CHAPTER 5 INTEGER ORDER SYSTEMS WITH FRACTIONAL …shodhganga.inflibnet.ac.in/bitstream/10603/16077/10/10_chapter5.pdf · Where q is the fractional order which can be a complex number,

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CHAPTER 5

INTEGER ORDER SYSTEMS WITH FRACTIONAL

ORDER CONTROLLERS

5.1 FRACTIONAL CALCULUS (FC)

Fractional calculus is three centuries old as is the conventional

calculus, but not very popular among science and/or engineering community.

The beauty of this subject is that fractional derivatives (and integrals) are not

a local (or point) property (or quantity). Thereby this considers the history

and non-local distributed effects. In other words, perhaps this subject

translates the reality of nature better. Therefore to make this subject available

as popular subject to science and engineering community, it adds another

dimension to understand or describe basic nature in a better way. Perhaps

fractional calculus is what nature understands, and to talk with nature in this

language is therefore efficient. For past three centuries, this subject was with

mathematicians, and only in last few years, this was pulled to several

(applied) fields of engineering and science and economics. However, recent

attempt is on to have the definition of fractional derivative as local operator

specifically to fractal science theory. Next decade will see several

applications based on this 300 years (old) new subject, which can be thought

of as superset of fractional differintegral calculus, the conventional integer

order calculus being a part of it. Differintegration is an operator doing

differentiation and sometimes integrations, in a general sense. Perhaps the

fractional calculus has become calculus of the twenty-first century.

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In a letter dated 30th September 1695, L’Hopital wrote to Leibniz

asking him a particular notation that he had used in his publication for the nth

derivative of a function. i.e., what would the result be if n = 1/2. Leibniz’s

response is “an apparent paradox from which one day useful consequences

will be drawn.” In these words, fractional calculus was born. Fractional

calculus does not mean the calculus of fractions, nor does it mean a fraction

of any calculus differentiation, integration, or calculus of variations. The

fractional calculus is a name of theory of integrations and derivatives of

arbitrary order, which unify and generalize the notion of integer order

differentiation and n-fold integration.

Several applications of fractional calculus can be found the area of

control systems. Fractional calculus allows the derivatives and integrals to be

of any real number. The fractional-order differentiator can be denoted by a

general fundamental operator qa tD as a generalization of the differential and

integral operators, which is defined as follows

qd ,R(q) 0qdtqD 1 ,R(q) 0a t

t q(d ) ,R(q) 0a

(5.1)

Where q is the fractional order which can be a complex number, the constant

‘a’ is related to the initial conditions. There are two commonly used

definitions for the general fractional differentiation and integration, i.e., the

Grünwald–Letnikov (GL) and the Riemann Liouville (RL) definitions

(Oldham 1974).

The GL definition is given below:

56

[(t q)/h] q1q jD f(t) lim ( 1) f(t jh)a qt h 0 jh j 0 (5.2)

Where (t q)h is an integer

While the RL definition is given by:

n t1 d f( )qD f(t) d (n 1) q na nt q n 1(n q) dt a (t ) (5.3)

Where n is an integer and q is real a number. (x) is the well known Euler’s

Gamma function. Also, there is another definition of fractional differ integral

introduced by (Caputo 1967).

Caputo’s definition can be written as:

(n)t1 f ( )qcD f(t) d (n 1) q na t q n 1(n q) a (t ) (5.4)

Fractional order differential equations are at least as stable as their

integer order counterparts. This is because systems with memory are typically

more stable than their memory-less alternatives.

The main reason for using the integer-order models was the

absence of solution methods for fractional differential equations. At present

time there are lots of methods for approximation of fractional derivative and

integral and fractional calculus can be easily used in wide areas of

applications (e.g.: control theory - new fractional controllers and system

models, electrical circuits theory - fractances, capacitor theory, etc.). For

closed loop control systems, there are four situations. They are 1) IO (integer

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order) plant with IO controller; 2) IO plant with FO (fractional order)

controller; 3) FO plant with IO controller and 4) FO plant with FO controller.

From control engineering point of view, doing something better is the major

concern. Existing evidences have confirmed that the best fractional order

controller can outperform the best integer order controller. It has also been

answered in the literature why to consider fractional order control even when

integer (high) order control works comparatively well. Fractional order PID

controller tuning has reached to a matured state of practical use. Since

(integer order) PID control dominates the industry, we believe FO-PID will

gain increasing impact and wide acceptance. Furthermore, we also believe

that based on some real time examples, fractional order control is ubiquitous

when the dynamic system is of distributed parameter nature.

5.2 FRACTIONAL ORDER CONTROLLERS (FOC)

The fractional order PID (FOPID) controller is the expansion of the

conventional PID controller based on fractional calculus. For many decades,

proportional - integral - derivative (PID) controllers have been very popular

in industries for process control applications. Their merit consists in

simplicity of design and good performance, such as low percentage overshoot

and small settling time (which is essential for slow industrial processes).

