PI Control Root Locus D(z) and Gc(s)Gs(s)

Cuthbert Nyack

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 negligible.
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 better.

Gif image below show how the applet should appear.

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