Fig. 5a–5c illustrates the graph between Ms and normalized derivative time d . As the normalized dead time To increases, the effect of Ms in terms of the tuning parameter , on the derivative time d slightly increases. Further, as the robustness degree is increased, the derivative time d decreases with respect to the normalized dead times in the range 0.1–0.9.

Fig. 5 – Normalized derivative time variation with Ms. (a) normalized dead times 0.1–0.3; (b) normalized dead times 0.4–0.6; (c) normalized dead times 0.7–0.9.

Using the information given in Figs. 3–5, a set of equations for the controller’s normalized parameters have been developed as a function of the robustness index Ms for different ranges of the normalized model dead times. Those specific tuning expressions are obtained as kc = a1 exp(b1 ∗ Ms) + c1 exp(d1 ∗ Ms)

(25)

i = a2 exp(b2 ∗ Ms) + c2 exp(d21 ∗ Ms)

(26)

d = a3 exp(b3 ∗ Ms) + c3 exp(d3 ∗ Ms)

(27)

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chemical engineering research and design 1 0 9 ( 2 0 1 6 ) 593–606

Table 1 – Constants values for calculating the normalized proportional gain kc . To

a1

b1

c1

d1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

7.64 4.611 2.475 2.469 2.029 1.756 1.763 1.559 1.354

0.123 0.07708 0.1184 0.04955 0.043282 0.03586 0.01296 0.0109 0.01304

−766.4 −56.02 −9297 −66.01 −34.38 −18.84 −4.176 −4.705 −6.461

−3.094 −1.762 −4.467 −1.897 −1.591 −1.322 −0.7112 −0.7219 −0.7983

for the normalized dead time in the range of (i) 0.1–0.3, (ii) 0.4–0.6 and (iii) 0.7–0.9 respectively for the above three cases. This broad classification allows a qualitative specification of the control system robustness.

3.2.

Simplified tuning rules

In Section 3.1, the lower and upper values for Ms are considered as 2 and 10.5. Note that Ms = 2 and Ms = 10.5 are two different significant cases where the robustness is of primary concern (Ms = 2) and where the aggressiveness is more important (Ms = 10.5). The controller parameters are plotted against

Table 2 – Constants values for calculating the normalized integral time  i .

14

b2

c2

d2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

4.024e+010 3028 2.122e+013 3.274e+005 3.074e+004 9049 889.9 8765 2.712e+004

−15.42 −4.433 −14.68 −5.087 −3.493 −2.572 −1.443 −1.718 −1.664

0.8914 1.328 3.66 3.559 5.278 7.589 7.475 12.16 22.25

−0.2878 −0.1651 −0.2664 −0.1305 −0.1296 −0.1296 −0.05575 −0.06257 −0.08384

b3

c3

d3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.02527 0.06492 0.08932 0.1446 0.1871 0.2327 0.2979 0.3459 0.3885

0.109 0.0532 0.0737 0.02943 0.02589 0.02119 0.00851 0.00726 0.008236

−4.873e+007 −38.96 −2.459e+008 −234.4 −41.49 −13.34 −1.228 −3.066 −7.318

−13.11 −3.96 −11.03 −3.79 −2.722 −2.04 −1.037 −1.122 −1.19

8

6

4

2 0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

To

(a) Ms=3.875 Ms=4.75 Ms=5.625 Ms=6.5

3.4 3.2 3

c

a3

=2.625 =3.25 =3.875 =4.5

10

Table 3 – Constants values for calculating the derivative time  d . To

Ms Ms Ms Ms

12

c

a2

k,

To

2.8 2.6 2.4 2.2

where ai , bi and ci (i = 1,2,3) are the constants and the values of these constants are given in Tables 1–3 for Eqs. (25)–(27) as a function of the corresponding To .

3.1.1.

