Second-Order System and Unit Step Response

Introduction

A second-order system is a system whose dynamics can be described by a second-order differential equation. It is commonly represented in the form: \[ \frac{d^2y(t)}{dt^2} + 2\zeta\omega_n \frac{dy(t)}{dt} + \omega_n^2 y(t) = \omega_n^2 u(t) \] where:

• \( y(t) \) is the output of the system,
• \( u(t) \) is the input to the system,
• \( \zeta \) (zeta) is the damping ratio,
• \( \omega_n \) (omega_n) is the natural frequency.

Transfer Function

The transfer function of a second-order system is given by: \[ H(s) = \frac{Y(s)}{U(s)} = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} \] where \( s \) is the Laplace variable.

Unit Step Response

The unit step response of a second-order system is the output \( y(t) \) when the input \( u(t) \) is a unit step function. The unit step function is defined as: \[ u(t) = \begin{cases} 0 & \text{for } t < 0, \\ 1 & \text{for } t \geq 0. \end{cases} \] The step response depends on the damping ratio \( \zeta \):

• Underdamped (\( 0 < \zeta < 1 \)): The system oscillates before settling to the steady-state value.
• Critically damped (\( \zeta = 1 \)): The system reaches the steady-state value without oscillation in the minimum possible time.
• Overdamped (\( \zeta > 1 \)): The system approaches the steady-state value slowly without oscillation.

Plot of Unit Step Response

Below is a plot of the unit step response for a second-order system with \( \omega_n = 1 \) and different values of \( \zeta \):

Conclusion

The unit step response of a second-order system provides insight into the system's behavior, such as its stability, speed of response, and oscillation characteristics. The damping ratio \( \zeta \) plays a crucial role in determining the nature of the response.