First, let us plot the root locus of the system without the time delay.

$$ \begin{equation} H(s)=\frac{147}{20.1s+1} \end{equation} $$

s = tf('s')
H = 147/(20.1*s+1)
rlocus(H)

Fig. 1 The system is unconditionally stable.

Next, let us add a time delay and plot the root locus. However, the “rlocus” command in MATLAB only works for a linear system. Thus, we first approximate the time delay such that we have a linear system with no time delay. For this purpose, we can use the Padé approximation.

$$ \begin{equation} G(s)=e^{-7s}\frac{147}{20.1s+1} \end{equation} $$

$$ \begin{equation}

\hat{G} = \text{pade}(G,2)= \frac{147 s^2 - 126 s + 36}{ 20.1 s^3 + 18.23 s^2 + 5.78 s + 0.2449}

\end{equation} $$

s = tf('s')
G = exp(-7*s)*147/(20.1*s+1)
G_hat = pade(G, 2) % 2nd-order approximation
rlocus(G_hat)

Fig. 2 When time delay is introduced, there are chances that the system becomes unstable. Notice how the root-locus lines extend to the RHP.

Therefore, we need to design a special control that can handle this unwanted behavior introduced by time delay.