Control Systems · 96 h · Topic 7 of 8

Frequency response in control

A delay does not change the gain of a loop, but it adds unbounded phase: its Nyquist plot is a spiral. That is why processes with delay are controlled with low gains.

Bode plot Nyquist criterion Gain margin Phase margin Delay

01Designing by looking at the loop gain in frequency

The root locus needs a model with poles and zeros. The frequency response works with what can be measured: the loop gain \( G(j\omega) \), with a generator and an analyzer, even if the plant has delays or parts that cannot be modeled. Its two representations are the Bode plot, magnitude and phase as a function of frequency, and the Nyquist plot, the curve traced by \( G(j\omega) \) in the complex plane. The fundamentals were covered in frequency response, in Signals and Systems, and their use in amplifiers in stability and compensation.

02The Nyquist criterion

The closed loop is unstable if \( 1 + G(s) \) has zeros in the right half-plane. Harry Nyquist showed in 1932 that they can be counted by looking at how many times the curve of \( G(j\omega) \), traced for ω from −∞ to +∞, encircles the point −1:

\[ Z = N + P \] \( Z \) is the number of unstable closed-loop poles, \( N \) the number of clockwise encirclements of −1 and \( P \) the number of unstable open-loop poles. If the plant is stable (P = 0), the closed loop is stable if the curve does not encircle −1.

The distance from the curve to the point −1 is the measure of relative stability, and is summarized in two margins:

\[ GM = \frac{1}{|G(j\omega_{180})|} \] \[ PM = 180° + \angle G(j\omega_c) \] \( \omega_{180} \) is where the phase equals −180° and \( \omega_c \), where the magnitude equals 1. The gain margin says how much the gain can grow before oscillating; the phase margin, how much additional lag is tolerated. Usual values: GM of 6 dB or more and PM between 30° and 60°.

03The delay

Many processes have a pure delay: the time hot water takes to travel along a pipe, or a measurement to be processed. A delay \( T \) does not change the magnitude but adds a phase \( -\omega T \) that grows without bound, and its Nyquist plot is a spiral. It cannot be represented with a finite number of poles, but the frequency response handles it without difficulty. It is the reason why processes with delay are controlled with low gains.

Lab · Nyquist plot and margins

The curve of \( K\,G(j\omega) \) for ω from 0 to infinity (the symmetric half, from −∞ to 0, is drawn faintly). The red dot is −1; the dotted circle is the unit-magnitude one, where the phase margin is read.

Gain K =

04From open loop to closed loop

The margins translate into the closed-loop response: a phase margin of 45° gives an overshoot close to 23 %, and one of 60°, close to 9 %, as in a second-order system. The crossover frequency \( \omega_c \) sets the bandwidth of the closed loop and, with it, the speed of response. Designing in frequency means shaping \( |G(j\omega)| \): high at low frequencies to reject disturbances and reduce the error, low at high frequencies so as not to amplify noise, and with a gentle slope, of about −20 dB per decade, at the crossover, so that the phase has margin.

05In the lab

Activity 1 · Bode plot of a plant

Measure with a generator and an oscilloscope the frequency response of a circuit of three RC stages separated by buffers, and draw its Bode plot. Calculate the gain that would drive the loop to oscillate.

Activity 2 · Nyquist

With the same measurements, draw the Nyquist plot and read the margins for a given gain.

Activity 3 · Digital delay

Close the loop with a microcontroller that adds one or two sampling periods of delay, and measure how the phase margin drops.

06Common mistakes

Looking at the closed-loop Bode plot to judge stability. The margins are read on the loop gain.
Forgetting the unstable open-loop poles. With P different from zero, the loop needs to encircle −1 to be stable.
Ignoring the delay of a digital controller. Conversion and computation add phase lag at high frequencies.
Relying on a single margin. A curve can have a good gain margin and pass very close to −1 elsewhere.

07Self-assessment

What is the gain margin of K/(s+1)³ with K = 4?

The phase reaches −180° at ω = √3, where \( |G| = 4/8 \): a margin of 2, that is, 6 dB.

How much phase does a 0.1 s delay add at 10 rad/s?

1 rad, about 57°.

What overshoot is expected with 45° of phase margin?

Around 23 %.

What does the Nyquist criterion say if the plant is stable?

That the closed loop is stable if the plot does not encircle the point −1.

08Further reading

Katsuhiko Ogata. Modern Control Engineering. 5th ed., Prentice Hall, 2010 (Spanish edition: “Ingeniería de control moderna”, Pearson). Bode and Nyquist plots and design by frequency response.
Karl J. Åström and Richard M. Murray. Feedback Systems. 2nd ed., Princeton University Press, 2021. The Nyquist criterion, robustness and the trade-offs of frequency-domain design.
Harry Nyquist. Regeneration Theory. Bell System Technical Journal, 11, 1932. The original paper on the stability criterion.

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