Control theory: Closed loop zeros, Root locus and its dynamic response

Question

Why the closed loop dominant poles of the root locus can show the response of the system? Wont it neglect the effect of the closed loop zeros?

As I read on the books, root locus method deal with the closed loop poles. It sketch the locus of the close-loop poles under an increase of one open loop gain(K) and if the root of that characteristic equation falls on the RHP. It means the close loop pole fall into RHP and make system unstable. But the exercises and examples also treat Root locus as a method of designing compensation or gain to fulfill the system requirement like damping factor and setting time. Isn't the root locus method for stability only?

I thought the system response should include the closeloop zero. Did I miss something?

Also, can Nyquist plot also used to show the system response and to design the system? As I thought its some how same as Root locus, with frequency being the varying K(gain).

Answer 1

But the exercises and examples also treat Root locus as a method of designing compensation or gain to fulfill the system requirement like damping factor and setting time. Isn't the root locus method for stability only?

What determines stability? zeros or poles? Poles on RHP always cause instability. That is what we would like to avoid. In many cases, as long as the system is stable, engineers do not care. They just like a system work either good or bad. But the system requirements is also important. But it is meaningless if the system is not stable. Therefore, stability is the first requirement. The system quality is a bonus.

Why the closed loop dominant poles of the root locus can show the response of the system?

Maybe what I am saying is not exactly what you mean by this question. However, here, I emphasize on the term 'dominant poles'. When the absolute value of a pole $||p_i||$ is small, it means that the effect of the pole in this fraction is high compared with $s$

$$\frac{1}{(s-p_i)}$$

Therefore, $p_i$ is called dominant. But, if the absolute value of the pole is very big, it means that the fraction is not very sensitive to $s$ and hence $s$ can be neglected. Therefore, nondominated poles can be approximated by a gain.

Wont it neglect the effect of the closed loop zeros?

Zeros are important as well. Zeros on RHP (nonminimum phase) are not desirable but they do not cause instability directly. But, since they attract the poles by the increase of the controller gain, they attract the poles to RHP if a high gain is used. Nonminimum phase systems are bad but not necessarily instable.

can Nyquist plot also used to show the system response and to design the system?

I personally do not use them because root locus does the job for me. In practice, It has not have any application for me but I am not sure about the others.

But the exercises and examples also treat Root locus as a method of designing compensation or gain to fulfill the system requirement like damping factor and setting time. Isn't the root locus method for stability only?

See page 2 of this slide.

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This paragraph is out of the scope of your question but worth mentioning. You mentioned RHP. For systems with fractional order such as $$\frac{(s^{0.6}+0.4)(s^{0.6}+0.7)}{(s^{0.6}+1.3)(s^{0.6}+0.9)}$$

The stability criteria could be more relaxed. See Figure (6-a) at page 13 of this article.

Answer 2

Why the closed loop dominant poles of the root locus can show the response of the system? Wont it neglect the effect of the closed loop zeros?

Yes , that is indeed considered as a weakness of rootlocus - its lack of capability to predict closed loop zero's effect.

Ref: http://lpsa.swarthmore.edu/Root_Locus/RLocusExamples.html

Quoting from the above reference "While the poles of a system (the roots of the denominator polynomial) are very important in determining the behavior of a system, the zeros of the system (the roots of the numerator polynomial) can also be important. After performing a root-locus design, it is critical to go back and test the closed loop system to ensure that it behaves as expected"

Answer 3

Late answer here. Zeros can have a significant effect on CL (closed loop) response. If they are within the bandwidth of the loop, they would cause overshoot in the step response. The CL stability is not affected as discussed elsewhere.

It may be related to note, that it's often easy to remove unwanted zeros from the CL transfer function.

• LHP zeros can be cancelled or approximately-cancelled directly at the input.
• It's also possible to accomplish the equivalent result by shifting zeros into the feedback path, and poles into the forward path. It can be a useful alternative to direct cancellation, in case the unwanted CL zero was created by a controller zero that can change. This will be shown below:

Consider a loop with the familiar CL transfer function $\displaystyle\frac{F}{1+FG}$

Let's break F and G into numerator and denominator (that is, zeros and poles).

$F:=\displaystyle\frac{N_F}{D_F}$ and $G:=\displaystyle\frac{N_G}{D_G}$

Thus

$\displaystyle\frac{F}{1+FG}=\frac{\frac{N_F}{D_F}}{1+\frac{N_FN_G}{D_FD_G}} = \frac{N_FD_G}{D_FD_G+N_FN_G}$

To avoid CL zeros (or more commonly, just to eliminate the stray one that falls into the CL bandwidth), we simply need to not place the unwanted zero into $N_FD_G$. We can usually do this while still obtaining the desired $(1+FG)=(D_FD_G+N_FN_G)$, by simply moving the "troublesome" zero out of $N_F$ and into $N_G$. (And less commonly, out of $D_G$ and into $D_F$).

The root locus analysis is completely unaffected.