Bang-bang control

# Bang-bang control

Discussion

Encyclopedia
In control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

, a bang–bang controller (on–off controller), also known as a hysteresis 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
Hysteresis is the dependence of a system not just on its current environment but also on its past. This dependence arises because the system can be in more than one internal state. To predict its future evolution, either its internal state or its history must be known. If a given input alternately...

. 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
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....

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 control
Variable structure control
Variable structure control, or VSC, is a form of discontinuous nonlinear control. The method alters the dynamics of a nonlinear system by application of a high-frequency switching control. The state-feedback control law is not a continuous function of time; it switches from one smooth condition to...

lers.

## Bang–bang solutions in optimal control

In optimal control
Optimal control
Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators in the Soviet Union and Richard Bellman in the United States.-General method:Optimal...

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, which can be "uncomfortable" for the passengers, is a bang–bang solution: maximum engine throttle followed by maximum braking.

A familiar everyday example is bringing water to a boil
Boiling
Boiling is the rapid vaporization of a liquid, which occurs when a liquid is heated to its boiling point, the temperature at which the vapor pressure of the liquid is equal to the pressure exerted on the liquid by the surrounding environmental pressure. While below the boiling point a liquid...

in the shortest time, which is achieved by applying full heat, then turning it off when the water reaches a boil.

Bang–bang solutions also arise when the Hamiltonian
Hamiltonian (control theory)
The Hamiltonian of optimal control theory was developed by L. S. Pontryagin as part of his minimum principle. It was inspired by, but is distinct from, the Hamiltonian of classical mechanics. Pontryagin proved that a necessary condition for solving the optimal control problem is that the control...

is linear in the control variable; application of Pontryagin's minimum principle
Pontryagin's minimum principle
Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It was formulated by the Russian mathematician Lev Semenovich...

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.

• Lyapunov's theorem
• Optimal control
Optimal control
Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators in the Soviet Union and Richard Bellman in the United States.-General method:Optimal...

• Robust control
Robust control
Robust control is a branch of control theory that explicitly deals with uncertainty in its approach to controller design. Robust control methods are designed to function properly so long as uncertain parameters or disturbances are within some set...

• Sliding mode control
Sliding mode control
In control theory, sliding mode control, or SMC, is a nonlinear control method that alters the dynamics of a nonlinear system by application of a discontinuous control signal that forces the system to "slide" along a cross-section of the system's normal behavior. The state-feedback control law is...

• Vector measure
Vector measure
In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of measure, which takes nonnegative real values only.-Definitions and first consequences:...