# stability of a closed-loop system

We have

$$G(s)H(s) = \frac{Ke^{-s}}{s(s^2+5s+9)}$$

We have to determine the maximum value of K for the closed-loop system to be stable . We have to do it using the Routh hurwitz criterion .

I don't have any idea of how to approach this one because it has an $e^{-s}$ term .

• Is using In verse of Laplace transformation useful here. I can find $f(t)$ such that $\mathcal{L}(f)=G(s)H(s)$. Jan 18 '14 at 19:18
• then what we can get from function $f(t)$? Jan 18 '14 at 19:26

Since a time delay of $T$ seconds has transfer function $e^{-sT}$ and since $e^{-sT}$ is not a rational function, the typical methods for assessing stability of transfer functions cannot be used. To resolve this issue, an approximation must be used. One approximation frequently in classical control theory is the Padé approximant.
Two common Padé approximants are $$e^{-sT} \approx \frac{1 - sT/2}{1 + sT/2}$$ and $$e^{-sT} \approx \frac{1 - sT/2 + (sT)^2/12}{1 + sT/2 + (sT)^2/12}.$$ Here, since $T = 1$, the first approximation will likely suffice since this is a relatively small delay. Using this approximation, the transfer function of this system will be in the form of a rational expression and the Routh-Hurwitz method for determining $K$ can be used.