The forward path transfer function of a control system with unity feedback is K s(s+ a)(s +30) G(s) = = where a and K are real constants. (a) Find the values of a and K so that the relative damping ratio of the complex roots of the characteristic equation is 0.5 and the rise time of the unit-step response is approximately 1 s. (b) With the values a and K found in part (a) determine the actual rise time using MATLAB simulation. With the values a and K found in part (a) determine steady-sate errors of the system when the reference input is: i. a unit-step function, and ii. a unit-ramp function

For the system shown in Figure 2, suppose that R(s) is the reference input signal and the D(s) is the disturbance input signal, with K and K, to be decided. D(s) R(s) E(s) u1(s) (s) Y(s) --2z + 2u K K Figure 2: 1 1). Find the closed-loop transfer functions G1(s) and G2(s) in E(s) = G1(s)R(s) + G2(s)D(s). 2). What are the damping ratio ( and natural frequency wn of this system ? Discuss the impact of K and K, on ç and wn. 3). Discuss the impact of K and K, on the tracking error e(0) for a unit-ramp reference signal and a unit-step disturbance signal.

Solve by hand and using matlab also as per quaestion

Q5 Sketch the root loci for the system with G(s)=K/ s(s + 0.5)(s2 + 0.6s + 10) , H\S) = 1. Q6 Consider a unity feedback control system with the following open-loop transfer function G(s) =K/s(s2 + s + 4). Determine the value of the gain K such that the phase margin is 50 What is the gain margin for this case?