  Observer design typically requires the observability of the underlying
system, which may be hard to verify for nonlinear systems, while guaranteeing
asymptotic convergence of errors, which may be insufficient in order to satisfy
performance conditions in finite time. This paper develops a method to design
Luenberger-type observers for nonlinear systems which guarantee the largest
possible domain of attraction for the state estimation error regardless of the
initialization of the system. The observer design procedure is posed as a two
step problem. In the the first step, the error dynamics are abstractly
represented as a linear equation on the space of Radon measures. Thereafter,
the problem of identifying the largest set of initial errors that can be driven
to within the user-specified error target set in finite-time for all possible
initial states, and the corresponding observer gains, is formulated as an
infinite-dimensional linear program on measures. This optimization problem is
solved, using Lasserre's relaxations via a sequence of semidefinite programs
with vanishing conservatism. By post-processing the solution of step one, the
set of gains that maximize the size of tolerable initial errors is identified
in step two. To demonstrate the feasibility of the presented approach two
examples are presented.
