A Novel Tuning Algorithm for Fractional Order

IMC Controllers for Time Delay Processes

Cristina I. Muresan, Eva H. Dulf, and Roxana Both Department of Automation, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Email: cristina.pop@aut.utcluj.ro

Abstract—This paper presents a novel tuning algorithm for

fractional order IMC controllers for time delay processes.

The tuning algorithm is based on computing the equivalent

controller of the IMC structure and imposing frequency

domain specifications for the resulting open loop system. A

second order time delay process is used as a case study. An

integer order IMC controller is designed, as well as a

fractional order IMC controller. The simulation results

show that the proposed fractional order IMC controller

ensures an increased robustness to modeling uncertainties.

Index Terms—fractional order controller, time delay

processes, tuning algorithm, robustness

I. INTRODUCTION

Fractional calculus represents the generalization of the

integration and differentiation to an arbitrary order. There

is currently a continuously increasing interest in

generalizing classical control theories and developing

novel control strategies that use fractional calculus. The

most commonly used method for controlling a great

range of processes is the PID controller, which is in fact a

special case of fractional order PIμDλs. The design

problem of fractional order controllers has been the

interest of many authors, with some valuable works, in

which the fractional order controllers have been applied

to a variety of processes to enhance the robustness and

performance of the control systems [1]-[4]. The choice of

fractional order PIμDλ controllers is based on their

potential to improve the control performance, due to the

supplementary tuning variables involved, μ and λ. Since

the fractional controller has more parameters than the

conventional controller, more specifications can be

fulfilled, improving the overall performance of the

system and making it more robust to modeling

uncertainties. Apart from the fractional order PIμDλ

controller, some extensions and generalizations of

advanced control strategies using fractional calculus have

been previously proposed, such as fractional optimal

control [5], fractional fuzzy adaptive control [6],

fractional iterative learning control [7], fractional

predictive control [8] and fractional model reference

adaptive control [9] to name just a few.

The internal model control (IMC) based PID controller

has gained widespread acceptance in the control

Manuscript received July 1, 2014; revised June 8, 2015.

community, mainly because of the simplicity in the

design that is based upon inverting the process model.

The IMC approach has been proposed as a method for

tuning fractional order PIμD

λ based on a bandwidth

specification [10], for first order plus time delay process

[11] and a class of fractional order systems [12-14]. In

this paper, a novel tuning algorithm is proposed for a

fractional order IMC (FO-IMC) controller, as compared

to the existing design approaches [10-14]. An integer

order IMC controller is first tuned to meet settling time

requirements. Next, the FO-IMC controller is tuned

according to the same performance specification.

However, due to the supplementary tuning parameter, the

fractional order of the FO-IMC filter, a second

performance criterion is imposed to increase the closed

loop performance and the robustness of the controller.

The simulation results, considering modeling errors,

show that the proposed FO-IMC controller offers better

closed loop results as compared to its integer order

version.

II. TUNING ALGORITHM

The basic structure of the IMC is shown in Fig. 1,

where Hp(s) is the process transfer function, Hm(s) is the

model of the process, HFO-IMC(s) is the fractional order

IMC controller transfer function and Hc(s) is the

equivalent fractional order controller.

Considering the second order time delay process,

described by the following transfer function:

s

2pme

cbsas

k)s(H

(1)

where m is the process time delay.

Figure 1. Basic IMC control structure

The proposed fractional order IMC controller is given

by:

1s

1

k

cbsas)s(H

2

IMCFO

(2)

218© 2015 Int. J. Mech. Eng. Rob. Res.

International Journal of Mechanical Engineering and Robotics Research Vol. 4, No. 3, July 2015

doi: 10.18178/ijmerr.4.3.218-221

with αϵ(0÷1), the fractional order. The equivalent

controller in Fig. 1 is computed as:

ssk

cbsas

)s(H)s(H1

)s(H)s(H

m

2

mIMCFO

IMCFOc

(3)

where a series approximation for the time delay was used,

with s1e msm

.

The open loop transfer function with the equivalent

controller and the process transfer function is described

by:

s

m

pcolme

ss

1)s(H)s(H)s(H

(4)

To tune the FO-IMC controller, a new tuning

technique is proposed, that allows the computation of the

time constant λ and the fractional order α based on two

imposed performance specifications, a specified gain

crossover frequency, ωgc, and a phase margin, γ. The

specified ωgc is given in order to ensure a certain closed

loop settling time, while an increased phase margin will

ensure increased stability of the closed loop system. The

phase margin condition is written in a mathematical form

as:

)j(H gcol (5)

which leads to:

2costan

2sin gcm

1gcm

(6)

where the following relation was used

2sinj

2cosj gcgc . The gain

crossover frequency condition is written in a

mathematical form as:

1)j(H gcol (7)

which leads to:

12

sin2 1gcm

2gc

2m

2gc

2

(8)

To compute the values for λ and α, a graphical

approach is used in which for different values of α,

relations (6) and (8) are used to compute λ and the

resulting values are plotted as a function of α. The

intersection point yields the final values for the fractional

order and the time constant.

