# Plotting root locus with variables in transfer function

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Sean on 25 Nov 2012
Hi. I've been using matlab to plot root loci for my control systems class but I ran into a problem with a recent assignment. This particular problem asks me to plot the root locus of a system in which the transfer function has a variable gain in addition to numeric terms. I just don't have the matlab experience to figure out how to do this on my own with the help command. It would be greatly appreciated if anyone could help me with this. Thank you.
Azzi Abdelmalek on 26 Nov 2012
Yes we can
g1=tf(1,[1 2]) % your GH(s) transfer function
g2=tf([1 1],[1 9]); % your second TF (s+1)/(s+9)
g=g1*g2;
rlocus(g)

Azzi Abdelmalek on 25 Nov 2012
Edited: Azzi Abdelmalek on 25 Nov 2012
If G(p) is your system transfer function, The transfer function of your system in a closed loop with gain k is Gc(p)=G(p)/(1+k.G(p))
%you are looking for closed loop system poles depending on k vakues
G=tf([1 10],[1 2 1]) % your system in open loop
k=0:0.1:100; % k values
rlocus(G,k); % your poles in closed loop
%or
rlocus(G);

Jonathan Epperl on 26 Nov 2012
Yeah, like Azzi is hinting: The regular Root Locus shows you where the solutions of the equation 1+k*G(s) = 0 go as you increase k from 0 to infinity.
Assuming you are interested in the transfer function from left to right here, it is
G_H/(1 + k_2*G_H*(s+1)/(s+9) ),
and so the closed-loop poles are precisely the solutions of 1 + k_2*G_H*(s+1)/(s+9) = 0. You will actually find, that this is the case for any transfer function in your loop, e.g. say from T_d to the controller output etc.
Anyway, you might be confused by the way the closed loop is written, probably you expect the controller in the forward path together with G_H, but you can just use the "normal" root locus techniques here, no trickery needed.
rlocus( series(G_H, tf([1 1],[1 9]) ) )