EF-ZRLS: Recursive Zonotopic Estimation

• EF-ZRLS is a set-membership estimation algorithm that produces zonotopic over-approximations of time-varying system parameters in the presence of bounded noise.
• It employs an exponential forgetting factor with recursive updates of the center, covariance, and generators to adapt to parameter drift while ensuring provable containment.
• The estimator supports safety-critical reachability analysis by processing streaming data and offering less conservative uncertainty bounds compared to traditional methods.

Exponentially Forgetting Zonotopic Recursive Least Squares (EF-ZRLS) is a set-membership estimation algorithm designed for online identification of time-varying system parameters and robust data-driven reachability analysis in dynamic systems, particularly under bounded noise and uncertain parameter drift. EF-ZRLS recursively computes over-approximating zonotopic sets that provably contain the true, slowly-varying system parameters, offering provable guarantees of containment and exponential adaptation under persistent excitation. The estimator provides an effective tool for safety-critical reachability analysis and verification when only streaming data is available and no accurate offline model identification can be assumed (Akhormeh et al., 21 Sep 2025).

1. Mathematical Framework

The EF-ZRLS estimator addresses the linear regression model with multiple outputs, time-varying parameters, and bounded additive noise, formulated as

$$y_k = \varphi_k \theta_{\mathrm{tr},k} + v_k, \qquad \theta_{\mathrm{tr},k} = \theta_{\mathrm{tr},k-1} + \delta\theta_{k-1},$$

where

• $y_k \in \mathbb{R}^{p \times m}$ is the observed output,
• $\varphi_k \in \mathbb{R}^{p \times n}$ is the regressor,
• $\theta_{\mathrm{tr},k} \in \mathbb{R}^{n \times m}$ is the unknown time-varying parameter matrix,
• $v_k$ is noise, $\|v_k\|_{\max} \leq \sigma_v$,
• $\delta\theta_{k-1}$ models parameter drift, $\|\delta\theta_{k-1}\|_{\max} \leq \sigma_\theta$.

At each time $k$, EF-ZRLS maintains a zonotope (set of parameter matrices) given by

$$\mathcal{M}_k = \left\langle C_k, G_k \right\rangle = \left\{ \Theta \in \mathbb{R}^{n \times m} \mid \Theta = C_k + G_k \beta,\, \|\beta\|_\infty \le 1 \right\},$$

with center $y_k \in \mathbb{R}^{p \times m}$0 and generator stack $y_k \in \mathbb{R}^{p \times m}$1 for some number of generators $y_k \in \mathbb{R}^{p \times m}$2.

2. Recursive Update Laws

The EF-ZRLS algorithm utilizes an exponential forgetting factor $y_k \in \mathbb{R}^{p \times m}$3 and a gain matrix $y_k \in \mathbb{R}^{p \times m}$4 to enable adaptation to parameter drift and noise:

• Center Update:

$y_k \in \mathbb{R}^{p \times m}$5

• Generator Update:

$y_k \in \mathbb{R}^{p \times m}$6

where $y_k \in \mathbb{R}^{p \times m}$7 are noise-generator matrices covering the admissible noise.

• Gain and Covariance Updates:

$y_k \in \mathbb{R}^{p \times m}$8

$y_k \in \mathbb{R}^{p \times m}$9

with $\varphi_k \in \mathbb{R}^{p \times n}$0, $\varphi_k \in \mathbb{R}^{p \times n}$1 initialized as a scaled identity.

The generator scaling by $\varphi_k \in \mathbb{R}^{p \times n}$2 ensures the zonotope can expand to encompass drift over time.

3. Initialization and Parameter Selection

• Center $\varphi_k \in \mathbb{R}^{p \times n}$3: Chosen as zero or, if possible, as the batch least-squares solution from an initial dataset.
• Covariance $\varphi_k \in \mathbb{R}^{p \times n}$4: Large multiple of the identity to ensure initial uncertainty covering the true parameter.
• Generators $\varphi_k \in \mathbb{R}^{p \times n}$5: Chosen to be large enough so that the initial parameter is guaranteed in the zonotope, a typical choice is one per unknown entry, each set to $\varphi_k \in \mathbb{R}^{p \times n}$6.
• Forgetting Factor $\varphi_k \in \mathbb{R}^{p \times n}$7: Selected in $\varphi_k \in \mathbb{R}^{p \times n}$8 to trade off between rapid adaptation ($\varphi_k \in \mathbb{R}^{p \times n}$9 small) and reduced noise sensitivity ($\theta_{\mathrm{tr},k} \in \mathbb{R}^{n \times m}$0 large).

