It is a fundamental building block in classical control theory.
Lead–lag compensators influence disciplines as varied as robotics, satellite control, automobile diagnostics, LCDs and laser frequency stabilisation.
Given the control plant, desired specifications can be achieved using compensators.
I, P, PI, PD, and PID, are optimizing controllers which are used to improve system parameters (such as reducing steady state error, reducing resonant peak, improving system response by reducing rise time).
The transfer function can be written in the Laplace domain as where X is the input to the compensator, Y is the output, s is the complex Laplace transform variable, z is the zero frequency and p is the pole frequency.
The pole and zero are both typically negative, or left of the origin in the complex plane.
This shifts the root locus to the left, which enhances the responsiveness and stability of the system.
The precise locations of the poles and zeros depend on both the desired characteristics of the closed loop response and the characteristics of the system being controlled.
Since their purpose is to affect the low frequency behaviour, they should be near the origin.
Both analog and digital control systems use lead-lag compensators.
The technology used for the implementation is different in each case, but the underlying principles are the same.
The transfer function is rearranged so that the output is expressed in terms of sums of terms involving the input, and integrals of the input and output.
In this case a lead-lag compensator will consist of a network of operational amplifiers ("op-amps") connected as integrators and weighted adders.
A possible physical realization of a lead-lag compensator is shown below (note that the op-amp is used to isolate the networks):
In digital control, the operations are performed numerically by discretization of the derivatives and integrals.
The reason for expressing the transfer function as an integral equation is that differentiating signals amplify the noise on the signal, since even very small amplitude noise has a high derivative if its frequency is high, while integrating a signal averages out the noise.
This makes implementations in terms of integrators the most numerically stable.
Remember that a complex function can be in general written as
If the total network phase angle has a combination of positive and negative phase as a function of frequency then it is a lead-lag network.
Depending upon the nominal operation design parameters of a system under an active feedback control, a lag or lead network can cause instability and poor speed and response times.