fractional order pid controller tuning based on imc

7/30/2019 Fractional Order PID Controller Tuning Based on IMC

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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 ORDERPID CONTROLLER TUNING

BASED ON IMC

Mohammad Reza Rahmani Mehdi Abadi1

and Ali Akbar Jalali2

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

[email protected]

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

Tehran, [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 delayelement. 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 timedelay [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 resonantcharacteristics [3, 4]. Although many processes have open-loop stable behavior, in someengineering fields (such as exothermic chemical processes, batch chemical reactors, biological

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


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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 stableprocesses 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 theprocess, 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 equivalentcontroller [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 tofractional 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 fractionalorder control [20]. Such controllers are the extended version of conventional integer ordercontrollers 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 offreedom, fractional order models can approximate processes by fewer parameters. Podlubny has

shown that fractional order PID controllers denoted by PI D , have a better response incomparison 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 ofFractional 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 offractional-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 studybetween the proposed FOPID and conventional PID has been made to show that FOPID hasbetter 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 timepart, 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 atypical 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 andfractional order systems.


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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 PC s G s F s

= %(1)

Where ( )p

G s% expresses minimum phase or invertible part of system ( )p

G 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)


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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 integerorder and they span a wider range of differential equations. Fractional calculus deals withfractional integration and differentiation. Therefore, a generalized differential and integral

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

where a and t

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

for negative , it denotes integral action as

( ) 0

1 ( ) 0

( ) 0( )

a t

t

a

d

Realdt

D Real

Reald

>

= =

, eq.(27) reformed to

1 0.1949490474 1 10.1

2 5 2 0.1263590714 2 0.125p

p p p p

+ =

(29)

In above equation, it is seen that with 5p

t = , arctan( )2

p pt

, ensuring that phase equation

will not be altered by variable pt . From above relation, =15.76574p 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

+ + = + + +

(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 t (31)

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

15.74512p = 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

+ + = + +

(32)

For selected 0.45333p

k = , 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 orderis a more robust than integer order for this typical system.


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0.20000, k 0.10179, m= 0.93774i d

k = = , and then for such a tuning 0 0.91239 = and

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

0.18943, k 7.68554, m=-0.48739e-1i d

k = = , result in 1.65716l = at 0 .912393 = . 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

output

0 10 20 30

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14load distorbance with value=.1 response

t

output

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 0.1in 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, 0.5pt l= = are defined.

From eq.(22) 0.4 = and low pass filter 2( ) ( 1) / ( 1)F s s s = + + 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 from below eq.(21). For this system with

( ) exp( 0.5 )P

G s s+

= , 1.6918= has been calculated.

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

k ,

to find uncertainty interval, gain crossover frequency 0 1 = 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 = . 2.00265pk = gives 0.91079e-1, k 0.44968i dk = =

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


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

results in 0.6135l = at same 0 . 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

out

put

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

0.06load distorbance with value=.1 response

t

out

put

desired response

integer order

fractional with kp=0.11566



integer order





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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 morerobust 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 betterthan conventional PID. Moreover, owning to its more degree of freedom, it is possible to addrobust conditions in fractional order controllers tunings. Latter can be future work.

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fractional order pid controller tuning based on imc

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