4. Algorithmic Workflow

The principal algorithm proceeds as follows (summarized from the cited pseudocode):

1. Innovation: $\theta_{\mathrm{tr},k} \in \mathbb{R}^{n \times m}$1
2. Compute $\theta_{\mathrm{tr},k} \in \mathbb{R}^{n \times m}$2: $\theta_{\mathrm{tr},k} \in \mathbb{R}^{n \times m}$3
3. Gain: $\theta_{\mathrm{tr},k} \in \mathbb{R}^{n \times m}$4
4. Update Center: $\theta_{\mathrm{tr},k} \in \mathbb{R}^{n \times m}$5
5. Update Covariance: $\theta_{\mathrm{tr},k} \in \mathbb{R}^{n \times m}$6
6. Update Generators:
   • Scale old: $\theta_{\mathrm{tr},k} \in \mathbb{R}^{n \times m}$7
   • Add noise generators: $\theta_{\mathrm{tr},k} \in \mathbb{R}^{n \times m}$8
   • Concatenate: $\theta_{\mathrm{tr},k} \in \mathbb{R}^{n \times m}$9
7. (Optional) Zonotope Reduction: Apply a generator-reduction operator $v_k$0 to control computational growth.
8. Output parameter set: $v_k$1.

5. Theoretical Guarantees

• Containment: For all $v_k$2, the true parameter $v_k$3 by construction (Proposition 1).
• Optimality: The gain $v_k$4 minimizes $v_k$5 for any positive semidefinite weighting matrix $v_k$6 (Theorem 1).
• Adaptation: Under persistent excitation (PE) of $v_k$7, the zonotope center $v_k$8 converges exponentially in mean-square to $v_k$9, and the zonotopic RLS error remains bounded even when drift $\|v_k\|_{\max} \leq \sigma_v$0.
• Forgetting Factor Role: The scaling of generators by $\|v_k\|_{\max} \leq \sigma_v$1 ensures comprehensive tracking of parameter drift. Taking $\|v_k\|_{\max} \leq \sigma_v$2 increases reliance on historical data (reducing sensitivity to noise, but slowing dynamic adaptation), while $\|v_k\|_{\max} \leq \sigma_v$3 closer to 0.95 boosts responsiveness to parameter changes at the cost of larger zonotope inflation.

6. Computational Complexity and Zonotope Reduction

At each iteration, operations scale as follows:

Computation Stage	Complexity	Notes
Gain computation	$\|v_k\|_{\max} \leq \sigma_v$4
Center/covariance	$\|v_k\|_{\max} \leq \sigma_v$5
Generator scaling	$\|v_k\|_{\max} \leq \sigma_v$6	$\|v_k\|_{\max} \leq \sigma_v$7: number of generators
Noise generator add	$\|v_k\|_{\max} \leq \sigma_v$8 blocks
Total per step	$\|v_k\|_{\max} \leq \sigma_v$9

Without generator reduction, the number of generators grows at each update, making zonotope reduction essential for long-term computational tractability. The reduction operator $\delta\theta_{k-1}$0 merges/prunes generators while preserving containment $\delta\theta_{k-1}$1.

7. Illustrative Example

For the scalar case ($\delta\theta_{k-1}$2): $\delta\theta_{k-1}$3 Initialization: $\delta\theta_{k-1}$4. Suppose measurements $\delta\theta_{k-1}$5, $\delta\theta_{k-1}$6. After two updates, the center evolves to approximately 1.028 and the zonotope parameters shrink, while always containing all parameter trajectories satisfying the drift and noise bounds. This demonstrates the estimator's ability to rapidly adapt and contract uncertainty, while ensuring containment of the true parameter (Akhormeh et al., 21 Sep 2025).

8. Applications and Significance

EF-ZRLS enables rigorous online reachability analysis for discrete-time linear time-varying and nonlinear Lipschitz systems, supporting safety verification under measurement noise and uncertain parameters. By explicitly characterizing bounded uncertainty in parameters and inputs with zonotopic over-approximations, the method offers less conservative reachability sets compared to classical interval or ellipsoidal approaches. It operates exclusively on real-time streaming data, obviating the need for pre-recorded offline experiments and is robust to slowly varying model dynamics. Empirical performance is validated through numerical simulations and real-world applications, highlighting its relevance for safety-critical cyber-physical systems in environments where accurate prior models are not available (Akhormeh et al., 21 Sep 2025).