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Koopman-Based Robust Model Predictive Control for Nonlinear Systems with Stochastic Intermittent Measurements
Guanhua Liu, Tong Wu, Lixian Zhang, Weifeng Du, Minghao Han
TL;DR
Intermittent measurements challenge constrained nonlinear MPC because dropouts increase prediction uncertainty and resets can compromise feasibility. The paper develops a Koopman-based stochastic MPC with Markov-jump error modeling and probabilistically truncated soft constraints. Under stated stability and disturbance conditions, it establishes bounded prediction and closed-loop errors together with recursive feasibility, with simulations corroborating tracking under measurement unavailability.
Problem
Intermittent measurements cause growing prediction uncertainty and measurement-triggered resets that can compromise stability and recursive feasibility in constrained nonlinear MPC.
Method
The paper combines a Lipschitz-constrained deep Koopman latent predictor, a unified Markov jump prediction-error model, probabilistic error-radius truncation, and exact-penalty soft constraints.
Results
Under terminal compatibility and bounded-disturbance conditions, the prediction error and closed-loop regulation error are mean-square ultimately bounded, while recursive feasibility is established.
Takeaways & Limitations
The framework supports convex online optimization for constrained nonlinear regulation with stochastic measurement unavailability and dropout-dependent constraint tightening.
Takeaways & Limitations
The probabilistic truncation uses axis-wise point-wise bounds rather than joint chance constraints, so exact multi-step joint probabilistic guarantees are sacrificed.
Abstract
from arXiv · showhide
Intermittent state measurements pose fundamental challenges to model predictive control of constrained nonlinear systems because prediction uncertainty grows during feedback outages and measurement-triggered resets disrupt nominal state propagation, potentially compromising closed-loop stability and recursive feasibility. This paper develops a Koopman-based stochastic MPC framework with probabilistically truncated soft constraints. Specifically, a Lipschitz-constrained deep Koopman model provides a linear latent predictor, enabling computationally efficient online optimization. The intermittent measurement process is modeled as a two-mode discrete-time Markov chain, yielding a unified Markov jump error model for open-loop propagation and measurement-triggered resets. Under numerically verifiable sufficient conditions, the prediction error is shown to be mean-square ultimately bounded, and an explicit uniform second-moment bound is obtained. A distribution-free probabilistic error radius is then constructed for a prescribed confidence level and used to truncate dropout-dependent constraint tightening. An exact-penalty soft-constraint mechanism accommodates reset-induced jumps and prolonged dropouts. Under the stated terminal compatibility and bounded-disturbance conditions, recursive feasibility and mean-square ultimate boundedness of the closed-loop regulation error are established. Numerical simulations on a visual-servoing tracking task corroborate these theoretical results and demonstrate effective tracking under stochastic measurement unavailability.
1 INTRODUCTION
The paper addresses constrained nonlinear control when intermittent measurements make uncertainty grow and disrupt nominal propagation. It combines Koopman modeling with stochastic, probabilistically truncated soft-constrained MPC to preserve safety and feasibility under such dropouts.
- Motivation: Intermittent measurements from communication dropouts or detection failures violate the uninterrupted-feedback assumption underlying conventional MPC.Visual servoing is especially affected by target occlusions and camera field-of-view limits.
- Motivation: Robust MPC is needed to keep constrained trajectories within safety boundaries during prolonged measurement dropouts.
- Background: Nonlinear MPC is theoretically suitable for constrained nonlinear systems but is hindered by the computational burden of non-convex real-time optimization.
- Background: Deep Koopman methods use neural networks to learn expressive latent dictionaries that improve representation accuracy for linearized control modeling.
- Gap: Under prolonged stochastic dropouts, worst-case or continuously measured tube-based MPC can produce highly conservative tightening and lose recursive feasibility.
- Related direction: Stochastic MPC reduces worst-case conservatism by using probabilistic bounds, while Koopman and Markov-jump approaches represent data-driven dynamics and stochastic modes.
- Contribution: The paper develops a Koopman-based stochastic MPC framework that models prediction-error resets and open-loop propagation, derives mean-square bounds, and uses truncated soft constraints.The formulation is designed to address the insufficiently explored preservation of recursive feasibility during prolonged dropouts.
2 PROBLEM FORMULATION
The problem formulation lifts constrained nonlinear dynamics into a Lipschitz-constrained latent space, where nominal MPC is linear and computationally tractable. It then models bounded disturbances, reconstruction residuals, and prediction errors for probabilistic error-envelope analysis.
- 2.1 Koopman-based System Modeling and Nominal MPC Formulation: The nonlinear system is represented through Koopman observables and an input-affine latent evolution with control treated as an exogenous forcing term.
- 2.1 Koopman-based System Modeling and Nominal MPC Formulation: Finite-dimensional Koopman truncation is needed for synthesis, but general nonlinear systems can produce approximation residuals without theoretical bounds.
- 2.1 Koopman-based System Modeling and Nominal MPC Formulation: A Lipschitz-constrained deep Koopman architecture globally lifts nonlinear dynamics into a finite-dimensional latent space.
- 2.1 Koopman-based System Modeling and Nominal MPC Formulation: An identity skip connection makes the first n_x latent coordinates equal the original state, while spectral normalization bounds the nonlinear feature map's Lipschitz constant by L_ψ ≤1.
- 2.1 Koopman-based System Modeling and Nominal MPC Formulation: The sparse projection C=[I_nx,0] directly reconstructs the original state from latent coordinates, avoiding nonlinear decoding overhead.
