Understanding Root Locus Method: Stability and Response of Control Systems

[thegreengineer]

Good afternoon people; recently I decided to give a study break from control engineering; now I'm back. I'm studying again control theory and I see some good results. By the way I got stuck in a topic called root locus method. Now my issue is not about how to sketch the root loci graph (BTW I got a PDF which explains it all as well as Brian Douglas's control engineering videos :) ) My issue is on the following. As far as I have seen the root locus method tell us how the stability of a control system is (i.e. whether if it is stable or unstable) and my teacher said for example: if K is between 1.5 and 1.78 then the step response of a system is overdamped or if K is less than 2.7 then the step response of the system won't present overshoot. Now you may have noticed that I underlined the words "STEP RESPONSE" on the previous quotes. Question: can root locus method tell us about the response of a system whose input is NOT the Heaviside step function (e.g. unit impulse function, sine function, ramp function) or does it only work for step input? I would appreciate your answers people. Thanks.

 

[FactChecker]

That's a good question. There are good reasons for using the step response as a standard test case for evaluating control laws. It is simply defined, contains all frequencies, is similar to several things that might occur in the real world, and allows a response that is relatively easy to characterize in several important ways.

The response of a system to a step function does tell us a lot about its response to other types of inputs. The response to any input function can be analysed in terms of the frequency content of the input versus output. Since the step function contains all frequencies, the response of a system to it can tell us, frequency-by-frequency, how it will respond to other inputs.

 

[thegreengineer]

That's a good question. There are good reasons for using the step response as a standard test case for evaluating control laws. It is simply defined, contains all frequencies, is similar to several things that might occur in the real world, and allows a response that is relatively easy to characterize in several important ways.

The response of a system to a step function does tell us a lot about its response to other types of inputs. The response to any input function can be analysed in terms of the frequency content of the input versus output. Since the step function contains all frequencies, the response of a system to it can tell us, frequency-by-frequency, how it will respond to other inputs.

Oh ya; I think I understand man. Thanks. BTW; this means that if for example if my root loci graph tells me that my response is unstable (will present exponential growth and oscilation) this means that no matter what input we feed into our system (whether if it is step, ramp, impulse, or even noise) our response will always be unstable because the stability is determined by the poles of the closed-loop transfer function (i.e. the zeros of the ch. equation), is that it?

 

[FactChecker]

Oh ya; I think I understand man. Thanks. BTW; this means that if for example if my root loci graph tells me that my response is unstable (will present exponential growth and oscilation) this means that no matter what input we feed into our system (whether if it is step, ramp, impulse, or even noise) our response will always be unstable because the stability is determined by the poles of the closed-loop transfer function (i.e. the zeros of the ch. equation), is that it?

Exactly. Almost any input disturbance, no matter how small, will have some of the problem frequency that grows exponentially. In fact, even with no input at all, the internal workings of the system will eventually have a tiny "hickup" that will act like a tiny input and start the frequency growing.
The only exception is an input that is specifically designed to suppress the problem frequency by countering it. That can be done with the right kind of feedback which places a zero at the pole and cancels it.

 

[FactChecker]

One further note: The response of a linear system is given by it's response to the unit impulse (Dirac delta function). The response to the unit impulse can be used as the kernal to directly represent the linear system (see http://lpsa.swarthmore.edu/Transient/TransInputs/TransImpulse.html). But the unit impulse is not a normal function. It is a "pseudo-function" (aka distribution) (see https://en.wikipedia.org/wiki/Dirac_delta_function ). It is much more "down-to-earth" to use it's related integral, the unit step function, to study the response of a linear system.