Sampled-data Kalman filtering and multiple model adaptive estimation for infinite-dimensional continuous-time systems

Kalman filtering and multiple model adaptive estimation (MMAE) methods have been applied by researchers in several engineering disciplines to a multitude of problems as diverse as aircraft flight control and drug infusion monitoring. MMAE methods have been used to adapt to an uncertain noise environment and/or identify important system parameters in these problems. All of the model-based estimation (and control) problems considered in this earlier research have at their core a linear (or mildly nonlinear) model based on finite-dimensional differential (or difference) equations perturbed by random inputs (noise). 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. Thus, we are motivated to extend existing finite-dimensional techniques, such as the Kalman filter, to allow the engineer to apply familiar tools to a larger class of problems. First, the infinite-dimensional sampled-data Kalman filter (ISKF), which is a mathematical extension of the finite-dimensional sampled-data Kalman filter, is derived. Next, an algorithm to create an essentially-equivalent finite-dimensional discrete-time model from an infinite-dimensional continuous-time model is constructed by modifying an extension of an existing technique for producing an equivalent discrete-time model for a finite-dimensional system. The resulting model completely captures the important qualities of the original infinite-dimensional description. Finally, an extended example featuring these new tools is presented for a stochastic partial differential equation. Specifically, the temperature profile along a slender rod is estimated using a Kalman filter for the case of a one-dimensional stochastic heat equation with Neumann boundary conditions. Additionally, the MMAE with a bank of Kalman filters is used to estimate the heat profile in the face of an unknown noise environment (zero-mean white Gaussian noises with uncertain covariances in the dynamics and/or measurement models) and to perform system identification (to determine the thermal diffusivity constant) in the face of an unknown noise environment.