Frequency response and Bode plots

Feed a sinusoid into a stable LTI system and, once the transients die away, a sinusoid comes out at exactly the same frequency, changed only in amplitude and phase. How the amplitude ratio and phase shift vary with frequency is the system's frequency response, and it is one of the most useful descriptions an engineer has. It can be measured directly on hardware with a signal generator or a chirp, without knowing any equations. It shows at a glance which frequencies a sensor passes, where a mechanism resonates, how much noise a filter removes and, through gain and phase margins, how close a feedback loop is to instability. The standard way to draw it is the Bode plot, with logarithmic frequency and gain in decibels, on which complicated systems become sums of simple straight-line pieces. This lesson derives the frequency response from the transfer function, teaches the asymptotic sketching rules, and applies them to bandwidth, resonance and stability margins.

Sinusoids in, sinusoids out

Lesson 2 noted that complex exponentials pass through LTI systems unchanged in shape. Here is why. If the input is \( x(t) = e^{j\omega t} \), convolution with the impulse response gives

The integral is just the transfer function evaluated on the imaginary axis, \( s = j\omega \). For a real sinusoid \( A\cos\omega t \), taking real parts gives the steady-state output

The magnitude \( |G(j\omega)| \) is the gain at that frequency, and the angle \( \angle G(j\omega) \) is the phase shift; a negative phase means the output lags the input. Together they form the frequency response. For a stable system it can be measured by injecting sinusoids of increasing frequency, or a swept sine (chirp), or broadband noise followed by spectral analysis (lesson 8), and comparing output with input. Robot engineers routinely measure the frequency response of servo axes this way to find resonances before tuning.

Decibels and the Bode plot

Gains in robotics span many orders of magnitude, from the near-unity gain of a sensor at low frequency to the tiny gain of a filter in its stopband, so they are expressed in decibels:

A gain of 10 is +20 dB, 100 is +40 dB, 1 is 0 dB, 0.1 is −20 dB, and \( 1/\sqrt{2} \) is −3 dB. Decibels turn multiplication into addition: the dB gain of blocks in series is the sum of their dB gains, and the phases also add. A Bode plot draws magnitude in dB and phase in degrees against frequency on a logarithmic axis. A factor of ten in frequency is a decade, and a factor of two an octave. On these axes most transfer-function factors become straight lines, so a complicated response can be sketched by adding a few simple pieces, which is why Hendrik Bode's 1940 invention is still used daily.

Sketching rules for the basic factors

Any rational transfer function factors into a gain, poles and zeros at the origin, real poles and zeros, and complex pairs. Each has a simple Bode shape.

• Constant gain K: a flat line at \( 20\log_{10}K \) dB, phase 0° (or −180° if K is negative).
• Integrator 1/s: a straight line falling at −20 dB per decade, passing 0 dB at ω = 1 rad/s, with a constant phase of −90°. A differentiator s rises at +20 dB per decade with +90°.
• Real pole \( 1/(s/\omega_c + 1) \): flat at 0 dB below the corner frequency \( \omega_c \), falling at −20 dB per decade above it. The true curve is 3 dB below the corner of the asymptotes at \( \omega_c \). The phase goes from 0° to −90°, passing −45° at \( \omega_c \), and changes mostly between \( \omega_c/10 \) and \( 10\omega_c \).
• Real zero \( s/\omega_c + 1 \): the mirror image, rising at +20 dB per decade with phase going to +90°.
• Complex pole pair (second-order, \( \omega_n \), \( \zeta \)): flat at 0 dB, then falling at −40 dB per decade above \( \omega_n \), with phase from 0° to −180° passing −90° at \( \omega_n \). For \( \zeta < 0.707 \) there is a resonant peak near \( \omega_n \) of height about \( 1/(2\zeta) \), or \( -20\log_{10}(2\zeta) \) dB, and the phase changes sharply. At \( \zeta = 0.1 \) the peak is +14 dB, a fivefold amplification.
• Pure delay \( e^{-sT} \): magnitude exactly 0 dB at all frequencies but phase \( -\omega T \) radians, growing without limit. This is why delay is so damaging to feedback loops.

Bandwidth and resonance

Bandwidth measures how fast a system can respond. For a low-pass system, the bandwidth \( \omega_B \) is the frequency at which the gain has fallen 3 dB below its low-frequency value: signals below it pass largely unchanged; signals above it are attenuated. For a first-order system the bandwidth equals \( 1/\tau \), and rise time and bandwidth are linked by the useful rule \( t_r \approx 2.2/\omega_B = 0.35/f_B \). A sensor with 100 Hz bandwidth cannot faithfully report events shorter than a few milliseconds; a joint servo with 10 Hz closed-loop bandwidth cannot track trajectories containing motion much faster than 10 Hz. Specifications for robot actuators and sensors therefore quote bandwidth constantly.

Resonance is the opposite danger: a lightly damped mode amplifies inputs near its natural frequency. A robot arm whose first structural mode is at 15 Hz will shake violently if the controller or the trajectory excites 15 Hz, which is why trajectory planners avoid exciting known modes and why notch filters (lesson 11) are inserted in servo loops to remove them.

Stability margins

Lesson 6 showed that a loop goes unstable when the signal returning around the loop comes back with enough gain and the wrong phase. On the Bode plot of the loop transfer function L(jω), this is easy to see. The gain crossover frequency \( \omega_{gc} \) is where \( |L| = 1 \) (0 dB). The phase crossover frequency \( \omega_{pc} \) is where \( \angle L = -180° \). Two numbers measure how far the loop is from instability:

• Phase margin: \( PM = 180° + \angle L(j\omega_{gc}) \), the extra phase lag that would make the loop unstable. Typical designs aim for 45° to 60°. Phase margin relates closely to damping: roughly \( \zeta \approx PM/100 \) for PM up to about 60°.
• Gain margin: \( GM = -20\log_{10}|L(j\omega_{pc})| \) dB, the extra gain that would make the loop unstable. Typical designs aim for at least 6 dB, a factor of two.

These margins come from the Nyquist stability criterion, which ROB-300 covers in full. For most robot servo loops, where L is stable and has a single gain crossover, the simple rule is: the closed loop is stable if the phase margin and gain margin are both positive.

import numpy as np
w = np.logspace(-1, 3, 4001)           # 0.1 to 1000 rad/s
s = 1j*w
for K in [100, 20]:
    L = K/(s*(s/10 + 1)) * np.exp(-0.01*s)   # motor plant plus 10 ms delay
    mag = 20*np.log10(np.abs(L))
    ph = np.degrees(np.unwrap(np.angle(L)))
    i = np.argmin(np.abs(mag))               # gain crossover
    j = np.argmin(np.abs(ph + 180))          # phase crossover
    print(f"K = {K:3d}: crossover {w[i]:5.1f} rad/s, phase margin {180 + ph[i]:5.1f} deg, gain margin {-mag[j]:5.1f} dB")

K = 100: crossover  30.8 rad/s, phase margin   0.3 deg, gain margin   0.1 dB
K =  20: crossover  12.5 rad/s, phase margin  31.5 deg, gain margin  14.1 dB

The script evaluates L(jω) on a fine logarithmic grid, finds the crossovers numerically and reports the margins. With K = 100, the plant sketched earlier plus a 10 ms delay, both margins are essentially zero: the loop sits on the edge of instability and would oscillate. Cutting the gain by a factor of five lowers the crossover frequency, where the plant and delay contribute less phase lag, and gives about 31° of phase margin and 14 dB of gain margin. That is stable but still short of the 45° target, so the response would overshoot noticeably; lead compensation, in ROB-300, recovers phase margin without giving up so much speed.