Recursive System Identification

• Recursive system identification is a class of online methods that update model parameters in real-time, ensuring robustness and strong convergence in dynamic systems.
• These techniques employ convex optimization with various loss functions such as least-squares, Huber, and quantile to balance efficiency, robustness, and computational cost.
• Empirical evidence shows convergence to errors below 10⁻³ in under 10,000 samples, making them practical for real-world applications like grid-edge fault detection.

Recursive system identification encompasses a class of methodologies for estimating the parameters of dynamic systems by processing data online, typically in a streaming fashion. These algorithms recursively update model parameters as new measurements arrive, enabling real-time adaptation and tracking of time-varying or stochastic system dynamics. This paradigm subsumes diverse approaches—least-squares, convex optimization, total least-squares, instrumental-variable, and even nonparametric direct weight optimization—all tailored to specific noise models, robustness requirements, and computational constraints. Theoretical foundations emphasize consistency, convergence, robustness, and efficiency under various stochastic assumptions.

1. Mathematical Formulation and Model Assumptions

The canonical model considered in recursive identification is the stochastic linear regression or ARX structure: $y_{k+1} = \theta^{*\,T} x_k + w_{k+1}$ where $x_k \in \mathbb{R}^d$ is the regressor (often formed from lagged outputs/inputs), $y_{k+1}$ is the scalar output, $\theta^*$ is the true parameter vector to be identified, and $w_{k+1}$ is additive noise. Assumptions typically include:

• The loss function $\Phi: \mathbb{R} \to \mathbb{R}$ is convex, nonnegative, and polynomially bounded.
• The regressor sequence $\{x_k\}$ is φ-mixing with geometric rate, possesses a strictly positive probability density over a ball, and sufficiently high moments.
• The noise sequence $\{w_k\}$ is i.i.d., has an even, continuous density at the origin, is independent of $x_k$, and possesses finite moments (Ding et al., 2024).

These conditions ensure that the identification problem is well-posed and admits consistent estimation with probability one.

2. Recursive Convex Optimization-Based Identification Algorithms

Recursive identification algorithms minimize empirical risk functions constructed from convex losses: $$J_n(\theta) = \frac{1}{n} \sum_{t=1}^n \Phi(y_t - x_t^T \theta)$$ where $\Phi$ may be least-squares, general $L_l(z) = |z|^l$ ($l \geq 1$), Huber, log-cosh, or quantile loss. Classical least-squares is a special case. The recursive estimation proceeds via stochastic approximation: $$\theta_{k+1} = \Pi_{\|\cdot\|\leq M_{\sigma_k}}\left( \theta_k + a_k\,x_k\,\phi(y_{k+1} - x_k^T \theta_k) \right)$$ with stepsize $a_k = 1/k$, derivative $\phi(z) = \Phi'(z)$ (or subgradient for nonsmooth cases), and truncation to the ball of radius $M_{\sigma_k}$ to control parameter growth.

For nonsmooth cases (e.g., $L_1$, quantile loss), the update uses the appropriate subgradient. The truncation sequence ensures asymptotic boundedness and consistency without a priori parameter bounds (Ding et al., 2024).

3. Convergence Properties and Theoretical Guarantees

The main convergence theorems establish strong consistency:

• The population risk $R(\theta) = E\,[\Phi(y_{k+1} - x_k^T\theta)]$ is strictly convex and uniquely minimized at $\theta^*$.
• Given the stated regularity conditions, the recursive update above yields $\theta_k \to \theta^*$ almost surely as $k \to \infty$.
• The proof leverages stochastic approximation with expanding truncations, Lyapunov arguments, and small-noise conditions on the martingale difference sequence.

For nonsmooth loss functions, analogous theorems cover subgradient recursions (Ding et al., 2024). The step-size schedule requires $a_k \to 0$, $\sum a_k = \infty$, which is satisfied by $a_k = 1/k$.

4. Robustness, Computational Complexity, and Special Structures

Recursive convex loss minimization offers powerful robustness to outliers by employing appropriate loss functions—Huber, log-cosh, quantile, or $L_1$—which downweight large residuals. Empirical studies in (Ding et al., 2024) demonstrate that when $1\%$ of observations are contaminated, robust losses incur significantly reduced error compared to kernel-regularized least squares.

