Lag compensation is a technique used in control systems to improve the stability and performance of a system by introducing a lag compensator. The transfer function of a lag compensator is given by:
$$ G(s) = \frac{s+1/T}{s+1/(\alpha T)}, \alpha > 1 $$
It is called lag compensation because the compensator introduces a phase lag in the frequency response of the system. This phase lag helps to increase the phase margin of the system, thereby improving its stability.
To understand how lag compensation works, let's analyze the frequency response and phase response of the lag compensator.
Frequency Response: At low frequencies ($s << 1/T$), the transfer function can be approximated as:
$$ ⁍ $$
At high frequencies ($s >> 1/(\alpha T)$), the transfer function can be approximated as: G(s) ≈ 1
This means that the lag compensator has a gain of \alpha at low frequencies and a gain of 1 at high frequencies. The transition between these two gains occurs at a frequency determined by the time constant T.
Phase Response: The phase response of the lag compensator can be derived from the transfer function:
$$
⁍ $$
At low frequencies, the phase lag introduced by the compensator is approximately:
$$ ⁍ $$
At high frequencies, the phase lag introduced by the compensator is approximately:
$$ ⁍ $$
The lag compensator introduces a maximum phase lag of $90°(1-1/\alpha)$ at a frequency determined by the time constant T.
MATLAB Code: Here's a MATLAB code snippet to plot the frequency response and phase response of a lag compensator:
% Define the parameters
T = 1;
alpha = 10;
% Create the transfer function
num = [T 1];
den = [alpha*T 1];
G = tf(num, den);
% Plot the frequency response and phase response
figure;
bode(G);
grid on;
title('Frequency Response and Phase Response of Lag Compensator');
Explanation: