In this paper we apply the infinite-dimensional sampled-data Kalman filter (ISKF) [1], [2] to a system characterized by the stochastic heat equation for the purpose of estimating the temperature distribution along a slender (one-dimensional) cylindrical rod using a simple linear measurement model. The key to applying the ISKF is the development of an essentially equivalent finite-dimensional discrete-time model from an infinite-dimensional continuous-time dynamics model. In addition to estimating the temperature of the rod, we employ a bank of elemental filters via the multiple model adaptive estimation (MMAE) technique to estimate unknown model parameters such as the thermal diffusivity constant of the slender cylindrical rod.

This paper contains a brief overview of the infinite-dimensional sampled-data Kalman filter (ISKF) derivation [1]. The ISKF is essentially a mathematical extension of the (finite-dimensional) sampled-data Kalman filter that applies to a larger class of problems that satisfy the strongly continuous semigroup property which includes certain partial and delay differential equations.

: Kalman filtering and multiple model adaptive estimation (MMAE) methods have been applied by researchers in several engineering disciplines to a multitude of problems featuring a linear (or mildly nonlinear) model based on finite-dimensional differential (or difference) equations perturbed by random inputs. However, many real-world systems are more naturally modeled using an infinite-dimensional continuous-time linear systems model, such as those most naturally modeled as partial differential equations or time-delayed differential equations along with a possibly infinite-dimensional measurement model. The Kalman filtering technique was extended to encompass infinite-dimensional continuous-time systems with sampled-data measurements and a technique to approximate an infinite-dimensional continuous-time system model with an essentially equivalent finite-dimensional discrete-time model upon which a filtering algorithm could be based was developed. The tools developed during this research were demonstrated using an estimation problem based on a stochastic partial differential equation with an unknown noise environment.