2 1.8 0.4

0.45

Robustness specific simple tuning rules

However, for unstable processes, it is difficult to achieve the robustness levels within small values of Ms. To develop the tuning rules for different robustness levels, three sets of Ms values are considered indicating different levels of robustness. Usually, for unstable time delay processes, Ms = [4.5, 6.5, 10.5] is recognized as the minimum acceptable robustness level or the minimum degree of robustness. In the present work, three ranges of Ms values are considered Ms = [3.875, 5.625, 9.125] for low level of robustness, Ms = [3.25, 4.75, 7.75] for medium level of robustness, Ms = [2.625, 3.875, 6.375] for high level of robustness

0.55

0.6

(b)

The above tuning equations allows the design of the control system with any desired robustness level in the following range to analyze the robustness-performance trade-off.

2.1 Ms=6.375 Ms=7.75 Ms=9.125 Ms=10.5

2 1.9 1.8

c

Ms = [2,4.5] with normalized dead times from 0.1 to 0.3, Ms = [3,6.5] with normalized dead times from 0.4 to 0.6, Ms = [5,10.5] with normalized dead times from 0.7 to 0.9

0.5

To

1.7 1.6 1.5 1.4 0.7

0.75

0.8

0.85

To (c) Fig. 6 – Normalized gain variation with To . (a) To between 0.1 and 0.3; (b) To between 0.4 and 0.6; (c) To between 0.7 and 0.9.

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chemical engineering research and design 1 0 9 ( 2 0 1 6 ) 593–606

0.13

Ms=2.625 Ms=3.25 Ms=3.875 Ms=4.5

1.8 1.6 1.4

0.12 0.11 0.1

1.2

i

0.09

d

1 0.08

0.8 0.07

0.6 0.4 0.2 0.1

Ms=2.625 Ms=3.25 Ms=3.875 Ms=4.5

0.06 0.05

0.15

0.2

0.25

0.3

0.04

To

(a)

0.03 0.1

5.5

4.5

0.14

0.16

0.18

0.2

0.24

0.26

0.28

0.3

(a) 0.3 Ms=3.875 Ms=4.75 Ms=5.625 Ms=6.5

4

i

0.22

To

Ms=3.875 Ms=4.75 Ms=5.625 Ms=6.5

5

0.12

3.5

0.25

d

3 2.5 2 1.5 0.4

0.2 0.45

0.5

0.55

0.6

To

(b)

0.15 0.4

14

12

0.45

0.5

0.55

0.6

To

Ms=6.375 Ms=7.75 Ms=9.125 Ms=10.5

(b) 0.44 0.42

i

10

0.4 8

d

0.38 6

4 0.7

0.36 Ms=6.375 Ms=7.75 Ms=9.125 Ms=10.5

0.34 0.75

0.8

0.85

To

(c)

0.32 0.3 0.7

0.75

Fig. 7 – Normalized integral time variation with To . (a) To between 0.1 and 0.3; (b) To between 0.4 and 0.6; (c) To between 0.7 and 0.9. Ms for different values of To and then the optimal coefficients are developed for different values of To based on regression analysis. In this section, the controller parameters are plotted against To for different values of Ms. Here, four different robustness levels such as Ms = {2.625, 3.25, 3.875, 4.5} for normalized dead times from 0.1 to 0.3, Ms = {3.875, 4.75, 5.625, 6.5} for normalized dead times from 0.4 to 0.6, Ms = {6.375, 7.75, 9.125, 10.5} for normalized dead times from 0.7 to 0.9,

0.8

0.85

To (c) Fig. 8 – Normalized derivative time variation with To . (a) To between 0.1 and 0.3; (b) To between 0.4 and 0.6; (c) To between 0.7 and 0.9. are considered and shown in Figs. 6–8. This information is used to obtain the new PID tuning rules as a function of normalized dead time based on regression techniques and are given below. 

kc = a1 + b1 Toc1 

i = a2 + b2 Toc2 

d = a3 + b3 Toc3

(28) (29) (30)

600

chemical engineering research and design 1 0 9 ( 2 0 1 6 ) 593–606

From the given UFOPDT model, note down the dead time to time constant ratio (T0)

Table 4 – Constants values for calculating the normalized proportional gain kc . a1

Ms 2.625 3.25 3.875 4.5 3.875 4.75 5.625 6.5 6.375 7.75 9.125 10.5

−1.361 −1.927 −1.571 −0.4606 −0.4157 −0.07606 0.03487 −2.619e−005 −0.1915 0.1079 0.4028 0.6546

b1

c1

To range

1.629 2.139 1.985 1.435 1.454 1.27 1.226 1.286 1.468 1.224 0.9682 0.7522

−0.8557 −0.7932 −0.8443 −0.9814 −0.9159 −1.006 −1.056 −1.064 −0.9516 −1.12 −1.365 −1.667

