# Partial fraction expansion deletes a pole ?

Dear all, I am trying to transform the discrete LTI sytem E(z) into its partial fraction expansion
E(z) = \frac,
N(z) = Mz^2 - Mz D(z) = (1+K)z^3+(-2K-3)z^2+(K+3)z -1
M and K are const. D(z) is of 3rd order. E(z) has 3 poles, {1,1, \frac }
My approach was to find the coefficients A,B,C of the partial fraction terms: \frac{ (z-1)^2 } + \frac + \frac{ z-\frac }
However, all my attempts ended up with A = 0, B = \frac C = - \frac
This is confusing. E(z) has 3 poles, while its partial fraction expansion counts only 2. Further, E(z) is not stable, while its partial fraction expansion is conditional stable, i.e. its impulse response will remain on a constant value.
What am I doing wrong here?
Best regards, Chris
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The numerator of the transfer function is M(z-1)(z+1), while the denominator is (K+1)[(z-1)^2(z-1/(1+K))]. So there is a pole zero cancelation. Maybe the real question is: Do you really want this?
Regards Mark Burke
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On Thu, 12 Feb 2009 07:58:49 -0800, Mark wrote:

And chances are the answer is "no!".
Unless the extra pole is a mathematical artifact that doesn't reflect the actual system dynamics, the pole-zero cancellation indicates that you do, indeed, have an unstable system with either an unobservable pole or an uncontrollable one. Either case will lead to grief in the real world.
--
http://www.wescottdesign.com

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Tim Wescott wrote:

Mark, Tim, thanks for your comments. Your pointer to zero-pole cancellation really helped me a lot to understand the effects.
In my analysis, E(z) is the control error signal in a closed loop PI system with a ramp input. I designed F(z)=\frac and the ramp R(z)=\frac. The control error of the closed loop system is: E(z) = R(z) - E(z) * F(z)
The zero-pole cancellation issue occurs because the (z-1) term from F(z) moved in the nominator. Because a ramp should result in a ramp I expected a stable, but not zero, error signal.
Up to my first posting, my numerical Matlab tests with the 3rd order E(z) system were not conclusive because Matlab has numerical difficulties with transfer functions containing multiple poles, which is indeed well documented. Long simulations runs resulted in totally weird signal responses. So, I came up with the partial fraction expansion.
I conclude the zero-pole cancellation can be applied. The whole thing is not a real world system. So, there is no inaccuracy in system modelling and perfect cancellation of a pole can be assumed.
Best regards, Chris
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