Stochastic Model Predictive Control

Updated 12 November 2025

SMPC is a predictive control method that minimizes the expected cost in discrete systems while satisfying state and input constraints with probabilistic guarantees.
It employs techniques like probabilistic reachable sets, scenario-based optimization, and convex relaxations to manage various disturbances including Gaussian mixtures and bounded uncertainties.
Practical implementations combine offline precomputation and online adaptive constraint tightening to ensure recursive feasibility and effective control in safety-critical applications.

Stochastic Model Predictive Control (SMPC) Framework

Stochastic Model Predictive Control (SMPC) refers to a class of predictive control methods for discrete-time dynamical systems that systematically account for uncertainty, often modeled as stochastic disturbances, through probability-aware optimization over a receding horizon. The SMPC framework designs control laws that seek to minimize a cost function in expectation while maintaining satisfaction of state and input constraints with prescribed probabilistic guarantees, typically encoded as chance constraints. Modern SMPC theory encompasses systems with unbounded noise (e.g., Gaussian or finite Gaussian mixtures), correlated disturbances, model epistemic uncertainty, and constraint specifications ranging from single to joint and two-sided chance forms. Core developments include tractable finite- and infinite-horizon formulations, recursive feasibility guarantees, convex relaxations through constraint tightening, scenario-based and distributionally robust approaches, and application-specific methodologies such as those for quantum filtering, distributed systems, and safety-critical vehicle control.

1. Stochastic System Modeling and Uncertainty Representations

SMPC frameworks assume discrete-time systems of the form

$x_{k+1} = A x_k + B u_k + w_k,$

where $x_k \in \mathbb{R}^n$ , $u_k \in \mathbb{R}^m$ , and $(A,B)$ is stabilizable. The modeling of the disturbance $w_k$ is central and varies:

Gaussian or sub-Gaussian noise: $w_k$ is i.i.d. with zero mean and covariance $\Sigma$ (Ao et al., 11 Mar 2025), allowing for unbounded support and tractable chance-constraint reformulations via CDF quantiles.
Finite Gaussian mixture disturbances: $w_k$ is drawn from a mixture $p(w) = \sum_{i=1}^M \pi_i\,\mathcal{N}(w; \mu_i, \Sigma)$ , with typically identical covariances (Engelaar et al., 12 Nov 2024). This form precludes analytic inversion of constraint probabilities and necessitates alternative approaches such as mixture decoupling.
Unknown, bounded-support distributions: Only the support $\mathbb{W}$ is available. Chance constraints are enforced via sampling-based sets and nonparametric quantiles (Lee et al., 2022).
Epistemic system/model uncertainty: Unknown parameters $\theta$ in $A(\theta), B(\theta)$ ; identified from data via bootstrapping or Bayesian/posterior inference. Joint epistemic–aleatoric uncertainty is handled via scenario-based methods (Micheli et al., 2022).
Non-Gaussian, central convex unimodal, or log-concave distributions: Key for recursive feasibility using PRS-based tightening (Hewing et al., 2018, Hewing et al., 2018).
Process and measurement noise in output-feedback setups: State-space models augmented with measurement equations and filter dynamics (e.g., Kalman filters) (Muntwiler et al., 2022).

This diversity in uncertainty modeling underpins the choice of constraint-handling, probabilistic set constructions, and optimization techniques throughout the SMPC literature.

2. Chance Constraints and Constraint Tightening

SMPC enforces state/input constraints in probabilistic terms, typically as linear or polyhedral forms: $\Pr(C x_k + D u_k \leq d) \geq 1-\epsilon$ or split marginally as

$\Pr(x_k \in \mathcal{X}) \geq p_x, \qquad \Pr(u_k \in \mathcal{U}) \geq p_u.$

Key formulations and tractable relaxations include:

