Question

new question as i don’t have extra information and better quality of the pic for previous question

Question 2 (20 marks) The feed forwand transfer function of an automated manufacturing process is given by K(s+3) 1-1S(5)9 where K is the proportional controller gain. The root locus is shown in Figure 6 a) Determine the range of values of K for which the closed loop system will exhibis a (10 marks) b) Determine a value of K that gives a damping ratio, = 0.707 and a settling time, T, S (6 marks) c) tsing the value of K found in (b), determine the peak overshoot and the peak time oscillatory response 2 seconds given a unit step reference inpat. (4 marks) g and 4. The transfer function of a temperature control system is given by 1 G (s) (0.5s1) (s1) (2s 1) a compensator such that the static position error constant Design Kp9 and the phase margin is at least 25°

Answer 1

2.:

a.:

Closed loop transfer function is given by:

G(s) K(s3) Ks3K T (s) 1G(s s2 1K(s +3) s2 Ks3K- 1 l

A system has oscillatory response when it has non-zero complex parts in its roots. Roots of the above sytem are:

K2 12K4 2 KIO

Therefore, the system has complex roots if:

K2-12K4 <0

=> K2 +4< 12K

This happens when 0.343 < K < 11.656 (These are the roots of the equation K² -12K+4 = 0)

Therefore, the system has oscillatory behaviour if 0.343 < K < 11.656

b.:

Settling time is given by:

In[0.02/1 TS wn

In the above system,

К wn VЗК -1 and - 2УЗК — 1

K In 0.021 K T K/2

= 0.707 K 0.707 => 23K 1 K 5.644 or 0.354

For K = 5.644,

In[0.02/1-2/35.014- T 5.644 1.509secs 5.644/2

For K = 0.354,

In[0.02/1-2/3.:354-T T 0.354 24.09secs 0.354/2

Therefore, K = 5.644

c.:

Peak overshoot is given by:

0.707 i-0.70720.0432 V1- =

Therefore, peak overshoot is 4.32%

Peak time is given by:

Tp wn V1-

For K = 5.644,

$\omega_n = \sqrt{3K-1} = 4$

$=>T_p = \frac{\pi}{4\sqrt{1-0.707^2}} = 1.11secs$