System Dynamics - The Effect of Pole Location on Response
A short treatise on the effect of pole location on second order response.

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Properties of the Responses of Second Order Systems

        It is important to understand the response of second order systems because the response of higher order systems is composed of responses that look like first order and second order responses.  Second order responses are more complex than first order responses, and you need to spend a little more time examining them.  We begin by examining the response of a particular second order system.  Here is the differential equation of the system we will use.

This system is the simplest second order system because the transfer function of this system has no zeroes.  (The RHS of the differential equation is a constant multiplying the input, u(t).)

        We are going to focus on the step response of this system.  If the input is a unit step, the expression for the step response is given below.  This expression is valid when the damping ratio, z, is less than one.

That response would look something like the one shown below.  The overshoot might be more or less, the oscillation frequency might not be what is shown, and the DC gain could change, but this shows the general form of this step response.

The things that you need to know about this response are the following. There are other  things that you need to know about this system.         Now, focus on a particular case, and let us see how we can use this information.  Imagine that we have a second order system.  It will have a denominator with this general form.

s2 + 2zwns + wn2 = 0

Let us examine a particular case.

s2 + 8s + 32 = 0

If you use the quadratic formula on this polynomial, you should find that the roots are:

Now, examine the conclusions we can draw from this.         So, now you might be wondering what happens if the poles are not complex.  (That would be when the damping ratio is greater that one.)  In that case, the form of the response is shown below, where the s-values are the poles - using the quadratic formula expression we had earlier. When you have that expression, you get a different form for the response.

That response could look something like the one shown below.  It looks something like a single time constant system, but if you examine it closely it starts out with zero derivatie.  That's something for you to explore.