Synthesizing the Optimal Luenberger-type Observer for Nonlinear Systems

摘要

Observer design typically requires the observability of the underlyingsystem, which may be hard to verify for nonlinear systems, while guaranteeingasymptotic convergence of errors, which may be insufficient in order to satisfyperformance conditions in finite time. This paper develops a method to designLuenberger-type observers for nonlinear systems which guarantee the largestpossible domain of attraction for the state estimation error regardless of theinitialization of the system. The observer design procedure is posed as a twostep problem. In the the first step, the error dynamics are abstractlyrepresented as a linear equation on the space of Radon measures. Thereafter,the problem of identifying the largest set of initial errors that can be drivento within the user-specified error target set in finite-time for all possibleinitial states, and the corresponding observer gains, is formulated as aninfinite-dimensional linear program on measures. This optimization problem issolved, using Lasserre's relaxations via a sequence of semidefinite programswith vanishing conservatism. By post-processing the solution of step one, theset of gains that maximize the size of tolerable initial errors is identifiedin step two. To demonstrate the feasibility of the presented approach twoexamples are presented.