Observer-Driven Error Dynamics

• Observer-driven state estimation error dynamics describe how the difference between true and estimated states evolves, accounting for both system and observer behaviors.
• The topic examines methodologies for linear, nonlinear, and Lie group systems using Lyapunov and ISS analyses to establish convergence and stability.
• Applications span robotics, networked systems, and learning-based observers, emphasizing robust performance under deterministic and stochastic disturbances.

Observer-driven state estimation error dynamics formalize the evolution of the state estimation error—typically the difference between the true system state and an observer’s estimated state—under explicit consideration of both the system and observer dynamics. These dynamics play a central role in establishing the convergence, robustness, and stability guarantees of state observers and are foundational in robust control, adaptive filtering, learning-based estimation, and geometric control on nonlinear manifolds.

1. Mathematical Definitions and General Structure

The observer-driven state estimation error is defined as $e(t) = x(t) - \hat x(t)$, where $x(t)$ is the true system state and $\hat x(t)$ is the observer estimate. For general nonlinear systems with process and measurement disturbances, the system and observer are described by

$$\begin{cases} \dot x = f(x, t) + B u + D d, \ y = C x, \end{cases}$$

$$\dot{\hat x} = f(\hat x, t) + B u + L(y - C \hat x).$$

The resulting error dynamics are

$\dot e = f(x, t) - f(\hat x, t) - L C e + D d.$

The specific structure of the error dynamics depends critically on system structure (linear/affine/nonlinear, time-varying or invariant, stochastic vs. deterministic), observer architecture (Luenberger, extended state, geometric, learning-based), and modeling of disturbances and uncertainties.

For systems on smooth manifolds or Lie groups $G$ (e.g., $SE(3)$, $SO(3)$, $SE_N(3)$), the error is often characterized by a group-invariant form, such as $E(t) = X^{-1}(t)\hat X(t)$ (left-invariant) or $E(t) = \hat X(t)X^{-1}(t)$ (right-invariant), with dynamics depending on the group structure and the observer’s equivariance properties (Li et al., 2023, Koldychev et al., 2013, Goor et al., 2023, Hopwood et al., 2024).

2. Observer Error Dynamics in Linear, Nonlinear, and Lie Group Systems

Linear Systems:

The canonical linear time-invariant error dynamics are $\dot e = (A - LC) e$, where $L$ is chosen so that $A - LC$ is Hurwitz, ensuring exponential decay (Pasand, 2019). Nonlinearities, disturbances, or unmodeled dynamics augment this with additional terms:

$\dot e = (A - LC) e + \Delta f(x, \hat x) + D d,$

where $\Delta f$ encapsulates model mismatch or higher-order nonlinearities (Nugroho et al., 2019, Brouillon et al., 2022).

Nonlinear Systems:

For nonlinear observers (including high-gain, Lipschitz, or learning-based designs), error dynamics take the general form

$\dot e = f(x, t) - f(\hat x, t) - L(t)C e,$

with Lyapunov or ISS analysis required to establish convergence (Farkane et al., 9 Jul 2025, Chakrabarty et al., 2020, Niazi et al., 2022). Advanced observers inject innovation terms inside nonlinearities or use adaptive/learning mechanisms for unmodeled dynamics.

Lie Group Systems:

Affine and linear group systems $X(t) \in G$ yield error dynamics naturally formulated on $G$ and its Lie algebra $\mathfrak{g}$. For linear group systems, two equivalent characterizations are:

• Homeomorphism of the group flow.
• Linearity of the lifted dynamics in $\mathfrak{g}$ (Li et al., 2023). With observer input, the error evolution is often expressed via a differential equation on $\mathfrak{g}$:

$$\dot \epsilon = A_\epsilon \epsilon + B_\epsilon d,$$

where $\epsilon = \log E$ is the Lie algebra error and $d$ represents external disturbances, which may be deterministic or stochastic (e.g., Brownian motion in tangent spaces) (Li et al., 2023, Goor et al., 2023).

3. Error Dynamics Under Disturbances: ODE/SDE and Robustness

Analysis of the error dynamics under external disturbances or random noise is central to robust state estimation:

• Deterministic disturbances: Projecting the group error onto $\mathfrak{g}$ yields an ODE:

$$\dot \epsilon(t) = A_\epsilon \epsilon(t) + B_\epsilon d(t).$$

For differentiable $d(t)$, exponential decay to a residual set is established via Lyapunov arguments; the residual depends on the disturbance’s smoothness (Li et al., 2023, Nguyen, 2024).

• Stochastic disturbances: Suppose system evolution in the tangent space is perturbed by a Brownian motion; then the projected error dynamics on $\mathfrak{g}$ are governed by:

$$d\epsilon_t = A_\epsilon \epsilon_t dt + \Sigma dW_t,$$

where $W_t$ is a Wiener process (Li et al., 2023). The long-term behavior is determined by the drift $A_\epsilon$ and the diffusion $\Sigma$.

• Structured noise and observer design: Extended state observers with delayed-difference approximations inject first-order disturbance models into the observer, yielding delay-differential error equations. Practical exponential bounds on the estimation error are obtained using Lyapunov–Razumikhin techniques and are optimal up to trade-offs between robustness and system bandwidth (Nguyen, 2024).

