Set-Membership Constraints for Data-Driven Control

• Set-membership constraints are nonparametric bounds that encapsulate all system behaviors consistent with measured data and known disturbance limits.
• They enable robust control synthesis in applications such as model predictive control by integrating polyhedral, ellipsoidal, or semialgebraic uncertainty sets into optimization problems.
• Methodologies employing these constraints guarantee stability, recursive feasibility, and performance while adapting to new data for improved system robustness.

Set-membership constraints for data-driven control formalize the requirement that, in the absence of a precise model, all predictions, decisions, and guarantees must hold uniformly for all systems (or trajectories, parameters, outputs, states) compatible with measured data and known disturbance bounds. This paradigm elevates robustness in data-driven control by replacing stochastic or nominal modeling with non-asymptotic, nonparametric characterizations of compatibility, often yielding polyhedral, ellipsoidal, or more generally semialgebraic sets of admissible behaviors. Theoretical and algorithmic developments ensure constraint satisfaction, stability, and performance across all admissible system realizations and provide systematic methodologies for learning, prediction, filtering, and controller synthesis directly from data.

1. Foundations of Set-Membership in Data-Driven Control

Set-membership (SM) interpretation in data-driven control is rooted in the idea of characterizing the uncertainty set as the collection of all systems (typically matrices, behaviors, or functions) that could have generated the observed data under prior knowledge of disturbance bounds. Formally, for a linear system,

$x_{t+1} = A_*x_t + B_*u_t + w_t$

with a known bound $w_t \in \mathcal{W}$, the set of admissible systems is

$$\Theta_T = \bigcap_{t=0}^{T-1} \left\{ \hat{\theta} : x_{t+1} - \hat{\theta} z_t \in \mathcal{W} \right\}$$

where $z_t = [x_t^\top,u_t^\top]^\top$. This uncertainty set enters the control synthesis as a constraint: all claimed properties (e.g., stability, output constraint satisfaction, etc.) must hold for every $(A,B)$ in $\Theta_T$ (Li et al., 2023).

Unlike probabilistic/moment-based regions, set-membership sets are built deterministically using the structure of process noise and input-output data. This approach underpins techniques for robust model predictive control (MPC), adaptive MPC, system identification and filtering, and global optimization without model identification.

2. Set-Membership Constraints in Data-Driven Model Predictive Control

Recent methods embed set-membership constraints directly into the MPC problem. Given noisy data, the system's behavior is parameterized not by an explicit model but by all consistent data-driven representations. For unknown LTI systems, trajectory prediction can be constructed via a persistently exciting input and data-derived Hankel matrices: $$\left[ H_L(u^d)\;\; H_L(y^d) \right] \alpha = \begin{bmatrix} \bar{u} \ \bar{y} \end{bmatrix}$$ Together with noise bounds, the predicted output sequence is not exact—the true output $y$ may differ from the predicted $\bar{y}$ by an error quantified by worst-case bounds as a function of the data, noise, and optimization variables. To guarantee satisfaction of hard output constraints, the MPC optimization employs a constraint tightening of the form

$$\| \bar{y}_k(t) \|_\infty + a_{1k}\| \bar{u}(t) \|_1 + a_{2k}\| \alpha(t) \|_1 + a_{3k}\| \sigma(t) \|_\infty + a_{4k} \leq y_{max}$$

where coefficients $a_{ik}$ are data-driven and system-constant-dependent (Berberich et al., 2020). These margins ensure that despite noise and modeling uncertainty, the true output remains within prescribed bounds.

Estimation procedures are required to compute critical system constants such as controllability ($\Gamma$) and observability bounds ($\rho_k$), which enter explicitly in the definition of constraint tightening coefficients.

Set-membership MPC has been extended to adaptive and dual frameworks (Parsi et al., 2022, Xie et al., 29 Apr 2024). In adaptive min-max MPC, the uncertainty set is iteratively updated with new input-state data (using quadratic matrix inequalities) ensuring that the controller is always robust with respect to all models consistent with both offline and online data. Strong duality is leveraged to reduce infinite families of robust MPC constraints to tractable LMIs involving set-membership multipliers.

3. Filtering, State Estimation, and Output Prediction

Set-membership identification naturally extends to filtering and state estimation, particularly under unknown but bounded disturbances. A key methodology is construction of Feasible Parameter Sets (FPSs) for multistep predictors: $$\Theta_p = \left\{ \theta_p : | \hat{y} - \phi_p^\top \theta_p | \leq \bar{d} \;\;\forall\,\text{data} \right\}$$ where each horizon $p$ yields an independent predictor with its local optimal error bound. The intersection of predicted uncertainty intervals from different horizon predictors forms a refined set guaranteed to contain the true output (Lauricella et al., 2020). The central value of this intersection is used as the filtered estimate, and its associated bound is minimal among all possible data-consistent models.

In the nonlinear context, computation of a minimum-volume bounding ellipsoid via semi-infinite programming (over (potentially) nonlinear images of the state set) provides accurate bounds for the feasible set of states, with constraints handled efficiently via parallelizable consensus-ADMM iterations (Li et al., 2022). This approach supports additional algebraic (equality) constraints on the state estimate.

