Quadratic Extended Kalman Filter

• QEKF is a filtering method that uses second-order Taylor expansions and Hessian corrections to capture curvature in nonlinear measurement functions.
• It integrates quadratic innovations and augmented cross-covariances to enhance the estimation accuracy over traditional EKF methods.
• The approach systematically reduces bias in state estimation, making it effective for systems with strong nonlinearities and non-Gaussian noise.

The Quadratic Extended Kalman Filter (QEKF) is a class of Gaussian state–space filtering algorithms that generalizes the classical Extended Kalman Filter by incorporating a quadratic, rather than merely linear, approximation of the conditional mean estimator. Utilizing second-order Taylor expansions of the measurement map, the QEKF retains the recursive, prediction-update structure of standard Kalman-type filters while systematically modeling curvature arising from nonlinear measurement functions. This results in a quadratic estimator of the minimum mean square error (MMSE) map from measurements to states, capturing features of the estimation problem overlooked by linear approximations (Servadio et al., 6 Jun 2025).

1. Problem Formulation and Notation

Consider the discrete-time, nonlinear state-space model: $$\begin{align*} x_k &= f(x_{k-1}) + w_{k-1}, & w_{k-1} \sim \mathcal{N}(0, Q_{k-1}) \ z_k &= h(x_k) + v_k, & v_k \sim \mathcal{N}(0, R_k) \end{align*}$$ where $x_k \in \mathbb{R}^n$ (state), $z_k \in \mathbb{R}^m$ (measurement), $w_{k-1}$, $v_k$ are independent Gaussian noises with given covariances. Key notations include:

• $x_{k|k-1} := \mathbb{E}[x_k | z_{1:k-1}]$
• $P_{k|k-1} := \mathrm{Cov}[x_k | z_{1:k-1}]$
• $x_{k|k} := \mathbb{E}[x_k | z_{1:k}]$
• $P_{k|k} := \mathrm{Cov}[x_k | z_{1:k}]$ These quantities respectively represent the posterior and prior state means/covariances.

2. Second-Order Expansion and Quadratic Approximation

Standard EKF approaches linearize the measurement map $h(x)$ at $x_k \in \mathbb{R}^n$0, but the QEKF carries the expansion to second order: $x_k \in \mathbb{R}^n$1 for each measurement component $x_k \in \mathbb{R}^n$2, where $x_k \in \mathbb{R}^n$3 and $x_k \in \mathbb{R}^n$4 is the Hessian of $x_k \in \mathbb{R}^n$5 at $x_k \in \mathbb{R}^n$6. These are assembled into

• $x_k \in \mathbb{R}^n$7, the Jacobian matrix,
• $x_k \in \mathbb{R}^n$8, a third-order tensor of Hessians.

The predicted measurement mean, carrying a Hessian-trace correction, is given by: $x_k \in \mathbb{R}^n$9 This explicitly incorporates the local curvature of the measurement function, a key distinction from EKF.

3. QEKF Update: Innovations and Augmented Gain

The QEKF update generalizes the Kalman update by including both first- and second-order innovations and their cross-moments. The core steps are:

• Innovation vectors:
  • $z_k \in \mathbb{R}^m$0
  • $z_k \in \mathbb{R}^m$1
• Central moments:
  • Third-order state central moment $z_k \in \mathbb{R}^m$2
  • Fourth-order kurtosis tensors for state and measurement noise

Augmented cross-covariances and measurement covariances are assembled: $z_k \in \mathbb{R}^m$3

The augmented Kalman gain and state update become: $z_k \in \mathbb{R}^m$4 where all “measurement-related” terms now account for both expectation and covariance up to quadratic order.

4. Relation to the Ordinary EKF

The EKF is recovered as the special case where only first-order (Jacobian-based) terms are retained:

• Predicted mean: $z_k \in \mathbb{R}^m$5
• Innovation covariance: $z_k \in \mathbb{R}^m$6
• Kalman gain: $z_k \in \mathbb{R}^m$7

In contrast, the QEKF adds:

• A trace correction $z_k \in \mathbb{R}^m$8 in predicted mean,
• State–measurement cross-covariances involving higher-order terms,
• Quadratic innovation $z_k \in \mathbb{R}^m$9,
• Contributions from third/fourth central moments.

This enables the QEKF to approximate the true conditional mean $w_{k-1}$0 with a parabolic estimator, reducing bias especially for regimes where $w_{k-1}$1 is strongly nonlinear or the noise is non-Gaussian.

5. Algorithmic Realization

A stepwise outline for the QEKF update is as follows:

Step	Action
1	Prediction: $w_{k-1}$2, $w_{k-1}$3, $w_{k-1}$4
2	Linearization and Hessians: $w_{k-1}$5, $w_{k-1}$6, Hessians $w_{k-1}$7 for all $w_{k-1}$8
3	Predicted measurement: $w_{k-1}$9
4	Innovations: $v_k$0, $v_k$1
5	High-order moments: Compute $v_k$2, $v_k$3 as needed
6	Augmented covariances: Form all required cross- and self-covariance blocks
7	Kalman gain and update: Compute $v_k$4, update $v_k$5, $v_k$6

Numerical studies demonstrate that the QEKF achieves markedly lower estimation error under nonlinear measurement mappings and/or non-Gaussian noise (Servadio et al., 6 Jun 2025).

6. Benefits, Limitations, and Generalizations

Benefits

• The QEKF yields a quadratic estimator for the MMSE map, rather than a linear one.
• Incorporating Hessian-trace terms and higher-order cross-moments systematically reduces bias in the state estimates, especially when $v_k$7 is strongly nonlinear or noise is non-Gaussian.

Limitations

• Second derivatives of $v_k$8 and third/fourth central moments are required, increasing coding and computational complexity.
• If the prior is strictly Gaussian (vanishing skewness), the QEKF reduces to the EKF, since the quadratic gain vanishes.

Generalizations

• The quadratic update can be generalized to other Gaussian-based filters. For the Quadratic Unscented Kalman Filter (QUKF), steps for linearization and moment calculation are replaced by sigma-point evaluation.
• The framework extends to cubic and higher-order polynomial estimators by propagating even higher derivatives and moments, at increased computational cost.

7. Summary and Outlook

The QEKF extends the EKF by embedding second-order curvature information into every measurement-related term, thereby effectuating a higher-fidelity, parabolic mapping from measurement to state. Its structure facilitates systematic bias reduction in high-curvature regimes, retains the familiar Kalman recursion, and offers a template for further higher-order estimator development. These features suggest its relevance for advanced nonlinear filtering applications where the limitations of linear estimators are significant (Servadio et al., 6 Jun 2025).