How can the Root Locus method in Julia be used to determine roots for a specific value of gain "K"?

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The Root Locus method in Julia can be used to determine roots for a specific value of gain "K" by setting the gain value to the desired value and then generating the root locus plot using the ControlSystems.jl package.

For example, suppose we have the transfer function:

G(s) = (s + 2) / (s^2 + 5s + 6)

and we want to find the roots of the closed-loop system for a gain of K = 3. We can use the following code in Julia:

using ControlSystems

G = tf([1, 2], [1, 5, 6]) # Define transfer function
K = 3 # Set gain value

rlocus(G*K) # Generate root locus plot for G*K

r = roots(G*K) # Get roots for G*K

The rlocus() function generates the root locus plot for the transfer function G*K (i.e., the closed-loop system with gain K). The vertical line on the plot represents the value of the gain K. We can use this plot to visualize the movement of the poles as we vary the gain.

The roots() function returns the roots of the closed-loop system with gain K. In this example, the roots are -3 and -2 (i.e., the poles of the closed-loop system). These roots represent the stability of the system for the specific value of gain K.