Renewable Energy Paper

International Journal in Electrical Engineering Systems and Renewable Energy 1Vol. 1, No. 1, March 2012Copyright © World Science Publisher, United Stateswww.worldsciencepublisher.org

Application of Optimal Control Pole-placement and Fuzzy PID Controller in the New Model of the Macpherson Suspension

System1S. Amir Ghoreishi, 1Mohammad-Ali Nekoui,

1Department of Engineering, Electrical Engineering Faculty, South Tehran Branch, Islamic Azad University, Tehran, IranEmail: [email protected]

Abstract – Suspension performance is expressed by ride quality in driving stability and dynamic tire force. The first goal of active suspension is improved ride quality driving stability. In this paper we will expressed a new model of the Macpherson suspension system that the system includes spring and damper. In this paper select a new model of the Macpherson suspension system and put it alongside the conventional model of the Macpherson suspension system, after introduction of parameters of conventional system we will investigated parameters of the new model and analyze this model with two different methods, that used methods are optimal pole-placement control and fuzzy control for the new model of Macpherson suspension system. In this model we examined vertical displacement of sprung mass is measured, while the angular displacement of the control arm is estimated. It is shown that the conventional model is special case of this model since the transfer function of this new model coincides with that of the conventional one if the lower support point of the damper is located at the mass center of the unsprung mass. It is also shown that the resonance frequencies of this new model agree better with experimental results. An optimal pole-placement control which combines the LQR control and pole-placement technique and also fuzzy logic is investigated using this new model. Eventually we will compare these two methods and expressed simulation results of them.

Keywords – PID Control, Optimal Pole Placement Control, LQR, Macpherson Suspension System

1. Introduction

Today, a rebellious race is taking place among the automotive industry so as to produce highly developed models. One of the performance requirements is advanced suspension systems which prevent the road disturbances to affect the passenger comfort while increasing riding capabilities and performing a smooth drive [3]. While the purpose of the suspension system is to provide a smooth ride in the car and to help maintain control of the vehicle over rough terrain or in case of sudden stops, increasing ride comfort results in larger suspension stroke and smaller damping in the wheel hop mode [9]. The roles of a suspension system are to support the weight of the vehicle to isolate the vibrations from the road and to maintain the traction between the tire and the road. The suspension systems are classified into passive and active systems according to the existence of a control input. The active suspension system is again subdivided into two types: a full active and a semi-active system In addition, the suspension systems can be divided, by their control methods, In particular, an adaptive suspension system is the type of suspension system in which controller parameters are continuously adjusted by adapting the time-varying characteristics of the system. Adaptive methods include a gain scheduling scheme, a model reference adaptive control, a self-tuning control [8]. The performance of a suspension system is characterized by the ride quality, the drive stability, the size of the rattle space, and the dynamic tire force. The prime purpose of adopting an active suspension system is to improve the ride quality and the drive stability. To improve the ride quality, it is important to isolate the vehicle body from road disturbances and to decrease the resonance peak at

or near 1 Hz which is known to be a sensitive frequency to the human body. So to achieve this goal we will design compound controllers such as Fuzzy PID Controller to reach better answer [5].2. Dynamic analysis and equations of the

conventional and new model of Macpherson suspension system

2.1. Conventional Model

Figure 1 shows the conventional model that depicts the vertical motions of the sprung and the unsprung masses. All coefficients in Fig 1 are assumed to be linear. The state

variables are defined as: the suspension

deflection, the velocity of the sprung mass,

the tire deflection, the velocity of the

unsprung mass. The equations of motion are:

(1) ) (

) ( ) ( ) (

d a u s p u s s s s

a r u t u s p u s s u u

f f z z c z z k z m

f z z k z z c z z k z m

��

��

Then, the state equation is:

And, the transfer function from the road input

(2)

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to the acceleration of the sprung mass is:

2.2. New Model

The schematic diagram of the Macpherson type suspension system is shown in Figure 2. In this model the mass of the control arm is ignored and the bushing is assumed to be a pin joint. As the mass of the control arm is much smaller than those of the sprung mass and the unsprung mass, it can be neglected and the vertical displacement of the

sprung mass and the rotation angle of the control arm

are chosen as the generalized coordinates.For this new model of Macpherson suspension system is

assumed that the horizontal movement of the sprung mass is neglected, i.e. the sprung mass has only the vertical

displacement . The unsprung mass is linked to the car

body in two ways. One is via the damper and the other is via

the control arm The values of and will be measured

from their static equilibrium points. The sprung and the unsprung masses are assumed to be particles. The coil spring deflection, the tire deflection and the damping forces are in the linear regions of their operating ranges.

