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Visualizing How Real and Imaginary Pole Parts relate to Step Response


Motivation : In recent lectures, we've learned that many systems are dominated by either: Therefore, it makes sense to learn (well!) how to find the transient response for each of these two, basic situation.

In either case, it's the slowest pole (or pole-pair) which dominates.
If you think about it, this make sense: the effects of faster poles happen, well, faster. So the slower part of the response is essentially a 'bottleneck' (and ends up being the part you actually notice - the part which dominates the whole response.)

Case #1: A single, real-value pole characterizes the speed of settling for a first-order system.
As we know, in one time constant, T, the system will have reached to about 63.2% of it's final value. In other words, it is within 36.8%.
After 2T, the system is within (.368)^2 = -e^(-2) = 15.3% of the final value.
After 3T, it's within (.368)^3 = -e^(-3) = 4.98% of the final value. (i.e. 95.02% of the way to final value)

If we are asked to find the number of time constants needed to come within x of the final value, we can just take the natural log (ln) of each side of the relation above.
So, for instance, to come to 90% of final value (in other words, within 10%), we need -ln(.10) = 2.3 time constants.
Case #2: a pole-pair characterizes the speed of settlling AND the frequency of oscillation for a second-order system.

At right, the red 'envelope' is just the first-order decay characterized by a single, real-valued pole (just located at the real value which is shared by both poles in the pole-pair). (This response is simply reflected about the final value on the y-axis to create the full envelope.)

The second order response, shown in magenta 'hits' this envelope at its maximum and minimum peaks of the oscillation. (These points are shown with blue + signs.)

Below, the frequency of 'peak hits' is that of a sinusoid described by just imaginary value of the pole-pair.

The time to the first 'peak' (which is the maximum overshoot, for a stable 2nd-order system) is 1/2 a full cycle (to go from a 'min peak' to a 'max peak', rather than from 'max peak' to 'max peak').
So in short:
  • Given a particular REAL value associated with a pole-pair, we can adjust the frequency of oscillation, which in turn determines what that first (maximum) overshoot value will be.
  • Changing the imaginary part alone does NOT change this envelope, however, which is solely determined by the real part.
gonzo@mit.edu page 4 (of 10) 2.010 Tutorial #7, 5-Nov-00