The recursive algorithms are online-efficient: each update requires $O(d)$ flops (inner product, multiplication, vector addition, truncation) per sample. In contrast, kernel-based regularization methods necessitate $O(n^2)$–$O(n^3)$ operations per new data for hyperparameter re-estimation and matrix inversion, unsuitable for real-time operation.

Practical schemes are compatible with widely used frameworks—batch convex optimization solvers (CVX), stochastic approximation, and streaming data interfaces.

5. Extensions: Model-Order Mismatch and Multi-Input Structures

For input/output models with order mismatch, recursive least-squares (RLS) with regularization converges to the unique equivalent higher-order model minimizing the regularization term, rather than the zero-padded true parameter vector (Lai et al., 2024). Explicit characterization is given for the asymptotic limit in terms of model equivalence constraints and regularization penalty minimization. Persistently exciting data is required for convergence, and practical guidelines are developed for initialization and hyperparameter tuning.

Distributed recursive identification for large multi-input-single-output FIR systems employs modular RLS recursions with minimal scalar exchange between local modules (each computing its own parameter vector). Lyapunov-based analysis establishes global asymptotic convergence. The approach scales efficiently with system size in terms of both computation and communication (Steentjes et al., 2018).

6. Practical Applications and Performance

Numerical examples in (Ding et al., 2024) demonstrate that recursive convex identification algorithms achieve error below $10^{-3}$ in under 10,000 samples, with total CPU time on the order of $0.02\text{–}0.05$ seconds, compared to kernel regularization methods which may require several hundred seconds under similar conditions. Robust losses outperform non-robust methods when data are contaminated by outliers.

Grid-edge fault detection employs recursive ARX (rARX) identification for real-time anomaly detection in power systems, demonstrating the ability to detect large faults rapidly and discriminate high-impedance faults from load changes using parameter deviation signatures (Yaagoubi et al., 24 Jun 2025).

7. Guidelines for Loss Function, Step-Size, and Initialization

Selection recommendations from (Ding et al., 2024):

• Use $L_2$ loss for purely Gaussian noise, but prefer robust losses for outlier-prone scenarios.
• Huber loss offers a good trade-off between efficiency and robustness; set the threshold near expected noise standard deviation.
• Quantile loss is necessary when estimating conditional quantiles under asymmetric noise distributions.
• Stepsize $a_k = 1/k$ is stable; $a_k = 1/(k^\alpha)$ with $\alpha \in (0.5, 1]$ permits accelerated convergence if summability criteria are met.
• Truncation bound $M_\ell = \ell^{1/(1+2l)}$ (for loss scaling as $|z|^l$) prevents unnecessary restarts.
• Initialize parameters using prior knowledge if available; otherwise, any finite value is acceptable.

Monitoring truncation events provides insight into algorithm stability and may indicate the need to adjust step-size or truncation rate.

Table: Loss Functions Supported by Recursive Convex Algorithms

Loss Name	Formula	Robust Outlier Handling
Least-squares	$\Phi(z) = z^2$	No
$L_l$	$\Phi(z) = |z|^l,\ l \geq 1$	Strong ($l \leq 1.5$)
Huber	See eq (Huber) above	Yes
Log-cosh	$\Phi(z) = \log \cosh(z)$	Yes
Quantile	See eq (Quantile) above	Yes (asymmetric)

A broad spectrum of convex loss functions can be handled within a unified recursive algorithmic framework, enabling efficient, robust, and strongly consistent identification in diverse stochastic systems (Ding et al., 2024).

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In summary, recursive system identification establishes a rigorous foundation for sequential parameter estimation with strong theoretical guarantees, broad model expressiveness, outlier robustness, and scalable real-time performance. The choice of loss, update schedule, and initialization determines the robustness and convergence properties of the algorithm. Extensions include treatment of order mismatch, distributed large-scale systems, and adaptive fault detection. Continued research focuses on integrating nonlinearity, quantized observations, structured models, and optimal regularization.