Owing to the paramount importance of PID controllers, continuous efforts are

being made to improve their quality and robustness. In the field of automatic

control, the fractional order controllers which are the generalization of

classical integer order controllers would lead to more precise and robust

control performances. Though it is reasonably true, that the fractional order

models require the fractional order controllers to achieve the best

performance, in most cases the fractional order controllers are applied to

regular linear or nonlinear dynamics to enhance the system control

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performances. Historically there are four major types of fractional order

controllers: (Xue and Chen, 2002)

CRONE Controller

Tilted Proportional and Integral (TID) Controller

Fractional Order PI D Controller

Fractional Lead-Lag Compensator

5.2.1 CRONE Controller

CRONE being the french acronym of Commande Robuste d Ordre

Non Entier means robust control of non integer order, represent the first

framework for non integer order systems application in the automatic control

area.

5.2.1.1 First generation CRONE Controller

The first generation CRONE controller is very suitable for gain-like

plant disturbance models and for constant plant phase around a frequency of

interest. Its transfer function is given by

C(s) = C0s (5.5)

where and C0 R.

The controller is defined within a frequency range ( b, h) around

the desired open-loop gain crossover frequency gc. The Oustaloup recursive

approximation can be used to implement this controller. However, any

approximating formula may be used as long as it allows us to obtain a rational

transfer function whose frequency response fits the frequency response of the

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original irrational-order transfer function in a desired frequency range ( b,

h). This type of controller is useful when the plant to be controlled already

has a constant phase, at least in a frequency range around the gain crossover

frequency (asymptotic plant frequency response within this band). In that

case, the loop will be robust to plant gain variations, since even though the

gain crossover frequency may change, the plant phase margin will not, and

neither will the controller phase.

5.2.1.2 Second generation CRONE Controller

For typical disturbed feedback control system its control

performance is fully characterized by the sensitivity function S(s), also known

as the transmittance in regulation, or the complementary sensitivity function

T(s), also known as the transmittances in tracking and we know that S (s) +

T(s)= 1. It is practically true that given the open loop behavior around the unit

gain frequency, one can determine the dynamic behavior in closed loop.

Therefore, we use the transmittance frequency template to define the desired

behavior of T(s) or S(s). The desired or ideal T(s) and S(s) are to set as

follows: In tracking, gain reaches a maximum for resonance frequency. The

resonance frequencies in tracking and in regulation are symmetrically

distributed with regard to the open loop unit gain frequency while the

resonance ratios in tracking and in regulation are identical.

Usually, descriptive specifications of the open loop behavior (for

the nominal plant) will be given such as

• the accuracy specifications at low frequencies

• the vertical template around unit gain frequency, u

• the input sensitivity specifications at high frequencies.

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For a stable minimum phase plant, it turns out that the behavior

thus defined can be described by a transmittance based on the frequency-

limited real non integer differentiator. In the particular case where transitional

frequencies b and h are sufficiently distant from frequency u, around this

frequency (i.e. b < h), selected frequency template (s) can be reduced

to transmittance (s) = ( u/s) . The order of relation, describes the

frequency truncation of the template defined by the transitional frequencies

b and h. This transmittance results from the substitution of the part raised at

power for the transmittance b/p which is used in the description of the

template between frequencies A and B. Finally, the controller C(s) in

cascade with the plant is synthesized from its frequency response according to

C(j ) = (j ) G0(j ) , where G0(j ) denotes the frequency response of the

nominal plant. There are a number of real life applications of CRONE

controller such as the car suspension control, flexible transmission, hydraulic

actuator etc. CRONE control has been evolved to a powerful nonconventional

control design tool with a dedicated MATLAB toolbox for it.

5.2.1.3 Third generation CRONE Controller

Third-generation Crone controllers can also be built so that the

open loop corresponds to different orders depending on the frequency. This

allows a more supple adjustment of the desired behavior, and presents

similarities with the well-established quantitative feedback theory (QFT)

controller design methodology. Third-generation Crone controllers ( just as

first- and second-generation ones) may be employed together with filters to

pre-compensate the plant, for example to attenuate resonant modes.

5.2.2 Tilted Proportional and Integral (TID) Controller

The object of TID is to provide an improved feedback loop

compensator having the advantages of the conventional PID compensator, but

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providing a response which is closer to the theoretically optimal response. In

TID scheme the proportional compensating unit is replaced with a

compensator having a transfer function characterized by 1/s1/n or s 1/n. This

compensator is herein referred to as a “Tilt” compensator, as it provides a

feedback gain as a function of frequency which is tilted or shaped with

respect to the gain/frequency of a conventional or positional compensation

unit. The entire compensator is herein referred to as a Tilt-Integral-Derivative

(TID) compensator. For the Tilt compensator, n is a nonzero real number,

preferably between 2 and 3. Thus, unlike the conventional PID controller,

wherein exponent coefficients of the transfer functions of the elements of the

compensator are either 0 or -1 or +1, TID scheme exploits an exponent

coefficient of 1/n. By replacing the conventional proportional compensator

with the tilt compensator of the invention, an overall response is achieved

which is closer to the theoretical optimal response determined by Bode.

5.2.3 Fractional Order PI D Controller

PI D controller, also known as PI D controller was studied in

time domain as well as in frequency domain . In general form, the transfer

function of PI D is given by

di

cs

s

11KC(s) (5.6)

where and are positive real numbers, Kc is the proportional gain, i is the

integration constant and d is the differentiation constant.

