Analysis of Fractional Order Control System with Performance and Stability

Analysis of Fractional Order Control

System with Performance and Stability

Parvendra Kumar*

Electrical Engineering Department, SunRise University, Alwar Rajasthan, India

[email protected]

+919205127282

Sunil Kumar Chaudhary

Electrical Engineering Department, SunRise University, Alwar Rajasthan, India

[email protected]

+919313426454

Abstract

In this paper, a new approach to stability for fractional order control system is proposed. Here a dynamic system whose behavior can be modeled by means of differential equations involving fractional derivatives. Applying Laplace transforms to such equations, and assuming zero initial conditions, causes transfer functions with no integer powers of the Laplace transform variable s to appear. In recent time, the application of fractional derivatives has become quite apparent in modeling mechanical and electrical properties of real materials. Fractional integrals and derivatives have originated wide application in the control systems. The measured system and the controller are termed by a set of fractional order differential equations.

Fractional calculus is the most useful tools for the description of hereditary properties and memory of various materials and processes. Fractional derivatives have better flexibility as the comparison to classical integer order models, in which system dynamics not taken into account. The rewards of fractional derivatives become more appealing in the modeling of electrical, mechanical and electro-mechanical properties of real resources, as well as in many other fields. Current times have a wide application in the field of fractional integrals and derivatives. In the concept of control systems, the controlled system and the controller are described by a set of fractional differential equations. Here the stability of fractional order system is checked at the different level and it is found that the stability region is large in the complex plane. This large stability region provides the more flexibility for system implementation in the control engineering.

Keywords: Fractional Order System; Fractional Order Calculus; Stability; Performance Analysis; MATLAB; Function Under Class.

1. Introduction

Many research and study using the notion of fractional-order may be a reliable step because a real process in the industry is generally fractional. However for many real processes fractional is very low. A typical example of fractional (non-integer) order system is, relationship of voltage-current in semi-infinite resistors with losses and capacitor (RC) line or the diffusion of heat in a semi-infinite solid, where the heat flow q(t) is naturally equal to the semi-derivative of temperature T(t) [1], as describe by the fractional order differential equation (FODE) which is given as

) ( ) ( 5 . 0 5 . 0

t q dt

t T d

= (1)

Clearly, an ordinary differential equation (ODE) for integer order describes the above system and it may differ significantly from the concrete situation.

controlled can also be modeled as a dynamic system describes by fractional order differential equations (FODE). A control system can contain both the fractional-order dynamic system or plant to be measured and the order controller. In the field of control engineering, it is a common practice to consider the fractional-order controller for analysis the system. The plant model may have already been obtained as an integer-fractional-order model in a usual sense. Generally, the objective for application of fractional-order control (FOC) is to enhance the system control performance. The key objective of this paper is to examine the stability and performance of the fractional order control system by clarifying some design examples. Some MATLAB function files are used in this paper to simulate the fraction order dynamic system [10]. The rest of the article is planned as follows: In section 2, a brief overview of fractional calculus and fractional order structure has been presented. Section 3 presents the stability analysis of fraction order systems. Section 4 present the analysis of three illustrative examples. Section 5 concludes this paper.

2. Fractional Order System Fundamentals

2.1 The introduction to fractional calculus.

The term “fractional-order calculus” is by no means new. It is a generalization of ordinary differentiation by non-integer derivatives. The theory of fractional-order derivatives was developed mainly in the 19th century [5, 7, 11 and 12]. In the development of fractional order calculus, there appeared different definitions of fractional-order differentiation and integration. To reduce to a general form fractional calculus from integration and differentiation to the fractional order fundamental operatorαD_tβf(t) , where α and t are the limit andβ∈R is

the directive of the task. The continuous integration differential operator is [10]

  

   

 

< =

> =



0 1

0 )

(

β τ

β β α

α β β

β

 



t t

d dt d

t f

D (2)

There are various definitions for fractional integration and differentiation. Some of the definitions spread out directly as of integer-order calculus. The deep-rooted descriptions include the Cauchy integral formula, the Grunwald–Letnikov (GL) definition and Riemann–Liouville (RL) definitions are given [10] as

