Technical Summary: Choose Wisely: Data-driven Predictive Control for Nonlinear Systems Using Online Data Selection

Problem Statement

The control community seeks to manage nonlinear systems where data is available but explicit system models are not. While Data-driven Predictive Control (DPC) methods, such as Data-enabled Predictive Control (DeePC), offer a "model-free" alternative to traditional Model Predictive Control (MPC) by bypassing system identification, standard DeePC relies on the Linear Time-Invariant (LTI) assumption. This assumption is severely violated in generic nonlinear systems, leading to arbitrary performance loss. Existing nonlinear extensions of DeePC face significant trade-offs:

• Specific Nonlinear Classes: Formulations tailored to specific nonlinear structures impose strong priors that may not exist for arbitrary systems.
• Time-Windowed Approaches: Methods using sliding windows of recent data are practical only for systems with benign nonlinearities and high signal-to-noise ratios.
• Kernelized/Lifted Approaches: Reformulating predictors using nonlinear kernels or Koopman operators often results in high-dimensional, non-convex optimization problems, increasing online computational complexity and leading to suboptimal outcomes.
• Sampling-Based MPC: While computationally cheap and requiring only zero-order information, these methods lack structured ways to incorporate explicit output constraints.

The core challenge is to develop a direct, data-driven predictive control method for nonlinear systems that maintains the convexity of the optimization problem (ensuring tractability and explicit constraint handling) while accurately capturing nonlinear dynamics without requiring an explicit model.

Methodology: Select-DPC

The paper proposes Select-DPC (Select-Data-driven Predictive Control), a method that implicitly linearizes system dynamics in "trajectory space" by selecting only the most relevant data at each decision moment.

Core Concept

Select-DPC operates in a receding-horizon fashion. Instead of utilizing a large, pre-collected dataset $D$ for every control step (which is computationally intractable and assumes global linearity), the algorithm iteratively solves a convex Quadratic Program (QP). At each iteration of the solver, it selects a subset of trajectories $\tilde{D} \subset D$ that are "relevant" to the current operating point.

Trajectory Space: Defined as a high-dimensional space where each dimension corresponds to an input or output over a discrete-time, finite-length trajectory. Linearization is achieved by selecting trajectories that are spatially "close" to the current open-loop solution estimate in this space.

Algorithmic Workflow

1. Initialization: The open-loop trajectory is warm-started (e.g., by replicating initial measurements).
2. Iterative Refinement:
   • Data Selection: A subset $\tilde{D}$ of size $N_{cols}$ is selected from the full dataset $D$ based on proximity to the current prediction $\tilde{\tau}$.
   • QP Solution: A standard DeePC optimization problem is solved using only $\tilde{D}$ to generate a new control action.
   • Convergence: The process repeats until a convergence criterion (tolerance or max iterations) is met.
3. Execution: The first control input is applied, and the horizon shifts.

Data Selection Strategies

The paper proposes and compares two methods for selecting the relevant subset $\tilde{D}$:

1. Norm-Based Selection: Computes the spatial distance (e.g., $L_1$ or $L_2$ norm) between the current open-loop trajectory and all trajectories in $D$. The $N_{cols}$ closest trajectories are selected.

   • Pros: Intuitive, efficient to implement, allows easy online updates.
   • Cons: Susceptible to the "curse of dimensionality" in high-dimensional trajectory spaces.

2. Manifold-Embedding-Based Selection: Utilizes manifold learning (specifically Isomap) to project high-dimensional trajectory data into a lower-dimensional embedding space that preserves geodesic distances.

   • Pros: Mitigates the curse of dimensionality; distances in the embedding space better reflect the intrinsic nonlinear structure of the data manifold.
   • Cons: Higher offline computational cost to compute the embedding (though this is a one-time cost); online embedding of the query trajectory is required.

Key Attributes

• Cost-Agnostic: The dataset generation is structurally independent of the control formulation. Cost functions and constraints can be modified a posteriori, allowing for zero-shot application to new tasks without new data collection.
• Convexity: By restricting the predictor to a locally linear subset of data, the resulting optimization remains a convex QP, unlike kernelized or lifted approaches which often become non-convex.
• Implicit Linearization: Similar to Sequential Quadratic Programming (SQP) in model-based MPC, Select-DPC performs iterative linearization, but does so in trajectory space using data rather than Jacobians of an explicit model.

