Bang–bang control.doc

Bangbang controlFrom Wikipedia, the free encyclopedia(Redirected fromOn-off control)Awater heaterthat maintains desired temperature by turning the applied power on and off (as opposed to continuously varying electricalvoltageorcurrent) based on temperaturefeedbackis an example application of bangbang control. Although the applied power switches from one discrete state to another, the water temperature will remain relatively constant due to the slow nature of temperature changes in materials. Hence, the regulated temperature is like asliding modeof thevariable structure systemsetup by the bangbang controller.Incontrol theory, abangbang controller(onoff controller), also known as ahysteresis controller, is afeedback controllerthat switches abruptly between two states. These controllers may be realized in terms of any element that provideshysteresis. 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 residentialthermostatsare bangbang controllers. TheHeaviside step functionin its discrete form is an example of abangbang controlsignal. Due to thediscontinuouscontrol signal, systems that include bangbang controllers arevariable structure systems, and bangbang controllers are thusvariable structure controllers.Bangbang solutions in optimal controleditInoptimal controlproblems, 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 (i.e., is strictly never in between the bounds), then that control is referred to as a bang-bang solution.Bangbang 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 uniqueswitching point, and then apply maximum braking to come to rest exactly at the desired position. This solution, which can be uncomfortable for the passengers, is a bangbang solution: maximum engine throttle followed by maximum braking.A familiar everyday example is bringing water to aboilin the shortest time, which is achieved by applying full heat, then turning it off when the water reaches a boil. A closed-loop household example is most thermostats, wherein the heating element or air conditioning compressor is either running or not, depending upon whether the measured temperature is above or below the setpoint.Bangbang solutions also arise when theHamiltonianis linear in the control variable; application ofPontryagins minimum or maximum principlewill then lead to pushing the control to its upper or lower bound depending on the sign of the coefficient ofuin the Hamiltonian.In summary, bangbang controls are actuallyoptimalcontrols in some cases, although they are also often implemented because of simplicity or convenience.See alsoedit Lyapunovs theorem Optimal control Robust control Sliding mode control Vector measure Euler equationReferencesedit Artstein, Zvi (1980). Discreteandcontinuous bang-bang and facialspaces, or: Look for the extreme points.SIAM Review22(2). pp.172185.doi:10.1137/1022026.JSTOR2029960.MR564562. Hermes, Henry; LaSalle, JosephP. (1969).Functional analysis and time optimal control. Mathematics in Science and Engineering56. New YorkLondon: Academic Press. pp.viii+136.MR420366. Kluvnek, Igor; Knowles, Greg (1976).Vector measures and control systems. North-Holland Mathematics Studies20(Notas de Matemtica, No. 58. Notes on Mathematics, No. 58). New York: North-Holland Publishing Co. pp.ix+180.MR499068. Rolewicz, Stefan (1987).Functional analysis and control theory: Linear systems. Mathematics and its Applications (East European Series)29(Translated from the Polish by Ewa Bednarczuk ed.). Dordrecht; Warsaw: D. Reidel Publishing Co.; PWNPolish Scientif