Question

Consider the first-order transfer function G(s)=τs+1K​ where τ>1. a Draw the block diagram for a proportional (P) controller for this system. b Find the closed-loop transfer function and put it in standard form (this transfer function is also first-order and should use the standard form for first-order systems, i.e., it should be in the form H(s)=αs+1L​ for some L and some α) c In terms of K and τ, what should Kp​ be to have a closed-loop time constant of 1 ?

Answer 1

a) The block diagram for a proportional (P) controller is constructed and shown. b) The closed-loop transfer function is derived and put in standard form. c) The value of Kp to achieve a closed-loop time constant of 1 is determined in terms of τ and K.

Step-by-step explanation:

a) The block diagram for a proportional (P) controller for the given transfer function G(s) = τs + 1/K can be represented as:

b) To find the closed-loop transfer function, we need to substitute the proportional controller in the feedback loop with a gain constant Kp.

The closed-loop transfer function H(s) is given by: H(s) = G(s) / (1 + G(s)Kp)

By substituting G(s) = τs + 1/K into the equation and simplifying, we get: H(s) = [τs + 1/(KKp)] / [τs + 1/(KKp) + Kp]

c) To have a closed-loop time constant of 1, we equate the denominator of the closed-loop transfer function H(s) to 1 and solve for Kp. This gives us: τs + 1/(KKp) + Kp = 1

By rearranging the equation and solving for Kp, we get: Kp = [1 - τs - 1/(KKp)] / (τs)