A Computational Method for $H_2$-optimal Estimator and State Feedback Controller Synthesis for PDEs

In this paper, we present solvable, convex formulations of $H_2$-optimal state estimation and state-feedback control problems for a general class of linear Partial Differential Equations (PDEs) with one spatial dimension. These convex formulations are derived by using an analysis and control framework called the `Partial Integral Equation' (PIE) framework, which utilizes the PIE representation of infinite-dimensional systems. Since PIEs are parameterized by Partial Integral (PI) operators that form an algebra, $H_2$-optimal estimation and control problems for PIEs can be formulated as Linear PI Inequalities (LPIs). Furthermore, if a PDE admits a PIE representation, then the stability and $H_2$ performance of the PIE system implies that of the PDE system. Consequently, the optimal estimator and controller obtained for a PIE using LPIs provide the same stability and performance when applied to the corresponding PDE. These LPI optimization problems can be solved computationally using semi-definite programming solvers because such problems can be formulated using Linear Matrix Inequalities by using positive matrices to parameterize a cone of positive PI operators. We illustrate the application of these methods by constructing observers and controllers for some standard PDE examples.