Stability of data-driven Koopman MPC with terminal conditions

1. Problem Statement

The paper addresses the challenge of designing Model Predictive Control (MPC) for unknown nonlinear dynamical systems using data-driven surrogate models.

• The Core Issue: Standard MPC stability proofs assume the model perfectly describes the plant. However, data-driven models (like those based on the Koopman operator) inherently contain approximation errors.
• The Gap: In the presence of model-plant mismatch, standard MPC formulations often fail to guarantee recursive feasibility (the ability to find a valid control sequence at every time step) and asymptotic stability (convergence to the equilibrium point).
• Specific Goal: The authors aim to derive rigorous conditions under which an MPC controller, utilizing a data-driven surrogate model with terminal conditions (terminal cost and terminal set), can asymptotically stabilize the true unknown system despite approximation errors. They specifically focus on ensuring the error bound is proportional to the state and input norms.

2. Methodology

A. System Setup and Assumptions

• Plant: A discrete-time nonlinear system $x^+ = f(x, u)$ with unknown dynamics, where the origin is a controlled equilibrium.
• Surrogate Model: The controller uses a data-driven approximation $x^+ = f^\varepsilon(x, u)$, where $\varepsilon$ represents the approximation accuracy.
• Key Assumption 1 (Error Bounds): The paper assumes the existence of proportional and uniform error bounds:
  $$\|f(x, u) - f^\varepsilon(x, u)\| \leq \min\{c_x^\varepsilon \|x\| + c_u^\varepsilon \|u\|, \eta^\varepsilon\}$$
  Crucially, the error must vanish as the state and input approach zero (proportionality), and the constants must converge to zero as the data density increases ($\varepsilon \to 0$).
• Assumption 2 (Terminal Conditions): The terminal cost $V_f^\varepsilon$ and terminal set $X_f^\varepsilon$ are designed based on the surrogate model to satisfy Lyapunov decrease conditions within the terminal set.

B. MPC Formulation with Constraint Tightening

To ensure recursive feasibility despite model errors, the authors incorporate constraint tightening:

• The state constraints in the optimization problem are tightened by a margin that grows with the prediction horizon $k$.
• The tightened constraint set at step $k$ is defined as $S \ominus B_{\bar{c}(k)\eta^\varepsilon}$, where $\bar{c}(k)$ depends on the Lipschitz constant of the surrogate model and the uniform error bound $\eta^\varepsilon$.
• This ensures that even if the real system deviates from the predicted trajectory due to model error, the actual state remains within the original feasible set $S$.

C. Stability Analysis

The authors prove Theorem 1, establishing that:

1. Recursive Feasibility: If the initial state is feasible, the optimization problem remains feasible for all future time steps.
2. Asymptotic Stability: The origin is asymptotically stable for the closed-loop system.

• Proof Logic: The proof uses the optimal value function $V_N^\varepsilon$ as a Lyapunov candidate. It demonstrates that the decrease in the value function is guaranteed because the proportional error bound ensures that the prediction error vanishes as the system approaches the origin. This allows the terminal cost to dominate the error terms near the equilibrium.

D. Data-Driven Implementation: Kernel EDMD (kEDMD)

The paper connects the theoretical stability conditions to Kernel Extended Dynamic Mode Decomposition (kEDMD):

• Koopman Operator: The method lifts the nonlinear system into a higher-dimensional (potentially infinite) linear space using the Koopman operator.
• PI-kEDMD: The authors utilize a specific variant called Physics-Informed kEDMD (PI-kEDMD).
  • Standard kEDMD is extended to control-affine systems.
  • Crucial Modification: To satisfy the proportional error bound requirement, the regression problem at the origin ($x=0$) is modified to enforce $f^\varepsilon(0, 0) = 0$.
• Theoretical Guarantee: Theorem 2 (citing [26]) confirms that PI-kEDMD models satisfy the required proportional error bounds ($c_x^\varepsilon \|x\| + c_u^\varepsilon \|u\|$) provided the data fill distance and cluster radius are sufficiently small.

3. Key Contributions

1. Rigorous Stability Proof for Terminal MPC: Unlike previous works that often relied on practical stability (convergence to a neighborhood) or required infinite horizons, this paper proves asymptotic stability for MPC with finite horizons and terminal conditions in the presence of model errors.
2. Proportional Error Bound Requirement: The paper identifies and proves that proportionality of the error bound (error $\propto$ state/input magnitude) is the critical condition for asymptotic stability, distinguishing it from uniform bounds which only guarantee practical stability.
3. Integration with kEDMD: The authors demonstrate how PI-kEDMD naturally satisfies these proportional error bounds, providing a constructive framework to generate stable data-driven MPC controllers.
4. Constraint Tightening Strategy: They provide a specific method for constraint tightening based on the Lipschitz constant and uniform error bound to guarantee recursive feasibility.

4. Results (Numerical Case Study)

The framework was tested on the Euler discretization of a controlled van der Pol oscillator:

• Setup: Two scenarios were compared using different numbers of virtual observation points ($d=352$ and $d=1327$).
• Comparison:
  • Standard kEDMD: Without the specific modification to enforce the proportional bound at the origin, the system exhibited practical asymptotic stability. The error stagnated at a non-zero constant value.
  • PI-kEDMD: With the modified regression enforcing $f^\varepsilon(0,0)=0$, the system achieved true asymptotic stability, driving the state error to zero.
• Performance: The PI-kEDMD approach allowed for successful stabilization with short prediction horizons ($N=4$ and $N=9$), demonstrating the effectiveness of the terminal condition design.

5. Significance and Limitations

Significance:

• This work bridges the gap between theoretical control stability and modern data-driven learning. It provides a mathematically rigorous path to using "black-box" or "grey-box" Koopman models in safety-critical control applications where asymptotic convergence is required.
• It validates that with the right data processing (PI-kEDMD) and control design (terminal conditions + tightening), data-driven MPC can be as robust as model-based MPC.

Limitations & Future Work:

• Conservative Constraint Tightening: The current method uses a constraint tightening based on the Lipschitz constant, which grows unbounded with the prediction horizon. This can render the optimization problem infeasible for long horizons or high-error models.
• Future Direction: The authors suggest replacing the Lipschitz-based tightening with more sophisticated techniques, such as those based on incremental stabilizability (as seen in [22]), to reduce conservativeness and allow for longer prediction horizons.