Adding bang-bang control algorithm

SUMMARY.md

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 * [Quantum Information](contents/quantum_information/quantum_information.md)
 * [Computus](contents/computus/computus.md)
 * [Approximate Counting Algorithm](contents/approximate_counting/approximate_counting.md)
+* [Bang-Bang Control](contents/bang-bang/bang-bang.md)
+

contents/bang-bang/bang-bang.md

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+# Bang-Bang Control
+
+A bang-bang controller is one of the simplest forms of feedback controller, used in systems where a simple on/off control signal is used to regulate some aspect of the system (temperature, for example).
+The controller compares the measured system state with user-defined min/max limits and toggles the input between on/off states as needed to keep the system within the specified range.
+Common everyday examples include water heaters and air conditioning controllers. 
+
+##### Simple Example: Water Heater Control
+Imagine a water heater that needs to regulate temperature, keeping within an allowable range of 45 to 50°C.
+The heater only has binary on/off control with no in-between states.
+In order to regulate the system temperature, the heater must cycle between on/off states to keep the temperature within the specified range.
+A basic bang-bang controller would implement the following logic:
+
+| System State | Action |
+| -------------- | ------------- |
+| Temperature is below desired range (<45°C) | Turn heater ON |
+| Temperature exceeds desired range (>50°C) | Turn heater OFF |
+
+Implementing just these two rules will keep the system temperature within the desired range.
+The overall system will oscillate between the two limits as the heater turns on and off: 
+<p>
+    <img  class="center" src="res/bang_bang_temp_history.png" style="width:70%" />
+</p>
+<p>
+    <img  class="center" src="res/bang_bang_control_history.png" style="width:70%" />
+</p>
+
+An important factor to consider when designing a bang-bang controller is the frequency at which a given controller will toggle the system state.
+In the example above, with a wide allowable temperature band of 45-50°C, the heater would be powered every ~18 minutes to maintain the overall system.
+If tighter control over the output is desired (for example, limiting temperature to a narrow range of 45 - 45.2°C), a bang-bang controller can still be used to regulate the system, but the heater would have to toggle on/off much more frequently to keep the temperature in the specified range.
+Depending on the system being controlled, this rapid on/off cycling may be undesirable:
+
+<p>
+    <img  class="center" src="res/bang_temp_fast.png" style="width:70%" />
+</p>
+<p>
+    <img  class="center" src="res/bang_control_fast.png" style="width:70%" />
+</p>
+
+##### Pros:
+1. Very simple to implement.
+2. Can be used to control a system that has only binary on/off states.
+
+
+##### Cons:
+1. The system is not controlled to a *specific* target value; it instead oscillates between specified upper and lower limits.
+2. Setting tight min/max limits on the system output may require the controller to toggle on/off very frequently to maintain the correct output.
+This may be undesirable depending on the system being controlled (for example, excess wear and tear caused by rapidly turning a pump on and off.)
+
+##### More Reading:
+Bang-bang controllers are only meant for systems controlled with binary on/off inputs.
+For systems with a continously variable input signal (for example, cruise control in a car), try PID Control.
+
+
+## License
+The text of this chapter was written by [wca747](https://github.com/wca747) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode).
+
+[<p><img  class="center" src="../cc/CC-BY-SA_icon.svg" /></p>](https://creativecommons.org/licenses/by-sa/4.0/)