Education

Abstract

I try to learn from my own mistakes, but when possible, I have found it's less painful to learn from the mistakes of others. Similarly, there are many problems in science and engineering where the state of a system can be estimated using noisy measurements from a variety of sources. For example, to guide a rocket into orbit, the navigation system may have access to measurements from many sources, including inertial data and radar measurements. Each source has errors associated with it, and the input control to the system may be noisy as well. The classic mathematical question is, “Is there a best way to synthesize these measurements to estimate the state of the system as it evolves in time?” One of the missions of the Education section is to approach classic topics from new perspectives. In this issue of SIAM Review, authors Jeffrey Humpherys, Preston Redd, and Jeremy West reintroduce faculty and students alike to Kalman filtering, which celebrated its fiftieth birthday last year. “A Fresh Look at the Kalman Filter” certainly lives up to its name. In its simplest form, the Kalman filter is a way to estimate the state of a discrete time system from noisy indirect measurements of the system and noisy inputs into the system. It is important to note that the filter estimates the evolving state of the system using all previous measurements and inputs. As the authors point out in their introduction, the Kalman filter is ubiquitous in guidance, navigation, image stabilization, time series analysis of economic systems, and many other application domains. This article puts a unique spin on the topic by deriving the classic filter as iterations of Newton's method to find the minimizer of a positive definite objective function. The same approach can be applied to several variations of the filter, including filters that update all previous states of the system variables as they evolve, and filters that take into account fading memory so that more recent inputs and measurements are weighted more heavily in the objective function. This issue's feature is ideal as a module for a modeling or numerical methods course, or as an application of linear algebra. The authors provide a self-contained derivation using only elementary knowledge of Newton's method and linear algebra, and they include a nice battery of exercises that students can implement in MATLAB. When students experience it for the first time, the Kalman filter may seem a bit like magic. In fact, the authors point out that one of Kalman's original papers was rejected because one referee felt “it cannot possibly be true.” Perhaps this topic will help students to come learn that when a mathematical discovery seems too good to be true, it might be true nonetheless.