RIAE-AKF: Innovation-based Adaptive Estimation

• RIAE-AKF is an adaptive filtering algorithm that dynamically estimates both process and measurement noise covariances using innovation and residual data.
• It leverages exponential averaging to update noise parameters, reducing reliance on hand-tuned settings and improving robustness in time-varying environments.
• Practical tuning via a forgetting factor ensures reliable performance, making it well-suited for applications such as dynamic state estimation in power systems.

Innovation-based Adaptive Estimation Adaptive Kalman Filter (RIAE-AKF) refers to a class of adaptive filtering algorithms that dynamically estimate the process noise covariance $Q$ and the measurement noise covariance $R$ within an Extended Kalman Filter (EKF) framework by leveraging the properties of the innovation and residual sequences. The objective is to achieve robust on-line state estimation in scenarios where the authentic noise characteristics are unknown or time-varying, as is common in complex dynamical systems such as synchronous machines in power networks. The RIAE-AKF systematically adjusts $Q$ and $R$ at each iteration using recurrence relations based on innovation and/or residual statistics, alleviating the need for ad hoc, hand-tuned noise parameters and providing greater resilience to initial mis-specification compared to conventional EKF approaches (Akhlaghi et al., 2017).

1. Innovation and Residual Concepts

At each time step $k$ within the EKF, the following variables are central:

• $\hat{x}_{k|k-1}$: a priori (predicted) state estimate
• $z_k$: actual measurement
• $h(\hat{x}_{k|k-1})$: predicted measurement
• $$H_k \triangleq \frac{\partial h}{\partial x}\Big|_{x_{k|k-1}}$$: Jacobian of the measurement function

The innovation vector $d_k$ quantifies the mismatch between the observed measurement and the predicted measurement:

$d_k = z_k - h(\hat{x}_{k|k-1})$

The residual vector $\varepsilon_k$ measures the mismatch after the posterior update:

$\varepsilon_k = z_k - h(\hat{x}_{k|k})$

The innovation contains information about unmodeled system variations, process noise, and measurement noise, and is thus foundational to adaptive covariance estimation.

2. Adaptive Laws for Process and Measurement Noise Covariances

The RIAE-AKF adaptively updates $R_k$ (measurement noise covariance) and $Q_{k-1}$ (process noise covariance) using exponential averaging and the current and prior filter innovations.

Residual-based $R$–update

The theoretical covariance of the residual is

$$E[\varepsilon_k \varepsilon_k^\top] = H_k P_{k|k-1} H_k^\top + R_k.$$

By rearrangement,

$$R_k = E[\varepsilon_k \varepsilon_k^\top] - H_k P_{k|k-1} H_k^\top.$$

Since $E[\varepsilon_k \varepsilon_k^\top]$ is not available, it is estimated via an instantaneous value combined with a forgetting factor $\alpha \in (0,1]$:

$$R_k = \alpha R_{k-1} + (1-\alpha)\left(\varepsilon_k \varepsilon_k^\top + H_k P_{k|k-1} H_k^\top\right).$$

Innovation-based $Q$–update

The process disturbance $w_{k-1}$ can be inferred retrospectively as

$w_{k-1} = x_k - x_{k|k-1} = K_{k-1} d_{k-1},$

where $K_{k-1}$ is the Kalman gain from the previous step. Therefore,

$$Q_{k-1} = E[w_{k-1} w_{k-1}^\top] = E[K_{k-1} d_{k-1} d_{k-1}^\top K_{k-1}^\top].$$

This is adaptively estimated as

$$Q_{k-1} = \alpha Q_{k-2} + (1-\alpha) K_{k-1} d_{k-1} d_{k-1}^\top K_{k-1}^\top.$$

The use of exponential averaging ensures gradual adaptation and, if $Q_0$ and $R_0$ are positive definite, guarantees $Q_k$ and $R_k$ remain positive definite throughout (Akhlaghi et al., 2017).

3. RIAE-AKF Algorithmic Workflow

The RIAE-AKF framework is structured as follows:

1. Initialization: Set $\hat{x}_{0|0}$, $P_{0|0}$, $Q_0$ (arbitrary but positive definite), and $R_0$ (from sensor specifications).
2. Prediction step:
   • $\hat{x}_{k|k-1} = f(\hat{x}_{k-1|k-1}, u_{k-1})$
   • $$F_{k-1} = \partial f/\partial x |_{\hat{x}_{k-1|k-1}}$$
   • $$P_{k|k-1} = F_{k-1} P_{k-1|k-1} F_{k-1}^\top + Q_{k-1}$$
3. Innovation and Kalman gain:
   • $d_k = z_k - h(\hat{x}_{k|k-1})$
   • $H_k = \partial h/\partial x |_{\hat{x}_{k|k-1}}$
   • $S_k = H_k P_{k|k-1} H_k^\top + R_{k-1}$
   • $K_k = P_{k|k-1} H_k^\top S_k^{-1}$
4. Update step:
   • $\hat{x}_{k|k} = \hat{x}_{k|k-1} + K_k d_k$
   • $P_{k|k} = (I - K_k H_k) P_{k|k-1}$
5. Residual computation:
   • $\varepsilon_k = z_k - h(\hat{x}_{k|k})$
6. Covariance adaptation:
   • $$R_k = \alpha R_{k-1} + (1-\alpha)\left(\varepsilon_k \varepsilon_k^\top + H_k P_{k|k-1} H_k^\top\right)$$
   • $$Q_k = \alpha Q_{k-1} + (1-\alpha) K_k d_k d_k^\top K_k^\top$$

