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Consider the following root locus:

enter image description here

which is obtained by the following code:

s = tf('s');
G = [2/(s+1) 3/(s+2);1/(s+1) 1/(s+1)];

k1 = 1;
k2 = 1;
D4 = [k1*(s+1)/s k2; -k1*(s+1)/s k2];
F4 = G * D4;

figure;
rlocus(F4(1,1))

how can I find the value of \$k1\$ that makes the closed loop stable from the root locus?

I have tried looking at the windows that opens in the plot:

enter image description here

and considered these valuse of gain as a bound for the gain to use,so I tried to use values of gain between \$0\$ and \$1.90\$, but the system is still unstable.

How can i solve this?

If I try to use the Routh criterion to find the value of \$k1\$ I have that the closed loop is:

\$\frac{F4(1,1)}{1+F4(1,1)}=\frac{-k(s-1)}{s^{2}+s(2-k)+k}\$

and then I use the Routh locus as follows:

\$s^{2} :\$ \$1\$ | \$1\$

\$s^{1} :\$ \$2-k\$ | \$0\$

\$s^{0} :\$ \$1\$ | \$0\$

where I have used the bar \$|\$ to indicate that the numbers belong to different columns in the table.

And so I should have that the closed loop is stable for \$k1<2\$, but if I use for example \$k1=1\$ the system is still unstable.

I have also done the Routh criterion for \$k2\$, which resulted in saying that I nedd \$k2>-0.28\$, but the system is still unstable.

What am I doing wrong?

also If I look for the gain margin, I have :

gm = margin(F4(1,1))

which gives as result:

gm =

2.0000

this if I consider \$k1=1\$ in \$F4(1,1)\$, so it agrees with my calculation using the Routh Criterion above, but why is it still unstable?

But for \$F4(2,2)\$ I have infinite gain margin, ans since I have done a partial decoupling, I have also the term \$F4(1,2)\$ for which also the gain margin is infinite. Could it be that the problem is this?

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I believe rlocus() only gives the stability of gain in the feedback loop (as in the form G(s)/(K*(G(s)+1)).) You can't find the root locus for any arbitrary gain that way.

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  • \$\begingroup\$ Thanks for answering, I have tried also using the Routh criterion to find the intervalo of \$k\$ , as I edited in my question, but I am still doing something wrong. \$\endgroup\$ – J.D. Jan 28 at 15:28
  • \$\begingroup\$ I don't want to submit this as a legitimate answer because it's my sort of dirty method of computing it, but I use a for loop that gives me a small interval and map every pole on the same map. \$\endgroup\$ – Jaywalk Jan 28 at 20:43

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