Response Time

One important measure of system performance is response time. That's a generic term, and what it means may depend very much on the system you are looking at. Here are some examples.

In a first order system, the time constant gives a good measure of how quickly a system responds. We have a few general rules.

In three time constants, the decaying portion of the response will die out to within 5% of the original value in three time constants. In other words, if a first order system with no input starts out with an output of 100, it will decay to 5 in three time constants. If a first order system with a step input is moving from 0 to 100, it will get to 95 in three time constants.

In higher order systems we need different measures of reponse time. (You may want to read the lesson on general time response.)

If we have a system responding to a step, the 10-90 rise time (the "ten to ninety rise time) is the time it takes the system to move from being 10% of the way to 90% of the way. If a system is moving from 0 to 20, 10% is 2 and 90% is 18, so you would look for those points on the step response and measure the time difference between those two points. Also note that if you are moving from 30 to 50, 10% is 32 and 90% is 48.)

If you have a second order system and the oscillations do not die out quickly, this may not be a very useful measure because the system may move through the 10% and 90% points only to overshoot, then undershoot, etc.

If we have a system responding to a step, the settling time is the time it takes for the system to get within some fraction of the final change and stay there.

Now, both of these measures - rise time and settling time - are performance measures that are difficult to predict in higher order systems, but which are often specified when designing control systems. In order to design systems with rise time and settling time specifications, you will probably need to be able to relate those measures to things like pole position (when using root locus) or bandwidth (when using Nyquist/Bode' analysis).

If you have a system, the response time is related to pole position. You need to distinguish the case when the pole is real, and when you have a pair of complex conjugate poles. (You may want to refer to the general lesson on second order response.

Since the root locus gives you pole location, you can use the root locus information to predict response times.

If you are designing using Nyquist/Bode' methods, you need to know that the bandwidth of the system is related to rise time and settling time.

Since we are mostly interested in estimating responses of closed loop systems, it will be helpful if you can estimate the closed loop bandwidth. The closed loop bandwidth is approximately the frequency at which the open looop zero-db crossing occurs.