Data-Driven Gain Scheduling Control

• Data-driven gain scheduling is a method that uses empirical data to design controllers, bypassing explicit parameter modeling through convex optimization techniques.
• It leverages tools like QMI, LMI, and neural operators to ensure stability, constraint satisfaction, and optimized performance across diverse operating regimes.
• This approach has been successfully applied to LPV, polytopic systems, and nonlinear PDEs, offering scalable and rigorous control solutions.

Data-driven gain scheduling is an emerging paradigm in control design wherein all controller synthesis steps—selection of scheduling variables, identification (if any), synthesis of gain laws, and verification of closed-loop properties—are driven directly by empirical data rather than explicit parametric modeling. This approach leverages measured trajectories, modern convex optimization tools (SDP, QMI, LMI), and, more recently, machine learning and neural operator techniques, to construct gain-scheduled controllers that are certified for stability, constraint satisfaction, and potentially performance, on all models and operating regimes supported by the observed data. While classical gain scheduling typically requires explicit local linearizations and model interpolation, data-driven methods seek to bypass or minimize traditional identification, working directly with informative data records and prior structural knowledge such as LPV, polytopic, or regionally linearized system templates.

1. Foundations and Scope

Data-driven gain scheduling solutions target a broad class of systems: discrete-time or continuous-time, finite-dimensional, LPV (linear parameter-varying), polytopic, or even PDEs with nonlinear state-dependent coefficients. The core scenario is control of a system with measurable, time-varying scheduling parameters θ(t) or ρ(k), where the parameter evolution is arbitrary or subject only to weak regularity conditions.

Key assumptions:

• Availability of a data record: inputs, states (or outputs), scheduling variables, and possibly noise bounds.
• Structural knowledge: parameter-affine or polytopic representation of A(θ), known polytope Θ or convex hull of parameter values.
• Goal: Synthesize a state-feedback law u = K(θ)x with K(θ) affine or convex in θ, guaranteeing closed-loop quadratic stability, constraint invariance, or H₂ performance, for all plants consistent with the data, for all admissible scheduling trajectories.

Prominent approaches include quadratic matrix inequality (QMI) characterizations (Miller et al., 2022), informativity-based convex synthesis (Modares et al., 2022), direct robust-invariance LMI frameworks (Mejari et al., 2023), data-driven distribution shift mitigation (Ramadan et al., 2024), regionally partitioned predictive control (Guerrero et al., 30 Sep 2025, Zieglmeier et al., 2 Dec 2025), and neural-operator-based gain scheduling for PDEs (Lamarque et al., 2024).

2. Data-driven Gain Scheduling via Convex Optimization

The principal methodology—exemplified in "Data-Driven Gain Scheduling Control of Linear Parameter-Varying Systems using Quadratic Matrix Inequalities" (Miller et al., 2022)—is to carve out a convex set of all plants compatible with the observed trajectory and prescribed noise bound by encoding the measurement residual as a single QMI: $[I ;\,Xᵀ]^T Φ [I ;\,Xᵀ] ⪰ 0,$ where X is the block matrix of unknown plant parameters, and Φ a data-driven noise shape matrix. All plants consistent with data and noise lie in this set.

Stabilization is then posed as:

• Find K(θ) = ∑_{v=1}^{N_v} c_v(θ) K_v (convex interpolation over Θ's vertices)
• For all plants in the QMI set and all θ∈Θ, enforce quadratic stability: there exists P ≻ 0 such that

$(A(θ)+B K(θ)) P (A(θ)+B K(θ))^T – P ⪯ 0.$

The S-lemma is employed to reduce the potentially infinite constraint set (in θ) to a finite collection at the vertices of Θ. Each resulting constraint is convex (LMI) in P, S_v = K_v P, and scalar multipliers α_v. The final optimization is a tractable SDP, with synthesis complexity scaling polynomially in system/state size and linearly in the number of parameter vertices.

The extracted controller K(θ) guarantees certified quadratic stability for every plant in the data-consistent set and every admissible scheduling evolution. Performance criteria (e.g., worst-case H₂ norm) can be enforced via additional LMIs, with decision variable Z ≻ 0 bounding the closed-loop norm.

3. Direct Data-driven Synthesis of Safe and Robust Controllers

Beyond stability, data-driven gain scheduling frameworks address safety and robust constraint satisfaction through λ-contractive set invariance, robust control invariant (RCI) sets, or explicit data-conformity constraints.

In (Modares et al., 2022), a λ-contractive gain-scheduling law for discrete-time LPV systems is synthesized directly from raw data, bypassing model identification:

• The closed-loop data matrices are constructed from measured trajectories and parameter assignments.
• Under "closed-loop richness", a right-inverse exists enabling recovery of A+B K in data coordinates.
• For polyhedral (resp. ellipsoidal) safe sets, an LP (SDP) in data variables yields K_i at each vertex; convex interpolation gives K(w).
• The resulting controller guarantees set invariance and asymptotic stability, with data requirements ($T\ge ns+1$) that are substantially weaker than full LTI identification's persistence-of-excitation conditions.

The direct computation of RCI sets and controllers via LMI hierarchies from single-trajectory data is developed in (Mejari et al., 2023), with explicit conditions for boundedness of the model set, reduced-vertex equivalence for invariance, and iterative convex maximization of set volume.

