What range of values of the parameters and p is the system

Question

CONTROL SYSTEMS

Solve all questions and show your step by step working

Q1: A system with unity feedback is shown in figure (1).

Use the Routh-Hurwitz stability criterion, determine if closed-loop system with

G(s) = 100(s +8)/(s +3)(s' +15s +10)(s +1) is stable or not.

Q2:

A closed-loop feedback system is shown in Figure. For what range of values of the parameters and p is the system stable?

FIGURE. Closed-loop system with parameters K and p.

Q3:

A closed-loop control system is shown in figure (3). where G(s) is the transfer function of the system. eG(s) is the transfer function of the controller, D(s) is a disturbance. R(s) and 17(r) are the input and output. respectively. The open-loop system is unstable with a transfer function of G(s) = 2/s-2

The objectives of the controller. G(s) are to make the closed-loop system stable and at the same thue to minimize the effect of the disturbance.

Consider a proportional-integral (P1) controller with a transfer function Gc_(s) = (k_s + 1)/s.

Answer the following questions:

(a) Determine the value of K so that the closed-loop system is stable and has a critically damped response

(b) Find the steady-state error for the case where R(s) = 1/s and D(s)= 0.

(c) Find the steady-state output of the system when R(s)= 0 and D(s)= 1/s.

(d) Base on the above analysis. explain whether the objectives of the controller have been met.

Q4:

A control system as shown in figure (4) has a transfer function of G(s) = 1/s(s-1)

(a) Let G_c(s) = K.By sketching the root-locus, show that the closed-loop system is always unstable.

(b) Let G_c(s)= K(s +2). Sketch the root-locus and show that the closed-loop system can be stabilized. Determine the range of K for the closed-loop system to remain stable.