Introduction SDPO

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MECHANICAL ENGINEERING,ITS 2010

ROBUST PID CONTROL OF A LINEAR MECHANICAL AXIS : A CASE STUDY

M Fajar Rohman (2107100174)

Abstract

This paper is a rebuild from Robust PID control of a linear mechanical axis. Inside, we try to find stability a system. From the

transfer function, we write in matlab to obtain signal response obviously. The transfer function, are combined gain of

proportional,integral,and derivative to stabilize signal quickly.

Keywords : Robust ; PID control ; linear mechanical axis.

INTRODUCTION

Nowadays, many industrial use robot to assembly the component to be product. It could be used for

welding,assembly,cutting,etc. it is controlled by controller and could move all of direction. When it

moving, it could be vibrations and this vibrations disadvantages. To overcome this trouble, PID

controller installed to controller to minimize error when the robot move. PID controller consist of

P(proportional), I(Integral), D(Derivative). They could use single or pairs even all it on one system. Theproportional term (gain) makes a change to the output that is proportional to the current error value.

The proportional response can be adjusted by multiplying the error by a constant Kp. If the

proportional gain is too high, the system can become unstable. In contrast, a small gain results in a

small output response to a large input error, and a less responsive (or sensitive) controller. If the

proportional gain is too low, the control action may be too small when responding to system

disturbances. The integral term (when added to the proportional term) accelerates the movement of

the process towards setpoint and eliminates the residual steady-state error that occurs with a

proportional only controller. However, since the integral term is responding to accumulated errors

from the past, it can cause the present value to overshoot the setpoint value (cross over the setpoint

and then create a deviation in the other direction). The magnitude of the contribution of the integral

term to the overall control action is determined by the integral gain, Ki. The derivative term slows the

rate of change of the controller output and this effect is most noticeable close to the controller

setpoint. Hence, derivative control is used to reduce the magnitude of the overshoot produced by the

integral component and improve the combined controller-process stability. However, differentiation

of a signal amplifies noise and thus this term in the controller is highly sensitive to noise in the error

term, and can cause a process to become unstable if the noise and the derivative gain are sufficiently

large.

ANALISYS

Now, from this figure below, we could analyze about response this system.

http://en.wikipedia.org/wiki/Setpoint





We could design block diagram for system as figure below

In design system,we use PID controller. A PID is the most commonly used feedback controller. A PID

controller calculate an error value as the difference between a measured process variable and a

desired setpoint. It could minimize error when system run.The PID controller consist of Proportional,

Integral, and Derivative. They could use single or pairs even all it on one system. The proportionalterm (gain) makes a change to the output that is proportional to the current error value. The

proportional response can be adjusted by multiplying the error by a constant Kp. If the proportional

gain is too high, the system can become unstable. In contrast, a small gain results in a small output

response to a large input error, and a less responsive (or sensitive) controller. If the proportional gain

is too low, the control action may be too small when responding to system disturbances. The integral

term (when added to the proportional term) accelerates the movement of the process towards

setpoint and eliminates the residual steady-state error that occurs with a proportional only controller.

However, since the integral term is responding to accumulated errors from the past, it can cause the

present value to overshoot the setpoint value (cross over the setpoint and then create a deviation in

the other direction). The magnitude of the contribution of the integral term to the overall control

action is determined by the integral gain, Ki. The derivative term slows the rate of change of the

controller output and this effect is most noticeable close to the controller setpoint. Hence, derivative

control is used to reduce the magnitude of the overshoot produced by the integral component and

improve the combined controller-process stability. However, differentiation of a signal amplifies noise

and thus this term in the controller is highly sensitive to noise in the error term, and can cause a

process to become unstable if the noise and the derivative gain are sufficiently large.

From the system,we get a equation for H1(s) and H2(s) as below.






mtot=m1+m2 ; ξ’, ξ as the damping ratio ; r= ωn’/ ωn as the inertia ratio.

Then, we determine transfer function for u and H(s)

H(s) = H1(s) x H2(s)

H(s) = (1 + a1s + a2s2)/{mtots

2(1 +b1s + b2s

2)

2)

From equation,we determine that

a1 as {2ξ’/ωn’}, a2 as {1/ωn’2}, b1 as {2ξ/ωn}, and b2 as {1/ωn

2}.

and value for variable

mtot = {80 ; 90} (kg), ωn=10π ; 15 π-(rad/s), ξ’=0.6 ;0.8-, ξ=0.6 ;0.8-, r=0.5 ;0.8-, r= ωn’/ ωn.

