Filtering and state estimation

Sensors do not reveal truth directly. A camera-derived position jitters, an ADC includes quantization and electrical noise, and a distance sensor may occasionally return an outlier. Filtering combines measurements over time to make a more useful estimate. The cost is usually delay or reduced ability to follow rapid change.

An N-sample moving average is ȳ_k = (1/N)Σx_{k-i}. For independent zero-mean noise with standard deviation σ, its standard deviation falls ideally to σ/√N, but the window delays changing signals. An exponential filter uses ŷ_k = αx_k + (1-α)ŷ_{k-1}, with 0 < α ≤ 1. Smaller α smooths more and responds more slowly. A state estimator goes further by combining a process model and measurement uncertainty; even a simple “predict, compare residual, correct” loop makes its assumptions explicit.

The residual r_k = measurement - predicted measurement is diagnostically valuable. Random residuals near zero may support the model; persistent bias suggests calibration error or an unmodelled force; abrupt large residuals may signal dropout or impact. Do not silently discard inconvenient readings. Record raw data, estimate, filter parameters, timestamp, saturation, and rejection decisions. For a camera mount, the build decision could be a modest filter that reduces pixel jitter while preserving enough bandwidth to track intentional pan motion.

A common misconception is that a smoother plot is more accurate. Heavy filtering can produce a clean but late estimate and hide a loose connector. Another is treating a filter's output as a physical measurement; it is a computed estimate based on assumptions. Averaging correlated drift does not remove the drift as σ/√N predicts. Choose filter bandwidth relative to system dynamics, examine residuals, and test known changes. If control stability worsens after filtering, added delay may outweigh reduced noise.