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In this work, we present a Linear Matrix Inequality (LMI) based method to synthesize an optimal H1 estimator for a large class of linear coupled partial differential equations (PDEs) utilizing only finite dimensional measurements. Our approach extends the newly developed framework for representing and analyzing distributed parameter systems using operators on the space of square integrable functions that are equipped with multipliers and kernels of semi-separable class. We show that by redefining the state, the PDEs can be represented using operators that embed the boundary conditions and input-output relations explicitly. The optimal estimator synthesis problem is formulated as a convex optimization subject to LMIs that require no approximation or discretization. A scalable algorithm is presented to synthesize the estimator. The algorithm is illustrated by suitable examples.
In this paper, we consider input-output properties of linear systems consisting of PDEs on a finite domain coupled with ODEs through the boundary conditions of the PDE. This framework can be used to represent e.g. a lumped mass fixed to a beam or a system with delay. This work generalizes the sufficiency proof of the KYP Lemma for ODEs to coupled ODE-PDE systems using a recently developed concept of fundamental state and the associated boundary-condition-free representation. The conditions of the generalized KYP are tested using the PQRS positive matrix parameterization of operators resulting in a finite-dimensional LMI, feasibility of which implies prima facie provable passivity or L2-gain of the system. No discretization or approximation is involved at any step and we use numerical examples to demonstrate that the bounds obtained are not conservative in any significant sense and that computational complexity is lower than existing methods involving finite-dimensional projection of PDEs.