Interactive Control Systems Tools
Introduction to Lead and Lag Compensators
The simplest and most common form of compensation is a filter with one zero and one pole. The general transfer function of such compensator is
If the 0<a<b, K(s) is called phase-lead compensator. If the pole occurs before the zero, i.e. if a>b>0, K(s) is a phase-lag compensator. The maximum phase contribution for the both compensators occurs at
How does the designer know whether to use a lead or a lag compensator? The answer depends on the system specifications. For example, if the only requirement is to stabilize the closed loop system, then either type of compensation can usually be used. For unstable plants, lead compensator must be used. However, if there are additional specifications, this will lead to a choice of one type of compensation over the other. For instance, if a certain crossover frequency is desired, the examination of the Bode plot would determine the type of the compensator.
The design of the lead and the lag compensator root locus, Bode and analytical techniques can be used. In most of the cases the actual closed loop response will differ from the that predicted from these designs. Always a close loop simulation of the total system have to be done and additional adjustments if necessary.
Lead compensator increases relative stability by increasing phase margin. For a given system gain, lead compensator increase steady state error (this is because a<b, which means that Kca/b<Kc). To decrease steady state error, large compensator gain must be used. Lead compensator increase also the gain crossover frequency. The additional effect of this is decrease if the step response settling time, i.e. increasing the system damping. This is desirable in many cases, but from other hand, the increase of the crossover frequency leads to increase if the bandwidth in the closed loop system. The increased bandwidth may result in undesirable signals and noise or instability in systems with time delay.
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