Clearly, taking = 1 and = 1, we obtain a classical PID

controller. If = 0 ( i = 0) we obtain a PD controller, etc. All these types of

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controllers are particular cases of the PI D controller. It can be expected that

PI D controller may enhance the systems control performance due to more

tuning knobs introduced. Actually, in theory, PI D itself is an infinite

dimensional linear filter due to the fractional order in differentiator or

integrator.

5.2.4 Fractional Lead-Lag Compensator

The transfer function of an FOLLC is given by

1x0,1s

1scKC(s) (5.6)

11sxcKC(s) s

(5.7)

Where is the fractional order of the controller, 1/ = zero is the

zero frequency, and 1/(x ) = pole is the pole frequency (when > 0). As can

be observed, this compensator corresponds to a fractional-order lead

compensator when > 0 and 0 < x < 1, and to a fractional-order lag

compensator when < 0 and 0 < x < 1. The condition 0 < x < 1 is maintained

in both cases. Assuming that the lead compensator behaves similarly to a

fractional-order PD controller and the lag compensator similarly to a

fractional-order PI controller, the first step for a latter generalization of these

structures to the fractional order PI D controller would be overcome.

In terms of readiness for real applications, CRONE method is the

best choice since it has a clear design interpretation with connections to the

familiar conventional controller design methods based on Bode plot and

Nichols chart. Compared to many well-proven PID parameter setting

techniques, development of setting or auto tuning techniques for the 5

parameters in PI D is strongly desired. Although TID can be regarded as a

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special type of PI D controller, it is observed that a systematic parameter

setting method has been proposed and tested. Therefore, TID should find its

wide applications in process control industry. Lead-lag compensator needs

more intuitive systematic design and parameter tuning method. However,

fractional order PI D controller is the most distinguished controller among

them. Many techniques have been proposed to tune the fractional order

controller. An elegant way of enhancing the performance of PID controllers is

to use fractional-order controllers (PI D ) where the integral and derivative

actions have, in general, non-integer orders. In a PI D controller, besides the

proportional, integral and derivative constants, denoted by Kc, i and drespectively, we have two more adjustable parameters: the powers of s in

integral and derivative actions, viz. and respectively. As such, this type of

controller has a wider scope of design, while retaining the advantages of

classical PID controllers. Finding the appropriate settings of the values of the

five parameters {Kc, i, d, , } to achieve optimal performance for a given

plant, as per user specifications, thus calls for real parameter optimization on

the five-dimensional space.

As shown in Figure 5.1, the FOPID controller generalizes the

conventional integer order PID controller and expands it from point to plane.

This expansion could provide much more flexibility in PID control design.

Point (0, 0) corresponds to P controller, point (0, 1) corresponds to PI

controller, point (1, 0) corresponds to PD controller and point (1, 1)

corresponds to PID controller where as the shaded portion between four

corners represent the FOPID controllers.

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Figure 5.1 Pictorial representation of FOPID controller

Figure 5.2 shows the schematic diagram of closed loop controlled

spherical tank experimental setup with FOPID controller.

Figure 5.2 Schematic diagram of FOPID controlled experimental setup

Storage tank

Rota meter

I/PConverterFOPID

DPT

R(s)

Pump Valve

Sphericaltank

ControlValve

_

Y(s)

65

5.3 RATIONAL FUNCTION OF PI D CONTROLLER

When fractional order controllers have to be implemented or

simulations using them are to be performed, fractional order transfer

functions are to be replaced by integer order transfer functions whose

behavior are close enough to the desired ones but much easier to handle.

Consider the transfer function of a PI D controller,

1C(s) K 1 sdc si (5.8)

The fractional order integrator 1/s and differentiator s can be

approximated by rational functions as follows:

5.3.1 Fractional Order Integrator

The problem of obtaining a continuous realizable model for a

fractional order controller can be viewed as a problem of obtaining a rational

approximation of the irrational transfer function, modeling the fractional

controller. Among other mathematical methods, two of them are particularly

interesting for this purpose, from a control theory point of view the continued

fraction expansion (CFE) method used for evaluation of functions, and the

rational approximation method used in interpolation of functions.

It is well known that the continued fraction expansions (CFE) is a

method for evaluation of functions, that frequently converges much more

rapidly than power series expansions, and converges in a much larger domain

in the complex plane. The result of such approximation for an irrational

function, G(s), can be expressed in the form:

66

b (s)1G(s) a (s)0 b (s)2a (s)1 b (s)3a (s)2 a (s)3 L

(5.9)

b (s)b (s) b (s) 31 2a (s)0 a (s) a (s) a (s)1 2 3L (5.10)

where a’s and b’s are rational functions of the variable s, or are constant.

