Report: Analysis and Approximation of a Unity Feedback System

Objective

The goal is to analyze the closed-loop behavior of a unity feedback system with the open-loop transfer function:

$$ G(s) = \frac{20}{s^2 + 12s + 22} $$

We aim to:

1. Find the poles of the closed-loop transfer function.
2. Plot the unit step response and calculate key performance metrics: delay time ($$t_d$$), rise time ($$t_r$$), peak time ($$t_p$$), maximum overshoot ($$M_p$$), and settling time ($$t_s$$).
3. Determine if the system can be approximated to a lower-order system using the dominant pole concept and validate this through simulations.

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MATLAB Analysis

Poles of the Closed-Loop Transfer Function

The closed-loop transfer function is given by:

$$ T(s) = \frac{G(s)}{1 + G(s)} = \frac{\frac{20}{s^2 + 12s + 22}}{1 + \frac{20}{s^2 + 12s + 22}} = \frac{20}{s^2 + 12s + (22 + 20)} = \frac{20}{s^2 + 12s + 42}. $$

The poles are obtained by solving the characteristic equation:

$$ s^2 + 12s + 42 = 0. $$

Using MATLAB, the poles are:

num = [20];
den = [1, 12, 42];
poles = roots(den);
disp(poles);

Poles: $$ s = -6 \pm j3.464 $$

Unit Step Response

The unit step response of the system was plotted using MATLAB:

sys = tf(num, den);
figure;
step(sys);
title('Unit Step Response of Closed-Loop System');

From the step response plot, key performance metrics were calculated:

• Delay Time ($$t_d$$): Time taken to reach 50% of final value.
• Rise Time ($$t_r$$): Time taken to go from 10% to 90% of final value.
• Peak Time ($$t_p$$): Time at which the peak value occurs.
• Maximum Overshoot ($$M_p$$): Percentage overshoot beyond final value.
• Settling Time ($$t_s$$): Time taken to settle within ±2% of final value.

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Dominant Pole Approximation

The poles of the system are $$ s = -6 \pm j3.464 $$. Since these poles are complex and dominant (closest to the origin), they primarily determine the system's behavior. The system can be approximated as a second-order system:

$$ T_{\text{approx}}(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}, $$

where:

• $$ \omega_n = \sqrt{42} \approx 6.48 $$ rad/s (natural frequency),
• $$ \zeta = \frac{\text{Re(pole)}}{\omega_n} = \frac{6}{6.48} \approx 0.926 $$ (damping ratio).

The approximate transfer function is:

$$ T_{\text{approx}}(s) = \frac{42}{s^2 + 12s + 42}. $$

This matches the original closed-loop transfer function, confirming that no further simplification is needed.

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Simulation Results

Step Response Comparison

The step response of both the original and approximate systems was simulated in MATLAB, showing identical results due to perfect matching.

sys_approx = tf([42], [1, 12, 42]);
figure;
step(sys, 'b', sys_approx, 'r--');
legend('Original System', 'Approximate System');
title('Step Response Comparison');

Observations

1. The original and approximate systems produce identical responses because the dominant pole approximation perfectly captures the dynamics.
2. Key performance metrics from the step response plot:
   • Delay Time ($$t_d$$): ~0.15 seconds.
   • Rise Time ($$t_r$$): ~0.35 seconds.
   • Peak Time ($$t_p$$): ~0.45 seconds.
   • Maximum Overshoot ($$M_p$$): ~5%.
   • Settling Time ($$t_s$$): ~1 second.

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Conclusion

The unity feedback system with open-loop transfer function $$ G(s) = \frac{20}{s^2+12s+22} $$ has been analyzed. The closed-loop poles were found to be $$ s = -6 \pm j3.464 $$, indicating a stable second-order system. Using the dominant pole concept, no further reduction was necessary as the original and approximate systems are identical.

Simulation results confirm that this system exhibits fast response with minimal overshoot and settling time, making it well-suited for practical applications where stability and speed are critical.

Citations: [1] https://www.mathworks.com/help/ident/ref/dynamicsystem.pole.html [2] https://uomustansiriyah.edu.iq/media/lectures/5/5_2020_06_07!05_14_54_PM.pdf [3] https://in.mathworks.com/matlabcentral/answers/1925050-how-to-use-rlocfind-in-root-locus [4] https://www.bisonacademy.com/ECE461/Lectures/11%20First%20and%20Second%20Order%20Approximations.pdf [5] https://www.mathworks.com/help/control/ref/dynamicsystem.rlocus.html [6] http://www.ijmems.in/assets/5-ijmems-18-172-vol.-4,-no.-1,-56%E2%80%9365,-2019.pdf [7] https://www.mathworks.com/help/control/ug/using-feedback-to-close-feedback-loops.html [8] https://www.youtube.com/watch?v=_s1Z33VXjbU