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Solution: Analyze the behavior of a simplified satellite attitude control system

Figure 1: Communication Satellite

Figure 2: Attitude Control Block Diagram

1. Determine the satellite's closed-loop transfer function with various system controllers

(General Closed-Loop Transfer Function Equation)

- When D(s) is a proportional gain
% Assume K = 5
% Note: There are a couple of techniques that you can use to solve this question.

%Technique One (using the ZPK function)	%Technique Two (using the TF function)
K = 5; z = []; % plant zeros p = [0 0]; % plant poles sys = zpk(z,p,K); sys1 = 0.1*sys; sysout1 = feedback(sys1,1); Answer: Zero/pole/gain: 0.5 ------------ (s^2 + 0.5)	K = 5; num1 = [1]; % numerator of plant den1 = [10 0 0]; % denominator of plant sysTF1 = tf(num1,den1); sysTF2 = K*sysTF1; sysout1 = feedback(sysTF2,1); Answer: Transfer function: 5 ------------- 10 s^2 + 5

- When D(s) is a PID controller
% Assume K = 25
K = 25;
zpid = [-0.5 -0.5];
ppid = [0];
syss = zpk(zpid,ppid,K);
sysg = tf(1,[10 0 0]);
sysss = series(syss,sysg);
sysout2 = feedback(sysss,1);

Answer:

Zero/pole/gain:
2.5 (s+0.5)^2
----------------------------------
(s+0.3622) (s^2 + 2.138s + 1.726)

- When D(s) is a lead-lag compensator
% Assume K = 25
K = 25;
zll = [-0.1];
pll = [-0.7];
sysserie1 = zpk(zll,pll,K);
systf = tf(1,[10 0 0]);
sysserie2 = series(sysserie1,systf);
sysout3 = feedback(sysserie2,1);

Answer:

Zero/pole/gain:
2.5 (s+0.1)
-----------------------------------
(s+0.1025) (s^2 + 0.5975s + 2.439)

2. Find the poles and zeros, plot them, and determine whether the system is stable
- When D(s) is a PID Controller
K = 25;
numpid = [4 4 1];
denpid = [4 0];
sys = tf(numpid,denpid);
syss = K*sys;
sysg = tf(1,[10 0 0]);
sysss = series(syss,sysg);
sysout2 = feedback(sysss,1);

% To find the poles and zeros of the TF
numfd = [100 100 25];
denfd = [40 100 100 25];
[zeros poles kgain] = tf2zp(numfd,denfd);

Answer:

zeros =
 -0.5000
 -0.5000 poles =
 -1.0689 + 0.7637i
 -1.0689 - 0.7637i
 -0.3622

% Plot the poles and zeros in the s-plane
pzmap(poles,zeros);
axis([-1.4 0.5 -1 1]);

Answer:

% Is the system stable?
The system is stable because all the poles are in the left-half plane of the s-plane.
3. Plot the system step response and find: Rise Time (tr), Settling Time (ts), Percent Overshoot (Mp)
- When D(s) is a lead-lag Compensator
K = 25;
numd = [1 0.1];
dend = [1 0.7];
sysd = tf(numd,dend);
sysserie1 = K*sysd;
systf = tf(1,[10 0 0]);
sysserie2 = series(sysserie1,systf);
sysout3 = feedback(sysserie2,1);

% plot the system step response
step(sysout3);

Answer:

Mp = 57.6772 (percent overshoot)

tr = 0.7550 sec (rise time)

ts = 11.2947 sec (settling time)

4. Plot the root locus and determine the gain value with pole values at: -1.4±0.4i
- when D(s) is a PID Controller
K = 25;
numpid = [4 4 1];
denpid = [4 0];
sys = tf(numpid,denpid);
syss = K*sys;
sysg = tf(1,[10 0 0]);
sysss = series(syss,sysg);
sysout2 = feedback(sysss,1);
rlocus(sysout2);

Answer:

(You can use either RLOCFIND or the RLOCUS mouse tracing to find the specified gain value) Gain = 0.27 with poles at -1.4±0.4i

5. Plot the Bode and Nyquist as another means of determining system stability
- when D(s) is a lead-lag Compensator
K = 25;
numd = [1 0.1];
dend = [1 0.7];
sysd = tf(numd,dend);
sysserie1 = K*sysd;
systf = tf(1,[10 0 0]);
sysserie2 = series(sysserie1,systf);
sysout3 = feedback(sysserie2,1);

% get numerator and denominator values from the closed-loop transfer function
numllc = [25 2.5]; denllc = [10 7 25 2.5];
bode(numllc,denllc); figure;
nyquist(numllc,denllc);

Answer:

There is a positive phase margin from the Bode plot, approximately 31 degrees, and the gain margin is infinity. Therefore, the system is considered stable.

There are no clockwise encirclements of -1 (N) in the Nyquist plot and no unstable poles (P) of G(s). Therefore, from the Nyquist Criterion (Z = N - P) Z = 0 and so the system is stable.