Control Systems — Time and Frequency Response

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

GATE Weightage: ~8–12 marks/year (Electrical/Instrumentation); Bode plot and steady-state error are extremely high-yield topics.

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First-Order System: G(s) = K/(τs + 1)

• Time constant τ; Step response: y(t) = K(1 – e^(–t/τ)); Settling time ≈ 4τ

Second-Order System: G(s) = ω_n²/(s² + 2ζω_n s + ω_n²)

ζ	Type	Response
ζ = 0	Undamped	Pure oscillation
0 < ζ < 1	Underdamped	Oscillatory decay
ζ = 1	Critically damped	Fastest no-overshoot
ζ > 1	Overdamped	Slow, no overshoot

Pole locations: s = –ζω_n ± jω_n√(1–ζ²); ω_d = ω_n√(1–ζ²) = damped frequency

Steady-State Error: e_ss = 1/(1 + K_p) for step, 1/K_v for ramp, 1/K_a for parabolic

Bode Plot Rules:

• Magnitude: –20 dB/decade per pole at origin; –20 dB/decade per pole away from origin; +20 dB/decade per zero
• Phase: –90° per pole at origin; –90° per pole (non-origin); +90° per zero

Nyquist: Encircle –1 + j0; Stability if # encirclements = # RHP poles of OLTF

Gain Margin (GM): Gain at phase crossover where ∠G(jω) = –180°; GM > 0 required Phase Margin (PM): Phase at gain crossover where |G(jω)| = 1; PM > 0 required; PM ≈ 100ζ

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

First-Order System Response

Standard form: G(s) = K/(τs + 1) or normalized: G(s) = 1/(τs + 1)

Unit Step Response

y(t) = K(1 – e^(–t/τ)) for t ≥ 0

Parameter	Value
Steady-state value	K
Time constant τ	Time to reach 63.2% of final
Rise time (10%–90%)	≈ 2.2τ
Settling time (2%)	≈ 4τ
Settling time (5%)	≈ 3τ

Unit Ramp Response

For input R(s) = 1/s²:

• y(t) = K(t – τ + τe^(–t/τ))
• Steady-state error e_ss = τ (ramp lag equals time constant)

Unit Impulse Response

y(t) = (K/τ)·e^(–t/τ) for t ≥ 0

Second-Order System — Detailed Analysis

Standard TF: G(s) = ω_n²/(s² + 2ζω_n s + ω_n²)

Key Specifications

• Peak time: T_p = π/ω_d = π/(ω_n√(1–ζ²))
• Percent overshoot: %OS = e^(–πζ/√(1–ζ²)) × 100
• Settling time (2% criterion): T_s ≈ 4/(ζω_n)
• Settling time (5% criterion): T_s ≈ 3/(ζω_n)
• Rise time (0–100%): T_r ≈ (π – θ)/ω_d, where θ = arctan(√(1–ζ²)/ζ)

GATE Formula: ζ from %OS: ζ = –ln(%OS/100) / √(π² + ln²(%OS/100))

Pole-Zero Map

Poles: s₁,₂ = –ζω_n ± jω_n√(1–ζ²)

• Real part = –ζω_n (determines settling time)
• Imaginary part = ω_n√(1–ζ²) = ω_d (determines oscillation frequency)

Steady-State Error Analysis

Error Constants

Input Type	Position (step)	Velocity (ramp)	Acceleration (parabola)
Input r(t)	u(t)	t	½t²
Laplace R(s)	1/s	1/s²	1/s³
Steady-state error	1/(1 + K_p)	1/K_v	1/K_a
K_p = lim G(s)	K_p	0	0
K_v = lim sG(s)	∞	K_v	0
K_a = lim s²G(s)	∞	∞	K_a

Type number = number of poles at s = 0 in G(s)

System Type and Compensation

• Higher type → better steady-state tracking, harder to stabilize
• Type 0: Tracks step, error to ramp/parabola
• Type 1: No error to step/ramp, error to parabola
• Type 2: No error to step/ramp/parabola (often used in position control)

Bode Plot Construction

Magnitude Plot

Element	Slope	Corner frequency
Pole at origin (1/s)	–20 dB/dec	ω = 0
Simple pole (1/(1+jω/ω_c))	–20 dB/dec	ω_c
Double pole (1/(1+jω/ω_c)²)	–40 dB/dec	ω_c
Zero (1 + jω/ω_c)	+20 dB/dec	ω_c
Second-order pole (denominator s²/ω_n² + 2ζs/ω_n + 1)	–40 dB/dec	ω_n

Phase Plot

Element	Phase contribution
Simple pole	0° → –90° over 2 decades around ω_c
Simple zero	0° → +90° over 2 decades around ω_c
Pole at origin	–90° constant
Second-order pole	0° → –180° depending on ζ

Minimum phase systems have a one-to-one correspondence between magnitude and phase.

