FRACTIONAL ORDER PID BASED ON IMC · FRACTIONAL ORDER PID CONTROLLER TUNING BASED ON IMC ... In this work, a class of fractional order controller (FOPID) is tuned based on internal

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International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.4, October 2012

DOI:10.5121/ijitca.2012.2403 21

FRACTIONAL ORDER PID CONTROLLER TUNING

BASED ON IMC

Mohammad Reza Rahmani Mehdi Abadi1 and Ali Akbar Jalali

2

1Electrical Engineering Department, Iran University of Science and Technology, Tehran,

Iran. [email protected]

2Electrical Engineering Department, Iran University of Science and Technology,

Tehran, Iran. [email protected]

ABSTRACT In this work, a class of fractional order controller (FOPID) is tuned based on internal model control

(IMC). This tuning rule has been obtained without any approximation of time delay. Moreover to show

usefulness of fractional order controller in comparison with classical integer order controllers, an

industrial PID controller tuned in a similar way, is compared with FOPID and then robust stability of both

controllers is investigated. Robust stability analysis has been done to find maximum delayed time

uncertainty interval which results in a stable closed loop control system. For a typical system, robust

stability has been done to find maximum time constant uncertainty interval of system. Two clarify the

proposed control system design procedure, three examples have been given.

KEYWORDS Fractional order PID, IMC, Robust Stability

1. INTRODUCTION

Many industrial processes can be modeled by a transfer function in which there is a time delay

element. Time delay in the model of a process appears because of measurement delay, actuator

delay, or approximating high order dynamics of processes by lower order dynamics plus time

delay [1]. In [2] a linear model of active queue management (AQM) router including time delay

has been obtained.

The process identification as a First-Order-Plus-Dead-Time (FOPDT) introduce a model which

represents the process behavior in efficient manner. FOPDT models have been used for

approximating industrial and chemical processes which do not have integral and resonant

characteristics [3, 4]. Although many processes have open-loop stable behavior, in some

engineering fields (such as exothermic chemical processes, batch chemical reactors, biological

reactors, waste treatment processes, etc.), processes have several steady states due to their

nonlinearity. Some of these steady states are unstable. On the other hand some specifications like

maximization of productivity, safety and reduction of economic costs need to model the processes

around an unstable steady state [5, 6]. When a collection of stable open-loop plants are connected,

the resulted open-loop process becomes unstable. Chemical irreversible exothermal reactor is an


22

example of such unstable processes [7].

Cvejn has proposed a method for tuning PI and PID controllers for FOPDT processes which deals

with time delay without approximation [8]. Roy and Iqbal have adopted a Hermite-Biehler

theorem based approach to design PID controllers for stabilization of FOPDT process models [9].

In [10] by employing integral squared time error standard forms, a PI-PD controller has been

designed to control unstable and integrating processes. The design of controllers for stable

processes is mostly based on three criteria, namely, error criteria, time domain and frequency

domain. Out of all these synthesis methods, designs based on desired closed loop specifications

have gained much attention by many researchers. In order to improve the performance of the

process, the model of the system can be incorporated in the design of controller that made huge

success through fabricating internal model control structure in the synthesis of equivalent

controller [11-14] .These equivalent controllers are robust in nature and even they are being used

for higher order systems [15-18] applying direct synthesis approach.

Conventional integer-order differentiation and integration can be extended to allow for orders that

are not necessarily integer. Non-integer differentiation and integration of real functions lead to

fractional differential equations which are dealt with in fractional calculus [19]. These concepts

have been transferred into control engineering as a new methodology of control called fractional

order control [20]. Such controllers are the extended version of conventional integer order

controllers that have some extra parameters which must be tuned more precisely and the control

system design procedure is more complicated than integer order controllers. Previously, fractional

derivative and integral have been used in many engineering fields. Having more degrees of

freedom, fractional order models can approximate processes by fewer parameters. Podlubny has

shown that fractional order PID controllers denoted by PI Dλ µ, have a better response in

comparison with standard PID controllers, when used for control of fractional-order systems [21].

