Chance-Constrained Control

As control engineering is applied to increasingly complex systems, learning-based control, which uses empirical data to reduce the reliance on first-principles models, has emerged as a promising paradigm. However, such approaches must cope with nonlinear and stochastic dynamics, the absence of state measurements, and scarce data, while ensuring safe operation under constraints, which requires rigorous uncertainty quantification. Whereas existing methods treat these challenges in isolation, this article proposes a Bayesian framework that jointly addresses them. A prior in state-space form is updated from input-output data using particle Markov chain Monte Carlo methods, yielding a representation of joint uncertainty over dynamics and latent states. Building on this representation, a validation procedure is developed that provides probabilistic guarantees for constraint satisfaction under arbitrary causal control laws. For control synthesis, a formulation penalizing the conditional value-at-risk of the constraint function is proposed, together with an adaptive calibration mechanism that adjusts the performance-robustness trade-off to achieve a desired robustness level. The probability of successful certification is characterized theoretically. The framework is extended to a receding-horizon setting, where posterior updates from new measurements allow the controller to progressively reduce uncertainty and improve performance. The approach is demonstrated on the optimal control of a vertical farm.

We study finite-horizon reinforcement learning with a safety constraint on the probability of reaching a failure set. In contrast to the well-studied safe reinforcement learning literature, we consider problems where the probability of reaching the failure set is vanishingly small, but the consequences of reaching the failure set are catastrophic. Direct simulation and estimation of the optimal policy is naturally ineffective when the failure probability is extremely small. Moreover, on a real system the controller can only choose actions, and this is not enough to make failures frequent. In contrast to the rare event simulation literature, our goal is not to estimate the probability of the rare event occurrence, but to upper bound it such that the failure constraint is satisfied. Thus, instead of sampling failures, we use the large deviations principle. We define a damage variable that on exceeding a certain threshold indicates failure. We study the cumulant generating function (CGF) as its positive root provides an upper bound on the failure probability via Cramer's theorem. We extend this approach to Markov decision processes with hitting times through state-action CGFs, and give finite-horizon and state-dependent versions of the bound. The state-action CGFs do not depend on the policy and can be estimated from regular transitions, so the bound can be checked for any policy a priori. This gives a policy improvement algorithm in which every policy satisfies the failure constraint with high probability.

Autonomous inspection of a target spacecraft is typically the initial step in on-orbit servicing. Planning close-proximity observation viewpoints involves a complex combinatorial problem constrained by coverage, fuel, and safety considerations. This paper models the viewpoint selection as a Constrained Markov Decision Process (CMDP) and applies Proximal Policy Optimization with Lagrangian relaxation to solve it. The system combines a high-level planner, which sequences camera viewpoints on an observation sphere around the target, with a low-level model that assesses each transfer using the Clohessy--Wiltshire (CW) equations for relative motion. The deterministic constraints on single-impulse reachability and $Δv$ cap are enforced directly on the action set. Safety margins are affected by navigation and manoeuvre errors, making the actual clearance stochastic. To address this, a chance constraint on the realised trajectory is implemented, with a Lagrangian multiplier governed by PID control to adapt the penalty during training. The approach is tested on a Galileo-class navigation satellite in medium Earth orbit, modelled as a point cloud where visibility from each viewpoint considers field-of-view and line-of-sight occlusion. Across various risk budgets, the trained policy maintains the keep-out violation rate within the specified limit, achieving coverage close to its maximum and staying within the mission $Δv$ budget. Compared to the usual practice of applying a fixed margin to the keep-out radius and planning deterministically, the chance-constrained policy attains less than half the violation rate at comparable coverage. This makes the collision risk an explicit design parameter, eliminating the need for re-tuning per case. The learned policy selects a viewpoint without the repeated Monte Carlo risk estimates required by a greedy, chance-constrained planner at every decision point.

In this article, we present socx, a GPU-accelerated solver for lifted convex reformulations of linear stochastic optimal control under chance constraints, implemented in JAX. We demonstrate that by exploiting the sparse and independent structure of the problem, distributed optimization can reduce it to simpler problems: independent linear-quadratic tracking (LQT) problems and Euclidean projections onto sets. These simpler problems can be solved in parallel and then coupled using the constraints. To validate our solver, we run it on a hover-linearized 3D quadrotor and show a speed-up of up to $98\times$ over MOSEK and up to $2224\times$ over SCS. socx also solves horizons up to $N=1120$, whereas MOSEK runs out of memory at $N=320$. We present ablations to isolate the key components that enable the speed-up.

