Data-Driven Output-Feedback Control

• Data-driven output-feedback control is a strategy that uses measured input–output data rather than full system models to directly design controllers.
• It leverages techniques such as delay embedding, data-dependent LMIs, and iterative optimization to create substitute states and achieve control objectives.
• This approach ensures robustness and efficiency in systems with unmeasurable states or high-dimensional outputs by bypassing traditional model identification.

A data-driven output-feedback control law refers to a feedback control strategy that is constructed directly from measured data—rather than explicitly identified parametric models—using output measurements (possibly also with input measurements) to achieve stabilization, regulation, or optimal performance objectives. In this context, the control law leverages input–output or output-only data and algorithmic frameworks designed to either replace, approximate, or sidestep system model identification. It is especially relevant in scenarios where state measurements are unavailable, model identification is impractical, or modeling errors are significant.

1. Principles of Data-Driven Output-Feedback Control

The essential characteristic of data-driven output-feedback control is that the controller is synthesized, tuned, or adapted directly via trajectories or data samples of inputs and outputs rather than from a full system model. This paradigm contrasts with classical model-based approaches that first identify (A, B, C, D) matrices or nonlinear models and then synthesize feedback.

Key principles include:

• Behavioral System Theory: Leveraging the idea that persistently excited input–output trajectories encapsulate the system’s dynamics and can substitute for explicit model identification through concepts like Willems’ Fundamental Lemma (Persis et al., 2019, Schmitz et al., 2022, Pan et al., 2022, Xie et al., 28 Aug 2025).
• Lifting/Embedding: Constructing an extended state or "information state" from delayed output and input trajectories (embedding) to provide a sufficient statistic for control design, especially for partially observed or output-only systems (Persis et al., 2019, Goyal et al., 2021, Xie et al., 28 Aug 2025).
• Parametrization via Data: Representing closed-loop or controller dynamics with data-driven variable substitutions and parameterizations, often constructed from Hankel matrices or delay vectors (Persis et al., 2019, Schmitz et al., 2022, Xie et al., 28 Aug 2025).
• Direct Controller Synthesis: Formulating control objectives (stabilization, regulation, optimality) as optimization or feasibility problems over the data domain—e.g., via Linear Matrix Inequalities (LMIs), semidefinite programming, or iterative optimization (Persis et al., 2019, Hu et al., 25 Aug 2024, Bajelani et al., 24 Feb 2025, Xie et al., 28 Aug 2025).

2. Core Methodologies for Output-Feedback Synthesis

Data-driven output-feedback controller synthesis employs combinations of the following methodologies, tailored to the system class, control objective, and available measurements:

(a) Delay Embedding and State Substitution

For linear systems, the unknown state $x_k$ can be expressed as a function of a finite window of past inputs and outputs ("substitute state"):

$x_k = M \xi_k,$

where $\xi_k$ is a stacked vector of delayed inputs and outputs constructed via trajectory-based or observer-based parameterizations (Xie et al., 28 Aug 2025, Lin et al., 23 Sep 2025). This representation enables one to "lift" the output-feedback control design into a problem resembling state-feedback, with data-derived states.

(b) Data-Dependent LMIs and SDP Problems

For stabilization and optimal control, data-driven feedback design is often cast as the search for matrices (controller gains, Lyapunov or storage functions) solving a data-dependent LMI or SDP:

$$\begin{bmatrix} X_0 Q & X_1 Q \ Q^\top X_1^\top & X_0 Q \end{bmatrix} \succ 0,$$

with $X_0, X_1$ constructed from delayed data and $Q$ a design variable yielding the controller expression (Persis et al., 2019, Hu et al., 25 Aug 2024, Liu et al., 6 Jun 2025).

• Stabilization: The aim is to find $Q$ ensuring closed-loop spectral properties (Schur, Hurwitz, or contractive).
• LQR/Optimality: The LQR performance index is recast in terms of the embedded state, and data-derived variables encode the Riccati equation or BeLLMan equation terms (Persis et al., 2019, Xie et al., 28 Aug 2025, Lin et al., 23 Sep 2025).
• Regulation and Tracking: Internal model augmentation, realized either through dynamic extension or via auxiliary variables, is included in the data-driven formulation to tackle disturbance rejection and tracking (Liu et al., 2018, Hu et al., 25 Aug 2024, Liu et al., 6 Jun 2025).