On the other hand, for interpolation purposes, rational functions are

sometimes superior to polynomials. This is, roughly speaking, due to their

ability to model functions with poles. (As it can be seen later, branch points

can be considered as accumulations of interlaced poles and zeros). These

techniques are based on the approximations of an irrational function, G(s),

by a rational function defined by the quotient of two polynomials in the

variable s:

P (s)G(s) Ri(i 1)(i 2) (i m) Q (s)L

(5.11)

P (s) P (s) P (s)0 1Q (s) Q (s) Q (s)0 1

L

L(5.12)

General CFE method for approximation of fractional integral

differential operators is

1G (s)h (1 sT)(5.13)

67

1G (s)l 1 s(5.14)

where Gh(s) is the approximation for high frequencies ( T >> 1), and Gl(s)

the approximation for low frequencies ( T

68

N 1 s1zK i 0 iIC (s) KI I N ss 11 pi 0 ic

(5.17)

Where

maxlogp0N 1 Integer

log ab (5.18)

K 1/ ( )I c (5.19)

[y/10(1 )]a 10 (5.20)

(y/10 )b 10 (5.21)

The poles pi’s and the zeros zi’s are given as

z ap0 0 ,iz z (ab)0i ,

ip p (ab)0i ,(y/20 )p 100 C

where y is error in dB.

5.3.2 Fractional Order Differentiator

The transfer function of the differentiation action of the fractional

PI D controller is a fractional order differentiator which is represented in the

frequency domain by the following irrational function:

69

C (s) sD (5.22)

With is a positive real number such that 0<

70

The poles pd’s and the zeros zd’s are given as

( /20 )0 10

ycz , 0 ( )

iip p ab , 0 ( )

iiz z ab , 0 0p az

y is error in dB

5.3.3 Fractional PI D Controller

Thus we showed how we can approximate the fractional order

integrator and differentiator by rational functions, for a given frequency band

of practical interest [ L, H] ; so Equation 5.5 becomes:

N 1 N 1I Ds sK 1 T K 1I D Dz zi 0 i 0i iC(s) Kp 1 N NI Ds sT 1 1I p pi 0 i 0i i

(5.29)

The poles pi’s, the zeros zi’s, the parameters KI and NI of the

rational function approximation of the fractional order integrator and the

zeros zDi’s, the poles pDi’s, the parameters KD and ND of the rational function

approximation of the fractional order differentiator can be easily found from

Equation 5.8 and Equation 5.12 respectively.

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5.4 TUNING OF FRACTIONAL ORDER PI CONTROLLER

Figure 5.3 Fractional order PI controller

The tuning parameters of fractional order PI controllers are Kc, iand . The transfer function of FOPI controller is

1( ) 1ci

C s Ks

(5.30)

Our tuning strategy, in the first place, is based on minimization of

ISE with minimum peak overshoot based PSO technique for selecting the

parameters Kc and i of the PI controller when =1 which means setting the

parameters of a simple conventional PI controllers. The fractional order PI

controller is designed around the gain crossover frequency of the open loop

transfer function of FOPDT model with integer order PI controller. With the

parameters Kc and i obtained in the first step, we use the minimization of ISE

technique to determine the optimum setting of the fractional integration order

of the PI controller. The ISE index J is given as:

2 2J [e(t)] dt [r(t) y(t)] dt0 0

(5.31)

Where e(t)=[r(t)-y(t)] is the error signal.

72

From Figure 4.1 the error signal E(s) is given as

R(s)E(s)1 C(s)G (S)p

(5.32)

1R(s)s

(5.33)

1 1E(s)s 1 C(s)G (S)p

(5.34)

5.4.1 Simulation Results

The FOPDT model developed about the operating point of 5.75 cm

is considered.

11.990.699( )33.247 1

s

peG s

s(5.35)

The is set as 1 and using PSO technique the parameters Kc and iare found. The resultant integer PI controller transfer function is

1C(s) 3.7588 121.891s

(5.36)

The open loop transfer function is

G(s) = C(s)Gp(s) (5.37)

For approximating the fractional power we need the gain cross over

frequency of the open loop transfer function G(s). was equated to 1 and

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hence the fractional controller would be a conventional PI controller. The

bode plot was drawn for the OLTF G(s).

At the gain cross over frequency the magnitude should be equal to

unity or 0 db.

20log G j 20log C j G j 0dbu u up (5.38)

From the Bode plot of the open-loop transfer function, the

crossover frequency was found to be u= 0.0846rad/s. Once Kc and i are set,

the fractional PI controller’s transfer function C(s) becomes

1C(s) 3.7588 121.891s

(5.39)

Bode Diagram

Frequency (rad/sec)

-50

0

50

10-3 10-2 10-1 100 1010

180

360

System: TPhase Margin (deg): 27.4Delay Margin (sec): 5.65At frequency (rad/sec): 0.0846Closed Loop Stable? Yes

Figure 5.4 Bode plot of open loop transfer function G(s) (5.75cm)

To set the parameter , the time delay e-11.99s of the plant’s transfer

function Gp(s) and the above fractional PI controller C(s) must be rational

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functions. Thus the time delay is approximated to a rational function using

Padé approximation and C(s) is approximated to a rational function using the

method proposed in section 5.3.2.