4. Constraint Tightening, Robustness, and Stability Guarantees

Constraint tightening derived from set-membership logic is essential for robust output constraint satisfaction in data-driven control. Data-driven approaches rigorously compute error bounds for finite-horizon prediction error, using them to steer the tightening margin: $$\| \hat{y}_{t+k} - \bar{y}_k^*(t) \|_\infty \leq \bar{\epsilon}\rho_{n+k} + \bar{\epsilon}(1 + \rho_{n+k})\| \alpha^*(t) \|_1 + (1 + \rho_{n+k}) \| \sigma^*(t) \|_\infty$$ This ensures that even in the presence of model-uncertainty and noise, the true closed-loop output is guaranteed to satisfy set-membership (“hard”) constraints (Berberich et al., 2020).

Adaptive and dual-MPC methods make this property explicit by enforcing the contractivity (or positive invariance) of state constraint sets for all systems in the current feasible set: $\psi_X(A x_k + B u_k + v) \leq \lambda \psi_X(x_k)$ for all $(A,B)$ in the updated set and for all $v \in \mathcal{V}$. The recursion and feasibility arguments underpinning recursive feasibility, robust invariance, and input-to-state stability rely on repeated set-membership (feasible set refinement) steps (Zheng et al., 7 Oct 2025).

5. Set-Membership Estimation: Non-Asymptotic and Ellipsoidal Methods

Set-membership estimation (SME) provides non-asymptotic bounds on the diameter of uncertainty sets that shrink as more data become available, often at a $1/T$ rate (up to dimension-dependent factors) (Li et al., 2023). Notably, these sets are typically smaller and less conservative than confidence ellipsoids arising from least-squares estimation with concentration inequalities. Minimal volume enclosing ellipsoid (MEE) algorithms have been pioneered to summarize high-dimensional, complex (possibly nonconvex) SME sets for practical use in control and perception (Tang et al., 2023). Using moment and sum-of-squares relaxations, redundant constraints are pruned, and tight ellipsoidal overapproximations are constructed—even directly on subspaces for projection or in non-Euclidean contexts (such as SO(3) for pose estimation).

Table: Key Computational Tools for Set-Membership Estimation

Objective	Method	Reference
Minimal error output interval	Multistep FPS intersection	(Lauricella et al., 2020)
Nonlinear state bounding	SIP + consensus-ADMM	(Li et al., 2022)
Volume-optimal ellipsoid	Hierarchy of SOS relaxations, pruning	(Tang et al., 2023)
Non-asymptotic convergence rate	Concentration, BMSB, block analysis	(Li et al., 2023)

Practicalities include the need for computationally efficient solvers (LP/SDP/SOS, parallel first-order methods), data-informativity (persistent excitation), and the scalability of set-representations in high dimensions.

6. Direct Data-Driven Control Synthesis and Applicability

Modern set-theoretic data-driven control formulations build explicit safety, convergence, and feasibility properties via constructions in the extended input-output (I/O) domain—eschewing the need for explicit state estimation or model parameterization. For instance, terminal set constraints are replaced by sample-based N-step input-output backward reachable sets (N-IOBRS) constructed from data and used to sequentially guide the system to a target set: $$\xi(t) = [u_{t-T_{ini}}, ..., u_{t-1}, y_{t-T_{ini}}, ..., y_{t-1}]^\top$$ The reachable set is recursively under-approximated as the convex hull of I/O segments generated by a data-driven safety filter. The resulting controller, built solely on empirical N-IOBRS and Hankel matrix data, guarantees recursive feasibility and finite-time convergence without a state model or explicit terminal cost (Bajelani et al., 1 Nov 2024).

Set-membership principles extend to the constraint design in safety filter and barrier-type controllers, where safe sets are represented implicitly via scalable convex sets (polytopes, zonotopes, predictive safe sets), and the Control Barrier Function (CBF) is defined by the minimal scaling required to include the state. The CBF is then enforced via an optimization constraint in safety filter design, which is particularly suited for high-dimensional or data-driven invariant sets (Wabersich et al., 10 Jul 2025, Bajelani et al., 24 Feb 2025).

7. Limitations, Open Directions, and Practical Considerations

While set-membership constraints in data-driven control enable robust, nonparametric formulation of robust performance and constraint satisfaction, several practical and theoretical considerations arise:

• Conservativeness: Tightening schemes, especially under high noise or limited excitation, may yield excessively conservative solutions, with input and output trajectories kept well inside the constraint boundary (Berberich et al., 2020). Computational procedures for system constants (e.g., for tightening margins) often scale poorly with dimension.
• Computational Burden: Construction of complex set representations (e.g., projection, MEE, high-dimensional polytopic sets) and online solution of robust optimization problems may be computationally intensive, although parallel and learning-based accelerations are emerging (Tang et al., 2023, Wabersich et al., 10 Jul 2025).
• Excitation and Informativity: Guaranteeing persistent excitation in closed-loop operation remains a challenge; recent work introduces explicit linear constraints to avoid non-exciting subspaces of input (Faro et al., 6 Apr 2025).
• Online Learning and Adaptation: Innovation-triggered learning and adaptive data-driven robust control strategies dynamically update the feasible set, balancing sample complexity and redundancy (Zheng et al., 29 Jan 2024). Data selection strategies control the growth and informativeness of the uncertainty set over time, and learning hyper-parameters provide trade-offs between conservatism and performance.

Overall, set-membership constraints in data-driven control provide a rigorous, flexible, and tractable foundation for safe, robust control with nonparametric model uncertainty quantification. The techniques are central in recent theoretical and algorithmic advances across robust MPC, adaptive control, output filtering, system identification, and safety-critical applications. These developments position set-membership methods as a cornerstone for data-centric control system design under strict safety and performance requirements.