Let denote the coordinates of

point , and , respectively, when the suspension system is at an equilibrium point. Let the sprung mass be

translated by upward and the unsprung mass be rotated

by in the counter-clockwise direction. Then, the following equations hold [1,2]:

Figure 1. Conventional Quarter Car Model for Macpherson Suspension System

Figure 2. A New Quarter Car Model of Macpherson Suspension System

Where is the initial angular displacement of the

control arm at an equilibrium point. Let . Then,

the following relations are obtained from the triangle :

1' 2 2 2

1' 2 2 ' 2

) soc 2 (

)) (soc 2 (

B A B A

B A B A

l l l l l

l l l l l

where is the initial distance from to at an equilibrium state, and is the changed distance from to

with the rotation of the control arm by . Therefore, the deflection of the spring, the relative velocity of the damper

and the deflection of the tire with considering of

and are [2]:

(5)

2 ' 2

2 ' '1 1 1 1 1

1' ' 2 ' ' 2

1

'' 1

1' 2

1 1

0 0

) ( ) (

. {2 )) (soc soc( 2

}) (soc soc ) (soc soc(

) (nis

)) (soc (2

)) (nis ) (nis(r C s r C

l l l

b a a b a

b

bi i l

b a

z l z z z

��

The equations of motion of the new model are now derived by the Lagrangian mechanics. Let , and denote the kinetic energy, the potential energy and the damping energy of the system, respectively. Then, these are:



(6)

2 2 2

2 2

.2

1 1) (

2 21 1

) ( ) (2 2

1) (

2

C C u s s

r C t s

p

z y m z m T

z z k l k V

l c D

��

Substituting the derivatives of )4( and )5( into )6( yields:

(7)

2 2 2

2 ' '1 1 1

1' ' 2 ' ' 2

1

20 0

' 2 21

'1 1

1 1soc ) (

2 21

(2 )) (soc soc( 2]2

[ )) (soc soc )) (soc soc

1[ ) (nis ) (nis( ]

2

) ( nis

)) (soc (8

s C u C u s u s

s

r C s t

p

z l m l m z m m T

ba a b a k V

b

z l z k

b cD

b a

��

�

Finally, for the two generalized coordinates and

, with considering of and

, the equations of motion are

obtained as follows:

20 0

0 0

2 '1 02

0 '1 1

0 0 0

'

) ( cos) ( sin) (

) )sin) ( sin) ( (

sin) (cos) (

4) cos) ((

cos) () )sin) ( sin) (( (

1sin) (

2

s u s u C u C

t s C r d

pu C u C s

t C s C r

s

m m z m l m l

k z l z f

c bm l m l z

a b

k l z l z

k

��

��

11 1

' 21 1

[ ]

) cos) ( (

B ad

b l f

c d

)8(

Now, introduce the state variables as

and with

linearization at the equilibrium state where Then, the resulting linear

equation is:

0 0

0 1 0 0

C p C s

s s

l c l k

mC m

3. Optimal Pole-Placement Control

In this section, an optimal pole-placement control which combines the control and the pole placement technique for the new model is presented [1]. The closed loop system is designed to have desired characteristics by means of the pole-placement technique, while minimizing the cost function, as defined by the weightings of the input, state and output of the system, as follows [8]. The considered linear time-invariant system and the performance index are [4,12,13,16]:

(10)

For given and , the optimal control law and the optimal closed loop system are:

(11)

where is the solution of the Riccati equation below:

The solution of the Riccati equation can be obtained in

another approach as follows. Let , introduce a

Hamiltonian matrix as:

The Jordan decomposition of is of the form where and contain the eigenvectors and

the eigenvalues of , respectively. Then, the following relationship is known:

where is the closed loop system matrix defined in )11(, denotes an eigen matrix in which the eigenvalues



of appear in diagonal terms, ,

consists of the eigenvectors of corresponding to the

eigenvalues of , and .

