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Adaptive Observer of Nonlinear One-Sided Lipschitz Systems Using Estimated State Regressors With Finite Excitation
Hamid Taghavifar, Brian Delgado Aguilar
TL;DR
The paper addresses regressor mismatch in finite-excitation adaptive observers for OSL-QIB nonlinear systems with unknown parameters and disturbances. It combines output-integral regression from measured outputs and estimated states with an OSL-QIB LMI observer and projected adaptive law. The analysis and numerical results show bounded disturbed behavior and improved learning with history-stack adaptation over no-stack adaptation.
Problem
Finite-excitation concurrent learning can avoid persistent excitation, but regressors depending on inaccessible states create mismatch that must be accounted for in observer and parameter-stability analysis.
Method
The paper uses output-integral regression with measured outputs and estimated states, bounds stored residuals and information-matrix perturbations, and designs an OSL-QIB LMI observer with projected adaptation.
Results
The proposed observer and parameter estimation outperform observers without history-stack learning, with stack activation minimizing parameter error while no-stack learning decays as excitation is lost.
Takeaways & Limitations
The analysis recovers exponential convergence in the disturbance-free, residual-free case while retaining stability with bounded disturbances and residuals.
Abstract
from arXiv · showhide
For systems with unknown parameters, finite excitation and concurrent learning can potentially yield parameter convergence without persistent excitation but the regressor may still depend on inaccessible states, leading to regressor mismatch. In this paper, this problem is addressed for a class of nonlinear systems with one-sided Lipschitz properties and quadratically inner-bounded nonlinearities with bounded disturbances and linearly parametrized uncertainties. To this aim, an output-integral regression is utilized by using measured outputs and estimated states, and history-stack residual is explicitly bounded in terms of state-estimation error and disturbance. Furthermore, a perturbation bound between the estimated-state and true-state information matrices is derived. Additionally, an OSL-QIB LMI condition is applied for the observer design and a projected adaptive law is designed without needing exact output matching. Stability analysis's results indicate the proposed observer and parameter estimation outperform observers without history-stack learning term.
I. Introduction
Persistent excitation can support parameter convergence but restricts applicability when regressors are not continuously rich. Finite-excitation and concurrent-learning methods address this limitation, while prior work leaves OSL-QIB regressor mismatch and related stability effects unresolved.
- Persistent excitation requires regressors to remain sufficiently rich over every moving time window, limiting the class of applicable systems.
- Leakage and projection can prevent parameter drift under disturbances and weak excitation, but do not provide the informative data needed for parameter convergence.
- Finite-excitation methods collect informative data over a finite interval, while concurrent learning stores data for parameter updates and integral concurrent learning avoids state-derivative estimation.
- Existing estimated-state concurrent-learning studies do not address OSL-QIB systems with explicit bounds on information-matrix differences and disturbance-dependent stored-data residuals.
- The paper proposes a finite-excitation adaptive observer for OSL-QIB nonlinear systems using estimated-state regressors and output-integral data.
II. Problem Formulation
The paper models a nonlinear system with unknown constant parameters and bounded disturbances, under regional OSL-QIB and Lipschitz-type conditions. An affine LMI observer design yields a gain that supports an error bound.
- The system includes known linear and nonlinear dynamics, an unknown constant parameter vector, and an unknown disturbance.
- Inputs and disturbances are locally bounded, trajectories remain in a known operating region, and unknown parameters lie in a known compact convex set.
- The incremental nonlinear conditions hold on a design domain determined by the estimation-error radius, with global inequalities allowing an unbounded domain.
- The observer design uses OSL-QIB-related inequalities and a Lipschitz bound on the parametrized nonlinearity increment.
- Choosing L=P^-1Y makes the observer condition affine in the decision variables and therefore an LMI.
- The resulting observer gain satisfies a bound in which estimation-error dissipation is balanced against disturbance magnitude.
III. Observer and Estimated-State Learning Identity
The observer uses estimated states and measured outputs to construct an implementable output-integral regression. Exact output matching is unnecessary because a projected design can minimize the associated mismatch bound.
- The observer estimates the state using the known model, estimated parameters, measured output, and observer gain.
- The projected matching design selects a least-restrictive auxiliary term, while exact matching is possible only when the projection condition vanishes.
- The regression data are constructed from the input, measured output, and estimated state over a finite window.
- The output-integral regression incorporates the true-state and estimated-state parametrized nonlinearities, with residual terms arising from estimation error and disturbance.
IV. Finite-Excitation Adaptive Law
The adaptive law freezes a finite history stack once its estimated-state information matrix is sufficiently informative, then uses projected updates with bounded residual effects. The analysis connects estimated-state excitation to true-state excitation through perturbation bounds, while noting that several true-state quantities remain unavailable online.