Consider a finite population of agents with decoupled Markov transition dynamics and empirical-density feedback, subject to the following constraints: with probability at least $1-δ_r$, at least a fraction $α_r$ of agents must reach a target region at some time $t^*$, while, at each time up to $t^*$, the unsafe population fraction must remain below $β_u$ with probability at least $1-δ_u$. However, standard mean-field methods enforce these constraints only in expectation, which fails to account for stochastic fluctuations at finite fleet size $N$. To address this control problem, we propagate the second-order moment (variance) of the empirical density alongside the mean-field trajectory via a discrete-time Lyapunov recursion, and apply the Cantelli inequality to convert chance constraints into tractable deterministic conditions on the moments of the empirical density. We then incorporate these moment-based surrogate constraints into a gradient-based sequential convex approximation procedure for density-feedback policy synthesis. We further introduce additional moment-error bounds to construct a rigorous finite-$N$ certificate. The method is evaluated on a gridworld environment and a power-system EV-charging aggregation problem and compared with a standard deterministic population-level LP baseline.

Safe reinforcement learning (RL) commonly enforces expected-cost constraints, but such expectation safety may fail to control the probability of rare high-cost trajectories. Chance-constrained MDPs (CCMDPs) impose a stronger probability-level requirement, but are widely viewed as harder because the chance constraint is nonconvex and depends on the full trajectory rather than a Bellman-linear expectation. In this paper, we reveal that this computational difficulty does not necessarily imply a higher statistical price. For tabular discounted CCMDPs with fixed bounded successor support and access to a certified planning oracle, we establish a model-based upper bound, with a matching lower bound up to logarithmic terms. Technically, our key idea is the Bellman distributional certificate, which constructs a Bellman recursion for constraint violation probabilities before policy selection. The certificate can be reused across candidate policies; combined with shared row-wise reverse-KL confidence sets, it gives a policy-uniform trajectory-KL transfer without a union bound over policies or time--budget Bellman tables. For stochastic policies, we give a model-free variance-reduced policy-gradient algorithm with a finite-sample expected KKT-residual guarantee and independent validation of every accepted policy. Numerical experiments on synthetic CCMDPs and an IEEE 14-bus energy storage control benchmark illustrate the safety and mechanism behavior of the proposed algorithms.

Safety evaluation in sampling-based stochastic model predictive control often requires numerical estimation of exit functionals. The approximation of first-exit times and exit indicators is therefore a key numerical bottleneck, and discretization error in these quantities directly affects the resulting controller. This paper studies how existing higher-order methods for strong approximation of exit times can be brought into safe control. Two cases are highlighted. For general noncommutative dynamics, an adaptive order-1 Milstein discretization is used together with Lévy-area simulation via Wiktorsson's method. For commutative dynamics, an adaptive order-1.5 construction achieves a stronger exit-time rate. Under a local anti-concentration condition on the exit-time law, we show that strong exit-time approximation transfers to strong approximation of the failure indicator. The methods are then studied in the context of chance-constrained path integral control, which provides an exact continuous-time representation of safety through exit events. Numerical experiments compare the two cases in terms of strong exit-time error, failure-indicator error, and closed-loop constraint satisfaction, showing improvement over Euler-Maruyama and thereby enabling existing and future techniques whose applicability depends on improved strong approximation.

Agent-based model predictive control (AMPC) has recently been proposed as a distributed scheme that collaborates with all agents to achieve optimal holistic performance. However, its optimality highly depends on the prediction accuracy that requires all agents or their contributions to be known, which is too idealistic for actual implementation. This research proposes a novel practical hybrid control scheme - learning agent-based MPC (LAMPC), combining the model-based AMPC approach and data-based learning methods to improve the holistic vehicle performance for multi-agent systems. The Gaussian process regression (GPR) enhanced by an online data management strategy serves as the learning core to predict unknown contributions. A novel multi-step prediction mechanism leverages the GPR learning potential along the horizon. The predicted mean, representing the learned unknown contributions, completes the system model in the MPC for more accurate control. Meanwhile, a stochastic framework is formulated to guarantee control safety and feasibility using soft chance constraints based on the prediction variance. Both simulations and experiments show that, with the learning capability, LAMPC outperforms the traditional AMPC. LAMPC can achieve higher tracking performance in well-learned scenarios and always guarantee constraint satisfaction even in less-learned scenarios. Moreover, the proposed hybrid control scheme is efficient for real-time implementation and is flexible to any control agent topology.

