# Routh-Hurwitz criterion for systems with delay

Is there an analogue of Routh-Hurwitz criterion for systems with time-delay?

In other words, given the open-loop transfer function

$$G(s) = e^{-sT} \frac{P(s)}{Q(s)}$$

where $P$ and $Q$ are polynomials, it is possible to decide if the closed-loop transfer function $G/(1+G)$ is stable or not in terms of $T$ and the coefficients of $Q$?

• Normally, the component of time delay $e^{-sT}$, which is a irrational function is converted to a rational function using a Pade approximant. As an example, $e^{-sT} \approx \frac{1 - \frac{sT}{2}}{1+\frac{sT}{s}}$. Then the poles can be analyzed using the Routh-Hurwitz criterion. Aug 18, 2018 at 19:27
• @WinterSoldier Do you know if there is a lower bound for the order of Padé approximant such that the approximate rational transfer function is guaranteed to be stable if and only if the original transfer function $G(s)$ is stable? Aug 18, 2018 at 19:49
• To my knowledge, I am not sure if a lower bound for a given order exists. However, note that the Padé approximant is usually chosen - no matter the degree - so that all poles (of the approximant) lie in the left half s-plane. Therefore, the Padé approximant only affects the stability (or instability) of the system if $G(s)$ is the loop transfer function and not the closed loop transfer function. Aug 18, 2018 at 20:06
• To be clear, you want to perform an approximate stability analysis using the Routh-Hurwitz on $1 + e^{-sT}{\frac{P(s)}{Q(s)}} = 0$. Using (as an example) a first order Pade approximation, the characteristic equation becomes $(1 + \frac{sT}{2})Q(s) + (1 - \frac{sT}{2})P(s) = 0$. If it is stable by Routh-Hurwitz, then it is approximately stable, since an approximation was used. Aug 19, 2018 at 12:30
• You can always use the Nyquist criterion. Aug 20, 2018 at 18:12