Now, we get coefficient for

a1 = {2*0.6/(0.510π)- = 0.76

a2 = 1/(0.510π)^2} = 0.004

b1 = {2*0.6/(10π)} = 0.04

b2 = 1/(10π)^2

} = 0.001

and then, we get transfer function for H1(s) is (1+0.76s +0.004s 2)/(80s2(1+0.004s+0.001s2));H2(s) is

1/(1+0.004s+0.001s2); and H(s) is (1 +0.76s +0.004s2)/{80s2(1+0.004s+0.001s2)2} or (1 +0.76s

+0.004s2)/(8*10^-5 s6+6.4*10^-3 s5+0.288 s4+6.4 s3+80 s2)

with matlab we can obtain figure for step response from transfer function above.To programme in

matlab,we can type in matlab

clear all;clc;

%tugas final project%

num =[0 0 0 0 0.004 0.76 1];

den =[0.00008 0.0064 0.288 6.4 80 0 0]

sys=tf(num,den);step(sys)

and then running the program to get figure step response as like below




Besides that,we could use program from internet in http://www.softintegration.com/cgi-

bin/chcgi/toolkit/control/ctk_step.ch. And we get step response

Rise time: 39.946189

Settling time: 61.865096

Peak value: 24.944638

Peak time: 62.500000

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

1

2

3

4

5

6x 10

-3 Step Response

Time (sec)

A m p l i t u d e

http://www.softintegration.com/cgi-bin/chcgi/toolkit/control/ctk_step.ch








And now,we try to use PID controller.Let’s type

clear all;clc;


num =[0 0 0 0 0.004 0.76 1];

den =[0.00008 0.0064 0.288 6.4 80 0 0]

t=0:0.0005:1;

plant=tf(num,den);

Kp=150;

Ki=10;

Kd=50;

PI=tf([Kp Ki],[1000 0]);

contr=feedback(PI*plant,0);

P=feedback(Kp*plant,0);

Q=tf([Kp Ki Kd],[1 0])

PID=feedback(Q*plant,0);

figure(1);




step(P,t);

figure(2);

step(plant,t);

figure(3);

step(contr,t);

figure(4);

step(PID,t);

and the figures from running programme are

figure 1

figure 2




figure 3

figure 4




Because the response increase continuously,we should reduce time range to be 0.4 s and we change

Kp,Ki,and Kd. Let’s try again to get new response with new time range.

clear all;clc;


num =[0 0 0 0 0.004 0.76 1];

den =[0.00008 0.0064 0.288 6.4 80 0 0]

t=0:0.0005:0.4;

plant=tf(num,den);

Kp=5000;

Ki=2;

Kd=100;

PI=tf([Kp Ki],[1000 0]);

contr=feedback(PI*plant,0);

P=feedback(Kp*plant,0);




Q=tf([Kp Ki Kd],[1 0])

PID=feedback(Q*plant,0);

figure(1);

step(P,t);

figure(2);

step(plant,t);

figure(3);

step(contr,t);

figure(4);

step(PID,t);

and the figures from the new running programme are

figure 1

Figure 2




Figure 3

Figure 4




After,trying the new range time. We get the new figure and the response nearly stable in 0.2-0.25 s. It

is too fast range time for robot to obtain stability. After all, we know that the response from transfer

function illustrate positive trend till infinitive.

CONCLUSION

This test try to find stable response from system control. A PID controller installed in system to

minimize error in system. Some gain Kp,Ki,and Kd adjusted to find stable performance. The response

from the test transfer function perform positive trend continuously. But the response for this systemtoo large. Many ways to solve it, the gain of proportional,integral,or derivative could be variant to

find stability in system. At last, the stability response found in range time 0.2-0.25 and then,the

graphic continuously rise.

References

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motion package using quantitative feedback theory. Mechatronics

2007;17:542 –50.

*2+ Alazard D. Comments on the benchmark for ‘‘Design and optimization of

restricted complexity controllers:towards a non-parametric model based

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law on the vibratory behaviour of high-dynamics systems. J Intell Robot Syst

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[4] Benson SJ, Ye Y. DSDP5 user guide – software for semidefinite programming,

<http://www-unix.mcs.anl.gov/DSDP/>; 2005.

[5] Bhattacharyya S, Chapellat H, Keel L. Robust control: the parametric

approach. Upper Saddle River, New Jersey: Prentice Hall; 2002.

[6] Boyd S, Ghaoui LE, Feron E, Balakrishnan V. Linear matrix inequalities in

system and control. Philadelphia: SIAM Studies in Applied Mathematics;

Introduction SDPO

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