0.1–0.3 0.1–0.3 0.1–0.3 0.1–0.3 0.4–0.6 0.4–0.6 0.4–0.6 0.4–0.6 0.7–0.9 0.7–0.9 0.7–0.9 0.7–0.9

Select desired level of robustness by choosing Ms

Use tables 1-3 & equations 25-27 OR tables 4-6 & equations 28-30 for design of the PID controller

Plot a graph between λ vs. Ms to understand the maximum and minimum robustness levels No

Satisfied with desired level of robustness selected Table 5 – Constants values for calculating the normalized integral time  i .

2.625 3.25 3.875 4.5 3.875 4.75 5.625 6.5 6.375 7.75 9.125 10.5

a2

b2

c2

To range

0.3049 0.2229 0.121 0.002252 1.35 0.9958 0.8769 0.8054 2.747 1.846 1.215 0.5584

25.8 17.11 9.834 5.801 27.28 16.54 14.56 13.5 20.1 16.15 14.05 12.52

2.355 2.13 1.759 1.38 3.855 3.155 3.124 3.202 5.752 4.701 4.079 3.491

0.1–0.3 0.1–0.3 0.1–0.3 0.1–0.3 0.4–0.6 0.4–0.6 0.4–0.6 0.4–0.6 0.7–0.9 0.7–0.9 0.7–0.9 0.7–0.9

Table 6 – Constants values for calculating the derivative time  d . Ms 2.625 3.25 3.875 4.5 3.875 4.75 5.625 6.5 6.375 7.75 9.125 10.5

a3

b3

c3

To range

−0.02617 −0.02253 −0.009337 0.001879 −0.06847 −0.1411 −0.01715 −0.02703 −0.1075 −0.0002367 0.06296 0.1012

0.327 0.3428 0.3764 0.4245 0.4709 0.5567 0.4601 0.4715 0.558 0.4618 0.4064 0.3752

0.7381 0.7676 0.895 1.032 0.7797 0.6588 0.9782 0.9243 0.7936 1.046 1.264 1.44

0.1–0.3 0.1–0.3 0.1–0.3 0.1–0.3 0.4–0.6 0.4–0.6 0.4–0.6 0.4–0.6 0.7–0.9 0.7–0.9 0.7–0.9 0.7–0.9

where constants ai , bi and ci (i = 1, 2, 3) of these new equations are shown in Tables 4–6. The method of obtaining the other controller parameters using Eqs. (28)–(30) are also shown in Figs. 7 and 8 which depicts the behaviour of controller parameters for different values of normalized dead times as function of a particular Ms value. The new tuning Eqs. (28)–(30) are more easy to use than the original ones (25)–(27). Fig. 6 implies that as the normalized dead time To increases, the controller gain kc decreases and as Ms increases, kc increases. From Fig. 7, it can be understood that as the normalized dead time To increases, the integral time i increases and as Ms increases, i decreases. Similarly, from Fig. 8 it can be understood that as the normalized dead time To increases, the derivative time d increases and as Ms increases, d decreases. For clear illustration, the present methodology is given in Fig. 9 as a flow sheet.

Analyse the closed loop responses Fig. 9 – Steps involved in the present design procedure.

Control acon, u Closed loop output, y

Ms

Yes

3 2 1 0

0

2

4

6

8

10

6

8

10

Time

10

0

-10

0

2

4

Time Fig. 10 – Closed loop and control action responses for Example 1 for To = 0.3, dashed line – Ms = 2.625, solid line – Ms = 3.25 and dotted line – Ms = 4.5.

4.

Simulation studies

The proposed analysis is explained in a more detailed manner by considering different examples in this section.

4.1.

