Control Systems Notes — B.Tech ECE Sem 6

Control Systems — Introduction

Control System: Input ke response mein desired output produce karne wala system.

Types:

Open Loop:   Input → [Controller] → [Plant] → Output
             No feedback, simple, inaccurate

Closed Loop: Input → [Σ] → [Controller] → [Plant] → Output
                      ↑___[Sensor/Feedback]____________|
             Feedback se error correct hota hai

Examples:

• Open loop: Washing machine timer, traffic light timer
• Closed loop: Thermostat, cruise control, autopilot

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1. Transfer Function & Block Diagrams

Transfer Function:

G(s) = C(s)/R(s) = Output Laplace / Input Laplace

For differential equation:
a₂ẏ + a₁ẏ + a₀y = b₀u

G(s) = b₀ / (a₂s² + a₁s + a₀)

Example — Mass-Spring-Damper:
Mẍ + Bẋ + Kx = F

G(s) = X(s)/F(s) = 1/(Ms² + Bs + K)

Block Diagram Reduction:

Series:     G1 → G2 = G1·G2

Parallel:   G1 ─┐
                 +─→ = G1 + G2
            G2 ─┘

Feedback:        R→[Σ]→[G]→C
                     ↑
                    [H]
Closed-loop TF = G/(1 + GH)  (negative feedback)
                 G/(1 - GH)  (positive feedback)

Mason's Gain Formula:

T = Σ(Pk·Δk) / Δ

Pk = forward path gain
Δ = 1 - Σ(loop gains) + Σ(non-touching loop products) - ...
Δk = Δ for part not touching kth forward path

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2. Time Domain Analysis

Standard Second-Order System:

G(s) = ωn² / (s² + 2ζωn·s + ωn²)

ζ (zeta) = damping ratio
ωn = natural frequency

Cases:
ζ = 0:      Undamped     → pure oscillation
0 < ζ < 1:  Underdamped  → oscillates, settles (most common)
ζ = 1:      Critically damped → fastest no overshoot
ζ > 1:      Overdamped   → slow, no oscillation

Time Domain Specifications:

Rise time (tr):    10% to 90% of final value
               tr ≈ 1.8/ωn (underdamped)

Peak time (tp):    Time to first peak
               tp = π/(ωd),  ωd = ωn√(1-ζ²)

Overshoot (%Mp): Mp = e^(-πζ/√(1-ζ²)) × 100%
               At ζ=0.5: Mp ≈ 16%

Settling time (ts): enter ±2% band and stay
               ts ≈ 4/(ζωn)  (2% criterion)

First-Order System:

G(s) = K/(τs + 1)

Step response: y(t) = K(1 - e^(-t/τ))
τ = time constant (63.2% of final value at t=τ)
Settling time ≈ 4τ (98%)

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3. Stability Analysis

Routh-Hurwitz Criterion

Characteristic equation: a₄s⁴ + a₃s³ + a₂s² + a₁s + a₀ = 0

Routh array:
s⁴ | a₄  a₂  a₀
s³ | a₃  a₁  0
s² | b₁  b₂  0     where b₁ = (a₃a₂ - a₄a₁)/a₃
s¹ | c₁  0         b₂ = (a₃a₀ - a₄·0)/a₃
s⁰ | d₁

Stable if: all first-column elements have same sign (positive)
Number of sign changes = number of RHP poles

Root Locus

Rules:
1. Starts at open-loop poles (K=0)
2. Ends at open-loop zeros (K→∞)
3. On real axis: to left of odd number of poles+zeros
4. Asymptotes: angles = (2q+1)×180°/n-m, q=0,1,2...
5. Centroid = Σpoles - Σzeros / (n-m)
6. Breakaway/break-in points: dK/ds = 0

Closed-loop poles move as K varies:
K=0: poles at open-loop pole locations
K→∞: poles at zero locations (or infinity via asymptotes)

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4. Frequency Domain Analysis

