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Distributed Adaptive Neural Interval Observers for Unknown Nonlinear Systems

Tien Dat Vu, My Nguyen Bach, Phuoc Vinh Nguyen, Minh Doan

arXiv:2610.00802v1eess.SYcs.SY

TL;DR

Distributed interval estimation is challenging when nonlinear dynamics are unknown and sensors have only partial local measurements. The paper develops adaptive neural observers with cooperative network dynamics and finite replay learning. Its simulation reports interval widths reduced by more than one order of magnitude relative to the nominal case after the initial transient.

  • Problem

    Distributed interval observers for unknown nonlinear dynamics are lacking an adaptive learning design, while conventional parameter convergence often requires persistent excitation.

    Source: §I · p. 1

  • Method

    Each node couples upper and lower neural observers using local measurements and neighboring estimates, with Lyapunov-derived adaptation and finite experience-replay integral concurrent learning.

    Source: §I · p. 1, C. Distributed Neural Adaptation · p. 4

  • Results

    More than one order of magnitude reduction in neural-compensated interval widths relative to the nominal case is reported after the initial transient.

    Source: B. Simulation Results · p. 8

  • Takeaways & Limitations

    The simulation supports the framework’s effectiveness for distributed state estimation and fault-detection applications in networked nonlinear systems.

    Source: §VI · p. 8

Abstract

from arXiv · show

This paper develops a distributed adaptive neural interval observer for unknown nonlinear systems with locally incomplete measurements. Each sensor node constructs lower and upper state estimates using its local output and neighboring observer information, while unknown nonlinear dynamics are approximated by adaptive neural models. A distributed adaptation mechanism guarantees bounded estimation and weight errors, whereas a cooperative realization preserves the componentwise interval property. To enhance neural-weight convergence without requiring persistent excitation, a finite experience-replay integral concurrent-learning mechanism is incorporated into the adaptation law. For non-Metzler error dynamics, a Sylvester-based coordinate transformation is introduced to recover a Hurwitz--Metzler distributed realization. The theoretical developments are validated through a nonlinear distributed estimation example.

I. INTRODUCTION

The paper targets distributed state enclosure when sensors have incomplete local measurements and the nonlinear dynamics are unknown. It combines adaptive neural interval observers with network information exchange under collective detectability.

I. INTRODUCTION

  • Distributed interval estimation for unknown nonlinear dynamics remains separated from adaptive neural learning, whose conventional parameter convergence typically requires persistent excitation.The paper identifies distributed interval designs as essentially model-based and neural interval observers as primarily centralized.Source: §I · p. 1

A. Graph-Theoretic Preliminaries

  • A weighted directed graph specifies which neighboring signals each sensor receives, alongside its own local measurement.The in-neighbor set contains nodes sending information to node i.Source: A. Graph-Theoretic Preliminaries · p. 1

B. Networked Nonlinear System

  • The plant has known matrices and an unknown nonlinear mapping, with bounded disturbance and adaptive neural models used to bound that mapping.The neural basis vectors are bounded, and their approximation bounds apply when the state and input lie in prescribed compact sets.Source: B. Networked Nonlinear System · pp. 1–2

C. Distributed Sensing and Communication

  • Each sensor uses local output and neighboring information; individual detectability is unnecessary when the sensing network is collectively detectable.The graph is assumed fixed and strongly connected, and neighboring information is transmitted continuously without delay or packet loss.Source: C. Distributed Sensing and Communication · p. 2

D. Problem Statement

The problem is to construct local upper and lower adaptive neural observers that use only admissible local and neighboring information. The architecture couples observers over the network and formulates their errors as distributed dynamics.