Example 1

Consider the following UFOPDT model for different values for the time delay. G(s) = kP e−s /(P s − 1);

␪ = {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9};

kP = P = 1 The PID controller parameters are derived based on Eqs. (28)–(30) for different robustness levels and are shown in Table 7. Simulation studies have been carried out by considering unit step change in the set-point followed by a disturbance of magnitude 0.5 at t = 5 s for different To values. The corresponding closed loop responses are shown in Fig. 10 for three different values of Ms for To = 0.3. It can be observed from Fig. 10 that for high value of Ms (= 4.5), the closed response becomes faster but provides some oscillations for both set point tracking and disturbance rejection where as low value of Ms (= 2.625) provides smooth set point tracking and

601

Table 7 – Controller parameters for different values of Ms & To for Example 1. To

Ms

kc

i

d

0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.8 0.8 0.8 0.8 0.9 0.9 0.9 0.9

2.625 3.25 3.875 4.5 2.625 3.25 3.875 4.5 2.625 3.25 3.875 4.5 3.875 4.75 5.625 6.5 3.875 4.75 5.625 6.5 3.875 4.75 5.625 6.5 6.375 7.75 9.125 10.5 6.375 7.75 9.125 10.5 6.375 7.75 9.125 10.5

10.3239 11.3618 12.3004 13.2878 5.0960 5.7411 6.1553 6.5026 3.3020 3.6320 3.9156 4.2166 2.9493 3.1161 3.2611 3.4069 2.3273 2.4742 2.5838 2.6871 1.9055 2.0468 2.1374 2.2134 1.8700 1.9324 1.9780 2.0176 1.6240 1.6789 1.7156 1.7456 1.4315 1.4847 1.5207 1.5512

0.4188 0.3498 0.2922 0.2441 0.8877 0.7782 0.7005 0.6317 1.8192 1.5398 1.3036 1.1037 2.1473 1.9148 1.7082 1.5238 3.2349 2.8537 2.5462 2.2731 5.1568 4.2980 3.8276 3.4366 5.3291 4.8649 4.4962 4.1630 8.3137 7.5020 6.8716 6.3040 13.7083 11.6863 10.3602 9.2267

0.0336 0.0360 0.0386 0.0413 0.0735 0.0771 0.0798 0.0825 0.1083 0.1135 0.1188 0.1244 0.1620 0.1633 0.1706 0.1751 0.2058 0.2115 0.2164 0.2214 0.2477 0.2565 0.2620 0.2670 0.3129 0.3178 0.3219 0.3257 0.3599 0.3654 0.3695 0.3733 0.4057 0.4134 0.4187 0.4236

disturbance rejection performances but are slow. Hence, it can be noted that Ms = 3.25 provides a trade-off between robustness and performance and a compromise can be achieved between robustness and performance for this value of Ms. Fig. 11 shows the closed loop responses for To = 0.6 and it can be observed from Fig. 11 that for a high value of Ms (= 6.5), the closed response becomes faster but provides some oscillations for both set point tracking and disturbance rejection where as low value of Ms (= 3.865) provides smooth set

Control acon, u Closed loop output, y

chemical engineering research and design 1 0 9 ( 2 0 1 6 ) 593–606

4

2

0

0

10

20

30

40

50

30

40

50

Time

5 0 -5 -10

0

10

20

Time

Fig. 12 – Closed loop and control action responses for Example 1 for To = 0.8, dashed line – Ms = 6.375, solid line – Ms = 7.75 and dotted line – Ms = 9.125. point tracking and disturbance rejection performances but are slow. Hence, it can be noted that Ms = 4.75 provides a trade-off between robustness and performance. Similarly, Fig. 12 shows the closed loop responses for To = 0.8 and it can be observed from Fig. 12 that high value of Ms (= 9.125) increases, the closed response becomes faster but provides some oscillations for both set point tracking and disturbance rejection where as low value of Ms (= 6.375) provides smooth set point tracking and disturbance rejection performances but are slow. Hence, it can be noted that Ms = 7.75 provides a trade-off between robustness and performance. It can be understood from these Figs. 10–12 that as To increases, the peak value increases for set point as well as for disturbance rejection responses which indicates that the model dead time to time constant ratio has an adverse effect over the closed-loop system performance and as this ratio increases the closed performance degrades. To analyze the robustness further, an analysis is carried out here based on Ms. Fig. 13 shows the variation of Ms with respect to  with normalized dead times To = 0.30, 0.6, 0.8. There exist a large value for Ms corresponding to  = 0.2, 0.48, 0.6 after which the Ms decreases. It can be observed from Fig. 13 that one should not select  without proper analysis. Hence  should be selected in the range of 0.25–0.7 for normalized dead time To = 0.30, 0.6–1.2 for normalized dead time To = 0.60 and in the range of 0.89–1.25 for normalized dead time To = 0.8. Within this range of , the maximum value for Ms will be 4.5, 6.5 and 10.5. If  is selected

