H₈-Optimal Estimator Synthesis for Linear 2D PDEs using Convex Optimization

Any suitably well-posed PDE in two spatial dimensions can be represented as a Partial Integral Equation (PIE) - with system dynamics parameterized using Partial Integral (PI) operators. Furthermore, L₂-gain analysis of PDEs with a PIE representation can be posed as a linear operator inequality, which can be solved using convex optimization. In this paper, these results are used to derive a convex-optimization-based test for constructing an H₈-optimal estimator for 2D PDEs. In particular, a PIE representation is first derived for arbitrary well-posed 2D PDEs with sensor measurements along boundaries of the domain. An associated Luenberger-type estimator is then parameterized using a PI operator L as the observer gain. Next, it is shown that an upper bound on the H₈-norm of the error dynamics for the estimator can be minimized by solving a linear operator inequality on PI operator variables. Finally, an analytical formula for inversion of a sub-class of 2D PI operators is derived and used to reconstruct the Luenberger gain L. Results are implemented in the PIETOOLS software suite - applying the methodology and simulating the estimator for an unstable 2D heat equation.