5. Frequency-Domain Analysis

Time-domain methods (step response, root locus) are intuitive. Frequency-domain methods (Bode, Nyquist) are the workhorse of practical design, especially when:

• The plant is partly empirical (you have measured frequency-response data, not an analytical model).
• The plant contains significant time delays (which are hard to handle in root locus but easy in Bode).
• You want to design for specific gain and phase margins.

5.1 Bode plots: the design environment

A Bode plot is two semi-log plots: $|G(j\omega)|$ in decibels vs $\log_{10} \omega$, and $\angle G(j\omega)$ in degrees vs $\log_{10} \omega$. Two reasons Bode is the workhorse of control engineering:

1. Pole and zero contributions add linearly. Each pole at $\omega = 1/\tau$ contributes $-20$ dB/decade rolloff above its corner frequency, and $-90°$ of phase shift swept across two decades around its corner. Zeros do the opposite: $+20$ dB/decade and $+90°$. So Bode plots can be sketched by hand from the pole-zero pattern.
2. Stability margins read directly. Gain margin and phase margin are visible at a glance.

Asymptotic Bode plot for $G(s) = K / ((1 + s/\omega_1)(1 + s/\omega_2))$, $\omega_1 < \omega_2$:

|G| dB   ─────────
              ╲
              ╲ -20 dB/dec
                 ╲
                 ╲────────
                  ╲ ω1
                  ╲ -40 dB/dec
                  ╲
                  ╲
        20 log K ─┼──────╲────╲──────  ω
                       ω1     ω2
 
phase (°)
   0° ───
       ╲
       ╲      \─\─
       ╲          ╲
       ╲           ╲
  -90° ╲            ╲
       ╲             ╲
       ╲              ╲
  -180°╲               ╲────  ω
                          ω2

Adding poles drags magnitude down and phase backward. Adding zeros pulls magnitude up and phase forward. Each integrator (pole at the origin) contributes $-20$ dB/decade across all frequencies and $-90°$ of constant phase.

5.2 Reading stability margins from a Bode plot

  |G| dB
        ╲
   0 ────╲────────────  ←── gain crossover ω_gc
         ╲
         ╲
         ╲
        ─40 dB
           ╲
   ─10 ────╲ ←── this dip below 0 dB at the phase crossover gives gain margin
            ╲
   ─20 ─────╲────────  ω
                ω_pc
 
  phase
     0 ────
            ╲
            ╲
            ╲
     -90    ╲
            ╲
            ╲
            ╲ ←── phase at gain crossover
   -180 ────╲────  ω
              ω_pc       (where phase = -180°)

• Gain crossover frequency $\omega_{gc}$: where $|G| = 0$ dB. Phase margin $\text{PM} = 180° + \angle G(j\omega_{gc})$.
• Phase crossover frequency $\omega_{pc}$: where phase $= -180°$. Gain margin $\text{GM} = -|G(j\omega_{pc})|_\text{dB}$.

Targets for typical designs: PM = 45° to 60°, GM = 6 to 12 dB. Larger margins are more robust but slower. Smaller margins give faster response but less safety against modeling errors.

5.3 Bandwidth, resonant peak, related metrics

For the closed-loop transfer function $T(s)$:

• Closed-loop bandwidth $\omega_B$: where $|T(j\omega)| = -3$ dB.
• Resonant peak $M_r$: maximum of $|T(j\omega)|$. For a 2nd-order system: $M_r = 1/(2\zeta\sqrt{1-\zeta^2})$ if $\zeta < 0.707$.
• Resonant frequency $\omega_r$: where the resonant peak occurs.

A useful empirical rule: PM° ≈ 100 × $\zeta$ for $\zeta < 0.7$. A 60° phase margin corresponds to roughly $\zeta = 0.6$, giving about 10% overshoot.

5.4 The Nyquist stability criterion

The Nyquist plot is $G(j\omega) H(j\omega)$ in the complex plane as $\omega$ varies from $-\infty$ to $+\infty$. The Nyquist stability criterion says:

$N = Z - P$

Where:

• $N$ = number of clockwise encirclements of the point $(-1, 0)$ by the Nyquist plot.
• $Z$ = number of closed-loop poles in the right half plane (zeros of $1 + GH$).
• $P$ = number of open-loop poles in the right half plane.

Stability requires $Z = 0$, hence $N = -P$.

For the most common case of stable open-loop systems ($P = 0$), no encirclements of $(-1, 0)$ means the closed-loop is stable. The Nyquist plot tells you stability and margins: the closer the curve passes to $(-1, 0)$, the smaller the margin.

For systems with delays or with unstable open-loop modes (such as inverted pendulums), Nyquist is the only frequency-domain test that works cleanly. Bode is generally easier when applicable, but Nyquist generalizes.

5.5 Conditional stability

Some systems are stable in a range of gains, unstable below and above. The Nyquist plot encircles $(-1, 0)$ in a way that depends nonmonotonically on the gain. Conditionally stable systems exist in saturating servo loops (where the gain may decrease during large transients) and lead-network designs.

Hardware-security implication. If you can attack a power regulator's gain (via a brownout, supply-line noise, or even thermal modulation), you can drive it from a stable operating point to a conditionally unstable one, where small perturbations get amplified and the regulator output starts oscillating wildly. This is a real attack vector for fault injection. Knowing your loop's PM, GM, and conditional-stability behavior tells you how robust it is.