(c) Observer and Filtering Construction

When only output is measured, a “model-free observer” may be constructed:

• An auxiliary state, e.g., $\zeta$ in (Lin et al., 23 Sep 2025), evolves according to an assigned polynomial or dynamics built from known or designed observer characteristics. The (unknown) mapping to the true state is captured by a matrix $M$.
• The output-feedback law is then $u = K M \zeta$ or similar, where $K$ is obtained via data-driven LQR or stabilization methods.

(d) Data-Efficient and Redundancy-Reduced Embeddings

Projection methods are used to reduce redundancy and improve numerical efficiency, especially in multi-output applications (see (Xie et al., 28 Aug 2025)), where the substitute state parameterization is projected onto the controllable (non-redundant) subspace to ensure minimal data demand and well-posedness.

(e) Iterative Optimization Algorithms

Policy-iteration (PI) and value-iteration (VI) algorithms are executed entirely in the data-embedded variable space. The framework guarantees quadratic or linear convergence (depending on the algorithm) and ensures, under mild conditions (such as controllability and data persistence of excitation), that the output-feedback solution is equivalent to the optimal state-feedback controller (Xie et al., 28 Aug 2025, Lin et al., 23 Sep 2025).

3. Performance Guarantees, Scalability, and Robustness

Performance and Feasibility

• Data-driven output-feedback controllers are rigorously shown to guarantee closed-loop exponential or practical stability under data richness and substitutability conditions (Persis et al., 2019, Xie et al., 28 Aug 2025, Lin et al., 23 Sep 2025).
• For the linear quadratic problem, the equivalence between the optimal state-feedback controller and the synthesized output-feedback controller is established under correct parameterization (Xie et al., 28 Aug 2025).
• The data requirements (data length and excitation order) are explicitly tied to the system's order and observability properties.

Scalability and Efficiency

• The projection-based parameterizations and ancillary system reduction (as in (Xie et al., 28 Aug 2025, Lin et al., 23 Sep 2025)) result in a lower-dimensional set of unknowns compared to traditional LS-based approaches, especially in high-dimensional or multi-output settings.
• The reduction from solving $O(n^2)$ or $O(n^2 + mn)$ unknowns to $O(n)$ or $O(n+m)$ directly impacts the computational and storage burden, rendering the approach practical for large systems.

Robustness and Handling of Practical Limitations

• When only a positive semi-definite solution exists for the underlying Riccati equation (e.g., in non-strictly controllable systems), the generalized PI/VI algorithms remain well-posed and converge (Lin et al., 23 Sep 2025).
• The output-feedback approach is robust to lack of explicit state observation and, in extensions, to measurement noise and modeling errors (when combined with robustification steps such as bootstrapped uncertainty or noise-informed filtering).
• Recursive feasibility and Lyapunov-based stability arguments are preserved in the presence of model uncertainty and even in stochastic settings, provided disturbances satisfy the required moment properties (Pan et al., 2022, Pan et al., 2022).

4. Representative Algorithms and Mathematical Formulations

A summary table below contrasts the main elements of recent data-driven output-feedback algorithms for linear systems:

Method	Substitute State/Observer	Controller Synthesis	Key Data Requirement
Trajectory-Based PI/VI (Xie et al., 28 Aug 2025)	Delay-embedded vector $\xi_k$; $x_k = M \xi_k$	Data-encoded PI/VI iteration; LQR via BeLLMan/data Q-function	Persistently excited input, $T \geq n$
Observer-Based PI/VI (Lin et al., 23 Sep 2025)	Model-free observer: auxiliary $\zeta$ with known dynamics, $x \approx M\zeta$	Similar PI/VI iteration with output-based embedding	Observer well-posedness, persistence, $M$ full (or via ancillary system)
Projection-Reduced Parameterization (Xie et al., 28 Aug 2025)	$\zeta_k = P v_k$ projects on controllable subspace	Policy iteration with reduced variables	Sufficient controllability, reduced data demand
Stochastic Lifting (Pan et al., 2022)	Data-embedded ARX/information state, PCE coefficients	Data-driven OCP with robustness constraints	Persistently excited data, disturbance moment info

Canonical forms and iterative updates, in the notation of (Xie et al., 28 Aug 2025) and (Lin et al., 23 Sep 2025), include:

• Substitute state update:

$$x_k = M \xi_k,\quad \xi_k = [u_{k-n}, ..., u_{k-1}, y_{k-n}, ..., y_{k-1}]^\top$$

• Output-feedback law:

$u_k = K v_k$

with $v_k$ the projected/non-redundant substitute state.

