Here’s how to approach the problem using MATLAB step by step:

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(a) Design a Lead Compensator:

Steps:

1. Define the transfer function $G(s)$ in MATLAB.
2. Use the bode function to analyze the phase margin of $G(s)$.
3. Design a lead compensator $C(s)$ to improve the phase margin to at least 40° while maintaining unity DC gain.
4. Tune the parameters of the lead compensator to achieve the desired phase margin.

matlabCopy code
% Define the transfer function G(s)
num = 50 * [1 3];  % Numerator: 50(s + 3)
den = conv([1 0], conv([1 2], [1 5]));  % Denominator: s(s + 2)(s + 5)
G = tf(num, den);

% Plot the Bode plot for G(s)
figure;
bode(G);
grid on;
title('Bode Plot of G(s)');

% Design a lead compensator C(s)
% Assume the compensator form: C(s) = K * (s + z) / (s + p)
K = 1;          % Unity DC gain
z = 10;         % Zero location (adjustable)
p = 20;         % Pole location (adjustable)
C = K * tf([1 z], [1 p]);

% Check the Bode plot with compensator
Gc = series(C, G);
figure;
bode(Gc);
grid on;
title('Bode Plot of Compensated System');

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(b) Verify Phase Margin:

Steps:

1. Use the margin function to verify the phase margin.
2. Check if the phase margin is at least 40°.

matlabCopy code
% Verify phase margin
[Gm, Pm, Wcg, Wcp] = margin(Gc);
fprintf('Phase Margin: %.2f degrees\n', Pm);
fprintf('Gain Margin: %.2f dB\n', 20*log10(Gm));
fprintf('Gain Crossover Frequency: %.2f rad/s\n', Wcg);
fprintf('Phase Crossover Frequency: %.2f rad/s\n', Wcp);

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(c) Find the Bandwidth of the Closed-Loop System:

Steps:

1. Create the closed-loop transfer function.
2. Use the bandwidth function to calculate the system bandwidth.

matlabCopy code
% Define the closed-loop transfer function
T = feedback(Gc, 1);  % Unity feedback system

% Calculate the bandwidth
BW = bandwidth(T);
fprintf('Closed-Loop System Bandwidth: %.2f rad/s\n', BW);

% Plot the Bode plot of the closed-loop system
figure;
bode(T);
grid on;
title('Bode Plot of Closed-Loop System');

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Key Explanations:

1. Lead Compensator Design: The lead compensator adds phase to improve the phase margin by introducing a zero-pole pair. The zero $z$ and pole $p$ are tuned iteratively.
2. Phase Margin Verification: The margin function provides the gain and phase margins directly.
3. Bandwidth: The bandwidth function measures the frequency range where the system remains effective.

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Expected Outputs:

• Bode plot of the uncompensated system.
• Bode plot of the compensated system with $C(s)$.
• Phase margin after adding the compensator (ensure $Pm \geq 40^\circ$).
• Bandwidth of the closed-loop system.

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Would you like further details on tuning the compensator or MATLAB code clarification?

Related Questions:

1. How does the lead compensator affect the phase margin?
2. What is the significance of the zero and pole locations in $C(s)$?
3. How can MATLAB's sisotool simplify the compensator design?
4. What happens if the phase margin is less than 40°?
5. How does the closed-loop bandwidth affect system performance?

Tip:

Use margin and bandwidth functions iteratively to fine-tune your compensator parameters for optimal results.