How does engineering implement a control action chosen by mathematical methods?

In the context of control theory, the implementation of a control action derived from mathematical methods is a technological task handled by specialized engineering fields. Once mathematical optimization determines the appropriate control law—a function that specifies the action to be taken based on the system's state—this law must be mechanized.1

In practice, this implementation is typically carried out by a controller, which functions as a special-purpose computer. The process generally involves two integrated steps:1

• State Determination: Because it is often impossible to measure every component of a system's state vector exactly, the controller uses an optimal filter (such as a Kalman filter) to calculate the best estimate of the current state.1
• Mechanization: The controller substitutes this estimated state vector into the optimal control law to generate the necessary control signals.1

These signals are then used to drive a motor or similar device, which draws power from a local source to vary the controlled quantity. While the specific engineering details vary depending on the application—such as a chemical plant versus an automobile factory—the underlying principles of using a controller to execute these mathematically derived actions remain consistent.21