Control Systems

Syllabus

Basic control system components; feedback principle; transfer function; block diagrams; signal flow graphs; transient and steady-state analysis of LTI systems; frequency response; Routh-Hurwitz and Nyquist stability criteria; Bode and root-locus plots; lag, lead and lag-lead compensation; state variable model and solution of state equations.

Key concepts and formulas

The GATE EE control systems lesson covers the same syllabus with more detail; the essentials are:

• Closed loop: T = G/(1 + GH). Mason: T = Σ P_kΔ_k/Δ.
• Second order: M_p = e^{−πζ/√(1−ζ²)}, t_p = π/(ω_n√(1 − ζ²)), t_s ≈ 4/(ζω_n) (2%), rise time decreases as ω_n increases.
• Steady-state error (unity feedback): step 1/(1 + K_p), ramp 1/K_v, parabola 1/K_a.
• Routh-Hurwitz: sign changes in the first column = right-half-plane roots. A row of zeros signals symmetric roots (for example, a pair on the jω axis).
• Root locus: P − Z asymptotes at angles (2k + 1)180°/(P − Z), centroid (Σ poles − Σ zeros)/(P − Z); break-away points from dK/ds = 0.
• Bode: each pole −20 dB/decade, each zero +20 dB/decade; a type-n system starts with slope −20n dB/decade. Gain and phase margins from the crossover frequencies.
• Nyquist: N = Z − P (encirclements of −1, counted clockwise as positive, give Z = closed-loop RHP poles).
• Compensators: lead (zero closer to the origin than the pole) adds phase margin and bandwidth; lag (pole closer to the origin) improves steady-state accuracy. Maximum phase lead of (1 + s/ω₁)/(1 + s/ω₂) is sin⁻¹((ω₂ − ω₁)/(ω₂ + ω₁)), at ω = √(ω₁ω₂).
• State space: ẋ = Ax + Bu; the eigenvalues of A are the poles; x(t) = e^Atx(0) for u = 0.

Worked example

Find the break-away point of the root locus of G(s)H(s) = K/(s(s + 4)), the maximum phase lead of a compensator with corners at 1 and 9 rad/s, and the state transition matrix of A = [[0, 1], [−2, −3]] at t = 1 s by series expansion.

import math

# K = -s(s + 4) -> dK/ds = -(2s + 4) = 0 -> s = -2
print("break-away point: s = -2, K there =", -(-2) * (-2 + 4))
w1, w2 = 1, 9
print(f"max phase lead = {math.degrees(math.asin((w2 - w1) / (w2 + w1))):.2f} deg at w = {math.sqrt(w1 * w2):.0f} rad/s")
A = [[0, 1], [-2, -3]]
def mul(X, Y):
    return [[sum(X[i][k] * Y[k][j] for k in range(2)) for j in range(2)] for i in range(2)]
term = [[1.0, 0.0], [0.0, 1.0]]
total = [[1.0, 0.0], [0.0, 1.0]]
for k in range(1, 40):
    term = [[v / k for v in row] for row in mul(term, A)]
    total = [[total[i][j] + term[i][j] for j in range(2)] for i in range(2)]
print("e^(A t) at t = 1:", [[round(v, 4) for v in row] for row in total])
e1, e2 = math.exp(-1), math.exp(-2)
print("exact first row:", [round(2 * e1 - e2, 4), round(e1 - e2, 4)])

break-away point: s = -2, K there = 4
max phase lead = 53.13 deg at w = 3 rad/s
e^(A t) at t = 1: [[0.6004, 0.2325], [-0.4651, -0.0972]]
exact first row: [0.6004, 0.2325]

Practice questions