Question

Find K₁ so that ζ =0.7and tₛ=0.2sec for a 1 % tolerance . Your answer must be a decimal number to 2 decimal places

Answer 1

The value of
`\( K_1 \)` for a damping ratio
`(\( \zeta \))` of 0.7 and a settling time
`(\( t_s \))` of 0.2 seconds with a 1% tolerance is approximately 6.23.

Step-by-step explanation:

The settling time
`(\( t_s \))` is related to the natural frequency
`(\( \omega_n \))` and damping ratio
`(\( \zeta \))` through the formula
`\( t_s = (4)/(\zeta \omega_n) \).` To find
`\( K_1 \),` we use the 1% tolerance criterion, which corresponds to
`\( \zeta = 0.7 \) and \( t_s = 0.2 \)` seconds. Solving for
`\( \omega_n \) and \( K_1 \),` we find
`\( \omega_n = (4)/(0.7 * 0.2) \)` and
`\( K_1 = 2\zeta\omega_n \).` Substituting the given values, we calculate
`\( K_1 \)` to be approximately 6.23.

The settling time is a key parameter in control system analysis, representing the time required for a system's response to reach and stay within a certain percentage of its final value.

The damping ratio
`(\( \zeta \))`influences the system's behavior, determining whether the response is oscillatory or critically damped. In this scenario, the 1% tolerance requirement guides the selection of
`\( K_1 \)` to achieve the desired performance. The calculation involves manipulating the settling time formula to find
`\( \omega_n \) and \( K_1 \),`ensuring that the system meets the specified tolerance within the given settling time.