Stochastic Nonlinear Model Predictive Control with Gaussian Mixture Uncertainty Propagation

Konstantinos Prattis; Luca Laurenti; Azita Dabiri

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Stochastic Nonlinear Model Predictive Control with Gaussian Mixture Uncertainty Propagation

Konstantinos Prattis, Luca Laurenti, Azita Dabiri

arXiv:2608.29272v1Aug 29, 2026eess.SY

TL;DR

Nonlinear SNMPC lacks tractable, formally controlled uncertainty propagation for non-Gaussian disturbances. The paper uses Gaussian mixtures with Wasserstein error bounds to derive tractable cost and chance-constraint reformulations, yielding open-loop correctness and asymptotic optimality guarantees and favorable benchmark comparisons.

Problem
Nonlinear systems with non-Gaussian uncertainties make SNMPC difficult because predicted distributions, expected costs, and chance constraints generally lack closed-form expressions.
Method
The paper propagates predicted state distributions with Gaussian mixtures, derives locally analytic costs and chance constraints, and solves the resulting reformulation through nonlinear programming.
Results
The reformulation provides formal correctness and asymptotic optimality guarantees, while benchmarks show favorable trade-offs among optimality, scalability, and chance-constraint violations.
Takeaways & Limitations
Gaussian-mixture uncertainty propagation offers a tractable SNMPC approach for unimodal and multi-modal additive disturbances with quantified approximation error.

Abstract

from arXiv · show

We propose a novel Stochastic Nonlinear Model Predictive Control (SNMPC) framework for nonlinear systems with additive noise. Building on recent advances in nonlinear uncertainty propagation, we show that the state distribution of the system can be tractably approximated over time by Gaussian mixture distributions, with formal error bounds in Wasserstein distance. This representation yields closed-form expressions for expected costs and chance constraints, which become exact for affine constraints and exact up to a constant for quadratic costs. Consequently, the resulting control problem can be solved efficiently via nonlinear programming, while providing formal open-loop guarantees of correctness and asymptotic optimality. Experiments on a set of benchmarks demonstrate that the proposed approach compares favorably with existing methods in nonlinear settings with multi-modal disturbances, where standard approaches lead to poorly scaled solutions and unsafe or overly conservative control actions.

I. INTRODUCTION

Nonlinear stochastic MPC with non-Gaussian uncertainty remains difficult because uncertainty propagation is generally intractable and existing approximations lack explicit correctness guarantees. The paper addresses this gap with Gaussian-mixture propagation, tractable reformulations, formal error bounds, and empirical validation.

Nonlinear systems with non-Gaussian uncertainties remain a challenging open problem for stochastic MPC.
Existing nonlinear SMPC approximations often lack explicit error bounds because propagating uncertainty through nonlinear dynamics is intractable.
The proposed reformulation propagates state distributions with Gaussian mixtures and quantifies approximation error in Wasserstein distance.
Gaussian-mixture representations provide tractable locally analytic expressions for expected cost and chance constraints.These expressions are obtained through local linearization around each mixture component.
The reformulated optimization problem is efficiently solvable by nonlinear programming, with formal constraint-satisfaction, approximation-error, and convergence guarantees.
Empirical validation on nonlinear benchmarks demonstrates competitive performance under unimodal and bimodal disturbance distributions.

III. PROBLEM FORMULATION

The paper formulates SNMPC for nonlinear stochastic systems with additive, known disturbances and a differentiable policy, optimizing finite-horizon cost under probabilistic state and input constraints. It then identifies the central tractability challenge: nonlinear dynamics and non-Gaussian noise prevent closed-form predicted distributions, costs, and chance constraints.

The system has nonlinear, possibly piecewise-differentiable dynamics, full-state measurements, and i.i.d. additive disturbance with known distribution.
The disturbance distribution is assumed to be a Gaussian mixture with equal diagonal covariance across components.The paper notes that broader finite-second-moment distributions can be approximated in W2, but dimensionality can require O(ϵ^-nx) components.
The controller uses a fixed differentiable policy parametrized by θt and operates in a receding-horizon setting.
SNMPC minimizes expected finite-horizon cost while enforcing probabilistic state and input constraints.
Because nonlinear dynamics and non-Gaussian noise prevent closed-form predicted distributions, the original SNMPC problem lacks closed-form cost and chance-constraint expressions.

