Consider the following root locus:
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:
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?
