Long-term Behavior

The questions below are due on Sunday February 24, 2019; 11:00:00 PM.

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All the intuition we will develop for poles stems from the observation that we can decompose complication systems into smaller systems of the form:

As we saw in lecture, this system's unit sample response is: [1, p_0, p_0^2, p_0^3, \ldots], or more precisely:

h[n] = \begin{cases} p_0^n, &\text{if}~n\geq 0\\ 0, &\text{otherwise} \end{cases}

This system's unit sample response is a geometric sequence with base p_0. From this, we could notice a few different behaviors, depending on the value of p_0:

If p_0 > 1, the output will tend toward infinity as n gets large.
If p_0 = 1, the output will remain constant.
If 0 < p_0 < 1, the output will tend toward 0 as n gets large.
If p_0 = 0, the system will be equivalent to a "wire," and its output will be the same as its input
If -1 < p_0 < 0, the output will tend toward zero as n gets large, and it will alternate signs on successive timesteps.
If p_0 = -1, the output will remain constant in magnitude as n gets large, and it will alternate signs on successive timesteps.
If p_0 < -1, the output will tend toward infinity as n gets large, alternating signs each timestep.

This is quite nice! Because the system's unit sample response is a geometric sequence, we can use our intuition about how geometric sequences with different bases behave to reason about the long-term behavior of the system without simulating it step-by-step! All we need to know is that one value, p_0 (the pole).

1) More Complicated Systems

As we saw in lecture, higher-order systems in general have an equivalent representation that looks like a scaled sum of systems of the form shown above. The unit sample response of such a system, then, can be viewed as a sum of scaled geometric sequences, whose bases are the poles of the system:

h[n] \sim \sum_i c_ip_i^n

The p_i values are the poles of the system, and the c_i values are constants that come from the partial fraction expansion of the system functional. We can still use the poles of the system to reason about the long-term behavior of the system!

In general, there will come a time n when one pole dominates this response (because its value raised to the n^{\text{th}} power is much larger than the other poles' values). We call this pole (which happens to be the pole with the largest magnitude) the dominant pole. If we are interested in the long-term behavior of a system, we will often consider only the dominant pole (since, in the long term, the contributions from the other poles will have died away).

So we will often say that, for large n, the unit sample response goes like p_0^n, where p_0 is the dominant pole of the system.

What property of the dominant pole guarantees that a system's unit-sample reponse will converge to 0 in the long term?

What property of the dominant pole guarantees that a system's unit-sample reponse will oscillate in the long term?

What property of the dominant pole guarantees that a system's unit-sample reponse will tend toward infinity in the long term?

What property of the dominant pole guarantees that a system's unit-sample reponse will exhibit monotonic behavior in the long term?

2) Dependence on Gain

Often, the poles of the system depend on a parameter of the system (such as the proportional controller gain k we have seen with the wall-finding robot). Often, we will be interested in determining which values of that parameter lead to different behaviors of the system. We can figure this out by noting the relationship between the parameter and the poles of the system, and figuring out at which values of the parameter the behavior changes (by looking at the magnitude and angle of the dominant pole as functions of that parameter).

You will get some practice with this notion in some of the remaining exercises this week, and in Design Labs 3 and 4.