Thursday 9 May 2013
PD CONTROLLER
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It is used to improve system stability without
effecting steady state error
Transfer function is given by KP + KDS
PD controller adds one finite zero on left side of
s-plane which improves the system stability. It doesn’t affect the TYPE of the
system
EXAMPLE
Characteristic equation is S2 + 5S + 1
From the above characteristic equation
ξ=
2.5
After applying controller
Characteristic equation is S2 + (5 + KD)
S + KP
From the above characteristic equation
ξ=
2.5 + (KD / 2)
We can infer that there is an improvement in the value
ξ
As ξ increases peak over shoot decreases
From above calculations
1) Damping factor increases.
2) Peak over shoot decreases.
3) Rise time increases.( To verify this substitute two
different values of ξ in Tr formula )
4) Bandwidth decreases, since Tr is
inversely proportion to bandwidth.
5) If KP is taken as 1, then ωn
remains unchanged.
6) Settling decreases as it is inversely proportional
to ξ.( To verify this substitute two different values of ξ in Ts
formula )
7) Affect on gain margin and phase margin
GAIN
MARGIN
Example
Without controller
Solution
For ωpc
-180 = -90 – tan-1(ω)- tan-1(ω/2)
By solving above equation we get
ωpc =
√2 rad/sec
Gain margin = -20 log | G(s) H(s) | at ωpc
| G(s) H(s) | at ωpc = 1/6
Gain margin= 15.5 db
For simplicity for calculation I assume controller (s +
1)
For ωpc
-180 = -90 – tan-1(ω) - tan-1(ω/2)
+ tan-1(ω)
ωpc = ∞
| G(s) H(s) | at ωpc = 0
Gain margin= ∞
So,
We Can Infer That There Is An Increase In Gain Margin
PHASE
MARGIN
Without controller
Solution
For ωgc
| G(s) H(s) | at ωgc = 1
ω4+2ω2-1 =0
finding roots we get ωgc = 0.6435
PM = 180 + angle (G(s) H(s) | at ωgc
= 180 -90 -
tan-1(ω/2)
= 90- tan-1(0.6435/2)
= 72.16
For simplicity for calculation I assume controller (s +
1)
Squaring on both sides and rearranging the equation we
get
ω4+ω2-1 =0
finding roots we get ωgc = 0.786
PM = 180 + angle (G(s) H(s))| at ωgc
= 180 -90 -
tan-1(ω/2) + tan-1(ω)
= 90- tan-1(0.786/2)
+ tan-1(0.786)
= 106.71
From
Above Two PM’s We Can Conclude That PM Increases
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