( )( )

( )p

pp

N sG s

D s(5.40)

( )( )( )

c

c

N sC sD s

(5.41)

Because the plant’s transfer function Gp(s) is without an integrator,

the fractional order integrator (1/s ) has to be implemented as (1/s ) = 1(s(1- )/s

to ensure the convergence of the Hall-Sartorius algorithm used to set the

parameter . Hence, the fractional PI controller’s transfer function C(s) will

be:

1sC(s) 3.7588 121.891s

(5.42)

In this case the fractional order integrator of C(s) is approximated

in the frequency band [ L, H]= [0.1 u, 10 u]= [0.00846 rad/sec, 0.846

rad/sec] with an approximation error y = 0.75dB and the frequency max =

100 H = 84.6rad/sec. (Chareff 2009)

The setting of the fractional integration action order of fractional

PI controller consists of finding this parameter that minimize the ISE index

J( ) of equation. For the minimization task, the ISE index J( ) values are

calculated when the parameters is varied from 0 to 1 each with a step of

0.1. Then, from the results obtained we can easily calculate the minimum ISE

index J( ) and its corresponding optimum setting of the fractional integration

75

action order and the fractional differentiation action order µ of the fractional

PI controller.

From the simulation results, the smallest ISE index J( ) obtained

corresponds to the value of = 0.9. Then the fractional PI controller’s

transfer function C(s) required is given as:

1C(s) 3.7588 1 0.921.891s (5.43)

0.1sC(s) 3.7588 121.891s

(5.44)

The rational function approximation of the irrational fractional

PI0.90 controller is

5 s1 ii 0 0.00393 6.812C(s) 3.7588 1 0.0287

6 s1 ii 0 0.00322 6.812

(5.45)

which has been obtained by substituting the values of u, max, a,b,N,po,Kp,

Ki, and y. Figure 5.4 shows the unit step response of FOPDT model

developed about the operating point of 5.75 cm with FOPI (tuned by

minimization of ISE based PSO) and IOPID (tuned by model based ZN

formula) controllers.

76

0 50 100 150 200 250 300 350 400 450 500-0.5

0

0.5

1

1.5

2

t(sec)

IOPIDFOPIr(t)

Figure 5.5 Unit step response of FOPDT system with FOPI and IOPID

controllers (5.75cm)

In the same way FOPI controllers are tuned for all other FOPDT

models developed about the remaining operating points around the gain cross

over frequencies of open loop transfer functions of FOPDT models with

integer order PI controllers and are listed in Table 5.1.

77

Table 5.1 Tuning parameters of FOPI controller and gain crossover

frequencies

Level(Cm)

TransferFunction u

Kc i

05.750133.247s

11.99s0.6995e 0.0846 3.7588 21.891 0.9

13.5001292.613s

11.02s1.97e 0.0307 3.8271 49.002 0.7

21.7501769.39s

10.03s3.031e 0.0318 7.7369 20.060 0.6

27.25011252.95s

8.98s-3.843e 0.0329 9.1817 49.982 0.4

32.00011709.465s

9.04s-4.3573e 0.0417 15.452 69.539 0.4

38.25012374.23s

8.95s-5.045e 0.0621 27.738 48.740 0.2

46.87513766.189s

7.97s-6.393e 0.0598 30.7706 30.067 0.2

The unit step closed loop responses of FOPDT model developed

about other operating points with FOPI and IOPID controllers are shown in

Figures 5.6 to 5.11.

78

0 100 200 300 400 500 600 700 800 900 1000-0.5

0

0.5

1

1.5

2

2.5

t(sec)

IOPIDFOPIDr(t)



0 100 200 300 400 500 600 700 800 900 1000-0.5

0

0.5

1

1.5

2

2.5

t(sec)

IOPIDFOPIr(t)



79

0 100 200 300 400 500 600 700 800 900 1000-0.5

0

0.5

1

1.5

2

2.5

t(sec)

IOPIDFOPIr(t)



0 100 200 300 400 500 600 700 800 900 1000-0.5

0

0.5

1

1.5

2

2.5

t(sec)

IOPIDFOPIr(t)


controllers (32cm)

80

0 100 200 300 400 500 600 700 800 900 1000-0.5

0

0.5

1

1.5

2

2.5

t(sec)

IOPIDFOPIr(t)



0 100 200 300 400 500 600 700 800 900-0.5

0

0.5

1

1.5

2

2.5

t(sec)

IOPIDFOPIr(t)


controllers (46.875 cm)

81

The comparison of time domain specifications like maximum peak

over shoot (Mp(in %)) and settling time, performance criteria measures like

ISE, IAE and ITAE of FOPDT models with FOPI and IOPID controllers are

summarized in Table 5.2.

