Question

Given a unity feedback system that has the forward-path transfer function G(s)= K/ (s+2)(s+4)(s+6) do the following:

Using a second-order approximation, design the value of K to yield 10% overshoot for a unit-step input.

Answer 1

To achieve a 10% overshoot in a unity feedback system with a second-order approximation
`(\(G(s) = (K)/((s+2)(s+4)(s+6))\))`, the value of
`\(K\)` is calculated using the damping ratio formula, resulting in a specific value for
`\(K\)`.

Step-by-step explanation:

To achieve a desired overshoot of 10% in a second-order system, we can use the following formula that relates the damping ratio
`(\(\zeta\))` and the overshoot
`(\%OS)`:

`\[ \%OS = 100 * e^{-(\zeta\pi)/(√(1-\zeta^2))} \]`

The damping ratio is related to the natural frequency
`(\(\omega_n\))` and the damping ratio
`(\(\zeta\))` by the equation:

`\[ \zeta = \frac{\text{ln}(\%OS/100)}{\sqrt{\pi^2 + [\text{ln}(\%OS/100)]^2}} \]`

For a second-order system with a transfer function
`\(G(s) = (K)/((s+a)(s+b))\)`, where
`\(a\)` and
`\(b\)` are the poles, the natural frequency
`(\(\omega_n\))` is given by:

`\[ \omega_n = √(ab) \]`

To achieve 10% overshoot, we can choose
`\(\zeta\)` such that
`\( \%OS = 10\%\)`, and then calculate
`\(K\)` accordingly.

**Detailed Calculation:**

1. **Given Information:**

- Forward-path transfer function
`\(G(s) = (K)/((s+2)(s+4)(s+6))\)`.

- Desired overshoot
`\( \%OS = 10\%\)`.

2. **Calculate Damping Ratio
`(\(\zeta\))`:**

- Use the formula
`\(\zeta = \frac{\text{ln}(\%OS/100)}{\sqrt{\pi^2 + [\text{ln}(\%OS/100)]^2}}\)`.

`- For \( \%OS = 10\%\), calculate \(\zeta\).`

3. **Calculate Natural Frequency
`(\(\omega_n\))`:**

- Use the transfer function to find
`\(a\)` and
`\(b\)`.

`- \(\omega_n = √(ab)\).`

4. **Determine
`\(K\)` for 10% Overshoot:**

- The desired damping ratio
`\(\zeta\)` is now known, use it to find
`\(K\)` by equating
`\(\zeta\)` to the damping ratio in the transfer function.

5. **Conclusion:**

- The value of
`\(K\)` required for a unity feedback system with a second-order approximation to achieve a 10% overshoot is now determined.

This calculation provides the necessary parameters for the second-order system to meet the specified overshoot criterion. The actual numerical values would depend on the specific calculations.