Furthermore, and are determined as follows:

Design procedure for optimal pole placement controller are as follows. At first we Select and and design a

controller then Evaluate the performance of the controller, and determine the eigenvalues that need to be shifted, Construct the Hamiltonian matrix and find the eigenvectors of corresponding to the eigenvalues that

need to be shifted, Obtain where is

the matrix that is composed of the unstable eigenvectors corresponding to the eigenvalues that need to be shifted and the stable eigenvectors corresponding to the eigenvalues that stay in their original locations. Let be the degree of relative stability of the eigenvalues that are to be shifted:

And eventually Solve the Riccati equation with the modified matrices or try the second method )equetion15(, to obtain the desired closed loop pole locations [6,10].4. fuzzy control and applying of fuzzy PID control

During the last decades fuzzy logic has implemented very fast hence the first paper in fuzzy set theory, which is now considered to be the influential paper of the subject, was written by Zadeh [3]. The aim of using fuzzy logic in the paper is to illustrate the application of fuzzy logic technique to the control of a continuously damping automotive suspension system. The ride comfort is improved by means of the reduction of the body acceleration caused by the car body when road disturbances from smooth road and real road roughness [7,11,15].

The fuzzy logic controller used for the conventional model has three inputs: body acceleration , body velocity

, body deflection velocity and one output: desired

actuator force .

Figure 3, 4, 5 and 6 show the membership functions of the conventional model.

The rule base used in the active suspension system showed in Table 2 with fuzzy terms derived by the designer’s knowledge and experience. The table consists of two parts, the left part has zero body acceleration so the control action was chosen to minimize the relative and the absolute body velocities only. The second part, the body acceleration has positive or negative values so important to modify the control action to minimize it also, which will lead to minimize the suspension working space and the dynamic tire load.

Interesting point is that applying of fuzzy controller to

suspension system with mentioned equations will not obtain acceptable results and states of the system are converging to zero in longer time. simulation results of designed fuzzy controller based on fuzzy rules that mentioned in table 2 are

Figure 3. Body Deflection Velocity

Figure 4. Body Velocity

Figure 5. Body Acceleration

shown in figure 7, 8. As we seen in figure 7 and 8, we can



conclude that fuzzy controller, singly can’t have good performance in control of conventional model of suspension system.

Figure 6. Force

0 10 20 30 40 50 60 70 80 90 100-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Cont

rol F

orce

Time (Sec)

Figure 8. States Response of The Conventional Model after Applying Fuzzy Controller

So we have to use compound controller. using of PID controller that its parameters are set by fuzzy controller recommended. Because we know that fuzzy PID controller has appropriate performance and robustness but because of not being easy on selecting the parameters such as )

(, it’s usually not encountered with significant

welcome. in this paper the goal is determination of these parameters by fuzzy logic. In this context, first we reimburse on how to design these parameters.

4.1. Fuzzy PID Controller

A fuzzy PID controller can be expressed as follows:

Figure 9. Block Diagram of Fuzzy PID Controller

So it’s enough that to give these rules to a fuzzy system till fuzzy system automatically adjust these parameters. Block diagram of fuzzy PID controller is shown in figure 9.Now we will design fuzzy system in figure 9. suppose that

interval changes of and are and

then we have:

- -¢ ¢= =

- -min min

max min max min

,p p d dp d

p p d d

K K K KK K

K K K K

With this method we normalized interval changes of

parameters such as and to . We assume that:

aa

= Þ =2p

i d id

KT T K

K

So parameters such as , and must be

determined. with considering , as input, final fuzzy

system consists of 3 fuzzy systems, 2 inputs and 1 output, and determine the PID parameters by using of equations )18( and )19(.We consider the IF- THEN rules as follows:

a¢ ¢ =

&

K

( ) ( ) ,

, , 1,2, ,

l l

l l lp d

IF e t isA ande t isB

THEN K isC K isD isE l M

Where , are fuzzy sets. fuzzy sets for

membership functions are shown in figure 10, 11 and 12. It

should be mentioned that and are shown

interval changes of and .