Table 5.2 Performance criteria of FOPDT models with FOPI and

IOPID controllers

Level(Cm)

TransferFunction

Controller Mp(%) ts(sec) ISE IAE ITAE

05.750133.247s

11.99s0.6995e FOPI 1.325 104 19.52 33.86 01990

IOPID 1.740 149 23.29 40.64 02548

13.5001292.613s

11.02s1.97e FOPI 05.50 099 25.70 45.56 05967

IOPID 225.0 318 57.16 102.3 16540

21.7501769.39s

10.03s3.031e FOPI 0.500 095 22.85 41.77 06096

IOPID 237.0 402 61.14 108.9 18160

27.250 11252.95s

8.98s-3.843e FOPI 00.00 089 23.10 40.66 03873

IOPID 239.0 341 57.69 101.2 13920

32.00011709.465s

9.04s-4.3573e FOPI 04.50 056 18.68 30.43 03930

IOPID 233.0 377 58.71 104.6 18740

38.25012374.23s

8.95s-5.045e FOPI 11.01 081 15.59 23.87 02529

IOPID 235.0 358 58.96 105.0 17160

46.87513766.189s

7.97s-6.393e FOPI 09.90 072 15.00 25.41 03216

IOPID 232.0 313 53.37 95.07 14770

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5.4.2 Test for Robustness

The main advantage of fractional order PI controllers is their

robustness to the changes in either the process parameters or controller

parameters. This is checked in simulation and the closed loop responses for

small changes in the process gain are shown in Figure 5.12.

The process model at the operating point of 5.75cm is considered.

Figure 5.12 Robustness checking of FOPI controller

5.4.3 Experimental Results

The tuning parameters of FOPI and IOPI controllers are used in

closed loop study of experimental setup of spherical tank at the operating

point of 5.75cm. The servo and regulatory responses of IOPI and FOPI

controllers are shown in Figure 5.13 and Figure 5.14 respectively.

85

Table 5.3 Comparison of performance criteria of FOPI and IOPID

controllers

Controller Mp(%) ISE IAE ITAE tS (sec)

FOPI 0.52 127 546 973 74

IOPI 21.2 2342 784 1347 142

5.5 CONCLUSIONS

From all the above simulation results it is observed that even

though both FOPI and IOPID controllers involve three tuning parameters the

performance of FOPI controller is far better than IOPID controller. It is

proved at all operating points qualitatively as well as quantitatively. In

addition the robustness of FOPI controller is proved.

From the experimental results it is observed that the FOPI

controller is very fast and it results almost zero percent overshoot.

5.6 TUNING OF FRACTIONAL ORDER PID CONTROLLER

The tuning parameters of fractional order PID (PI D ) controllers

are Kc, i, d, and µ. Model based Ziegler-Nichols tuning rule is applied for

tuning the parameters Kc, i and d. Having obtained the parameters Kc, i and

d in the first step, the optimum settings of the fractional integration order

and the fractional differentiation order µ of the PI D controller are

determined by using minimization of ISE technique. For implementation of

PI D controller, the rational function approximation of PI D controller C(s)

86

is made in the frequency band [ L, H]=[0.1 u, 10 u], where u is the unity

gain crossover frequency of the open-loop transfer function C(s)Gp(s) when

C(s) is a classical PID controller. For the minimization task, the ISE index

J( ,µ) values are calculated when the parameters and µ are varied from 0 to

1 each with a step of 0.1. Then, from the results obtained we can easily

calculate the minimum ISE index J( ,µ) and its corresponding optimum

setting of the fractional integration action order and the fractional

differentiation action order µ of the PI D controller.

The transfer function developed about the operating point of 27.25

cm is considered as an example.

8.983.843( )

1252.95 1

seG sp

s (5.46)

First the parameters and µ of PI D controller are set to 1

=µ=1). Then using the model based Ziegler- Nichols tuning method, the

parameters Kc, i and d are found to be Kc= 43.568, i = 17.96 and d =4.49.

Figure 5.15 shows the Bode plot of the open loop transfer function G(s) =

C(s)Gp(s). Here C(s) is the integer order PID controller. From the Bode plot

of the open-loop transfer function, the crossover frequency is found to be u=

0.136rad/s.

87

Bode Diagram

Frequency (rad/sec)

-200

-100

0

100

200

10-5

10-4

10-3

10-2

10-1

100

101

180

270System: TPhase Margin (deg): 38.9Delay Margin (sec): 4.98At frequency (rad/sec): 0.136Closed Loop Stable? Yes

Figure 5.15 Bode plot of open loop transfer function G(s) = C(s)Gp(s)

Once Kc, i and d are set, the fractional PI Dµ controller’s transfer

function C(s) becomes

1C(s) 43.568 1 4.49s17.96s

(5.47)

Because the plant’s transfer function Gp(s) is without an integrator, the

fractional order integrator (1/s ) has to be implemented as (1/s ) = (1 s(1- )/s) to

ensure the convergence of the Hall-Sartorius algorithm used to set the

parameters and µ. Hence, the fractional PI Dµ controller’s transfer function

C(s) will be:

sC(s) 43.568 1 4.49s17.96s

(5.48)

88

The fractional order integrator and differentiator of C(s) are

approximated in the frequency band [ L, H]= [0.1 u, 10 u]= [0.0136rad/s,

1.36rad/s] with an approximation error y=0.75dB and the frequency max=100

H =136.00rad/s.

From the simulation results, the smallest ISE index J( ,µ) obtained

corresponds to the couple ( ,µ)=(0.10,0.90). Then the fractional PI D

controller transfer function C(s) is given as:

0.9s 0.9C(s) 43.568 1 4.49s17.96s

(5.49)

The transfer function of PI D controller with expansion of

fractional power is

5 5s s1 0.011 1i ii 0 i 00.00393 6.812 0.00149 6.8120.0287C(s) 43.568 1s 6 6s s1 1i ii 0 i 00.00322 6.812 0.00841 6.812

Similarly FOPID controllers are designed for all FOPDT models

developed at different operating points of the spherical tank and the tuning

parameters of both IOPID and FOPID controllers are listed in Table 5.4.