- The history stack adds or replaces data to increase the information matrix’s minimum eigenvalue and freezes when the finite-excitation condition is first satisfied.
- Finite-data informativity can be assessed directly from the recorded estimated-state regressors.
- A sufficient eigenvalue condition relates estimated-state excitation to a lower bound on true-state information.
- The estimated-state and true-state information matrices differ by a perturbation generated by state-estimation error.
- The recorded stack data are computable online, but the true-state information matrix, perturbation bound, aggregate residual, and real state-error terms are available only for analysis or offline verification.
- The projected parameter update preserves the parameter estimate within its known compact set and activates history-stack learning after stack formation.
- After activation, parameter-error dynamics include contraction from the information matrix and a residual term bounded using state error and disturbance.
V. Stability Analysis
The stability analysis establishes regional forward invariance and ultimate boundedness of observer and parameter errors under fixed-stack excitation and an LMI condition, with stronger conclusions in disturbance-free or global settings.
- Stability Analysis: Under the theorem assumptions, fixed-stack excitation and condition (43), the observer error remains in the regional design domain for t ≥ T_F.Conservative bounds ̄e_s = r_e and ̄d_s = ̄d can be used on the stated data domain.
- Stability Analysis: The resulting observer and parameter errors are ultimately uniformly bounded, while global assumptions and a global LMI implication remove the regional condition.The global conclusion applies when the stated assumptions and design inequalities hold globally.
- Stability Analysis: The Lyapunov analysis uses z = col(e, ˜θ) and bounds the coupled state-parameter dynamics after history-stack activation.The derivative analysis explicitly accounts for state-error cross terms and imperfect cancellation through the measurable output-error update.
- Stability Analysis: Forward invariance follows because the Lyapunov function prevents the observer error from reaching the boundary ∥e∥ = r_e.The continuation argument combines the Lyapunov bound with the regional condition to preserve the design domain.
- Stability Analysis: With zero disturbance, the theorem specializes to disturbance-free bounds; with zero aggregate residual, V(t) decays exponentially after T_F.The disturbance-free proof gives V(t) ≤ V(T_F)e^−μ(t−T_F).
- Stability Analysis: The excitation test remains achievable when the state-error bound is unknown, but a small output residual need not bound the full state error when C has a nontrivial nullspace.An output-residual threshold therefore confirms the required condition only through a state-error certificate.
VI. Numerical Results and Discussion
The numerical study evaluates the proposed observer with finite-excitation history-stack learning under nominal and disturbed conditions. Results show finite-excitation stack activation, loss of excitation without the stack, bounded errors under disturbance, and satisfaction of regional design conditions.
- Simulation setup: Only the first state is measured, while the nonlinear functions, true parameters, initial conditions, and simulation settings define the numerical test.The initial conditions are x(0) = [1.20, −0.75]^⊤, ˆx(0) = [−0.60, 0.20]^⊤, and ˆθ(0) = [0.75, −1.00]^⊤, with ¯θ = 1.5.
- Excitation and stack activation: The no-stack regressor’s Gramian minimum eigenvalue decreases from 2.7376 × 10−2 to 5.1913 × 10−4, indicating loss of excitation.The diagnostic uses a sliding-window Gramian for the no-stack state estimate.
- Excitation and stack activation: At T_F = 6 s, the history stack reaches finite excitation with five stored points and λmin(Ŝ_N) = 0.05.The reported coupled gain margin is 9.9662 × 10−2.
- Validation and stability: The offline validation reports λmin(S_x^N) = 0.0503, ρ_G = 6.0992 × 10−4, ¯e_s = 5.2388 × 10−4, and ∥R_N∥ = 2.8138 × 10−4.The regional invariance condition is satisfied, and max_t∥e(t)∥ = 2.03 remains below r_e = 4.
- Comparison: Stack activation minimizes parameter error, whereas the no-stack observer loses learning as excitation decays.The comparison uses the same observer and adaptation gains, projection, and initial conditions with and without the history-stack term.
- Disturbed case: With d(t) = [0, 0.03 sin(1.1(t−T_d))]^⊤ applied from T_d = 8 s, state and parameter estimation errors remain bounded.The stack freezes earlier at T_F = 6 s, before the disturbance begins.
VII. Conclusion
The paper designs a finite-excitation adaptive observer for OSL-QIB nonlinear systems with estimated-state regressors. By bounding regressor mismatch and stored-data residuals, it derives bounded-disturbance stability and recovers exponential convergence for disturbance-free, residual-free stored data.
- A finite-excitation adaptive observer is designed for OSL-QIB nonlinear systems with estimated-state regressors.
- The stability analysis incorporates regressor mismatch in output-integral data and establishes boundedness under disturbances.
- Exponential convergence is recovered when disturbances and stored-data residuals are absent.