Frequency Response, Bode Plots & Minimum-Phase Structure

For a stable LTI system driven by $u(t)=A\cos\omega t$, transients eventually decay and the steady output is

$$y_{ss}(t)=A|G(j\omega)|\cos\!\left(\omega t+\angle G(j\omega)\right).$$

The frequency response $G(j\omega)$ therefore records gain and phase at each angular frequency. It is an input/output property under small-signal linear assumptions; saturation, amplitude-dependent dynamics, switching, and time variation require additional analysis.

Magnitude slopes add in dB and phases add in degrees. Delay leaves magnitude unchanged while continuously removing phase.

Why logarithmic coordinates work #

Factor a SISO transfer function as

$$G(s)=K\,s^{n_z-n_p}\frac{\prod_i(1+s/\omega_{z_i})}{\prod_k(1+s/\omega_{p_k})}e^{-sT_d}.$$

Then

$$20\log_{10}|G(j\omega)|$$

is a sum of factor magnitudes, and $\angle G(j\omega)$ is a sum of factor phases. A logarithmic frequency axis displays dynamics spanning orders of magnitude and turns power laws into straight slopes.

For a left-half-plane real zero $1+s/\omega_z$:

• magnitude transitions from 0 to $+20$ dB/decade around $\omega_z$;
• phase transitions from $0^\circ$ toward $+90^\circ$.

A real pole contributes the opposite. An integrator contributes $-20$ dB/decade and $-90^\circ$ at every frequency. A complex pole pair eventually contributes $-40$ dB/decade and $-180^\circ$, with resonant peaking when damping is low.

Straight-line asymptotes are sketches. At a first-order break, the exact magnitude differs by about 3.01 dB from the intersecting asymptotes and the phase is halfway through its transition.

Delay is pure phase loss #

For $G_d(s)=e^{-sT_d}$,

$$|G_d(j\omega)|=1,\qquad \angle G_d(j\omega)=-\omega T_d.$$

Thus a magnitude-only fit cannot identify pure delay, yet the delay can destroy phase margin near crossover. At crossover $\omega_c$, an extra delay $\Delta T$ removes $\omega_c\Delta T$ radians of phase. This is why raising bandwidth makes timing uncertainty more dangerous.

Open-loop crossover and closed-loop bandwidth #

The gain crossover frequency satisfies $|L(j\omega_c)|=1$. Closed-loop bandwidth is normally defined from a particular closed-loop transfer, often the $-3$ dB point of $T=L/(1+L)$. They are related for well-behaved loops but not identical definitions.

At low frequency, large $|L|$ makes $|S|=|1/(1+L)|$ small and supports tracking/disturbance rejection. Around crossover, phase and uncertainty determine robustness. At high frequency, loop gain should usually fall to avoid measurement noise, unmodeled modes, and excessive control activity. A single “bandwidth” number hides this three-region design problem.

Minimum phase and hidden phase #

A stable, causal, rational SISO system is minimum phase when its zeros are also stable and it has no pure delay. Among stable systems with a given magnitude response, the minimum-phase system has the least phase lag. A right-half-plane zero

$$1-\frac{s}{z},\qquad z>0,$$

has the same magnitude as the left-half-plane zero $1+s/z$ but contributes phase lag instead of lead. An all-pass factor can likewise alter phase without magnitude.

Therefore magnitude does not generally determine phase. Bode gain–phase relations require minimum-phase and regularity assumptions. If measured phase contains more lag than a minimum-phase model predicts, suspect delay, right-half-plane zeros, all-pass dynamics, incorrect synchronization, or an unmodeled pole.

Worked factor reading #

Consider

$$G(s)=\frac{10(1+s/2)}{s(1+s/10)(1+s/50)}e^{-0.02s}.$$

Before calculating a plot, predict:

1. The integrator starts the magnitude slope at $-20$ dB/decade and phase near $-90^\circ$.
2. The zero at $2$ rad/s flattens the magnitude slope to $0$ and adds phase lead.
3. The pole at $10$ returns the slope to $-20$ dB/decade.
4. The pole at $50$ makes the high-frequency slope $-40$ dB/decade.
5. Delay changes no magnitude but adds $-0.02\omega$ radians; at $50$ rad/s that is about $-57.3^\circ$.

This prediction is a diagnostic. If software shows a different final slope or phase trend, check factor signs, units, and whether frequency is in rad/s or Hz.

Measuring a frequency response #

A swept sine estimates one frequency at a time after transients decay. Multisine and broadband excitation can cover many frequencies faster but require leakage, periodicity, crest-factor, and nonlinear-distortion checks. For each frequency:

• keep excitation inside the linear operating region yet above noise;
• wait or window long enough to separate transient and steady response;
• measure the input actually applied, not only the commanded input;
• preserve timing synchronization for phase;
• report coherence or repeated-trial uncertainty;
• avoid frequencies that violate actuator, safety, or unmodeled-mode limits.

Frequency response is empirical even when calculated from a model. Comparing measured and predicted $G(j\omega)$ directly reveals missing resonances, delay, and gain scheduling needs.

Reading a Bode plot as a control engineer #

1. Locate integrators and low-frequency gain: what error/disturbance classes can be rejected?
2. Locate crossover: what dynamics contribute phase there?
3. Estimate gain and phase margins, then inspect sensitivity peaks rather than relying on margins alone.
4. Check resonances and lightly damped modes on both sides of crossover.
5. Identify delay and nonminimum-phase zeros that cap useful bandwidth.
6. Inspect high-frequency roll-off and the controller-to-actuator/noise paths.
7. Compare the plot over plant uncertainty and operating points.

A Bode plot is most valuable as a causal accounting sheet: every slope and phase transition should have a physical or modeled source.