We introduce a framework for the control of discrete-time switched stochastic systems with uncertain distributions. In particular, we consider stochastic dynamics with additive noise whose distribution lies in an ambiguity set of distributions that are ɛ−close, in the Wasserstein distance sense, to a nominal one. We propose algorithms for the efficient synthesis of distributionally robust control strategies that maximize the satisfaction probability of reach-avoid specifications with either a given or an arbitrary (not specified) time horizon, i.e., unbounded-time reachability. The framework consists of two main steps: finite abstraction and control synthesis. First, we construct a finite abstraction of the switched stochastic system as a robust Markov decision process (robust MDP) that encompasses both the stochasticity of the system and the uncertainty in the noise distribution. Then, we synthesize a strategy that is robust to the distributional uncertainty on the resulting robust MDP. We employ techniques from optimal transport and stochastic programming to reduce the strategy synthesis problem to a set of linear programs, and propose a tailored and efficient algorithm to solve them. The resulting strategies are correctly refined into switching strategies for the original stochastic system. We illustrate the efficacy of our framework on various case studies comprising both linear and non-linear switched stochastic systems.@en

This paper builds Wasserstein ambiguity sets for the unknown probability distribution of dynamic random variables leveraging noisy partial-state observations. The constructed ambiguity sets contain the true distribution of the data with quantifiable probability and can be exploited to formulate robust stochastic optimization problems with out-of-sample guarantees. We assume the random variable evolves in discrete time under uncertain initial conditions and dynamics, and that noisy partial measurements are available. All random elements have unknown probability distributions and we make inferences about the distribution of the state vector using several output samples from multiple realizations of the process. To this end, we leverage an observer to estimate the state of each independent realization and exploit the outcome to construct the ambiguity sets. We illustrate our results in an economic dispatch problem involving distributed energy resources over which the scheduler has no direct control.

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The aim of this paper is to compare two classes of structured ambiguity sets, which are data-driven and can reduce the conservativeness of their associated optimization problems. These two classes of structured sets, coined Wasserstein hyperrectangles and multi-transport hyperrectangles, are explored in their trade-offs in terms of reducing conservativeness and providing tractable reformulations. It follows that multi-transport hyperrectangles lead to tractable optimization problems for a significantly broader range of objective functions under a decent compromise in terms of conservativeness reduction. The results are illustrated in an uncertainty quantification case study.@en

This paper provides a data-driven solution to the problem of coverage control by which a team of robots aims to optimally deploy in a spatial region where certain event of interest may occur. This event is random and described by a probability density function, which is unknown and can only be learned by collecting data. In this work, we hedge against this uncertainty by designing a distributionally robust algorithm that optimizes the locations of the robots against the worst-case probability density from an ambiguity set. This ambiguity set is constructed from data initially collected by the agents, and contains the true density function with prescribed confidence. However, the objective function that the robots seek to minimize is non-smooth. To address this issue, we employ the so-called gradient sampling algorithm, which approximates the Clarke generalized gradient by sampling the derivative of the objective function at nearby locations and stabilizes the choice of descent directions around points where the function may fail to be differentiable. This enables us to prove that the algorithm converges to a stationary point from any initial location of the robots, in analogy to the well-known Lloyd algorithm for differentiable costs when the spatial density is known.

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We present a novel framework for formal control of uncertain discrete-time switched stochastic systems against probabilistic reach-avoid specifications. In particular, we consider stochastic systems with additive noise, whose distribution lies in an ambiguity set of distributions that are ε−close to a nominal one according to the Wasserstein distance. For this class of systems we derive control synthesis algorithms that are robust against all these distributions and maximize the probability of satisfying a reach-avoid specification, defined as the probability of reaching a goal region while being safe. The framework we present first learns an abstraction of a switched stochastic system as a robust Markov decision process (robust MDP) by accounting for both the stochasticity of the system and the uncertainty in the noise distribution. Then, it synthesizes a strategy on the resulting robust MDP that maximizes the probability of satisfying the property and is robust to all uncertainty in the system. This strategy is then refined into a switching strategy for the original stochastic system. By exploiting tools from optimal transport and stochastic programming, we show that synthesizing such a strategy reduces to solving a set of linear programs, thus guaranteeing efficiency. We experimentally validate the efficacy of our framework on various case studies, including both linear and non-linear switched stochastic systems. Our results represent the first formal approach for control synthesis of stochastic systems with uncertain noise distribution.

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This paper introduces a spectral parameterization of ambiguity sets to hedge against distributional uncertainty in stochastic optimization problems. We build an ambiguity set of probability densities around a histogram estimator, which is constructed by independent samples from the unknown distribution. The densities in the ambiguity set are determined by bounding the distance between the coefficients of their Haar wavelet expansion and the expansion of the histogram estimator. This representation facilitates the computation of expectations, leading to tractable minimax problems that are linear in the parameters of the ambiguity set, and enables the inclusion of additional constraints that can capture valuable prior information about the unknown distribution.@en

Ambiguity sets of probability distributions are a prominent tool to hedge against distributional uncertainty in stochastic optimization. The aim of this paper is to build tight Wasserstein ambiguity sets for data-driven optimization problems. The method exploits independence between the distribution components to introduce structure in the ambiguity sets and speed up their shrinkage with the number of collected samples. Tractable reformulations of the stochastic optimization problems are derived for costs that are expressed as sums or products of functions that depend only on the individual distribution components. The statistical benefits of the approach are theoretically analyzed for compactly supported distributions and demonstrated in a numerical example.@en