7.2.2: Fundamentals of control

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Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The external input of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. The usual objective of control theory is to calculate solutions for the proper corrective action from the controller that result in system stability, that is, the system will hold the reference state values and not oscillate around them.

The input and output of the system are related to each other by what is known as a transfer function (also known as the system function or network function). The transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system.

Example

Consider an aircraft’s autopilot, with one of its functionalities being to maintain altitude at a reference value provided by the pilot. The controller is the autopilot, the plant is the aircraft (the equations of motion), and the system is together the aircraft and the autopilot.

The system output is the aircraft’s altitude, and the control itself is the pitching which determines the deflection of the elevator needed.

In a closed-loop control system, a sensor monitors the system output (the aircraft’s altitude, in this case the barometric altitude) and feeds back to a controller that adjusts the control (the elevator) to maintain the desired system output (the reference altitude). Now when the aircraft flies above the desired altitude (there is an error measured/estimated by the aircraft), the elevator position changes to pitch down, speeding the vehicle, and descending to the desired value. Feedback from measuring the aircraft’s altitude has allowed the controller to dynamically compensate for changes to the altitude.