write mathematical representation of square root information filter
时间: 2024-05-30 07:10:06 浏览: 8
The Square Root Information Filter is a variant of the Kalman filter that uses the square root of the information matrix to represent the state covariance matrix.
The state vector at time step k is denoted as x(k), and the measurement vector at time step k is denoted as z(k). The state update equation is given by:
x(k) = F(k-1) * x(k-1) + B(k-1) * u(k-1) + w(k-1)
where F(k-1) is the state transition matrix, B(k-1) is the control input matrix, u(k-1) is the control input vector, and w(k-1) is the process noise.
The measurement update equation is given by:
y(k) = z(k) - H(k) * x(k)
where H(k) is the measurement matrix.
The information vector at time step k is denoted as v(k), and the information matrix at time step k is denoted as V(k). The information vector and matrix are defined as:
v(k) = inv(P(k)) * x(k)
V(k) = inv(P(k))
where P(k) is the state covariance matrix.
The Square Root Information Filter updates the information vector and matrix using the following equations:
v(k|k) = v(k|k-1) + K(k) * y(k)
V(k|k) = V(k|k-1) - K(k) * H(k) * V(k|k-1)
where K(k) is the Kalman gain matrix.
The state estimate at time step k is then computed as:
x(k|k) = inv(V(k|k)) * v(k|k)
The state covariance matrix at time step k is computed as:
P(k|k) = inv(V(k|k))^2
Overall, the Square Root Information Filter provides a numerically stable and computationally efficient implementation of the Kalman filter.
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