89

Table 5.4 Tuning parameters of IOPID and FOPID controllers

Level(Cm)

TransferFunction

Controller Kc i d

05.750133.247s

11.99s0.6995eIOPID 04.76000 23.98 5.995 1.0 1.0

FOPID 04.76000 23.98 5.995 0.9 0.9

13.500 1292.613s

11.02s1.97e IOPID 16.17435 22.04 5.510 1.0 1.0

FOPID 16.17435 22.04 5.510 0.5 0.9

21.750 1769.39s

10.03s3.031e IOPID 30.37065 20.06 5.015 1.0 1.0

FOPID 30.37065 20.06 5.015 0.1 0.9

27.250 11252.95s

8.98s-3.843e IOPID 43.56806 17.96 4.490 1.0 1.0

FOPID 43.56806 17.96 4.490 0.1 0.9

32.000 11709.465s

9.04s-4.3573e IOPID 52.08173 18.08 4.520 1.0 1.0

FOPID 52.08173 18.08 4.520 0.1 0.9

38.250 12374.23s

8.95s-5.045e IOPID 63.09860 17.90 4.475 1.0 1.0

FOPID 63.09860 17.90 4.475 0.1 0.9

46.875 13766.189s

7.97s-6.393e IOPID 88.69900 15.94 3.985 1.0 1.0

FOPID 88.69900 15.94 3.985 0.1 0.9

The unit step closed loop responses of FOPDT models with

corresponding tuned IOPID and FOPID controllers are shown in figures from

5.16 to 5.22.

90

0 50 100 150 200 250 300-1.5

-1

-0.5

0

0.5

1

1.5

2

t(sec)

r(t)IOPIDFOPID

Figure 5.16 Unit step response of FOPDT system with FOPID and

IOPID controllers (5.75cm)

0 50 100 150 200 250 300 350 400 450 500-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

t(sec)

IOPIDFOPIDr(t)



91

0 50 100 150 200 250 300 350 400 450 500-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

t(sec)

IOPIDFOPIDr(t)



0 100 200 300 400 500 600 700 800 900 1000-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

t(sec)

IOPIDFOPIDr(t)



92

0 100 200 300 400 500 600 700-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

t(sec)

IOPIDFOPIDr(t)


IOPID controllers (32cm)

0 50 100 150 200 250 300 350 400 450 500-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

t(sec)

IOPIDFOPIDr(t)



93

0 100 200 300 400 500 600 700-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

t(sec)

r(t))IOPIDFOPID



The Bode plots of FOPDT models with both tuned IOPID and

FOPID controllers are shown in figures from 5.23 to 5.29.

Bode Diagram

Frequency (rad/sec)

-40

-20

0

20

40

60

80

10-4

10-2

100

102

104

90

135

180

225

270

315

C(s)=FOPIDC(s)=IOPID


(5.75cm)

94

-100

0

100

200Bode Diagram

Frequency (rad/sec)10

-610

-410

-210

010

210

40

180

360



(13.5 cm)

-50

0

50

100

150Bode Diagram


-610

-410

-210

010

210

490

180

270

360



(21.75cm)

95

10-6

10-4

10-2

100

102

104

90

180

270

360

Bode Diagram

Frequency (rad/sec)

-50

0

50

100

150

200

C(s)=FOPIDTC(s)=IOPID


(27.25cm)

Bode Diagram

Frequency (rad/sec)

-50

0

50

100

150

10-6 10-4 10-2 100 102 10490

180

270

360



(32cm)

96

-50

0

50

100

150Bode Diagram


-510

010

590

180

270

360



(38.25cm)

Bode Diagram


-510

010

590

135

180

225

270

315-50

0

50

100

150



(46.875cm)

97

The time domain specifications like maximum peak overshoot and

settling time (for 2% tolerance band), frequency domain specifications like

gain cross over frequency, phase margin and gain margin, performance

criteria measures like ISE, IAE and ITAE of unit step closed loop response of

FOPDT systems with IOPID and FOPID controllers are compared in

Table 5.5.

Table 5.5 Quantitative Comparison of FOPDT systems with IOPID and

FOPID controllers

Level(Cm)

Controller Mp(%)

ts for2% of

TB(sec)

ISE IAE ITAEgc

(rad/sec)

PM(deg)

GM(dB)

