# What is Root locus: Definition and 29 Discussions

In control theory and stability theory, root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system. This is a technique used as a stability criterion in the field of classical control theory developed by Walter R. Evans which can determine stability of the system. The root locus plots the poles of the closed loop transfer function in the complex s-plane as a function of a gain parameter (see pole–zero plot).
A graphical method that uses a special protractor called a "Spirule" was once used to determine angles and draw the root loci.

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1. ### Engineering Can I use root locus when the input is the negative feedback?

I have used root locus before but my confusion now is that the input is the negative feedback. Usually when I have negative feedback I consider the the error between the input (ideal) signal and the observed signal. Also, in this case what is the tranfer function since u = -k*y, and what does...
2. ### PID - Root Locus (Sisotool) for Transfer Function with Zeros and Poles

Hi everyone! I have a 8th order transfer function, you can see it in the first image: % Transfer function num = [2.091,0,203.3,0,-2151,0,-1.072e05]; den = [1,0,-830.4,0,-1.036e05,0,-5.767e05,0,2.412e07]; tf = tf(num, den) I need to use a PID, so I'm trying to use a compensator, adding poles...
3. ### Root Locus of Negative Feedback System

From my understanding, the root locus is only concerned with open loop gain. I figured this means you would ignore the negative feedback loop and calculate the root locus from just the plant's function Workings: zeros: -1 poles: 0, -2, -2, relative degree = 2 => 90-degree asymptotes meeting...
4. ### How to find breakaway points in root locus

Homework Statement For an exam question i need to be able to sketch the root locus of a system, for example the following: g(S) = 200(S+3) / ((S+2)(S+4)(S+6)(S+8)(S+10) The Attempt at a Solution So i counted number of poles and zeroes and calculated no. of asymtodes: p-z = 4 and calculated the...
5. ### Understanding Root Locus Method: Stability and Response of Control Systems

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...
6. ### Finding points on root-locus by hand

Homework Statement [/B] I am trying to answer two questions: 1. Solve part (b) in the image above by hand. 2. What are the differences between transfer function (a) and transfer function (b). Homework Equations cos(Im(s) / Re(s)) = ζ The Attempt at a Solution 1. For part one given the...
7. ### Calculate gain of a transfer function without root locus

Hi, Having a bit of trouble with this question: "Assuming a proportional controller is used, determine the gain to achieve a damping ratio of 0.5, for the following transfer function. Hence calculate the associated natural frequency and oscillation period. G(s) = -4(s+0.4) / s^2+1.6s+14." I...
8. ### Alternative Methods vs Iterative Design — Control Theory

QUESTION (2/2) 1) What are some alternatives to iterative design in control theory? 2) I have a certain plant transfer function PTF(s) that is higher order than two and non-unity numerator. I want certain characteristics such as a certain damping ratio (zeta). So I want approximate it as...
9. ### Root Locus Question: Unraveling Intuition for Stability Analysis

Homework Statement I've been trying for a week to understand root locus and how it works but what i got is a big confusion that's why I'm asking here for help. Why do we only look at open loop transfer function of the system when we are supposed to know the stability of the closed loop system...
10. ### Sketching Root Locus: n=2, m=1, Angle?

The characteristic equation is 1+K(S+1)/S2 Below is the root locus diagram. I don't get why there are two branches when there is only one pole, at 0. Does it count as having two poles even if it is the same because of s^2 ? If it does then why doesn't the angle of departure make sense? number...
11. ### Root Locus Sketching: Why Is -0.435 the Breakaway Point?

Hi guys. Firstly the answer to the requirement of the post is all in the picture (problem statement, relevant equations etc.). I'm just wondering if someone could tell me why they use the root -0.435 as the breakaway point? Like I know there's two real roots; -0.435 and -1.61, so obviously one...
12. ### How Does Unity Position Feedback Affect System Dynamics in Control Theory?

Homework Statement With unity position feedbck, i.e. make K2=0, plot root locus as a function of pitch gain (K1). By imposing 2nd order system approximation, estimate settling time, rise time, peak time of the closed-loop system with 20% overshoot. Pic of system...
13. ### Finding roots for a root locus

Hi, In the attached image the roots are shown for the characteristic equation. I don't know how the roots were found. Anyone able to help? Thanks Splint
14. ### Root Locus - Why it is not possible to locate poles arbitrarilly?

