For the open loop system in Fig. 2.51 (a) find the percentage change in the steady-state value of C(s) for a unit step input and for a 10% change in K. For the closed loop system in Fig. 2.51 (b), find the percentage change in the steady-state value of C(s) for a unit step input for the same increase in the value of K.

a) For the open loop system: Step 1: Determine the transfer function C(s)/R(s). From Fig. 2.51 (a), the open-loop transfer function is G(s) = (K)/(0.1s+1). For a unit step input, R(s) = (1)/(s). The output C(s) is given by: C(s) = G(s)R(s) = (K)/(0.1s+1) · (1)/(s) Step 2: Calculate the original steady-state value of C(s) using the Final Value Theorem. C_ss1 = _s 0 sC(s) = _s 0 s ( (K)/(0.1s+1) · (1)/(s) ) C_ss1 = _s 0 (K)/(0.1s+1) = (K)/(0.1(0)+1) = K Step 3: Calculate the new value of K and the new steady-state value. K changes by 10%, so K_new = K + 0.1K = 1.1K. The new steady-state value C_ss2 is: C_ss2 = _s 0 s ( K_new0.1s+1 · (1)/(s) ) = _s 0 K_new0.1s+1 = K_new = 1.1K Step 4: Calculate the percentage change in the steady-state value. Percentage Change = C_ss2 - C_ss1C_ss1 × 100\% Percentage Change = (1.1K - K)/(K) × 100\% = (0.1K)/(K) × 100\% = 0.1 × 100\% Percentage Change = 10% b) For the closed loop system: Step 1: Determine the closed-loop transfer function C(s)/R(s). From Fig. 2.51 (b), this is a unity feedback system with G(s) = (K)/(0.1s+1) and H(s)=1. The closed-loop transfer function T(s) is: T(s) =