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 .

  • $\begingroup$ Is using In verse of Laplace transformation useful here. I can find $f(t)$ such that $\mathcal{L}(f)=G(s)H(s)$. $\endgroup$
    – Mikasa
    Jan 18 '14 at 19:18
  • $\begingroup$ then what we can get from function $f(t)$? $\endgroup$ 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 \begin{equation} e^{-sT} \approx \frac{1 - sT/2}{1 + sT/2} \end{equation} and \begin{equation} e^{-sT} \approx \frac{1 - sT/2 + (sT)^2/12}{1 + sT/2 + (sT)^2/12}. \end{equation} 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.

  • $\begingroup$ can't it be tested using poles and if they are inside unit circle?or it is different situation? $\endgroup$ Jan 18 '14 at 18:56
  • $\begingroup$ @datodatuashvili This is a different situation since it is a system in continuous time. If it were discrete time, then the test for poles inside the unit circle would apply. $\endgroup$ Jan 18 '14 at 18:57
  • $\begingroup$ here is tutorial for closed loop systems faculty.ksu.edu.sa/alhajali/ChE323_CourseNotes/ChE323/… $\endgroup$ Jan 18 '14 at 18:59
  • 2
    $\begingroup$ They do mention that the Routh-Hurwitz test doesn't (directly) apply to systems with time delay, though oddly they don't provide the Padé approximant as a tool. This is a very common method in classical controls, however. $\endgroup$ Jan 18 '14 at 19:05
  • $\begingroup$ yes like this one en.wikibooks.org/wiki/Control_Systems/Stability $\endgroup$ Jan 18 '14 at 19:10

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