Zonotopic Recursive Least Squares (ZRLS)

• ZRLS is a recursive parameter estimation framework that encapsulates uncertainty using zonotopes to provide online adaptive control and uncertainty quantification.
• It leverages affine invariance and recursive propagation to update both parameter estimates and their uncertainty sets efficiently.
• Its continuous and discrete formulations, including nonlinear extensions, support robust safety-critical control and data-driven reachability analysis.

Zonotopic Recursive Least Squares (ZRLS) is a set-based, recursive parameter estimation framework in which the evolving uncertainty about unknown parameters is represented as a zonotope. Through affine invariance and recursive propagation, ZRLS and its variants efficiently encapsulate both parameter estimates and their corresponding uncertainty sets, thereby enabling online uncertainty quantification, adaptive control synthesis, and formal reachability analysis in dynamic systems characterized by parametric and measurement uncertainty. The algorithm admits both continuous-time and discrete-time formulations, including nonlinear extensions and mechanisms for robustness to bounded disturbances and time-variance.

1. Foundations: RLS and Zonotopic Parameter Representation

ZRLS builds upon the recursive least squares (RLS) estimation framework for models affine in unknown parameters. For a measurement model

$y(t) = \phi(t)^\top \theta + w(t),$

where $\theta \in \mathbb{R}^p$ is the unknown, $\phi(t) \in \mathbb{R}^p$ is the regressor, and $w(t)$ is additive noise, the continuous-time RLS estimator minimizes an exponentially discounted quadratic cost over historical data, resulting in the ODEs: $$\begin{align*} \dot{\hat\theta}(t) &= \Gamma(t)\,\Phi(t)^\top\big(Y(t) - \Phi(t)\,\hat\theta(t)\big), \ \dot{\Gamma}(t) &= \beta\,\Gamma(t) - \Gamma(t)\,\Phi(t)^\top\Phi(t)\,\Gamma(t), \end{align*}$$ where $\Gamma(t)$ evolves as an inverse covariance matrix and initial conditions model prior beliefs (Cohen et al., 2023).

The zonotope, a centrally symmetric polytope, is defined as

$$\mathcal{Z}(c, G) = \big\{ x \in \mathbb{R}^p : x = c + G \lambda,\, \|\lambda\|_{\infty} \leq 1 \big\}$$

for center $c \in \mathbb{R}^p$ and generator matrix $G \in \mathbb{R}^{p\times q}$. Affine closure ensures

$$A\,\mathcal{Z}(c,G) + b = \mathcal{Z}(Ac + b,\, AG).$$

This property is central to ZRLS, enabling propagation of uncertainty sets.

2. Zonotopic Recursive Least Squares: Dynamics and Construction

ZRLS interprets the RLS update as an affine transformation of the initial parameter set. Beginning with an initial zonotope of prior uncertainty $\theta \in \mathbb{R}^p$0, the parameter set at time $\theta \in \mathbb{R}^p$1 propagates as

$\theta \in \mathbb{R}^p$2

Here, the zonotope’s center tracks the point estimate, and the generator matrix corresponds to the evolving uncertainty (covariance). Under mild excitation conditions, the true parameter $\theta \in \mathbb{R}^p$3 remains in $\theta \in \mathbb{R}^p$4 for all $\theta \in \mathbb{R}^p$5 (Cohen et al., 2023).

The ODEs for the center and generator are: $\theta \in \mathbb{R}^p$6

3. Algorithmic Implementation and Computational Aspects

For continuous-time ZRLS, practical implementation requires numeric integration via methods such as Euler or Runge–Kutta, using small steps $\theta \in \mathbb{R}^p$7. The canonical workflow comprises:

1. Initialization: Select prior $\theta \in \mathbb{R}^p$8, weighting $\theta \in \mathbb{R}^p$9, set $\phi(t) \in \mathbb{R}^p$0, $\phi(t) \in \mathbb{R}^p$1, and form a history stack $\phi(t) \in \mathbb{R}^p$2.
2. Recursive Update: At each step, update the history, build regressor and output matrices, then compute
   • $\phi(t) \in \mathbb{R}^p$3,
   • $\phi(t) \in \mathbb{R}^p$4, followed by integration.
3. Complexity Management: Zonotope order reduction may be applied to restrain generator growth, with complexity $\phi(t) \in \mathbb{R}^p$5 if invoked. The per-step computational cost remains $\phi(t) \in \mathbb{R}^p$6, matching standard RLS (Cohen et al., 2023).

In discrete time and multivariate settings, the Exponentially Forgetting ZRLS (EF-ZRLS) extends standard ZRLS. EF-ZRLS employs a forgetting factor $\phi(t) \in \mathbb{R}^p$7 to weight recent data and accommodate time-varying parameters, with recursive updates for both center and generator sets: $\phi(t) \in \mathbb{R}^p$8 where $\phi(t) \in \mathbb{R}^p$9 is chosen to minimize the weighted-trace of the new covariance (Akhormeh et al., 21 Sep 2025).

