fractional-order filter design for set-point weighted pid

International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:17 No:05 173

170605-9494-IJMME-IJENS © October 2017 IJENS I J E N S

Fractional-Order Filter Design for Set-point

Weighted PID Controlled Unstable Systems

Kishore Bingi, Rosdiazli Ibrahim, Mohd Noh Karsiti and Sabo Miya Hassan

Department of Electrical & Electronic Engineering,

Universiti Teknologi PETRONAS, 32610, Perak, Malaysia

[email protected], [email protected], [email protected], [email protected]

Abstract— Control of unstable systems with conventional PID

controllers gives poor set-point tracking and disturbance

rejection performance. The use of set-point weighted PID

controllers (SWPID) to improve the control performance with

respect to set-point tracking and disturbance rejection have been

attempted. This is due to the fact that, SWPID will reduce

proportional and derivative kicks in the control action. However,

the control signal of SWPID controller is still inheriting the PID’s

undesired oscillations in the control signal. This leads to faster

degradation of actuators. In this work, a fractional-order low-

pass filter is designed alongside SWPID controller for unstable

systems. Incorporating such filter will help to reduce undesired

oscillations. The result’s comparison shows that the performance

of SWPID with fractional-order filter is better compared to its

performance with an integer-order filter. This is true for all the

three unstable systems considered.

Index Term— Fractional-order Filter, PID Controller, Set-point

Weighting, Unstable System, Noise

I. INTRODUCTION

Control of stable systems using PID controller gives satisfactory performance for disturbance rejection and set-point tracking. However, for unstable systems the PID controller effectively works either for disturbance rejection or set-point tracking [1, 2]. This is because, the proportional and derivative kick in the PID control actions results in large overshoot and high settling time [1-3]. Therefore, to mitigate this issue the set-point is weighted for proportional and derivative actions. This set-point weighted PID (SWPID) has a capability of responding set-point changes and external disturbance rejections separately.

On the other hand, reducing oscillations in the control signal is another common control objective [4]. This is because, the undesired oscillations in the control signal increases the deterioration rate or failure of the control valves [5-8]. Therefore, filtering the control signal is an effective method to reduce oscillations in the control signal [5, 6, 9]. In recent years fractional order filters have been increasingly used in stable and unstable systems [10-16]. This is due to fact that, the tunable order in the fractional order filter makes the filter

more flexible in comparison with the integer-order filter.

The advantages of fractional-order filter are [10, 11, 14, 17]:

fractional-order filters are less sensitive to parameter changes of the controller and the control system,

fractional-order filter can perform as a higher or lower order integer-order filter based on the value of ,

the design degree of freedom is high when compared to integer-order filter,

the conversion of fractional-order filter from low-pass to high-pass, band-pass and vice versa is simple and reliable.

In this paper, a set-point weighted PID controller is designed for first, second and third order unstable systems for effective disturbance rejection and set-point tracking. Furthermore, for the control of undesired oscillations in the control signal a fractional-order low-pass filter is designed. The performance of the fractional-order filter is compared with the performance of the integer-order filter in the presence of noise signal.

The remainder of this paper is divided into four sections:

the approximation of fractional-order operator s

through

refined Oustaloup filter is given in Section II. The design of set-point weighted PID controller and fractional-order filter is given in Section III. Results by applying the SWPID controller and the designed fractional order filter on three unstable systems in the presence of noise and disturbance is presented and discussed in Section IV. Finally, concluded in Section V.

II. REFINED OUSTALOUP FILTER

The approximation of fractional-order operator

s

through refined Oustaloup filter in the frequency range

( ),l h [11, 15] is given as

2

2(1 )

Nh k

k Nh k

qs p s ss K

q s p s q s

(1)

where,

,l h are the lower and higher order frequency

bounds

is the fractional-order parameter, R

N is the order of approximation

,p q are constants, 0 , 1p q

The gain K , zeros k and poles k of s in Eq. (1)

are given as follows:

mailto:[email protected]






hqK

p

0.5(1 )

2 1

k N

Nh

k ll

(2)

0.5(1 )

2 1

k N

Nh

k ll

III. METHODOLOGY

A. Set-point Weighted PID Controller (SWPID)

The PID control signal ( )u t given in Eq. (3) is a sum of P, I

and D actions with gains ,p iK K and dK .

