State Estimation for diffusion systems using a Karhunen-Loeve-Galerkin Reduced-Order Model

This thesis focuses on generating a continuous estimate of state using a small number of sensors for a process modeled by the diffusion partial differential equation(PDE). In biological systems the diffusion of oxygen in tissue is well described by the diffusion equation, also known by biologists as Fick\u27s first law. Mass transport of many other materials in biological systems are modeled by the diffusion PDE such as CO2, cell signaling factors, glucose and other biomolecules. Estimating the state of a PDE is more formidable than that of a system described by ordinary differential equations (ODEs). While the state variables of the ODE system are finite in number, the state variables of the PDE are distributed in the spatial domain and infinite in number. Reduction of the number of state variables to a finite small number which is tractable for estimation will be accomplished through use of the Karhunen-Loeve-Galerkin method for model order reduction. The model order reduction is broken into two steps, (i) determine an appropriate set of basis functions and (ii) project the PDE onto the set of candidate basis functions. The Karhunen-Loeve expansion is used to decompose a set of observations of the system into the principle modes composing the system dynamics. The observations may be obtained through numerical simulation or physical experiments that encompass all dynamics that the reduced-order model will be expected to reproduce. The PDE is then projected onto a small number of basis functions using the linear Galerkin method, giving a small set of ODEs which describe the system dynamics. The reduced-order model obtained from the Karhunen-Loeve-Galerkin procedure is then used with a Kalman filter to estimate the system state. Performance of the state estimator will be investigated using several numerical experiments. Fidelity of the reduced-order model for several different numbers of basis functions will be compared against a numerical solution considered to be the true solution of the continuous problem. The efficiency of the empirical basis compared to an analytical basis will be examined. The reduced-order model will then be used in a Kalman filter to estimate state for a noiseless system and then a noisy system. Effects of sensor placement and quantity are evaluated. A test platform was developed to study the estimation process to track state variables in a simple non-biological system. The platform allows the diffusion of dye through gelatin to be monitored with a camera. An estimate of dye concentration throughout the entire volume of gelatin will be accomplished using a small number of point sensors, i.e. pixels selected from the camera. The estimate is evaluated against the actual diffusion as captured by the camera. This test platform will provide a means to empirically study the dynamics of diffusion-reaction systems and associated state estimators