PI Control Root Locus D(z) and Gc(s)Gs(s)
Applet on this page compares the Closed Loop Pole Zero locations for the 2 methods, shown below, of deriving a sampled version of a type 0 first order system with PI control.
In the first case, the system below has a controller transfer function Gc(s) = K(1 + 1/(Tis)) and the system
transfer function Gs(s) = 1/(T1s + 1).
The forward sampled transfer function G(z) is obtained by a step
invariant transformation and is given by:-
In the second system below G1(z) is obtained as above using only Gs(s) and
D(z) is obtained from Gc(s) by making a trapezoidal approximation
to the integral part of the PI controller.
The forward transfer function in this case is:-
In both cases G(z) can be expressed as:-
and the closed loop transfer function G(z)/(1 + G(z)) as:-
This system has 2 poles which may be real or complex conjugate
depending on the coefficients and 2 zeros at 0 and -b2/b1.
The denominator coefficients for G(z) are the same for both
approaches and are given by:-
In the first case the denominator coefficients are:-
And in the second case of D(z) they are:-
Although the coefficients look different the applet below show
that for most cases of interest, the difference in the pole zero
locations is minimal.
eg parameters (4.0, 1.0, 2.0, 0.1). Here the difference is
eg parameters (4.0, 1.0, 2.0, 0.24). Here the difference is also
negligible with both systems becoming unstable for K > 8.36.
eg parameters (K, 0.3, 0.2, 0.24). Here the difference is noticeable
with the first system becoming unstable for K > 2.44 and the
second becoming unstable for K > 2.64.
The applet indicates that only small differences occur when T1 and Ti
are unusually small and then the second system performs marginally
Gif image below show how the applet should appear.
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COPYRIGHT © 2006 Cuthbert Nyack.