4. Geometric, Symmetry-Preserving, and Reduced-Order Observer Error Systems

Observers exploiting system symmetries are able to define error coordinates invariant under group actions. Consider the reduced-order observer where the invariant error is constructed as $$\eta = \varphi_{\gamma(y)}(\hat x) - \varphi_{\gamma(y)}(x)$$ using a moving frame $\gamma(y)$ of the Lie group, leading to invariant error dynamics:

$\dot\eta = F(\eta, X, Y, U),$

where $F$ is defined via the group action, system, and observer structures (Hopwood et al., 2024, Koldychev et al., 2013, Goor et al., 2023). This form enables uniform exponential stability proofs via Lyapunov methods and recasts observer convergence as a property of the invariance structure.

In examples such as velocity estimation for rigid bodies, the invariant error reduces directly to a linear ODE $\dot\eta = -L \eta$ with global exponential convergence guaranteed for $L > 0$ (Hopwood et al., 2024).

5. Stability Analysis and Performance Guarantees

Stability, convergence rate, and robustness of the observer-driven error dynamics are established using several tools:

• Lyapunov-based analyses: Quadratic (or more general) Lyapunov functions are constructed to bound the error dynamics and derive sufficient conditions—often LMIs or Riccati inequalities—for global or local asymptotic stability (Niazi et al., 2022, Ascencio et al., 2019, Nugroho et al., 2019).
• Razumikhin approach for delay systems: Ensures exponential convergence in the presence of artificial delays introduced by Taylor-approximation-based observers, with residual estimation error scaling with disturbance smoothness (Nguyen, 2024).
• Input-to-state stability (ISS): For nonlinear and infinite-dimensional cases (e.g., continuum robots), ISS-type bounds guarantee convergence in the presence of bounded disturbances or unmodeled dynamics, with energy dissipation established through boundary injection or innovation terms in the observer (Zheng et al., 2023).
• Robustness under impulsive/distorting noise: Proximal observers and $\mathcal{L}_\infty$ observers are designed to maintain bounded error under arbitrarily large but sparse measurement attacks or process uncertainties, with error bounds independent of the noise amplitude (Bako et al., 2024, Nugroho et al., 2019).

Explicit formulas frequently define the convergence rate:

$$\|e(t)\| \leq Ce^{-\lambda t} \|e(0)\| + e_{ss}(t),$$

where $e_{ss}$ is a residual term determined by disturbance or modeling uncertainties.

6. Application Domains and Representative Examples

• Robotics and navigation: Explicit error dynamics on $SE_N(3)$ for GNSS/INS observers provide almost-global asymptotic and local exponential stability, with convergence rates determined independently of gain selection and robustness to extreme initial errors, provided excitation conditions are met (Li et al., 2023, Goor et al., 2023).
• Distributed and networked systems: The structure of error dynamics under time-varying graphs and dynamic communication topologies is determined by “age-of-information” (freshness-index) indices, yielding finite-time or exponential convergence when the error propagation is tightly controlled via the network structure (Mitra et al., 2018).
• Nonlinear and learning-based observers: In neural-network-based observers, the error dynamics are controlled through adaptive (learned) gains, with provable uniform convergence under mild observability and detectability assumptions by minimizing the ODE residual in a physics-informed loss, although explicit exponential rates may not be available (Farkane et al., 9 Jul 2025).
• Systems with unmodeled dynamics: Modular learning observers deploy a safe initial design, identify unmodeled nonlinearities (e.g., via Gaussian processes), and then redesign the observer for improved convergence, with input-to-state bounds at every stage (Chakrabarty et al., 2020).

7. Summary Table: Key Error Dynamics Forms

System/Observer	Error Dynamics Form	Stability Mechanism
Linear (Luenberger)	$\dot e = (A-LC)e$	Hurwitz spectrum via $L$
Lie Group (local linear)	$\dot e = -a_0 e$ (after logarithmic linearization)	Exponential (log-coordinates) (Koldychev et al., 2013)
Affine Group, stochastic	$d\epsilon = A_\epsilon\epsilon\,dt + \Sigma dW_t$	Exponential in mean-square (Li et al., 2023)
Nonlinear (Lipschitz)	$\dot e = f(x)-f(\hat x) - L(t)C e$	Lyapunov, ISS, or detectability
Disturbed (ESO, Taylor)	Delay-diff.: $$\dot e = (A+LC) e + \tfrac1h D_2[e-e(t-h)]-\Gamma_2$$	Razumikhin, delay-Lyapunov (Nguyen, 2024)
Symmetry-preserving	$\dot\eta = F(\eta, X, Y, U)$	Lyapunov, invariant error system (Hopwood et al., 2024)
Infinite-dimensional (PDE)	$$\partial_t \varepsilon + \Lambda \partial_z \varepsilon=0$$, etc.	Dissipative PDE Lyapunov (Ferrante et al., 2021, Zheng et al., 2023)

The formulation and analysis of observer-driven error dynamics remain central in advancing robust, high-performance state estimation across linear, nonlinear, geometric, distributed, and learning-based settings (Li et al., 2023, Nguyen, 2024, Koldychev et al., 2013, Hopwood et al., 2024, Mitra et al., 2018, Goor et al., 2023, Farkane et al., 9 Jul 2025).