Question

PROBLEM 1 A closed loop control system is depicted in Figure 1 where K 2 and the nominal value of a is 1. The goal in this problem is to examine the sensitivity of the whole system in Figure 1 to the parameter a (a) If a = 1, demonstrate explicitly that the steady state value of output is equal to 2 when input is unit step function. Confirm that, after 4 seconds, the unit step response is within 2% of the steady state value. (b) For a 0.5, 2, and 5, obtain the graph of the unit step response. Associate the results to the sensitivity of the closed loop system to deviation of parameter a. Pos Figure 1: A system with uncertain parameter a.

Please solve them on MATLAB

Answer 1

matlab code:

for a=[1 0.5 2 5]
k=2;
z=[];
p=[a];
s1=zpk(z,p,k)
sys=feedback(s1,1)
figure
step(sys),grid;
stepinfo(sys)
end

output:

s1 =

2
-----
(s-1)

Continuous-time zero/pole/gain model.

sys =

2
-----
(s+1)

Continuous-time zero/pole/gain model.

ans =

struct with fields:

RiseTime: 2.1970
SettlingTime: 3.9121
SettlingMin: 1.8090
SettlingMax: 1.9999
Overshoot: 0
Undershoot: 0
Peak: 1.9999
PeakTime: 10.5458

s1 =

2
-------
(s-0.5)

Continuous-time zero/pole/gain model.

sys =

2
-------
(s+1.5)

Continuous-time zero/pole/gain model.

ans =

struct with fields:

RiseTime: 1.4647
SettlingTime: 2.6080
SettlingMin: 1.2060
SettlingMax: 1.3333
Overshoot: 0
Undershoot: 0
Peak: 1.3333
PeakTime: 7.0306

s1 =

2
-----
(s-2)

Continuous-time zero/pole/gain model.

sys =

2
-
s

Continuous-time zero/pole/gain model.

ans =

struct with fields:

RiseTime: NaN
SettlingTime: NaN
SettlingMin: NaN
SettlingMax: NaN
Overshoot: NaN
Undershoot: NaN
Peak: Inf
PeakTime: Inf

s1 =

2
-----
(s-5)

Continuous-time zero/pole/gain model.

sys =

2
-----
(s-3)

Continuous-time zero/pole/gain model.

ans =

struct with fields:

RiseTime: NaN
SettlingTime: NaN
SettlingMin: NaN
SettlingMax: NaN
Overshoot: NaN
Undershoot: NaN
Peak: Inf
PeakTime: Inf

Step Response 1.6 1.2 0.8 0.6 0.4 0.2 5 Time (seconds)

Step Response 1.2 0.8 0.6 0.4 H 0.2 Time (seconds)

Step Response 100 15 10 Time (seconds) 5

Step Response 25 x 10 20 10 12 14 16 18 Time (seconds)