3 2 1 0

0

5

10

15

20

25

Time

5

0

-5

Maximum Sensivity (Ms)

Control acon, u

Closed loop output, y

600 500 400 300 200 100 0 0

5

10

15

20

25

Time Fig. 11 – Closed loop and control action responses for Example 1 for To = 0.6, dashed line – Ms = 3.875, solid line – Ms = 4.75 and dotted line – Ms = 6.5.

0

0.5

1

1.5

λ

2

2.5

Fig. 13 – Variation of Ms for different values of  for Example 1 with normalized dead times To = 0.30 (dashed line), 0.6 (solid line), 0.8 (dash dotted line).

3

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4.5

Maximum Sensivity (Ms)

Maximum Sensivity (Ms)

10

4

3.5

3

2.5

2 0.25

9

8 7

6

5

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.9

0.95

1

1.05

λ

Maximum Sensivity (Ms)

6.5 6 5.5 5 4.5 4 3.5 0.7

0.8

0.9

λ

1

1.1

1.2

1.25

3 2 1 0

0

5

10

15

10

15

Time

5

0

-5

1.2

0

5

Time

Fig. 15 – Maximum and minimum values of Ms for selected range of  for Example 1 with normalized dead times To = 0.6.

outside this range the closed loop performance is not good or is not stable. Note that the minimum value of Ms achievable in this range of  is 2, 3, 5 as shown in Figs. 14–16. Hence the Ms ranges were selected as 2–4.5 for To = 0.3, 3–6.5 for To = 0.6, 5–10.5 for To = 0.8.

4.2.

1.15

Fig. 16 – Maximum and minimum values of Ms for selected range of  for Example 1 with normalized dead times To = 0.8.

Control acon, u Closed loop output, y

Fig. 14 – Maximum and minimum values of Ms for selected range of  for Example-1 with normalized dead times To = 0.3.

3 0.6

1.1

λ

Example 2

Here, UFOPDT models with different process gains, time constants and time delay are considered whose parameters are given below.

(i) kp = 4,  p = 4,  = 0.4 (To = 0.1) (ii) kp = 1.8,  p = 1.049,  = 0.298 (To = 0.284) (iii) kp = 2,  p = 1.247, ␪ = 0.691 (To = 0.554)

Fig. 17 – Closed loop and control action responses for Example 2 for To = 0.1, dashed line – Ms = 2.625, solid line – Ms = 3.25, dotted line – Ms = 3.875 and dash-dotted line – Ms = 4.5. The controller parameters are calculated from Eqs. (28)–(30) and are given in Table 8. Simulation studies have been carried out by considering unit step change in the set-point followed by a disturbance of magnitude 0.5 at t = 7.5 s for different To values. The corresponding closed loop responses are shown in Fig. 17 for four levels of robustness (high, medium, low, minimum) for different values of Ms for To = 0.1. It can be observed from Fig. 17 that for high value of Ms (= 4.5), the closed response becomes faster but provides more oscillations for both set point tracking and disturbance rejection where as low value of Ms (= 2.625) provides smooth set point tracking, disturbance rejection and also control action performances but are slow. Hence, Ms = 4.5 is less robust and Ms = 2.625 is more robust. It can be noted that Ms = 3.25 provides a tradeoff between robustness and performance. Fig. 18 shows the closed loop responses for To = 0.284 and it can be observed from

Table 8 – Controller parameters for different values of Ms & To for Example 2. To = 0.1