Process

ConventionalController)e.g., PID Controller(

Fuzzy Systems

First-Level Control

Second-Level Control



Figure 10. Fuzzy Sets for

Figure 11. Fuzzy Sets for

Figure 12. Fuzzy sets for

Now we settle on extraction of fuzzy rules. Figure 13 shows the step response of system:

Figure 13.

At first in point we need a big control in order to reach

the system response to the desired point with high speed. In

order to have a big control we should increase parameter

and decrease and also increase the integral gain. But

integral gain is proportional to in reverse so we should

decrease . Therefore we can write fuzzy rule as follows:

Also in point we need small control in order to escape

from big overshot. so we have to decrease parameter and

increase and . Therefore we can write fuzzy rule as

follows:

Table 3.

Table 4.

Table 5.

NB NM NS ZO PS PM PB

0

&( )( )et ore t+Me or+Mde

-Me or-Mde

0

p

d

K or

K1

Small1 Big

0

MS

2

S1

3 4 5

M B

a

0 1a

t

output

setpoint

1b

1c

1d

2a 2

b



(22)a¢ ¢

&( ) ( ) ,

, ,p d

IF e t isZOandet isNB

THEN K isSmall K isBig isB

Figure 14. First Input )error(

Figure 15. Second Input)derivative error(

Figure 16. Output for

We can use this method for other points, so fuzzy rules for 3 fuzzy systems are shown in table 3, 4, 5.

Figure 14, 15, 16, 17 show membership functions of

inputs for and outputs for .

5. Simulation results

In this paper, it is assumed that the main purpose of the control system design is to improve the ride quality. Thus, to reduce the vertical acceleration of the sprung mass at the resonance frequency near , more weights are put on the

state variables and that correspond to the displacement

and velocity of the sprung mass. The weighting matrices initially selected are:

Figure 17. Output for

5 5 1 1

2

)10 10 10 10 (

10

Q diag

R

The closed loop eigenvalues with )24( and values of parameters of table 1 are:

} 3.2042 7.1971 , 10.8560 48.2377 {c i i

Compared to the open loop system, the resonance peak near of the controlled system is lower. In this section, the damping ratios of the two dominant poles are raised for the purpose of increasing the rise time. The damping ratio of the first resonance frequency is increased from to

by shifting the dominant pole, by , to the left. Therefore, the eigenvalues of the closed loop system are:

} 11.2042 7.1971 , 10.8560 48.2377 {opt i i

Figure 18, 19 show the comparison of frequency response with )closed loop system( and without )open

loop system( controller for outputs and we can

compare these two systems in two condition at peak. As we

can see the peak gain for output at frequency for

open loop system is and at frequency for closed

loop system is and also we have peak gain for output

At frequency for open loop system is

and at

frequency for closed loop system is .For better

comparison we compare step response of outputs and

With controller )closed loop system( and without controller )open loop system( in figure 20, 21.

Figure 20 and 21 show that when we use controller



overshot decrease and system reaches to steady state faster. figure 24 shows response of the Macpherson suspension system states after applying optimal pole placement controller, as we can see system reaches to steady state faster.

Simulation results of conventional model after applying PID controller which explains on section IV are shown in figure 22 and 23. Comparing figure 22 and 23 with figure 7 and 8, we can see that fuzzy PID controller has better performance than fuzzy controller. In figure 22 we can see that conventional model states are stable when we apply fuzzy PID controller than we use fuzzy controller singly.

