Robust Control and Ambiguity

• Robust control under ambiguity is defined as designing strategies that remain effective despite incomplete system models by using ambiguity sets to capture uncertainty.
• Methodologies such as moment-based, Wasserstein, KL, and Gelbrich sets transform worst-case analysis into tractable optimization problems.
• Empirical applications in LQG, MPC, and data-driven settings demonstrate that these approaches offer reliable performance with strong theoretical guarantees.

Robust control and ambiguity addresses the design and analysis of control strategies that maintain good performance when key system parameters, disturbance statistics, or even the governing probability laws are incompletely known. Approaches in this field formalize and exploit "ambiguity sets"—collections of distributions or models deemed plausible—then synthesize control policies that hedge against the worst-case member of these sets. The interplay between tractable ambiguity set specification, computational reformulation, and theoretical guarantees defines the technical landscape of robust control under ambiguity. The following exposition surveys core formulations, ambiguity modeling techniques, computational methods, and representative results as established in contemporary research.

1. Formalization of Ambiguity in Robust Control

Uncertainty in robust control is modeled via ambiguity sets: collections of probability measures or models that capture the controller’s incomplete information. The controller then solves a min–max or saddle-point problem, seeking a policy that achieves the best guaranteed performance (e.g., minimal cost or loss) against the worst-case realization of the unknown from within the ambiguity set.

Let $\mathcal{M}$ denote the set of policies (controllers) and $\mathcal{A}\subset\mathcal{P}$ an ambiguity set of distributions on the uncertainty space. The robust control problem takes the canonical form: $$\inf_{M\in\mathcal{M}} \sup_{\mu\in\mathcal{A}} \mathbb{E}_{\mu}\left[ J(M, w) \right]$$ where $J$ is the cost, $w$ are the exogenous uncertainties, and the supremum is over all laws $\mu$ in $\mathcal{A}$. In dynamic settings, this is embedded in dynamic programming or receding-horizon (MPC) frameworks.

Ambiguity sets can take numerous forms:

• Moment-based sets: $$\mathcal{A} = \{ \mu : \mathbb{E}_{\mu}[w] = m, \operatorname{Cov}_{\mu}[w]=\Gamma \}$$
• Wasserstein balls: $$\mathcal{A} = \{ \mu : W_p(\mu, \hat\mu) \leq \rho \}$$ for empirical law $\hat\mu$
• Kullback–Leibler (KL) balls: $$\mathcal{A} = \{ \mu : D_{KL}(\mu\|\hat\mu) \leq \rho \}$$
• Gelbrich sets: All Gaussian laws whose mean/covariance is within a Gelbrich (closed-form Wasserstein-2) distance $\rho$ from an empirical pair $(\bar m, \bar \Gamma)$
• Projection-constrained sets: $$\mathcal{A} = \{P: \mathbb{E}_{P}[\phi_j(w_j^\top \xi)] \leq b_j, \forall j\}$$ (constraints on 1D projections)

The choice and tractability of $\mathcal{A}$ are critical for both computational and robustness properties (Pan et al., 2023, Fochesato et al., 13 May 2025, Cescon et al., 26 Mar 2025, Tzikas et al., 9 Aug 2025, Ye et al., 11 Oct 2025).

2. Canonical Robust Control Formulations and Theoretical Guarantees

In standard robust optimal control, the controller maximizes the worst-case expected payoff (or minimizes the cost) over all distributions in the ambiguity set. Given a stochastic LTI system with unknown disturbance law $\mu_W$: $$x_{k+1} = A x_k + B u_k + E w_k, \quad y_k = C x_k + D u_k + F w_k,$$ where only an ambiguity set $\mathcal{A}$ is known for $\mu_W$, robust control seeks

$$\min_{u_{0:N-1}} \max_{\mu \in \mathcal{A}} \mathbb{E}_{\mu}\Bigl[\sum_{k=0}^{N-1}(y_k^\top Q y_k + u_k^\top R u_k)\Bigr]$$

subject to dynamics and constraints.

Key robustness theorems establish that, for appropriate choices of $\mathcal{A}$ (e.g., moment-based, Wasserstein, or Gelbrich), the robust and min–max value coincide, and saddle-point optimal policy $\pi^*$ exists (Pan et al., 2023, Fochesato et al., 13 May 2025, Taşkesen et al., 2023). For well-structured ambiguity sets (e.g., continuous-moment sets, Wasserstein balls), minimax duality or strong duality applies, and any optimal robust policy achieves its guarantee not only in the worst-case law within $\mathcal{A}$ but also for nearby distributions in the weak topology—a property sometimes called "robust robustness" (Ball et al., 29 Aug 2024).

