Discrete Kalman Process Noise Covariance Derivation from Inertial Bench Test Metrics
Bench Allan variance parameters convert to discrete Kalman process noise matrices by integrating state transition matrices over the sampling interval.
Van Loan Method
Matrix Computation
Numerical techniques provide a reliable way to compute the discrete-time equivalents of continuous-time system and noise matrices. The specific van loan method utilizes a single matrix exponential to solve for both the state transition matrix and the discrete-time process noise covariance simultaneously. This approach avoids the need for complex numerical integration of the matrix differential equations.
Computational Efficiency
Computational efficiency is achieved by forming a block-triangular matrix from the continuous-time dynamics and the noise intensity. When this augmented matrix is exponentiated, the desired discrete matrices appear as specific blocks in the result.
Precision Advantage
Precision advantages make this method preferable over simple approximations when the time step is large or the system dynamics are highly coupled. The van loan method accounts for the continuous-time noise being filtered by the system dynamics over the entire sampling interval. This results in a more accurate representation of the uncertainty propagation in a digital filter.
Because it relies on standard matrix exponential routines, it benefits from the high-performance algorithms available in modern numerical libraries.
Practical Context
Practical applications for this technique are found in the initialization of kalman filters for inertial navigation and aerospace control. It is particularly useful when the noise characteristics are specified in the continuous domain but the processing occurs in discrete steps. The method provides a consistent transition between the two domains.