Gain-Scheduling Data-Enabled Predictive Control for Nonlinear Systems with Linearized Operating Regions

1. Problem Statement

Data-Enabled Predictive Control (DeePC) is a model-free control strategy that uses measured input-output trajectories to predict future system behavior, relying on Willems' Fundamental Lemma. While effective for Linear Time-Invariant (LTI) systems, standard DeePC struggles with strongly nonlinear systems.

• The Limitation: A single global Hankel matrix constructed from data covering the entire operating range often fails to capture local nonlinear dynamics accurately. This leads to poor prediction accuracy, degraded closed-loop performance, and high computational costs due to the need for massive data matrices.
• The Challenge: Existing nonlinear extensions (e.g., LPV-DPC) often require complex model embeddings or precise scheduling signals, while other data-driven approaches (e.g., Select-DeePC) lack systematic region handling. There is a need for a framework that combines the simplicity of gain-scheduling (using local linear models) with the data-driven nature of DeePC, while ensuring smooth transitions between regions to avoid control chattering.

2. Methodology: Gain-Scheduled DeePC (GS-DeePC)

The authors propose GS-DeePC, a framework that partitions the operating range of a measurable scheduling variable ($\rho$) into multiple local regions. Instead of one global predictor, the controller switches between multiple locally linear data representations.

A. Data Collection and Regional Partitioning

1. Scheduling Variable: A measurable variable $\rho(k)$ (e.g., output position) is selected to parameterize the dominant nonlinearity.
2. Partitioning: The range of $\rho$ is divided into $n$ uniform (or non-uniform) regions $\{M_i\}$.
3. Data Extraction: Input-output trajectories are segmented based on which region $\rho(k)$ occupies. Only subsequences long enough to satisfy the prediction horizon ($T_{ini} + N$) are retained.
4. Mosaic-Hankel Matrices: For each region, a Hankel matrix is constructed by concatenating valid subsequences. To ensure uniformity, all regional matrices are forced to have the same number of columns.

B. Composite Regions and Switching Logic

A critical innovation is the introduction of Composite Regions to handle the discontinuity issues inherent in switching between discrete linear models.

• Construction: A composite region $C_i$ is formed by merging two neighboring linear regions ($M_i \cup M_{i+1}$). The corresponding Hankel matrices are concatenated.
• Hysteresis Effect: Because composite regions overlap, the controller has a "buffer" zone. When the scheduling variable crosses a boundary, the controller does not immediately switch matrices if the state remains within the current composite region's bounds.
• Switching Strategies: The paper compares three strategies:
  1. Simple Switching: Immediate switch based on current $\rho$ (prone to chattering).
  2. Dwell Time: Enforces a minimum time before switching back (reduces but does not eliminate chattering).
  3. Composite Region Switching: Uses the overlapping data sets to ensure smooth transitions. The paper demonstrates this is the most robust method, effectively suppressing chattering without complex dwell-time tuning.

C. Optimization Formulation

The online optimization problem retains the standard DeePC structure (minimizing tracking error and control effort subject to constraints). The only change is the selection of the active Hankel matrix ($H_i$ or $H_{C,i}$) based on the current operating region. This preserves the computational structure of the solver while adapting the data source.

3. Key Contributions

1. Systematic Framework: A structured methodology for collecting data, partitioning operating regions, and constructing regional/composite Hankel matrices for nonlinear systems.
2. Composite Regions: The introduction of overlapping composite regions that merge neighboring data sets. This acts as a data-level hysteresis mechanism, ensuring smooth transitions and eliminating the "chattering" often seen in gain-scheduled systems.
3. Computational Efficiency: By using short, locally informative data sequences rather than one massive global dataset, GS-DeePC reduces the size of the Hankel matrices, leading to lower computational complexity compared to standard DeePC applied to nonlinear systems.
4. Empirical Validation: Extensive experiments on a nonlinear DC-motor with an unbalanced disc, including sensitivity analyses on the number of regions and comparisons with state-of-the-art benchmarks.

4. Experimental Results

The framework was tested on a nonlinear DC-motor with an unbalanced disc, a system with significant sinusoidal nonlinearity.

• Performance vs. Standard DeePC:
  • Standard DeePC (using a single global matrix) exhibited high steady-state tracking error ($RMSE_{SS} \approx 0.057$) because it could not accurately model the local nonlinearities.
  • GS-DeePC significantly reduced steady-state error. With 23 composite regions, it achieved $RMSE_{SS} \approx 0.003$.
• Performance vs. LPV-DPC (Benchmark):
  • LPV-DPC (Linear Parameter-Varying DPC) is a sophisticated benchmark that embeds the scheduling variable directly into the Hankel matrix.
  • GS-DeePC achieved performance nearly identical to LPV-DPC ($RMSE_{SS} \approx 0.002$ for LPV-DPC vs. $0.003$ for GS-DeePC) but with a simpler formulation (fixed regional matrices vs. time-varying LPV embedding) and lower modeling effort.
• Chattering Suppression:
  • Switching strategies using dwell times reduced chattering but did not eliminate output fluctuations.
  • Composite Region switching completely eliminated observable chattering, resulting in smooth, stable closed-loop trajectories even when the reference signal crossed region boundaries.
• Sensitivity Analysis:
  • Increasing the number of regions ($n_C$) improved performance up to a point (plateauing around $n_C > 25$).
  • There is a trade-off: too many regions require longer data collection times per region to satisfy persistence of excitation, while too few regions fail to capture local dynamics.

5. Significance and Conclusion

This paper bridges the gap between classical gain-scheduling control and modern data-driven predictive control.

• Practicality: It offers a "model-free" alternative to gain-scheduling that does not require deriving local linear models analytically; instead, it learns them directly from data.
• Robustness: The composite region approach solves the critical issue of switching instability (chattering) in a data-driven context, making the method viable for real-world applications.
• Efficiency: It demonstrates that decomposing a nonlinear problem into smaller, locally linear data-driven sub-problems is computationally more efficient and more accurate than attempting to solve it with a single global model.

The authors conclude that GS-DeePC provides a coherent, robust, and computationally efficient framework for controlling nonlinear systems, achieving performance comparable to complex LPV-based methods while remaining fully data-driven. Future work suggests automating the region partitioning and hyperparameter tuning.