Lead and Lag compensator
Introduction to compensators in control systems¶
Compensators are used to improve the stability and accuracy of a system. It is added to a controller as a plugin to tune the system to meet the system's dynamic performance requirements.
Lead compensator¶
The lead compensator is an network (typically an RC network) which produces a sinusoidal output having phase lead when a sinusoidal input is applied. It is added to improve the phase margin of a system, which reduces excessive ringing / overshoot in a system. It increases system's bandwidth.
Here, the capacitor is parallel to the resistor R1 and the output is measured across resistor R2. The transfer function of this lead compensator is -
$$\cfrac{V_o(s)}{V_i(s)}=\left(\cfrac{1+\alpha{}\tau{}s}{1+\tau{}s}\right)$$
Where,
$$\tau{}=R_1C$$ $$\alpha{}=\cfrac{R_2}{R_1+R_2}$$
From the transfer function, we can observe that the lead compensator has pole at \(s=1/\tau{}\) and zero at \(s=1/\alpha{}\tau{}\). \(\alpha{}\) is less than unity. This means the zero will appear earlier than the pole in the bode plot.
To find the phase relation, substituting, \(s=j\omega{}\) in the transfer function -
$$\cfrac{V_o(j\omega{})}{V_i(j\omega{})}=\left(\cfrac{1+j\omega{}\alpha{}\tau{}}{1+j\omega{}\tau{}}\right)$$
Phase angle,
$$\phi{}=\tan^{-1}\left(\alpha{}\omega{}\tau{}\right)-\tan^{-1}\left(\omega{}\tau{}\right)$$
Maximum phase is added at frequency,
$$\omega{}_m=\cfrac{1}{\tau{}\sqrt{\alpha{}}}$$
Maximum phase,
$$\phi{}_m=\cfrac{\alpha{}-1}{\alpha{}+1}$$
Lag compensator¶
The lead compensator is an analog circuit network which produces a sinusoidal output having phase lead when a sinusoidal input is applied. It is added to improve the phase margin of a system, which reduces excessive ringing / overshoot in a system.
Here, the capacitor is in series with the resistor R2 and the output is measured across this combination. The transfer function of this lead compensator is -
$$\cfrac{V_o(s)}{V_i(s)}=\left(\cfrac{1+\alpha{}\tau{}s}{1+\tau{}s}\right)$$
Where,
$$\tau{}=R_2C$$
$$\alpha{}=\cfrac{R_1+R_2}{R_2}$$
\(\alpha{}>1\). From the transfer function, we can conclude that the lag compensator has one pole at \(s=1/\tau{}\) and one zero \(s=1/\alpha{}\tau{}\). This means, the pole will appear earlier than zero in the bode plot for the pole-zero configuration of the lag compensator.
To find the phase relation, substituting, \(s=j\omega{}\) in the transfer function -
$$\cfrac{V_o(j\omega{})}{V_i(j\omega{})}=\cfrac{1}{\alpha{}}\left(\cfrac{j\omega{}+\cfrac{1}{\tau{}}}{j\omega{}+\cfrac{1}{\alpha{}\tau{}}}\right)$$
Phase angle,
$$\phi{}=\tan^{-1}\left(\omega{}\tau{}\right)-\tan^{-1}\left(\alpha{}\omega{}\tau{}\right)$$
Maximum phase is subtracted at frequency,
$$\omega{}_m=\cfrac{1}{\tau{}\sqrt{\alpha{}}}$$
Maximum phase,
$$\phi{}_m=\cfrac{\alpha{}-1}{\alpha{}+1}$$
Lead and Lag compensator¶
We can also combine the two types of compensators listed above to yield a lead-lag compensator. Sometimes, both effects are needed — faster rise/fall (lag-compensator) and reaching steady-state faster without ringing (lead compensator). This is similar to PID tuning.