Reference Recursive Recipe (RRR) for EKF Tuning

• Reference Recursive Recipe (RRR) is a recursive framework for adaptive EKF tuning that uses iterative forward/backward passes and cost function assessments to adjust filter parameters.
• It systematically updates the initial state, process noise, and measurement noise covariances by converging eight generalized cost functions to their nominal values.
• RRR has demonstrated enhanced performance in simulation and real-world aerospace applications by achieving statistical equilibrium and improved uncertainty quantification.

The Reference Recursive Recipe (RRR) is a systematic, fully recursive framework for adaptive tuning of the initial state, initial covariance, process-noise covariance, measurement-noise covariance, and fixed parameters within the Extended Kalman Filter (EKF). Developed by Ananthasayanam et al., RRR utilizes repeated forward/backward passes through available data to iteratively adjust all EKF statistics by balancing generalized cost functions that monitor the internal statistical consistency of the filter. This approach aims to achieve a “statistical equilibrium” where the cost functions stabilize at theoretically expected values, providing reliable, low-bias, and uncertainty-quantified parameter and state estimates in both simulated and real-world domains, including aerospace system identification (Ananthasayanam et al., 2015, M et al., 2015).

1. EKF Tuning Problem and RRR Formulation

The fundamental challenge in Kalman filtering is tuning the statistics $\{\mathbf{X}_0, \mathbf{P}_0, \Theta, R, Q\}$—the initial state-mean and covariance, process and measurement noise covariances, and unknown model parameters—based solely on observed data without recourse to extensive off-line optimization. For a nonlinear discrete-time system,

$$\begin{aligned} X_k &= f(X_{k-1}, \Theta) + w_k,\;\quad w_k \sim \mathcal{N}(0, Q), \ Z_k &= h(X_k, \Theta) + v_k,\;\quad v_k \sim \mathcal{N}(0, R), \qquad k=1,\dots,N, \end{aligned}$$

the EKF augments $X_k \in \mathbb{R}^{n+p}$ to include both system states $x_k$ and parameters $\Theta$ treated as pseudo-states. Selection of the aforementioned statistics markedly impacts filter convergence, bias, Cramér–Rao bounds (CRB), and the risk of divergence (Ananthasayanam et al., 2015).

The RRR addresses this problem via iterative, recipe-like passes: each involves a forward EKF run, Rauch–Tung–Striebel (RTS) smoothing, and cost-based updates to all statistics. Convergence is declared when all cost functions approach theoretically predicted values.

2. Algorithmic Structure of RRR

The RRR operates as an outer loop over iterations $i=1,2,\ldots$, with each iteration comprising:

1. Forward EKF Pass: Utilizing the current statistics, advance through $k=1, \ldots, N$, producing prior, posterior, and innovation estimates.
2. Backward RTS Smoothing: Apply the RTS algorithm over $k=N-1,\ldots,0$ to obtain smoothed state and covariance estimates.
3. Generalized Cost Function Computation: Evaluate eight cost functionals ($J_0$–$J_8$) reflecting filter consistency in both measurement and state spaces.
4. Statistical Updates:
   • Initial Covariance Scaling: Set $$\begin{aligned} X_k &= f(X_{k-1}, \Theta) + w_k,\;\quad w_k \sim \mathcal{N}(0, Q), \ Z_k &= h(X_k, \Theta) + v_k,\;\quad v_k \sim \mathcal{N}(0, R), \qquad k=1,\dots,N, \end{aligned}$$0, and optionally trim to parameter block-diagonal.
   • Parameter Update: Extract current $$\begin{aligned} X_k &= f(X_{k-1}, \Theta) + w_k,\;\quad w_k \sim \mathcal{N}(0, Q), \ Z_k &= h(X_k, \Theta) + v_k,\;\quad v_k \sim \mathcal{N}(0, R), \qquad k=1,\dots,N, \end{aligned}$$1 from the final state or solve a normal equation derived from a negative log-likelihood cost.
   • Noise Covariance Updates: Utilize reference-based EM-type statistics on smoothed residuals to update $$\begin{aligned} X_k &= f(X_{k-1}, \Theta) + w_k,\;\quad w_k \sim \mathcal{N}(0, Q), \ Z_k &= h(X_k, \Theta) + v_k,\;\quad v_k \sim \mathcal{N}(0, R), \qquad k=1,\dots,N, \end{aligned}$$2 and $$\begin{aligned} X_k &= f(X_{k-1}, \Theta) + w_k,\;\quad w_k \sim \mathcal{N}(0, Q), \ Z_k &= h(X_k, \Theta) + v_k,\;\quad v_k \sim \mathcal{N}(0, R), \qquad k=1,\dots,N, \end{aligned}$$3.
5. Convergence Test: If all $$\begin{aligned} X_k &= f(X_{k-1}, \Theta) + w_k,\;\quad w_k \sim \mathcal{N}(0, Q), \ Z_k &= h(X_k, \Theta) + v_k,\;\quad v_k \sim \mathcal{N}(0, R), \qquad k=1,\dots,N, \end{aligned}$$4-costs are within a prescribed tolerance of $$\begin{aligned} X_k &= f(X_{k-1}, \Theta) + w_k,\;\quad w_k \sim \mathcal{N}(0, Q), \ Z_k &= h(X_k, \Theta) + v_k,\;\quad v_k \sim \mathcal{N}(0, R), \qquad k=1,\dots,N, \end{aligned}$$5 (measurement dimension) or $$\begin{aligned} X_k &= f(X_{k-1}, \Theta) + w_k,\;\quad w_k \sim \mathcal{N}(0, Q), \ Z_k &= h(X_k, \Theta) + v_k,\;\quad v_k \sim \mathcal{N}(0, R), \qquad k=1,\dots,N, \end{aligned}$$6 (state dimension), halt; else, iterate (Ananthasayanam et al., 2015, M et al., 2015).