0

( ) ( ) ( ) ( )

t

p i d

du t K e t K e d K e t

dt (3)

These P, I and D actions are functions of error signal ( )e t ,

given as

( ) ( ) ( )e t r t y t (4)

where, ( )r t is the reference signal or set-point signal and

( )y t is the output signal or controlled signal.

From Eq. (3), the control signal of SWPID is obtained by

weighting proportional action with b and derivations action

with c as given in Eq. (5).

0

( ) ( ( ) ( )) ( ( ) ( ))

( ( ) ( ))

t

p i

d

u t K br t y t K r y d

dK cr t y t

dt

(5)

As shown, the error associated to integral action is not weighted to avoid steady-state control error. The range of set-

point weights ,b c are usually between 0 and 1.

The Laplace transform of control signal ( )U s from Eq. (5)

is given as:

( ) ( ) ( )i ip d p d

K KU s K b K cs R s K K s Y s

s s

(6)

The parameters of various controller derived from Eq. (6)

are given in Table I. In order to avoid excessive transient

response in the control signal, c is set to zero. Therefore, Eq.

(6) is simplified to

( ) ( ) ( )i ip p d

K KU s K b R s K K s Y s

s s

(7)

TABLE I

PARAMETERS OF VARIOUS CONTROLLER CONFIGURATIONS

Controller Structure Set-point parameters

b c

I-PD 0 0

ID-P 0 1

PI-D 1 0

PID 1 1

Thus, the control signal is a sum of two control actions,

one for set-point tracking ability and other for external

disturbance rejection capability as shown in the Fig. 1. That it

has an ability to respond for external disturbance rejection and

set- point tracking separately.

Kp

Ki

Kd s

+

U(s)

1

s

Kp

Ki +R(s) 1

s

b

-

Y(s)

Fig. 1. Block diagram of SWPID controller.

B. Fractional-order Filter (FF)

The transfer function of integer-order low-pass filter for a

cut-off frequency c is given as:

1

( )1 / c

IF ss

(8)

From Eq. (8), the transfer function of the fractional-order

low-pass filter is obtained by replacing integer order by

fractional-order as follows:

1

( )1 / c

FF ss

(9)

where, R is the fractional-order parameter.

The magnitude and phase of the fractional-order low-pass

filter in Eq. (9) is given as:

2 2

| ( ) |

2 cos2

c

c

FF

(10)

1

sin2

( ) tan

cos2

c

FF

(11)



From Eq. (9), the fractional-order high-pass filter is

obtained by scaling the low-pass filter transfer function by1

s.

Therefore, the transfer function of fractional-order high-pass

filter is given as:

1

( )1 1/ c

FF ss

(12)

1

( )1 / c

FF ss

(13)

By comparing Eqs. (9) and (13), it can be clear that the

conversion from fractional-order low-pass to high-pass and

vise-versa can be done simply by changing the sign of . The

magnitude and phase of the fractional-order high-pass filter in

Eq. (13) is given as:

2 2

| ( ) |

2 cos2

c

c

FF

(14)

1

sin2

( ) tan

cos2

c

FF

(15)

Therefore, the block diagram of SWPID controller ( )C s

for unstable systems ( )G s with a fractional-order filter

( )FF s in the presence of external disturbance ( )D s and noise

( )N s is shown in Fig. 2. From the figure, it can be seen that

the oscillatory SWPID control signal to the unstable system is

filtered using fractional-order filter.

+-

( )d s

++

( )n s

( )R s ( )U s( )F s( )C s ( ) ( ) s

pG s G s e ( )Y s

Fig. 2. Block diagram of system with SWPID controller and fractional filter in

the presence of noise and disturbance

IV. RESULTS AND DISCUSSIONS

In this section, the effectiveness of filtered controller in the presence of noise and disturbances is evaluated for first, second and third order unstable systems. The noise signal of power 0.0001 shown in Fig. 3 is used to simulate for possible noise scenarios.

The parameters of refined Oustaloup filter (see Section II)

used in the simulation are:310l ,

310h , 5N ,

10p and 9q .

Fig. 3. White noise signal

A. First-order System

The first order plus delay time transfer function model of the unstable process given in Eq. (16) is considered.

21

4( )

4 1

sG s es

(16)

The plant has an unstable pole at 0.25 and has a delay of 2 sec. The open loop bode response of the system is shown in Fig. 10A. For the control of this system, the tuned controller and filter parameters are given in Table II. From the table, the transfer function of integral-order and fractional low-pass filters are given in Eq. (17) and (18) respectively.

