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control theory Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the applic ...
, a bang–bang controller (hysteresis, 2 step or on–off controller), is a feedback controller that switches abruptly between two states. These controllers may be realized in terms of any element that provides
hysteresis Hysteresis is the dependence of the state of a system on its history. For example, a magnet may have more than one possible magnetic moment in a given magnetic field, depending on how the field changed in the past. Plots of a single component of ...
. 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 The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
in its discrete form is an example of a bang–bang control signal. Due to the discontinuous control signal, systems that include bang–bang controllers are variable structure systems, and bang–bang controllers are thus variable structure controllers.


Bang–bang solutions in optimal control

In
optimal control Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations ...
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 (i.e., is strictly never 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 for a car starting at rest to arrive at a certain position ahead of the car in the shortest possible time, 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. A familiar everyday example is bringing water to a
boil A boil, also called a furuncle, is a deep folliculitis, which is an infection of the hair follicle. It is most commonly caused by infection by the bacterium ''Staphylococcus aureus'', resulting in a painful swollen area on the skin caused by ...
in 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. Bang–bang solutions also arise when the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
is linear in the control variable; application of Pontryagin's minimum or maximum 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. In summary, bang–bang controls are actually ''optimal'' controls in some cases, although they are also often implemented because of simplicity or convenience.


Practical implications of bang–bang control

Mathematically or within a computing context there may be no problems, but the physical realization of bang–bang control systems gives rise to several complications. First, depending on the width of the hysteresis gap and inertia in the process, there will be an oscillating error signal around the desired set point value (e.g., temperature), often saw-tooth shaped. Room temperature may become uncomfortable just before the next switch 'ON' event. Alternatively, a narrow hysteresis gap will lead to frequent on/off switching, which is often undesirable (e.g. an electrically ignited gas heater). Second, the onset of the step function may entail, for example, a high electrical current and/or sudden heating and expansion of metal vessels, ultimately leading to
metal fatigue In materials science, fatigue is the initiation and propagation of cracks in a material due to cyclic loading. Once a fatigue crack has initiated, it grows a small amount with each loading cycle, typically producing striation (fatigue), striati ...
or other wear-and-tear effects. Where possible, continuous control, such as in PID control, will avoid problems caused by the brisk state transitions that are the consequence of bang–bang control.


See also

* Double-setpoint control *
Optimal control Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations ...
*
PID controller PID or Pid may refer to: Medicine * Pelvic inflammatory disease or pelvic inflammatory disorder, an infection of the upper part of the female reproductive system * Primary immune deficiency, disorders in which part of the body's immune system is ...
* Robust control *
Sliding mode control In control systems, sliding mode control (SMC) is a nonlinear control method that alters the dynamic system, dynamics of a nonlinear system by applying a discontinuous control signal (or more rigorously, a set-valued control signal) that forces th ...


References


Further reading

* * * * * * {{DEFAULTSORT:Bang-Bang Control Optimal control