Fractional order controllers have been applied to FOPDT processes. In [22] a fractional-order

controller has been applied to an FOPDT model. In [23] a method for practical tuning of

Fractional Order Proportional Integral (FOPI) controller in which the system to be controlled has

been modeled by an FOPDT transfer function has been given. In [24-29] recent applications of

fractional-order controllers have been given.

This work gives a FOPID tuning rule for Stable/Unstable- Plus-Dead-Time processes, based on

IMC. Then robust stability of the proposed FOPID has been investigated. A comparison study

between the proposed FOPID and conventional PID has been made to show that FOPID has

better performance than PID. Here, the proposed tuning rule uses delayed time part without any

approximation. However, when controller has a simple pole at origin and system has delayed time

part, this approach will be applicable. Robust stability analysis has been done to find maximum

delayed time uncertainty interval which results in a stable closed loop control system. For a

typical system, which shows a nearly constant phase around phase crossover frequency, robust

stability has been done to find maximum time constant uncertainty interval of the system.

Organization of this paper is as follows: section 2 explains IMC and fractional order controllers.

Section 3 describes tuning rules for a class of fractional order controllers and robust stability is

investigated. In section 4 this tuning rule is applied to three systems. Finally section 5 concludes

and gives some future work suggestions.

2. PRINCIPLES

This section gives preliminaries for next sections, covers IMC approach control design and

fractional order systems.


23

2.1. IMC

IMC controller has structure shown in fig. 1.

Where ( )P

G s% is identified system and ( )PG s is actual system. IMC controller defined as

( ) ( ) ( )C P

C s G s F s−= %

(1)

Where ( )pG s−% expresses minimum phase or invertible part of system ( )pG s% , containing all

stable and unstable poles of system and stable zeros, but not delayed time and unstable zeros of it.

F(s) is a low pass filter designed so that IMC controller can be realizable as well as to reduce

effect of uncertainty of system at high frequency.

Figure 1. IMC control structure

From IMC structure, equivalent and well-known control system structure with unit feedback can

be obtained (fig. 2).

Figure 2. Equivalent IMC control structure

Controller of such a structure is

( )( )

1 ( ) ( )

C

C P

C sC s

C s G s=

− % (2)


24

2.2. Basic definitions in fractional control

Fractional calculus has been used as a mathematical tool for modelling physical systems and

designing controllers. Fractional calculus is an extension of integer order calculus in which

ordinary differential equations have been replaced by fractional order differential equations. In

fractional order differential equations, derivatives and integrals are not necessarily of integer

order and they span a wider range of differential equations. Fractional calculus deals with

fractional integration and differentiation. Therefore, a generalized differential and integral

operator has been introduced as a single fundamental operator represented by a tDλ

where a and t

are the limits and ( )Rλ λ ∈ the order of the operation. For positive λ , it denotes derivative and

for negative λ , it denotes integral action as

( ) 0

1 ( ) 0

( ) 0( )

a t

t

a

d

Realdt

D Real

Reald

λ

λ

λ

λ

λ

λ

λτ −

>

= = <∫

. (3)

Several ways exist to define fractional-order derivatives and integrals. The mostly used

definitions for fractional derivatives are Riemann–Liouville, Grunwald–Letnikov and Caputo

definitions [19]. The Caputo fractional derivative of order λ with respect to the variable t is

defined as

=

<<−−−Γ=

∫ −+

ntfdt

d

nnt

df

ntfD

n

n

t

n

n

t

λ

λτ

ττ

λ λλ

),(

1,)(

)(

)(

1

)(0 1

0 (4)

where n is the first integer not less than λ , and ( )ZΓ is Euler’s Gamma function which is given

by

1

0( ) z tZ t e dt

∞− −Γ = ∫ . (5)

The Laplace transform of the Caputo fractional derivative is

1

1 ( )

0

0

{ ( )} ( ) (0),

1 .

nk k

t

k

L D f t s F s s f

n n N

λ λ λ

λ

−− −

=

= −

− < ≤ ∈

∑ (6)

The main advantage of using the Caputo definition is that, only integer order derivatives of

function ( )f t at t=0 appear in the Laplace transform of the Caputo fractional derivative [19].

For zero initial conditions in (6), a straightforward result is obtained as

0{ ( )} ( )

tL D f t s F sλ λ= . (7)


25

3. TUNING BASED ON IMC

In eq. (2), it can be seen that for a delayed time system, C(s) has a simple pole at s=0.

