Nonlinear Sampled-Data Observer Analysis

• Nonlinear sampled-data observers are constructs that estimate the continuous state of a nonlinear system from discrete measurements, effectively handling challenges like quantization, asynchronous sampling, and discretization errors.
• They employ methodologies such as observer emulation, inter-sample prediction, and hybrid architectures with Lyapunov-based stability analysis to ensure robust performance and bounded error convergence.
• These observers are pivotal in distributed consensus and adaptive control scenarios, facilitating exponential convergence in multi-agent systems even under digital communication constraints.

A nonlinear sampled-data observer is a system-theoretic construct that estimates the state of a nonlinear continuous-time plant from measurements available only at discrete time instants. In the context of sampled-data systems and distributed control, observers for nonlinear dynamics arise in scenarios where inter-agent communication or measurement happens at isolated instants or under digital constraints. Unlike linear settings, nonlinear sampled-data observers require addressing system nonlinearities, discretization-induced errors, and the impact of asynchronicity or quantization on estimation convergence and robustness.

1. Nonlinear Sampled-Data Observer Fundamentals

A sampled-data observer for a nonlinear system seeks to reconstruct the state $x(t)$ of a system,

$$\dot{x}(t) = f(x(t),u(t),t), \quad y(t) = h(x(t),t),$$

from samples $y(t_k)$ at time instances $\{t_k\}$, possibly under constraints such as quantization or packet dropouts. The observer must generate an estimate $\hat{x}(t)$ in continuous or hybrid time with provable convergence properties. Challenges compared to continuous observers include handling inter-sample evolution, time-varying delays, and maintaining robustness to non-idealities inherent in digital implementations.

Sampled-data observer designs often rely on emulation (discretization of a continuous observer), inter-sample predictors, or hybrid observer architectures. For nonlinear systems, observer convergence typically leverages Lyapunov stability theory, input-to-state stability, or incremental observers, with stability analysis adapting to hybrid or impulsive systems frameworks.

2. Sampled-Data Observers in Distributed Nonlinear Consensus

Distributed nonlinear consensus and tracking problems serve as a core application domain for nonlinear sampled-data observers. In adaptive consensus settings, each agent possesses nonlinear dynamics and communicates only intermittent state or output information. For example, in adaptive consensus under quantized or sampled communication, agents exchange quantized or delayed versions of their states, which mandates observer structures that can compensate for information loss between samples.

In "Adaptive Consensus with Exponential Decay" (Choi et al., 8 Jun 2025), sampled-data and quantized communication complicate estimation and adaptation. The estimation of unknown parameters via concurrent learning leverages stored samples $\{x_{i,k}\}$ to achieve exponential convergence in parameter and consensus errors. Sampled-data observer concepts underlie how each agent utilizes both instantaneous (possibly quantized or sampled) and historical data to update estimates robustly, circumventing the need for persistent excitation.

3. Observer Structure and Parameter Adaptation with Sampled-Data

In sampled-data nonlinear observer schemes for multi-agent systems, the observer structure must accommodate asynchronous updates, quantized measurements, or irregular sampling. The estimation algorithms often augment classical continuous-time adaptive update laws with terms constructed from sampled or stored data: $$\dot{\hat{\theta}}_i(t) = \Phi_i(t,x_i)^T B^T P \sum_{j=1}^n a_{ij}[q_u(x_i) - q_u(x_j)] - \sum_{k=1}^r \Phi_i(x_{i,k})^T \Phi_i(x_{i,k}) [\hat{\theta}_i(t)-\theta_i],$$ where $q_u(\cdot)$ denotes a uniform quantizer, and $(x_{i,k})$ are stored samples (Choi et al., 8 Jun 2025). This structure explicitly fuses real-time sampled input with historical snapshots, forming a "concurrent learning" observer. It thus achieves parameter convergence even when only sampled data deliver finite excitation.

Sampled-data observer convergence analysis adapts continuous Lyapunov methods (with terms involving stored samples) to the hybrid/sampled-data context, using inequalities tailored for quantized and sampled signals. Robustness to sampling errors or quantization is achieved via uniform bounds relating the sampling period or quantization resolution to estimation error neighborhoods.

4. Impact of Quantization and Sampling on Observer Performance

Sampling and quantization impinge directly on observer performance by inducing discretization noise and impairing excitation. In the quantized scenario detailed in (Choi et al., 8 Jun 2025), Krasovskii solutions are employed to rigorously analyze discontinuities due to digital communication. The Lyapunov stability analysis shows that convergence to consensus and precise parameter values is replaced by convergence to a bounded neighborhood, with the asymptotic error $O(\sigma)$ scaling linearly in the quantization level $\sigma$,

$$V(t) \leq e^{-\mu t} V(0) + \frac{J}{\mu}n(\sigma/2)^2.$$

This establishes that the sampled-data observer architecture is robust in the presence of digital channel imperfections: the estimation error can be made arbitrarily small by reducing sampling intervals or quantization coarseness.

5. Lyapunov Analysis and Exponential Stability

Sampled-data observers in nonlinear adaptive consensus contexts rely on composite Lyapunov functions accounting for both state and parameter errors, including terms generated from sampled or stored data. In (Choi et al., 8 Jun 2025), the Lyapunov candidate,

$$V(x,\tilde{\Theta}) = x^T (L \otimes P)x + \frac{1}{2}\sum_{i=1}^n \|\tilde{\Theta}_i\|^2,$$

enables the derivation of exponential convergence bounds for both continuous and sampled-data cases. The observer analysis provides explicit exponential decay rates, governed by the spectral properties of the Laplacian and the "rank" (excitation) induced by sampled regressor matrices. Robustness is further reinforced by parameter tuning thresholds dependent on graph connectivity and the richness of the sampled trajectory set.

6. Extensions and Variants in Nonlinear Sampled-Data Observer Design

Contemporary research broadens the nonlinear sampled-data observer paradigm to encompass switched systems, high-order agent dynamics, and heterogeneous networks. For instance, in distributed formation tracking and adaptive control frameworks, each agent applies a sampled-data observer or learning rule using local and sampled neighbor information to estimate unmeasured states or unknown dynamics while compensating for nonlinearities, time-variations, and digital constraints. Techniques such as separation-based adaptive backstepping, prescribed performance transformations, and robust learning-aided neurodynamics further enhance observer performance in nonlinear, networked, and sampled-data settings (Choi et al., 8 Jun 2025, Yan et al., 2023, Lv et al., 2020).

7. Significance and Future Directions

Nonlinear sampled-data observers constitute an essential tool for advanced distributed control, particularly under digital communication, quantization, and computation constraints. The convergence guarantees, explicit performance bounds, and robustness to sampling artifacts provided by these observers are foundational for modern cyber-physical systems, cooperative robotics, and multi-agent adaptive learning. Further development is anticipated in integration with event-triggered sampling, learning-based observer design, and observer-based control for large-scale nonlinear networked systems, extending the rigor and practical applicability highlighted in current literature (Choi et al., 8 Jun 2025).