Probabilistic Reachable Sets (PRS): For noise with known or quantifiable moments, construct sets $\mathcal{R}$ $R$ such that $\Pr(e \in \mathcal{R}) \geq p$ $Pr (e \in R) \geq p$ , where $e$ $e$ is the deviation from a nominal trajectory:
- Ellipsoidal PRS: Used for sub-Gaussian or Gaussian disturbances with covariance propagation (Ao et al., 11 Mar 2025, Hewing et al., 2018).
- Variable-size tubes: Tube cross sections adjusted online via mean/variance interpolation and error propagation (e.g., via Lyapunov recursion for $V[e_k]$ ) (Schlüter et al., 2023).
Mixture Model Decoupling: For Gaussian mixture noise, disturbance is separated into a discrete component (the mixture mean) and a continuous Gaussian component. Constraints are tightened separately for each disturbance branch, yielding sets $\mathcal{Z} = \mathcal{X} \ominus \mathcal{R}_x$ , $\mathcal{V} = \mathcal{U} \ominus K \mathcal{R}_u$ (Engelaar et al., 12 Nov 2024).
SOC/SDP reformulations: For distributionally robust approaches, chance constraints under moment ambiguity are expressed as second-order cone programs (SOC) using results such as those of Zymler et al. and Zhang et al., with explicit slack/auxiliary variables to capture both upper and lower constraint bounds (Tan et al., 2022).
Scenario-based constraints: Sample-based enforcement using a finite set of uncertainty/parameter scenarios; validity and coverage rates are rigorously linked to the number of samples and optimization variables through scenario theory and support rank (Schildbach et al., 2013, Micheli et al., 2022).

Constraint tightening is applied at each prediction step in the horizon, ensuring that nominal states/inputs (and possibly their affine disturbance-feedback extensions) remain well inside original constraints with a margin dictated by the probabilistic sets.

3. Optimization Problem Structures and Solvers

The SMPC finite-horizon control problem is formulated as a constrained optimization that, depending on the uncertainty and policy parameterization, takes the form of a QP, SOCP, or SDP. Main architectures include:

Nominal + Error Decomposition: Predict nominal states and use PRS-tightened constraints; closed-loop input is $u_k = v_k + K e_k$ , often yielding convex QPs if sets are polyhedral or ellipsoidal (Hewing et al., 2018, Hewing et al., 2018).
Interpolation over initial state: The first nominal predicted state is optimized as a convex combination $z_0 = (1-\alpha)x_k + \alpha z_{1}(k-1)$ , enabling a continuum between full measurement feedback and pure prediction and forming a single QP in joint variables (Köhler et al., 2022, Schlüter et al., 2023, Schlüter et al., 2022).
Branch Model Predictive Control (BMPC): For Gaussian mixtures, generate all possible branches over the disturbance means across the horizon, optimizing over this tree (size $M^N$ ), with a binary variable $\xi$ toggling the nominal initialization between state reset and recursive shifting (Engelaar et al., 12 Nov 2024).
Affine disturbance–feedback policies (“SADF”): For problems demanding convexity under disturbance feedback (e.g., in cyberphysical or uncertain systems), inputs parameterized as affine functions of past disturbances (Wang et al., 2022, Micheli et al., 2022).
Scenario and sample-based programs: Direct scenario enumeration with convex constraints, possibly followed by sample-and-remove strategies or scenario reduction for tractability (Schildbach et al., 2013).
Distributionally robust/conic programming: SOC or SDP constraints encode worst-case probability violations under moment ambiguity, supporting two-sided constraint forms (Tan et al., 2022).
Quantum filtering case: Infinite-horizon cost for quantum systems is reduced to a single-step deterministic problem via eigenstate reduction, yielding deterministic Lindblad propagations and one-step fidelity cost minimizations (Lee et al., 8 Nov 2025).

Computational complexity scales with the horizon, the number of scenarios in sampling-based approaches, or the number of branches in BMPC; typically, the per-step solve is a convex program.

