Adaptive Observers and Parameter Estimation for a Class of Systems Nonlinear in the Parameters

We consider the problem of asymptotic reconstruction of the state and parameter values in systems of ordinary differential equations. A solution to this problem is proposed for a class of systems of which the unknowns are allowed to be nonlinearly parameterized functions of state and time. Reconstruction of state and parameter values is based on the concepts of weakly attracting sets and non-uniform convergence and is subjected to persistency of excitation conditions. In absence of nonlinear parametrization the resulting observers reduce to standard estimation schemes. In this respect, the proposed method constitutes a generalization of the conventional canonical adaptive observer design.

💡 Research Summary

The paper addresses the long‑standing problem of simultaneously estimating the state and unknown parameters of continuous‑time dynamical systems when the parameters appear nonlinearly in the model. Classical adaptive observer designs assume linear parameterization, which limits their applicability to a narrow class of systems. To overcome this limitation, the authors introduce two novel concepts: weakly attracting sets and non‑uniform convergence. These ideas allow them to construct an observer‑parameter adaptation scheme that guarantees asymptotic convergence under a persistency of excitation (PE) condition, even though the parameter dependence is nonlinear.

The considered system is described by
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