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The Kalman filter computes the optimal variable-gain using prior knowledge ofthe initial state and random (process and measurement) noise distributions,which are assumed to be Gaussian with known variance. However, when thesedistributions are unknown, the Kalman filter is not necessarily optimal andother simpler state-estimators, such as fixed-gain ({\alpha}, {\alpha}-\b{eta}or {\alpha}-\b{eta}-{\gamma} etc.) filters may be sufficient. When such filtersare used as low-complexity state-estimators in embedded tracking systems, thefixed gain parameters are usually set equal to the steady-state gains of thecorresponding Kalman filter. An alternative procedure, that does not rely priordistributions, based on Luenberger observers, is presented here. It issuggested that the arbitrary placement of closed-loop state-observer poles is asimple and intuitive way of tuning the transient and steady-state response of afixed-gain tracking filter when prior distributions are unknown. All poles areplaced inside the unit circle on the positive real axis of the complex z-planeat p for a well damped response and a configurable bandwidth. Transient biaserrors, e.g. due to target manoeuvres or process modelling errors, decrease asp=0 is approached for a wider bandwidth. Steady-state random errors, e.g. dueto sensor noise, decrease as p=1 is approached for a narrower bandwidth. Thusthe p parameter (with 0<p<1) may be interpreted as a dimensionless smoothingfactor. This tutorial-style report examines state-observer design by poleplacement, which is a standard procedure for feedback controls but unusual fortracking filters, due to the success and popularity of the Kalman filter. AsBayesian trackers are designed via statistical modelling, not by pole-zeroplacement in the complex plane, the underlying principles of lineartime-invariant signals and systems are also reviewed.

On the design of fixed-gain tracking filters by pole placement: Or an introduction to applied signals-and-systems theory for engineers