4. Recursive Feasibility and Closed-Loop Guarantees

Recursive feasibility—ensuring the SMPC optimization admits a feasible solution at every k once feasibility is established at k=0—is foundational for stability and safety:

Terminal sets/invariant sets: A terminal set $\mathcal{Z}_f$ is chosen such that it is invariant under the feedback policy and absorbing with respect to disturbance mean or support, guaranteeing that once the terminal constraint is reached, feasibility will be maintained for all future steps (Engelaar et al., 12 Nov 2024, Hewing et al., 2018, Hewing et al., 2018).
Initialization strategies: Indirect-feedback (e.g., always initializing with previously predicted nominal state) or interpolated initialization (using an optimization variable between last prediction and measured state) ensure recursive feasibility even under unbounded noise (Köhler et al., 2022).
Relaxation and state-dependent scaling: Enlarging the constraint sets dynamically when a violation would otherwise occur (e.g., using scaling variables $\gamma_x, \gamma_u$ penalized in the objective), with the property that in expectation relaxation vanishes asymptotically, restoring tight enforcement (Fiacchini et al., 24 Apr 2025).
Realization-adaptive constraint tightening: Recompute the enforcement sets at each step based on actual disturbances realized thus far for lower conservatism (particularly under bounded-support uncertainty) (Lee et al., 2022).
Scenario-based guarantees: Support rank and scenario count trade off constraint violation rate, complexity, and conservatism, ensuring mean or almost sure bounds on violation frequency (Schildbach et al., 2013).
BMPC guarantee via SSR: Stochastic simulation relations provide a policy-lifting argument demonstrating that the BMPC solution ensures equivalent (distributionally matched) closed-loop satisfaction for the true system (Engelaar et al., 12 Nov 2024).
Mean-square and average cost bounds: Under standard settings, long-run average (stage) cost is bounded by steady-state costs of the (unconstrained) LQR or similar linear feedback law (Köhler et al., 2022, Lee et al., 8 Nov 2025, Fiacchini et al., 24 Apr 2025).

The recursive feasibility and chance-constraint satisfaction are intertwined; enforceable through a combination of carefully designed initialization/terminal constraints, probabilistic set design, and, when necessary, dynamic constraint relaxations.

5. Specialized Extensions: Mixtures, Robustness, and Quantum Systems

Recent SMPC strategies have addressed a range of advanced applications and modeling settings:

Gaussian Mixture Disturbances: The BMPC/SSR framework extends classic SMPC to cases where disturbance is a finite mixture, handling strong multimodality and discrete jumps in mean (Engelaar et al., 12 Nov 2024). Performance guarantees (recursive feasibility and closed-loop chance constraints) are shown to hold via a policy simulation relation, and empirical studies exhibit success in applications such as vehicle control on rough terrain.
Scenario-Based and Distributionally Robust Methods: Scenario-based SMPC, particularly through the scenario approach and support-rank concepts, allows for explicit trade-offs between violation rate, computational complexity, and sample count (Schildbach et al., 2013, Micheli et al., 2022). Distributionally robust SMPC uses worst-case probability optimization over all distributions matching moments, providing explicit SOC-convex programs and two-sided constraint handling (Tan et al., 2022).
Quantum Filtering and Eigenstate Reduction: In quantum SMPC for finite-dimensional filtered systems, infinite-horizon stochastic objectives collapse (under almost-sure eigenstate reduction) to a closed-form one-step fidelity term, requiring only deterministic propagation of the Lindblad equation and optimizing over the resulting cost (Lee et al., 8 Nov 2025). This yields significant computational savings and scalability for high-dimensional quantum systems.
Non-Gaussian, Sub-Gaussian, and Data-Driven Uncertainty: Sub-Gaussian noise models enable high-probability tail bound construction and propagation through linear dynamics, supporting both process and measurement noise and resulting in provable containment rates for probabilistic reachable sets (Ao et al., 11 Mar 2025). Distributional model learning and bootstrapping are leveraged in scenario-based SMPC with epistemic uncertainty (Micheli et al., 2022).
Interpretable SMPC (RL-based disturbance estimation): Distributional RL is nested within SMPC to estimate complex, non-parametric aerodynamic disturbances for quadrotor trajectory-tracking, with convex disturbance-feedback parameterizations ensuring tractable controller optimization and proven stability (Wang et al., 2022).