3. Specification and Reformulation of Ambiguity Sets

The specification of $\mathcal{A}$ and its reformulation into computational primitives are pivotal. Modern approaches employ:

• Moment and Gelbrich sets: The set of Gaussian laws with $(m, \Gamma)$ in a Gelbrich ball about $(\bar m, \bar \Gamma)$ provides a closed, convex, and tractable set. This set is bijective to an ellipsoid in the space of polynomial chaos (PCE) coefficients, enabling exact finite-dimensional reductions for robust stochastic control in unknown LTI systems (Pan et al., 2023).
• Wasserstein and Sinkhorn balls: Wasserstein ambiguity sets offer non-parametric, data-driven ambiguity for distributionally robust optimization (DRO). The Sinkhorn-regularized variant introduces an entropic regularization encouraging closeness to a reference law $\nu$ and interpolates between pure Wasserstein DRO (as $\epsilon\to 0$) and classical stochastic control ($\epsilon\to\infty$) (Cescon et al., 26 Mar 2025).
• KL-balls and relative entropy: KL-based ambiguity sets are natural for robust statistical decision-theory and LQG robustification, often leading to convex programs indexed by relative-entropy radius (Fochesato et al., 13 May 2025).
• One-dimensional projections: Ambiguity sets enforcing constraints on expectations of $\phi_j(w_j^\top \xi)$ produce highly interpretable yet expressive ambiguity. Such sets can encode means, variances, probabilities of halfspaces, and admit exact dualization and finite convex reformulations under convexity (Tzikas et al., 9 Aug 2025).

Tabulated summary:

Ambiguity set type	Example definition	Reformulation tractability
Moment-based/Gelbrich	$(m,\Gamma)$ within Gelbrich ball	Convex, PCE/ellipsoidal (Pan et al., 2023)
Wasserstein/Sinkhorn ball	$d_W(\mu, \hat\mu) \leq \rho$ (Sinkhorn: $W_c^\epsilon$)	Duality (LMIs), convex (Cescon et al., 26 Mar 2025, Taşkesen et al., 2023)
KL-ball (relative entropy)	$D_{KL}(\mu \| \hat\mu) \leq \rho$	Duality, Riccati for LQG (Fochesato et al., 13 May 2025)
1D projection/functional constraints	$E_P[\phi_j(w_j^\top \xi)] \leq b_j$	Semi-infinite/finite convex (Tzikas et al., 9 Aug 2025)

4. Data-Driven and Scenario-Based Solution Methods

Robust control under ambiguity is frequently implemented in data-driven or model-free situations, relying on historical data or sampled experiments:

• Stochastic fundamental lemma: Trajectories of unknown LTI systems can be equivalently expressed as linear combinations (in the PCE basis) of columns of a Hankel matrix built from past data. This framework enables system description without explicit model identification in data-driven robust control (Pan et al., 2023).
• Scenario sampling and convex relaxations: Infinite-dimensional uncertainty over $\mathcal{A}$ can be approximated by considering a convex hull of finitely many sampled "extreme points" of the ambiguity set. Scenario-based SOCPs or MILPs, using randomly sampled vertices, produce approximate solutions with probabilistic guarantees on constraint satisfaction given established sample complexity bounds (Pan et al., 2023, Ma et al., 8 Feb 2025).
• Numerical optimization algorithms: For large-scale or structured ambiguity sets (e.g., in robust LQG), projection-free Frank–Wolfe schemes efficiently compute least-favorable noise covariances, capitalizing on the smooth and concave nature of the inner cost functions over covariance matrices (Taşkesen et al., 2023). Sinkhorn ambiguity sets lead to convex conic programs with LMI and log-det constraints, solvable efficiently for moderate horizon, sample size, and dimension (Cescon et al., 26 Mar 2025).
• Reinforcement learning and robust Q-learning: When ambiguity sets consist of a finite collection of plausible Markov kernels, robust Q-learning with proper update rules converges almost surely to the robust Bellman optimum, scaling linearly in the ambiguity set cardinality. This permits full control over which uncertainties are hedged (Decker et al., 5 Jul 2024).

5. Theoretical Robustness, Performance, and Limitations

The robust control literature investigates both exact and approximate performance guarantees:

• Strong minimax duality: For ambiguity sets with convexity and stability (e.g., moment sets, Wasserstein balls), robust control objectives admit strong duality and saddle-points, i.e., the robust-optimal policy is linear (in LQG) and the adversarial worst-case distribution is Gaussian with appropriately perturbed covariance and mean. Convexification through the PCE or LMI/SDP relaxations preserves exactness under suitable mappings (Pan et al., 2023, Taşkesen et al., 2023, Fochesato et al., 13 May 2025).
• Global payoff robustness and topological regularity: Recent work sharpens the classical max–min criterion by requiring the payoff guarantee to extend in a weak-topology neighborhood of the ambiguity set—i.e., small misspecification in $\mathcal{A}$ should not cause discontinuous loss in guaranteed performance. Continuity holds for continuous-moment and Wasserstein-ball sets, but fails for support, quantile, or total-variation ambiguity sets (Ball et al., 29 Aug 2024).
• Probabilistic guarantees under finite sampling: Scenario-based relaxations inherit finite-sample guarantees on constraint satisfaction, often leveraging bounds from random convex programming. The sample size $s$ can be chosen to achieve prescribed risk $\epsilon$ and confidence $1-\beta$ (Pan et al., 2023, Ma et al., 8 Feb 2025).
• Nonconservatism and tightness: Bayesian and value-targeted algorithmic construction of ambiguity sets (e.g., RSVF, Bayesian credible regions) adapt their size and location to the posterior, often sharply reducing worst-case regret relative to classical confidence-region approaches (Russel et al., 2018, Petrik et al., 2019).
• Limitations: Non-convex ambiguity sets may not admit finite convex reduction; convergence of iterative solution dynamics (e.g., best-response for projection-constrained sets) is not always guaranteed; scenario approximations may underperform for high-dimensional or highly non-linear systems.

