Frequency response and Bode plots

Lucie has spent her career looking at vibration spectra, and she thinks about systems in terms of frequency. When Owen shows her the root locus of the PI pressure loop, she asks a question it cannot answer directly: what happens if the oil temperature sensor, mounted a few meters down the return line, adds a fraction of a second of pure delay? Delays do not fit the root locus rules, because \(e^{-sT}\) is not a ratio of polynomials. The frequency response handles them without difficulty, and it has another advantage that suits a test laboratory: it can be measured. Feed a sine wave into a stable system, wait for the transient to die, and the output is a sine wave of the same frequency, scaled and shifted. Those two numbers at every frequency are the frequency response.

Frequency response

For a stable linear system G(s), a steady sinusoidal input \(u = U\sin\omega t\) produces a steady output

The amplitude ratio is the magnitude of G evaluated at \(s = j\omega\), and the phase shift is its angle. Lucie recognizes this at once as the frequency response function of MEC-380, measured there with a hammer and an accelerometer; for a control loop it is measured with a sine sweep from the controller or computed from the model. For a first-order lag \(1/(\tau s + 1)\), \(|G(j\omega)| = 1/\sqrt{1 + (\omega\tau)^2}\) and \(\angle G = -\arctan\omega\tau\): at low frequencies the output follows the input; at \(\omega = 1/\tau\), the corner frequency, the amplitude has fallen to 0.707 and the phase lags by 45°; at high frequencies the amplitude falls in proportion to 1/ω and the lag approaches 90°.

Bode plots

Hendrik Bode, working on telephone amplifiers in the 1930s and 1940s, introduced the plots that carry his name: magnitude in decibels, \(20\log_{10}|G|\), and phase in degrees, both against frequency on a logarithmic axis. The logarithms turn products into sums, so the Bode plot of a loop made of many blocks is the sum of the plots of the blocks. Each standard factor has a simple straight-line approximation, the asymptotes, that can be drawn by hand.

The magnitude asymptote of a lag is accurate to within 3 dB, the worst error falling at the corner. The phase changes over about two decades centered on the corner. The time delay is unusual: it leaves the magnitude untouched but adds a phase lag proportional to frequency, so on a logarithmic axis its phase curve falls ever more steeply.

The pressure loop's Bode plot

For the PI loop designed in Lesson 10 (\(K_c = 1.46\), \(T_i = 0.72\) s), the loop transfer function contains an integrator, the gain \(1.46 \times 0.708/0.72 = 1.44\) after cancellation, and lags with corners at 2 and 10 rad/s. The magnitude falls at 20 dB/decade from low frequencies, crosses 0 dB at 1.22 rad/s, then steepens to 40 and finally 60 dB/decade beyond the two corners. The phase starts at −90° because of the integrator and falls toward −270° as the lags add their 90° each.

Gain and phase margins

Stability of the closed loop can be read from the open-loop Bode plot. At the gain crossover frequency \(\omega_c\), where \(|L(j\omega)| = 1\), the phase margin is how far the phase is above −180°: \(PM = 180^\circ + \angle L(j\omega_c)\). At the phase crossover frequency \(\omega_{180}\), where the phase is −180°, the gain margin is how far the magnitude is below 0 dB: \(GM = -20\log_{10}|L(j\omega_{180})|\). If both margins are positive, the closed loop is stable (for loops whose open-loop poles are all stable, which covers every loop in this course). The gain margin agrees with the Routh result of Lesson 8; the phase margin is new, and it is the margin that delays consume.

The margins measure robustness. The gain margin says how much the plant's gain may grow before the loop goes unstable; the phase margin says how much extra lag, from a delay, a slow sensor or a neglected dynamic, the loop can tolerate at its crossover frequency. Typical targets are a gain margin of at least 6 dB and a phase margin of 45° to 60°. A phase margin of about 50° to 60° usually gives a step response with modest overshoot, roughly 10 to 20 percent for loops containing an integrator.

# Frequency response of the pressure loop: Bode values, crossover, gain and phase margins, and the effect of dead time
import math, cmath
def L(w, Kc, Ti=None, delay=0.0):
    s = 1j*w
    G = 0.708/((0.5*s + 1)*(0.72*s + 1)*(0.1*s + 1))
    C = Kc*(1 + 1/(Ti*s)) if Ti else Kc
    return C*G*cmath.exp(-s*delay)
def margins(Kc, Ti=None, delay=0.0):
    ws = [10**(k/2000) for k in range(-6000, 4001)]          # 0.001 to 100 rad/s
    wc = wg = None; prev = None
    for w in ws:
        v = L(w, Kc, Ti, delay); mag = abs(v); ph = math.degrees(cmath.phase(v))
        ph_unw = ph if prev is None else prev[1] + ((ph - prev[1] + 180) % 360 - 180)
        if prev and prev[0] >= 1 > mag and wc is None: wc, pm = w, 180 + ph_unw
        if prev and prev[1] > -180 >= ph_unw and wg is None: wg, gm = w, -20*math.log10(mag)
        prev = (mag, ph_unw)
    return wc, pm, wg, gm
print("Bode values of the PI loop (Kc = 1.46, Ti = 0.72 s)")
print("   w (rad/s)   |L| (dB)   phase (deg)")
ph_prev = None
for w in (0.1, 0.3, 1.0, 1.39, 2.0, 3.0, 6.0, 10.0, 30.0):
    v = L(w, 1.46, 0.72); ph = math.degrees(cmath.phase(v))
    if ph > 0: ph -= 360
    print(f"   {w:8.2f}   {20*math.log10(abs(v)):7.1f}    {ph:8.1f}")
print("\n  loop                                crossover (rad/s)  PM (deg)  phase crossover (rad/s)  GM (dB)")
for lab, Kc, Ti, d in (("P, Kc = 4", 4.0, None, 0.0), ("PI, Kc = 1.46, Ti = 0.72 s", 1.46, 0.72, 0.0),
                       ("PI, same, 0.2 s dead time", 1.46, 0.72, 0.2), ("PI, Kc = 3.0, 0.2 s dead time", 3.0, 0.72, 0.2)):
    wc, pm, wg, gm = margins(Kc, Ti, d)
    print(f"  {lab:34s}  {wc:8.2f}           {pm:6.1f}       {wg:8.2f}             {gm:6.1f}")
# Dead time costs phase: 0.2 s at the crossover frequency of the PI loop
wc = margins(1.46, 0.72)[0]
print(f"\nA 0.2 s delay removes {math.degrees(0.2*wc):.1f} deg of phase at w = {wc:.2f} rad/s without changing |L|")

