Closed-Loop Synthesis Framework

• Closed-loop synthesis frameworks are methodological systems that incorporate output feedback to mitigate bias and ensure robust control performance.
• They leverage instrumental variable augmentation and multi-step predictor synthesis to correct for feedback-induced bias in data-driven control settings.
• Empirical evaluations show improved tracking cost, noise sensitivity, and sample efficiency, outperforming traditional open-loop methods in real-time applications.

Closed-Loop Synthesis Framework

A closed-loop synthesis framework is any methodological system for constructing control or decision systems in which feedback from outputs is systematically used to generate new actions, predictions, or designs, with specific mechanisms to enforce performance, safety, or consistency objectives in the presence of feedback-induced bias or uncertainty. Modern closed-loop synthesis frameworks span data-driven control, system identification, planning, layout synthesis, and simulation-based design. These frameworks are essential for achieving provably consistent, robust, and efficient closed-loop behavior in scenarios where open-loop (one-way) methods are inadequate.

1. Theoretical Foundations and Problem Setup

Closed-loop synthesis frameworks are distinguished by their explicit treatment of feedback-induced bias and dynamic correlation between control inputs and noise/disturbance signals. A canonical example is given by CL-DeePC, which extends Data-enabled Predictive Control (DeePC) to closed-loop contexts, accounting for the issue that, under feedback, control inputs $u_k$ are correlated with past measurement noise $e_j$ $(j<k)$. This "closed-loop identification bias" impairs standard regressors trained on closed-loop data. The system model is typically characterized in generalized innovation form: $$\begin{aligned} x_{k+1} &= A x_k + B u_k + K e_k \ y_k &= C x_k + D u_k + e_k \end{aligned}$$ where $e_k$ is zero-mean innovation noise, and control objectives are posed as finite-horizon receding-obstacle optimal control with quadratic regularization and increment penalties.

Closed-loop synthesis replaces unavailable parametric models with data-driven predictors, subject to the constraint that the collected data—segments from the closed-loop system—are persistently exciting.

2. Instrumental Variable Augmentation

The core innovation in rigorous closed-loop synthesis is the adoption of instrumental variables (IVs) constructed from measured data that are uncorrelated with closed-loop innovation noise. To eliminate feedback-induced bias, CL-DeePC introduces an IV matrix $Z \in \mathbb{R}^{n_z \times N}$ satisfying: $$\begin{aligned} \text{rank}\left(\lim_{N\to\infty}\mathbb{E}[\Psi Z^T]\right) &= (p+1)r + pl \ \lim_{N\to\infty}\mathbb{E}[E Z^T] &= 0 \end{aligned}$$ where $E$ collects innovation noise blocks, and $\Psi$ is the Hankel block of past/future I/O. A practical choice (when the closed-loop controller has no direct feedthrough) is to set $Z$ as the one-step Hankel $\Psi_{i,1,N}$.

The predictor parameter $G^{IV}$ is solved from an augmented equation: $$[ \Psi_{i,1,N}\;\;Y_{i+p,1,N} ] Z^T G^{IV} = [\overline{\Psi}_{k,1,f};\, \hat{y}_{k+p,1,f}^{IV}]$$ ensuring asymptotic unbiasedness under closed-loop data as shown in Theorem 1: $$\mathbb{E}[\hat{y}^{IV} - y_{\text{true}}] \rightarrow 0\quad \text{as}\;N,p\rightarrow\infty$$ This mechanism, unique to closed-loop synthesis frameworks, systematically corrects closed-loop bias by regression on IVs.