4. Theoretical Guarantees and Containment Properties

ZRLS and its EF-ZRLS extension guarantee, under bounded measurement noise and parameter variation, the true parameter is always contained within the maintained zonotope. Formally, if the initial set contains the true parameter and noise bounds are accurate, containment holds at all future times (Akhormeh et al., 21 Sep 2025).

For regressor sequences satisfying a persistent excitation (PE) condition, the center estimate converges exponentially (in mean-square sense) to the true, possibly slowly varying parameter. The limiting size of generators is set by measurement noise and parameter drift, modulated by the forgetting factor. Tightness is optimal in a weighted-trace sense among recursive zonotopic estimators using the derived gain $w(t)$0 (Akhormeh et al., 21 Sep 2025).

Robustness extends to unmodeled parameter jumps (up to the scaling permitted by $w(t)$1), and any noise realized within the bounded zonotope is admissible.

5. Applications in Adaptive Safety-Critical Control

ZRLS enables online adaptive control and safety-critical control synthesis by furnishing set-valued uncertainty quantification. In systems of the form: $w(t)$2 ZRLS delivers a dynamic zonotope $w(t)$3 containing the true parameter $w(t)$4.

Control Barrier Functions (CBFs) utilize these sets to ensure forward invariance of safety sets $w(t)$5 under worst-case affine decompositions. For a Robust Adaptive CBF (RaCBF), the following constraint is imposed: $w(t)$6 where the right side utilizes zonotope parameters and norm bounds. The resulting safety filter is cast as a quadratic program ensuring that, even under all admissible realizations of $w(t)$7 and disturbances, the safety constraint is enforced (Cohen et al., 2023).

6. Reachability Analysis and Extensions to Uncertain Dynamical Systems

The EF-ZRLS construction extends naturally to data-driven reachability analysis for both discrete-time linear time-varying (LTV) systems and nonlinear systems locally linearized at each operating point. In the LTV case, EF-ZRLS maintains a matrix zonotope enclosing the true, time-varying linear dynamics matrix, enabling guaranteed over-approximation of future reachable states. In nonlinear Lipschitz systems, EF-ZRLS estimates both the Jacobian and drift offset via regression, while a separate zonotope captures higher-order remainder terms (Akhormeh et al., 21 Sep 2025).

The overall reachability update incorporates the parameter zonotope, input and state sets, and process/noise zonotopes, thus delivering provable over-approximations robust to bounded uncertainties and measurement errors.

7. Computational Considerations and Practical Implementation

Each recursive EF-ZRLS update involves $w(t)$8 operations due to a $w(t)$9 matrix inversion and $$\begin{align*} \dot{\hat\theta}(t) &= \Gamma(t)\,\Phi(t)^\top\big(Y(t) - \Phi(t)\,\hat\theta(t)\big), \ \dot{\Gamma}(t) &= \beta\,\Gamma(t) - \Gamma(t)\,\Phi(t)^\top\Phi(t)\,\Gamma(t), \end{align*}$$0 generator updates, where $$\begin{align*} \dot{\hat\theta}(t) &= \Gamma(t)\,\Phi(t)^\top\big(Y(t) - \Phi(t)\,\hat\theta(t)\big), \ \dot{\Gamma}(t) &= \beta\,\Gamma(t) - \Gamma(t)\,\Phi(t)^\top\Phi(t)\,\Gamma(t), \end{align*}$$1 is the number of generators. Generator count may grow linearly unless reduced using zonotope order-reduction schemes (e.g., retaining the generators with largest norm and absorbing others), which controls complexity at $$\begin{align*} \dot{\hat\theta}(t) &= \Gamma(t)\,\Phi(t)^\top\big(Y(t) - \Phi(t)\,\hat\theta(t)\big), \ \dot{\Gamma}(t) &= \beta\,\Gamma(t) - \Gamma(t)\,\Phi(t)^\top\Phi(t)\,\Gamma(t), \end{align*}$$2.

Initialization without offline data employs a large, identity-weighted prior and wide generators to guarantee set-inclusion; with offline identification, initial center and covariance can be chosen for reduced conservatism.

Parameter vectorization or direct matrix updates can be used, with matrix-based updates generally being more computationally efficient in multivariate regression settings (Akhormeh et al., 21 Sep 2025).

• "Uncertainty Quantification for Recursive Estimation in Adaptive Safety-Critical Control" (Cohen et al., 2023)
• "Online Data-Driven Reachability Analysis using Zonotopic Recursive Least Squares" (Akhormeh et al., 21 Sep 2025)