Exponential Convergence Rate and Oscillatory Modes of the Asymptotic Kalman Filter Covariance

The Kalman filter is an iterative state estimation algorithm employed extensively, including in electricity generation, aerospace, robotics, etc. Inputting noisy measurements on a dynamical system, it outputs a state estimate and associated covariance. This work focuses on the time evolution of the covariance matrix given regular, uniform measurements. Prior work has derived important results for the covariance at $t\rightarrow \infty $ , but has inadequately described the approach to that fixed point. That behavior is determined by the Jacobian of the Kalman iteration, evaluated at the fixed point. I show that the Jacobian factors into the Kronecker product of a “square root” matrix times itself, resulting in special convergence properties for the Kalman filter. When the leading eigenvalues of the square-root are a complex conjugate pair, the Jacobian matrix has 3 leading eigenvalues of equal magnitude, producing a triplet of modes all decaying at the same exponential rate. One has simple exponential decay and the other two oscillate sinusoidally with exponentially decaying amplitude. I demonstrate the process using two example kinematic systems, with Gaussian white noise in acceleration or jerk, respectively. The examples parameterize the trade-off between the sensor’s measurement rate versus its spatial precision. Interestingly, the first example toggles from triplet-dominated to singlet-dominated by increasing the process noise. In sum, this work provides needed analytical insight into the behavior of Kalman filters and algebraic Riccati equations in general.