In this paper, we consider stochastic optimal control problems with infinite-horizon joint chance constraints. By means of an appropriate state augmentation, we reformulate the original problem as a constrained Markov decision process, in which both the cost and the constraint function exhibit an additive structure. We then prove that this formulation enjoys strong duality, thereby enabling us to reformulate the problem as an equivalent unconstrained one in the Lagrange dual framework. We propose a dual-ascent algorithm to solve the resulting problem and show that it converges to a deterministic Markov policy defined over the augmented state space that is both optimal and feasible. To accommodate continuous state-input spaces, we propose a dedicated learning algorithm to approximate the value function in an offline training setting, thereby significantly reducing the computational complexity of the online control phase. We then test our approach on a numerical example and demonstrate its effectiveness compared to online predictive control methods in terms of performance and computational complexity.

Motion planning algorithms compute control sequences that drive autonomous robots to goal regions while avoiding unsafe states. Existing methods, from sampling-based planning to deep reinforcement learning, typically provide task-completion guarantees only with respect to a nominal model or simulator, which may be invalidated when the true dynamics are unknown or difficult to model accurately. This letter addresses this limitation for systems with unknown dynamics and an available approximate nominal model, contributing a planner-agnostic constraint-tightening procedure that equips existing planners with a probabilistic task-completion guarantee on the true system. We leverage conformal prediction to provide a probabilistic bound on the nominal-to-true trajectory deviation over a distribution of planning problems. We tighten the planning constraints using that bound, and show that solving the tightened problem under the nominal model is a sufficient condition for solving the original problem on the true system with a prescribed probability. We validate the theoretical guarantees empirically and demonstrate substantially improved task completion relative to nominal-model planning.

We consider defining risk probability in stochastic control problems under distribution ambiguity. Current approaches for chance-constrained control typically assume that the true state distribution is known and Gaussian distributed. These assumptions are not amenable to many real-world engineering applications where system dynamics are nonlinear and only approximately modeled. In this work, we define a distribution ambiguity set and, with a variational expression for exponential integrals, bound the expected risk value under an unknown distribution that resides within a relative entropy distance of a nominal Gaussian reference distribution. Our bound recovers the reference risk value in the zero-divergence limit. A method is presented to determine the relative entropy distance defining the ambiguity set that is a function of the reference covariance evolution and second-order dynamical truncation errors. The resulting contributions provide a framework for handling distributional ambiguity in nonlinear covariance steering problems. A stochastic spacecraft guidance example is presented to demonstrate our contributions.

This paper presents a distribution-agnostic robust trajectory-optimization framework based on chance-constrained reinforcement learning. The uncertainty is represented here through initial conditions and process noise, with the only requirement being that it can be sampled. A deterministic nominal trajectory is first computed offline, and reinforcement learning is then used only to robustify that baseline through a structured affine closed-loop correction law comprising a feedforward control adjustment and time-varying feedback gains. Probabilistic feasibility is enforced empirically through rollout-based upper-tail quantiles, while terminal dispersion is regulated through covariance-feasibility penalties. The framework is assessed on two materially different trajectory design problems. The flagship case study is a three-dimensional multi-impulse Earth-Mars transfer, where the learned policy is benchmarked against a recent robust trajectory-optimization reference under Gaussian uncertainty and then evaluated under bounded uniform uncertainty and under process disturbances not seen during training. The second case study is a stochastic atmospheric pinpoint rocket landing problem, used to assess portability to a short-horizon continuous-thrust setting with drag, mass depletion, and glide-slope constraints. The results show that the proposed framework can remain competitive in upper-tail fuel cost while preserving probabilistic feasibility, and that the same robustification scaffold can be carried across heterogeneous spacecraft trajectory planning problems without redesign of its core stochastic-control structure.

