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Compensators: lead and lag networks

From specifications to the shape of the loop function, the lead network and maximum phase lead with proof, a complete design with verification, the lag network and lead-lag.

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Frequency-domain design consists of giving the loop function L(j\omega) the right shape: high gain at low frequency for steady-state errors, a suitable crossover frequency for speed, enough phase margin for damping. The tools are compensators: the lead network adds phase near crossover, the lag network raises the low-frequency gain without touching the margin. In this lesson we derive them and complete the example from the lesson on the Nyquist criterion.

From specifications to the shape of L

Time-domain specifications are translated into frequency with approximate but widely used rules:

  • phase margin \varphi_m → damping \zeta \approx \varphi_m/100, hence overshoot;
  • crossover frequency \omega_c → speed: the rise time is about 1.8/\omega_c and the closed-loop bandwidth is of the same order as \omega_c;
  • low-frequency gain → steady-state error (lesson on errors).

The Bode magnitude plot should cross 0 dB with a slope of about −20 dB/decade: steeper slopes at crossover mean a phase close to −180° and small margins.

The lead network

    \[ C_a(s) = \frac{1 + \alpha T s}{1 + T s},\qquad \alpha \gt 1 \]

has a zero at -1/(\alpha T) and a pole further left, at -1/T. Between the two frequencies the magnitude rises by 20\log_{10}\alpha dB and the phase is positive: it is a limited derivative action, which does not amplify noise without bound.

Bode plots of the lead network
The magnitude goes from 0 dB to 20 dB between the frequencies 1/(αT) and 1/T, here with α = 10. The phase is positive in that interval, with a maximum φ_max at the geometric mean of the two frequencies, 1/(T√α).

Theorem. The phase lead is largest at the frequency \omega_m = \frac{1}{T\sqrt\alpha}, the geometric mean of zero and pole, and equals

    \[ \sin\varphi_{max} = \frac{\alpha - 1}{\alpha + 1} \]

There the magnitude is \sqrt\alpha, that is 10\log_{10}\alpha dB.

Proof. The phase is \varphi(\omega) = \arctan(\alpha\omega T) - \arctan(\omega T). Setting x = \omega T and using the tangent of a difference:

    \[ \tan\varphi = \frac{\alpha x - x}{1 + \alpha x^2} = \frac{(\alpha - 1)x}{1 + \alpha x^2} \]

The maximum is where the derivative with respect to x vanishes: (1 + \alpha x^2) - 2\alpha x^2 = 0, that is x = 1/\sqrt\alpha. There \tan\varphi_{max} = \frac{\alpha - 1}{2\sqrt\alpha}, and from the right triangle with legs \alpha - 1 and 2\sqrt\alpha the hypotenuse is \sqrt{(\alpha - 1)^2 + 4\alpha} = \alpha + 1: so \sin\varphi_{max} = \frac{\alpha - 1}{\alpha + 1}. The magnitude at x = 1/\sqrt\alpha is \sqrt{(1 + \alpha)/(1 + 1/\alpha)} = \sqrt\alpha. ∎

\alpha461020
\varphi_{max}36.9°45.6°54.9°64.8°

Beyond \alpha = 20 the phase gain grows little and the noise a lot: for larger leads two networks in cascade are used.

A complete design

Plant and gain from the lesson on the Nyquist criterion: L(s) = \frac{10}{s(s + 1)(s + 5)}, with a phase margin of 25°, too little. Goal: at least 45 to 50°.

  1. Choosing α. About 25 to 30° of lead is needed, plus a margin for the fact that the network moves crossover to the right, where the plant phase is worse. Try \alpha = 6 (\varphi_{max} \approx 46°).
  2. Placement. It is best to put the phase maximum right at the new crossover. There the network adds 10\log_{10}6 \approx 7.8 dB, so the new \omega_c is where the plant alone is at -7.8 dB, that is |L| = 1/\sqrt6: this happens at \omega \approx 2.02 rad/s.
  3. Computing T. \omega_m = 1/(T\sqrt\alpha) gives T \approx 0.20 s, and the network is C_a(s) = \frac{1 + 1.21s}{1 + 0.20s}.
  4. Check. With the network the phase margin becomes about 50° and crossover moves from 1.23 to 2.02 rad/s: a better damped response (in simulation the overshoot drops from about 49% to 17%) and a faster one too.

With \alpha = 4 one would get about 46°, with \alpha = 10 about 54°: the final choice is made by simulating and also checking the effect on noise.

The lag network

    \[ C_r(s) = \frac{1 + T s}{1 + \alpha T s},\qquad \alpha \gt 1 \]

is the opposite: it attenuates high frequencies \alpha times, leaving low ones unchanged. Used together with an \alpha-fold increase in gain, it has the effect of multiplying the low-frequency gain by α (and dividing the steady-state error by α) without changing the behaviour near crossover. Its phase lag must be kept away from \omega_c: the zero 1/T is placed at least a decade below crossover.

Lead and lag networks together form a lead-lag compensator, the frequency-domain equivalent of a PID: the lag part does the work of the integral, the lead part that of the derivative.

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