Continuous–Discrete EKF

• Continuous–Discrete EKF is a recursive estimator for nonlinear stochastic systems that uses continuous-time state propagation and discrete measurement updates.
• It employs linearization and sigma-point methods to handle nonlinear dynamics and to maintain robustness in stiff or ill-conditioned scenarios.
• Advanced implementations, such as invariant EKF on Lie groups and square-root techniques, enhance numerical stability and estimation accuracy in complex systems.

A continuous–discrete Extended Kalman Filter (CD-EKF) is a recursive minimum-variance estimator for nonlinear stochastic systems in which the state evolves according to a continuous-time stochastic differential equation (SDE), while observations are made only at discrete time points. The CD-EKF generalizes the Kalman filter to nonlinear dynamics and nonlinear observation models, using linearization (or, in some variants, derivative-free sigma-point approximations) for the propagation and update steps. This framework supports state-space models involving smooth manifolds (such as matrix Lie groups), can be adapted for both standard and highly nonlinear or nondifferentiable systems, and admits robust implementations for stiff and ill-conditioned scenarios.

1. Theoretical Formulation of the Continuous–Discrete EKF

The CD-EKF addresses systems modeled as: $$dx(t) = f(t, x(t))\,dt + G(t)\,d\beta(t),\quad x(0)\sim\mathcal{N}(\bar x_0, \Pi_0)$$ where $x(t)\in\mathbb{R}^n$ is the (possibly manifold-valued) state, $f$ is a smooth drift, $G$ encodes diffusion, and $\beta(t)$ is a Brownian process with increment covariance $Q(t)\,dt$. Discrete noisy measurements

$$y_k = h\left(x(t_k)\right) + v_k, \quad v_k \sim \mathcal{N}(0, R_k)$$

are acquired at times $\{t_k\}$ with observation map $h$.

Classically, the CD-EKF consists of two main cycles:

• Prediction (Time Update): Propagate the mean $\hat x$ and covariance $x(t)\in\mathbb{R}^n$0 of the state estimate from $x(t)\in\mathbb{R}^n$1 to $x(t)\in\mathbb{R}^n$2 using a local linearization of the SDE. For systems on manifolds or matrix Lie groups, evolution is modeled so that solutions remain on the manifold, e.g., via Stratonovich SDEs (Maurer et al., 2 Jun 2025).
• Measurement Update (Correction): Incorporate the new measurement at $x(t)\in\mathbb{R}^n$3 using a first-order Taylor expansion of $x(t)\in\mathbb{R}^n$4, yielding a linearized innovation, Kalman gain, updated mean and covariance. For invariant filtering on Lie groups, this involves state correction in the group using the exponential map, and a full-order covariance reset reflecting the geometry of the underlying manifold (Maurer et al., 2 Jun 2025).

2. Invariant EKF on Matrix Lie Groups

When the system state evolves on a matrix Lie group $x(t)\in\mathbb{R}^n$5, the CD-EKF is adapted to exploit left- or right-invariant error structures in both the process and measurement models (Maurer et al., 2 Jun 2025). The continuous-time dynamics are represented as a Stratonovich SDE with group-valued noise injection: $x(t)\in\mathbb{R}^n$6 where $x(t)\in\mathbb{R}^n$7 is a white Gaussian noise in the Lie algebra, ensuring that $x(t)\in\mathbb{R}^n$8 for all $x(t)\in\mathbb{R}^n$9.

The local error is expressed, for left-invariant EKF, as $f$0, with $f$1. The mean and covariance propagate via: $f$2 where $f$3 incorporates the system Jacobian and adjoint action. Discrete-time measurements are linearized in the local Lie algebra chart. After measurement update, the covariance is reset via a full-order expression involving the right Jacobian $f$4 of the group exponential (critical to maintain consistency and invariance between different error definitions). Numerical experiments on inertial navigation confirm that only with full-order reset do left- and right-invariant formulations yield identical estimates to numerical precision, and that reduced-order resets break filter invariance and worsen empirical performance (Maurer et al., 2 Jun 2025).

3. Derivative-Free and Square-Root Implementations

For nonlinear or nondifferentiable $f$5 and $f$6, derivative-free CD-EKF variants have been developed (Kulikova et al., 2024, Kulikova et al., 2024). Instead of calculating explicit Jacobians, these schemes propagate the mean and covariance using deterministic sampling via sigma-point sets:

• The covariance $f$7 is decomposed as $f$8, with sigma-points $f$9 for a large $G$0.
• The dynamic and observation functions are evaluated at each sigma-point; empirical means and covariances are constructed from these samples to close the moment differential equations.

