---
title: Data-Driven Output-Feedback Control
url: https://www.emergentmind.com/topics/data-driven-output-feedback-control-law
type: topic
---

# Data-Driven Output-Feedback Control

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 [1903.06842], [2202.07930], [2211.17074], [2508.20748].
- **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 [1903.06842], [2107.08086], [2508.20748].
- **Parametrization via Data:** Representing closed-loop or controller dynamics with data-driven variable substitutions and parameterizations, often constructed from Hankel matrices or delay vectors [1903.06842], [2202.07930], [2508.20748].
- **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 [1903.06842], [2408.13914], [2502.17688], [2508.20748].

## 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 [2508.20748], [2509.18819]. 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 [1903.06842], [2408.13914], [2506.06079].

- **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 [1903.06842], [2508.20748], [2509.18819].
- **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 [1803.01694], [2408.13914], [2506.06079].

### (c) Observer and Filtering Construction
When only output is measured, a “model-free observer” may be constructed:

- An auxiliary state, e.g., $\zeta$ in [2509.18819], 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 [2508.20748]), 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 [2508.20748], [2509.18819].

## 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 [1903.06842], [2508.20748], [2509.18819].
- For the linear quadratic problem, the equivalence between the optimal state-feedback controller and the synthesized output-feedback controller is established under correct parameterization [2508.20748].
- 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 [2508.20748], [2509.18819]) 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 [2509.18819].
- 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 [2211.17074], [2212.07661].

## 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 [2508.20748] | 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 [2509.18819]    | 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 [2508.20748] | $\zeta_k = P v_k$ projects on controllable subspace | Policy iteration with reduced variables | Sufficient controllability, reduced data demand |
| Stochastic Lifting [2211.17074]   | 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 [2508.20748] and [2509.18819], 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 [2508.20748].
- 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 [1902.04480], [2506.06079], [2506.09273].

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 [2509.18819].

## 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 [2506.06079], [2506.09273], [2211.17074].
- 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 [2211.17074], [2212.07661].

## 7. Summary Table: Algorithmic Features

| Aspect                    | PI/VI Approach [2508.20748]         | Ancillary System-based [2509.18819] | 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     |

## References

Key contributions referenced are found in [1903.06842], [2107.08086], [2202.07930], [2211.17074], [2212.07661], [2408.13914], [2506.06079], [2506.09273], [2508.20748], and [2509.18819]. 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.

Source: https://www.emergentmind.com/topics/data-driven-output-feedback-control-law