# Stability of a system with poles and zeros at infinity

I have a system transfer function

H(s) = 1/(e^s + 10)

This system has both poles and zeros at infinity and -infinity.

Can anybody tell me if this is a stable system. Thanks.

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berkeman
Mentor

mheslep
Gold Member
As shown H(s) has no zeros, and it has a pole at e^s=-10, not at +/-infinity

e^s = -10 is the pole. You cannot solve this equation. ln(-10) does not exist. That is why i concluded the pole is at infinity. Is my conclusion wrong?

The Electrician
Gold Member
Yes you can solve it. The solutions are complex. One solution is s = i*pi+Ln(10)

So the pole is at 2.3 (ln 10) +j 3.14. SO this is an unstable system. Am I correct? Also fourier transform does not exist for this am I right? Since it not abolutely integrable within - infinity and infinity.

mheslep
Gold Member
So the pole is at 2.3 (ln 10) +j 3.14. SO this is an unstable system. Am I correct? Also fourier transform does not exist for this am I right? Since it not abolutely integrable within - infinity and infinity.
In continuous time, the Laplace transform is used to obtain the transfer function. A system is stable if the poles of this transfer function lie strictly in the closed left half of the complex plane (i.e. the real part of all the poles is less than zero).
http://en.wikipedia.org/wiki/Control_theory

CEL
So the pole is at 2.3 (ln 10) +j 3.14. SO this is an unstable system. Am I correct? Also fourier transform does not exist for this am I right? Since it not abolutely integrable within - infinity and infinity.
As The Electrician said, this is ONE of the solutions. There is an infinity of them, in the form ln(10) + j(2k+1)pi.

So stability depends om the value of k? Can somebody point me to a good link which teaches how to solve equations like e^x = -a;

CEL
So stability depends om the value of k? Can somebody point me to a good link which teaches how to solve equations like e^x = -a;
k is an integer variable that can take any value between minus infinity and plus infinity. Since for an infinity of values of k the poles lie on the RHP, the system is unstable.
For an explanation on solving your equation see
http://mathforum.org/library/drmath/view/61830.html