First-Order Control System: Unit Step Response

 

First-Order Control System: Unit Step Response

A first-order control system is typically represented by the transfer function:

$$G(s)=\frac{K}{\tau s+1}$$

where:

• $K$ = system gain

• $\tau$ = time constant

Unit Step Input Analysis

For a unit step input $R(s)=\frac{1}{s}$, the output is:

$$C(s)=G(s)\cdot R(s)=\frac{K}{\tau s+1}\cdot \frac{1}{s}$$

Taking the inverse Laplace transform:

$$c(t)=K(1-e^{-\frac{t}{\tau }}),t\ge 0$$

Key Characteristics

• Initial value: $c(0)=0$

• Final value (steady state): $c(\mathrm{\infty })=K$

• Exponential rise: The response gradually approaches the final value.

Concept of Time Constant ($\tau$)

• The time constant is the time taken for the response to reach 63.2% of its final value.

• Physically, it indicates the speed of response:

  • Smaller $\tau$ → faster system response.

  • Larger $\tau$ → slower system response.

• After about 5τ, the system is considered to have reached steady state (≈99% of final value).

Intuitive Understanding

Think of the time constant as a measure of "system sluggishness." For example:

• An RC circuit has $\tau =RC$.

• If resistance or capacitance increases, the circuit takes longer to charge → larger $\tau$.