05.750IOPID 61.70 94.8 22.19 36.11 1702 0.095 55.7 4.55

FOPID 0.000 38.5 13.69 13.98 509.8 0.129 56.1 4.02

13.500IOPID 104.7 193 36.97 64.53 6063 0.111 40.3 4.44

FOPID 4.000 19.0 13.45 13.52 1423 0.178 44.5 3.79

21.750IOPID 125.0 248 46.92 77.68 7971 0.122 39.2 4.44

FOPID 3.500 11.5 12.23 13.13 1366 0.195 43.6 3.84

27.250IOPID 115.0 308 40.89 81.24 13420 0.136 38.9 4.44

FOPID 5.000 12.0 10.96 12.19 2390 0.216 43.2 3.92

32.000IOPID 126.3 245 45.05 76.34 9151 0.135 38.8 4.44

FOPID 5.000 12.0 11.04 10.41 1152 0.215 43.1 3.91

38.250IOPID 113.3 177 33.07 55.03 4935 0.137 38.7 4.44

FOPID 5.500 12.0 10.93 9.377 619.4 0.217 43.1 3.92

46.875IOPID 128.8 216 39.75 66.7 7467 0.154 38.7 4.44

FOPID 6.900 11.5 9.73 8.179 647 0.242 42.7 4.0

98

From the above comparison, it is clearly observed that addition of

two more tuning parameters improves the performance of closed loop control

system in all aspects.

5.6.1 Test for Robustness

The main advantage of fractional order PID controllers is their

robustness to the changes in either the process parameters or controller

parameters. This is checked in simulation and the closed loop responses for

small changes in the controller gain are shown in Figure 5.30.

0 50 100 150 200-1.5

-1

-0.5

0

0.5

1

Time(sec)

Response for Loop Gain Variations

K = 0.9K = 1K = 0.8

Figure 5.30 Robustness test of FOPID controller at the operating point

of 12 cm

5.6.2 Experimental results

For experimental validation, the tuning parameters of FOPID and

IOPI controllers were used to study the closed loop behavior of the control

system for two different set points 12 cm and 27.25 cm. The servo response

99

of FOPI controller for the set point of 12 cm is shown in Figure 5.31. The

corresponding error graph is shown in Figure 5.32. The servo and regulatory

responses of FOPID and IOPI control systems are shown in Figures 5.33 and

5.34 respectively. The performance specifications of the FOPID and IOPI

control systems are compared and listed in Table 5.6. The servo and

regulatory responses of FOPID and IOPI control systems for the set point of

27.25 cm are shown in Figures 5.35 and 5.36 respectively. Table 5.7 lists the

performance specifications of the FOPID and IOPI control systems.

0 100 200 300 400 500 600 700 8000

2

4

6

8

10

12

14

16

18

20

t(sec)

r(t)c(t)

Figure 5.31 Servo response of FOPID controller when the set point is 12cm

100

0 100 200 300 400 500 600 700 800-4

-2

0

2

4

6

8

10

12

t(sec)

error

Figure 5.32 Error graph when the set point is 12 cm

0 200 400 600 800 1000 12000

2

4

6

8

10

12

14

16

18

20r(t)y(:,2)

d(t)

Figure 5.33 Servo and regulatory response of FOPID controller when

the set point is 12 cm

101

0 500 1000 1500 2000 2500 30000

2

4

6

8

10

12

14

16

18

20

t(sec)

r(t)c(t)

d(t)

Figure 5.34 Servo and regulatory response of IOPI controller when the

set point is 12 cm

Table 5.6 Quantitative comparison of FOPID and IOPI control systems

when the set point is 12 cm

Controller

Rise

time

(sec)

Peak

time (sec)

Peak

overshoot

(%)

Settling

time(sec)ISE

PI 142 278 33.5 948 5249

PI0.09D0.90 134.5 205 41 372 1982

102

0 100 200 300 400 500 6000

10

20

30

40

50

60

t(sec)

d(t)r(t)c(t)

Figure 5.35 Servo and regulatory response of FOPID control system

when the set point is 27.25 cm

0 200 400 600 800 1000 1200 14000

10

20

30

40

t(sec)

r(t)c(t)d(t)

Figure 5.36 Servo and regulatory response of IOPI control system when

the set point is 27.25 cm

103

Table 5.7 Quantitative comparison of FOPID and IOPI control systems

when the set point is 27.25 cm

Controller

Rise

time

(sec)

Peak

time (sec)

Peak

overshoot

(%)

Settling

time(sec)ISE

PI 72.2 132 37.77 478 4146

PI0.10D0.90 61.5 115 39.44 181 2617

From the obtained experimental results it was observed that

FOPDT systems with FOPID controllers were performing well in some

aspects like rise time and peak over shoot; performing very well in some

other aspects like settling time and ISE. The difference between simulation

and real time results motivated us to improve the quality of control system

either by including some modifications in the structure of control system or

by modeling the system in a more effective way.

Vijayan and Panda (2012) proposed a simple set point filter which

requires only peak overshoot and peak time of the system response regardless

of type and order of the system with arbitrary PID parameters to reduce the

peak overshoot to a desired/tolerable limit.

A double-feedback loop/method is also proposed (Vijayan and

Panda 2012) to achieve stability and better performance of the low order

stable and unstable processes.

Double-feedback closed loop structure with set point filter and

fractional order modeling were attempted and are explained in chapters 6 and

7 respectively.

Documents

CHAPTER 5 INTEGER ORDER SYSTEMS WITH FRACTIONAL …shodhganga.inflibnet.ac.in/bitstream/10603/16077/10/10_chapter5.pdf · Where q is the fractional order which can be a complex number,