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 ordinary differential equation (ODE) models when the parameters appear nonlinearly. Classical adaptive observers work well only for systems that are linear in the unknown parameters; extending them to nonlinear parameterizations has been challenging because existing methods either require monotonicity, one‑to‑one mappings, or rely on offline optimization that is computationally intensive and prone to local minima.

The authors propose a unified adaptive observer framework that encompasses both linearly and nonlinearly parameterized systems. The considered class of forward‑complete single‑input‑single‑output systems is written as

  ẋ = A x + B ϕᵀ(t, λ, y) θ + g(t, λ, y, u) + ξ(t), y = Cᵀx,

where A, B, C are known matrices satisfying a Lyapunov inequality (ensuring observability), θ ∈ ℝᵐ enters linearly, λ ∈ ℝᵖ enters nonlinearly through known smooth functions ϕ and g, u(t) is a known input, and ξ(t) is a bounded disturbance. The parameter sets Ω_θ and Ω_λ are compact hyper‑rectangles.

Key assumptions: (i) the triple (A, B, C) admits a matrix P > 0 and a gain ℓ such that P(A+ℓCᵀ)+(A+ℓCᵀ)ᵀP ≤ –Q for some Q > 0 (the standard adaptive‑observer condition); (ii) ϕ and g are bounded, continuously differentiable in time and state, and Lipschitz in λ; (iii) the time‑derivatives of ϕ and g along system trajectories are uniformly bounded (technical condition for exponential stability).

The observer consists of two coupled subsystems:

1. Linear‑parameter estimator – identical to the canonical adaptive observer:

  ẋ̂ = A x̂ + ℓ(Cᵀx̂ – y) + B ϕᵀ(t, λ̂, y) θ̂ + g(t, λ̂, y, u)
  θ̂̇ = –γ (Cᵀx̂ – y) ϕ(t, λ̂, y),

where γ > 0 is a design constant. If the regressor β(t)=ϕᵀ(t, λ̂, y) is persistently exciting (PE), Theorem 3 guarantees exponential convergence of the combined error e =