Considering this, if C(s) is rewritten as

( )( )

f sC s

s=

(8)

For tuning a specified controller, the controller must have a simple pole at origin, consequently.

Here it is assumed that ( ) ( )P P

G s G s=% .

Controllers that are used to tune are a well-known PID, and a class of fractional order controller,

as formulated below

11( ) 1

1

di p

i d

k sC s k

k s ak s

+= +

+

(9)

( )( ) 1 1mi

f p d

kC s k k s

s

= + +

(10)

Where m may be any real number. Now tuning rules are as following

3.1 Integer order PID tuning rule

As it is seen from eq.(9), ( )iC s can be written in form of

( ) (1 ) ( )( ) ,

(1 ) (1 )i d

g s s f sC s ak

s s s s

ββ

β β

+= = =

+ +

(11)

From equations (2, 8), it is obvious that f(s) is only a function of system parameters. To determine

a value for β , As a result of previous work [13], one can guess that 0.25max( , )d pk l t= ,

where l and p

t denote delayed time and slow pole of system respectively, and a=0.1 is chosen.

Thus in eq.(11), ( )1 ( )s f sβ+ is known. On the other hand, from eq.(9) it is seen that equivalent

term to ( )1 ( )s f sβ+ is ( )( )1 1p

i d

i

kk s k s

k+ + . To find out values of three unknown

parameters , ,p i dk k k ,Taylor series of g(s) at point s=0 is used. Comparing coefficients of s

terms of this Taylor series and eq.(9) gives values of unknowns. Results have been stated below

[13].


26

(0 ) (0 )

(0) (0) (0 )

(0) (0 ) 2 (0)

2 ! 2

p

i

c

c d

kg f

k

k g f f

g f fk k

β

β

= =

′ ′= = +

′′ ′′ ′+= =

(12)

This PID tuning rule has some disadvantages, that are as follows. First of all, it is required to

guess the value of β , in turn value of dk , and then in tunig rules eq.(12) find the value of

dk again. Secondly, it is needed to define low pass filter F(s) in eq.(1) in a complex form, that is

not a easy design of filter. Finally and worst of all is that there is one and only one set of solution

for eq.(12); this, makes achievement to a more robust stability solution hard. With fractional

definition, as given below, these disadvantages will be eliminated.

3.2 Fractional order controller tuning rule

Considering eq.(10), there are four unknowns that must be determined. To find these parameters,

( )FC s will be rewritten as

( )( )F

g sC s

s=

(13)

Where

( )( )( ) 1m

p i dg s k s k k s= + +

(14)

On the other side, from eqs. (2, 8) ( )FC s can be defined as function of system parameters. From

relation ( ) ( )f s g s= , where f(s) and g(s) are defined in eq.(8) and eq.(14) respectively, it is

possible to determine unknowns , , ,p i dk k k m . This tuning can be done by using Taylor series

as done below.

2 3 2 3(0) (0) (0) (0)

( ) (0) (0) (0) (0)2! 3! 2! 3!

g s g s f s f sg s g g s f f s

′′ ′′′ ′′ ′′′′ ′= + + + + = + + + +L L

(15)

Because of four unknowns, one way to tune is that first four Taylor series terms of g(s) and f(s)

are used. This results in following relation.

2 2 2

2 2 2 3 3 2 3 3

2( ( )) ( )( )

(0) 3( ( ))

(0)

(0) ( )( )

(0)

( )( 3 2 )

p d p i d d

p d d p i d d d

p i

p p i d

g k k

g k k k

k mk k k m k mk

g k m k mk k k m k m k

m

g

mk

k

+ −

=

′ = +

′′ =

′′′ = − + − +

(16)

To find solutions of this set of nonlinear equations with 4 equations and 4 unknowns, procedure

below is suggested.


27

first and second equations of eq.(16) leads to relations below

1 ( ) (0)

(0)( )

(0)

p i

p

a d

C k k f

f kC mk

f

= =

′ −= =

(17)

By replacing p ik k and dmk with their equivalent term from eq.(17), in two other equations in

eq.(16), It can be seen that relations (18) will be obtained.

