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Session S01 - Modeling and Computation for Control and Optimization of Biological and Physical Systems

Wednesday, July 14, 17:50 ~ 18:15 UTC-3

Real Time Estimation of Advection Diffusion Equations

John Burns

Virginia Tech, US   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Consider the convection diffusion equation \begin{equation} \label{eq:CDeq} \frac{\partial z(t,x)}{\partial t} = \nabla \cdot\left[K \nabla z(t,x))\right] - \nabla \cdot \left[K z(t,x)\right] +g(x) \eta(t) \end{equation} on a domain $\Omega \subset \mathbb{R}^{3}$. We assume the sensed output is given by \begin{equation} y_{i}(t)=\iiint\limits_{B_{\delta}(x_{i}(t))\cap\Omega}h_{i}(x)z(t,x)dx +E_{i}v(t), \end{equation} where $B_{\delta}(x_{i}(t))$ is a $\delta-$neighborhood of the trajectory $x_{i}(t)$ of a moving sensor platform and and $h_{i}(x)$ is a kernel function.

Let $\boldsymbol{y}(t)= [y_{1}(t) \ y_{2}(t) \ ... y_{p}(t)]^{T}$ and \[ C(t) = \iiint\limits_{B_{\delta}(\boldsymbol{\boldsymbol{{x}}}_{i}(t))\cap\Omega }h_{i}(\boldsymbol{\boldsymbol{{x}}})z(t,\boldsymbol{\boldsymbol{{x}}})d\boldsymbol{\boldsymbol{{x}}} \]

Given the system above, a stable (full) state estimator (Luenberger observer) will have the form% \begin{equation} \frac{\partial\hat{z}(t,x)}{\partial t} = \nabla \cdot\left[K \nabla z(t,x))\right] - \nabla \cdot \left[K z(t,x)\right] +g(x) \eta(t) +\mathcal{F}(t,x)[\boldsymbol{y}(t)-C(t)\hat{z}(t,x)],\label{eq:abstractobserver} \end{equation} where $\mathcal{F}(t,\cdot):\mathbb{R}^{p}\rightarrow Z=L_{2}(\Omega)$ is a bounded linear operator called the observer gain operator. The goal is to find an observer gain operator so that the error $e(t)=z(t)-\hat{z}(t)$ goes to zero as $ t$ approaches $ + \infty $ and the data driven dynamical estimator can be realized in real time. In this talk, we discuss this problem and show that real time implementation is possible.

Joint work with James Cheung (Virginia Tech), Michael Demetriou (WPI), Nikolaos Gatsonis (WPI), Weiwei Hu (University of Georgia) and Xin Tian (WPI).

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