Definition 1: - Cauchy integral formula



− Γ =

c

d t f

j t

f

D τ

τ τ π

γ

γ γ

1 ) (

) ( 2

) 1 ( )

( (3)

Where c is the smooth curve encircling the single value function f (t) Definition 2: - Grunwald–Letnikov (GL) definition

) ( ) 1 ( 0

) (

0

jh t f j h

h Lim t

f D

h t

j

j

t  −

    

− →

=



   −

= −

α

β

β β

α (4)

The following function given below is obtained by Laplace Transform of the GL and RL fractional differential-integral. The zero initial conditions and order β gives the following result

) ( ]

); (

[αDt±βf t s =s±βF s

 ₍₆₎

2.2 Fractional order system

The fractional-order system is the extension form of the traditional integer order systems. Fractional order system is gained from the fractional-order differential equations. A classic n-term linear fractional order differential equation (FODE) is assumed by

0 ) ( ) ( ... )

(t + + 1D _t1 yt + 0D _t0 yt =

y D n t n β α β α

α β (7)

Let considering the control function on which input signalu(t) is applied to FODE system (7) as follows: ) ( ) ( ) ( ... )

(t 1D _t1 yt 0D _t0 yt u t

y

D n

t

n + + + =

β α β

α

α β (8)

After Laplace transform of equation (8), we get ) ( ) ( ) ( ... ) ( 0 0 1

1s _t Y t s _t Y t U t

t Y s n

t

n + + + =

β α β

α

α β (9)

From Eq. (9), we can obtain a fractional-order transfer function as

n s s s s U s Y s G n β β β _{α α} α + + + = = .. 1 ) ( ) ( ) ( 1 0 1 0 (10)

In broad, for a dynamic system with single variable and fractional order transfer function of a system can be defined as n m s a s a s a s b s b s b s G n y m β β β γ γ + + + + + + = ... ... ) ( 1 0 1 0 1 0 1

0 ₍₁₁₎

Hereb_i(i=01,...m),a_i(i=01,...n)are constant and γ_i(i=01,...m),βi(i=01,...n)are random real or rational

number and without lacking generality, can be prescribed as γm>γm−1 >...γ0 andβm>βm₋1 >...β0.

The incommensurable fractional order system Eq. (11) can also be spoken incommensurable method by the multi-valued transfer function

). 1 ( , ... ... ) ( ₁ 1 0 1 1 0 _> + + + + + + = v s a s a s a s b s b s b s H v n n v v m m v (12)

Note that all fractional order system may be represented in the form of Eq. (12) and domain of H(s) explanation is a Riemann sheets.

3. Stability of Fractional Order System

Fig.1. Stable and unstable region of LTI fractional order system

Theorem 3.1. According to Matignon’s stability theorem the fractional order transfer function

) (

) ( ) (

s D

s N s

G = is

stable if and only if

2 )

arg(σi =qπ , where

q

s =

σ , (0<q<2) with∀σ_i∈C, ith root of D(σ)=0.

Ifs=0is a single root of D(s), the system cannot be stable.

Above theorem stability region is shown in Fig. (1), Indicate the wholes s plane whereq=0. It shows the Routh-Hurwitz stability when q=1and tends to negative real axis forq=2.

As we know that only the poles play an important role in the stability of a system. So the stability assessment is done by denominator only and numerator does not affect the stability of an FOTF. The stability of fractional order system can be analyzed in another way also. Let considering here, the characteristic equation of a general fractional order system as:

0 ...

0 1

0 0 + 1+ + =



= i s s

s s

n

i i n

n β β

β

β _{α α α}

α (13)

For ,

v vi i =

β we can transform the Eq.(13) into σ-plane.

0 0

=

=





= =

i i

v n

o i

i v

v n

i

is α σ

α (14)

Here m

k

s =

σ and m is the least common multiple of ѵ.

For a given αi, if the absolute phase of all roots of transform equation (14) isφσ = arg(σ), we can close the following points for stability of fractional order systems.

