Synthesizing the optimal luenberger-type observer for nonlinear systems

Observer design typically requires the observability of the underlying system to guarantee asymptotic convergence of errors. Unfortunately verifying the observability of an underlying nonlinear system may be challenging. Moreover, ensuring asymptotic convergence may be insufficient to certify satisfaction of performance constraints in finite time. This paper develops a method to design Luenberger-type observers for nonlinear systems that guarantee the largest possible domain of attraction for the state estimation error regardless of the initialization of the system without requiring a priori certification of observability. 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. Two examples are presented to demonstrate the feasibility of the presented approach.