🎯 CONTROL SYSTEMS PREREQUISITES - COMPLETE CHEAT
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GATE 2026 | EE & ECE | Exam-Ready Format
1️⃣ DIFFERENTIAL EQUATIONS
General LTI System
dn y dy dm u
an + ⋯ + a1 + a0 y = bm m + ⋯ + b0 u
dtn dt dt
Order = highest derivative of output (not input)
Control systems: mostly 1st & 2nd order
Characteristic Equation
d
Replace dt → s
an sn + ⋯ + a1 s + a0 = 0
Roots = System Behavior:
Root Type Behavior Stability
Real negative Exponential decay ✅ Stable
Real positive Exponential growth ❌ Unstable
Complex (−ve real) Damped oscillation ✅ Stable
Complex (+ve real) Growing oscillation ❌ Unstable
Pure imaginary Sustained oscillation ⚠️ Marginally stable
Repeated at origin Ramp-like growth ❌ Unstable
Standard 2nd Order System ⭐
s2 + 2ζωn s + ωn2 = 0
, ζ (Damping Ratio) Response Type
ζ=0 Undamped (pure oscillation)
0<ζ<1 Underdamped (oscillatory)
ζ=1 Critical damping
ζ>1 Overdamped
Key Points:
↑ ζ → ↓ overshoot, ↑ settling time
↑ ωₙ → ↑ speed (faster response)
Additional: 1st Order System
dy
τ + y = Ku(t)
dt
Transfer Function: G(s) K
= τs+1
Time constant τ = time to reach 63.2% of final value
Settling time ≈ 4τ (2% criterion)
2️⃣ LAPLACE TRANSFORM
Core Concept
Time domain → s-domain (frequency domain)
Converts differential equations → algebraic equations
Essential Transforms
Time Domain s-Domain
1 (step) 1
s
t (ramp) 1
s2
e−at 1
s+a
sin(ωt) ω
s2 +ω2
SHEET
GATE 2026 | EE & ECE | Exam-Ready Format
1️⃣ DIFFERENTIAL EQUATIONS
General LTI System
dn y dy dm u
an + ⋯ + a1 + a0 y = bm m + ⋯ + b0 u
dtn dt dt
Order = highest derivative of output (not input)
Control systems: mostly 1st & 2nd order
Characteristic Equation
d
Replace dt → s
an sn + ⋯ + a1 s + a0 = 0
Roots = System Behavior:
Root Type Behavior Stability
Real negative Exponential decay ✅ Stable
Real positive Exponential growth ❌ Unstable
Complex (−ve real) Damped oscillation ✅ Stable
Complex (+ve real) Growing oscillation ❌ Unstable
Pure imaginary Sustained oscillation ⚠️ Marginally stable
Repeated at origin Ramp-like growth ❌ Unstable
Standard 2nd Order System ⭐
s2 + 2ζωn s + ωn2 = 0
, ζ (Damping Ratio) Response Type
ζ=0 Undamped (pure oscillation)
0<ζ<1 Underdamped (oscillatory)
ζ=1 Critical damping
ζ>1 Overdamped
Key Points:
↑ ζ → ↓ overshoot, ↑ settling time
↑ ωₙ → ↑ speed (faster response)
Additional: 1st Order System
dy
τ + y = Ku(t)
dt
Transfer Function: G(s) K
= τs+1
Time constant τ = time to reach 63.2% of final value
Settling time ≈ 4τ (2% criterion)
2️⃣ LAPLACE TRANSFORM
Core Concept
Time domain → s-domain (frequency domain)
Converts differential equations → algebraic equations
Essential Transforms
Time Domain s-Domain
1 (step) 1
s
t (ramp) 1
s2
e−at 1
s+a
sin(ωt) ω
s2 +ω2
