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Robust Tube-based Model Predictive Control with Koopman Operators--Extended Version
Xinglong Zhang, Wei Pan, Riccardo Scattolini, Shuyou Yu, Xin Xu
TL;DR
Koopman MPC offers computationally lighter linear optimization for nonlinear systems, but its robustness under approximation errors and disturbances remains unresolved. The paper proposes r-KMPC, combining nominal MPC based on a lifted Koopman model with nonlinear feedback, and proves robust closed-loop properties under stated assumptions.
Problem
The robustness of Koopman MPC under modeling approximation errors and exogenous disturbances, including constraint satisfaction, remains unresolved.
Method
r-KMPC combines online nominal linear MPC using a lifted Koopman model with an offline nonlinear static state-feedback policy.
Results
The paper proves recursive feasibility, closed-loop robustness, and asymptotic stability without exogenous disturbance under stated assumptions, with simulations verifying effectiveness.
Takeaways & Limitations
The approach permits robust MPC design with a limited-order Koopman model without assuming convergence of the approximated Koopman operator.
Takeaways & Limitations
The Koopman predictor inevitably contains modeling errors from non-orthonormal observables, disturbance-estimation errors, and regularization.
Abstract
from arXiv · showhide
Koopman operators are of infinite dimension and capture the characteristics of nonlinear dynamics in a lifted global linear manner. The finite data-driven approximation of Koopman operators results in a class of linear predictors, useful for formulating linear model predictive control (MPC) of nonlinear dynamical systems with reduced computational complexity. However, the robustness of the closed-loop Koopman MPC under modeling approximation errors and possible exogenous disturbances is still a crucial issue to be resolved. Aiming at the above problem, this paper presents a robust tube-based MPC solution with Koopman operators, i.e., r-KMPC, for nonlinear discrete-time dynamical systems with additive disturbances. The proposed controller is composed of a nominal MPC using a lifted Koopman model and an off-line nonlinear feedback policy. The proposed approach does not assume the convergence of the approximated Koopman operator, which allows using a Koopman model with a limited order for controller design. Fundamental properties, e.g., stabilizability, observability, of the Koopman model are derived under standard assumptions with which, the closed-loop robustness and nominal point-wise convergence are proven. Simulated examples are illustrated to verify the effectiveness of the proposed approach.
1 Introduction
MPC needs prediction models, but nonlinear formulations can be computationally intensive. Koopman-based finite-dimensional predictors offer a linear MPC route for nonlinear dynamics while retaining a broad operation range.
- Data-driven models are presented as a promising direction for constructing effective MPC prediction models.
- Nonlinear MPC can produce nonlinear, nonconvex optimization problems that are computationally intensive for highly nonlinear systems.
- Koopman operators represent nonlinear dynamics through linear evolution in a lifted observable space, despite being infinite-dimensional.
- Finite-dimensional Koopman truncations can form linear predictors for designing MPC of nonlinear systems.
2 Control problem and preliminaries
The paper formulates nonlinear systems with bounded additive disturbances and develops a data-driven Koopman predictor for linear MPC. The preliminary model exposes both the computational benefit of lifting and the unresolved robustness impact of approximation errors.
- 2.1 Control problem: The control problem concerns nonlinear discrete-time systems with bounded additive disturbances, state and control constraints, and quadratic infinite-horizon cost.
- 2.2 Preliminary Koopman MPC: Koopman theory lifts nonlinear dynamics into linear evolution of observable functions, then extends the construction to controlled systems using an augmented state containing input and disturbance sequences.
- 2.2 Preliminary Koopman MPC: EDMD computes a finite-dimensional Koopman approximation from input-state data, with regularization and a separate linear map recovering the original state from observables.
- 2.2 Preliminary Koopman MPC: The setup assumes sampled data satisfy a probability or ergodicity condition for constructing the data-driven Koopman model.
- 2.2 Preliminary Koopman MPC: The lifted predictor uses an abstract observable state, while the original state is obtained as an output through a mapping matrix and constrained in the MPC problem.
- 2.2 Preliminary Koopman MPC: Finite Koopman modeling can reduce MPC computation, but approximation and disturbance-related errors leave constraint satisfaction and closed-loop robustness unresolved.
3 Robust Koopman MPC
The paper develops r-KMPC by modeling Koopman approximation errors and disturbances as bounded uncertainties, then combines nominal linear MPC with nonlinear error-feedback. Under stated assumptions, it establishes recursive feasibility, closed-loop robustness, and point-wise asymptotic convergence without requiring Koopman-operator convergence.
- Robust Koopman model: r-KMPC represents Koopman-model approximation errors and disturbances through bounded uncertainty sets in the lifted dynamics.Boundedness, rather than convergence, is sufficient for the theoretical guarantees and permits lower-order Koopman models.