• BeLLMan equation update (Q-function-based PI iteration):

$$\Psi_0^\top \Theta \Psi_0 = Y_0^\top Q Y_0 + U_0^\top R U_0 + V_1^\top [I, K]^\top \Theta [I, K] V_1,$$

where all data matrices are constructed from input–output data and $K$ is updated greedily.

5. Applications and Impact

Data-driven output-feedback control design is especially valuable in scenarios characterized by:

• Unmeasurable or partially observed states where classic separation or observer-based methods are difficult or require prior identification.
• High-dimensional or multi-output systems with latent redundancy, for which the developed projection methods ensure feasibility and data efficiency (Xie et al., 28 Aug 2025).
• Situations where model identification is unreliable or expensive, and measurement data is more accessible.

Published results document application to multi-output LQR, robust regulation under model uncertainty, flight control, robotics, networked control, and PDE-governed systems (Wang et al., 2019, Liu et al., 6 Jun 2025, Harry et al., 10 Jun 2025).

A characteristic implication of these developments is that, provided persistently excited and sufficiently rich data is available, closed-loop performance and stability can be achieved with quantifiable data requirements, even in the absence of state measurements and explicit models. Structural modifications (e.g., output-based ancillary systems) reduce computational complexity and relax feasibility requirements compared to earlier approaches (Lin et al., 23 Sep 2025).

6. Limitations and Ongoing Research Directions

While the data-driven output-feedback frameworks have substantially improved the practical and computational feasibility of model-free optimal control, some limitations and open directions remain:

• Sensitivity to noisy or corrupted data, particularly for ill-conditioned or marginally observable systems, is still an active research topic; robust and regularized variants are being explored.
• Extensions to nonlinear, time-varying, descriptor, or networked systems are under active development, leveraging ideas from passivity and kernelized learning (Liu et al., 6 Jun 2025, Harry et al., 10 Jun 2025, Pan et al., 2022).
• The balance between data demand (length, excitation order) and closed-loop performance remains a core issue, with a need for adaptive or data-informativity-guided algorithms.
• For stochastic environments or non-Gaussian disturbances, lifting and robustification via polynomial chaos or distributionally robust optimization continue to be developed (Pan et al., 2022, Pan et al., 2022).

7. Summary Table: Algorithmic Features

Aspect	PI/VI Approach (Xie et al., 28 Aug 2025)	Ancillary System-based (Lin et al., 23 Sep 2025)	Observer-Based/Projection
Measurement types	Outputs, Inputs	Outputs, Inputs	Outputs, Inputs
State estimation	Substitute/embedded state	Model-free observer (auxiliary $\zeta$)	Filter/projection
Data requirement	Persistence, $T \ge n$	Less stringent with new method	Reduced via projection
Computational scaling	Scalable via projection/ancillary	Significantly fewer unknowns	Efficient for MO cases
Riccati solution property	Handles semidefinite solutions	Handles semi-definite, not just definite	Both
Robustness	Rank and data-driven feasibility	Eliminates unknown rank restriction	Robust to redundancy

Key contributions referenced are found in (Persis et al., 2019, Goyal et al., 2021, Schmitz et al., 2022, Pan et al., 2022, Pan et al., 2022, Hu et al., 25 Aug 2024, Liu et al., 6 Jun 2025, Harry et al., 10 Jun 2025, Xie et al., 28 Aug 2025), and (Lin et al., 23 Sep 2025). These works collectively delineate the mathematical, algorithmic, and practical foundations of data-driven output-feedback controller design for both linear and nonlinear systems, establishing a comprehensive paradigm for output-feedback synthesis from input–output data.