Model-Free Dynamic Mode Adaptive Control for Data-Driven Control Synthesis

1. Problem Statement

The paper addresses the control of complex engineering systems (e.g., fluid flows, combustion processes, flexible structures) where:

• Mathematical models are unavailable or unsuitable: High-fidelity physics-based models are often too computationally intensive for real-time control, and first-principles models may not exist.
• High-dimensional sensing: Systems often have distributed sensors generating high-dimensional data, while control objectives are defined on lower-dimensional physical variables.
• Limitations of existing methods:
  • Classical adaptive control (e.g., MRAC) often assumes a known parametric model structure.
  • Optimization-based model-free methods (e.g., RCAC, RL) can be computationally burdensome, sensitive to tuning, or lack stability guarantees in continuous-time settings.
  • Many data-driven methods (e.g., standard DMD) are offline and do not integrate directly with real-time controller synthesis.

Goal: Develop a unified, model-free framework that performs online system identification and real-time controller synthesis directly from streaming measurement data, preserving physical interpretability without relying on batch data or complex optimization.

2. Methodology: Dynamic Mode Adaptive Control (DMAC)

The proposed DMAC framework operates in a recursive loop at each time step $k$, combining two main modules:

A. Dynamics Approximation (Online Identification)

Instead of using vectorized parameters or input-output models, DMAC estimates a local linear state-space representation directly in the measurement space.

• Model Form: It assumes a local linear approximation: $\xi_{k+1} = A_k \xi_k + B_k u_k$, where $\xi_k$ is the measured state and $u_k$ is the control input.
• Algorithm: It employs a Matrix Recursive Least Squares (RLS) algorithm with a forgetting factor ($\lambda$).
  • Unlike standard RLS which vectorizes parameters, DMAC updates the matrix $\Theta_k = [A_k \quad B_k]$ directly.
  • Cost Function: Minimizes an exponentially weighted regularized least-squares cost:
    $$J_k(\Theta) = \sum_{i=0}^k \lambda^{k-i} \|\xi_i - \Theta \phi_{i-1}\|_2^2 + \lambda^k \text{tr}(\Theta^T R_\Theta \Theta)$$
  • Benefits: This structure-preserving approach avoids the dimensional explosion associated with vectorization and yields state-space matrices ($A, B$) directly, maintaining physical interpretability.

B. Controller Synthesis

Once the local dynamics ($A_k, B_k$) are estimated, a controller is synthesized immediately.

• Control Law: A full-state feedback controller with integral action is used for reference tracking:
  $$u_k = K_{\xi, k} \xi_k + K_{q, k} q_k + v_k$$
  • $K_{\xi, k}$ and $K_{q, k}$ are time-varying gains computed via Linear Quadratic Regulator (LQR) design based on the current estimates $(A_k, B_k)$.
  • $q_k$ is an integrator state for error accumulation.
  • $v_k$ is an excitation signal (i.i.d. zero-mean noise) added to ensure persistence of excitation (PE) in the regressor, which is critical for the convergence of the RLS estimator.

3. Key Contributions

1. Unified Architecture: Introduces DMAC, a framework that seamlessly integrates dynamic mode approximation (data-driven modeling) with adaptive control synthesis, enabling real-time controller design from streaming data.
2. Matrix-Valued RLS: Proposes a recursive identification algorithm that estimates low-order state-space models directly in the measurement space using a matrix-valued RLS formulation. This avoids vectorization, lifting, and input-output reconstruction, reducing computational complexity and preserving physical structure.
3. Theoretical Guarantees:
   • Proves convergence of the recursive identification algorithm (estimation error converges to zero or is bounded).
   • Establishes Uniform Ultimate Boundedness (UUB) of the closed-loop system state, proving that the system remains stable despite the excitation signal required for identification.
4. Robustness: Demonstrates that the method works for linear, nonlinear, and distributed-parameter systems without requiring a priori knowledge of the system model.

4. Results and Numerical Examples

The authors validated DMAC on three systems of increasing complexity:

• Unstable Linear System:
  • Result: Successfully stabilized an open-loop unstable system.
  • Observation: The state converged to a bounded neighborhood of the origin. The size of this neighborhood was proportional to the magnitude of the excitation signal ($v_k$), confirming theoretical bounds. Parameter estimates converged to true values.
• Van der Pol Oscillator (Nonlinear):
  • Result: Achieved reference tracking for a nonlinear oscillatory system.
  • Sensitivity: The controller maintained stable tracking performance across wide variations in hyperparameters ($\lambda$, regularization, LQR weights) and system parameters (nonlinear damping $\mu$).
• Burgers' Equation (Distributed-Parameter PDE):
  • Result: Controlled a 1D fluid dynamics system governed by a nonlinear PDE using only sparse sensor measurements (7 nodes out of 100).
  • Significance: Demonstrated the ability to handle high-dimensional, distributed systems by identifying a low-order local linear model directly from sparse measurements.
  • Robustness: Showed robustness to variations in the viscosity coefficient ($\nu$).

5. Significance and Impact

• Bridging the Gap: DMAC bridges the gap between data-driven modeling (DMD) and adaptive control, moving from offline analysis to online, real-time control synthesis.
• Physical Interpretability: By identifying state-space matrices directly rather than abstract input-output maps, the resulting controller retains a clear physical meaning, which is crucial for engineering applications.
• Computational Efficiency: The matrix-valued RLS approach is computationally efficient, making it suitable for systems with high-dimensional sensing but low-dimensional control objectives.
• Theoretical Rigor: Unlike many learning-based control methods that rely on heuristics, DMAC provides rigorous Lyapunov-based proofs for stability and boundedness.
• Future Directions: The authors identify future work in reducing the reliance on persistent excitation (to minimize performance degradation from noise) and enforcing structural constraints (e.g., stabilizability) during the identification phase.

In summary, this paper presents a robust, theoretically grounded, and computationally efficient method for controlling complex, unknown dynamical systems using only measured data, offering a viable alternative to traditional model-based control and heavy optimization-based learning methods.