3. Multi-Step and Sequential Predictor Synthesis

Closed-loop synthesis is extended sequentially for multi-step-ahead prediction. Multi-step prediction is realized by recursively applying single-step IV-augmented predictors or constructing an $s$-step predictor applied $q$ times over horizon $f = s \cdot q$. The generalized predictor equation is: $$[ \Psi_{i,s,N}\;\;Y_{i+p,s,N} ]Z^T G^{IV} = [\overline{\Psi}_{k,s,q}^m; \hat{y}_{k+p,s,q}^{IV,m}]$$ Theorem 2 guarantees consistency of multi-step IV-based outputs. The final receding-horizon optimization constraint replaces the model-based evolution with a block-Toeplitz data-driven predictor: $$\hat{y} = \mathcal{L}_u[u_{\hat{k}-p:\hat{k}-1}] + \mathcal{L}_y[y_{\hat{k}-p:\hat{k}-1}] + \mathcal{G}_u[u_{\hat{k}:\hat{k}+f-1}]$$ This direct-data construction is critical for closed-loop policy design in predictive control.

4. Equivalence to Closed-Loop Subspace Predictive Control

A salient feature of CL-DeePC is its mathematical equivalence (in the noise-consistent limit) to Closed-Loop Subspace Predictive Control (CL-SPC). CL-SPC provides a least-squares IV estimate of one-step Markov parameters: $$\widehat{L}_1 = Y_{i+p,1,N} Z^T (\Psi_{i,1,N} Z^T)^{+}$$ From these estimates one recovers $\widehat{\Gamma}_f u$, $\widehat{\mathcal{T}}_f^u$, $\widehat{\Gamma}_f y$ and the corresponding block-Toeplitz predictors. By matching block structures and parameter correspondences $$(\beta_j = \widehat{C A^{p-j}B}, \theta_j = \widehat{C A^{p-j}K})$$, CL-DeePC and CL-SPC yield identical predictor matrices, optimal control problems, and closed-loop implementations.

5. Computational Implementation and Efficiency

Efficient closed-loop synthesis frameworks avoid explicit optimization over large regression variables. For CL-DeePC, if $\Psi Z^T$ is square/invertible, the IV parameter is eliminated: $G^{IV} = (\Psi Z^T)^{-1} \overline{\Psi}$ so that single-step prediction reduces to: $$\hat{y} = \Sigma_{yz} \Sigma_{\psi z}^{-1} \overline{\psi}$$ By decomposing into block-rows, one exploits strictly lower-triangular Toeplitz structure in sequential recursion: $$\begin{aligned} \alpha_0 &= \tilde{\alpha}_0 \ \alpha_j &= \tilde{\alpha}_j + \sum_{m=1}^p \theta_m \alpha_{j+m-p-1},\quad j \geq 1 \end{aligned}$$ Matrices $\mathcal{L}_u, \mathcal{L}_y, \mathcal{G}_u$ are read off from the recursion without full storage of $G^{IV}$. This efficient realization is essential for deployment in real-time or resource-limited closed-loop predictive control applications.

6. Performance Metrics and Empirical Evaluation

Closed-loop synthesis frameworks directly rectify the limitations of model-free open-loop approaches, especially under closed-loop identification bias. Performance benchmarking (e.g., on weakly stable 5th-order plants) demonstrates:

• 48% lower sensitivity to noise-induced reference tracking performance degradation versus DeePC
• 49% lower tracking cost with $p=f=100$
• Superior sample efficiency (near-oracle performance with fewer samples)

CL-DeePC eliminates persistent bias in learned block evolution matrices (e.g., $\mathcal{T}_f^u$) that is otherwise observed in DeePC trained on closed-loop data. Causality and block-Toeplitz structure guarantee strict robustness and feasibility under sequential feedback.

7. Current Impact and Implications

CL-DeePC typifies the new generation of closed-loop synthesis frameworks that unify consistency, robustness, sample efficiency, and exact data-driven implementation. These designs—underpinned by IV augmentation, sequential block-Toeplitz structure, and equivalence to subspace control—advance the capability for rigorous closed-loop predictive control in noise-correlated, nonparametric settings, extending to broader domains such as simulation-based planning, integrated system identification, and hybrid learning-control architectures (Dinkla et al., 2024). Their adoption provides a critical advance for reliable, unbiased, and theoretically supported policies in modern, data-rich feedback systems.