How to design a PI, PD, or PID controller for a first or second order system: For first and second order systems, we can design PI, PD and PID controllers by imposing a reference transfer function to the closed loop system: From the above diagram we can see that: E ( s) = Y c ( s) ! Y ( s) Y ( s) = K ( s)G( s) E ( s) (1) If we eliminate E(s) from the equation we have: Y ( s) = K ( s)G( s)(Y c ( s) ! Y ( s)) (2) so the transfer function of the closed loop system is: Y ( s) Y c ( s) = K ( s)G( s) 1 + K ( s)G( s) = T ( s) (3) having T(s) we can get K(s): ! ! = !(!) !(!)(1 − !(!)) (4) Assume that the desired closed loop is a first order system with a time constant ! ! : ! ! = 1 1 + ! ! ! (5) PI Controller If the Plant (G(s)) is a first order system with the gain ! and time constant !: ! ! = ! 1 + !" Then combining equations 4 and 5 with this Plant we have a controller: K(s) G(s) E(s) U(s) Y(s) Y c (s)
3
Embed
PI Controller - la.epfl.ch · How to design a PI, PD, or PID controller for a first or second order system: For first and second order systems, we can design PI, PD and PID controllers
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
How to design a PI, PD, or PID controller for a first or second order system:
For first and second order systems, we can design PI, PD and PID controllers by imposing a reference transfer function to the closed loop system:
From the above diagram we can see that:
E(s) = Yc (s) !Y (s)Y (s) = K (s)G(s)E(s) (1)
If we eliminate E(s) from the equation we have:
Y (s) = K (s)G(s)(Yc (s) !Y (s)) (2) so the transfer function of the closed loop system is:
Y (s)Yc (s)
= K (s)G(s)1+ K (s)G(s)
= T (s) (3)
having T(s) we can get K(s):
! ! = ! !(!)!(!)(1 − !(!)) (4)
Assume that the desired closed loop is a first order system with a time constant !!:
! ! = 11 + !!!
(5)
PI Controller
If the Plant (G(s)) is a first order system with the gain ! and time constant !:
! ! = !1 + !"
Then combining equations 4 and 5 with this Plant we have a controller:
K(s) G(s) E(s) U(s) Y(s) Yc (s)
! ! =1
1 + !!!!
1 + !" (1 −1
1 + !!!)= !" + 1!!!!
= !!!!
(1 + 1!")
This is a PI transfer function with:
!! =!!!!
!! = !
PD Controller
We can consider then a second order system with an integrator term:
! ! = !!(1 + !")
In order to have a first order system as the closed loop, the controller is:
! ! = 1!!!
(1 + !")
which means the controller is a PD with:
!! =1!!!
!! = !
PID controller
Finally, we can consider a second order system with two poles: