With the aforementioned equations, it is now possible to calculate the gains of the Kalman filter and FSFB controller. For the purpose of discretization, there are 3 constraints that were imposed on the optimization when the original controller was designed.
- The sampling time is set to 800 µs. This restriction is based on the maximum measured time to execute a single loop of the program on the original computer.
- The maximum closed-loop natural frequency is restricted to 1000 rad/sec. The natural frequency is the frequency at which the control system will oscillate without any external input. The reason why we have this restriction is because of the Nyquist frequency, which is π/T = 3926 rad/sec ≅ 4000 rad/sec. If a digital sensor attempts to read a signal that has a frequency above the Nyquist frequency, it will construct an aliased, or distorted, version of the signal. If the displacement signal from the motion plant is aliased, the controller won't be able to operate the plant correctly. Thus, the natural frequency is limited to ¼ of the Nyquist frequency.
- The minimum closed-loop natural frequency is limited to 100 rad/sec.
A Matlab script was written that would calculate the optimum gains that would minimize payload travel overshoot. It adjusted the poles of the FSFB controller and the process spectral density Q while ensuring that the 3 constraints were adhered to. With this script, the following gains were calculated.