6. Implementation Procedures and Numerical Benchmarks

A typical implementation pipeline for an SMPC scheme is:

Offline computations:
- Synthesize feedback matrices K, compute steady-state or horizon-specific PRS for error dynamics (via Lyapunov or Riccati equations), and precompute tightened constraint sets.
- For BMPC or multi-branch/multi-scenario approaches, perform all branch or scenario enumerations, and predefine terminal sets.
- For distributionally robust or scenario-based designs, select the number of samples or scenario counts per support-rank requirements and risk bounds.
Online (at each control step):
- Measure the current system state (and, if needed, uncertainty realization or model/parameter estimates).
- For initialization, optimize over an interpolant between measurement and last nominal prediction, or directly set nominal to predicted/previous value (Köhler et al., 2022, Fiacchini et al., 24 Apr 2025).
- Propagate nominal trajectories, error variances/covariances, and compute dynamic or realization-adaptive constraint-tightening sets.
- Solve the QP/SOCP/SDP or robust/semi-infinite program (policies may be nominal + error feedback, affine disturbance feedback, or explicit scenario branching).
- Apply the first input; update any nominal trajectory buffers or scenario-branching indices.
- Monitor and adapt scaling/relaxation variables as needed to recover feasibility in unanticipated realizations.

Empirical studies demonstrate that, when properly tuned, SMPC frameworks with the above pipeline achieve empirical violation rates at (or well below) the specified thresholds, with significantly improved cost profiles and constraint boundary utilization compared to robust or overly conservative schemes. Practical applicability is demonstrated in tasks as diverse as vehicle control on rough roads, surgical drill guidance with vision feedback, high-rate quadrotor tracking, quantum state transfer, and large-scale distributed building energy systems.

7. Theoretical and Practical Limitations

Scalability: Branching or scenario-tree approaches (such as BMPC in finite-M Gaussian mixture) incur exponential growth with horizon length, limiting the practical horizon for high M or N. Scenario-based approaches demand sample sizes scaling with decision dimension and risk parameters.
Conservatism: Chebyshev- or Bonferroni-tightened sets, as well as union-bound risk allocation for multi-dimensional or two-sided constraints, can be significantly conservative—leading to reduced feasible regions or suboptimal cost (Tan et al., 2022, Schildbach et al., 2013).
Recursive feasibility vs. constraint tightness: Relaxation and scaling (γ-parameters) recover feasibility but may degrade closed-loop optimality; framework design seeks to ensure these relaxations vanish in expectation as trajectories enter invariant regions (Fiacchini et al., 24 Apr 2025).
Implementation requirements: Fast convex solvers (for QP/SDP/SOCP), online covariance propagation, and scenario/branch enumeration necessitate parallelization or careful precomputation for high-dimensional or fast-sampling applications.
Terminal set, invariant set, and tail policy design: The choice and computation of terminal ingredients critically affect guarantee proofs for recursive feasibility and stability—numerous schemes are built fundamentally upon the properties of these sets under noise and control laws.

Ongoing research addresses scalability, non-Gaussian heavy-tailed disturbances, improved scenario reduction, more efficient realization-adaptive strategies, and frameworks for SMPC under partial information or in nonconvex settings.

The synthesis above reflects the breadth and technical depth of SMPC research, covering the major theoretical models, constraint formulations, optimization structures, analytical guarantees, and implementation paradigms as developed in the modern literature, with particular emphasis on the handling of intricate stochastic complexities and the design of tractable, high-probability safe control policies (Engelaar et al., 12 Nov 2024, Köhler et al., 2022, Micheli et al., 2022, Lee et al., 8 Nov 2025, Ao et al., 11 Mar 2025, Fiacchini et al., 24 Apr 2025, Schildbach et al., 2013, Lee et al., 2022, Hewing et al., 2018, Hewing et al., 2018, Muntwiler et al., 2022, Schlüter et al., 2023, Schlüter et al., 2022, Wang et al., 2022).