Question

Question# 1 (25 points) For a unity feedback system with open loop transfer function K(s+10)(s+20) (s+30)(s2-20s+200) G(s) = Do the following using Matlab: a) Sketch the root locus. b) Find the range of gain, K that makes the system stable c) Find the value of K that yields a damping ratio of 0.707 for the system's closed-loop dominant poles. d) Obtain Ts, Tp, %OS for the closed loop system in part c). e) Find the value of K that yields closed-loop critically damped dominant poles. [5 pts] 15 pts] (5 pts] [5 pts (5 pts]

Answer 1

matlab code:

clc;clear all;close all;
z=input('enter zeros of the system =')
p=input('enter the poles of the system =')
k=input('enter the value of the gain =')
sys=zpk(z,p,k)
rlocus(sys)

output:
enter zeros of the system =[-10 -20]

z =

-10 -20

enter the poles of the system =[-30 10+10*i 10-10*i]

p =

Columns 1 through 2

-30.0000 + 0.0000i 10.0000 +10.0000i

Column 3

10.0000 -10.0000i

enter the value of the gain =1

k =

1

sys =

(s+10) (s+20)
------------------------
(s+30) (s^2 - 20s + 200)

Continuous-time zero/pole/gain model.

a):rootlocus:

Root Locus 20 0.88 0.62 0.35 0.968 15 F 0.986 10 0.997 30 10 50 Аг 70 60 5 0:997 -10 0986 15 0968 0.88 0.935 0.8 0.62 0.35 -70 60 50 40 -30 20 10 10 20 Real Axis (seconds1) Imaginary Axis (Se seconds

B): range of K for stability:

sys =

(s+10) (s+20)
------------------------
(s+30) (s^2 - 20s + 200)

Continuous-time zero/pole/gain model.

k = 23.9298 -------> it the K_max for stable system..

Root Locus 20 0.88 0.62 0.35 0.968 System: sys 15 F Gain: 24 ... 38+17.8i Dampha: 0.000772 0.986 Overshoot (% ): 99.8 Frequency (rad/'s) 17.8 10 0.997 30 10 50 Аг 70 60 5 0:997 -10 0986 15 0968 0.88 0.935 0.8 0.62 0.35 -70 60 50 40 -30 20 10 10 20 Real Axis (seconds1) Imaginary Axis (Se seconds

Root Locus 20 0.88 0.62 0.35 0968 System: sys Gain: 24 15 F Dampina: 0.000772 Overshoot (%): 99.8 Frequency (rad/s) 17.8 0.986 10 System: sys System: sys System: sys 04 e 64.9 0.997 Damping: 1 Overshoot (% ) : 0 Frequency (rad/s) 64.9 Damping Overshoot ( % ): 0 Frequency (rad/s) 48.7 Damping: 1 Overshoot ( % ): 0 Frequency (rad/s): 20 70 50 1 Sys Ca Gain: Inf Pole: -10 Pole: -15+3.28e-07 5 0:997 Damping: 1 Overshoot (%): 0 Frequency (rad/s Damping: 1 Overshoot (%): 0 Frequency (rad/'s) 10 15 -10 0986 15 0968 0.88 0.935 0.35 0.8 0.62 -70 60 50 40 -30 20 10 10 20 Real Axis (seconds1) Imaginary Axis (seconds

c):

Root Locus 20 0.88 0.8 0.62 0.35 0.968 15 F 0.986 System: sys 10 Bole:13+13 Damping: 0.707 Overshoot (%): 4.34 Frequency (rad/s): 18.4 0.997 10 50 Аг 30 70 60 5 0:997 -10 0986 15 0968 0.88 0.935 0.8 0.62 0.35 -70 60 50 40 -30 20 10 10 20 Real Axis (seconds1) Imaginary Axis (Se seconds

d):

4 T, =

=

4 0.707 18.4

=0.3075 sec

pi Tp P w l 3

= 0.2417.

e):

closed loop critcally damped poles exit at breaking points...

Root Locus 20 0.88 0.62 0.35 0.968 15 F 0.986 10 System: sys Pola:15+0.0783 0.997 Damping: 1 Overshoot (%): 0 Frequency (rad/s): 15 50 Аг 30 70 60 5 0:997 -10 0986 15 0968 0.88 0.935 0.8 0.62 0.35 -70 60 50 40 -30 20 10 10 20 Real Axis (seconds1) Imaginary Axis (Se seconds