Kalman Filter

KFState{V, MU}

A type for the Kalman Filter State, which is parameterized by the types of the mean estimate and the upper triangular cholesky component of the covariance matrix.

KFState(; μ, Σ, make_symmetric=true)

A constructor for the Kalman Filter State, which is parameterized by the mean estimate and the covariance matrix. If make_symmetric is true, the covariance matrix is made symmetric internally. This is useful for numerical stability.

M = cholesky(M::Cholesky)

This is a dummy method to allow for the cholesky method to be called on a cholesky decomposition.

U = chol_sqrt(A)

returns an upper-triangular matrix $U$ such that $A = U^T U$.

s_{k+1|k+1} = correct(s_{k+1|k}, y_{k+1}, C, V)

Uses the system model

\[y_{k+1} = C x_{k+1} + v\]

where $v \sim \mathcal{N}(0, V)$ to correct the predicted state.

diag(M::Cholesky)

is a fast method for getting the diagonal of a cholesky matrix.

This will eventually be included into the Julia standard library. https://github.com/JuliaLang/julia/pull/53767

s_{k+1} = filter(s_k, y_{k+1}, u_k, A, B, C, V, W)

Runs both the prediction and the correction steps. Assumes a system model

\[ \begin{align} x_{k+1} &= A x_k + B u_k + w, \\ y_k &= C x_k + v \end{align}\]

where $w ∼ \mathcal{N}(0, W)$, $v ∼ \mathcal{N}(0, V)$.

get_Σ(s::S) where {S <: KFState}

Get the covariance matrix of the Kalman Filter State.

get_μ(s::S) where {S <: KFState}

Get the mean estimate of the Kalman Filter State.

get_σ(s::S) where {S <: KFState}

Get a vector of the standard deviation of the Kalman Filter State

s_{k+1|k} = predict(s_{k|k}, A, W)

Uses the system model

\[ x_{k+1} = A x_k + w\]

where $w ∼ N(0, W)$ to predict the next state.

  s_{k+1} = predict(s_k, A, B, u_k, W)

Uses the system model

\[ x_{k+1} = A x_k + B u_k + w\]

where $w ∼ \mathcal{N}(0, W)$ to predict the next state.

R = qrr(A, B)

returns

\[R = \sqrt{A^TA + B^TB}\]

The result is an UpperTriangular matrix.