The celebrated Kalman filter finds applications in many fields of engineering and economics. While many are familiar with the basic concepts of the Kalman filter, almost equally many find the “tuning parameters” associated with a Kalman filter nonintuitive and difficult to choose. While there are several parameters that can be tuned in a real-world application of a Kalman filter, we will focus on the most important ones: the process and measurement noise covariance matrices.

The Kalman filter

The Kalman filter is a form of Bayesian estimation algorithm that estimates the state $$x$$ of a linear dynamical system, subject to Gaussian noise $$e$$ acting on the measurements as well as the dynamics, $$w$$. More precisely, let the dynamics of a discrete-time linear dynamical system be given by

$$x_{k+1} = {Ax_k + Bu_k + w_k}$$

$$y_k = {Cx_k + Du_k + e_k}$$

where $$x_k \in \Bbb R^{n_x}$$ is the state of the system at time $$k, u_k \in \Bbb R^{n_u}$$ is an external input, $$A$$ and $$B$$ are the state transition and input matrices respectively, $$c$$ an output matrix and $$w_k \backsim N(0, R_1) $$ and $$e_k \backsim N(0, R_1) $$, are normally distributed process noise and measurement noise terms respectively. A state estimator like the Kalman filter allows us to estimate $$x$$ given only noisy measurements $$y \in \Bbb R^{n_y}$$, i.e., without necessarily having measurements of all the components of $$x$$ available.[obs] For this reason, state estimators are sometimes referred to as virtual sensors, i.e., they allows use to estimate what we cannot measure.

The Kalman filter is popular for several important reasons, for one, it is the optimal estimator in the mean-square sense if the system dynamics is linear (can be time varying) and the noise acting on the system is Gaussian. In most practical applications, neither of these conditions hold exactly, but they often hold sufficiently well for the Kalman filter to remain useful.[nonlin] A perhaps even more useful property of the Kalman filter is that the posterior probability distribution over the state remains Gaussian throughout the operation of the filter, making it efficient to compute and store.

What does “tuning the Kalman filter” mean?

To make use of a Kalman filter, we obviously need the dynamical model of the system given by the four matrices $$A,B,C$$ and $$D$$. We furthermore require a choice of the covariance matrices $$R_1$$ and $$R_2$$, and it is here a lot of aspiring Kalman-filter users get stuck. The covariance matrix of the measurement noise is often rather straightforward to estimate, just collect some measurement data when the system is at rest and compute the sample covariance, but we often lack any and all feeling for what the process noise covariance, $$R_1$$, should be.

In this blog post, we will try to give some intuition for how to choose the process noise covariance matrix $$R_1$$. We will come at this problem from a disturbance-modeling perspective, i.e., trying to reason about what disturbances act on the system and how, and what those imply for the structure and value of the covariance matrix $$R_1$$.