Second-Order Control System: Unit Step Response

 

Second-Order Control System: Unit Step Response

A standard second-order system is represented by the transfer function:

$$G(s)=\frac{{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}$$

where:

• ${\omega }_n$ = natural frequency

• $\zeta$ = damping ratio

Unit Step Response (Conceptual Form)

For a unit step input, the time response depends on the damping ratio $\zeta$:

• Underdamped ($0<\zeta <1$) Oscillatory response with decaying amplitude.

$$c(t)=1-\frac{e^{-\zeta {\omega }_{n}t}}{\sqrt{1-{\zeta }^2}}\sin ({\omega }_{d}t+\varphi )$$

where ${\omega }_d={\omega }_n\sqrt{1-{\zeta }^2}$.

• Critically damped ($\zeta =1$) Fastest non-oscillatory response.

$$c(t)=1-(1+{\omega }_{n}t)e^{-{\omega }_{n}t}$$

• Overdamped ($\zeta >1$) Slow, non-oscillatory response with two exponential terms.

Effect of Damping

• Underdamped ($0<\zeta <1$) → Oscillations occur before settling.

• Critically damped ($\zeta =1$) → No oscillations, fastest approach to steady state.

• Overdamped ($\zeta >1$) → No oscillations, but slower response compared to critical damping.

• Zero damping ($\zeta =0$) → Sustained oscillations (pure sinusoidal).

Key Performance Measures

• Rise time: Time to go from 0% to 100% of final value.

• Peak time: Time to first maximum overshoot.

• Maximum overshoot: How much the response exceeds final value.

• Settling time: Time to stay within a tolerance band (usually 2% or 5%).

Think of damping as the "brake" on oscillations:

• Too little → system vibrates.

• Too much → system becomes sluggish.

• Just right (critical) → smooth and fast response.