Lead–lag compensator

Introduction

A lead-lag compensator is a type of electronic circuit that corrects undesirable frequency effects in the signal processing elements of a feedback and control system. A typical lead-lag compensator consists of a network of operational amplifiers (or "op-amps") in a closed-loop feedback system that is designed to improve stability, speed, and response in an active control system where a steady-state is desired. Good multi-disciplinary knowledge is required for reliable design and implementation of a lead-lag compensator, including physics, calculus, systems engineering, and electrical engineering. Lead-lag compensators are everywhere, influencing disciplines as varied as robotics, satellite control, automobile diagnostics, laser frequency stabilization, and many more.

1. Lead-Lag Networks

To begin designing a lead-lag compensator, an engineer must consider whether the system needing correction can be classified as a lead-network, a lag-network, or a combination of the two: a lead-lag network (hence the name "lead-lag compensator"). The electrical response of this network to an input signal is expressed by the network's Laplace-domain transfer function, a Complex mathematical function which itself can be expressed as one of two ways: as the Current-gain ratio transfer function or as the Voltage-gain ratio transfer function. Remember that a complex function can be in general written as F(x) = A(x) + i*B(x), where A(x) is the "Real Part" and B(x) is the "Imaginary Part" of the single-variable function F(x).

The ratio of the Imaginary part, over the Real part, of the Current- or Voltage-gain ratio transfer function yields the Network Phase Angle. If the Network Phase Angle is negative for all signal frequencies in the network then the network is classified as a Lag Network. If the Network Phase Angle is positive for all signal frequencies in the network then the network is classified as a Lead Network. 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.

2. Design

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. A Lead-Lag Compensator is then a component of the active feedback control system responsible for correcting the phase of a Lead-Lag Network as desired to obtain a steady-state.

Sources:

[1] Nise, Norman S. (2004); Control Systems Engineering (4 ed.); Wiley & Sons; ISBN 0471445770 [2] Horowitz, P. & Hill, W. (2001); The Art of Electronics (2 ed.); Cambridge University Press; ISBN 0521370957 [3] Cathey, J.J. (1988); Electronic Devices and Circuits (Schaum's Outlines Series); McGraw-Hill ISBN 0070102740