Use the Routh-Hurwitz criterion to find the range ofKfor

Step 1 of 4

The feedback of the system is \(H(s)=\frac{1}{s^{2}+2 s+4}\)

The open-loop transfer function of the system is \(G(s)=K\left(s^{2}-2 s+2\right)\)

Now, the transfer function of the closed-loop transfer function will be

\(\begin{aligned} T(s) & =\frac{G(s) H(s)}{1+G(s) H(s)} \\ & =\frac{\left(K\left(s^{2}-2 s+2\right)\right)\left(\frac{1}{s^{2}+2 s+4}\right)}{1+\left(\left(K\left(s^{2}-2 s+2\right)\right)\left(\begin{array}{c} 1 \\ s^{2}+2 s+4 \end{array}\right)\right)} \\ & =\frac{K\left(s^{2}-2 s+2\right)}{s^{2}+2 s+4+K\left(s^{2}-2 s+2\right)} \\ & =\frac{K\left(s^{2}-2 s+2\right)}{(1+K) s^{2}+(2-2 K) s+(4+2 K)} \end{aligned}\)

Hence, the transfer function of the system is \(T(s)=\frac{K\left(s^{2}-2 s+2\right)}{(1+K) s^{2}+(2-2 K) s+(4+2 K)}\)