Optional: To avoid using current data that may depend on updated covariances (circularity), use $K_{k-1}$ and $d_{k-1}$ for updating $Q_k$ (Akhlaghi et al., 2017).

4. Design Parameters and Practical Tuning

The filter’s agility and stability are managed through the forgetting factor $\alpha$:

• $\alpha \rightarrow 1$: slower adaptation, reduced variance in $Q$ and $R$ estimates.
• $\alpha \ll 1$: faster tracking of changes, increased variance.

Empirical results suggest $\alpha \approx 0.3$ achieves a satisfactory trade-off in typical dynamic state estimation tasks for power systems (Akhlaghi et al., 2017).

• $R_0$ can be initialized from sensor noise characteristics.
• $Q_0$ can be conservatively over- or under-estimated; the RIAE-AKF will adapt appropriately.
• Monitoring the estimated $R_k$ online can reveal measurement anomalies or abrupt changes in sensor noise; sudden spikes indicate deteriorating measurement integrity.
• Positive definiteness of $Q$ and $R$ is maintained assuming positive definite seeds and strict exponential averaging.

Routine sanity checks, such as bounding the largest eigenvalue, are recommended to prevent transient divergence or instability caused by abrupt covariance changes.

5. Numerical Performance and Comparative Analysis

In benchmarks with the IEEE two-area, four-machine system, RIAE-AKF demonstrates:

• Robust convergence when $Q_0$ is orders of magnitude mis-specified; the conventional EKF can diverge under such conditions.
• Comparable or improved mean-squared error (MSE) in state estimates relative to fixed-covariance EKF.
• Stable and consistent operation across Monte Carlo trials ($N=200$), indicating resilience to stochastic modeling perturbations (Akhlaghi et al., 2017).

Table: Summary of RIAE-AKF versus Conventional EKF (CEKF) Performance

$Q_0$ Scenario	CEKF Convergence	RIAE-AKF Convergence	Relative MSE
Extremely small $Q_0$	No	Yes	Lower
Extremely large $Q_0$	Yes	Yes	Comparable
$Q_0$ near true	Yes	Yes	Comparable

Under all tested scenarios, RIAE-AKF achieves at least comparable, and often superior, filtering accuracy and robustness compared to CEKF (Akhlaghi et al., 2017).

6. Relationship to Other Innovation-based Adaptive Schemes

Classical innovation-based adaptive estimation (IAE) strategies, including RIAE-AKF and its extensions, rely fundamentally on the innovation covariance gap (i.e., the discrepancy between the theoretical and empirical innovation covariances) as an indicator of noise model mismatch. This is the foundation for the covariance-matching philosophy prevalent across adaptive Kalman filtering literature (Havangi et al., 2010).

Alternative approaches, such as the Adaptive Neuro-Fuzzy EKF (ANFEKF) (Havangi et al., 2010), use nonlinear mappings (e.g., adaptive neuro-fuzzy inference systems) to learn more complex adaptation laws from data, trading off higher computational cost for potentially finer-grained and nonlinear covariance adjustments. These schemes may be justified when the statistics of $Q$ and $R$ are state-dependent or subject to abrupt, nonstationary fluctuations. However, in scenarios with limited data or stringent computational constraints, RIAE-AKF remains advantageous due to its simplicity, low overhead, and proven effectiveness under a broad regime of modeling errors.

7. Practical Recommendations and Implementation Guidelines

• Use a moderate forgetting factor ($\alpha \in [0.2, 0.5]$) for general power system state estimation tasks.
• Initialize $R_0$ based on metrology or PMU technical specifications; $Q_0$ may be overestimated to guarantee initial stability.
• Continuous monitoring of adaptive $R_k$ can serve as an online sensor-health diagnostic.
• The adaptive law’s exponential-averaging form, combined with initial positive definiteness, reliably maintains filter statistical properties over time.
• Embedding safety guards (e.g., eigenvalue constraints) further enhances operational robustness.
• This method provides an effective and computationally efficient alternative to hand-tuning, especially in high-reliability, real-time environments with uncertain or drifting noise profiles (Akhlaghi et al., 2017).

A plausible implication is that RIAE-AKF can serve as the baseline for more advanced adaptive and learning-based Kalman filter enhancements, particularly in power systems and other engineering domains where accurate, real-time dynamic state estimation is essential.