3 2 2

1

1

C (2k +C (-k +C ))=f (

3 (

0),

C C C C C) ( 3 2 ) (0)Cp d d d

a p d a

a a a a ak k k k f ′′′+ =

′

+

′

− + −

(18)

Now, from the set of 2 equation-2 unknown in (18), ,d p

k k can be obtained. From first equation

of eq.(18) dk is defined, then putting it in second relation of eq.(18), it can be seen that a fourth

order equation of pk will be achieved. Thus there will be four choices to define

pk . These

choices make a more powerful robust stability design.

3.3 Defining low pass filter F(s)

Low pass filter for all stable pole systems is defined as

1( )

( 1)nF s

sλ=

+

(19)

And for unstable systems defined as

1( )

( 1)n

sF s

s

γ

λ

+=

+ (20)

Where γ is chosen so that the following relation is satisfied.

( ) 11/P

p

F s Gs t

+ ==

(21)

Where ( )P

G s+

is noninvertible part of system and n is an integer number that is selected so that

IMC controller becomes realizable (proper). To select λ tradeoffs between speed of response and

stability of the closed loop must be considered. Here, following rules explained will be used [13].

FOPDT system: max(0.2 ,1.7 )

SOPDT system: max(0.2 ,0.25 )

IPDT system: 10

p

p

for t l

for t l

for l

λ

λ

λ

=

=

=

(22)


28

3.4 Robust stability investigation

Robust stability will be investigated from two aspects, from phase margin and gain margin. Using

phase margin, makes it possible to investigate uncertainty occurred in phase of closed loop. On

the other side, using gain margin, one can investigate uncertainty occurred in gain of system.

These two robust stability methodologies will be clarified below.

3.4.1 Robust stability from phase margin point of view

Using phase margin, it is necessary for magnitude of open loop bode diagram not to change by

uncertainty. To provide this condition, it is supposed that there exists uncertainty in delayed time

of system. For stability of closed loop control system with open loop stable system, it is

necessary that phase of open loop transfer function at gain crossover frequency, not to reach π−

radian. According to this, it is possible to find maximum uncertainty interval, at which closed

loop is stable. Thus to find maximum uncertainty resulting stable closed loop response, procedure

is as follows. At first, gain crossover frequency is determined from magnitude relation of open

loop transfer function. Then from phase relation of loop gain at this frequency, delayed time

uncertainty will be achieved.

3.4.2 Robust stability from gain margin point of view

When using gain margin, it is necessary for phase of open loop bode diagram not to change by

uncertainty. For some typical cases it is reasonable that with uncertainty in time constant of

system, this condition approximately is provided. For stability of closed loop control system with

open loop stable transfer function, it is necessary that magnitude of open loop transfer function at

phase crossover frequency, is less than 1. According to this, it is possible to find maximum

uncertainty interval for time constant of system, at which closed loop is stable.

4. SOME EXAMPLES

4.1 Example 1

Consider a FOPDT system ( ) 10exp( 0.1 ) / (5 1)PG s s s= − + . Here for 5, 0.1, 10p

t l k= = = ,

from eq.(22) 1λ = for first order low pass filter ( ) 1 / ( 1)F s sλ= + and in eq.(11) 0.125β = is

chosen. According to tuning rules for integer order PID,

k = .46632,p

= 5.12952, k = 0.12636i dk . For fractional order controller, there will be three

reasonable selections (that result in positive pk ), 0.40371e-3, 0.45333, 0.45455 , which

provide desired response to step input and load disturbance and very similar to integer order one.

But difference of each selection will become important, when stability robustness gets critical.

This point has been discussed below.

4.1.1 Delayed time Uncertainty interval for integer order PID

To find uncertainty interval, from open loop magnitude equation, gain crossover frequency 0ω

obtained.


29

( )( )( )

2 2

0

2 2 2 2 2 20 0 0

111 1

1 1

d

p

i p

kkk

k t

ω

ω ω β ω

+ + =

+ +

(23)

From above equation 0 0.93176ω = . Now equality of phase equation of open loop with π− , will

give maximum uncertainty in delayed time that provide stable response.