D. Problem Statement

  • At every node, the design must keep the true state between the lower and upper estimates while bounding estimation and neural-weight errors.The estimates are initialized to enclose the state, and the design uses only admissible local and neighboring information.Source: D. Problem Statement · p. 2
  • Bounded estimation dynamics and preservation of nonnegative interval errors are separate requirements; Hurwitz stability alone does not ensure componentwise enclosure.The paper assigns boundedness to network Lyapunov analysis and interval preservation to a cooperative error realization.Source: D. Problem Statement · p. 2

A. Observer Architecture

  • Each node transforms the plant coordinates and couples its upper and lower observers through local output residuals and neighboring estimates.If the original-coordinate realization already has the required Hurwitz and Metzler properties, the identity transformation can be used.Source: A. Observer Architecture · pp. 2–3
  • The stacked upper and lower estimation errors evolve under a shared distributed network matrix, with neural approximation and disturbance residuals entering their dynamics.The matrix blocks encode local dynamics on the diagonal and graph-weighted transformed couplings off diagonal.Source: A. Observer Architecture · p. 3

B. Cooperative Realization

A cooperative realization supplies the Metzler structure needed for interval preservation while imposing stability on the full distributed error matrix. When the direct realization lacks that structure, a Sylvester-based coordinate transformation can recover it.

B. Cooperative Realization

  • The distributed error matrix must be both Metzler and Hurwitz for componentwise interval preservation and stable error dynamics.The coupling blocks are order preserving when the transformed neighbor maps are nonnegative; the local realization remains the key design issue.Source: B. Cooperative Realization · p. 3
  • For each node, the coordinate transformation is obtained from a Sylvester relation with a prescribed Metzler local matrix and injection gain.Spectral separation between A and the prescribed matrix gives a unique solution; the transformation must also be nonsingular.Source: B. Cooperative Realization · p. 3
  • The assembled realization is admissible when transformed couplings are nonnegative and the complete network matrix is Hurwitz.Individual local matrices need not be Hurwitz, allowing network coupling to compensate for local sensing deficiencies.Source: B. Cooperative Realization · p. 3
  • Once transformed estimates enclose the transformed state, the physical-state interval is reconstructed through the inverse coordinate transformation.The paper uses interval bounds for the transformed state to reconstruct the physical-state bounds.Source: B. Cooperative Realization · p. 3

C. Distributed Neural Adaptation

The adaptation laws are derived from a network Lyapunov function, and finite replay data supplement online adaptation to improve neural-weight convergence without persistent excitation. Approximation residuals bound how closely the weights converge.

C. Distributed Neural Adaptation

  • The adaptive laws follow from the Lyapunov derivative rather than an independent instantaneous neural loss or backpropagation criterion.The derivation uses a network Lyapunov function for the distributed estimation and weight errors.Source: C. Distributed Neural Adaptation · p. 4
  • The replay matching condition imposes structural compatibility between the uncertainty channels and the output-related matrices, independently of collective detectability.The passage states this compatibility condition follows from the existence of the matrices in Assumption 8.Source: C. Distributed Neural Adaptation · p. 4
  • A finite-rank replay information matrix provides the data-richness condition used by the composite experience-replay corrections.The replay stack is constructed from previously recorded informative data.Source: C. Distributed Neural Adaptation · p. 4
  • Finite informative replay data replace persistent excitation of the online regressor, and replay approximation residuals determine the resulting neural-weight error neighborhood.The stored samples are retained and repeatedly reused once the finite-rank condition is met.Source: C. Distributed Neural Adaptation · p. 4

A. Boundedness Analysis

The network Lyapunov analysis establishes bounded estimation and neural-weight errors under finite replay-stack richness. In the ideal residual-free case, that richness also yields exact weight convergence without persistent excitation, while bounded transformed errors imply bounded physical errors and interval widths.

