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

TL;DR

This paper introduces a pole placement method for designing fixed-gain tracking filters as an alternative to Kalman filters, emphasizing intuitive tuning of transient and steady-state responses without prior distribution knowledge.

Contribution

It presents a simple pole placement approach for fixed-gain filters, providing an intuitive tuning method based on the placement of poles on the real axis, contrasting with Bayesian statistical methods.

Findings

Pole placement allows intuitive tuning of filter response.

Adjusting pole position balances transient bias and steady-state noise.

The method offers a low-complexity alternative to Kalman filtering.

Abstract

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