0 0 0 0 0

0

arctan( ) arctan(1/(k ))+arctan(k )-arctan( )+p i dl tl

ω ω ω ω βω π

ω

− − −∆ =

(24)

Uncertainty interval for delayed time 1.59275l∆ = has been calculated.

4.1.2 Delayed time Uncertainty interval for Fractional order controller

Procedure is as same as for integer order case, but equations are changed. Magnitude equation is.

( )( )

2 220

2 2 20 0

11 1

1

m

dip

p

kkkk

t

ω

ω ω

+ + =

+

(25)

Whereas said before, there are three choices for , , ,p i dk k k m . For all of these choices,

0 0.90911ω = that is approximately as same as integer order case. But, for these three selections,

uncertainty interval changes. Uncertainty interval is obtained from relation below.

0 0 0 0

0

arctan( ) arctan(k / )+marctan(k )+p i dl tl

ω ω ω ω π

ω

− − −∆ =

(26)

For selected 0.45333p

k = , from eq.(16) 7.47426dk = , 0.20054ik = , m=0.00240 are

calculated. Thus 1.62745l∆ = is obtained. If k =0.00040p is selected, thus 225.18636ik = ,

4.99995dk = , 1.00003m = , and 1.62880l∆ = is determined. It is seen that if 0.40e-3p

k =

is chosen, maximum uncertainty in delayed time is obtained, but for all of choices uncertainty

intervals are approximately the same. Besides, remembering, integer order uncertainty interval, it

becomes obvious that fractional order is a more robust than integer order for this typical system.

4.1.3 Time constant Uncertainty interval for integer order PID

To find uncertainty interval, at first from open loop phase equation, phase crossover frequency

pω obtained.

arctan( ) arctan(1/(k ))+arctan(k ) arctan( )=p p p i p d p p

l tω ω ω ω βω π− − − − − (27)

Because of existence of function arctan(.) in eq.(27), it is not possible to find an analytical

solution for pω from above equation. But it is possible to find p

ω graphically, or use following

approximation.


30

, 1

arctan( ) 1 , 1

2

x x

xx

x

π

<

= − >

(28)

Considering values of , , , ,p i d pk k k l t , and knowing that0 1

pω ω> � , eq.(27) reformed to

1 0.1949490474 1 1

0.12 5 2 0.1263590714 2 0.125

p

p p p p

π π πω π

ω ω ω ω

− − − − + − − − =−

(29)

In above equation, it is seen that with 5p

t = , arctan( )2

p pt

πω → , ensuring that phase equation

will not be altered by variable p

t . From above relation, =15.76574p

ω is calculated.

Now equality of magnitude equation of loop gain with 1, will give maximum uncertainty in time

constant of system that provide stable response.

( )( )( )

2 2

2 2 2 2 2 2

111 1

1 ( ) 1

d p

p

i p p p p p

kkk

k t t

ω

ω ω β ω

+ + = + + ∆ +

(30)

Uncertainty interval for time constant -4.70845pt∆ = is calculated. This means that up to about

30% faster system can be stable in closed loop.

4.1.4 Time constant Uncertainty interval for Fractional order controller

Procedure is as same as for integer order case, but equations are changed. Phase equation is

arctan( ) arctan(k / )+marctan(k )=p p p i p d pl tω ω ω ω π− − − − (31)

Whereas said before, there are three choices for , , ,p i dk k k m . Almost, for all of these choices,

15.74512pω = that is approximately as same as integer order case. But, for these three

selections, uncertainty interval changes very little and is bigger than integer order. Uncertainty

interval is obtained from relation below:

( )( )

2 22

2 2 2

11 1

1 ( )

m

d pip

p p p p

kkkk

t t

ω

ω ω

+ + = + + ∆

(32)

For selected 0.45333pk = , 7.47426dk = , 0.20054ik = , m=0.00240 and -4.71575pt∆ =

are obtained. But if k =0.00040p

is selected, 225.18636ik = , 4.99995dk = , 1.00003m = ,

and -4.73209pt∆ = . It is seen that if 0.40e-3pk = is selected, maximum uncertainty in time

constant will be obtained, but for all of choices uncertainty intervals are approximately the same.

Besides, remembering, integer order uncertainty interval, it becomes obvious that fractional order

is a more robust than integer order for this typical system.


