Wasserstein Ambiguity Sets for Robust Control

• Wasserstein ambiguity sets are defined as balls of probability measures around a nominal distribution based on a prescribed Wasserstein distance.
• They enable robust control in LTI systems by characterizing uncertainty propagation and enforcing risk constraints such as distributionally robust CVaR.
• Applications in systems like quadrotor landing demonstrate significant improvements in state dispersion control and constraint satisfaction under uncertainty.

A Wasserstein ambiguity set is a set of probability measures within a prescribed Wasserstein distance—typically $W_2$ with a Euclidean cost—from a nominal or reference distribution. These ambiguity sets play a central role in robust control and optimization problems where system noise or parameter distributions are only partially known. They enable the formulation of distributionally robust control objectives and constraints, wherein performance and safety must be guaranteed for all probability measures inside the set. Recent developments in robust density control exploit the properties of Wasserstein balls to propagate distributional uncertainty through linear time-invariant (LTI) systems and to enforce risk constraints, such as distributionally robust Conditional Value-at-Risk (CVaR) inequalities, through tractable convex programming.

1. Mathematical Definition of Wasserstein Ambiguity Sets

Let $P$ be a nominal probability distribution on $\mathbb{R}^d$ and $\mathcal{P}_2(\mathbb{R}^d)$ the set of probability measures with finite second moment. The type-2 Wasserstein distance between $P$ and $Q$ is

$$W_2(P, Q) = \left( \inf_{\pi \in \Pi(P, Q)} \int_{\mathbb{R}^d \times \mathbb{R}^d} \| \xi - \xi' \|^2 d\pi(\xi, \xi') \right)^{1/2}$$

where $\Pi(P, Q)$ is the set of couplings of $P$ and $Q$. The Wasserstein ambiguity set of radius $\epsilon$ is the closed ball

$$B_{\epsilon,2}^{\|\cdot\|}(P) = \{ Q \in \mathcal{P}_2(\mathbb{R}^d) : W_2(Q, P) \leq \epsilon \}$$

as formalized in [Definition 2, (Pilipovsky et al., 19 Mar 2024)].

This set captures all distributions with mass transport cost at most $\epsilon$ (measured in the Euclidean norm) from $P$.

2. Propagation Through Linear Dynamical Systems

When the disturbance noise $w_k$ in a stochastic LTI system

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

is modeled as lying within a Wasserstein ambiguity set around a nominal (often Gaussian) law, its effect propagates through the system as a pushforward ambiguity set. For an affine control law, the error state induced by the noise can be described by a linear map applied to the disturbance trajectory. The key result (Theorem 1, (Pilipovsky et al., 19 Mar 2024)) asserts that if $A$ is full row-rank, then for a ball $B_{\epsilon}^c(P)$,

$$A_{\#} B_{\epsilon}^c(P) = B_{\epsilon}^{c \circ A^\dagger}(A_{\#} P)$$

(where $A_{\#}$ is the pushforward by $A$ and $A^\dagger$ the Moore–Penrose pseudoinverse). This means the structure of the ambiguity set is preserved under the system flow: propagation of ambiguity can be characterized tightly and often closed-form.

3. Distributionally Robust CVaR Path Constraints

Risk constraints over uncertain state distributions are handled with distributionally robust CVaR constraints of the form

$$\sup_{P_k \in \mathcal{S}_k} \mathrm{CVaR}_{1 - \gamma}^{P_k} \left( \max_j \{ \alpha_j^\top x_k + \beta_j \} \right) \leq 0, \quad \forall k$$

where $\mathcal{S}_k$ is the ambiguity set for the state at time $k$. For $P_k$ inside a Wasserstein ball centered at a nominal Gaussian with mean $\hat{\mu}_k$ and covariance $\hat{\Sigma}_k$, these constraints are outer-approximated by tractable convex inequalities (Theorem 4, (Pilipovsky et al., 19 Mar 2024)): $$\beta_j + \alpha_j^\top \hat{\mu}_k(v) + \tau \sqrt{\alpha_j^\top \hat{\Sigma}_k(L) \alpha_j} + \tilde{\epsilon}_k(L) \|\alpha_j\| \leq 0$$ where $\tau = \sqrt{(1-\gamma)/\gamma}$ and $\tilde{\epsilon}_k(L)$ is a computable inflation term depending on the ambiguity radius and system gains.