Nyquist Stability Criterion

Nyquist plot: Plot G(jω) as ω goes from 0 to ∞ (complex plane).

Stability test: Z = N + P, where:

• Z = number of closed-loop poles in RHP
• N = number of encirclements of –1+j0 (clockwise direction)
• P = number of open-loop poles in RHP

For stability: Z = 0 → N = –P (no net encirclements if P = 0 and system stable)

Gain and Phase Margin from Nyquist

• Gain Margin: GM = 1/|G(jω_p)| where ω_p = phase crossover (∠G = –180°)
• Phase Margin: PM = 180° + ∠G(jω_g) where ω_g = gain crossover (|G| = 1)
• Both must be positive for stability
• Typical desired: GM > 6 dB, PM > 30°–45°

Nichols Chart

A Nichols chart combines:

• M-contours: constant closed-loop magnitude (|G/(1+G)| = constant, in dB)
• N-contours: constant closed-loop phase

Used to read closed-loop response from open-loop Nyquist data.

Key use: Find resonant peak M_r (maximum |T(jω)|) and resonant frequency ω_r from the highest M-contour tangent to the Nyquist plot.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Delay Time, Rise Time, Peak Time — Exact Formulas

For underdamped second-order (ζ < 1):

Delay time (10% to 50% response): T_d ≈ (1 + 0.7ζ) / ω_n

Rise time (0% to 100%): T_r = (π – φ) / ω_d, where φ = arctan(ω_d/ζω_n)

Peak time: T_p = π / ω_d

Maximum (peak) overshoot: M_p = |G(jω_d)| at ω = ω_d = exp(–πζ/√(1–ζ²))

Compensation Techniques

Lag Compensator

G_c(s) = (1 + αT s) / (1 + T s), where α > 1

• Increases K_p (improves steady-state error)
• Small negative impact on phase margin
• Adds low-frequency gain without significantly altering high-frequency response

Lead Compensator

G_c(s) = (1 + T s) / (1 + αT s), where α < 1

• Increases phase margin (improves transient response)
• Increases bandwidth (faster response)
• May amplify high-frequency noise

Lead-Lag Compensator

Combines both: G_c(s) = (1 + T₁s)(1 + T₂s) / ((1 + αT₁s)(1 + αT₂s))

• T₁ = lead time constant (α < 1)
• T₂ = lag time constant (α > 1)

Frequency Response — Resonance Peak

For standard second-order:

Resonant frequency: ω_r = ω_n√(1 – 2ζ²) (only exists if ζ < 1/√2 ≈ 0.707)

Resonant peak: M_r = 1/(2ζ√(1–ζ²)) (only if ζ < 0.707)

GATE Trick: If asked “what is the nature of system with given ζ?” — check: ζ < 0 → unstable oscillations grow; ζ = 0 → sustained oscillation; 0 < ζ < 1 → decaying oscillations.

Bode Plot — Asymptotic vs Actual

Asymptotic Bode is approximate (straight lines at corner frequencies). Actual magnitude at corner frequency for a simple pole:

• 20·log|G(jω_c)| = –20·log√2 = –3 dB (exactly)

The asymptotic approximation error is:

• Simple pole/zero: max 3 dB error at corner
• Double pole/zero: max 6 dB error at corner

Closed-Loop Frequency Response

|T(jω)| = |G(jω)| / |1 + G(jω)|

Using Nichols chart, you can determine:

• Closed-loop bandwidth (ω_BW where |T| drops to –3 dB of low-frequency gain)
• Bandwidth ≈ ω_n for second-order systems
• Higher ζ → narrower bandwidth, less ringing

Relationship Between Time and Frequency Domain

Parameter	Time Domain	Frequency Domain
Settling time	T_s ≈ 4/(ζω_n)	Bandwidth related
Peak time	T_p = π/ω_d	Resonant peak M_r
Overshoot	%OS = e^(–πζ/√(1–ζ²))	M_r (higher M_r → higher %OS)
Natural frequency	ω_n	Crossover frequency ω_c

Approximate relations:

• PM ≈ 100ζ (for ζ between 0.2 and 0.8)
• GM depends on high-frequency gain of G(s)
• ω_c (crossover) ≈ ω_n for well-damped systems

GATE Exam Strategy — Time and Frequency Response

Expected question types:

1. Find %OS, T_s, T_p from given ζ, ω_n
2. Draw Bode plot magnitude/phase from given G(s)
3. Find GM and PM from Bode or Nyquist
4. Determine steady-state error for step/ramp/parabolic input
5. Find closed-loop transfer function from Bode data
6. Stability analysis using Nyquist

Common GATE mistakes:

• Using settling time formula T_s = 4/(σ) where σ = real part of dominant pole (not ζω_n)
• Confusing ω_d (damped frequency) with ω_n (natural frequency)
• Drawing Bode with wrong slopes for repeated poles/zeros
• Forgetting that GM/PM must be positive — negative means unstable
• Miscounting poles at origin for type number

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