Question

The transfer function of an open loop system is: $G_o(s) = \frac{5Ks}{s^2 + 2s + 4}$ The open loop system consists of a controller, a differentiator and a plant whose transfer functions are $G_c(s)$, $G_a(s)$ and $G_p(s)$ respectively. All three components are connected in a series. e) Determine the steady state error of the closed loop system if the input is (i) a unit step function, or (ii) a unit ramp function of magnitude 10. Assume H(s)=1. (2 marks) f) Sketch the root locus of the closed loop system for 0<K<? indicating clearly the starting and ending points (if any), asymptotes (if any), intersections between the root locus and the imaginary axis (if any), break-away and break-in points (if any) as well as the departure angles from the open-loop poles (if any). (6 marks) g) Comment on the types of stability of the closed loop system that are achieved for different positive values of K. Make reference to the values of the damping ratio. (3 marks) h) What are the values of the poles and K if the system is critically stable? (1 mark) i) What values of K would ensure $\zeta$=0.707? What would be the values of poles in this case? (2 marks) Figure 1

          The transfer function of an open loop system is:
$G_o(s) = \frac{5Ks}{s^2 + 2s + 4}$
The open loop system consists of a controller, a differentiator and a plant whose transfer functions are $G_c(s)$, $G_a(s)$ and $G_p(s)$ respectively. All three components are connected in a series.
e) Determine the steady state error of the closed loop system if the input is (i) a unit step function, or (ii) a unit ramp function of magnitude 10. Assume H(s)=1.
(2 marks)
f) Sketch the root locus of the closed loop system for 0<K<? indicating clearly the starting and ending points (if any), asymptotes (if any), intersections between the root locus and the imaginary axis (if any), break-away and break-in points (if any) as well as the departure angles from the open-loop poles (if any).
(6 marks)
g) Comment on the types of stability of the closed loop system that are achieved for different positive values of K. Make reference to the values of the damping ratio.
(3 marks)
h) What are the values of the poles and K if the system is critically stable?
(1 mark)
i) What values of K would ensure $\zeta$=0.707? What would be the values of poles in this case? (2 marks)
Figure 1

The transfer function of an open loop system is:
Go(s) = (5Ks)/(s^2 + 2s + 4)
The open loop system consists of a controller, a differentiator and a plant whose transfer functions are Gc(s), Ga(s) and Gp(s) respectively. All three components are connected in a series.
e) Determine the steady state error of the closed loop system if the input is (i) a unit step function, or (ii) a unit ramp function of magnitude 10. Assume H(s)=1.
(2 marks)
f) Sketch the root locus of the closed loop system for 0<K<? indicating clearly the starting and ending points (if any), asymptotes (if any), intersections between the root locus and the imaginary axis (if any), break-away and break-in points (if any) as well as the departure angles from the open-loop poles (if any).
(6 marks)
g) Comment on the types of stability of the closed loop system that are achieved for different positive values of K. Make reference to the values of the damping ratio.
(3 marks)
h) What are the values of the poles and K if the system is critically stable?
(1 mark)
i) What values of K would ensure ζ=0.707? What would be the values of poles in this case? (2 marks)
Figure 1

Added by Heather P.

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