A. Boundedness Analysis

  • Finite-data richness after a finite collection interval removes the need for persistent excitation of online regressors; with bounded replay approximation error, weight errors converge to a residual neighborhood.The replay stack supplies additional coercive decay, and exact exponential convergence follows in the ideal replay and residual-free case.Source: A. Boundedness Analysis · pp. 5–6
  • Under the stated assumptions and finite replay-stack richness, transformed estimation errors and neural-weight errors are uniformly ultimately bounded; in the ideal residual-free case, weights converge to zero without persistent excitation.Exact convergence additionally requires zero approximation residuals, zero replay-data errors, and zero leakage terms.Source: A. Boundedness Analysis · p. 5
  • The composite ICL laws cancel neural state–weight cross terms, while replay terms provide additional negative finite-data corrections to the weight-error dynamics.The Lyapunov analysis bounds residuals using the prescribed operating region and bounded disturbances.Source: A. Boundedness Analysis · pp. 5–6
  • Bounded transformed errors imply bounded physical estimation errors and interval widths, with the inverse transformation’s conditioning and sign structure governing width amplification.The residual bounds are stacked across nodes to obtain network-wide bounds.Source: A. Boundedness Analysis · pp. 5–6

B. Distributed Interval Preservation

Interval preservation is separate from bounded adaptive estimation: it requires the online neural models to respect nonlinear ordering and the cooperative error dynamics to preserve nonnegative errors. Under those conditions, each node’s lower and upper estimates enclose the state.

B. Distributed Interval Preservation

  • The online neural models must preserve the nonlinear ordering whenever the transformed state lies between its bounds; this interval-validity condition is independent of weight convergence.Bounded neural-weight errors alone do not guarantee this ordering.Source: B. Distributed Interval Preservation · p. 6
  • If the transformed upper and lower errors start nonnegative and the cooperative error matrix is Metzler, both errors remain nonnegative for all time.The nonnegative error terms and Metzler dynamics preserve the ordering.Source: B. Distributed Interval Preservation · p. 6
  • The resulting estimates satisfy xi(t) ≤ x(t) ≤ x̄i(t) at every node, and the upper and lower trajectories cannot cross.Boundedness from Theorem 1 does not require Assumption 9, but interval preservation does; violating the neural ordering removes the positive-system guarantee.Source: B. Distributed Interval Preservation · p. 6

A. Simulation Setup

The simulation uses a nonlinear control-affine plant observed by five scalar-output sensors, each with an incomplete local measurement. Their aggregate sensing is observable, and a directed ring connects the nodes.

A. Simulation Setup

  • The example uses a four-state nonlinear control-affine plant with bounded disturbance, whose observers know only the disturbance bounds.The simulation runs over 0–25 s.Source: A. Simulation Setup · pp. 6–7
  • Five scalar-output nodes sense individual state components; each local pair is nondetectable, but the aggregate sensing pair is observable.Thus, collective detectability holds for the sensing network.Source: A. Simulation Setup · p. 7
  • The nodes communicate over a strongly connected directed ring, and the chosen cooperative realization has a Hurwitz–Metzler network matrix.The setup uses Ti = I4, Ki = 4Ciᵀ, and κ = 1.5, with max Re(λ) = −0.2026.Source: A. Simulation Setup · p. 7
  • Each node uses 15 Gaussian radial basis functions, with adaptation gains Γi+ = Γi− = 8INϕ and leakage rates σi+ = σi− = 0.03.The neural weights are initialized to zero.Source: A. Simulation Setup · p. 7

B. Simulation Results

Neural compensation produces tight interval estimates for all four states, while the nominal observer retains wider bounds; the neural-compensated widths are reduced by more than one order of magnitude after the initial transient.

B. Simulation Results

VI. CONCLUSION

The conclusion reports bounded interval-estimation errors and adaptive neural weights, with stored-data learning improving parameter convergence without persistent excitation. Simulations show tighter intervals than the nominal observer and support distributed estimation and fault-detection applications.

VI. CONCLUSION

  • The design guarantees bounded transformed and physical-state interval-estimation errors together with bounded adaptive neural weights.Source: §VI · p. 8
  • Stored-data learning improves parameter convergence without requiring persistent excitation of the online regressors.Source: §VI · p. 8
  • Simulations show neural-compensated intervals contract to significantly tighter bounds than the nominal observer and weights approach steady values under finite excitation.Source: §VI · p. 8
  • The results support the framework for distributed state estimation and fault-detection applications in networked nonlinear systems.Source: §VI · p. 8
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