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The open loop transfer function of a feedback control system incorporating a dead time element is given by

$$ G(s)=\frac{K e^{-T s}}{s(s+1)} $$

Where $\mathrm{K}>0$, and $\mathrm{T}>0$ are variable scalar parameters.

(a) For a given value of $T$ show that the closed loop system is stable for all value of $K<K_0$ where $K_0=\omega_0 \operatorname{cosec} \omega_0 T$ and $\omega_0$ is the smallest value of $\omega$ satisfying the equation $\omega=\cot \omega T$