- 2.1 Koopman-based System Modeling and Nominal MPC Formulation: The training loss combines prediction, eigenvalue, and orthogonal regularization terms to support stable and non-normality-controlled latent dynamics.The eigenvalue penalty uses an envelope β∈(0,1), while orthogonal regularization suppresses transient growth.
- 2.1 Koopman-based System Modeling and Nominal MPC Formulation: Nominal MPC optimizes a quadratic cost subject to latent linear dynamics, state constraints, input bounds, and a terminal invariant-set condition.The formulation uses weighting matrices Q and R, a DARE-derived terminal penalty P_f, horizon N_p, and reference state z_ref.
3 MAIN RESULTS
The framework bounds prediction error under Markovian measurement dropouts and uses a distribution-free radius to truncate constraint tightening. Under terminal compatibility and bounded-disturbance conditions, it preserves recursive feasibility and establishes mean-square bounded closed-loop tracking error.
- Probabilistic constraint tightening: After continuous dropouts, the high-dimensional error envelope is projected into coordinate-wise low-dimensional radii for constraint tightening.The projection uses an L∞-norm outer hypercube envelope.
- Probabilistic constraint tightening: The L∞-norm relaxation provides a conservative inner approximation that replaces non-polyhedral set differences with axis-aligned linear constraints.This preserves theoretical safety while supporting real-time computation.
- Probabilistic constraint tightening: As dropout duration increases, tightening can shrink the admissible domain enough to compromise recursive feasibility.The issue motivates truncating deterministic dropout-dependent margins.
- Prediction-error boundedness: A Schur-stable latent matrix and p11 < 1 are sufficient for global mean-square ultimate boundedness of the Markov jump prediction error.The result applies despite open-loop propagation and measurement-triggered resets.
- Prediction-error boundedness: The prediction error avoids unbounded mean-square growth statistically, although a single prolonged dropout can cause severe local envelope inflation.The transition probability p11 cannot be arbitrarily close to one because of instantaneous state-constraint considerations.
- Closed-loop guarantees: Bounded prediction and nominal-state jump errors yield a finite global bound on the expected squared error, preventing finite-time escape.The closed-loop result follows by decomposing true tracking error into nominal tracking and prediction-error components.
- Closed-loop guarantees: A distribution-free radius supports probabilistically truncated tightening, while exact-penalty soft constraints preserve feasibility under stochastic resets and prolonged dropouts.With initial feasibility and terminal compatibility, recursive feasibility holds for every measurement mode, and the closed-loop tracking error is globally mean-square ultimately bounded with marginal violation probability at most 1 − p.
4 NUMERICAL VALIDATION
The numerical validation evaluates probabilistic safety, tracking under stochastic dropouts, baseline performance, and computational feasibility in a visual-servoing task.
- Simulation setup: The 2-DOF flexible-joint gimbal model uses nonlinear pan and tilt dynamics with elastic, damping, Coriolis, centrifugal, and gravitational terms.The pan and tilt equations define the simulated plant and its physical parameters.
- Stochastic measurement and constraints: The measurement-availability mode is modeled as a homogeneous discrete-time Markov chain with transition probabilities p01 = 0.15 and p11 = 0.8.The reference trajectory is a three-petal rose curve, and the safe state space is constrained by camera field-of-view limits.
- Safety-boundary validation: At confidence level p = 0.95, the probabilistic safety radius is Rprob = 10.33°, while the empirical 95th percentile is 9.63° across 300 Monte Carlo trials.The overall empirical violation rate is 4.10%, below the predefined 5% risk tolerance.
- Tracking performance: The proposed controller achieves RMSEs of 1.5° and 1.8° for the pan and tilt axes, respectively, under stochastic measurement dropouts.During measurement availability, the MAEs are 1.1° and 1.5°; during unavailability, they are 1.2° and 1.6°.
- Baseline comparison: Compared with the baseline, the proposed method reduces RMSE by 82.1% for pan and 70.0% for tilt.The reported reductions during unavailable measurements are 82.6% and 68.6% for the respective axes.
- Computational feasibility: The convex optimization solver has a median single-step computation time of 2.0 ms, satisfying the 50 Hz real-time control requirement.The simulations and validations ran in Python on an AMD Ryzen AI 9 H 365 CPU with 32 GB RAM.
5 CONCLUSION
The paper addresses stability and recursive feasibility for constrained nonlinear systems with stochastic intermittent measurements. It combines deep Koopman lifting, probabilistic safety bounds, and exact-penalty soft constraints in an efficient MPC framework.
- The paper targets stability and recursive feasibility under stochastic intermittent measurements in constrained nonlinear dynamical systems.
- A Lipschitz-constrained deep Koopman operator lifts the nonlinear dynamics into a computationally tractable linear latent space.
- A switched linear error formulation yields mean-square ultimate boundedness and a distribution-free probabilistic safety radius.
- An exact-penalty soft-constrained MPC framework guarantees recursive feasibility and mean-square ultimate boundedness while maintaining high computational efficiency.
A NETWORK PARAMETERS AND CONFIGURATION
Table A1 presents the parameters and training configuration of the Deep Koopman operator network.
- Table A1 lists the parameters of the Deep Koopman operator network and its training configuration.
- The table is intended to document the network setup used for the Deep Koopman model.
- The listed configuration supports the paper’s use of a Deep Koopman operator network for nonlinear-system modeling.