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:

! ! = !(1 + !!!)(1 + !!!)

the controller for such a system is:

! ! = !! + !!!!!

(1 + 1!! + !! !

+ !!!!!! + !!

!)

which is clearly a PID controller with:

!! =!! + !!!!!

!! = !! + !!

!! =!!!!!! + !!

PID$controller$could$be$represented$with$a$parallel$structure:$$

$PID$transfer$function:$! ! = !! + !!

! + !!!$$Or$with$a$serial$structure:$

$PID$transfer$function:$! ! = !!(1+ !

!!!+ !!! + !!

!!)$

The$serial$PID$can$be$translated$to$a$parallel$structure$by$transforming$the$gains:$$!!! = !!

!! + !!!!

$

!!! =$!! + !! $

!!! = !!!!!!! + !!

$