Bang-bang control

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A bang-bang controller is a controller that switches abruptly between two states. These controllers may be realized in terms of any element that provides hysteresis. They are often used to control a plant that accepts a binary input, for example a furnace that is either completely on or completely off. Most common residential thermostats are bang-bang controllers. The Heaviside step function in its discrete form is an example of bang-bang control.

[edit] Bang-bang solutions in optimal control

In optimal control problems, it is sometimes the case that a control is restricted to be between a lower and an upper bound. If the optimal control switches from one extreme to the other at certain times (i.e., is never strictly in between the bounds) then that control is referred to as a bang-bang solution.

Bang-bang controls frequently arise in minimum time problems. For example if it is desired to stop a car in the shortest possible time at a certain position sufficiently far ahead of the car, the solution is to apply maximum acceleration until the unique switching point, and then apply maximum braking to come to rest exactly at the desired position. This solution (a rather uncomfortable one for the passengers) is a bang-bang solution: maximum engine throttle followed by maximum braking. Bang-bang solutions also arise when the Hamiltonian is linear in the control variable; application of Pontryagin's minimum principle will then lead to pushing the control to its upper or lower bound depending on the sign of the coefficient of u in the Hamiltonian.

[edit] References

Sonneborn, L., and F. Van Vleck (1965): The Bang-Bang Principle for Linear Control Systems, SIAM J. Control 2, 151-159.