img Transient Response of a DEQ

The graph below shows the response of a system with n poles (denoted by an x) to a non-zero initial conditions. This response is characteristic of the transient response of the system. You can add poles to the system by changing the number of poles. You can change pole locations by dragging them on the complex plane. Note that the DEQ is given in transfer function form. You should be able to convert between transfer function and DEQ form (remind me).

The transfer function of the form:
bn-1 sn-1 + bn-2 sn-2 + ... + b1s + b0____________an sn + an-1 sn-1+ ... + a1s + a0
can be written as a differential equation of the form
x(n)+an-1x(n-1)+an-2x(n-2)+ ... +a1x(1) + +a0x     
     = bn-1y(n-1)+bn-2y(n-2)+ ... +b1y(1) + +b0y

where x(n) = dnx/dtn

Notice the a coefficients are with the output, x, and the b coefficients are with the input y
1____________ s3 + 10 s2+ 12s + 24
poles: 2


  1. How does the system respond when any pole is positive (in the right half plane of the complex plane)? (answer)
    The system response grows exponentially, is unstable, when any pole has a positive real component.
  2. For a 1st order system, how does the distance from the origin affect settling time? Try again with a 2nd order system. (answer)
    The system response decays faster the more negative the pole.
  1. How does the response of a 2nd order system change when you keep the real component the same but change the imaginary component? (answer)
    Changing the imaginary component of a 2nd order system changes the frequency of oscillation, but it doesn't affect the time for the response to decay (also called settling time). Experiment by moving a complex pair of poles vertically.
  2. How does the response for a pair of complex poles change as you move away from the origin at a constant angle? Try a 45° angle: -2±2i; -4±4i; -12±12i, then try a different angle (its OK to approximate). (answer)
    The shape of the response is the same, but scaled. Poles that lie along a constant angle from the origin are said to have a constant "damping ratio".
  3. Given a system with 3 or more poles, how do poles near the origin affect the overall response as opposed to poles further away from the origin? (answer)
    The response is dominated by the effect of the poles closest to the origin. They are called the dominate poles. This happens because poles closer to the origin produce a slower response and do don't decay (die out) as fast as poles further from the origin.