Control Systems

Syllabus

Mathematical modelling, feedback, transfer functions, block diagrams and signal flow graphs (Mason's gain formula); transient and steady-state analysis of LTI systems; stability by Routh-Hurwitz and Nyquist criteria; Bode plots; root loci; lag, lead and lead-lag compensators; P, PI and PID controllers; state space models and solution of state equations.

Key concepts and formulas

Feedback. Closed-loop transfer function T = G/(1 + GH) for negative feedback. Mason's gain formula: T = Σ P_kΔ_k/Δ.

Second-order system G(s) = ω_n²/(s² + 2ζω_ns + ω_n²):

• Damped frequency ω_d = ω_n√(1 − ζ²).
• Peak overshoot M_p = e^{−πζ/√(1−ζ²)}; peak time t_p = π/ω_d.
• Settling time t_s ≈ 4/(ζω_n) (2% band) or 3/(ζω_n) (5% band).
• ζ < 1 underdamped, = 1 critically damped, > 1 overdamped.

Steady-state error (unity feedback, open-loop G):

The type of a system is the number of poles of G at the origin. A type-1 system has zero step error and a finite ramp error.

Routh-Hurwitz. The number of sign changes in the first column of the Routh array equals the number of roots in the right half-plane. For s³ + a₂s² + a₁s + a₀, stability needs all coefficients positive and a₂a₁ > a₀.

Root locus. Starts at open-loop poles (K = 0) and ends at zeros or infinity; the number of asymptotes = P − Z, at angles (2k + 1)180°/(P − Z) from the centroid σ = (Σ poles − Σ zeros)/(P − Z). A real-axis point is on the locus if the number of poles and zeros to its right is odd.

Frequency response. Gain margin GM = 1/|G(jω_pc)| at the phase crossover (phase = −180°); phase margin PM = 180° + ∠G(jω_gc) at the gain crossover (|G| = 1). A stable minimum-phase system has positive GM (in dB) and PM. In Bode plots, each pole adds −20 dB/decade, each zero +20 dB/decade. Nyquist criterion: Z = P + N (closed-loop RHP poles = open-loop RHP poles + clockwise encirclements of −1).

Compensators and controllers. A lead compensator adds phase (improves transient response and PM); a lag compensator improves steady-state error. Integral action removes the steady-state error to a step; derivative action adds damping.

State space. ẋ = Ax + Bu, y = Cx + Du; transfer function C(sI − A)⁻¹B + D; the eigenvalues of A are the poles. State transition matrix e^At = L⁻¹{(sI − A)⁻¹}. Controllable if [B AB … Aⁿ⁻¹B] has full rank.

Worked example

For G(s) = 25/(s(s + 6)) with unity feedback, find ω_n, ζ, the peak overshoot, peak time, 2% settling time and the ramp error. Then find the range of K for which s³ + 6s² + 11s + 6 + K = 0 is stable.

import math

# Closed loop: s^2 + 6s + 25 -> wn^2 = 25, 2 zeta wn = 6
wn = 5.0
zeta = 6 / (2 * wn)
wd = wn * math.sqrt(1 - zeta ** 2)
Mp = math.exp(-math.pi * zeta / math.sqrt(1 - zeta ** 2))
print(f"wn = {wn}, zeta = {zeta}, Mp = {Mp * 100:.2f}%, tp = {math.pi / wd:.4f} s, ts = {4 / (zeta * wn):.4f} s")
Kv = 25 / 6
print(f"Kv = {Kv:.4f}, ramp error = {1 / Kv:.2f}")
# Routh for s^3 + 6s^2 + 11s + (6 + K): need 6*11 > 6 + K and 6 + K > 0
print(f"stable for {-6} < K < {6 * 11 - 6}")

wn = 5.0, zeta = 0.6, Mp = 9.48%, tp = 0.7854 s, ts = 1.3333 s
Kv = 4.1667, ramp error = 0.24
stable for -6 < K < 60

Practice questions