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

TL;DR

This paper introduces a novel method using occupation measures and LMI relaxations to compute inner and outer approximations of the set of all consistent initial conditions and parameters for nonlinear continuous-time systems, accommodating uncertainties and constraints.

Contribution

The paper develops a new approach combining occupation measures with LMI relaxations to efficiently approximate the consistent parameter and initial condition sets for nonlinear systems, including uncertain constraints.

Findings

The method provides converging approximations to the true consistent set.

It can incorporate uncertain and bounded state/output constraints.

Demonstrated on a biochemical reaction network example.

Abstract

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