control - Find the largest value of $K_1$ and $K_2$ for instability of the system

A cascade control system with proportional controller is shown below

Theroritically the largest values of the gain K1,and K2 that can be set withot causing instability of the closed loop sytem are

Given \$G1=\frac{1}{(s+1)(2s+1)}\$ and \$G_2=\frac{1}{(3s+1)}\$

\$(A)\$10 and 100 \$(B)\$100 AND 10 \$(C)\$10 AND 10\$(D)\$\$\infty\$ and \$\infty\$

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The closed loop T.F will be

\$\frac{C(s)}{R(s)}=\frac{K_1K_2}{(s+1)(2s+1)(3s+1)+K_2(3s+1)+K_1K_2}\$

Now C.E will be $$6s^3+11s^2+(6+3K_2)s+1+K_2+K_1K_2=0$$

Now after applying R.H criteria I got two conditon for stability

that is $$20+9K_2-2K_1K_2>0$$ and $$1+K_2(1+K_1)>0$$

Now after satifying Options one by one all options becoming unstable for this equations.

Is the options are wrong or I am doing it wrong?

Answer

You seem to be doing things right. I verified them using Mathematica. So either you have \$G_1\$ and \$G_2\$ wrong, or the question is wrong.

There are two possibilities:

1. The max value of \$K1\$ is \$\frac{9}{2}\$, and the max value of \$K2\$ is \$\infty\$.



2. \$K1\$ is some finite value greater than \$\frac{9}{2}\$, and \$0<\text{K2}<\frac{20}{2 \ \text{K1}-9}\$.