31

Figure 3, shows step response of two controllers. IAE measure for integer order is 0.1 but for

fractional order is 0.0. Load disturbance rejection for disturbance value 0.1 applied in time

interval [4,5] seconds, is also shown with IAE=0.28 for I.O., and 0.29 for F.O.

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1step response

t

outp

ut

0 5 10 15 20 25 30-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14load distorbance with value=.1 response

t

outp

ut

desired response

integer order

fractional

integer order

fractional

Figure 3. step response and load distorbance rejection of exa. 1

4.2 Example 2

Considering system ( ) 10(0.1 1) exp( 0.1 ) / (5 1)PG s s s s= + − + , thus 5, 0.1pt l= = are defined.

From eq.(22) 1λ = and low pass filter ( ) 1 / ( 1)F s sλ= + is selected.

Now for case of integer order, calculating k = .45723, = 5.02955, t = 0.28871e-1p i dk , to

find uncertainty interval, from

( )( )( )( )

2 2 2 2

1 0 0

12 2 2 2 2 20 0 0

1 111 1, 0.1, 5

1 1

d

p p

i p

t kkk t t

k t

ω ω

ω ω β ω

+ + + = = =

+ +

(33)

gain crossover frequency 0 .91239ω = is determined. From Phase equation at 0ω , uncertainty

0 0 0 0 0 1 0

0

arctan( ) arctan(1/(k ))+arctan(k )-arctan( )+arctan(t )+p i dl tl

ω ω ω ω βω ω π

ω

− − −∆ =

(34)

1.62710l∆ = is calculated. Now considering fractional order controller, there are two

reasonable choices 0.45455, 0.47992p

k = . Selected .45455p

k = , gives


32

0.20000, k 0.10179, m= 0.93774i dk = = − , and then for such a tuning 0 0.91239ω = and

1.62617l∆ = are calculated. If .47992pk = is chosen, calculated

0.18943, k 7.68554, m=-0.48739e-1i dk = = , result in 1.65716l∆ = at 0 .912393ω = . It is

seen that for this typical system, robust stability is the same for both integer and fractional order.

But difference in their step responses is obvious. Figure 4 shows this.

0 5 10-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2step response

t

outp

ut

0 10 20 30-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12


t

outp

ut

desired response

integer order

fractional

integer order

fractional

Figure 4. step response and load distorbance rejection of exa. 2

As it is seen, transient response of integer order is not as good as fractional ones. For I.O. step

response IAE=0.1, but for F.O.=0.0 And in load disturbance rejection for disturbance value of 0.1

in time interval [4,5] seconds, for I.O. IAE=0.28 and for F.O. IAE=0.29 are achieved.

4.3 Example 3

Considering system ( ) exp( 0.5 ) / (2 1)(0.5 1)PG s s s s= − − + , thus 2, 0.5p

t l= = are defined.

From eq.(22) 0.4λ = and low pass filter 2( ) ( 1) / ( 1)F s s sγ λ= + + is selected. Reason of

selecting a second order filter it to make IMC controller realizable. On the other side, to obtain

positive unknowns, it is needed to choose γ from below eq.(21). For this system with

( ) exp( 0.5 )P

G s s+ = − , 1.6918γ = has been calculated.

Now for the case of integer order, by calculating k = 2.12342, = 11.64156, t =0.63135p i dk ,

to find uncertainty interval, gain crossover frequency 0 1ω = is obtained, then from phase

relation, maximum 1l∆ = − is calculated.

Now considering fractional order controller, there are three reasonable choices for

0.11566, 1.68415, 2.00265pk = . 2.00265p

k = gives 0.91079e-1, k 0.44968i dk = =

, m=1.36119 , thus for such a tuning, 0 1ω = and 0.6279l∆ = is determined. If


33

0.11566pk = is selected, 1.57701, k 10.98993, m=0.99704i dk = = will be calculated, and

results in 0.6135l∆ = at same 0ω . It is seen that for this typical system, robust stability for

different fractional order are similar and is greater than for integer order controller. However,

their step responses are not as much different as their robust stability. Figure 5 shows this.

Figure5 step response and load distorbance rejection of exa. 3

Table 1 shows IAE measure for different tunings.

TABLE 1.IAE FOR EXA. 3.