1. The stability condition is as arg( ) .

2m m

π σ π _{< <}

2. The oscillation condition is as . 2 ) arg(

m

π σ =

The characteristics equation for the above Eq. (15) is given as 0 1 5 . 0 8 . 0 )

( 2.2 0.9

1 s = s + s + =

D (16)

Equation (16) can be written as

0 1 5 . 0 8 . 0 ) ( 10 9 10 22

1 s = s + s + =

D (17)

The transformed equation for Eq. (17) in σ-plane is as 0 1 5 . 0 8 . 0 )

( 22 9

1 σ = σ + σ + =

D (18)

For solving the Eq. (18), MATLAB function solve ( ) is used here and the obtained roots by this function is as 023 . 3 ) arg( ; 9772 . 0 3080 .

0 1,2

2 ,

1 = ± σ =

σ i 010 . 1 ) arg( ; 8359 . 0 5243 .

0 _3,4

4 ,

3 = ± σ =

σ i 698 . 2 ) arg( ; 4414 . 0 9297 .

0 5,6

6 ,

5 =− ± σ =

σ i 595 . 1 ) arg( ; 0111 . 1 0254 .

0 _7,8

8 ,

7 =− ± σ =

σ i 834 . 1 ) arg( ; 9625 . 0 2596 .

0 9,10

10 ,

9 =− ± σ =

σ i 023 . 3 ) arg( ; 1182 . 0 9970 .

0 _11,12

12 ,

11 =− ± σ =

σ i 717 . 0 ) arg( ; 6795 . 0 7793 .

0 13,14

14 ,

13 = ± σ =

σ i 151 . 2 ) arg( ; 8633 . 0 5661 .

0 _15,16

16 ,

15 =− ± σ =

σ i 431 . 2 ) arg( ; 6420 . 0 7465 .

0 17,18

18 ,

17 =− ± σ =

σ i 1661 . 0 ) arg( ; 1684 . 0 0045 .

1 _19,20

20 ,

19 = ± σ =

σ i 411 . 0 ) arg( ; 3960 . 0 9084 .

0 21,22

22 ,

21 = ± σ =

σ i (19) After analysis, the roots obtained in Eq. (19), it is find out that the complex conjugate roots

1661 . 0 ) arg( ; 1684 . 0 0045 .

1 19,20

20 ,

19 = ± σ =

σ i , satisfy the stability condition − < < 

m m

π σ

π _{arg( ) -0.3 <}

0.1661 < 0.3 and _>  m 2 )

arg(σ π 0.1661 > 0.157. Hence it is shown that the fractional order system G(s) is stable.

To test approximately the stability of a given fractional order transfer function model Eq. (15), the MATLAB function isstable defined under @fotf class [10] is used. The return argument K is shown the stability of the system. If it returns the value 1, means the system is stable and if it returns the value 0, means the system is unstable. The function isstable checked the denominator of G1(s), 0.8s2.2+0.5s0.9+1 and it is found that K=1,

indicate the system is stable, with q=0.1. Stability region for fractional order system Eq. (15) shown in Fig.(2). The region of stability depends on the value of q. since q=0.1, the angle is around 90_{. Here we can see that G}

1(s)

Fig.2. Poles position in complex plane for G1(s)

Example2

The fractional-order transfer function [14] is given by

3 4 .2 2 .1 7 .5 2 .1 96 .0 4 .2 3 .3

3 4 .2 2 .1 96 .0 )

( _{.227 .177 .1}.159_{59 .1}.1₂2 _.0.0₉₇39 _{.047 .039} 2

+ + + + + +

+

+ + + =

s s s s s s

s

s s s s

G (20)

The characteristics equation for the above Eq. (20) is given as

0 3 4 . 2 2 . 1 7 . 5 2 . 1 96 . 0 4 . 2 3 . 3 )

( 2.27 1.77 1.59 1.2 0.97 0.47 0.39

2 s = s + s + s + s + s + s + s + =

D (21)

Equation (21) can be written as

0 3 4 . 2 2 . 1 7 . 5 2 . 1 96 . 0 4 . 2 3 . 3 )