Bode values of the PI loop (Kc = 1.46, Ti = 0.72 s)
   w (rad/s)   |L| (dB)   phase (deg)
       0.10      23.1       -93.4
       0.30      13.5      -100.2
       1.00       2.1      -122.3
       1.39      -1.5      -132.7
       2.00      -6.1      -146.3
       3.00     -11.9      -163.0
       6.00     -23.8      -192.5
      10.00     -34.0      -213.7
      30.00     -59.9      -247.8

  loop                                crossover (rad/s)  PM (deg)  phase crossover (rad/s)  GM (dB)
  P, Kc = 4                               2.19             62.4           6.06               15.4
  PI, Kc = 1.46, Ti = 0.72 s              1.22             51.7           4.48               18.5
  PI, same, 0.2 s dead time               1.22             37.8           2.36                8.3
  PI, Kc = 3.0, 0.2 s dead time           2.03              9.8           2.36                2.1

A 0.2 s delay removes 14.0 deg of phase at w = 1.22 rad/s without changing |L|

The program evaluates the loop at many frequencies and finds the crossovers numerically. The PI design has a crossover at 1.22 rad/s, a phase margin of 51.7° and a gain margin of 18.5 dB, comfortably inside the targets. For comparison, the proportional loop at \(K_c = 4\) crosses over faster, at 2.19 rad/s, with a phase margin of 62° and the 15.4 dB gain margin found in Lesson 8; its larger margin does not save it, because it fails the accuracy requirement.

The cost of dead time

Now Lucie's question. Suppose the measurement arrives 0.2 s late, because the transmitter's averaging or a long impulse line delays it. The delay does not change the magnitude, so the crossover stays at 1.22 rad/s, but it removes \(0.2 \times 1.22\) rad, or 14°, of phase there. The program shows the phase margin falling from 51.7° to 37.8°, and the phase crossover moving down from 4.48 to 2.36 rad/s, which cuts the gain margin from 18.5 to 8.3 dB. The loop is still stable but noticeably more oscillatory. Doubling the controller gain to \(K_c = 3.0\) with the delay present would leave a phase margin of under 10° and a gain margin of 2 dB, a loop close to the edge.

The general lesson, which Owen writes at the top of his design notes, is that delay limits how fast any loop can be made. Since a delay of T costs ωT radians at crossover, keeping its cost below, say, 30° requires \(\omega_c < 0.5/T\). A loop with 0.2 s of delay cannot reasonably be given a crossover above about 2.5 rad/s, however clever the controller. Long sensor lines, slow analyzer sampling and transport delays in pipes are therefore design issues, not details.

Bandwidth and the closed-loop frequency response

The closed-loop frequency response \(T(j\omega)\) is close to 1 at low frequencies, where the loop gain is high, and falls away above the crossover. The bandwidth, the frequency at which \(|T|\) has fallen 3 dB below its low-frequency value, is usually close to the crossover frequency, and it is the frequency-domain measure of speed: rise time is roughly \(2/\omega_{BW}\) for well-damped loops. A resonant peak in \(|T|\) corresponds to poorly damped closed-loop poles and to overshoot in the step response; the peak grows as the phase margin shrinks. The sensitivity function \(|S(j\omega)|\) is small below crossover, where disturbances are rejected, and has a peak just above it, where disturbances are amplified; a large peak, typically more than 2, is another sign of an aggressive design.

Nyquist's view

Harry Nyquist, a colleague of Bode, gave the general stability test from which the margins come. Plotting \(L(j\omega)\) in the complex plane as ω runs from zero to infinity traces the Nyquist curve. For a loop whose open-loop poles are all stable, the closed loop is stable if the curve does not encircle the point −1. The gain margin is how far the curve passes from −1 along the negative real axis; the phase margin is the angle, measured where the curve crosses the unit circle, between the curve and the negative real axis. The Nyquist criterion also handles open-loop unstable plants, by counting encirclements, and loops with delays, which simply wind the curve into a spiral. For the loops in this course, the Bode margins carry the same information in a more convenient form.

Measuring frequency response

Because frequency response can be measured, it is a natural way to validate a model in a laboratory. Gilles will run a sine sweep on the valve actuator during commissioning: the controller's output is oscillated by a few percent at frequencies from 0.5 to 50 Hz, and the stem position recorded. The measured magnitude should fall at 20 dB/decade, the signature of the integrator, until the spool lag near 10 Hz steepens it; a peak near 166 Hz would confirm the hydraulic resonance of Lesson 4. Differences between model and measurement point straight at the element that is wrong, which is harder to see in a step response.