Data-Driven Structured Controller Design

Updated 30 January 2026

Data-driven structured controller design is a method that synthesizes controllers with prescribed structures directly from finite, noisy data without requiring explicit system identification.
It leverages convex optimization techniques—including matrix ellipsoids and the S-procedure—to handle model uncertainty and enforce performance criteria like robust stabilization and H2/H∞ regulation.
The approach integrates structural constraints (such as sparsity and decentralization) via iterative linearization or convex relaxations, enabling scalable, robust controller synthesis for complex systems.

Data-driven structured controller design encompasses algorithms and formulations for synthesizing controllers of prescribed structure (e.g., sparsity, subspace, or decentralized architectures) directly from finite, noisy experimental data, without requiring explicit identification of a system model. This research area enables robust, high-performance regulation or stabilization, even under uncertainty, by exploiting convex optimization, matrix ellipsoid set-membership, the matrix S-procedure, and iterative linearization techniques. Typical objectives include robust stabilization, $H_2$ or $H_\infty$ performance, and structural properties such as sparsity, subspace, or decentralized gain patterns.

1. Problem Formulation and Structural Constraints

The canonical setting is the continuous- or discrete-time LTI system, whose true matrices $(A,B)$ are unknown: $x_{k+1} = A x_k + B u_k + G d_k, \quad y_k = C x_k + D u_k + H d_k,$ where $d_k$ or $w_k$ is a bounded disturbance or process noise. Rather than identification of $(A,B)$ , one collects a single trajectory of finite-length state and input samples $(X,U)$ , and possibly state-derivative samples in continuous time. The controller is structured, typically static state feedback $u = K x$ , with $K$ satisfying explicit constraints such as:

Sparsity pattern: $K \circ I_{S^c} = 0$ for some binary mask $I_S$ (Yang et al., 28 Jan 2026, Yang et al., 19 Mar 2025).
Subspace: $K \in \mathcal{S}\subseteq \mathbb{R}^{m \times n}$ (Miller et al., 2024).
Decentralization: block-diagonal or prescribed zero subblocks (Ma et al., 2019, Alemzadeh et al., 2021).
Output or partial state structure: target-output feedback $u = KF x$ (Zhang et al., 27 May 2025).

The goal is a controller $K$ that (i) stabilizes all plants consistent with data and disturbance bounds, (ii) respects the structure constraint, and (iii) attains specified performance (e.g., low $H_2$ / $H_\infty$ norm, pole placement, or output regulation).

2. Data-driven Model Uncertainty Sets

Central to data-driven control is the construction of a tractable uncertainty set of plant matrices consistent with observed data and known noise/disturbance bounds. For discrete time, from $x_{k+1}=A x_k + B u_k + w_k$ with $|w_k| \leq \varepsilon$ , the feasible $(A,B)$ satisfy for all $i$ : $[I;A^\top;B^\top]^\top \Psi_i [I;A^\top;B^\top] \succeq 0,$ where $\Psi_i$ is derived from $(x_i, u_i, x_{i+1})$ and the noise bound (Yang et al., 19 Mar 2025). Collecting all such inequalities over a trajectory yields an intersection set $\Sigma = \bigcap_{i} \Sigma_i$ (Yang et al., 28 Jan 2026), generally intractable.

To enable convex formulation, this is relaxed to a minimal-volume matrix ellipsoid: $\Phi = \{(A,B): (Z-\delta)^\top \mathbf{A} (Z-\delta) \preceq I \}$ whose parameters are optimized via dual multipliers and an SDP (e.g., minimizing $-\log\det(\mathbf{A})$ ), constrained so the ellipsoid covers all feasible $(A,B)$ (Yang et al., 28 Jan 2026).

For $H_2$ regulation, an alternative is the “matrix ellipsoid” from quadratic matrix inequalities constructed on the observed data blocks $(X_-, X_+, U_-)$ (Miller et al., 2024).

3. Data-driven Structured Controller Synthesis Methods

Multiple frameworks have been developed for direct, structured controller design. Key distinctions arise in the structural constraint treatment and the embedding of data uncertainty.

Matrix S-procedure and Convexification

Robustness to data uncertainty is embedded via a matrix S-Lemma (S-procedure), which lifts the infinite family of LMI constraints—one per feasible $(A,B)$ —to a single tractable condition. For example, robust $H_2$ synthesis requires: $\left[ \begin{array}{cc} P & (A+BK)^\top \ (A+BK) & ... \end{array} \right] \succ 0 \quad \forall (A,B)\in\Sigma$ which, by S-procedure, is replaced with an LMI involving sum of dual variables times the data-induced inequalities and a multipliers set (Yang et al., 19 Mar 2025, Miller et al., 2024).

Structural Imposition

The structured constraint (sparsity/subspace) on $K$ is typically non-convex under standard variable substitutions (e.g., $L=KP$ destroys sparsity). Two principal remedies arise:

Penalty relaxations and iterative linearization: Introduce slack variables and linearize non-convex terms (e.g., $Y \preceq P^{-1}$ ) around an iterate, resulting in an ILMI algorithm guaranteeing feasibility and monotonic improvement (Yang et al., 28 Jan 2026, Yang et al., 19 Mar 2025).
Convex sufficient parametrizations: Using a change of variables $L=KR$ and requiring $L\in\mathcal{S}$ and $R\in \mathcal Y(\mathcal{S})$ (convex), guarantees $K\in\mathcal{S}$ when $R$ invertible (Miller et al., 2024).

