Kalman filtering and estimation error

Let x_{k+1}=Ax_k+Bu_k+w_k and y_k=Cx_k+v_k. Assume zero-mean process and measurement noises, independent across time and mutually independent, with covariances W≥0 and R>0; the initial state is independent of all noises with specified mean and covariance. Inputs are deterministic and known to the estimator. For conditional-mean optimality among all estimators, assume the initial state and noises are jointly Gaussian under these independence assumptions. With only second moments, the same recursion gives the best linear least-squares estimate, which need not equal the unrestricted conditional mean.

After processing data through time k−1, the prediction for time k propagates the previous updated estimate through A and adds the known input. Its covariance becomes A P_{k−1}^+ Aᵀ+W. Process noise is added because it represents new uncertainty, not a measurement one can subtract. Covariance propagation accounts for correlations between state coordinates; propagating only separate variances can lose important information about which combinations are well known. A deterministic control input affects the predicted mean but does not itself add covariance in this model.

The innovation is the actual measurement minus its predicted value. Its covariance S includes both projected state uncertainty and sensor uncertainty. The gain K=P^-CᵀS^{-1} maps this discrepancy back into state coordinates. The updated estimate adds K times the innovation, and its error covariance decreases according to the update formula. In exact arithmetic the simplified covariance update is equivalent to the Joseph form, which explicitly includes both remaining prediction error and injected measurement noise. The latter is often preferable numerically because its positive-semidefinite structure is more visible.

For scalar direct measurement, K=P^-/(P^-+R). A more uncertain prediction gets a larger correction; a noisier sensor gets a smaller one. This is a model-based weighting, not an adaptive verdict that the larger observed number must be more reliable. Correlated process and measurement noise require modified cross-covariance terms. Reusing these independent-noise equations in that setting can understate or misallocate uncertainty. Similarly, repeatedly assimilating the same measurement as if it were independent counts the same information more than once.

Under the Gaussian model, P⁺ is the actual covariance conditional on the observed data. With only second moments it is the unconditional error covariance of the best linear estimator, generally not the data-conditional covariance. For independent equally likely signs X and v, observing Y=X+v gives the linear-error variance 1/2, but Var(X|Y=2)=0 and Var(X|Y=0)=1. An unmodeled sensor bias can also make an estimate systematically wrong despite a small reported covariance. Innovation checks compare residual scale and correlation with the assumed model; they do not identify the source of every mismatch or independently measure ground-truth error.

Detectability is relevant to whether hidden unstable modes can be estimated over long runs. Under standard time-invariant assumptions, suitable detectability and process-noise stabilizability conditions support a stabilizing steady-state Riccati filter, but the finite recursion here does not require one to assert a constant gain prematurely. Time-varying sensors or noise covariances can keep the gain time dependent. Combining estimation with feedback also introduces performance and robustness questions beyond this estimator calculation. The full-state LQR gain and an estimator gain solve different Riccati problems and should not be interchanged merely because their algebra looks similar.