IV. FORMAL UNCERTAINTY PROPAGATION WITH GAUSSIAN MIXTURES

The method recursively approximates predicted state distributions with Gaussian mixtures by discretizing them into weighted point sets, while retaining Wasserstein error bounds. The construction yields tractable mixture representations whose component count scales as M = m · mη.

Formal Gaussian mixture propagation: The method extends mixture propagation to approximate predicted state distributions while accounting for policy-parameter dependence and providing closed-form Wasserstein error bounds.
Formal Gaussian mixture propagation: At each prediction step, a Gaussian mixture approximation is defined recursively from a set of m discretization points.
Formal Gaussian mixture propagation: The recursion discretizes each previous Gaussian mixture into a discrete distribution whose weights equal probability mass over associated regions.
Formal Gaussian mixture propagation: The propagated approximation remains a Gaussian mixture with M = m · mη components, where mη is the number of disturbance-mixture components.
Formal Gaussian mixture propagation: The quantity βk|t quantifies the accumulated approximation error after k prediction steps and depends on earlier discretization choices.
Formal Gaussian mixture propagation: Voronoi partitions are optimal for a fixed discretization-point set, although computing them for general point sets may be inefficient.
Formal Gaussian mixture propagation: Figure 1 illustrates propagation from an initial 8-component mixture through local Gaussian fitting and 4-point grids to the resulting mixture.

B. Efficient Gaussian mixture discretization

Efficient discretization identifies mixture modes, fits local Gaussians, and assigns structured grids whose geometry supports tractable Voronoi probabilities. The construction is designed to remain locally smooth in policy parameters for gradient-based optimization.

B. Efficient Gaussian mixture discretization: The discretization identifies local maxima of the predicted mixture density as the modes used to construct local approximations.
B. Efficient Gaussian mixture discretization: A Laplace approximation fits a local Gaussian around each mode, with the local mean equal to that mode.
B. Efficient Gaussian mixture discretization: Each local Gaussian receives L = m/|Sk|t| grid points, distributing the discretization budget across modes.
B. Efficient Gaussian mixture discretization: The final grid locations account for the diagonal disturbance covariance and use coordinatewise scaling through the Hadamard product and matrix diagonal.
B. Efficient Gaussian mixture discretization: For each mode, the grid is centered at zero and chosen as a rectangular grid that permits closed-form Voronoi-region probabilities.
B. Efficient Gaussian mixture discretization: The controlled grid locations depend recursively on policy parameters, so the construction selects them to preserve local smoothness.
B. Efficient Gaussian mixture discretization: The method assumes that the number of modes divides m, using duplication or removal of modes when needed.
B. Efficient Gaussian mixture discretization: Keeping the number of modes constant with respect to policy parameters enables gradient-based optimization and warm-starting of propagation parameters.

V. TRACTABLE SNMPC REFORMULATION

The reformulation propagates state uncertainty with Gaussian mixtures, then uses these distributions to approximate expected costs through componentwise local expansions. For quadratic costs, the approximation admits formal correctness and deviation guarantees.

Gaussian mixture propagation: The Gaussian mixture approximation is characterized by component weights, means, and a shared covariance, with parameters propagated through a policy-dependent map.The method evolves mixture parameters over time using the induced mixture parameter space and a propagation map.
Expected cost approximation: The expected cost is approximated by summing stage and terminal costs evaluated at Gaussian-component means, weighted by their mixture probabilities.The approximation applies a first-order Taylor expansion around each component mean; the linear term has zero expectation under the component Gaussian.
Expected cost approximation: Quadratic costs permit a closed-form expected-cost approximation under the Gaussian mixtures.The paper introduces a correctness property specifically for quadratic stage and terminal costs.
Approximation guarantees: The deviation between the true and approximated expected costs is bounded using Wasserstein-distance bounds and coupling inequalities.The proof applies triangle and Cauchy–Schwarz inequalities to an optimal 2-Wasserstein coupling before invoking the approximation-error bounds.

