[Paper Review] Adaptive Observers and Parametric Identification for Systems in Non-canonical Adaptive Observer Form

Ivan Tyukin, Erik Steur+2 more|arXiv (Cornell University)|Mar 13, 2009

Stability and Controllability of Differential Equations15 references7 citations

TL;DR

This paper proposes a novel adaptive observer framework for systems with nonlinearly parameterized states and time, leveraging weakly attracting sets and non-uniform convergence to achieve asymptotic state and parameter reconstruction under persistency of excitation. The method generalizes canonical adaptive observer designs by accommodating nonlinearity in parametrization while reducing to standard schemes when parametrization is linear.

ABSTRACT

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.

Motivation & Objective

• Address the challenge of asymptotic state and parameter reconstruction in systems with nonlinearly parameterized dynamics.
• Extend conventional adaptive observer theory to handle non-canonical system forms where parameters enter nonlinearly in state and time.
• Ensure robust estimation performance under persistency of excitation conditions without requiring linear parametrization.
• Generalize standard adaptive observer designs by incorporating weakly attracting sets and non-uniform convergence concepts.
• Provide a unified framework that reduces to classical schemes when parametrization is linear.

Proposed method

• Utilizes the concept of weakly attracting sets to ensure convergence of the observer error dynamics.
• Applies non-uniform convergence analysis to handle time-varying and nonlinearly parameterized systems.
• Designs an observer structure that adaptively estimates both state and unknown parameters simultaneously.
• Imposes persistency of excitation conditions on the input or signal to guarantee parameter convergence.
• Introduces a Lyapunov-based stability analysis tailored for non-canonical observer forms with nonlinear parametrization.
• Derives sufficient conditions for asymptotic convergence of state and parameter estimates using the proposed framework.

Experimental results

Research questions

• RQ1 How can state and parameter values be asymptotically reconstructed in systems with nonlinearly parameterized dynamics?
• RQ2 What conditions ensure convergence of the observer when parametrization is nonlinear in state and time?
• RQ3 In what way does the proposed method generalize standard canonical adaptive observer designs?
• RQ4 How do weakly attracting sets and non-uniform convergence contribute to observer stability in non-canonical forms?
• RQ5 Under what signal conditions (e.g., persistency of excitation) is parameter identification guaranteed?

Key findings

• The proposed observer ensures asymptotic convergence of both state and parameter estimates under persistency of excitation.
• When parametrization is linear, the method reduces to standard adaptive observer schemes, confirming consistency with established results.
• The use of weakly attracting sets enables convergence even in the absence of uniform convergence, broadening applicability.
• Non-uniform convergence analysis allows the framework to handle time-varying and nonlinearly parameterized systems effectively.
• The method provides a systematic approach to observer design for systems not in canonical form, extending the scope of adaptive estimation.
• The theoretical framework is validated through stability analysis based on Lyapunov-like techniques tailored for non-canonical systems.