Designing of PID for Ball and Beam system Part#2

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State space model:

State space model of this system is defined as we consider two state variables ball position ‘r’ and velocity

 

 

 

 

Code    s = tf('s');P_ball = -m*g*d/L/(J/R^2+m)/s^2H = -m*g/(J/(R^2)+m);

A = [0 1 0 0

   0 0 H 0

   0 0 0 1

   0 0 0 0];

B = [0 0 0 1]';

C = [1 0 0 0];

D = [0];

ball_ss = ss(A,B,C,D)

 

System response using Matlab Command:

In this section we will check the transfer function,system response at step input and then plot them to check stability.we see that our system is not stable so first we stable it then design PID controller in order to improve the response of our system as efficient as possible.

Stability:Response of this system was open loop so by giving unity gain feedback our system becomes stable.    Code     sys_cl=feedback(C*P_ball,1);

                                                                                                               

                 

Designing of PID controller:

We design PID using Ziegler Nichols method. Transfer function of PID is given below

                      C(s)=Kp +KI/s+Kds

                             =Kp(1+1/T1s+Tds)

                                   

Parallel Form of the PID

 where: Kp := Proportional Gain     KI := Integral Gain     TI := Reset Time =Kp/Ki   Kd :=Derivative gain     Td := Rate time or derivative time

The  proportional  term  in  the  controller  generally  helps  in  establishing  system  stability  and improving the transient response while the  derivative term  is often used when it is necessary to improve the closed loop response speed even further. Conceptually the effect of the derivative term  is  to  feed  information  on  the  rate  of  change  of  the  measured  variable  into  the  controller action. The most important term in the controller is the integrator term  that introduces a pole at  s =  0  in  the  forward  loop  of  the  process.The  proportional  term  in  the  controller  generally  helps  in  establishing  system  stability  and improving the transient response while the  derivative term  is often used when it is necessary to improve the closed loop response speed even further. Conceptually the effect of the derivative term  is  to  feed  information  on  the  rate  of  change  of  the  measured  variable  into  the  controller action. The most important term in the controller is the integrator term  that introduces a pole at  s =  0  in  the  forward  loop  of  the  process.

 

Steps to determine PID controller parameters:

1.  Reduce the integrator and derivative gains to 0.

2.  Increase Kp from 0 to some critical value Kp=Kc at which sustained oscillations occur.we achieve such ocilations at KP=1.then check step response

3.  Note the value Kc and the corresponding period of sustained oscillations

In this system we apply KP=1 we get ocillations then change then increase KP=10 and introduce Kd=10 for the removal of overshoot and then check response we found that over shoot reduces but not at desired level

 

Simulink        

Block Diagram of Ball and Beam System

Output of Ball and Beam System

 

References

http://www.purduecal.edu/cpmi/NSF%20Courses/ECET-462/LABORATORIES/8-TuningofaPIDcontrollerusingZiegler-NicholsMethod.pdf

http://www.s2is.org/Issues/v5/n1/papers/paper2.pdf

Values of Kp Kd Ki (R,C):

Kp = 10

Rf = 1K

Ri = 10K

Kd = 10

R = 100K

C = 100 micro Farad

 



About the author

furqan-azam

Name: Furqan Azam
Education: BS electronic engineering (In Progress)
Profession: : freelance writer and blogger.
Hobby: Coins Collection.
I am crazy about playing video games like FIFA 14.

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