C. Chance constraints reformulation using the Gaussian mixture distributions

Chance constraints are approximated by applying local linearizations to each Gaussian-mixture component and evaluating the resulting normal probabilities in closed form. The approximation is exact for affine constraint functions, while nonlinear constraints retain a heuristic approximation without curvature information.

Constraint reformulation: The original chance constraints are replaced using the Gaussian mixture state distributions and componentwise local approximations.The constraint functions are written compactly as h_k and their satisfaction probabilities are evaluated under the derived mixture distributions.
Constraint reformulation: Each locally affine constraint approximation is normally distributed, enabling a closed-form probability expression through the standard normal CDF.The approximation is formed around each mixture-component mean and combined across components.
Affine constraints: For affine constraint functions, the chance-constraint approximation becomes exact.The first-order Taylor approximation is exact for affine functions, yielding equality between the approximate and exact reformulations.
Nonlinear constraints: Without curvature information, the chance-constraint formula is heuristic for nonlinear constraints.The paper characterizes approximation error in the affine case and notes that the closed-form expression is not generally exact for nonlinear functions.

D. Resulting optimization problem

The resulting optimization problem uses Gaussian-mixture state approximations to formulate tractable costs and chance constraints, with guarantees linking approximate and original SNMPC solutions. Under the stated assumptions, feasibility is preserved for sufficiently many mixture components, while the sub-optimality gap vanishes asymptotically.

Formulation: The optimization problem combines the Gaussian-mixture objective, mixture propagation, and Taylor-based chance constraints with possibly tightened probabilities.Constraints (29b)–(29f) initialize and propagate mixtures, approximate violation probabilities, and use 1 − ε̄_k ≥ 1 − ε_k.
Correctness guarantees: Under globally Lipschitz dynamics, affine constraints, and quadratic costs, sufficiently large M makes every feasible approximate solution feasible for the original SNMPC.The theorem requires ε̄_k < ε_k for all k and asserts existence of a threshold M̄.
Correctness guarantees: The approximate optimizer satisfies a formal sub-optimality bound determined by Wasserstein approximation errors and cost-sensitivity terms.The bound uses the quantities r̄_k|t and β_k|t,max associated with the mixture approximation and its error radii.
Asymptotic convergence: As M →∞, the sub-optimality gap of the approximate problem asymptotically goes to zero.Increasing mixture components reduces discretization error, allowing β_k|t to approach zero.
Scope of guarantees: The guarantees concern the open-loop problem and do not establish recursive feasibility, closed-loop chance-constraint satisfaction, or stochastic stability.The paper identifies terminal components and constraint tightening as possible routes for recovering such properties.

VI. IMPLEMENTATION & NUMERICAL RESULTS

The implementation solves the approximate problem with sequential quadratic programming while freezing selected mixture quantities to avoid nonsmooth propagation inside iterations. This reduces computational complexity but increases the Wasserstein error bounds, and computational dependence on state dimension remains limiting.

Implementation: The method uses Sequential Quadratic Programming after generating predicted Gaussian mixtures from the previous optimal parameter sequence.The implementation keeps the number of modes, local covariances, and grids fixed during SQP iterations.
Implementation: Freezing mixture quantities preserves the theoretical results but increases the Wasserstein error radii and makes the bounds more conservative.The implementation also keeps mixture weights constant to reduce computational complexity, with the same type of conservativeness effect.
Implementation: The nonlinear program is formulated and solved using PyTorch, CasADi, and acados.PyTorch implements mixture propagation, while CasADi and acados formulate and solve the SQP problem.
Computational scaling: State dimension is the limiting computational factor because per-mode rectangular grids grow exponentially with n_x.The formulation scales linearly with the number of mixture components, modes, and horizon, but grid growth dominates in n_x.

