Exponentially Forgetting ZRLS

• EF ZRLS is an adaptive estimation framework that computes set-valued zonotopic bounds on time-varying parameters using exponential weighting of recent data.
• It employs recursive updates and uncertainty propagation with scalable generators to robustly track slowly varying dynamics amidst bounded noise.
• The method underpins real-time reachability analysis and safety verification, and it is enhanced by variable-rate and directional forgetting schemes for improved adaptability.

Exponentially Forgetting Zonotopic Recursive Least Squares (EF ZRLS) is an adaptive estimation framework that recursively computes set-valued (zonotopic) bounds for time-varying model parameters from online noisy measurements, with exponential weighting assigned to recent data. EF ZRLS generalizes classical recursive least squares by integrating zonotopic uncertainty propagation, exponential forgetting (via a scalar or vector forgetting factor), and, where applicable, extensions such as variable-rate or direction-dependent forgetting. This approach achieves robust parameter estimation and model set bounding for systems with slowly-varying dynamics and bounded noise, forming the basis for real-time reachability analysis and safety-critical verification tasks (Akhormeh et al., 21 Sep 2025, Lai et al., 2023).

1. Mathematical Foundations of EF ZRLS

EF ZRLS operates on a regression model of the form $y_k = \phi_k \theta_{\text{tr}, k} + v_k$, where $y_k$ is the measured output, $\phi_k$ is a persistently exciting regressor, $\theta_{\text{tr}, k}$ is the (possibly time-varying) true parameter vector, and $v_k$ is bounded measurement noise. The parameter evolution, $$\theta_{\text{tr}, k} = \theta_{\text{tr}, k-1} + \delta\theta_{k-1}$$, admits slow variation.

Instead of producing a point estimate $\theta_k$, EF ZRLS maintains a matrix zonotope $\mathcal{Z}_k = \langle C_k, G_k \rangle$ as a set-valued over-approximation:

• $C_k$ is the zonotope center (nominal estimate).
• $G_k$ are generator matrices encoding uncertainty from noise, model variation, and prior estimate propagation.

The recursive update combines exponential forgetting (parametrized by $\lambda$: $0 < \lambda \leq 1$) and zonotopic set propagation: $$\begin{aligned} C_{k+1} &= (I - K_k \phi_k) C_k + K_k y_k \ G^{(i)}_{k+1} &= \lambda^{-\tfrac{1}{2}} (I - K_k \phi_k) G^{(i)}_k, \qquad\forall\, i \ G_{k+1} &= [G^{(1)}_{k+1}, \ldots, G^{(N_G)}_{k+1}, G_{v, k}] \end{aligned}$$ where $K_k$ is the correction gain, and $G_{v, k}$ is the generator associated with measurement noise.

The optimal gain is specified by (Akhormeh et al., 21 Sep 2025): $$K_k^* = P_k \phi_k^\top (\phi_k P_k \phi_k^\top + \lambda Q)^{-1}$$ with $P_k = G_k G_k^\top$ and $Q$ derived from the noise description. Exponential forgetting is realized by scaling past uncertainty generators by $\lambda^{-\frac{1}{2}}$; thus, older uncertainties receive exponentially less weight.

2. Exponential Forgetting Mechanism

The core of exponential forgetting in EF ZRLS is the recursive covariance update: $P_{k+1} = \lambda^{-1} (I - K_k^* \phi_k) P_k$ Older data's influence is diminished by a factor of $\lambda$ at each recursion. Lower $\lambda$ enables rapid adaptation to parameter changes, while $\lambda \approx 1$ preserves memory of earlier samples—see (Fraccaroli et al., 2015, Lai et al., 2023).

Variants exist for parameter-wise forgetting. If parameters exhibit distinct dynamics, vector forgetting schemes ($$\boldsymbol{\lambda} = [\lambda_1, \cdots, \lambda_p]^\top$$) generalize the recursive update using a kernel-inspired forgetting map: $R_t = F_{\lambda}(R_{t-1}) + \phi(t)\phi(t)^\top$ with $F_\lambda$ encoding diagonal, tuned/correlated, or cubic spline updating (see (Fraccaroli et al., 2015)). This design enables each parameter or direction to "forget" at distinct rates, enhancing tracking in multi-rate systems.

3. Zonotopic Set Propagation

EF ZRLS represents parameter estimates as matrix zonotopes: $$\mathcal{Z}_k = \left\{ C_k + \sum_{i=1}^{N_G} \alpha_i G^{(i)}_k\;:\; |\alpha_i| \leq 1 \right\}$$ At each step:

• The central update $(I - K_k \phi_k) C_k + K_k y_k$ incorporates new information.
• Each generator propagates via $\lambda^{-\tfrac{1}{2}}$ scaling and multiplication by $(I-K_k \phi_k)$. Noise generators are appended to maintain boundedness under measurement uncertainty.

Zonotopic propagation guarantees that the set $\mathcal{Z}_k$ contains all plausible $\theta_{\text{tr}, k}$, given bounded noise and dynamic variation. This is essential for set-membership identification and for reachability over-approximation (Akhormeh et al., 21 Sep 2025, Lai et al., 2023).

