Question

Consider the closed loop system given by:

C(s)/ R(s)= ωₙ ²/ s² +2ζωₙ s+ωₙ²

Determine the values of ζ and ωₙ so that the system responds to a step input with approximately 5% overshoot and a settling time of 2 seconds (+/−2%)

Answer 1

The student is asked to calculate the damping ratio (ζ) and the natural frequency (ωₙ) of a closed-loop control system to match specified overshoot and settling time criteria.

Step-by-step explanation:

The subject of the question is a control systems problem that requires the determination of the damping ratio (ζ) and the natural frequency (ωₙ) for a closed-loop system given in terms of its transfer function. To achieve approximately 5% overshoot, one needs to find the corresponding ζ using the standard overshoot formula in control theory. The settling time (Ts) of 2 seconds is related to ζ and ωₙ through another standard formula associated with the natural response of a second-order system.

Using the standard relationship between overshoot (OS) and damping ratio (ζ), we have OS ≈ e(-ζπ)/(1-ζ2)0.5 for a step response. For an overshoot of 5%, ζ can be calculated accordingly. The settling time formula is Ts ≈ 4/(ζωₙ), where 2 seconds is the target settling time. By solving these two equations, we can find the values of ζ and ωₙ.