This reduction leverages the Gelbrich distance and linear image properties of Wasserstein balls, allowing deterministic equivalent reformulations via semidefinite programming.

4. Optimal Feedback Law Synthesis

Affine state feedback laws of the form $u_k = K_k(x_k - \bar{x}_k) + v_k$ are employed, where $\bar{x}_k$ is a nominal trajectory. Upon change-of-variables, policy optimization is conducted over both $v$ (open-loop input) and $L$ (parameterizing feedback gain via $K = L(I + \mathcal{B}L)^{-1}$).

The optimization minimizes a cost function—including the worst-case quadratic loss over the ambiguity set and DR-CVaR constraints—subject to convex constraints on the mean and covariance of the terminal state as well as on ambiguity set size (through appropriate LMIs). This yields a convex semidefinite program amenable to standard interior-point solvers.

5. Terminal Ambiguity Set Constraints and Trajectory Shaping

Terminal safety and performance requirements are incorporated by enforcing that, at the final time step, the ambiguity set for the state is nested within a prescribed terminal set $S_f$, characterized by a desired mean $\mu_f$, covariance envelope $\Sigma_f$, and maximum ambiguity radius $\delta$. This is enforced through:

• Equality constraints on nominal mean,
• Linear matrix inequalities ensuring $\hat{\Sigma}_N(L) \preceq \Sigma_f$,
• An LMI constraining the maximum singular value of the state transformation applied to the ambiguity radius: $\epsilon \sigma_{\max}^2(\tilde{L}_N) \leq \delta$.

Thus, both the center and "shape" (dispersion, ambiguity) of the terminal state distribution are controlled.

6. Advantages over Moment-Based and Nominal Approaches

Wasserstein ambiguity sets generalize classical Chebyshev-type ellipsoidal sets: they are geometrically aware, capable of enforcing constraints for all distributions within a specified transportation distance from the nominal law—rather than just matching low-order moments. Propagation rules and concentration inequalities guarantee that such sets tightly enclose the true state distribution with high probability, provided the radius is selected based on sample size and system properties. Relaxations (e.g., Gelbrich distance-based outer approximations) yield convex reformulations. Compared to approaches that enforce path or terminal constraints for a single nominal distribution, Wasserstein-based robust density control reduces the risk of constraint violation under distributional shifts, heavy-tailed disturbances, or model misspecification.

7. Applications and Simulation Insights

The framework is applied to quadrotor landing under stochastic wind turbulence ((Pilipovsky et al., 19 Mar 2024), Section 6). The system is linearized about a reference trajectory, and the wind disturbance process is modeled as nominal Gaussian with spatially-varying covariance. With Wasserstein ambiguity sets for the wind, the controller synthesizes affine feedback to steer the mean and shape the spread of the state distribution. DR-CVaR constraints enforce safety requirements along the trajectory. Results show that for nominal and severe or heavy-tailed wind conditions, the distributionally robust controller yields tighter containment of the final state distribution and lower violation probability than covariance-steering (moment-based) methods. The robust solution achieves up to 15-fold reduction in dispersion for critical state components at high wind speeds, confirming the practical value of Wasserstein ambiguity sets for safety-critical distributed control under uncertainty.

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Aspect	Wasserstein Ambiguity Set	Moment-Based Set
Geometric structure	Metric ball in distribution space, geometry-aware	Ellipsoid (Chebyshev), moment-based
Propagation through LTI	Closed-form (pushforward, scaling)	Closed-form (for mean/covariance)
Path constraint enforcement	DR-CVaR constraints, outer-approximation by LMI/SOCC	Approximated via Boole's inequality
Finite-sample guarantees	Explicit (concentration-of-measure, high probability)	At moment level, possibly less tight

This framework systematically combines pushforward propagation of ambiguity, tractable convex reformulations, and rigorous probabilistic coverage for robust density control. Its effectiveness is demonstrated for stochastic systems with significant uncertainty in noise distribution.