To address numerical instability in ill-conditioned or stiff scenarios, square-root (Cholesky or SVD) implementations propagate the factor $G$1 (where $G$2 or $G$3), using numerically robust orthogonal or singular value decompositions for both propagation and update. These methods are significantly more stable and preserve semi-definiteness of $G$4 even under machine roundoff limits (Kulikova et al., 2024, Kulikova et al., 2024).

4. Extensions: Stiffness, Ill-Conditioning, and Robust Bayesian Updates

Continuous–discrete EKFs face challenges with stiff SDEs and highly ill-conditioned measurement models. For stiff SDEs, implicit or stiff ODE solvers (e.g., Radau, ode15s) are recommended for the time-update phase, ensuring stable propagation of $G$5 (Behera et al., 4 Nov 2025, Kulikova et al., 2024). For ill-conditioned innovation covariance (e.g., nearly rank-deficient measurement Jacobians), traditional matrix inversion in the Kalman gain amplifies numerical errors and can break positive-definiteness of $G$6. Gaussian Bayesian Network EKFs (GBN-EKF) perform conditioning via local arc reversals and one-dimensional variance updates, avoiding global matrix inversion; they demonstrate lower RMSE than standard EKF and Unscented KFs in extreme ill-conditioning, while preserving robust performance for stiff systems (Behera et al., 4 Nov 2025).

5. Specialized Geometric and Structure-Preserving Filters

For systems whose state naturally lies on Riemannian manifolds—for example, in robotics, kinematics, or geometric control (e.g., SE(3)-valued state for UAV navigation)—CD-EKF variants have been constructed so that all filter operations (propagation, update, and retraction) occur in a coordinate-free manner compatible with the manifold's geometry. The exponential map is used to represent attitude correction; the covariance is propagated in the tangent space. These filters avoid singular local charts and couple translational and rotational dynamics consistently. For quadrotor UAVs performing aggressive maneuvers, this approach yields improved estimation accuracy and operational robustness (Goodarzi et al., 2016).

6. Algorithmic Summary and Computational Characteristics

A representative CD-EKF recursion proceeds as:

1. Propagate mean and covariance using continuous-time ODEs (or SDE discretizations for derivative-free variants) over the interval $G$7, using either the system’s Jacobian or empirically from sigma-points.
2. At discrete measurement time $G$8, linearize $G$9 (or use sigma-points), compute Kalman gain, evaluate innovation, correct state (in the tangent space if manifold-valued), and update covariance—including full-order reset for invariant and geometrically consistent algorithms.
3. Square-root updates maintain factored covariance representation for stability.

For high-dimensional and stiff systems, the computational cost scales as $\beta(t)$0 per step, dominated by factorizations and sigma-point evaluation. Square-root and SVD-based updates are essential when numerical robustness is a concern (Kulikova et al., 2024, Kulikova et al., 2024, Behera et al., 4 Nov 2025).

7. Empirical Performance and Quantitative Benchmarks

Extensive Monte Carlo studies on medium and large-scale nonlinear filtering problems validate the theoretical properties and compare diverse algorithmic variants:

• Invariant EKF on Lie groups: Full-order covariance reset is indispensable for invariance with respect to error definitions, and yields the best empirical accuracy in inertial navigation (4.1 m position error, 0.28 rad attitude error) (Maurer et al., 2 Jun 2025).
• Derivative-free and square-root CD-EKFs: Accuracy (ARMSE) matches standard EKF on well-conditioned problems; square-root variants remain numerically stable under ill-conditioning ($\beta(t)$1 down to $\beta(t)$2), where standard approaches fail at $\beta(t)$3 (Kulikova et al., 2024, Kulikova et al., 2024).
• GBN-EKF: Under ill-conditioned measurement models ($\beta(t)$4), the EKF diverges while the GBN-EKF maintains bounded errors and lower ARMSE, outperforming UKF and CKF which typically break down before EKF (Behera et al., 4 Nov 2025).

The continuous–discrete EKF, with these adaptations, comprises a family of estimators accommodating nonlinear, stiff, ill-conditioned, and geometric dynamical systems, with rigorous algorithmic implementations for robust, accurate, and efficient state estimation in continuous-time stochastic environments.