# Control Systems/Common Nonlinearities

There are some nonlinearities that happen so frequently in physical systems that they are called "Common nonlinearities". These common nonlinearities include Hysteresis, Backlash, and Dead-zone.

## Hysteresis

Continuing with the example of a household thermostat, let's say that your thermostat is set at 70 degrees (Fahrenheit). The furnace turns on, and the house heats up to 70 degrees, and then the thermostat dutifully turns the furnace off again. However, there is still a large amount of residual heat left in the ducts, and the hot air from the vents on the ground may not all have risen up to the level of the thermostat. This means that after the furnace turns off, the house may continue to get hotter, maybe even to uncomfortable levels.

So the furnace turns off, the house heats up to 80 degrees, and then the air conditioner turns on. The temperature of the house cools down to 70 degrees again, and the A/C turns back off. However, the house continues to cool down, and then it gets too cold, and the furnace needs to turn back on.

As we can see from this example, a bang-bang controller, if poorly designed, can cause big problems, and it can waste lots of energy. To avoid this, we implement the idea of Hysteresis, which is a set of threshold values that allow for overflow outputs. Implementing hysteresis, our furnace now turns off when we get to 65 degrees, and the house slowly warms up to 75 degrees, and doesn't turn on the A/C unit. This is a far preferable solution.

## Backlash

•Backlash refers to the angle that the output shaft of a gearhead can rotate without the input shaft moving. Backlash arises due to tolerance in manufacturing; the gear teeth need some play to avoid jamming when they mesh. An inexpensive gearhead may have backlash of a degree or more, while more expensive precision gearheads have nearly zero backlash. Backlash typically increases with the number of gear stages. Some gear types, notably harmonic drive gears (see Section 26.1.2), are specifically designed for near-zero backlash, usually by using flexible elements.

E.g.: Mechanical gear.