These approaches yield a sequence of SDPs, each imposing the structure via linear or convex constraints on $K$ , $L$ , $Y$ , and penalizing any structural violation (e.g., $\|K\circ I_{S^c}\|_F^2$ ).

Algorithmic Outline

Typical algorithms proceed as follows (Yang et al., 28 Jan 2026, Yang et al., 19 Mar 2025, Miller et al., 2024):

Construct the data-consistent uncertainty ellipsoid from sample trajectories and disturbance bounds.
Formulate the robust control objective (stabilization, $H_2$ , $H_\infty$ ) as an LMI robust over the ellipsoid/S-procedure relaxed uncertainty set.
Handle structural constraints by linearization (ILMI) or convex embeddings.
Iteratively solve the SDP, updating the linearization anchor or penalty weight until convergence.

The LMI block sizes depend only on the system order, not the number of data points, ensuring scalability.

4. Extension to Decentralized, Distributed, and Partial-State Control

Decentralized/distributed architectures employ block-sparsity or graph-induced patterns on $K$ (e.g., block-diagonal, prescribed off-diagonal zeros, or adjacency-driven blocks). Algorithms like D2SPI (Alemzadeh et al., 2021) use local clique decomposition and block-monoid structure, with learning performed on local neighborhoods with auxiliary communication in the learning phase only, before restricting to the imposed pattern.

Partial-state/target-output feedback: Direct data-driven pole placement using rank conditions and data factorizations enables controller design for subsystem outputs or aggregate features, even without full state controllability (Zhang et al., 27 May 2025). Existence and synthesis reduce to data-induced rank tests and low-dimensional factorization on partial-state or output trajectories, and the separation principle holds when observer design is embedded (Zhang et al., 27 May 2025).

5. Nonlinear, Polynomial, and Safety-Oriented Structured Control

Data-driven structured synthesis is not restricted to LTI cases. For safety synthesis in nonlinear or polynomial systems, sum-of-squares (SOS) programming is adopted:

The system is abstracted as a differential or difference inclusion consistent with observed data (Ahmadi et al., 2018, Akbarzadeh et al., 28 Jan 2026).
A structured polynomial barrier certificate (e.g., a parameterized Lyapunov or control barrier function) is sought to enforce safety separation from unsafe sets, guaranteeing robust invariance under all consistent dynamics and disturbances.
The resulting SOS problem enforces structural constraints on the barrier and/or controller (e.g., sparsity, decentralized dependence, function class).
Robust safety controllers are certified over uncertainty sets using data-driven algebraic certificates (Ahmadi et al., 2018, Akbarzadeh et al., 28 Jan 2026).

6. Performance, Robustness, and Computation

The data-driven structured optimization approaches yield robust, often conservative, controllers—guaranteed stabilizing and certifiable in performance for all plants consistent with the data and modeling assumptions.

Empirical comparisons (Yang et al., 19 Mar 2025, Yang et al., 28 Jan 2026, Miller et al., 2024) demonstrate that:

Data-driven structured controllers can offer tighter guarantees and less conservatism than earlier set-membership or energy-bound approaches.
Performance gaps to unstructured (full-information) or model-based controllers can be small as data length increases or noise decreases.
Computational complexity is dominated by SDP solves whose size scales with system order and controller structure, independent of data length.

In nonlinear/polynomial SOS regimes, practical solvability is achieved for moderate-dimensional systems (e.g., $n\leq4$ , polynomial degree $\leq6$ ) using modern SDP solvers (Ahmadi et al., 2018).

7. Open Challenges and Extensions

Current limitations are anchored in:

Local optimality and conservatism due to convex relaxations and linearization.
Scalability to high-dimensional MIMO cases with very sparse structures.
Lack of global optimality certificates due to NP-hardness of generic structured feedback.
Extensions to output feedback, dynamic/delayed structure, and non-convex parametric dependence remain ongoing challenges (Yang et al., 28 Jan 2026, Yang et al., 19 Mar 2025).

Promising directions include integration with neural parameterizations for nonlinear identification (Ecker et al., 2023), adaptive online mechanisms for non-stationary dynamics, and synthesis under switching or time-varying structural patterns (Alemzadeh et al., 2021, Akbarzadeh et al., 28 Jan 2026).

The state of the art in data-driven structured controller design employs convex optimization, matrix S-procedure, and iterative linearization to synthesize structured static and, in some cases, dynamic controllers directly from noisy finite data. The field encompasses LTI regularization, robust $H_2$ / $H_\infty$ control, decentralized/distributed architectures, as well as polynomial safety certification, illustrating a unified, scalable methodology for networked, uncertain, and high-dimensional control synthesis (Yang et al., 28 Jan 2026, Yang et al., 19 Mar 2025, Miller et al., 2024, Ma et al., 2019, Alemzadeh et al., 2021, Ahmadi et al., 2018, Akbarzadeh et al., 28 Jan 2026, Ecker et al., 2023, Zhang et al., 27 May 2025).