4. Stability, Robustness, and Convergence

Stability of parameter estimation under EF ZRLS, both pointwise and zonotopically, follows from generalized forgetting RLS theory (Lai et al., 2023). With persistently exciting regressors and bounded noise:

• If $P_k$ is uniformly positive definite and properly bounded ($aI \preceq P_k \preceq bI$), global uniform exponential stability is obtained via Lyapunov analysis: $$V_k = (\theta_k - \theta_{\text{tr}, k-1})^\top P_k^{-1} (\theta_k - \theta_{\text{tr}, k-1})$$ and

$$V_{k+1} - V_k \leq -\alpha \|\theta_k - \theta_{\text{tr}, k-1}\|^2$$

• Under time-varying parameters and bounded noise, a global uniform ultimate bound for the estimation error is established via (Lai et al., 2023, Akhormeh et al., 21 Sep 2025): $$\epsilon = \epsilon^* [ \delta_\theta + b \sqrt{\overline{\beta}} ( \sqrt{\overline{\delta}_\phi} \theta_{\max} + \overline{\delta}_y ) ]$$ for suitable bounds $\delta_\theta$, $\overline{\beta}$, $\theta_{\max}$, etc.

The zonotopic idea ensures robust tracking: the true parameter remains within the set $\mathcal{Z}_k$ despite drift, noise, and modeling error. Numerical experiments confirm less conservative over-approximations compared to batch LS and non-adaptive methods (Akhormeh et al., 21 Sep 2025).

5. Variable-Rate and Directional Forgetting

Extensions to EF ZRLS allow adaptation of the forgetting rate in time or direction:

• Variable-rate forgetting (VRF): The forgetting factor $\lambda$ becomes time-varying, $\beta_k$, with cost function weights $\rho_k = \prod_{i=0}^k \beta_i$ (Bruce et al., 2020). This improves responsivity to abrupt system changes.
• Variable-direction forgetting: A matrix-valued forgetting factor $\Lambda_k$ applies forgetting only to directions actively excited by new data (Goel et al., 2020). This reduces over-conservatism in the absence of persistent excitation.

Both mechanisms can be embedded within EF ZRLS by modifying generator scaling and update rules to accommodate these adaptive schemes, yielding tighter zonotopic bounds, especially in multi-rate or partially excited systems.

6. Applications: Real-Time Reachability Analysis

EF ZRLS has demonstrated efficacy in reachability analysis of uncertain and time-varying systems:

• Discrete-time linear time-varying systems (LTV): EF ZRLS recursively computes a set of plausible $(A_k, B_k)$ matrices for $x_{k+1} = A_k x_k + B_k u_k + w_k$, with $w_k$ and $u_k$ uncertain and/or noisy. Zonotope-based reachable sets are propagated forward, leveraging the model set at each step (Akhormeh et al., 21 Sep 2025).
• Nonlinear Lipschitz systems: EF ZRLS estimates a family of local linearizations. The nonlinear remainder is over-approximated via Lipschitz bounds—again using zonotopic arithmetic.
• Safety-critical systems: The method operates online, on real measurements, and adapts to parameter changes without offline data. Real-world applications include autonomous vehicle safety envelope computation.

Compared to classical model-based reachability methods, EF ZRLS yields less conservative over-approximations and responds efficiently to dynamic changes.

7. Comparative Performance and Trade-offs

Table: Key EF ZRLS Features and Advantages

Feature/Method	Effect	Source
Exponential forgetting	Rapid adaptation to parameter changes	(Fraccaroli et al., 2015, Akhormeh et al., 21 Sep 2025)
Zonotopic uncertainty bounds	Guaranteed over-approximation, robustness	(Akhormeh et al., 21 Sep 2025, Lai et al., 2023)
Multiple/variable forgetting	Enhanced tracking of multi-rate/time-varying parameters	(Fraccaroli et al., 2015, Bruce et al., 2020)
Directional forgetting	Non-conservatism under nonpersistent excitation	(Goel et al., 2020)
Lyapunov stability & boundedness	Strong theoretical stability guarantees	(Lai et al., 2023)
Less conservative reachability	Improved safety analysis in online settings	(Akhormeh et al., 21 Sep 2025)

In simulation and experimental studies, EF ZRLS adapts more quickly than constant-rate approaches, and set-based propagation consistently retains the true parameter set within the zonotope, even under bounded noise and drifting models.

Summary

Exponentially Forgetting Zonotopic Recursive Least Squares combines recursive least squares with exponential data weighting and zonotopic set-membership estimation to deliver robust, real-time identification and reachability certification in dynamic, uncertain systems. Variants with multiple or variable forgetting factors further enhance adaptability and reduce conservatism. Lyapunov-based analyses confirm exponential stability and ultimate boundedness of the parameter estimation error under broad conditions—including the presence of bounded noise and slowly time-varying models. EF ZRLS stands as a rigorous and practical solution for online, safety-critical parameter identification and reachability analysis, with superior performance and reduced conservatism compared to batch and classical methods (Akhormeh et al., 21 Sep 2025, Lai et al., 2023, Fraccaroli et al., 2015).