( 100

39 100

47 100

97 100

120 100

159 100

177 100

227

2 s = s + s + s + s + s + s + s + =

D (22)

Fig.3. Step response of fractional order system, G1(s)

The transformed equation for Eq. (22) in σ-plane is as

0 3 4 . 2 2 . 1 7 . 5 2 . 1 96 . 0 4 . 2 3

. 3 )

(_{σ = σ}227_{+ σ}177_{+ σ}159_{+ σ}120_{+ σ}97_{+ σ}47_{+ σ}39_{+ = (23)}

-1 -0.5 0 0.5 1 1.5 -1.5

-1 -0.5 0 0.5 1 1.5

Pole-Zero Map

Real Axis

Im

ag

in

ar

y

A

x

is

unstable region stable region

0 10 20 30 40 50 60 70 80 90 100 0

0.2 0.4 0.6 0.8 1 1.2 1.4

Time (seconds)

A

m

pl

it

ud

3 4 . 2 2 . 1 7 . 5 2 . 1 96 . 0 4 . 2 3 .

3 _s2.27_{+ s}1.77_{+ s}1.59_{+ s}1.2_{+ s}0.97_{+ s}0.47_{+ s}0.39_{+ and it is found thatK=1, indicate the} system is stable Fig. (4), withq=0.001. Here we can see that G₂(s) has more stability region. Now it is easy to summarize here that the system is stable even if its pole lies on the right side of the imaginary axis. Fig. (5), Shows that the step response of the system is converging which indicate, that the system is stable.

Example3

Consider the fractional order system

1 5 . 1

1 )

( _0.5

3

+ −

= s s

G the characteristic equation for the above transfer

function is written as 0 1 5 . 1 )

( 0.5

3 s =− s + =

D ₍₂₄₎

The transformed equation for Eq. (24) in σ-plane for 10 1

s =

σ and m=10 is given by 0

1 5 . 1 )

( 5

3 σ =− σ + =

D ₍₂₅₎

The roots of equation are 9221

. 0 1 =

σ σ2,3 =−0.7460±0.5420i;arg(σ2,3) =0.628

Fig.4. Poles position in complex plane for G2(s)

Fig.5. Poles position in complex plane G2(s) (zoomed graph)

-1.5 -1 -0.5 0 0.5 1 1.5 -1.5

-1 -0.5 0 0.5 1 1.5

Pole-Zero Map

Real Axis

Im

a

g

in

a

ry

A

xis

stable region unstable region

0.7 0.8 0.9 1 1.1 1.2 1.3 -0.3

-0.2 -0.1 0 0.1 0.2 0.3

Pole-Zero Map

Real Axis

Im

ag

in

ar

y

A

x

is

Fig.6. Step response of fractional order system, G2(s)

25 . 1 ) arg( ; 8769 . 0 2849 .

0 _3,4

5 ,

4 = ± σ =

σ i (26) Thus no roots of Eq. (25) satisfy the stability condition

m m

π σ π _{< <}

− arg( ) and

m

2 )

arg(σ > π . It indicates that the above system G3(s) is unstable. The function isstable checked the denominator of G₃(s),−1.5s0.5+1and it is

found thatK=0, indicate the system is unstable. Step response of G₃(s) is diverging Fig. (7) It shows the instability in the system.

Fig.7. Step response of fractional order system, G3(s)

5. Conclusions

This paper investigated, three different fractional order control system for the stability and performance analysis. All basic ideas of fractional calculus, the stability of fractional order system and MATLAB function are presented here. The main purpose of the paper is to draw attention to fractional order system stability and analysis in a non-conventional way. It concludes here that the fractional order system has a large region for stability which improves the performance of the system. We believe that the technique used in this paper is useful for stability analysis in the industry where the obtained model is fractional in nature.

0 10 20 30 40 50 60 70 80 90 100 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (seconds)

A

m

pl

it

ud

e

0 10 20 30 40 50 60 70 80 90 100 -4

-3.5 -3 -2.5 -2 -1.5 -1 -0.5

0x 10 25

Time (seconds)

A

m

pl

it

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