- Koopman-model properties: The lifted Koopman model is stabilizable when the original nonlinear model is stabilizable, under the stated assumptions.In the unperturbed case, the corresponding closed-loop lifted and original states can converge point-wise to the origin.
- r-KMPC design: The proposed controller applies nominal MPC together with nonlinear state-error feedback u = ˆu + K(s −ˆs), with F = A + BK Schur stable.Both the nominal input and correction term are nonlinear control laws in the original state because the lifted state depends on Ψ(x).
- Closed-loop properties: Under Assumptions 1–5, initial feasibility implies recursive feasibility of the online MPC problem.The result is stated for the nominal optimization problem used to compute the lifted nominal state and control.
- Closed-loop properties: Under Assumptions 1–5, the lifted nominal system converges asymptotically to the origin and the actual closed loop remains within robust tubes.The theorem separately characterizes nominal convergence and the bounded deviation of the disturbed system from its nominal trajectory.
- Closed-loop properties: With zero disturbance and the theorem’s additional condition, the lifted and original closed-loop systems converge asymptotically to the origin.The proof uses a Schur-stability small-gain condition and the relation x_k = Cs_k + v(s_k).
4 Simulation results
Simulations evaluate r-KMPC on a Van der Pol oscillator, an inverted pendulum, and a non-affine system under disturbances and constraints. Across these examples, trajectories remain within designed tubes, while comparisons with KMPC show convergence and lower cumulative cost for r-KMPC.
- Simulation setup: The simulations cover a Van der Pol oscillator, an inverted pendulum, and a non-affine system with bounded state and control constraints.The first two systems use sinusoidal disturbances; the Van der Pol experiment also compares controllers under wo = 0.
- Simulation setup: r-KMPC uses data-driven Koopman models constructed from sampled trajectories and lifted observable functions for each nonlinear system.The Van der Pol model uses M = 8·105 samples and thinplate basis functions; the inverted-pendulum model uses M = 5 · 104 samples and Gaussian kernels.
- Robustness under disturbances: Under sinusoidal noise, the real state and control trajectories lie within grey state and control tubes centered on their nominal trajectories.This tube behavior is reported for the Van der Pol oscillator, inverted pendulum, and non-affine system.
- Performance comparison: With wo = 0, r-KMPC state and control converge asymptotically to the origin, whereas KMPC does not converge within 400 steps for the Van der Pol oscillator and diverges for the inverted pendulum.The reported comparisons concern Figures 3 and 6.
- Performance comparison: r-KMPC achieves lower cumulative cost than KMPC in the Van der Pol and inverted-pendulum comparisons, despite using more observable functions.The cumulative costs are reported in Tables 1 and 2.
- Non-affine system: For the non-affine system, the reported r-KMPC simulation shows state and control convergence to the origin under wo = 0.The corresponding performance is summarized in Figure 9.
5 Conclusions
The paper proposes r-KMPC, a robust tube-based controller that combines online linear MPC with a nominal Koopman model and an offline nonlinear feedback policy. The authors verify robustness under modeling errors and disturbances, prove asymptotic stability without exogenous disturbance under mild assumptions, and support effectiveness through simulations.
- r-KMPC addresses nonlinear systems with additive disturbances, state constraints, and control constraints using a lifted global linear Koopman model.
- The closed-loop controller combines online linear MPC based on the nominal Koopman model with a nonlinear static state-feedback policy.
- The paper verifies closed-loop robustness under internal modeling errors and exogenous disturbances.
- Under no exogenous disturbance and mild assumptions, the controlled system is proven asymptotically stable.
- Simulation results verify the effectiveness of the proposed approach.
A Statistical validations of sets ¯
The appendix validates uncertainty sets used by r-KMPC through empirical-risk estimates and a scaling procedure based on local Lipschitz bounds and data coverage. The resulting sets contain the real uncertainty sets under the stated bounds.
- Statistical validation: The uncertainty-set construction uses statistical learning theory to estimate the true risk from empirical risk and finite datasets.The validation relies on the law of large numbers and a relaxed Hoeffding-based risk condition.
- Validation losses: The uncertainty-set loss evaluates whether lifted prediction errors and state reconstruction errors belong to the sets W̄ and V.The learned predictor supplies lifted error terms, while vi = xi − CΨ(xi) represents reconstruction error.
- Uncertainty-set bounds: Proposition 6 gives W̄ ⊆ W̄I ⊕ ΔW and V ⊆ VI ⊕ ΔV, with scaling terms determined by Lipschitz constants, sampling distances, and disturbance-estimation deviation.The bounds include LΨ, Lv, Ls, Lu, Lδw, and L ŵ together with data-neighborhood distances.