The root locus design method is used to locate poles at desired locations. However, it is not possible to locate poles arbitrarily. Provide reasons for this statement. K = 1/│G(s)H(s)│ K G(s)H(s) = (2k+1)π I've formulated the answer for this as, The root locus gives the path of...
15. ### Control Systems - Root Locus with proportional control

Homework Statement Homework Equations See below The Attempt at a Solution Root Locus? Easy Peasy right? Based on provided equation G(s), i solved the coordinates of the root locus diagram using quadratic equation. For the 2nd part of the question, i have to find gain at which rise time...
16. ### Sketching Root Locus of System: K Varies 0 to ∞

Hello guys, I need help to sketch the root locus of the system when K is varying from 0 to infinity. (K>=0) The system: http://www.freeimagehosting.net/newuploads/anspe.jpg I'm stuck. I really don't know how proceed. Thanks in advance.
17. ### Control Systems - Root Locus

Homework Statement Given the transfer function: R(s)=\frac{K}{s(s+2)(s^2+4s+5)} Find the following (needed to sketch the root locus: a) Number of branches b) Symmetry c) Starting and ending points d) behavior at infinity (asymptotes) e) Real axis breakaway and break-in points. f)...
18. ### Root Locus: Identifying Asymptotes for Negative Feedback System

When sketching a root locus of a simple closed loop negative feedback system (with positive gain K)... if you have more poles than zeros, we know that they will tend towards infinity along some asymptotes. How do you know which pole will travel along which asymptote? For example in the...
19. ### What is the centroid of the given root locus?

i just want to make sure I'm doing this correctly as there is some discrepancy in my textbook. i know the centroid is equal to (Ʃpoles - Ʃzeros) / (# of poles - # of zeros) the equation i have is s2 + 2s + 8 / s(s2 + 2s + 10) so the zeros are -1 + i√7 and -1 - i√7 and the poles are 0, -1...
20. ### Root Locus Sketching: Learn How to Draw and Analyze

Hello, I do not understand very well how to sketch a root locus plot. I am able to find poles/zeros, where the root locus exist on the real axis, the break away/in points, intersection of asymptotes, crossing of the imaginary axis, etc., but I do not understand how to actually draw the root...
21. ### Root Locus Sketching: Re-arrange & Sketch for Parameter "a

Homework Statement Re-arrange the open-loop transfer function and sketch the root locus for the system for variation of the parameter 'a' from 0 to infinity. Homework Equations G(s)H(s) = 10(s+a) / [s(s+1)(s+8)] The Attempt at a Solution my problem is that i don't know how to re-arrange...
22. ### Root Locus: Plotting for k=1.33 & Finding Equivalent Damping Coefficient

I was told to plot a root locus(by hand) of the open loop transfer function: H(s)G(s)= (s+2)/(s^2 + 2s + 3) and for a value of k=1.33 determine the location of the root, then find the equivalent damping coefficient. After several attempts i was able to draw the...
23. ### Parameters for Root Locus Method using Lead Compensator

Hi everyone, I would please like some suggestions regarding where my error is, designing a lead compensator with the root locus method for a unity feedback control system where G = [12.5] / [2.5 1 2500]. I want to set the overshoot zeta = 0.5 for an overshoot of 16%, and settling time Ts =...
24. ### Designing a Cruise Control System Controller with Root Locus Method | Coursework

Homework Statement I have to design a controller for "Cruise Control System" using Root Locus method. Homework Equations We havnt studied that method yet and nor do I think it is in the course outline. Actually that is why I selected root locus so that I can learn something beyond the...
25. ### Root Locus of Unity Feedback Transfer Function: Find K Range for Stability

I want a MATLAB code to draw the root locus for a characteristic equation of a transfer function of unity feedback, also what is the range of K that keeps the system stable , here is the characteristic equation: s^5+600s^4+50000s^3+ks^2+24ks+80k=0 please help, thanks in previous
26. ### How to Analyze Root Locus in Matlab for DC Motor Control?

Homework Statement I'm trying to analyze the root locus of this transfer function in Matlab to control a DC motor: mysys=tf([0.022],[0.00000000007 0.0000000626 0.000493]) The Attempt at a Solution Using: %[Kp,P]=rlocfind(mysys) %rltool(mysys) rlocus(mysys) I get a root locus...
27. ### Root Locus: 4 Open Loop Poles at s=-2

If have a system with 4 open loop poles, all at say s=-2, how would the locus approach each of the four asymptotes? thanks
28. ### Root Locus: Get Help with State-Variable Feedback Design

I had to use state-variable feedback to design a position control system for a motor. I have to design the controller using root locus methods. The gain that is varied is located in the feedback loop as shown in the picture. I am completely lost as to how to generate a root locus with the...
29. ### Help with Root Locus: Get into General Form for Drawing

I had to use state-variable feedback to design a control system for a motor. I first designed a deadbeat controller, which was pretty easy. But now I have to design another controller using root locus methods. The gain that is varied is located in the feedback loop as shown in the picture...