6. Representative Applications and Empirical Insights

Robust control under ambiguity has been instantiated in regimes including:

• Data-driven LTI control: Double integrator systems with unknown, non-Gaussian disturbance are robustly controlled by deriving Hankel data from random open-loop trials, computing empirical moments, and solving convex programs over the derived Gelbrich ambiguity set. Robust policies substantially reduce constraint violations compared to optimistic (single-moment) designs (Pan et al., 2023).
• Classical examples—LQG and MPC: Redesigns of LQG via Wasserstein or KL ambiguity sets yield robust controllers with optimal linear structure. In distributionally robust convex formulations, policies outperform classic, non-robust LQG under model misspecification, with computational viability established via Frank–Wolfe or dynamic programming approximations (Fochesato et al., 13 May 2025, Taşkesen et al., 2023, Cescon et al., 26 Mar 2025).
• Reinforcement learning and MDPs: Bayesian and RSVF approaches in robust MDPs realize dramatic reductions in safe-return regret, empirically matching or outperforming classic union-bound-based RMDPs (Russel et al., 2018, Petrik et al., 2019). Finite-ambiguity robust Q-learning achieves tractable and reliable hedging in regime-switching environments (Decker et al., 5 Jul 2024).
• Portfolio and trajectory planning: Projection-constrained ambiguity enables modeling mean, variance, and quantile band uncertainty in asset returns or initial state. Cutting-set and best-response algorithms solve robust portfolio and trajectory planning under such ambiguity, with cycle-free convergence observed in practice for convex subproblems (Tzikas et al., 9 Aug 2025).
• Hybrid and integer control: Piecewise-linear control via lifted policies—augmented by discretized disturbance segments—gains asymptotic optimality as segmentation increases, with tight non-asymptotic suboptimality bounds, enabling robust hybrid MPC under Wasserstein uncertainty (Ma et al., 8 Feb 2025).

Empirical outcomes consistently indicate that robust control under meticulously specified ambiguity sets delivers nonconservative performance (cost increase is mild relative to the robustification), enforces constraints with high reliability, and is computationally tractable within typical application scales.

7. Behavioral and Decision-Theoretic Foundations

Beyond algorithmic and performance guarantees, the behavioral underpinnings of robust control under ambiguity have been formalized:

• Continuity axioms and max–min preferences: A robust policy’s guarantee is behaviorally robust if, loosely, action-graphs are upper semicontinuous at every act—formalized in terms of graphical limits. This axiom links topological continuity of state space with epistemic structure on ambiguity, strengthening classical Gilboa–Schmeidler max–min expected utility (Ball et al., 29 Aug 2024).
• Comparative statics and ambiguity set selection: Mechanisms and control designs based on continuous-moment or Wasserstein ambiguity sets inherit topological robustness of guarantees; those based on support, quantile, or total variation constraints may display discontinuous drops in worst-case performance under small model misspecification.

This decision-theoretic lens guides both the mathematical formalization and the practical construction of ambiguity sets in robust control synthesis.

---

References:

• (Pan et al., 2023) Distributionally robust uncertainty quantification via data-driven stochastic optimal control
• (Ball et al., 29 Aug 2024) Robust Robustness
• (Taşkesen et al., 2023) Distributionally Robust Linear Quadratic Control
• (Fochesato et al., 13 May 2025) Distributionally Robust LQG with Kullback-Leibler Ambiguity Sets
• (Cescon et al., 26 Mar 2025) Data-driven Distributionally Robust Control Based on Sinkhorn Ambiguity Sets
• (Tzikas et al., 9 Aug 2025) Distributionally Robust Control with Constraints on Linear Unidimensional Projections
• (Ma et al., 8 Feb 2025) Data-Driven Distributionally Robust Mixed-Integer Control through Lifted Control Policy
• (Decker et al., 5 Jul 2024) Robust Q-Learning for finite ambiguity sets
• (Russel et al., 2018) Tight Bayesian Ambiguity Sets for Robust MDPs
• (Petrik et al., 2019) Beyond Confidence Regions: Tight Bayesian Ambiguity Sets for Robust MDPs