Bode Plot

G(jω) evaluated at jω (s=jω, sinusoidal input)

Magnitude: |G(jω)|dB = 20 log₁₀|G(jω)|
Phase: ∠G(jω) degrees

Standard factors and their Bode plots:

1. Gain K: horizontal line at 20log₁₀K dB, 0° phase

2. Integrator 1/s: -20dB/decade slope, -90° phase

3. First-order pole 1/(τs+1):
   Corner frequency ωc = 1/τ
   Below ωc: 0dB, flat
   Above ωc: -20dB/decade
   At ωc: -3dB, -45°

4. Second-order:
   Below ωn: flat
   Above ωn: -40dB/decade, -180°
   At ωn: resonance peak (for small ζ)

Gain Crossover & Phase Crossover:

Gain crossover (ωgc): frequency where |G| = 0 dB (unity gain)
Phase margin = 180° + ∠G(jωgc)    [positive → stable]

Phase crossover (ωpc): frequency where ∠G = -180°
Gain margin = -|G(jωpc)|dB = 20log₁₀(1/|G(jωpc)|) [positive → stable]

Nyquist Criterion

Nyquist Stability: Z = N + P
Z = closed-loop RHP poles (must be 0 for stability)
N = clockwise encirclements of (-1, j0)
P = open-loop RHP poles

For stable open-loop (P=0):
  Stable if Nyquist plot doesn't encircle (-1,0) → N=0

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5. PID Controller Design

PID Transfer Function:
Gc(s) = Kp + Ki/s + Kd·s
      = Kp(1 + 1/(Ti·s) + Td·s)

Effects:
Increase Kp: faster response, reduce steady-state error, may oscillate
Increase Ki: eliminate steady-state error, may overshoot more
Increase Kd: reduce overshoot, dampen oscillations, noise sensitive

Ideal step response target:
- Small overshoot (<10%)
- Fast rise time
- Zero steady-state error
- Short settling time

Ziegler-Nichols Tuning:

Method 1 — Step response:
L = delay, T = time constant
Kp = 1.2T/L, Ti = 2L, Td = 0.5L

Method 2 — Sustained oscillation:
Increase Kp until sustained oscillation (Ku)
Measure period Tu
PID: Kp=0.6Ku, Ti=0.5Tu, Td=0.125Tu

Digital PID (Discrete):

float pid_compute(float setpoint, float measured) {
    float error = setpoint - measured;
    integral += error * dt;
    float derivative = (error - prev_error) / dt;
    prev_error = error;
    return Kp*error + Ki*integral + Kd*derivative;
}

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Exam Important Questions

1. Block diagram reduction rules — series, parallel, feedback
2. Second-order system ke time specifications explain karo
3. Routh-Hurwitz criterion se stability check karo: s³+2s²+3s+4=0
4. Root locus ke rules aur sketch karo G(s)=K/s(s+2)
5. Bode plot draw karo G(s) = 10/(s(s+10))
6. PID controller mein P, I, D ka individual effect explain karo
7. Gain margin aur phase margin kaise calculate karte hain?
8. Closed-loop transfer function nikalo feedback system ke liye

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Viva Questions

Q: Type 0, 1, 2 system mein kya fark hai? A: System type = number of open-loop integrators (poles at origin). Type 0: finite position error to step. Type 1: zero position error, finite velocity error. Type 2: zero velocity error. Higher type → better tracking but harder to stabilize.

Q: Lead aur Lag compensator mein kya fark hai? A: Lead: phase advance, increases bandwidth, improves transient (fast response). Lag: phase lag, reduces high-frequency gain, improves steady-state accuracy. Lead-Lag: combines both benefits.

Q: State space representation kya hai? A: ẋ = Ax + Bu, y = Cx + Du. A=system matrix, B=input, C=output, D=feedthrough. Controllability (can reach any state), Observability (can estimate all states from output). Modern control theory ka basis.