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3step response

t

ou

tpu

t

0 5 10 15 20 25 30-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05


t

ou

tpu

t

desired response

integer order

fractional with kp=0.11566



integer order





34

5. CONCLUSION In this work a class of fractional order controller has been tuned. Results show that without any

robust condition there is at least one set of solution for fractional order controller that is more

robust than conventional PID controller. Besides, it is seen that in desired response tracking,

when fractional order has robust stability near to integer order, F.O. with smaller IAE is better

than conventional PID. Moreover, owning to its more degree of freedom, it is possible to add

robust conditions in fractional order controllers tunings. Latter can be future work.

REFERENCES

[1] Shamsuzzoha, M. and Lee, M., (2009) "Enhanced disturbance rejection for open-loop unstable

process with time delay", ISA Transactions, Vol. 48, No. 2, pp 237-244.

[2] Testouri, S.; Saadaoui, K. and Benrejeb, M., (2012) "Analytical design of first-order controllers for

the TCP/AQM systems with time delay", International Journal of Information Technology, Control

and Automation (IJITCA), Vol. 2, No.3, pp 27-37.

[3] Marlin, T.E., (2000) Process Control, Designing Processes and Control Systems For Dynamic

Performance, 2nd Ed., McGraw Hill.

[4] Luyben, W.L., (1990) Process Modeling: Simulation and Control For Chemical Engineers, 2nd Ed.,

McGraw Hill.

[5] Pasgianos, G.D.; Syrcos, G.; Arvanitis, K.G. and Sigrimis, N.A., (2003) "Pseudo-derivative feedback-

based identification of unstable processes with application to bioreactors", Computers and Electronics

in Agriculture, Vol. 40, No. 1-3, pp 5-25.

[6] Panda, R.C., (2009) "Synthesis of PID controller for unstable and integrating processes", Chemical

Engineering Science, Vol. 64, No. 12, pp 2807-2816.

[7] Rojas, R.; Camacho, O. and Gonzalez, L., (2004) "A sliding mode control proposal for open-loop

unstable processes", ISA Transactions, Vol. 43, No. 2, pp 243-255.

[8] Cvejn, J., (2009) "Sub-optimal PID controller settings for FOPDT systems with long dead time",

Journal of Process Control, Vol. 19, No. 9, pp 1486-1495.

[9] Roy, A. and Iqbal, K., (2005) "PID controller tuning for the first-order-plus-dead-time process model

via Hermite-Biehler theorem", ISA Transactions, Vol. 44, No. 3, pp 363-378.

[10] Kaya, I., (2003) "A PI-PD controller design for control of unstable and integrating", ISA

Transactions, Vol. 42, No. 1, pp 111-121.

[11] Morari, M. and Zafiriou, E., (1989) Robust Process Control. Prentice-Hall, Englewood Cliffs, NJ.

[12] Rivera, D.E.; Morari, M. and Skogestad, S., (1986) "Internal model control. 4. PID controller design",

Ind. Eng. Proc. Des. Dev., Vol. 25, pp 252-265.

[13] Chen, D. and Seborg, D.E., (2002) "PI/PID controller design based on direct synthesis and

disturbance rejection", Ind. Eng. Chem. Res., Vol. 41, No. 19, pp 4807–4822.

[14] Skogestad, S., (2003) "Simple analytical rules for model reduction and PID controller tuning", J. Proc.

Control, Vol. 13, No. 4, pp 291-309.

[15] Lee, Y.; Lee, J. and Park, S., (2000) "PID controller tuning for integrating and unstable processes with

time delay", Chem. Eng. Sci. Vol. 55, No. 17, pp 3481-3493.

[16] Panda, R.C.; Yu, C.C. and Huang, H.P., (2004) "PID tuning rules for SOPDT systems: review and

some new results", ISA Trans., Vol. 43, No. 2, pp 283-295.

[17] Panda, R.C., (2008) "Synthesis of PID tuning rule using desired closed-loop response", Ind. Eng.

Chem. Res., Vol. 47, No. 22, pp 8684-8692.

[18] Tan, W.; Marquez, H.J. and Chen, T., (2003) "IMC design for unstable processes with time delays",

Journal of Process Control, Vol. 13, No. 3, pp 203-213.

