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The damping ratio, denoted by \( \zeta \), is a key parameter in a second-order system as it determines how oscillations in the system decay over time. When analyzing the overshoot in a control system, we use this ratio to describe the extent to which the system oscillates around its final value before settling.
The formula relating overshoot (\( \text{OS} \)) to the damping ratio is given by: \[ \text{OS} \approx 100 \cdot e^{\frac{-\pi \zeta}{\sqrt{1-\zeta^{2}}}} \].
Overshoot is the maximum peak value of the response curve as a percentage over the final value of the response.
For a desired overshoot of approximately 5%, we set up the equation: \[ 5 = 100 \cdot e^{\frac{-\pi \zeta}{\sqrt{1-\zeta^{2}}}} \].
Then, we solve for \( \zeta \) by rearranging the terms:
First, convert the overshoot percentage into a decimal: \[ 0.05 = e^{\frac{-\pi \zeta}{\sqrt{1-\zeta^{2}}}} \].
Next, take the natural logarithm on both sides to remove the exponential: \[ \ln(0.05) = \frac{-\pi \zeta}{\sqrt{1-\zeta^{2}}} \].
Rearrange the expression to isolate \( \zeta \): \[ \zeta \approx 0.69 \].
This calculation shows that a damping ratio of around 0.69 is required for achieving the desired overshoot of 5%.

The natural frequency, denoted as \( \omega_n \), represents the frequency at which the system oscillates in the absence of damping. It's fundamental to assessing how quickly a system responds.
The settling time (\( \text{ST} \)) is the time it takes for the system to settle within a certain percentage of its final value. For our purposes, we use the 2% criterion, which means the system should settle within 2% of the final value.
The formula relating settling time to natural frequency and damping ratio is given by: \[ \text{ST} \approx \frac{4}{\zeta \omega_{n}} \].
With a desired settling time of 2 seconds, we substitute the known values: \[ 2 \approx \frac{4}{0.69 \omega_{n}} \].
Solving for \( \omega_{n} \), we rearrange the formula: \[ \omega_{n} \approx \frac{4}{2 \cdot 0.69} \approx 2.90 \, \text{rad/s} \].
This shows that the system's natural frequency must be around 2.90 radians per second to meet the settling time requirement.

For any control system, especially a second-order system, performance is typically assessed based on specific criteria such as overshoot, settling time, steady-state error, and rise time. These parameters help determine how well a system behaves in response to a given input.

• Overshoot: Refers to how much the system output exceeds the desired response. It's commonly expressed as a percentage.
• Settling Time: Defined as the time it takes for the system response to remain within a certain percentage (2%) of its final value.
• Natural Frequency: Indicates how fast the system oscillates naturally without damping. It's critical in defining the speed of the system's response.
• Damping Ratio: Gives insight into how oscillations in the system decay. It's essential for ensuring the system doesn't oscillate excessively and achieves a smooth response.

By carefully analyzing and calculating these parameters, we can design and tweak control systems to achieve desired performance criteria. For instance, ensuring a damping ratio of approximately 0.69 and a natural frequency of 2.90 rad/s, as calculated earlier, helps ensure the system meets the required 5% overshoot and 2-second settling time specifications.