Frequency Response of Alpha-Beta-Gamma Trackers

The alpha-beta-gamma tracking equations are written in compact form and then Z transformed. The resulting expressions for estimated position, estimated velocity, estimated acceleration, and predicted position are in terms of the measured position and Z (the Z transform variable). These equations are first solved for the ratio of output (an estimated or predicted quantity) to input (the measured position) and then Z is replaced by exp(-j (omega) T) (omega is the radian frequency and T is the sample interval). After this change, the magnitude and phase for various output-input ratios is plotted as a function of omega T. Linear plots and Bode plots (especially useful for real-time control system design) are presented. Equations are given for the frequency at which the output is 0.707 of the input, for the minimum of the output to input ratio and for the overshoot (amount by which the output exceeds the input). A synopsis of previously available statistical properties of alpha-beta-gamma trackers is also included.