Until 1980, drives were controlled by analog regulators using Proportional, Integral and Derivative control. From 1980 to 2000, the regulators were digital, but the program simulated the analog functions for Proportional, Integral and Derivative. In about 2000, State-Space Control Regulators were used in drives for the vector, torque and speed control. State-Space is taught in controls classes for engineering students.
Matrices are populated with differential equations representing the drive system. These matrices can be manipulated with math programs to obtain desired control system response. Most Control Theory textbooks use MatLab along with and understanding of Control Theory to assist with the control optimization. Finally, the Continuous State-Space Control Matrices are converted to Discrete Time State Equations. These discrete time state equations are also in the form of matrices. Computers are great at solving matrix equations.
The auto-tune in a drive is used to populate the matrices with values that represent the drive, motor and desired gains.
State-Space regulators can have much better performance than old-style PID control. A higher skill set is needed to apply them correctly. To date, State-Space regulators are not used for the outer tension loops in web handling lines.