To = 0.284

To = 0.554

Ms

kc

i

d

Ms

kc

i

d

Ms

kc

i

d

2.625 3.25 3.875 4.5

2.58 2.84 3.07 3.32

1.67 1.40 1.17 0.97

0.13 0.14 0.15 0.16

2.625 3.25 3.875 4.5

1.90 2.15 2.31 2.48

1.71 1.46 1.25 1.07

0.10 0.11 0.11 0.12

3.875 4.75 5.625 6.5

1.04 1.11 1.16 1.20

5.17 4.44 3.96 3.54

0.28 0.29 0.30 0.30

603

3 2 1 0

0

5

10

15

Time

5

0

-5

0

5

10

Control acon, u Closed loop output, y

Control acon, u

Closed loop output, y

chemical engineering research and design 1 0 9 ( 2 0 1 6 ) 593–606

3 2 1 0

0

5

10

2

25

15

20

25

0

-4

0

5

10

Time

Time

Fig. 18 that for high value of Ms (= 4.5), the closed response becomes faster but provides some oscillations for both set point tracking and disturbance rejection where as low value of Ms (= 2.625) provides smooth set point tracking and disturbance rejection performances but are slow. Here, it can be noted that Ms = 3.875 provides a trade-off between robustness and performance. Similarly, Fig. 19 shows the closed loop responses for To = 0.554 and it can be observed from Fig. 19 that a high value of Ms (= 6.5) provides faster closed response becomes with more oscillations for both set point tracking and disturbance rejection, whereas low value of Ms (= 3.875) provides smooth set point tracking and disturbance rejection performances but are slow. Hence, it can be noted that Ms = 4.75 provides a trade-off between robustness and performance. It can be understood from these Figs. 17–19 that as To increases, the peak value increases for set point as well as for disturbance rejection responses which indicates that

20

-2

15

Fig. 18 – Closed loop and control action responses for Example 2 for To = 0.284, dashed line – Ms = 2.625, solid line – Ms = 3.25, dotted line – Ms = 3.875 and dash-dotted line – Ms = 4.5.

15

Time

Fig. 19 – Closed loop and control action responses for Example 2 for To = 0.554, dashed line – Ms = 3.875, solid line – Ms = 4.75, dotted line – Ms = 5.625 and dash-dotted line – Ms = 6.5.

the model dead time to time constant ratio has an adverse effect over the closed-loop system performance and as this ratio increases the closed performance degrades. To analyze closed loop system robustness, simulation studies are carried out for uncertainty of +10% in dead time and the corresponding closed loop responses are shown in Fig. 20. It can be observed that less value of Ms provides more robust closed loop responses than those of high Ms values. To analyze the robustness further, an analysis is carried out here based on Ms. Fig. 21 shows the variation of Ms with respect to ␭ with normalized dead times To = 0.1, 0.284, 0.554. There exist a large value for Ms corresponding to  = 0.17, 0.15, 0.48 after which the Ms decreases. The range of Ms for selected ranges of  are calculated from this analysis (figures not given). Note that the maximum and minimum values of Ms for selected range of  with normalized dead times To = 0.1 and 0.224 are the same as already discussed in Example 1.

Closed loop output, y

3.5 3 2.5 2 1.5 1 0.5 0

0

5

10

15

20

25

15

20

25

Time

Control action, u

2 1 0 -1 -2 -3 -4

0

5

10

Time Fig. 20 – Closed loop and control action responses for Example 2 for To = 0.554 for uncertainty of +10% in dead time, dashed line – Ms = 3.875, solid line – Ms = 4.75, dotted line – Ms = 5.625 and dash-dotted line – Ms = 6.5.

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200

150

160

Maximum Sensitivity (Ms)

Maximum Sensivity (Ms)

180

100

50

0

140 120 100 80 60 40

0

0.5

1

λ

1.5 20

Fig. 21 – Maximum sensitivity (Ms) for different values of  for Example 2 with normalized dead times To = 0.1 (dotted line), 0.284 (dashed line), 0.554 (solid line).

