Can you derive the equation of the frequency response of the closed loop control system?

Sure, I can help you derive the equation of the frequency response of the closed loop control system. Here are the steps:

1. Start with the transfer function of the closed loop control system. Let's denote it as H(s).

2. The transfer function H(s) can be written as the product of the transfer function of the plant, G(s), and the transfer function of the controller, C(s). So, we have H(s) = G(s) * C(s).

3. Express G(s) and C(s) in terms of their numerator and denominator polynomials. Let G(s) = Ng(s) / Dg(s) and C(s) = Nc(s) / Dc(s), where Ng(s), Dg(s), Nc(s), and Dc(s) are polynomials in s.

4. Substitute G(s) and C(s) into the equation H(s) = G(s) * C(s). This gives us H(s) = (Ng(s) / Dg(s)) * (Nc(s) / Dc(s)).

5. Multiply the numerator and denominator polynomials to obtain the overall numerator and denominator polynomials of H(s). Let's denote them as Nh(s) and Dh(s) respectively.

6. Simplify the expression H(s) = Nh(s) / Dh(s) to obtain the transfer function of the closed loop control system.

7. The frequency response of the closed loop control system can be obtained by evaluating the transfer function H(s) at the desired frequency. Let's denote the frequency as ω.

8. Substitute s = jω into the transfer function H(s) and simplify the expression to obtain the frequency response of the closed loop control system.

By following these steps, you can derive the equation of the frequency response of the closed loop control system.