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Question 6.4: Consider a process model: Y(s)/X(s) = K(τas + 1)/(τ1s + 1)(τ...

Consider a process model:

\frac{Y(s)}{X(s)}=\frac{K\left(\tau_{a} s+1\right)}{\left(\tau_{1} s+1\right)\left(\tau_{2} s+1\right)}\left(\tau_{1}>\tau_{2}\right)

For a step input, show that:

(a) y(t) can exhibit an extremum (maximum or minimum value) in the step response only if

\frac{1-\tau_{a} / \tau_{2}}{1-\tau_{a} / \tau_{1}}>1

(b) Overshoot occurs only for \tau_{a} / \tau_{1}>1.

(c) Inverse response occurs only for \tau_{a}<0.

(d) If an extremum in y exists, the time at which it occurs can be found analytically. What is it?

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