Q. Effect of feedback on stability?

Let us consider the direct transfer function of the unstable first-order system of Figure (a). The transfer function is given by

where p = d/dt is the differential operator. If a unit-step function is applied as the input quantity E, the output becomes

for which the corresponding time solution is given by

c(t) = K(1 - e^t/τ)

which is clearly unstable, since the response increases without limit as time passes. By placing a negative feedback path H around the direct transfer function, as shown in Figure (b), the closed-loop transfer function is then

where c_f(t) denotes response with feedback. The system is now clearly stable with a_K > τ . Thus, the insertion of feedback causes the unstable direct transmission system of Figure (a) to become stable. Such a technique is often used to stabilize space rockets and vehicles, which are inherently unstable because of their large length-to-diameter ratios. Note that the direct transfer function of Equation has a pole located in the right half p-plane at p = 1/τ , whereas the pole of the closed-loop transfer function of Equation with a_K > τ is located in the left half p-plane.