img Describing the Transient Response of a DEQ

The graph below shows the response of a 2nd order system to a non-zero initial conditions. The system's poles are denoted by an x. You can change pole locations by dragging them on the complex plane. As you change the pole location the corresponding system descriptor are updated. In addition to grid lines on the graph show lines of constant damping (radial lines) and lines of constant natural frequency (arcs). Experiment with how pole locations and notice how a change in the pole location affect the time domain response and response descriptors.

After experiment with this system, go back to this experiment and notice how the addition of poles and zeros affect the response. Our descriptors are valid for only a first or second order systems but can be used to approximate higher order systems that have dominate poles (remind me).

img Total Response of a DEQ
In higher order systems the slower poles tend to dominate the response - hence dominate poles. Slower poles are those closer to the origin.

Settling Time, ts (sec)1Damping Ratio, ζ1
Damped Frequency, ωd (rad/s)1Natural Frequency, ωn (rad/s)1


  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. How do you move the poles to keep the damped frequency constant and change the natural frequency? (answer)
    The damped frequency is related to the imaginary component of the pole. As long as that is constant the damped frequency will not change.
  1. At what value of damping ratio does the system have no overshoot? (answer)
    Although there is always some overshoot unless the ζ > 1, there is negligible overshoot around ζ = 0.8;
  2. How does the response change as you move along a line of constant damping (radial lines)? (answer)
    The shape of the response is the same, but scaled in the time domain axis.
  3. What is significant about a damping ratio of 1? (answer)
    ζ = 1 occurs when there are repeated real poles.
  4. When is the natural frequency equal to the damped frequency? (answer)
    When there is no damping (ζ=0). At this point the settling time is infinite because the response never dies out.