  This paper considers the H\infty-optimal estimation problem for linear
systems with multiple delays in states, output, and disturbances. First, we
formulate the H\infty-optimal estimation problem in the Delay-Differential
Equation (DDE) framework. Next, we construct an equivalent Partial Integral
Equation (PIE) representation of the optimal estimator design framework. We
then show that in the PIE framework, the H\infty-optimal estimator synthesis
problem can be posed as a Linear PI Inequality (LPI). LPIs are a generalization
of LMIs to the algebra of Partial Integral (PI) operators and can be solved
using the PIETOOLS toolbox. Finally, we convert the PIE representation of the
optimal estimator back into an ODE-PDE representation - a form similar to a
DDE, but with corrections to estimates of the infinite-dimensional state (the
time-history). Numerical examples show that the synthesis condition we propose
produces an estimator with provable H\infty-gain bound which is accurate to 4
decimal places when compared with results obtained using Pade-based
discretization.
