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Nonlinear Moving-Horizon Estimation Using State- and Control-Dependent Models

Mohammadreza Kamaldar

Abstract

This paper presents a state- and control-dependent moving-horizon estimation (SCD-MHE) algorithm for nonlinear discrete-time systems. Within this framework, a pseudo-linear representation of nonlinear dynamics is leveraged utilizing state- and control-dependent coefficients, where the solution to a moving-horizon estimation problem is iteratively refined. At each discrete time step, a quadratic program is executed over a sliding window of historical measurements. Moreover, system matrices are consecutively updated based upon prior iterates to capture nonlinear regimes. In contrast to the extended Kalman filter (EKF) and the unscented Kalman filter (UKF), nonlinearities and bounds are accommodated within a structured optimization framework, thereby circumventing the reliance on local Jacobian matrices. Furthermore, theoretical analysis is presented to establish the convergence of the iterative sequence, and bounded estimation errors are mathematically guaranteed under uniform observability conditions. Finally, comparative numerical experiments utilizing a quadrotor vertical kinematics system demonstrate that the SCD-MHE achieves superior estimation accuracy relative to the EKF, the UKF, and a fully nonlinear moving-horizon estimator, while reducing per-step computational latency by over an order of magnitude.

Nonlinear Moving-Horizon Estimation Using State- and Control-Dependent Models

Abstract

This paper presents a state- and control-dependent moving-horizon estimation (SCD-MHE) algorithm for nonlinear discrete-time systems. Within this framework, a pseudo-linear representation of nonlinear dynamics is leveraged utilizing state- and control-dependent coefficients, where the solution to a moving-horizon estimation problem is iteratively refined. At each discrete time step, a quadratic program is executed over a sliding window of historical measurements. Moreover, system matrices are consecutively updated based upon prior iterates to capture nonlinear regimes. In contrast to the extended Kalman filter (EKF) and the unscented Kalman filter (UKF), nonlinearities and bounds are accommodated within a structured optimization framework, thereby circumventing the reliance on local Jacobian matrices. Furthermore, theoretical analysis is presented to establish the convergence of the iterative sequence, and bounded estimation errors are mathematically guaranteed under uniform observability conditions. Finally, comparative numerical experiments utilizing a quadrotor vertical kinematics system demonstrate that the SCD-MHE achieves superior estimation accuracy relative to the EKF, the UKF, and a fully nonlinear moving-horizon estimator, while reducing per-step computational latency by over an order of magnitude.

Paper Structure

This paper contains 11 sections, 4 theorems, 46 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Statement
  4. State- and Control-Dependent Moving-Horizon Estimation
  5. Warm-Starting the Iterative Solver
  6. Stopping Criteria
  7. Arrival Cost Update
  8. Quadratic Programming Formulation
  9. Theoretical Analysis
  10. Numerical Results
  11. Conclusion

Key Result

Lemma 1

For all $k\ge\ell-1$, let $\hat{z}_{k,*} \triangleq \text{argmin}_{\zeta \in {\mathbb R}^{n_z}} J_k(\zeta)$. Assume assum:bounded_noise is satisfied, and that there exist $\underline{q}, \underline{r} \in (0, \infty)$ such that, for all $k \ge 0$, $Q_k \succeq \underline{q} I_n$ and $R_k \succeq \un

Figures (3)

  • Figure 3: Moving-horizon estimation sliding window at time step $k$. The statistical confidence of all past data prior to the current horizon is compressed by the arrival cost.
  • Figure 4: Flowchart of the SCD-MHE algorithm executing at a single time step $k$. The iterative quadratic programming steps dictated by the SCDC parameterization are enclosed by the dashed box.
  • Figure 5: State estimation trajectories for quadrotor vertical kinematics subject to rangefinder saturation. The EKF maintains a persistent bias due to measurement Jacobian collapse. The UKF escapes the unobservable region at $t \approx 2$ s but incurs large transient error. The N-MHE recovers at $t \approx 0.5$ s with residual transient delay. The SCD-MHE tracks the true state immediately post-horizon via the SCDC factorization. The lower panel illustrates the sensor saturation at $h_{\max} = 30$ m.

Theorems & Definitions (9)

  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof