Chapter 13

Lead-Lag Control combines the benefits of both the Lead and the Lag Controllers. The transfer function of the Lead-Lag Controller is as follows:

 $G_c(s) = K_c \cdot \frac{s\tau_1 +1}{s\alpha_1\tau_1+1}\cdot\frac{s\alpha_2\tau_s +1}{s\tau_2+1}$ Equation 13-27

Kc corresponds to the DC gain of the controller and both $\alpha_1 < 1$, $\alpha_2 <1$. There are two zeros, at $s_1 = -\frac{1}{\tau_1}$, and $s_2 = -\frac{1}{\alpha_2\tau_2}$, and two poles, at $s_3 = -\frac{1}{\alpha_1\tau_1}$ and $s_4 = -\frac{1}{\tau_2}$. Note that what makes this compensator “tick,” is its sequence: POLE-ZERO-ZERO-POLE, as shown in Figure 13‑29. In the frequency domain, the four corner frequencies are:

$\omega_1 = \frac{1}{\tau_1}$, $\omega_2 = \frac{1}{\alpha_2\tau_2}$, $\omega_3 = \frac{1}{\alpha_1\tau_1}$, $\omega_4 = \frac{1}{\tau_2}$

A frequency response plot of the lead-lag compensator is shown in Figure 13‑30. Again, note the sequence: POLE-ZERO-ZERO-POLE, as shown in Figure 13‑29. This structure is sometimes also referred to as the Lag-Lead Controller – the Lag block comes first on the frequency plot, followed by the Lead block as Figure 13‑30 shows. We will however use the name Lead-Lag Controller, based on the sequence in which its components are used in the design – the Lead component is used first, then the Lag component.

Figure 13‑31 shows the values significant for the design procedure which is as follows: choose the compensator gain $K_c$, based on the steady state error requirements for the closed loop operation. Re-plot the open loop frequency response, including the required “gain lift”:

 $G_{open}(j\omega)=K_cG(j\omega)H(j\omega)$ Equation 13-28

Assume the necessary phase margin $\Phi_m$, based on the required Percent Overshoot. Determine the crossover frequency $\omega_{cp}$, from the settling time requirement.

Determine the necessary phase lead lift $\theta$ at this frequency (add an extra 5 degrees, since the Lag Controller block will be used):

 $\theta= \phi_{max} = -180^{\circ} + \Phi_m + 5^{\circ}-\angle GH(\omega_{cp})$ Equation 13-29

Calculate the Lead parameter $\alpha_1$:

 $\alpha_1 = \frac{1-\sin{\phi_{max}}}{1+\sin{\phi_{max}}}$ Equation 13-30

Calculate the Lead time constant $\tau_1$ from:

 $\omega_{cp}=\omega_0=\frac{1}{\sqrt{\alpha_1}\tau_1}$ Equation 13-31

Calculate (or measure from the plot) the total open loop gain at the crossover frequency:

 $M_{open}(j\omega_{cp})=\left | G(j\omega_{cp})H(j\omega_{cp}) \right | \cdot K_c\cdot\frac{1}{\sqrt{\alpha_1}}$ Equation 13-32

Calculate the Lag parameter $\alpha_2$ from a necessary gain reduction at this frequency:

 $\alpha_2 = \frac{1}{M_{open}}$ Equation 13-33

Calculate the Lag time constant $\tau_2$ from:

 $\omega_{cp}=\frac{10}{\alpha_2\tau_2}$ Equation 13-34

Comment:

This design theoretically meets all three typical performance requirements – accuracy, speed, and lack of oscillations. Whether it will work well, depends on how closely the compensated closed loop transfer function resembles our standard second order under-damped model, on which the design was based. Always run simulations of the closed loop system response under this compensation scheme – the design may require iterations to improve its performance.