# "HW6 Question 3"

This questions asks you to examine the effects of a time delay in the loop of a control system. The time delay is modeled by the pade approximation which converts it into a rational polynomial in \(s\). The time delay transfer function \(G(s) = e^{s\tau}\) (well, its pade approximation) should be entered in the feedback portion of the control system.

Recall that if \(P(s)C(s)\) is the forward transfer function and \(F(s)\) is the feedback transfer function that feeds from the output back into the summation block, then the closed loop transfer function is \(\frac{PC}{1+PCF}\).

And perhaps as importantly, the open loop transfer function is \(F(s)P(s)C(s)\) (you can work out the math if you want to see why it is in this order, which is different from the order in the equation in the previous paragraph). The reason for this is that the open loop transfer function is what is used in the Nyquist criterion, and the Nyquist criterion decides whether the input into the summation block is going to cause the system to be unstable.

--Fuller 17:26, 15 November 2007 (PST)