Second Order Underdamped Response

The underdamped second-order system is the most widely analyzed case in control systems. When the damping ratio ζ lies strictly between 0 and 1, the step response oscillates while decaying toward the final value. This oscillatory transient defines all the key time-domain performance specifications such as overshoot, peak time, and settling time, which are directly asked in GATE and form the design basis for most real controllers.

Core Concept: Underdamped Behavior

When 0 < ζ < 1, the characteristic equation has complex conjugate poles s = -ζωn ± jωd, where ωd = ωn√(1-ζ²) is the damped natural frequency. The unit step response in the time domain is c(t) = 1 - (e^(-ζωnt)/√(1-ζ²))·sin(ωdt + β), where β = cos⁻¹(ζ). This expression shows that the response is a sinusoid at frequency ωd whose amplitude decays as an exponential envelope with time constant 1/(ζωn). The output oscillates about the final value of 1 before settling.

The oscillation frequency ωd is always less than ωn because the exponential decay is caused by the real part -ζωn. The quantity ζωn is called the damping coefficient or attenuation, and its reciprocal 1/(ζωn) is the time constant of the decaying envelope. The larger the product ζωn, the faster the oscillations die out. This means two systems can have the same ωn but very different settling times if their ζ values differ.

Time Domain Specifications

The percent overshoot (%OS) is the maximum amount by which the response exceeds its final value, expressed as a percentage. It is given by: %OS = exp(-πζ/√(1-ζ²)) × 100. This formula is a function of ζ alone, independent of ωn. At ζ = 0.707, %OS = 4.3%, which is the standard design target for many systems. At ζ = 0.5, %OS = 16.3%. The overshoot increases steeply as ζ decreases toward zero.

The peak time Tp is the time at which the first peak (maximum overshoot) occurs. It equals Tp = π/ωd. This is derived by differentiating c(t) with respect to time and setting it to zero; the first solution is t = π/ωd. The settling time Ts is the time after which the response remains within ±2% of final value. The standard formula is Ts = 4/(ζωn). For the ±5% band, Ts = 3/(ζωn). These follow from solving the decaying envelope condition |e^(-ζωnt)| < 0.02.

The rise time Tr for underdamped systems is Tr = (π - β)/ωd where β = cos⁻¹(ζ) in radians. This is valid for 0 < ζ < 1. Note that rise time decreases as ωn increases, meaning a higher natural frequency gives faster initial rise. However, a faster rise also typically means more overshoot if ζ is kept constant, revealing the inherent speed-accuracy tradeoff in control system design.

Practical Understanding

In real systems, ζ = 0.707 is widely used as a design target because it provides a good compromise: the 4.3% overshoot is acceptable in most applications, and the settling time is near-optimal. This value comes from the maximally flat (Butterworth) criterion. Systems with ζ below 0.3 are considered poorly damped and will oscillate noticeably, which is undesirable in mechanical and process control applications. Automobile suspension systems, for instance, are designed with ζ near 0.6 to 0.7 to prevent bouncing while responding adequately to road inputs.

Given:
Second-order system: T(s) = 25 / (s² + 4s + 25)
Input: Unit step

Why this formula applies:
Identify ωn and ζ first, then apply underdamped response formulas

Formula:
ωn² = 25 → ωn = 5 rad/s
2ζωn = 4 → ζ = 4/(2×5) = 0.4
ωd = ωn√(1-ζ²)
%OS = exp(-πζ/√(1-ζ²)) × 100
Tp = π/ωd
Ts = 4/(ζωn)

Substitution:
ωd = 5×√(1-0.16) = 5×0.9165 = 4.58 rad/s
%OS = exp(-π×0.4/√(1-0.16)) × 100 = exp(-1.452) × 100
Tp = π/4.58
Ts = 4/(0.4×5)

Calculation:
%OS = e^(-1.452) × 100 = 0.2339 × 100 = 23.4%
Tp = 3.1416/4.58 = 0.686 s
Ts = 4/2 = 2 s (2% settling)

Final Answer:
ωn=5 rad/s, ζ=0.4, ωd=4.58 rad/s, OS=23.4%, Tp=0.686 s, Ts=2 s

Exam Tip: Memorize %OS = exp(-πζ/√(1-ζ²))×100 and Tp = π/ωd. These two formulas are the most frequently tested in GATE numerical questions. Also note: overshoot depends ONLY on ζ, not on ωn. Settling time depends on the product ζωn. If a GATE question changes ωn but keeps ζ fixed, overshoot does not change but settling time does.

Oscillatory Response Characteristics

• The response envelope decays as e^(-ζωnt). The time constant of this decay is τ = 1/(ζωn).
• The period of damped oscillations is Td = 2π/ωd. The response crosses the final value at every π/ωd seconds.
• At ζ = 0.707, %OS = 4.3% and ωd = ωn/√2. This is the most commonly used design point.
• As ζ approaches 1 from below, %OS approaches 0 and the response transitions to critically damped behavior.
• As ζ approaches 0, %OS approaches 100% and the number of visible oscillations increases sharply.

Quick Revision

• Underdamped: 0 < ζ < 1. Poles are complex conjugates at -ζωn ± jωd.
• Damped frequency: ωd = ωn√(1-ζ²). This is the actual frequency of oscillation in the step response.
• Percent overshoot: %OS = exp(-πζ/√(1-ζ²)) × 100. Depends only on ζ, not ωn.
• Peak time: Tp = π/ωd. Settling time (2%): Ts = 4/(ζωn). Rise time: Tr = (π-β)/ωd.
• Design standard: ζ = 0.707 gives 4.3% OS and near-optimal settling. Called ITAE-optimal for many systems.
• Trap: Tp = π/ωd uses ωd (damped), NOT ωn (natural). Using ωn gives wrong answer.
• Trap: Ts = 4/(ζωn) uses the product ζωn, not just ωn or just ζ separately.