Adaptive Observer Design for Partially Measured Nonlinear Cascade Systems with Unknown Parameters

Abstract

This paper addresses the design of an adaptive observer for a class of nonlinear cascade systems with partially measured states and unknown constant parameters. The considered systems have a cascade structure involving an unmeasured-state subsystem and a measured-state subsystem. The unknown parameter vector enters the measured state dynamics through a known distribution matrix, while the unmeasured state dynamics are described by a lower triangular subsystem satisfying suitable conditions. An adaptive observer is proposed to reconstruct the unmeasured state variables and to compensate for the effect of the unknown parameters using only the measured output and the known input. Sufficient gain conditions are derived through a Lyapunov-based analysis to guarantee asymptotic convergence of the state estimation errors in the absence of uncertainties, while ensuring boundedness of the parameter estimation error. The effect of bounded model uncertainties is then analyzed, and a σ-modified adaptive law is introduced to prevent parameter drift and guarantee uniform ultimate boundedness of the estimation errors. The proposed framework is illustrated through a reduced-order marine vessel model involving slowly varying environmental bias states and an unknown constant disturbance term. Numerical simulations illustrate the convergence properties and the robustness of the observer under bounded model uncertainties. Additional numerical tests with noisy output measurements are also reported to assess sensitivity to measurement noise.