III. CASE STUDY: SECOND ORDER TIME DELAY PROCESS

To exemplify the tuning procedure described in the

previous section, the following second order process is

used as a case study:

s10

2p e10s2s

5)s(H

(9)

For comparison purposes, an integer order IMC

controller is designed:

2

2

IMC1s2.3

1

5

10s2s)s(H

(10)

with the time constant λ=3.2. The open loop transfer

function is computed similarly to (4) as:

s

m22ol

mes2s

1)s(H

(11)

-100

-50

0

Ma

gn

itu

de

(d

B)

Bode Diagram

Frequency (rad/sec)

10-2

10-1

100

101

102

-5.76

-4.608

-3.456

-2.304

-1.152

0x 10

4

System: HPhase Margin (deg): 52.9Delay Margin (sec): 15.2At frequency (rad/sec): 0.0609Closed Loop Stable? Yes

Ph

ase

(d

eg

)

Figure 2. Bode diagram of the open loop transfer function with integer

order IMC controller

0.8 0.85 0.9 0.950

2

4

6

8

10

eq. (6)

eq. (8)

Figure 3. Graphical solution for the tuning algorithm

-80

-60

-40

-20

0

20

40

Ma

gn

itu

de

(d

B)

Bode Diagram

Frequency (rad/sec)

10-4

10-2

100

102

104

-7.0779

-4.7186

-2.3593

0x 10

7

System: HdPhase Margin (deg): 61Delay Margin (sec): 17.9At frequency (rad/sec): 0.0595Closed Loop Stable? Yes

Ph

ase

(d

eg

)

Figure 4. Bode diagram of the open loop transfer function with

fractional order IMC controller

The Bode diagram of (11) is given in Fig. 2, yielding a

gain crossover frequency ωgc=0.06 and a phase margin,

219© 2015 Int. J. Mech. Eng. Rob. Res.

International Journal of Mechanical Engineering and Robotics Research Vol. 4, No. 3, July 2015

γ=53o. To tune the FO-IMC controller, based on (6) and

(8), the same gain crossover frequency is imposed of 0.06

rad/sec, while the phase margin is increased for improved

performance γ=60o. Fig. 3 presents the graphical

representation of the solutions of (6) and (8) for α ranging

from 0 to 1. The solution is found at the intersection point:

α=0.88 and λ=4.87. Fig. 4 shows the Bode diagram of the

open loop transfer function with the resulting FO-IMC

controller. As noted, the obtained gain crossover

frequency and the phase margin meet the design

requirements previously mentioned.

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (sec)

Outp

ut

FO-IMC controller

Integer order IMC controller

Figure 5. Closed loop simulation results using the two controllers under nominal conditions

0 20 40 60 800

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (sec)

Outp

ut

FO-IMC controller

Integer order IMC controller

Figure 6. Closed loop simulation results using the two controllers under +30% variation of the process time delay

0 20 40 60 800

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (sec)

Outp

ut

FO-IMC controller

Integer order IMC controller

Figure 7. Closed loop simulation results using the two controllers under +50% variation of the process gain

Fig. 5 presents the simulation results considering both

the integer order, as well as the proposed FO-IMC

controllers. Under nominal conditions, assuming

Hp(s)=Hm(s), the two controllers achieve similar results,

with the settling time equal to 30 seconds, and no

overshoot. To test the robustness of the designed FO-IMC

controller, two case scenarios are considered. Fig. 6

presents the simulations results considering a +30%

variation of the time delay, while Fig. 7 shows the

simulation results considering a+50% variation of the

process gain.

The comparative results in terms of settling time and

overshoot are given in Table I, showing the increased

robustness of the proposed FO-IMC controller.

TABLE I. COMPARATIVE CLOSED LOOP PERFORMANCE RESULTS

UNDER PARAMETER VARIATIONS

+30% variation of the time delay

Settling time Overshoot

Integer

order IMC

73 seconds 20%

Fractional

order IMC

51 seconds 12%

+50% variation of the process gain

Settling time Overshoot

Integer

order IMC

64 seconds 37%

Fractional

order IMC

42.4 seconds 21%

IV. CONCLUSIONS

The paper presents a novel approach for tuning FO-

IMC controllers in the frequency domain that is based on

improving the closed loop performance and robustness of

a classical integer order IMC controller. The simulation

results considering a second order time delay system

show that the proposed FO-IMC controller provides

improved closed loop performance, despite modeling

uncertainties, if compared to the integer order IMC.

ACKNOWLEDGMENT

This work was supported by a grant of the Romanian

National Authority for Scientific Research, CNDI–

UEFISCDI, project number PNII-RU-TE-2012-3-0307.

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Cristina I. Muresan received the degree in Control Systems in 2007, and the Ph.D. in

2011 from Technical University of Cluj-

Napoca, Romania. She is currently lecturer within the Automation Department of the same

university. She has published over 50 papers and book chapters. Her research interests

include modern control strategies, such as

predictive algorithms, fractional order control, time delay compensation methods and

multivariable systems.

Eva-Henrietta Dulf received her B. Eng.

degree in Automation from Technical University of Cluj-Napoca, Romania, in 1997,

and her M.Sc. and Ph.D. degrees from the same university in 1998 and 2006,

respectively. In November 1998, she joined

Technical University of Cluj-Napoca as teaching assistant and she is now Associate

Professor. Current research interests include various areas in process control, including

mathematical modeling, controller design, optimization methods and

practical implementation. She’s published more than 100 technical journal papers and conference talks, and she is a member of IEEE.

Roxana Both received her B.Eng. degree in

Control Systems in 2008 and the Ph.D. in

2011 from Technical University of Cluj-Napoca, Romania. She is currently lecturer at

the same university at the Automation Department. Her current research interests

include process modelling and simulation,

process control strategies design and implementation: robust control, model

predictive control, fractional control. She has published more than 30 papers in journal and conferences.

221© 2015 Int. J. Mech. Eng. Rob. Res.

International Journal of Mechanical Engineering and Robotics Research Vol. 4, No. 3, July 2015