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, andcan also be used in digital control.

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

:$frac\left\{Y\right\}\left\{X\right\} = frac\left\{s-z\right\}\left\{s-p\right\}$

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 Argand 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

:$frac\left\{Y\right\}\left\{X\right\} = frac\left\{\left(s-z_1\right)\left(s-z_2\right)\right\}\left\{\left(s-p_1\right)\left(s-p_2\right)\right\}.$

Typically $|p_1| > |z_1| > |z_2| > |p_2|$, where "z"1 and "p"1 are the zero and pole of the lead compensator and "z"2 and "p"2 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 reducesthe 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.

Implementation

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. For example,

:$Y = X - \left(z_1 + z_2\right) frac\left\{X\right\}\left\{s\right\} + z_1 z_2 frac\left\{X\right\}\left\{s^2\right\}+ \left(p_1+p_2\right)frac\left\{Y\right\}\left\{s\right\} - p_1 p_2 frac\left\{Y\right\}\left\{s^2\right\}.$

In analog control systems, where integrators are expensive, it is common to group termstogether to minimise the number of integrators required:

:$Y = X + frac\left\{1\right\}\left\{s\right\}left\left(\left(p_1+p_2\right)Y - \left(z_1+z_2\right)X + frac\left\{1\right\}\left\{s\right\}\left(z_1 z_2 X - p_1 p_2 Y\right) ight\right).$

In analog control, the control signal is typically an electrical voltage or current(although other signals such as hydraulic pressure can be used).In this case a lead-lag compensator will consist of a network of operational amplifiers ("op-amps") connected as integrators and
weighted adders. In digital control, the operations are performed numerically.

The reason for expressing the transfer function as an integral equation is thatdifferentiating signals amplifies the noise on the signal, since even very smallamplitude noise has a high derivative if its frequency is high, while integrating asignal averages out the noise. This makes implementations in terms of integratorsthe most numerically stable.

Intuitive explanation

To begin designing a lead-lag compensator, an engineer must consider whether the systemneeding correction can be classified as a lead-network, a lag-network, or a combinationof the two: a lead-lag network (hence the name "lead-lag compensator"). The electricalresponse of this network to an input signal is expressed by the network's Laplace-domaintransfer function, a complex mathematical function which itself can be expressed as oneof two ways: as the Current-gain ratio transfer function or as the Voltage-gain ratiotransfer function. Remember that a complex function can be in general written as$F\left(x\right) = A\left(x\right) + i B\left(x\right)$, where $A\left(x\right)$ is the "Real Part" and $B\left(x\right)$ is the "Imaginary Part" ofthe single-variable function $F\left(x\right)$.

The "phase angle" of the network is the argument of $F\left(x\right)$; in the left half plane this is $tan^\left\{-1\right\}\left(B\left(x\right)/A\left(x\right)\right)$. If the phase angleis negative for all signal frequencies in the network then the network is classifiedas a "lag network". If the phase angle is positive for all signal frequenciesin the network then the network is classified as a "lead network". If the total networkphase angle has a combination of positive and negative phase as a function of frequencythen it is a "lead-lag network".

Depending upon the nominal operation design parameters of a system under an activefeedback control, a lag or lead network can cause instability and poor speed andresponse times.

References

#Nise, Norman S. (2004); "Control Systems Engineering" (4 ed.); Wiley & Sons; ISBN 0-471-44577-0
#Horowitz, P. & Hill, W. (2001); "The Art of Electronics" (2 ed.); Cambridge University Press; ISBN 0-521-37095-7
#Cathey, J.J. (1988); "Electronic Devices and Circuits (Schaum's Outlines Series)"; McGraw-Hill ISBN 0-07-010274-0

ee also

* Proportional control
* PID controller

* [http://www.mathpages.com/home/kmath249/kmath249.htm Lead-Lag Frequency Response] at MathPages
* [http://www.mathpages.com/home/kmath198/kmath198.htm Lead-Lag Algorithms] at MathPages

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