Soft projections for robust data-driven control

1. Problem Statement

The paper addresses the challenges in Direct Data-Driven Predictive Control (DDPC), specifically within the framework of Behavioral Systems Theory.

• Context: Traditional DDPC methods (like DeePC) formulate control as a projection of a reference trajectory onto the system's behavior (a subspace of valid trajectories). However, in practice, data is noisy, causing the data matrix to be full rank, which makes the exact projection ill-defined or trivial (identity).
• Current Limitations:
  • Existing methods rely on regularization (e.g., adding a penalty on the decision variable $g$) to handle noise. While effective, the theoretical interpretation of this regularization is often heuristic, especially when constraints are present.
  • Explicitly estimating the system's subspace (low-dimensional model) is sensitive to the choice of system order; if the estimated dimension differs from the true dimension, the error is maximal.
  • Incorporating online data updates into these schemes is computationally expensive and theoretically difficult when relying on explicit subspace estimation.

2. Methodology

The authors propose a shift from "learning a model" to "learning the solution map" directly using Soft Projections.

A. Soft Projections

Instead of projecting strictly onto the column space of the data matrix $H$ (which is impossible with noisy, full-rank data), the authors introduce a soft projector $\tilde{P}_H$.

• Definition: The soft projector is defined as $\tilde{P}_H = H(H^\top H + \delta I)^{-1}H^\top$, where $\delta > 0$ is a regularization parameter.
• Mechanism: This operator acts as a "denoising" filter. It scales the singular values of the data matrix $H$. Large singular values (signal) are preserved, while small singular values (noise) are suppressed.
• Connection to DeePC: In the unconstrained case, the standard DeePC solution is mathematically equivalent to applying a weighted soft projection to the reference trajectory. The authors generalize this concept to handle constraints and provide a rigorous theoretical foundation.

B. Theoretical Analysis (Approximation Error Bounds)

The paper derives a rigorous upper bound on the approximation error between the soft projector and the true behavioral projector ($P_B$).

• Bias-Variance Trade-off: The error bound $\gamma(\delta)$ consists of two terms:
  1. Variance: Proportional to the noise level ($\|E\|$).
  2. Bias: Proportional to the regularization parameter ($\delta$).
• Key Insight: The bound is independent of the true system order. This is a significant advantage over methods that explicitly estimate a low-dimensional subspace, where order selection errors lead to catastrophic failure (gap metric = 1).

C. Novel Control Formulations

The authors propose two new control formulations that explicitly utilize soft projections to enforce the behavior constraint $w \in \mathcal{B}_L$:

1. Formulation (9): Minimizes the tracking error plus a penalty term $\alpha \| (I - \tilde{P}_H)w \|^2$. This treats the constraint as a soft penalty.
2. Formulation (10): Minimizes the tracking error plus a penalty $\hat{\alpha} w^\top (I - \tilde{P}_H) w$.

• Comparison: Theoretical analysis and eigenvalue distribution show that Formulation (9) acts as a higher-order filter, separating signal from noise more effectively than Formulation (10) (and standard DeePC), leading to better robustness.

D. Online Adaptation (Rank-One Updates)

To handle streaming data, the authors derive efficient rank-one update formulas for the soft projector and the regularization parameter $\delta$.

• Instead of recomputing the inverse of a large matrix at every time step, the projector is updated recursively using the new data trajectory.
• The update rule for $\delta$ is tied to the trace of the data matrix ($\delta_t = \epsilon \cdot \text{tr}(H_t H_t^\top)$), ensuring the regularization adapts to the current data richness without manual tuning.

3. Key Contributions

1. Conceptual Shift: Reinterpreting regularized data-driven control as a soft projection problem, providing a clear geometric and theoretical interpretation that holds even with constraints.
2. Order-Independent Robustness: Proving that the approximation error bound of soft projections does not depend on the system order, avoiding the sensitivity issues of explicit subspace identification.
3. Novel Algorithms: Proposing two new control formulations (specifically Formulation 9) that generalize DeePC and offer superior noise filtering capabilities.
4. Efficient Online Learning: Developing computationally efficient recursive update formulas that allow the controller to adapt to streaming data in real-time with $O((qL)^2)$ complexity per step.

4. Results

The methods were validated on a double spring-mass-damper system (two rotating discs) under varying Signal-to-Noise Ratios (SNR: 10, 5, and 3).

• Performance: The proposed method (Formulation 9) consistently outperformed standard DeePC (Formulation 4).
• Metrics:
  • Mean Realized Cost: Formulation 9 achieved significantly lower costs, especially as noise increased (e.g., at SNR=3, cost was 6.87 vs. 9.43 for DeePC).
  • Prediction Error: The proposed method exhibited lower mean prediction error and lower variance.
• Robustness: The gap in performance between the proposed method and DeePC widened as the noise level increased, confirming the superior robustness of the soft projection approach.

5. Significance

This paper bridges the gap between heuristic regularization in data-driven control and rigorous mathematical theory.

• Theoretical Impact: It provides a unified view of regularization as soft projection, offering error bounds that are independent of system order—a major hurdle in behavioral control.
• Practical Impact: The proposed online update formulas make robust, data-driven control feasible for time-varying systems and streaming data applications, removing the need for computationally expensive re-identification or manual retuning of regularization parameters.
• Future Direction: It opens avenues for stochastic interpretations of behavioral projections and further generalizations for nonlinear systems.