Time Response of First Order System - Control Systems - Electrical Engineering (EE)

In a transfer function, when the highest power of s appears only in the denominator and that power is one, the system is a first-order system. A standard, widely used canonical form of a stable first-order linear time-invariant (LTI) system transfer function is

G(s) = K / (τ s + 1)

where K is the steady-state (DC) gain and τ (often written as T) is the time constant. The time constant characterises how quickly the system responds: a smaller τ gives a faster response; a larger τ gives a slower response. By definition, the time constant τ is the time required for the response to a unit step to reach approximately 63.2% of its final (steady-state) value.

Response of first-order system to a unit step input

Consider the system with transfer function G(s) = K / (τ s + 1) and a unit step input r(t) = u(t). The Laplace transform of the unit step is R(s) = 1/s. The output Y(s) is

Y(s) = G(s) · R(s)

Y(s) = K / (τ s + 1) · 1/s

Use partial fraction expansion to express Y(s) in terms whose inverse Laplace transforms are standard.

Y(s) = K / [s(τ s + 1)]

Assume

Y(s) = A/s + B/(τ s + 1)

Multiply both sides by s(τ s + 1) to find constants

K = A(τ s + 1) + B s

Equate coefficients of like powers of s:

Coefficient of s: 0 = A τ + B

Constant term: K = A

Thus A = K and B = -K τ.

Therefore

Y(s) = K/s - Kτ/(τ s + 1)

Taking inverse Laplace term by term gives the time-domain response

y(t) = K [1 - e^(-t/τ)] · u(t)

For the common unity-gain case (K = 1) the response simplifies to

y(t) = 1 - e^(-t/τ)

At t = τ, y(τ) = 1 - e^(-1) ≈ 0.632 × final value, which is the origin of the 63.2% definition of the time constant.

Practical timing measures for first-order systems

The following approximate relations are useful in design and analysis:

• Time constant: τ (1 time constant)
• Rise time (10%-90%): approximately 2.2 τ
• Settling time (to within 2%): approximately 4 τ
• Settling time (to within 1%): approximately 5 τ

Time response of a first-order system to a unit ramp input

Let the input be a unit ramp r(t) = t · u(t). Its Laplace transform is R(s) = 1 / s². For the same forward transfer function G(s) = K / (τ s + 1), the output is

Y(s) = G(s) · R(s) = K / (τ s + 1) · 1 / s²

Perform partial fraction expansion of Y(s):

Y(s) = K / [s²(τ s + 1)]

Assume

Y(s) = A/s + B/s² + C/(τ s + 1)

Multiply both sides by s²(τ s + 1) and equate coefficients to solve for A, B, C.

Result (after equating coefficients):

B = K

A = -K τ

C = K τ²

So

Y(s) = -K τ / s + K / s² + K τ² / (τ s + 1)

Take inverse Laplace term by term to obtain y(t):

y(t) = -K τ + K t + K τ e^(-t/τ)

Rearrange

y(t) = K t - K τ [1 - e^(-t/τ)]

Define the tracking error e(t) = r(t) - y(t). For r(t) = t,

e(t) = t - [K t - K τ + K τ e^(-t/τ)]

e(t) = (1 - K) t + K τ [1 - e^(-t/τ)]

For the unity forward-gain case (K = 1), the transient term proportional to t cancels and the error reduces to

e(t) = τ [1 - e^(-t/τ)]

The steady-state error as t → ∞ is therefore

e_ss = τ (for K = 1)

Thus, for a first-order system without an integrator in the forward path, the steady-state error to a unit ramp is finite and equals the time constant τ (in the unity-gain forward path case). A smaller τ gives a smaller steady-state ramp error.

Response of first-order system to a unit impulse input

For a unit impulse input r(t) = δ(t), the Laplace transform is R(s) = 1. The output transform is

Y(s) = G(s) · R(s) = K / (τ s + 1)

Take inverse Laplace:

y(t) = (K / τ) e^(-t/τ) · u(t)

This is the impulse response of the first-order system. The impulse response magnitude at t = 0+ is K/τ and then decays exponentially with time constant τ.

Summary 

A stable first-order LTI system with transfer function G(s) = K/(τ s + 1) shows simple exponential behaviour for canonical inputs:

• Unit step: y(t) = K [1 - e^(-t/τ)]; final value = K; reaches 63.2% of final value at t = τ.
• Unit ramp: y(t) = K t - K τ [1 - e^(-t/τ)]; for K = 1, steady-state error to unit ramp = τ.
• Unit impulse: y(t) = (K/τ) e^(-t/τ); decays with time constant τ.

Key practical design points:

• Make τ small to obtain fast responses and small steady-state ramp error (for the forward-path unity case).
• Adding integrators (poles at the origin in the open-loop transfer function) changes the system type and eliminates or reduces steady-state errors to certain input types; for example, a system of type 1 has zero steady-state error to a ramp.
• Use the approximate relations for rise time and settling time to relate τ to performance specifications: rise time ≈ 2.2 τ (10%-90%), settling time ≈ 4 τ (2% criterion), or ≈ 5 τ (1% criterion).