Key Contributions

1. Introduction of Select-DPC: A direct data-driven predictive controller for nonlinear systems that achieves implicit linearization in trajectory space via online data selection, maintaining convex optimization.
2. Data Selection Methods: The proposal and comparison of two distinct selection mechanisms: a norm-based method and a manifold-embedding-based method (Isomap) to handle high-dimensional nonlinear data.
3. Empirical Validation: Demonstration of Select-DPC's superiority over standard DeePC and Time-Windowed DeePC across three benchmark nonlinear systems:
   • Planar rocket landing (open-loop unstable).
   • Planar robotic manipulator (Reacher).
   • Cart-pole inverted pendulum swing-up.

Results and Evaluation

Experiments were conducted on three simulators (Rocket, Reacher, Cart-Pole) comparing Select-DPC against Standard DeePC (using the full dataset) and Time-Windowed DeePC.

1. Reusable Rocket Landing

• Performance: Standard DeePC and Time-Windowed DeePC failed to stabilize the system or diverged. Select-DPC successfully stabilized the rocket at the desired setpoint.
• Data Sets: Tested on "IID Data" (predominantly linear behavior) and "Random Walk Data" (highly nonlinear excitation).
  • On IID data, Select-DPC reduced closed-loop cost by a factor of 1.9 compared to Standard DeePC.
  • On Random Walk data, Standard DeePC failed to stabilize, while Select-DPC succeeded.
• Prediction Accuracy: Both selection methods (Norm and Isomap) yielded lower cumulative residuals on holdout validation sets compared to using the full dataset or random sketching.
• Computation: Select-DPC reduced computational cost by an order of magnitude compared to Standard DeePC by solving smaller QPs, with solve times scaling quadratically with $N_{cols}$.

2. Planar Robotic Manipulator (Reacher)

• Performance: Select-DPC successfully tracked end-effector references, whereas Standard DeePC and Time-Windowed DeePC failed to converge.
• Constraints: The method demonstrated zero-shot generalizability by handling a priori unknown output constraints (e.g., avoiding specific regions) without structural changes to the controller or new data.
• Ablation: Random sketching failed completely in this environment, while selecting a specific subset size ($N_{cols}$) revealed a "sweet spot" for performance, balancing prediction accuracy and computational load.

3. Cart-Pole Inverted Pendulum Swing-Up

• Task: Transitioning from a stable downward equilibrium to an unstable upward equilibrium.
• Performance: Standard DeePC oscillated around the stable equilibrium and failed to swing up. Select-DPC successfully learned from suboptimal demonstrations in the dataset to perform the swing-up and stabilize the pendulum.
• Mechanism: The controller did not simply copy a single trajectory but linearly combined multiple trajectory segments at each solver iteration to construct the swing-up maneuver.

Significance and Claims

The paper claims that Select-DPC bridges the gap between the tractability of convex data-driven control and the performance requirements of nonlinear systems. Its significance lies in:

• Explicit Constraint Handling: Unlike sampling-based MPC or RL methods which struggle with structured output constraints, Select-DPC retains the ability to enforce hard constraints via convex optimization.
• Scalability: It scales favorably with large datasets by subsampling relevant data, avoiding the superlinear solve time of using the full dataset.
• Nonlinear Capability: It extends DPC to nonlinear domains without requiring explicit models, complex kernelizations, or non-convex optimization.
• Zero-Shot Adaptability: The decoupling of data collection from control design allows the same dataset to be reused for different tasks (e.g., constrained vs. unconstrained tracking) without retraining.

Limitations

The authors acknowledge several limitations:

• Computational Cost: The current implementation requires 200–400 ms per step, limiting its use in highly dynamic applications.
• Memory Footprint: Large prediction horizons and datasets may violate memory constraints.
• Hyperparameter Tuning: Achieving satisfactory performance requires tuning hyperparameters (e.g., $N_{cols}$, horizon lengths), and the paper notes that a large-scale empirical study of its capabilities across diverse tasks is subject to future work.