Unit Impulse Response of 2nd Order System - Control Systems - Electrical Engineering

Time Domain Characteristics

When we describe the transient response of a control system to a standard input (commonly a unit step), several time-domain specifications are used to quantify performance. These specifications help compare systems and guide design choices.

• Delay time (t_d): Time required for the response to reach 50% of the final value on its first attempt.

• Rise time (t_r): Time required for the response to rise from 0% to 100% of the final value for an under-damped second-order system.

• Peak time (t_p): Time required for the response to reach the first maximum (peak) of the time response (first overshoot).
  Expression: t_p = nπ / ω_d seconds.
  For the first peak (first maxima), n = 1, so t_p = π / ω_d.
• Settling time (t_s): Time required for the response to enter and remain within a specified tolerance band (commonly 2% or 5%) of the final value.


• Peak overshoot (M_p): Normalised difference between the peak value of the response and the steady-state (final) value.



Relation with damping ratio: For a standard under-damped second-order system, the peak overshoot is given by

M_p = e^{-ζπ / √(1 - ζ²)}

Where tan φ =


• Steady-state error (e_ss): The difference between the desired output and the actual output as t → ∞.

Unit Impulse Response of a Second-Order System

This section derives and summarises the unit impulse response of a standard linear time-invariant second-order system. The unit impulse response is the system output when the input is a unit impulse δ(t). It is the inverse Laplace transform of the system transfer function when the input Laplace transform equals 1.

Standard second-order transfer function

The commonly used standard form is

G(s) = ω_n² / (s² + 2ζω_ns + ω_n²)

Here ω_n is the natural frequency and ζ is the damping ratio. The damped natural frequency is

ω_d = ω_n √(1 - ζ²)

Impulse response: general relation

For a unit impulse input, Y(s) = G(s) · 1 = G(s). Therefore the impulse response y(t) is the inverse Laplace transform of G(s).

Underdamped case (0 < ζ < 1)

Poles are complex conjugates and the inverse Laplace gives a decaying sinusoid.

The impulse response is

y(t) = (ω_n / ω_d) e^-ζω_nt sin(ω_d t), for t > 0

Critically damped case (ζ = 1)

Poles are repeated at s = -ω_n; partial fraction inversion yields

y(t) = ω_n² t e^{-ω_n t}, for t > 0

Overdamped case (ζ > 1)

Poles are distinct real negatives, s₁ and s₂, where

s₁ = -ω_n(ζ + √(ζ² - 1)), s₂ = -ω_n(ζ - √(ζ² - 1))

Using partial fractions:

y(t) = (ω_n² / (s₁ - s₂)) [ e^{s₁ t} - e^{s₂ t} ], for t > 0

Connection between impulse and step responses

The unit step response is the time integral of the impulse response.

If c(t) denotes the step response and g(t) the impulse response, then

c(t) = ∫₀ᵗ g(τ) dτ

This relation is often used to derive step-response specifications from the impulse response and vice versa.

Time-domain specifications in terms of ζ and ω_n

• Peak time: t_p = π / ω_d = π / (ω_n √(1 - ζ²)).
• Peak overshoot: M_p = e^{-ζπ / √(1 - ζ²)}.
• Rise time (under-damped): t_r ≈ (π - φ) / ω_d, where φ = arccos(ζ) = arctan(√(1 - ζ²) / ζ).
• Settling time: For common tolerance bands,
For 2% tolerance, t_s ≈ 4 / (ζ ω_n).
For 5% tolerance, t_s ≈ 3 / (ζ ω_n).

Effect of Adding a Zero to a System

If a zero at s = -z is added to a second-order system, the closed-loop/numerator polynomial changes and the transient response is modified. Consider the modified transfer function with a zero at s = -z. The mathematical form and algebraic manipulation are shown in the figures below.

• The multiplicative factor is often adjusted so that the steady-state gain of the modified system remains unity.

• The added zero introduces a derivative-like term in the time response which tends to produce an early pronounced peak (peaking) in the transient response.
• The closer the zero is to the origin, the more pronounced the peaking phenomenon becomes.
• Because of this effect, real-axis zeros close to the origin are generally avoided unless deliberately used to improve a sluggish system. Introducing a suitably placed zero can improve rise time and reduce steady lag in some designs.

Types of Feedback Control System

The type of a feedback system is determined by the number of pure integrators (poles at the origin) in the open-loop transfer function. If the open-loop transfer function is written as:

• If n = 0, the system is called a type-0 system.
• If n = 1, the system is called a type-1 system.
• If n = 2, the system is called a type-2 system, and so on.

Steady-State Error and Error Constants

The steady-state performance of a stable feedback control system is judged by its steady-state error in tracking standard inputs: step, ramp and parabolic. For a unity feedback system, the block diagram relation gives an error transfer function which determines e(s) in terms of the input R(s) and the open-loop transfer function G(s)H(s).

For a unity feedback configuration, the error E(s) is

Thus, steady-state error depends on the input R(s) and the forward (open-loop) transfer function G(s). The following error constants are defined for standard inputs:

1. Unit step input (R(t) = u(t))
   R(s) = 1 / s.
   The steady-state error is
   

   where

   
   K_p is the positional error constant.
2. Unit ramp input (R(t) = t u(t))
   R(s) = 1 / s².
   The steady-state error is
   

   e_ss = 1 / K_v

   where

   
   K_v is the velocity error constant.
3. Unit parabolic input (R(t) = 0.5 t² u(t))
   R(s) = 1 / s³.
   The steady-state error is
   

   e_ss = 1 / K_a

   where

   
   K_a is the acceleration error constant.

The values of K_p, K_v and K_a and the system type determine whether the steady-state error is finite or zero for the respective input types. Increasing system type (adding integrators) improves steady-state accuracy for ramp and parabolic inputs but affects transient behaviour and stability; design requires a trade-off.

Summary

The unit impulse response of a second-order system reveals the essential transient behaviour: exponential decay, oscillation frequency and the effect of damping. Time-domain specifications such as rise time, peak time, settling time and overshoot can be expressed in terms of the damping ratio ζ and natural frequency ω_n. Adding zeros alters transient response (often producing peaking) and must be used with care. Steady-state tracking error for standard inputs is determined by system type and error constants K_p, K_v, K_a, which guide the selection of controller structure and gains.