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Obtaining initial conditions and parameterizations leading to a modelconsistent with available measurements or safety specifications is importantfor many applications. Examples include model (in-)validation, prediction,fault diagnosis, and controller design. We present an approach to determineinner- and outer-approximations of the set containing all consistent initialconditions/parameterizations for nonlinear continuous-time systems. Theseapproximations are found by occupation measures that encode the system dynamicsand measurements, and give rise to an infinite-dimensional linear program. Weexploit the flexibility and linearity of the decision problem to incorporateuncertain-but-bounded and pointwise-in-time state and output constraints, afeature which was not addressed in previous works. The infinite-dimensionallinear program is relaxed by a hierarchy of LMI problems that providecertificates in case no consistent initial condition/parameterization exists.Furthermore, the applied LMI relaxation guarantees that the approximationsconverge (almost uniformly) to the true consistent set. We illustrate theapproach with a biochemical reaction network involving unknown initialconditions and parameters.

Estimation of consistent parameter sets for continuous-time nonlinear systems using occupation measures and LMI relaxations