FORMULA / 018

Kalman filter

Interactive derivation module for Kalman filter.

Ready

\[K_k=P_k^-H^T(HP_k^-H^T+R)^{-1}\]

Connects the formula to engineering interpretation, limits, and computation.

Problem Definition

Problem statement and variable definition

A noisy IMU or encoder stream becomes usable when model prediction and measurement correction are balanced.

Core variable legend

Symbol	Quantity	Unit	Meaning
$Q$	Process noise covariance	依系統而定	Trust penalty on the model.
$R$	Measurement noise covariance	依系統而定	Trust penalty on the sensor.
$K_k$	Kalman gain	無	Weight applied to measurement residual.

Analytical Derivation

Model assumptions and analytical derivation

Model simplifications

Item	Assumption
Assumption 1	Linear time-invariant or locally linear behavior is assumed when using transfer functions.
Assumption 2	Parameters are treated as constant over the analysis interval.
Assumption 3	Numerical plots are educational approximations, not full circuit or field solvers.

Start from the governing relation

P_k^-=AP_{k-1}A^T+Q

This is the smallest equation that preserves the physics needed for the formula.

Choose the model state or domain.

Apply engineering assumptions

P_k^-=AP_{k-1}A^T+Q

The model is simplified so the dominant mechanism is visible.

Drop second-order parasitics unless they are the topic.

Rearrange to the working formula

K_k=P_k^-H^T(HP_k^-H^T+R)^{-1}

The final form is the one used for design, simulation, or measurement review.

Algebraic rearrangement.

parameter $\to0$dominant term remains

The formula reduces to its simplest physical behavior.

parameter $\to\infty$parasitics and implementation limits dominate

The closed-form equation becomes a warning rather than a full implementation model.

Parametric Verification

Parametric simulation and numerical verification

Use the sliders to change the physical parameters and watch the formula expose its behavior. The metric panel keeps sample count and compute latency visible so the visualization remains an engineering instrument, not only a picture.

INTERACTIVE MODULE

Measurement and filtered estimate

Open interactive view

Engineering Application

Engineering application and implementation notes

Design rules

Check units before trusting numeric output.
Sweep the key parameter through the critical region, not only the nominal value.
Validate the simplified formula against measurement or simulation before committing hardware.

Core algorithm

python

def kalman_1d(zs, q, r):
    x, p = 0.0, 1.0
    out = []
    for z in zs:
        p += q
        k = p / (p + r)
        x = x + k * (z - x)
        p = (1 - k) * p
        out.append(x)
    return out