Lead-lag controller

In control theory, two important types of system controllers are lead controllers and lag controllers. They are sometimes used in cascade with each other, in lead-lag configurations.

Lead controllers take the form:

$$C(s)=k\frac{T+1}{\alpha T+1}$$

Lag controllers take the form:

$$C(s)=\alpha \frac{T+1}{\alpha T+1}$$

where $k$, $\alpha$, and $T$ are parameters to be designed.

Lead controllers are used to increase the phase margin and crossover frequency.

Lag controllers are mainly used to increase the low-frequency gain of the system without significantly affecting the phase at the crossover frequency. We usually use them if:

• We need better steady-state accuracy, but the phase margin and crossover frequency are already satisfactory.
• High-frequency attenuation is needed, especially for noise or unmodelled behaviour.
• We need to meet steady-state specifications, but can’t increase the overall gain $k$ because this would affect crossover frequency or stability margins. This is for systems where increasing $k$ would push crossover frequency too high or reduce phase margin too much.

Parameters

Several relevant parameters are given by:

$${\Phi }_{\text{max}}=\arcsin \left(\frac{1-\alpha }{1+\alpha }\right)$$

where ${\Phi }_{\text{max}}$ is the maximum argument (which we can find from the Bode plot of the controller).

$$\alpha =\frac{1-\sin ({\Phi }_{\text{max}})}{1+\sin ({\Phi }_{\text{max}})}$$

The maximum frequency is given by:

$${\omega }_{\text{max}}=\frac{1}{\sqrt{\alpha }T}$$

Design

Lead controllers are designed as follows:

• Determine a value $k$ that satisfies steady-state tracking error or crossover frequency ${\omega }_c$ specifications.
• Determine ${\Phi }_{\text{max}}$ to obtain the desired phase margin. In a design context, the formula is given by (where ${\text{PM}}_d$ is the desired phase margin, ${\text{PM}}_{og}$ is the original PM, and extra margin is typically a value like $30°$):

$${\Phi }_{\text{max}}={\text{PM}}_d-{\text{PM}}_{og}+\text{extra margin}$$

• Find $\alpha$ from ${\Phi }_{\text{max}}$.
• Find ${\omega }_{\text{max}}$ with the above formula.
• Then use this formula:

$$|G(j{\omega }_{\text{max}})|=10\log \alpha$$

• With this value, we check the Bode plot (usually the $k$ adjusted one) to find ${\omega }_{\text{max}}$. This can be done by finding the value of $\omega$ for which $|G|=10\log \alpha$.
• Then, we can compute $\frac{1}{T}$.
• These are all the parameters we need to find.

Lag controllers are easier to design:

• Choose some $\alpha$ to meet steady-state specifications.
• Choose some $\frac{1}{T}$ such that $C(s)$ doesn’t affect the PM and ${\omega }_c$. In the majority of cases, it suffices to find $T$ with the inequality:

$$\frac{1}{T}\le 0.1{\omega }_c$$