1

1( )

0.5 1IF s

s

(17)

1 1.15

1( )

0.5 1FF s

s

(18)

The open loop bode response of filters 1( )FF s and 1( )IF s

is shown in Fig. 10D. From the figure, it can be seen that

amplitude-frequency curve of 1( )FF s is much flatter

than 1( )IF s .

The comparison of closed loop response and the control signal for set-point tracking, disturbance rejection and the control action in the presence of disturbance and noise with integer order filer (IF), fractional-order filter (FF) and without filter is shown in Fig. 4. In figure, the regions of interest A, B, C and D are zoomed in Fig. 5.

From Fig. 4, it can be seen that the set-point tracking ability of SWPID controller is better than PID. While, the disturbance rejection capability of both the controllers is same and satisfactory. However, in both cases the non-filtered control signal of PID and SWPID is more oscillatory while the filtered

control signal of SWPID using 1( )IF s and 1( )FF s is

smoother. Moreover, the control action using fractional- order filter is smoother than integer-order filter. This is because, the fractional-order filter is less sensitive to parameter changes in the controlled system and control system.



Fig. 4. Comparison of controller and filter performance on first order

unstable system 1( )G s .

Fig. 5. Zoomed-in view of regions A, B, C and D of Fig. 4.

The numerical assessment of rise time (s), settling time (s) after the disturbance and overshoot (%) for non-filtered and filtered control signal of SWPID controller is given in Table. II. From the table, it can be seen that SWPID controller produced less overshoot of 9.6742% while IF-SWPID and FF-SWPID has an overshoots of 13.4246% and 10.1412%. However, FF-SWPID controller responds faster than IF-SWPID and SWPID with a rise time of 3.6401s as against the 3.7941 and 5.9880s of the latter. Furthermore, FF-SWPID controller settles faster than IF-SWPID and SWPID with a settling time of 136.0194s as against the 136.1253s and 146.9023s of the latter.

B. Second-order System

The second order transfer function model of the unstable process given in Eq. (19) is considered.

2

1( )

(2 1)(0.5 1)

sG s es s

(19)

The plant has an unstable pole at 0.5, stable pole at - 2 and has a delay of 1 sec. The open loop bode response of the system is shown in Fig. 10B. For the control of this system, the

tuned controller and filter parameters are given in Table II. From the table, the transfer function of integral-order and fractional low-pass filters are given in Eq. (20) and (21) respectively.

2

1( )

0.05 1IF s

s

(20)

2 1.9

1( )

0.05 1FF s

s

(21)

The open loop bode response of filters 2 ( )FF s and 2 ( )IF s

is shown in Fig. 10E. From the figure, it can be seen that

magnitude curve and octave selectivity of 2 ( )FF s is much

better than 2 ( )IF s .

Fig. 6. Comparison of controller and filter performance on second order


Fig. 7. Zoomed-in view of regions A, B, C and D of Fig. 6

Similarly, the comparison of closed loop response, filtered and non-filtered control signal for set-point tracking, disturbance rejection and the control action in the presence of disturbance and noise is shown in Fig. 6. In figure, the regions of interest A, B, C and D are zoomed in Fig. 7.

From Fig. 6, it can be seen that the set-point tracking ability of SWPID controller is better than PID. While, the disturbance



rejection capability of both the controllers is similar and satisfactory. However, in both cases the non-filtered control signal of PID and SWPID is more oscillatory while the filtered

control action of SWPID using 2 ( )IF s and 2 ( )FF s is

smoother. Moreover, the fractional-order filtered control signal is smoother than integer-order filter.

The numerical assessment of rise time (s), settling time (s) after the disturbance and overshoot (%) for non-filtered and filtered control signal of SWPID controller is given in Table. II. From the table, it can be seen that SWPID controller produced less overshoot of 48.8942% while IF-SWPID and FF-SWPID has an overshoots of 56.6002% and 52.1263%. However, FF- SWPID controller responds faster than IF-SWPID and SWPID with a rise time of 1.9076s as against the 1.9758 and 2.0769s of the latter. Furthermore, FF-SWPID controller settles faster than IF-SWPID and SWPID with a settling time of 129.1116s as against the 129.1566 and 129.1876s of the latter.