3. Generalized Cost Functions and Their Roles

The RRR uniquely employs eight cost components reflecting the agreement of the EKF with the data and its own underlying assumptions: $$\begin{aligned} X_k &= f(X_{k-1}, \Theta) + w_k,\;\quad w_k \sim \mathcal{N}(0, Q), \ Z_k &= h(X_k, \Theta) + v_k,\;\quad v_k \sim \mathcal{N}(0, R), \qquad k=1,\dots,N, \end{aligned}$$7

For correct tuning,

$$\begin{aligned} X_k &= f(X_{k-1}, \Theta) + w_k,\;\quad w_k \sim \mathcal{N}(0, Q), \ Z_k &= h(X_k, \Theta) + v_k,\;\quad v_k \sim \mathcal{N}(0, R), \qquad k=1,\dots,N, \end{aligned}$$8

with $$\begin{aligned} X_k &= f(X_{k-1}, \Theta) + w_k,\;\quad w_k \sim \mathcal{N}(0, Q), \ Z_k &= h(X_k, \Theta) + v_k,\;\quad v_k \sim \mathcal{N}(0, R), \qquad k=1,\dots,N, \end{aligned}$$9 the observation dimension and $X_k \in \mathbb{R}^{n+p}$0 the number of state variables. These costs serve as diagnostic indicators: their stabilization signals statistical equilibrium and proper filter tuning, while oscillation or deviation indicates parameter or covariance mismatch or unobservable system structure (Ananthasayanam et al., 2015, M et al., 2015).

4. Covariance Update Formulas and Iterative Heuristics

A core feature of RRR is the iterative update of $X_k \in \mathbb{R}^{n+p}$1 and $X_k \in \mathbb{R}^{n+p}$2 based on EM-type smoothed-residual statistics:

• Measurement-Noise Covariance:

$X_k \in \mathbb{R}^{n+p}$3

• Process-Noise Covariance:

$X_k \in \mathbb{R}^{n+p}$4

where $X_k \in \mathbb{R}^{n+p}$5. The initial covariance $X_k \in \mathbb{R}^{n+p}$6 is heuristically reset via scaling: $X_k \in \mathbb{R}^{n+p}$7, countering the shrinkage of posterior covariance to zero through repeated smoothing (Ananthasayanam et al., 2015).

When sensor scale or bias is unknown, the augmented state approach is directly embedded within the RRR loop, and all steps above apply unaltered (M et al., 2015).

5. Simulation and Real-World Evaluation

RRR’s performance is demonstrated on both simulated and real datasets:

• Spring–Mass–Damper Simulation: With weak nonlinear spring, RRR achieves parameter estimates within $X_k \in \mathbb{R}^{n+p}$8 of Newton–Raphson solutions in 2–3 iterations (noise-free) and approaches theoretical CRBs within 5–6 iterations. In the presence of process noise ($X_k \in \mathbb{R}^{n+p}$9), typical convergence occurs in 20–30 iterations with $x_k$0 estimated to within $x_k$1 of ground truth, outperforming classical Myers–Tapley (MT) and Mohamed–Schwarz (MS) covariance-matching approaches (Ananthasayanam et al., 2015).
• Flight Data (Airplane Test Cases): Three real-data experiments with high-dimensional states, measurements, and time-varying or colored noise illustrate that RRR yields lower bias, CRBs, and correlation between estimated parameters than MT/MS. Importantly, RRR’s use of smoothed-residual cost functions allows the discrimination of “definitive” (well-tuned) from “deceptive” fits (proper cost values not obtained), a critical consideration in system identification (M et al., 2015).

Quantitative comparison reveals that RRR consistently produces smaller absolute percent errors in parameter estimates and better uncertainty quantification via CRB than MT or MS. The convergence of all $x_k$2-costs to nominal values provides a built-in termination criterion and diagnostic for result reliability.

6. Practical Guidance, Limitations, and Extensions

The RRR is applicable in both batch (offline) and limited-window (near real-time) scenarios, although its requirement for multiple forward/backward passes through the entire dataset naturally favors offline or batch usage. For real-time adaptation, a moving-window approach with several recursions can be adopted (Ananthasayanam et al., 2015).

Recommendations and caveats explicitly stated:

• Always use $x_k$3 in practical/nearly-realistic applications due to the ubiquity of colored, non-Gaussian, or time-varying noise.
• Use smoothed-residual EM statistics for $x_k$4 update and either smoothed-state or DSDT (difference between stochastic and zero-noise dynamics) statistics for $x_k$5 update.
• Innovations-only or filtered-residual-only updates are unstable when both $x_k$6 and $x_k$7 are unknown.
• The scale-up of $x_k$8 is mandatory to prevent covariance shrinkage.
• Convergence is robust when the EKF is a reasonable local approximation and initial statistics are not badly chosen. For pronounced nonlinearity or multimodal posteriors, more sophisticated filtering or global optimization may be required.
• Large off-diagonal values in the $x_k$9-correlation matrix highlight parameter unobservability or ill-conditioning (M et al., 2015).

The RRR remains a heuristic approach, without a proof of global optimality, but is empirically validated against both synthetic and real datasets and offers a theoretically motivated, statistically consistent solution to EKF statistics tuning problems.