🧠 TIME DOMAIN ANALYSIS — COMPREHENSIVE GATE CHEAT
SHEET
1️⃣ SYSTEM TYPE & STEADY-STATE ERROR
System Type = Number of poles at origin in open-loop G(s)
Type Poles at s=0 Step Error Ramp Error Parabolic Error
0 0 finite ∞ ∞
1 1 0 finite ∞
2 2 0 0 finite
Static Error Constants (Unity Feedback):
Kₚ = lim(s→0) G(s)
Kᵥ = lim(s→0) s·G(s)
Kₐ = lim(s→0) s²·G(s)
Steady-State Error Formulas:
eₛₛ(step) = 1/(1+Kₚ)
eₛₛ(ramp) = 1/Kᵥ
eₛₛ(parabolic) = 1/Kₐ
⚠️ KEY POINTS:
For non-unity feedback: Use ess = R(s)/(1+G(s)H(s)) approach
Disturbance input: Use superposition principle
Higher type = better tracking, but harder to stabilize
2️⃣ FIRST-ORDER SYSTEM
Transfer Function:
G(s) = K/(τs+1)
Time Response Characteristics:
, Final Value = K
At t = τ: Output reaches 63.2% of final value
At t = 3τ: Output reaches 95% (settling approx)
At t = 5τ: Output reaches 99.3%
Key Formulas:
Settling Time (2%): Tₛ = 4τ
Settling Time (5%): Tₛ = 3τ
Rise Time: Tᵣ ≈ 2.2τ
Time Constant: τ = 1/pole location
Step Response:
c(t) = K(1 - e^(-t/τ))u(t)
💡 QUICK TRICK: If graph shows time to 63.2% → that's your τ!
3️⃣ SECOND-ORDER STANDARD FORM ⭐
Closed-Loop Transfer Function:
G(s) = ωₙ²/(s² + 2ζωₙs + ωₙ²)
OR in pole-zero form:
G(s) = ωₙ²/((s - p₁)(s - p₂))
System Parameters:
ωₙ = Natural frequency (rad/s)
ζ = Damping ratio (dimensionless)
ωd = Damped frequency = ωₙ√(1-ζ²)
Pole Locations:
s = -ζωₙ ± jωₙ√(1-ζ²) = -σ ± jωd
where σ = ζωₙ (real part)
From Pole Location:
SHEET
1️⃣ SYSTEM TYPE & STEADY-STATE ERROR
System Type = Number of poles at origin in open-loop G(s)
Type Poles at s=0 Step Error Ramp Error Parabolic Error
0 0 finite ∞ ∞
1 1 0 finite ∞
2 2 0 0 finite
Static Error Constants (Unity Feedback):
Kₚ = lim(s→0) G(s)
Kᵥ = lim(s→0) s·G(s)
Kₐ = lim(s→0) s²·G(s)
Steady-State Error Formulas:
eₛₛ(step) = 1/(1+Kₚ)
eₛₛ(ramp) = 1/Kᵥ
eₛₛ(parabolic) = 1/Kₐ
⚠️ KEY POINTS:
For non-unity feedback: Use ess = R(s)/(1+G(s)H(s)) approach
Disturbance input: Use superposition principle
Higher type = better tracking, but harder to stabilize
2️⃣ FIRST-ORDER SYSTEM
Transfer Function:
G(s) = K/(τs+1)
Time Response Characteristics:
, Final Value = K
At t = τ: Output reaches 63.2% of final value
At t = 3τ: Output reaches 95% (settling approx)
At t = 5τ: Output reaches 99.3%
Key Formulas:
Settling Time (2%): Tₛ = 4τ
Settling Time (5%): Tₛ = 3τ
Rise Time: Tᵣ ≈ 2.2τ
Time Constant: τ = 1/pole location
Step Response:
c(t) = K(1 - e^(-t/τ))u(t)
💡 QUICK TRICK: If graph shows time to 63.2% → that's your τ!
3️⃣ SECOND-ORDER STANDARD FORM ⭐
Closed-Loop Transfer Function:
G(s) = ωₙ²/(s² + 2ζωₙs + ωₙ²)
OR in pole-zero form:
G(s) = ωₙ²/((s - p₁)(s - p₂))
System Parameters:
ωₙ = Natural frequency (rad/s)
ζ = Damping ratio (dimensionless)
ωd = Damped frequency = ωₙ√(1-ζ²)
Pole Locations:
s = -ζωₙ ± jωₙ√(1-ζ²) = -σ ± jωd
where σ = ζωₙ (real part)
From Pole Location:
