Moving Horizon Extended State Observer

• MH-ESO is an observer architecture that augments the state vector with unknown disturbances, parameters, and unmodeled dynamics for enhanced estimation.
• It employs receding-horizon optimization to jointly recover system states and extended states, delivering robust performance under uncertainty and constraints.
• Advanced extensions of MH-ESO include distributed, multi-rate, and constraint-enforcing methods, broadening its applications in control and real-time monitoring.

A Moving Horizon Extended State Observer (MH-ESO) is an observer architecture that combines the moving horizon estimation (MHE) paradigm with explicit incorporation of “extended” states—including unknown disturbances, constant or time-varying parameters, or unmodeled dynamics—into the estimation process. MH-ESO schemes leverage receding-horizon optimization to reconstruct both the system’s state and the extended states, providing robust performance under uncertainty, disturbances, and constraints. MH-ESO designs occupy the intersection of nonlinear observer theory, constrained optimization, and robust estimation, and have applications across control, fault detection, distributed systems, and real-time monitoring.

1. Fundamental Principles and Frameworks

The core mechanism of MH-ESO is the extension of the traditional state vector to include unmeasured or uncertain quantities required for compensation or monitoring tasks. For a general nonlinear system,

$x_{t+1} = f(x_t, u_t, w_t, p),$

$y_t = h(x_t, u_t, w_t, p),$

an MH-ESO augments the state either with unknown constant parameters $p$ (Schiller et al., 2023), time-varying parameters or disturbances $z_t$ (Schiller et al., 15 Apr 2024), or lumped mismatch/disturbance terms $d(t)$ (Nguyen, 5 May 2024). The observer aims to jointly estimate both $x_t$ and the extended states by solving an optimization problem at each step over a moving window.

The general MHE optimization used in MH-ESO can be formulated as: $$\begin{aligned} &\min \quad J_t(\hat{x}_{t-N|t}, \hat{z}_{t-N|t}, \hat{w}_{\cdot|t}) = \gamma(N_t)\|\hat{x}_{t-N_t|t} - \bar{x}_{t-N_t}\|_{W_{t-N_t}}^2 \ &+ \eta_1^{N_t} \|\hat{z}_{t-N_t|t} - \bar{z}_{t-N_t}\|_{V_{t-N_t}}^2 \ &+ \sum_{j=1}^{N_t} \left[\eta_2^{j-1}\|\hat{w}_{t-j|t}\|_{Q_{t-j}}^2 + \|\hat{y}_{t-j|t} - y_{t-j}\|_{R_{t-j}}^2\right], \end{aligned}$$ subject to extended system dynamics and constraints (Schiller et al., 2023, Schiller et al., 15 Apr 2024).

This structure allows off-line or on-line optimization and is compatible with nonlinear constraints, time-varying weights, and adaptation of regularization to account for time-varying parameter observability or excitation conditions.

2. Robustness, Stability, and Theoretical Guarantees

MH-ESO inherits and extends multiple forms of robust stability from MHE and nonlinear observer theory:

• RGAS and RGES: Robust global (asymptotic/exponential) stability guarantees can be obtained under incremental input-output-to-state stability (i-IOSS) or δ-Lyapunov frameworks, provided that the cost function is carefully tuned to be compatible with system detectability (Knuefer et al., 2021, Schiller et al., 2022).
• Suboptimality Tolerance: Robust stability is inherited under suboptimal computation—i.e., whenever the solution achieves a cost improvement over a feasible candidate (typically based on an auxiliary observer trajectory), the overall estimator remains robust (Schiller et al., 2020, Schiller et al., 2021). This removes dependence on solver-specific contraction or full optimization at each step.
• Handling Insufficient Excitation: By incorporating adaptive regularization on parameter estimates, the MH-ESO maintains bounded error for all times, even when persistent excitation—which is necessary for parameter observability—is not guaranteed. When excitation is insufficient, parameter updates are frozen or regularized, avoiding estimator divergence (Schiller et al., 2023, Schiller et al., 15 Apr 2024).
• Error Bounds: The estimation error admits explicit, often exponential, bounds. For instance,

$$\|e_t\| \leq \alpha_2 \|e(0)\| e^{-\beta_2 t} + \gamma_2 d_2 h,$$

(for disturbance estimation with Taylor approximation (Nguyen, 5 May 2024)), or

$$\|x_t - \hat{x}_t\| \leq C_1 \|x_0 - \hat{x}_0\| \lambda^t + \sum_{\tau=1}^t \lambda^\tau (C_2|w_{t-\tau}| + C_3|v_{t-\tau}|),$$

for suboptimal MHE (Schiller et al., 2021, Schiller et al., 2022).

