// y: Measurments
// u: Actuator thrust
func (k *ExtKalmanFilter) Step(y, u matrix.Matrix) (x matrix.Matrix) {
var y_ = k.h(k.x)
var H = k.dhdx(k.x)
var Ht = matrix.Transpose(H)
// -----------------
// Estimation
// Kalman gain matrix
// K = P * H^t * (H*P*Ht + R)⁻¹
var inv = matrix.Inverse(matrix.Sum(matrix.Product(H, k.mP, Ht), k.mR))
var K = matrix.Product(k.mP, Ht, inv)
var Kt = matrix.Transpose(K)
// State estimation update
// x_ = x + K * (y - h(x))
var ydiff, _ = y.Minus(y_)
if k.normalize_ydiff != nil {
ydiff = k.normalize_ydiff(ydiff)
}
var x_ = matrix.Sum(k.x, matrix.Product(K, ydiff))
// Error covariance update
// P_ = (I - K*H) * P * (I - K*H)^t + K*R*K^t
var IKH, _ = matrix.Eye(H.Cols()).Minus(matrix.Product(K, H))
var IKHt = matrix.Transpose(IKH)
var P_ = matrix.Sum(matrix.Product(IKH, k.mP, IKHt), matrix.Product(K, k.mR, Kt))
// -----------------
// Propagation
var F = k.dfdx(x_, u)
var Ft = matrix.Transpose(F)
// State estimation propagation
// x(k+1) = A*x_(k) + B*u
k.x = k.f(x_, u)
// Error covariance propagation
// P(k+1) = A*P_*A^t + W*Q*W^t
k.mP = matrix.Sum(matrix.Product(F, P_, Ft), k.cWQWt)
return k.x
}
func householder(a *matrix.DenseMatrix) *matrix.DenseMatrix {
m := a.Rows()
s := sign(a.Get(0, 0))
e := unitVector(m)
u := matrix.Sum(a, matrix.Scaled(e, a.TwoNorm()*s))
v := matrix.Scaled(u, 1/u.Get(0, 0))
// (error checking skipped in this solution)
prod, _ := v.Transpose().TimesDense(v)
β := 2 / prod.Get(0, 0)
prod, _ = v.TimesDense(v.Transpose())
return matrix.Difference(matrix.Eye(m), matrix.Scaled(prod, β))
}
func Kalman(W, R, Q, x0, P0, A, B, H matrix.Matrix) (k *ExtKalmanFilter) {
f := func(x, u matrix.Matrix) matrix.Matrix {
// f(x) = Ax + Bu
return matrix.Sum(matrix.Product(A, x), matrix.Product(B, u))
}
dfdx := func(x, u matrix.Matrix) matrix.Matrix {
// df/dx = A
return A
}
h := func(x matrix.Matrix) matrix.Matrix {
// h(x) = H*x
return matrix.Product(A, x)
}
dhdx := func(x matrix.Matrix) matrix.Matrix {
// dh/dx = H
return H
}
return ExtendedKalman(W, R, Q, x0, P0, f, dfdx, h, dhdx, nil)
}
