Question

Consider a unity feedback system with a closed loop transfer function Determine the open loop transfer function G(s), and find the steady state error for Unit Ramp input.

C(s)/R(s) = ks + b / s² + as + b
Determine the open loop transfer function G(s), and find the steady state error for Unit Ramp input.

Answer 1

The open loop transfer function for the unity feedback system can be determined using the relationship T(s)=G(s)/(1+G(s)), and the steady state error for a Unit Ramp input can be found using the final value theorem.

Step-by-step explanation:

For a unity feedback system, the relationship between the closed loop and open loop transfer function is given by T(s)=G(s)/(1+G(s)H(s)), where H(s) is the feedback transfer function. Since it is a unity feedback system, H(s)=1 and thus the open loop transfer function is G(s)=T(s)/(1-T(s)). Substituting the given closed loop transfer function T(s) = (ks + b) / (s² + as + b), we can solve for G(s).

To calculate the steady state error for a Unit Ramp input, we use the final value theorem which states that the steady state error ess is the limit as s approaches 0 of s * R(s) * (1 - T(s)), where R(s) is the Laplace transform of the ramp function, which is 1/s². The steady state error for a ramp input is thus ess = limit as s->0 of s * (1/s²) * (1 - ((ks + b)/(s² + as + b))).