4.3. Case study: control of exit concentration in a chemical reactor A isothermal continuous stirred tank reactor is considered here which exhibits multiple steady state solutions. The mathematical model of the reactor is given as (Sree and Chidambaram, 2006)

0

where Cf is the inlet concentration, Q is the inlet flow rate, V is the volume of the reactor, C is the exit concentration, k1 & k2 are the kinetic parameters. The operating parameters are given as Q = 0.0333 L/s, V = 1 L, k1 = 10 L/s, and k2 = 10 L/mol. This process exhibits three steady states. By considering a nominal value of Cf = 3.288 mol/L, the steady states are obtained as C = 1.7673, 1.316 and 0.01424 mol/L. Out of the three steady states, there is one unstable steady state at C = 1.316 mol/L. Feed concetration is considered as the manipulated variable

0

5

10

15

λ

20

25

30

Fig. 23 – Maximum sensitivity (Ms) for different values of . and exit concentratio is the controlled variable. Linearization of the manipulated variable around this operating condition C = 1.316 gives the unstable transfer function model as 3.433/(103.1 s). For this particular case, the time delay is considered as 20 s. Hence, the unstable transfer function model is obtained as, Gp =

k1 C Q dC = (Cf − C) − 2 dt V (k2 C + 1)

X: 21.79 Y: 2.624

3.433e−20s 103.1s − 1

For this UFOPDT model, the proposed methodology is applied. Ms is considered as 2.62 and corresponding to this Ms, the controller is designed using Eqs. (28)–(30) and the PID controller settings are obtained as kc = 1.5342,  i = 87.35,  d = 7.35. With these controller settings the proposed method is simulated for a unit step change in the set point at time t = 0 and a disturbance of magnitude 0.5 at t = 350 s respectively. The corresponding closed loop response and control action responses are shown in Fig. 22 for perfect model conditions. To analyze the effect of uncertainties, perturbation of +20%

Closed loop output, y

2.5 2 1.5 1 0.5 0

0

100

200

300

400

500

600

700

400

500

600

700

Time

Control action, u

2

1

0

-1

-2

0

100

200

300

Time Fig. 22 – Closed loop and control action responses, solid line – perfect model, dashed line – uncertainty of +20% in dead time.

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chemical engineering research and design 1 0 9 ( 2 0 1 6 ) 593–606

Exit Concetration, C

1.42 1.4 1.38 1.36 1.34 1.32 1.3

0

100

200

300

400

500

600

700

400

500

600

700

Time Feed Concentration, Cf

3.4 3.35 3.3 3.25 3.2

0

100

200

300

Time Fig. 24 – Closed loop and control action responses obtained by simulation on the nonlinear model, solid line – perfect model, dashed line – uncertainty of +20% in dead time.

in dead time is considered and the corresponding closed loop responses are also shown in Fig. 22. It can be observed that the proposed method works well and able to provide good closed loop responses. The normalized dead time is 0.194. For understanding the maximum and minimum values of Ms values, Ms values are plotted against the tuning parameter, . Fig. 23 shows the variation of Ms with . It can be observed that low value of  gives higher values of Ms and vice versa. Further, the designed controller is applied by simulation directly on the original nonlinear system by giving a step change in the set point from 1.316 to 1.34 at time t = 0 and a step disturbance of magnitude 0.034 at t = 350 s respectively. Fig. 24 shows the closed loop responses for perfect model parameters. Again the closed loop responses are good for both set point tracking and disturbance rejection. To analyze the performance of the designed controller for uncertainties, perturbations of +20% in dead time is considered and the corresponding closed loop responses are also shown in Fig. 24. It can be observed that the proposed method performs well even for uncertainties on the nonlinear system simulation.

5.

Discussion

The proposed methodology and tuning rules are useful for design of PID controllers for UFOPDT processes. The tuning rules obtained in Eqs. (25)–(27) are useful for the cases where To varies between 0.1 and 0.9 and to design the PID controller for desired Ms values. However, the tuning rules obtained in Eqs. (28)–(30) are useful for the cases where the PID controller need to be designed for different rages of To and to achieve any of the four Ms. In this paper, the tuning rules are provided for specific ranges of To and Ms. Moreover, the proposed methodology can be extended to other ranges of To and Ms. Also, the methodology can be extended to second order unstable systems after validation with multiple plants with uncertainty.

6.