10-3

10-2

10-1

100

101

102

103

-220

-200

-180

-160

-140

-120

-100

-80

-60

-40

Mag

(dB)

Open loop and closed loop system based on New model (Vertival Acceleration of Sprung Mass)

Freq (rad/sec)

Open loop

Closed loop

Figure 18. Frequency Response for Output

10-3

10-2

10-1

100

101

102

103

-400

-350

-300

-250

-200

-150

-100

-50

Mag

(dB)

Open loop and closed loop system based on New model (Angular Displacement)

Freq (rad/sec)

Open loop

Closed loop

Figure 19. Frequency Response for Output

0 0.5 1 1.5 2 2.5 3-20

-15

-10

-5

0

5

10

x1do

tdot

Time (Sec)

Vertical acceleration of Sprung Mass

Closed loop

Open loop

Figure 20. Step Response for Output

Now according to the considered fuzzy PID controller, step response of new model will be figure 24. According to these figures we can see that there is little overshot and system reaches to steady state faster.

In figure 24 we will see new model of Macpherson suspension system states that optimized by fuzzy PID controller and optimal pole placement controller that have similar performance.

0 0.5 1 1.5 2 2.5 3-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

x3

Time (Sec)

Angular Displacement of Control Arm

Closed Loop

Open Loop

Figure 21. Step Response for Output

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

-80

-60

-40

-20

0

20

40

Cont

rol F

orce

Time (Sec)

Figure 23. Response of Force Control of Conventional Model of Macpherson Suspension System after Applying Fuzzy PID Controller



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x1

Time (Sec)

PID Fuzzy

LQR

Figure 25.

For example in figure 25 we can see one of the new model of Macpherson suspension system states that is compared in fuzzy PID controller and optimal pole placement and indicates that it’s similar in both methods and has similar performance. Figure 26 shows response of new model control force after applying both controller and it represents that fuzzy PID controller has better performance.

0 1 2 3 4 5 6 7 8 9 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

4

Cont

rol F

orce

Time (Sec)

PID Fuzzy

LQR

Figure 26.

6. Conclusion

PID controller due to the integral term which increases the system type has ability to remove disturbances and step type uncertainties. On the other hand fuzzy systems based on defined membership functions have inherent robustness that will ensure the power of these systems against various disturbances and uncertainties. So combine PID controllers with fuzzy systems offers a controller with acceptable robustness. But in contrast LQR controller is type of optimal controllers, will ensure good performance in case system model is linear and well known, but in the case that parameter uncertainty and external disturbances are exist or system model is nonlinear )it’s usually like this(, there is no guarantee for optimal performance and even closed loop system stability based on this controller.

References[1] M. M. M. Salem and A. A. Aly, “Fuzzy control of a quarter-car

suspension system,” World Academy of Science, Engineering and Technology, 2009.

[2] K. S. Hong, D. S. Jeon and H. C. Sohn, “A New Modeling of the Macpherson Suspension System and its Optimal Pole-Placement Control,” Proceedings of the 7th Mediterranean Conference on Control and Automation (MED99) Haifa, Israel, June 28-30, 1999.

[3] L. X. Wang, A Course in Fuzzy Systems and Control. Prentice Hall, New Jersey, August 1996.

[4] J. V. da Fonseca Neto and C. P. Bottura, “Parallel Genetic Algorithm Fitness Function Team for Eigenstructure Assignmentvia LQR Designs,” in Proceedings of IEEE Congress on Evolutionary Computation, 2003.

[5] K. S. Lee, M. W. Suh, and T. I. Oh )2006(. "A Robust Semi-active Suspension Control Law )withEnglish abstract(," Korea Society of Automotive Engineers, Vol. 2, No. 6, pp.117-126.

[6] J. V. da Fonseca Neto, et al., “Modelos e Convergência de um AlgoritmoGenéticoparaAlocação de Auto-estrutura via RLQ,” in IEEE Latin America Transactions, vol. 6, no. 1, pp. 1-9, March 2008.

[7] H. Chen, Z. -Y. Liu, P.-Y. Sun, “Application of Constrained H_Controlto Active Suspension Systems on Half-Car Models”, Journal ofDynamic Systems, Measurement, and Control, Vol. 127 / 353, SEP.2005.

[8] C. Wongsathan and C. Sirima, “Application of GA to Design LQR Controller for an Inverted Pendulum System,” in Proceedings of the IEEE International Conference on Robotics and Biomimetics, 2009.

[9] A. H. Zaeri, M. BayatiPoodeh, and S. Eshtehardiha, “Improvement of Cûk Converter Performance with Optimum LQR Controller Based on Genetic Algorithm,” in Proceedings of International Conference on Intelligent and Advanced Systems, 2007.