[19] Oldham, K.B. and Spanier, J., (1974) The fractional calculus, integrations and differentiations of

arbitrary order NewYork, Academic Press.

[20] Podlubny, I., (1999a) Fractional differential equations New York, Academic Press.

[21] Podlubny, I., (1999b) "Fractional-order systems and PI Dλ µ -controllers", IEEE Trans Automatic

Control, Vol. 44, No. 1, pp 208-222.


35

[22] Mukhopadhyay, S.; Chen, Y.Q.; Singh, A. and Edwards, F., (2009) "Fractional Order Plasma Position

Control of the STOR-1M Tokamak", 48th IEEE Conference on Decision and Control and 28th

Chinese Control Conference Shanghai, China, pp. 422-427.

[23] Bhaskaran, T.; Chen, Y.Q. and Xue, D., (2007) "Practical tuning of fractional order proportional and

integral controller (1): tuning rule development", Proceedings of the ASME International Design

Engineering Technical Conferences & Computers and Information in Engineering Conference,

IDETC/CIE, Las Vegas, Nevada, USA.

[24] Bouafoura, M.K. and Braiek, N.B., (2010) " PI Dλ µ controller design for integer and fractional plants

using piecewise orthogonal functions", Commun Nonlinear Sci Numer Simulat, Vol. 15, No. 5, pp

1267–1278.

[25] Chao, H.; Luo, Y.; Di, L. and Chen, Y.Q., (2010) "Roll-channel fractional order controller design for

a small fixed-wing unmanned aerial vehicle", Control Engineering Practice, Vol. 18, No. 7, pp 761–

772.

[26] Melicio, R.; Mendes, V.M.F. and Catalão, J.P.S., (2010) "Fractional-order control and simulation of

wind energy systems with PMSG/full-power converter topology", Energy Conversion and

Management, Vol. 51, No. 6, pp. 1250–1258.

[27] Luo, Y.; Wang, C.Y. and Chen, Y.Q., (2009) "Tuning fractional order proportional integral

controllers for fractional order systems", Control and Decision Conference, CCDC '09. Chinese, pp

307-312.

[28] Padula, F. and Visioli, A., (2011) "Tuning rules for optimal PID and fractional-order PID controllers",

Journal of Process Control, Vol. 21, No. 1, pp 69–81.

[29] Zamani, M.; Karimi-Ghartemani, M.; Sadati, N. and Parniani, M., (2009) "Design of a fractional

order PID controller for an AVR using particle swarm optimization", Control Engineering Practice,

Vol. 17, No. 12, pp 1380–1387.

Authors

Mohammad Reza Rahmani Mehdi Abadi was born in Yazd, Iran, in 1987. He received

his B.S. degree in electronics engineering from Yazd University in 2008, and the M.S.

degree in control engineering from Iran University of science and technology, in 2011. His

main area of interest includes robust control. Ali Akbar Jalali was born in Damghan, Iran, in 1954. He received his B.Sc. degree in

Electronics Engineering from Khajeh Nasiredin Toosi University of Technology, Tehran,

Iran, May 1985. He obtained his M.Sc. degree in Electrical Engineering from Oklahama

University, Norman, US, in 1988. Then, he earned his Ph.D. and Post Doctoral in

Electrical Engineering, from West Virginia University, Morgantown, US, in 1993 and

1994 respectively.Dr. Jalali became a member of the Lane Department of Computer

Science and Electrical Engineering, Collage of Engineering and Mineral Resources, West Virginia

University, as an adjunct Professor in 2002. Currently, he is working in the Department of Electrical

Engineering, Iran University of Science and Technology (IUST) where he has been since 1994 as an

associate professor. His research field interests include mainly Extended Kalman Filtering, Robust

Control, H-infinity and Fractional order control. Furthermore, study of Information Technology and its

applications like: Virtual Learning, Virtual Reality, Internet City, Rural ICT developments and Designing

ICT Strategic Plan are his other research interests.

Documents

FRACTIONAL ORDER PID BASED ON IMC · FRACTIONAL ORDER PID CONTROLLER TUNING BASED ON IMC ... In this work, a class of fractional order controller (FOPID) is tuned based on internal