It has also been recognized that for systems with strong nonlinearities and state-input coupling, naive data-driven robust control may induce distributional shifts between training and closed-loop regimes, undermining performance. The methodology in (Ramadan et al., 2024) augments standard robust/gain-scheduling SDPs with trace-based regularizers on the closed-loop empirical covariance, penalizing deviation from data-driven statistics, and convexifying the distribution shift penalty via relaxations on covariance slack variables.

4. Gain-Scheduled Data-Enabled Predictive Control (GS-DeePC)

Data-driven gain scheduling extends to predictive control and receding-horizon architectures as in GS-DeePC (Zieglmeier et al., 2 Dec 2025) and DeePC-GS (Guerrero et al., 30 Sep 2025):

• The operating range of a measurable scheduling variable is partitioned into regions (Ω_i) according to observed data spread and persistency of excitation.
• Regional (or composite) Hankel matrices are built solely from data located within each region.
• Online, the controller selects the appropriate Hankel bank based on real-time measurement of the scheduling variable, and solves a standard DeePC quadratic program reflecting only locally informative data.
• Switching between regions employs composite regions with robust overlap and dwell-time logic to guarantee feasibility and suppress chattering.
• The result preserves the advantages of DeePC (direct data-driven predictive control), reduces computational cost (by splitting data into smaller local banks), and restores high performance on strongly nonlinear or regime-varying systems.

Formal guarantees (feasibility, recursive feasibility, practical stability) for each regional DeePC QP are inherited from the LTI DeePC literature, provided persistency and informativity conditions are met locally. Extensive simulation for examples such as nonlinear DC motors and ship steering show marked improvements in constraint satisfaction, tracking, and computational efficiency over single-bank DeePC and Koopman-MPC alternatives.

5. Advanced Scheduling: Dual Control, Uncertainty Sets, and Neural Operators

The frontier of data-driven gain scheduling includes integration with dual control (Venkatasubramanian et al., 2020), reinforcement learning, and operator-theoretic techniques:

• Robust dual control via gain scheduling computes optimal exploration strategies and explicit affine gain schedules in the (randomized) parameter-mean update direction, using rigorous ellipsoidal uncertainty propagation, offline convex SDPs, and probabilistic performance certificates.
• For nonlinear PDEs (e.g., transport with nonlinear recirculation), neural operators (DeepONet) are trained offline to approximate the mapping from parameter functions to feedback-gain kernels. After training, the NO supports real-time evaluation of the scheduled gain for each observed state, yielding local exponential stabilization when approximation error and region-of-attraction conditions are met (Lamarque et al., 2024). This approach achieves orders of magnitude speedup over classical PDE backstepping gain scheduling.
• Mode-Aware Probabilistic Scheduling (MAPS (Kim et al., 16 Sep 2025)) exploits real-time state estimation to infer mode probabilities, which then become interpolation weights for the gain-scheduled LPV controller. This probabilistic approach enables seamless, data-driven adaptation to mode switches and uncertainties without requiring a-priori parameter models.

6. Computational and Theoretical Perspectives

Data-driven gain scheduling synthesizes controllers by formulating the full closed-loop requirements as convex optimization (LP/SDP/LMI/QMI) problems. Major results include:

• Reduction of infinite-dimensional robust/stability constraints to finite vertex checks via S-lemma, polytope vertex enumeration, and informativity analysis.
• Convexity in noise-consistent plant parameterizations, closed-loop invariance, or distribution shift objectives enables the direct use of mature SDP solvers (e.g., Mosek, SCS, JuMP).
• The complexity of synthesis typically scales polynomially in the state/input/scheduling parameter dimensions and linearly in the number of scheduling regions or parameter vertices. Partitioning the data by region or mode ameliorates cubic scaling in Hankel-based DeePC approaches.
• Theoretical guarantees extend to global quadratic stability, λ-contractivity, worst-case H₂ norm, practical constraint invariance, and upper-bounding of closed-loop covariances for all models and scheduling parameters supported by the data and structural assumptions.

7. Limitations, Guidelines, and Practical Considerations

Data-driven gain scheduling frameworks require:

• Sufficiently informative data: trajectory length/coverage relative to the state-input-scheduling dimension and the target operating regimes.
• Persistency of excitation or rank/informativity conditions on extracted data blocks.
• Correct a priori knowledge of affine/LPV structure, scheduling variable meaningfulness, and, where applicable, careful design of switching logic to avoid chattering or infeasibility.
• Realistic constraints on the size of the scheduling polytope (number of vertices), particularly for high-dimensional systems.

These methods do not remove the need for modeling insights—selection of region boundaries, dwell-time, or composite region overlapping—nor can they always guarantee global properties (for nonlinear or PDE systems, only local stability may be certifiable due to quasi-static or region-of-attraction assumptions). For physical interpretability, tuning, or certification outside the data-covered regime, explicit modeling may still be necessary.

Nonetheless, across a range of linear, LPV, nonlinear, and distributed-parameter systems, data-driven gain scheduling provides a scalable, non-conservative, and theoretically rigorous alternative to classic gain-scheduling practices, with minimal reliance on explicit plant identification and model-reduction (Miller et al., 2022, Modares et al., 2022, Mejari et al., 2023, Ramadan et al., 2024, Guerrero et al., 30 Sep 2025, Zieglmeier et al., 2 Dec 2025, Lamarque et al., 2024, Kim et al., 16 Sep 2025, Venkatasubramanian et al., 2020).