C. Third-order System

The third order transfer function model of the unstable process considered here is given in Eq. (22).

0.5

3

1( )

(5 1)(0.5 1)(2 1)

sG s es s s

(22)

The plant has two stable poles at -0.5, -2 and has an unstable pole at 0.2. The plant also has a delay of 0.5 sec. The open loop bode response of the system is shown in Fig. 10C. For the control of this system, the tuned controller and filter parameters are given in Table II. From the table, the transfer function of integral-order and fractional low-pass filters are given in Eq. (23) and (24) respectively.

3

1( )

0.25 1IF s

s

(23)

3 1.5

1( )

0.25 1FF s

s

(24)

The open loop bode response of filters 3( )FF s and 3( )IF s

is shown in Fig. 10F. From the figure, it can be seen that

magnitude curve of 3( )FF s is much flatter than 3( )IF s .

Similarly, the comparison of closed loop response, filtered and non-filtered control signal for set-point tracking, disturbance rejection and the control action in the presence of disturbance and noise is shown in Fig.8. In figure, the regions of interest A, B, C and D are zoomed in Fig. 9.

From Fig. 8, it can be seen that the set-point tracking ability of SWPID controller is better than PID. While, the disturbance rejection capability of both the controllers is similar and satisfactory. However, in both cases the non-filtered control signal of PID and SWPID is more oscillatory while the filtered

control action of SWPID using 3( )IF s and 3( )FF s is

smoother. Moreover, fractional-order filtered control signal is smoother than integer-order filter. This is because, the fractional-order filter is less sensitive to parameter changes in the controlled system and control system.

Fig. 8. Comparison of controller and filter performance on third order


Fig. 9. Zoomed-in view of regions A, B, C and D of Fig. 9.

The numerical assessment of rise time (s), settling time (s) after the disturbance and overshoot (%) for non-filtered and filtered control signal of SWPID controller is given in Table. II. From the table, it can be seen that SWPID controller produced less overshoot of 5.8287% while IF-SWPID and FF-SWPID has an overshoots of 14.3958% and 8.1932%. However, FF- SWPID controller responds faster than IF-SWPID and SWPID with a rise time of 3.1583s as against the 3.3629 and 3.8415s of the latter. Furthermore, FF-SWPID controller settles faster than IF-SWPID and SWPID with a settling time of 125.8280s as against the 127.9643 and 129.7340s of the latter.

V. CONCLUSION

In this paper, a fractional order filter is designed for the control of undesired oscillations in the control signal of SWPID. The performance of the designed filter is evaluated on three unstable systems for set-point tracking, disturbance rejection and control action in the presence of noise and disturbance. The results show that the performance of SWPID with the designed fractional-order filter is better compared to its performance with an integer-order filter.



Fig. 10. Bode responses of: unstable systems (A) 1( )G s ; (B) 2( )G s ; (C) 3( )G s ; (D) 1 1( ), ( )FF s IF s ; (E) 2 2( ), ( )FF s IF s ; (F) 3 3( ), ( )FF s IF s .

TABLE II

CONTROLLER, FILTER PARAMETERS AND PERFORMANCE ANALYSIS OF FILTERED SWPID CONTROLLER

System Controller Parameters Filter Parameters

Method Performance Analysis

pK iK dK b

c Rise Time (s) Settling Time (s) Overshoot (%)

1( )G s 0.5318 0.0447 0.5515 0.1107 1.1500 2.0000

SWPID 5.9880 146.9023 9.6742

IF-SWPID 3.7941 136.1253 13.4246

FF-SWPID 3.6401 136.0194 10.1412

2( )G s 1.5874 0.2016 1.2094 0.2136 1.3500 12.500

SWPID 2.0769 129.1876 48.8942

IF-SWPID 1.9758 129.1566 56.6002

FF-SWPID 1.9076 129.1116 52.1263

3( )G s 3.2484 0.1443 5.0198 0.5079 1.500 4.0000

SWPID 3.8415 129.7340 5.8287

IF-SWPID 3.3629 127.9643 14.3958

FF-SWPID 3.1583 125.8280 8.1932

ACKNOWLEDGMENT

The authors would like to acknowledge the support of Universiti Teknologi PETRONAS (UTP) through the Award of Yayasan UTP Fundamental Research under Grant 0153AA-H16.

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fractional-order filter design for set-point weighted pid

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