Bode, Nyquist, Compensators and State Space

Frequency-response methods judge stability and robustness from G(jω). State-space methods describe a system with first-order matrix equations.

Bode plots

Magnitude in dB (20 log|G|) and phase against log ω.

Stability margins

• Gain crossover ω_gc: |G| = 1 (0 dB). Phase margin PM = 180° + ∠G(jω_gc).
• Phase crossover ω_pc: ∠G = −180°. Gain margin GM = 1/|G(jω_pc)| (in dB: −20 log|G|).
• Stable (minimum-phase systems) if both margins are positive. For a second-order system, PM ≈ 100ζ degrees (for ζ < 0.7).
• Resonant peak M_r = 1/(2ζ√(1 − ζ²)) at ω_r = ω_n√(1 − 2ζ²) (for ζ < 0.707). Bandwidth grows with ω_n.

Nyquist criterion

Plot G(jω)H(jω) for ω from −∞ to ∞. Z = N + P:

• P = open-loop poles in the RHP; N = clockwise encirclements of (−1, 0); Z = closed-loop RHP poles.
• For stability Z = 0, so the plot must encircle −1 counter-clockwise P times (no encirclement if P = 0).

Compensators and controllers

• Lead compensator (zero before pole): adds phase, improves transient response and margins; acts like a PD controller (high-pass).
• Lag compensator (pole before zero): raises low-frequency gain, reduces steady-state error; acts like PI (low-pass).
• PID: K_p + K_i/s + K_d s. Integral action removes steady-state error (raises type by 1); derivative action adds damping.

State space

ẋ = Ax + Bu, y = Cx + Du

• Transfer function: G(s) = C(sI − A)⁻¹B + D.
• Eigenvalues of A = poles.
• State transition matrix φ(t) = e^(At) = L⁻¹[(sI − A)⁻¹].
• Controllable if [B AB … Aⁿ⁻¹B] has rank n. Observable if [C; CA; …; CAⁿ⁻¹] has rank n.

Worked example

G(s) = 10/(s(s + 1)) at ω = 3 rad/s.

• |G| = 10/(3 × √10) = 1.054; ∠G = −90° − tan⁻¹3 = −161.6°.

Practice