Conclusions

In the present work, analytical tuning rules have been developed for PID controllers for UFOPDT processes. The proposed tuning rules allow the designer to design closed loop control systems with a specified low, medium, or high robustness level measured using the maximum sensitivity. The performancerobustness trade-off is addressed. Once the UFOPDT model is known, based on the dead time to tome constant ratio, the controller parameters can be selected from the developed tuning relations for a desired level of robustness. The applicability of the present method is discussed with two examples for different values of time delay to time constant ratios and also with a case study of CSTR.

References Alfaro, V.M., Vilanova, R., Arrieta, O., 2010. Maximum sensitivity based robust tuning for two-degree-of-freedom proportional-integral controllers. Ind. Eng. Chem. Res. 49 (11), 5415–5423. Anusha, A.V.N.L., Seshagiri Rao, A., 2012. Design and analysis of IMC based PID controller for unstable systems for enhanced closed loop performance. In: IFAC Conference on Advances in PID control, Brescia, Italy. Arrieta, O., Vilanova, R., 2012. Simple servo/regulation proportional-integral-derivative (PID) tuning rules arbitrary Ms-based robustness achievement. Ind. Eng. Chem. Res. 51 (6), 2666–2674. Arrieta, O., Vilanova, R., Visioli, A., 2011. Proportional-integral-derivative tuning for servo/regulation control operation for unstable and integrating processes. Ind. Eng. Chem. Res. 50 (6), 3327–3334. Liu, T., Zhang, W., Gu, D., 2005. Analytical design of two degree of freedom control scheme for open-loop unstable processes with time delay. J. Process Control 15 (5), 559–572. Morari, M., Zafiriou, E., 1989. Robust Process Control. Prentice Hall, Englewood Cliffs, NJ.

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Nasution, A.A., Cheng, J., Huang, H.-P., 2011. Optimal H2 IMC-PID controller with set point weighting for time-delayed unstable process. Ind. Eng. Chem. Res. 50, 4567–4578. Normey-Rico, J.E., Camacho, E.F., 2008. Simple robust dead-time compensator for first-order plus dead-time unstable processes. Ind. Eng. Chem. Res. 47, 4784–4790. Sree, R.P., Chidambaram, M., 2006. Control of Unstable Systems. Narosa Publishers, New Delhi. Park, J.H., Sung, S.W., Lee, I.B., 1998. An enhanced PID control strategy for unstable process. Automatica 34 (6), 751–756. Rao, A.S., Chidambaram, M., 2006. Control of unstable processes with two RHP poles, a zero and time delay. Asia Pacific J. Chem. Eng. 1, 63–69. Rao, A.S., Chidambaram, M., 2012. PI/PID controllers for integrating and unstable systems. In: Vilanova, R., Visioli, A. (Eds.), PID Control in the Third Millennium. Springer, pp. 75–111. Rao, A.S., Rao, V.S.R., Chidambaram, M., 2007. Simple analytical design of modified Smith predictor with improved performance for unstable first order time delay (FOPTD) processes. Ind. Eng. Chem. Res. 46, 4561–4571.

Shamsuzzoha, M., Lee, M., 2008. Analytical design of PID controller for integrating and first order unstable processes with time delay. Chem. Eng. Sci. 63 (10), 2717–2731. Skogestad, S., Postlethwaite, I., 2005. Multivariable Feedback Control: Analysis and Design. John Wiley & Sons, New York. Sree, R.P., Srinivas, M.N., Chidambaram, M., 2004. A simple method of tuning PID controllers for stable and unstable FOPTD systems. Comput. Chem. Eng. 28, 2201–2218. Tan, W., 2010. Analysis and design of a double two-degree of freedom control scheme. ISA Trans. 49 (3), 311–317. Tan, W., Marquez, H.J., Chen, T., 2003. IMC design for unstable processes with time delays. J. Process Control 13 (3), 203–213. Uma, S., Chidambaram, M., Rao, A.S., 2010. Set-point weighted modified Smith predictor with PID filter controllers for non-minimum phase (NMP) integrating processes. Chem. Eng. Res. Des. 88 (5), 592–601. Vanavil, B., Anusha, A.V.N.L., Perumalsamy, M., Rao, A.S., 2014. Enhanced IMC-PID controller design with lead-lag filter for unstable and integrating processes with time delay. Chem. Eng. Commun. 201, 1468–1496.