Batch processes are inherently transient and typically nonlinear, motivating nonlinear model predictive control (NMPC). However, adopting NMPC is hindered by the cost and unavailability of dynamic models. Thus, we propose to use Gaussian Processes (GP) in a model-learning NMPC scheme (GP-MLMPC) for batch processes. We initialize the GP-MLMPC using data from a single initial trajectory, e.g., from a PI controller. We iteratively apply the NMPC embedded with GPs to run batches and update the GP with new observations from each iteration, thereby achieving batch-wise improvements. Using uncertainty quantification from the GPs, we formulate chance constraints to enforce safe operation to the required confidence levels. We demonstrate our approach in silico on a semi-batch polymerization reactor for tracking and economic objectives over durations of two hours, and the reactor temperature is constrained in a range of $\pm2^\circ C$ around its setpoint. After only four batch iterations, tracking error from the GP-MLMPC scheme converged to a reduction of $83\%$, compared to the initial trajectory. Furthermore, under an economic objective, the GP-MLMPC resulted in a 17-fold increase in final product mass by iteration 8, compared to the initial trajectory. In both cases, the resulting GP-MLMPC performance is on par with the full-model NMPC, which shows that the optimal controller can be learned by the approach. By collecting samples around the optimal trajectory, the GP-MLMPC remains sample-efficient across iterations and achieves quick convergence. Thus, the proposed GP-MLMPC scheme presents a promising data-efficient approach for the control of nonlinear batch processes without mechanistic knowledge.

We study the problem of adaptive control of the stochastic linear quadratic regulator (LQR) with constraints that must be satisfied at every time step. Prior work on the multidimensional problem has shown $\tilde{O}(T^{2/3})$ regret and satisfaction of robust constraints, leaving open the question of whether $\tilde{O}(\sqrt{T})$ regret can be attained in the constrained LQR setting. We contribute to this problem by showing $\tilde{O}(\sqrt{T})$ regret and satisfaction of chance constraints. This type of constraints allow us to handle unbounded noise and also enable analytical techniques not directly applicable to robust constraints. Our proposed algorithm for this problem uses an SDP to select an optimistic policy, and then "scales back" this policy until it is verifiably-safe. Our theoretical analysis establishes regret and constraint guarantees via a key lemma that bounds the system covariance in terms of the chosen policy. This covariance-based analysis is in contrast with the cost-to-go based analysis that is typically used in adaptive LQR.

In this paper, we present an equivalent convex optimization formulation for discrete-time stochastic linear systems subject to linear chance constraints, alongside a tight convex relaxation for quadratic chance constraints. By lifting the state vector to encode moment information explicitly, the formulation captures linear chance constraints on states and controls across multiple time steps exactly, without conservatism, yielding strict improvements in both feasibility and optimality. For quadratic chance constraints, we derive convex approximations that are provably less conservative than existing methods. We validate the framework on minimum-snap trajectory generation for a quadrotor, demonstrating that the proposed approach remains feasible at noise levels an order of magnitude beyond the operating range of prior formulations.

This paper presents novel method for distribution-free robust trajectory optimization and control of discrete-time, nonlinear, and non-Gaussian stochastic systems, with closed-loop guarantees on chance constraint satisfaction. Our framework employs conformal inference to generate coverage-based confidence sets for the closed-loop dynamics around arbitrary reference trajectories, by constructing a joint nonconformity score to quantify both the validity of contraction (i.e., incremental stability) conditions and the impact of external stochastic disturbance on the closed-loop dynamics, without any distributional assumptions. Via appropriate constraint tightening, chance constraints can be reformulated into tractable, statistically valid deterministic constraints on the reference trajectories. This enables a formal pathway to leverage and validate learning-based motion planners and controllers, such as those with neural contraction metrics, in safety-critical real-world applications. Notably, our statistical guarantees are non-diverging and can be computed with finite samples of the underlying uncertainty, without overly conservative structural priors. We demonstrate our approach in motion planning problems for designing safe, dynamically feasible trajectories in both numerical simulation and hardware experiments.

This paper proposes a strictly convex chance-constrained stochastic control framework that accounts for uncertainty in control specifications such as reference trajectories and operational constraints. Control inputs and risk allocation are jointly optimized under general, possibly non-Gaussian, uncertainties through sufficient deterministic conditions for probabilistic constraint satisfaction. A decreasing risk regularizer ensures existence and full use of the allowable risk. Under the stated regularity assumptions, the joint solution is unique and locally Lipschitz-continuous, supporting stable numerical optimization. The formulation is extended to learning-based exactly linearizable (EL) models through monotone state and input transformations. A hybrid powertrain example compares deterministic control, equal risk allocation, and optimized risk allocation with and without regularization in terms of charge sustaining, constraint violations, closed-loop cost, and solver time.