Root Locus of Unity Feedback Transfer Function: Find K Range for Stability

[muh_j18]

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

 

[MATLABdude]

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

The Control Toolbox rlocus function:
http://www.mathworks.com/access/helpdesk/help/toolbox/control/ref/rlocus.html

...And a brief tutorial on its use:
http://www.engin.umich.edu/group/ctm/rlocus/rlocus.html

 

[oswaler]

The root locus is a graphical method used to analyze the stability of a control system by plotting the locations of the closed-loop poles as a function of a parameter, in this case, the gain K. The root locus is a powerful tool for understanding the behavior of a system and determining the range of values for K that will result in a stable system.

To draw the root locus in MATLAB, you can use the "rlocus" function. This function takes the transfer function of the system as an input and plots the root locus. In this case, the transfer function would be:

G(s) = 1 / (s^5 + 600s^4 + 50000s^3 + ks^2 + 24ks + 80k)

To plot the root locus, you can use the following code:

% Define the transfer function
num = 1;
den = [1 600 50000 k 24*k 80*k];
G = tf(num, den);

% Plot the root locus
rlocus(G)

The resulting plot will show the locations of the closed-loop poles as K varies. The root locus will be stable if all the poles are located in the left half of the complex plane.

To determine the range of K that keeps the system stable, you can use the following steps:

1. Find the breakaway and break-in points: These are the points on the root locus where the poles move from the real axis to the complex plane or vice versa. The breakaway points can be found by solving the characteristic equation for the values of s that result in a zero denominator. Similarly, the break-in points can be found by solving the characteristic equation for the values of s that result in a zero numerator.

2. Draw asymptotes: The root locus asymptotes are lines that approximate the behavior of the root locus at high values of K. They can be drawn by finding the number of poles and zeros in the transfer function and using the following formula:

N = number of poles - number of zeros
Asymptotes angle = (180 * (2k+1)) / N

3. Find the range of K for stability: The range of K that keeps the system stable is the region between the breakaway and break-in points.

In conclusion, the root locus is a useful tool for analyzing the stability of a control system. By using the "rlocus" function in MATLAB, you can plot the root locus

 

[oswaler]

The root locus is a graphical representation of the poles and zeros of a transfer function as the gain parameter, K, varies. It is a useful tool for analyzing the stability and performance of a feedback control system.

To find the range of K that keeps the system stable, we can use the Routh-Hurwitz stability criterion. This criterion states that for a system to be stable, all the coefficients of the characteristic equation must have the same sign. In this case, since the characteristic equation is a polynomial of degree 5, we need to ensure that all the coefficients of s^5, s^4, s^3, s^2, s, and the constant term are positive.

Using this criterion, we can determine the range of K that keeps the system stable by setting the coefficients of s^5 and s^4 to 0 and solving for K. This will give us the boundary values of K that divide the stable and unstable regions of the root locus.

To draw the root locus using MATLAB, we can use the "rlocus" command. This command takes the transfer function as input and plots the root locus on the complex plane for a range of K values. We can also specify the range of K using the "rlocus" command to see how the root locus changes as K varies.

In this case, the MATLAB code to draw the root locus for the given characteristic equation would be:

% Define the transfer function with unity feedback
num = [0 0 0 1 24 80];
den = [1 600 50000 K];
G = tf(num, den);

% Specify the range of K values
K = 0:0.01:100;

% Plot the root locus
rlocus(G, K);

This code will plot the root locus for K values ranging from 0 to 100 with a step size of 0.01. From the plot, we can see the regions of stability and instability for different values of K.

In conclusion, the range of K that keeps the system stable for the given characteristic equation is between the two boundary values obtained using the Routh-Hurwitz stability criterion. We can use MATLAB to visualize the root locus and analyze the stability and performance of the system for different values of K.