Electrical Understanding

What is a compensator, and how does it operate ?

• A lead–lag compensator is a component in a control system that improves an undesirable frequency response in a feedback and control system. It is a fundamental building block in classical control theory.

Applications

Lead–lag compensators influence disciplines as varied as robotics, satellite control, automobile diagnostics, laser frequency stabilization, and many more. They are an important building block in analog control systems, and can also be used in digital control.
Given a control plant, desired specifications can be achieved using compensators. I, D, 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). All these operations can be done by compensators as well.

Theory

Both lead compensators and lag compensators introduce a pole–zero pair into the open loop transfer function. 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. In a lead compensator, the pole is left of the zero in the complex plane, | z | < | p | , while in a lag compensator | z | > | p | .
A lead-lag compensator consists of a lead compensator cascaded with a lag compensator. The overall transfer function can be written as

Typically | p1 | > | z1 | > | z2 | > | p2 | , where z1 and p1 are the zero and pole of the lead compensator and z2 and p2 are the zero and pole of the lag compensator. The lead compensator provides phase lead at high frequencies. This shifts the poles to the left, which enhances the responsiveness and stability of the system. The lag compensator provides phase lag at low frequencies which reduces the steady state error.
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. However, the pole and zero of the lag compensator should be close together so as not to cause the poles to shift right, which could cause instability or slow convergence. Since their purpose is to affect the low frequency behaviour, they should be near the origin.
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