Difference between repeated poles and distinct poles?

An important concept in dsp is marginal stability where we often see the term" repeated roots " or "repeated poles "? What are they?

Does the term repeated means that two or more poles occur at exactly same coordinates?

Please kindly give answer along with examples for both s domain and z domain?

• abjt, you completely shifted the focus of that question. Don't do that. A) it's not your question, B) even if it was, suddenly asking something different is frowned upon. Reverted your edit. Dec 3 '19 at 17:27
• Hey, @abjt, are you perhaps Man's second account and I was mistaken to revert your edit? Dec 3 '19 at 18:41

Repeated poles simply means there are more than one pole at the same location. If a pole is not repeated then it is a Distinct Pole. Consider the simple case of a cascade of two integrators (in s) which is similar to the cascade of two accumulators (in z).

The Laplace transform of an integrator Is $$\frac{1}{s}$$, which is one pole at the origin. Two integrators in cascade would be $$\frac{1}{s^2}$$ so has repeated poles at the origin. The same with two accumulators in cascade would be $$\frac{1}{(1-z^{-1})^2}$$, which has repeated poles at z=1.

If there were two poles at z=1 and a third pole at z= 0.5, then such a system would have two distinct poles.

• Though, it can be noted that in the MIMO case (so a transfer function matrix) this does become a little more complicated. Dec 3 '19 at 19:24
• If we have two poles lying on imaginary axis but both these poles have same value of x but different values of y, will these two poles be considered as repeated poles?and our system will be unstable or marginally stable in this case? Dec 11 '19 at 17:17
• I don’t follow what you mean by x and y? Do you mean x is real and y is imaginary so if they are on the imaginary axis x=0? Are you on the s plane or z plane? Dec 11 '19 at 18:04
• @DanBoschen yes i mean,x is real and y is imaginary. i am on s plane
– engr
Feb 26 '20 at 16:48
• If they do not have the same complex value then they are not repeating. Tone repeating x and y would need to match Feb 26 '20 at 16:59