Find the overall transfer function of the system represented by the block diagram developed in Q2)

4- Represent the system shown in Figure P4 in state space where the output is OL(t). T(t) N₁ = 30 2 N-m/rad 3 N-m-s/rad oooo N-m/rad N3 = 10 farm FIGURE P4 10 N₂= 300 I▬▬▬▬▬▬▬▬ N4 = 100 200 N-m-s/rad

Problem 3. Consider the electrical circuit shown in Figure 3 (left). Assume that R₁ = R2 = 1 and C = 1F. 1) (15 points) Derive the transfer function G(s) = Vo(s)/Vs(s) of the electrical circuit. 2) (15 points) If a controller is used to control the system, then the block diagram of the closed-loop system is given by Fig. 3 (right). Suppose that the reference is a ramp signal, i.e., r(t) = t 1(t). Design a controller De(s) such that the closed-loop system achieves perfect tracking (e(t) = r(t) - y(t) → 0). Justify your design. a -> + US R₁i w C: iz iz i3 + R2 Vo RO Σ Do(s) G(s) OY Figure 3: Electrical system in Problem 3.

QUESTION 5 Consider the system whose block diagram is shown below. D(s) R(s) + + C(s) K G(s) G(8) R(s) is the reference input or target, and D(s) is the disturbance input. Gp(t) is the plant (house, motor, car,...) transfer function, Gc(t) is the compensator transfer function, and K is the compensator gain given by 1 G (s) G (s) = P s+10 1 3s+2 and K=5. Find the transfer function, C(s)/R(s). 3 s²+16s+14 8 3s²+32s+25 S 382+8+6 8(s+ 10) 3s² + 32s+25 3s+2 3s²+32s+25 5 3x²+32x+25