3. Off-Line Approximator Identification and Computational Efficiency

The conventional MH-ESO paradigm is computationally demanding due to the nonconvexity of the trajectory optimization problem. A significant enhancement is off-line pre-computation of the measurement-to-state map via nonlinear identification, as proposed in (Alamir, 2012): $$x(k) \approx \mathcal{F}(Z(k)), \qquad F(Z) = \Gamma^{-1}(L^T Z),$$ where $Z(k)$ summarizes past input-output data. Identification reduces to a constrained quadratic program (QP) by choosing monotonic nonlinear basis expansions, permitting efficient off-line mapping determination. This mapping can be used in lieu of on-line optimization or as a fast, analytic initial guess for the full MH-ESO, with an explicit, quantifiable error bound: $\|\hat{x} - x\| \leq \varepsilon_x + O(\gamma),$ where $\varepsilon_x$ captures the fit residual and $\gamma$ reflects the regressor’s noise sensitivity (Alamir, 2012).

4. Adaptive Regularization and Parameter Observability

When MH-ESO is used for joint state and parameter estimation, excitation-dependent adaptive regularization in the cost function is central (Schiller et al., 2023, Schiller et al., 15 Apr 2024). The estimator employs an update law: $\bar{p}_t = \begin{cases} \hat{p}_t, & \text{if sufficient excitation is detected over %%%%0%%%%}, \ \bar{p}_{t-N_t}, & \text{otherwise,} \end{cases}$ enabling “freezing” of the parameter estimate when informative measurements are absent. Lyapunov-based arguments ensure boundedness and, under frequent excitation, exponential convergence of the combined state-parameter errors.

The choice of horizon length, windowing, and discount factors is informed by contraction conditions and excitation analysis, while online or scenario-based observability certification is used to ensure practical feasibility and error guarantees (Alamir, 2020).

5. Advanced Design Extensions: Distributed, Multi-Rate, and Constrained MH-ESO

MH-ESO frameworks can be adapted for large-scale, distributed, or partitioned systems (Farina et al., 31 Jan 2024), joint state-parameter estimation with multi-rate structure (Desai et al., 2023), and fully constrained estimation:

• Distributed/Partitioned Estimation: Partition-based MH-ESO divides the extended state among subsystems interacting via neighbor-to-neighbor or all-to-all architectures. Each subsystem solves a reduced-order MH-ESO problem, exchanging local estimates or covariances. Stability and convergence depend on Schur properties of system observability matrices as determined by subsystem partitioning.
• Multi-Rate and Parallel Estimation: In systems with fast and slow dynamics (e.g., batteries), a multi-rate MH-ESO can be realized using non-uniform down-sampling, coupled fast (state-focused) and slow (parameter-focused) MHEs, and parallelized computation for real-time feasibility (Desai et al., 2023).
• Physical Constraints: Explicit upper/lower bounds on the extended state or disturbance can be enforced through the optimization constraints, yielding improved robustness and estimation accuracy.

6. Novel Observer Structures and Enhanced Disturbance Estimation

Recent advances incorporate improved modeling of disturbance dynamics into the MH-ESO design:

• Taylor (backward difference) approximation is used to represent the disturbance derivative more accurately: $$\dot{d}(t) \approx \frac{d(t)-d(t-h)}{h} - R(t, h),$$ with the residual error term $R(t, h)$ quantified and the delay parameter $h$ chosen to balance stability and estimation accuracy (Nguyen, 5 May 2024). The observer and extended system are thus augmented with explicit delay or integral approximation dynamics. This structure enables compensation for ramp or sinusoidal disturbances beyond the constant or piecewise-constant model underpinning standard ESOs.

7. Representative Applications

MH-ESO and its variants have been applied to:

• Nonlinear chemical reactors (joint state-parameter estimation, fast process monitoring) (Schiller et al., 2021, Schiller et al., 2022).
• Battery management systems (joint SOC and parameter estimation, multi-rate MHE for state and parameter separation) (Desai et al., 2023).
• Distributed consensus in multi-agent systems with delays and disturbances, using predictive ESOs with LMI-based stability guarantees (Jiang et al., 2018).
• Large-scale mechanical or process systems using distributed and constraint-exploiting estimation methods (Farina et al., 31 Jan 2024).
• Practical drive systems under unknown ramp-type or mismatched disturbances, employing ESO with Taylor integral approximation (Nguyen, 5 May 2024).

A summary of core features across representative MH-ESO designs is shown below:

Feature	Reference	Structural Mechanism
Off-line mapping	(Alamir, 2012)	Constrained QP, Monotonic function
Distributed ESO	(Jiang et al., 2018)	Neighbor-based, LMI stability
Constrained MHE	(Farina et al., 31 Jan 2024)	Partitioned, Riccati arrival cost
Joint param. est.	(Schiller et al., 2023)	Adaptive regularization, PE check
Multi-rate MHE	(Desai et al., 2023)	Parallel fast/slow horizon MHE
Taylor approx.	(Nguyen, 5 May 2024)	Delay-augmented dynamics

The breadth and customization flexibility of the MH-ESO design allow robust estimation and compensation for a wide class of practical, uncertain, and nonlinear dynamic systems. The theoretical underpinnings—stability, contraction, boundedness under weak observability—are now well developed and inform parameter tuning, window length selection, and constraint enforcement in modern observer development.