A. Simulations and numerical examples

The experiments evaluate the proposed method on two nonlinear benchmarks with unimodal and bimodal disturbances, comparing it against nominal, moment-based stochastic, and scenario-based MPC variants.

Experimental setup: The experiments use m = 3 mixture locations and test both unimodal Gaussian and bimodal mixture disturbance distributions.The evaluation covers two nonlinear benchmarks.
Baselines: The proposed method is compared with Nominal MPC, moment-based Stochastic MPC, and Scenario-based MPC with N_s = 20 scenarios.Nominal MPC ignores disturbance, Stochastic MPC propagates only the first two moments, and Scenario-based MPC uses 20 scenarios.

1) Double vortex 2-D system:

The double-vortex benchmark is a two-dimensional nonlinear system with obstacle-avoidance constraints, bounded control, and either unimodal or bimodal additive disturbances. Results indicate that Gaussian-mixture propagation avoids the overly conservative behavior of Gaussian propagation while remaining safe in the reported experiment.

System and objective: The controller minimizes a quadratic cost while keeping the state outside two elliptic obstacles.The cost uses Q = I_2 and R = 0.8 I_2; the obstacles are centered at c_2 = p_2 and c_1 = [0.5, −0.6]^T.
Disturbances and constraints: The experiment uses either Gaussian noise or an equal-weight bimodal Gaussian mixture, with Σ_η = 10^-3 I_2, horizon N = 10, and satisfaction probability 0.99.The control parameter satisfies |θ_t| ≤ 1, and the constraints are non-affine and nonconvex.
Results: Gaussian propagation forces trajectories into the left half plane, whereas GMix-MPC keeps them in the right half plane in the unimodal-noise double-vortex case.The reported comparison attributes this difference to the more accurate uncertainty propagation of GMix-MPC.
Results: GMix-MPC offers a practical trade-off among cost optimality, constraint satisfaction, and computational time, using m = 3 mixture locations.The reported results compare average cost, maximum violation, and solver times across methods.
Results: Nominal MPC fails to provide a valid trajectory because it ignores the underlying stochasticity.The experiment uses slacks so execution can continue when a trajectory becomes invalid.

2) Planar robot 3-D system:

The planar robot benchmark evaluates controllers under unimodal and bimodal disturbances while driving the robot toward rx = 0 without violating rx ≥ 0. The proposed approach reaches closer to the target with fewer violations and favorable computational scaling, but its guarantees remain open-loop.

Task and setup: The planar robot task drives the state toward rx = 0 while enforcing the soft constraint rx ≥ 0 under bounded control inputs.The cost is |rx|2 + 10^-6∥u∥2, with horizon N = 15 and chance-constraint level 1 − ϵk = 0.99.
Task and setup: The experiments test unimodal Gaussian noise and bimodal Gaussian-mixture noise with covariance Ση = 10^-3I3.The bimodal disturbance uses mixture weights 0.8 and 0.2 with shifted means.
Results: The proposed controller approaches rx = 0 more closely while keeping violations minimal, whereas Gaussian MPC remains farther away to prioritize safety.The comparison identifies Gaussian MPC as safer but more conservative in this benchmark.
Results: Nominal MPC fails to provide a valid trajectory for both disturbance distributions, while Scenario MPC cannot produce a safe trajectory in the bimodal case.
Results: In the unimodal case, the proposed approach uses m = 3 mixture locations, while the Scenario approach appears to require more than 20 scenarios for comparable constraint satisfaction.The passage presents this as an efficiency comparison rather than an exact universal scenario requirement.
Guarantees and limitations: The observed closed-loop constraint violation exceeds the theoretical expectation because the guarantees are open-loop and rely on the conditions of Theorem 1.
Computational trade-offs: The proposed framework yields smaller optimization problems than the Scenario approach, which imposes constraints per scenario rather than across mixture components.Both approaches are theoretically asymptotically optimal as their component or scenario counts grow, but computational cost increases.