[10] M. BayatiPoodeh, et al., “Optimizing LQR and Pole placement to Control Buck Converter by Genetic Algorithm,” in Proceedings of International Conference on Control, Automation and Systems, 2007.

[11] M. Jonsson )2008(. "Simulation of Dynamical Behaviour of a Front Wheel Suspension," Vehicle

System Dynamics, 20, pp. 269-281.

[12] D. Ali, L. Hend, and M. Hassani, “Optimized Eigenstructure Assignment by Ant System and LQR Approaches,” in International Journal of Computer Science and Applications, vol. 5, no. 4, pp. 45-56, 2008.

[13] P.G. Wright and D.A. Williams, "The application of active suspension to high performance road vehicles", Proceedings of IMecE Conference on Microprocessors in fluid power engineering, Mechanical Engineering Publications, London, C239/84:23–28, 1984.

[14] R.S. Sharp and S.A. Hassan, "On the performance capabilities of active automobile suspension systems of limited bandwidth", Vehicle System Dynamics, 16:213–225, 2006.

[15] R.J. Dorling. Integrated Control of Road Vehicle Dynamics. PhD thesis, Cambridge University, April 1996.

[16] V. Sukontanakarn, and M. Parnichkun, “Real-Time Optimal Control for Rotary Inverted Pendulum,” in American Journal of Applied Sciences, vol. 6, no. 6, 2009.

Table 1. Quarter-Car Model Parameters

Parameters Symptoms Value

Body mass

Wheel mass

Body roughness factor

Wheel roughness factor

Damper roughness factor



Table 2. Rule Base

NSP or NPMPMZEZEPMPM

NMP or NPMPSNSZEPMPS

NBP or NPMZENMZEPMZE

NBP or NPMNSNMZEPMNS

NVP or NPMNMNBZEPMNM

NSP or NPSPMZEZEPSPM

NMP or NPSPSNSZEPSPS

NMP or NPSZENSZEPSZE

NBP or NPSNSNMZEPSNS

NBP or NPSNMNMZEPSNM

PMP or NZEPMPSZEZEPM

PSP or NZEPSZEZEZEPS

ZEP or NZEZEZEZEZEZE

NSP or NZENSZEZEZENS

NMP or NZENMNSZEZENM

PBP or NNSPMPMZENSPM

PBP or NNSPSPSZENSPS

PMP or NNSZEPSZENSZE

PMP or NNSNSZEZENSNS

PSP or NNSNMPBZENSNM

PVP or NNMPMPMZENMPM

PBP or NNMPSPSZENMPS

PBP or NNMZEZEZENMZE

PMP or NNMNSNSZENMNS

PSP or NNMNMNMZENMNM



0 20 40 60 80 100-1

-0.5

0

0.5

1x1

Time (Sec)0 20 40 60 80 100

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

x2

Time (Sec)

0 20 40 60 80 100-0.3

-0.2

-0.1

0

0.1

0.2

x3

Time (Sec)0 20 40 60 80 100

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

x4

Time (Sec)

Figure 7. Step Response of The Conventional Model after Applying Fuzzy Controller

0 20 40 60 80 100-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x1

Time (Sec)0 20 40 60 80 100

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

x2

Time (Sec)

0 20 40 60 80 100-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

x3

Time (Sec)0 20 40 60 80 100

-0.2

-0.1

0

0.1

0.2

0.3

x4

Time (Sec)

Figure 22. Step response of Conventional Model of Macpherson Suspension System after Applying Fuzzy PID Controller



0 2 4 6 8 10-0.2

0

0.2

0.4

0.6

0.8

1

1.2

x1

Time (Sec)0 2 4 6 8 10

-8

-6

-4

-2

0

2

x2

Time (Sec)

0 2 4 6 8 10-4

-3

-2

-1

0

1

x3

Time (Sec)0 2 4 6 8 10

-100

-50

0

50

100

x4

Time (Sec)

PID Fuzzy

LQR

Figure 24. Response of New Model of Macpherson Suspension System after Applying


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