Given the following open loop transfer function
$$G_{OL} = \frac{k}{(s+1)^2(0.5s + 1)^2}e^{-\tau s}$$
Let $k = k_{max}/2$. Let $\tau_{max}$ denote the value of the parameter $\tau$ such that the system in a unity negative feedback loop is stable for $\tau < \tau_{max}$. Find $\tau_{max}$
I know $k_{max} = 9/2$ from applying the Routh Hurwitz stability criterion with $\tau = 0$, so $k = 9/4$ in this case.
Questions:
How do I determine $\tau_{max}$? I can expand $G_{OL}$ find the magnitude and phase, approximate a Bode plot or make a Nyquist plot, but how does this help? I should also be able to find the closed-loop transfer function since its unity negative feedback. I guess I just don't see the connection between the tools (Bode, root locus, Nyquist) used to analyze stability and actually finding $\tau_{max}$.
Any help would be appreciated!
Best Answer
You can find $\tau_{max}$ as the delay margin. Set $\tau=0$, find the phase margin and the corresponding frequency. Then compute for which $\tau$ the phase shift induced by the delay will be equal to the phase margin.