Bart P. G. Van Parys

Generalized Gauss inequalities via semidefinite programming

A sharp upper bound on the probability of a random vector falling outside a polytope, based solely on the first and second moments of its distribution, can be computed efficiently using semidefinite programming. However, this Chebyshev-type bound tends to be overly conservative since it is determined by a discrete worst-case distribution. In this paper we obtain a less pessimistic Gauss-type bound by imposing the additional requirement that the random vector’s distribution must be unimodal. We prove that this generalized Gauss bound still admits an exact and tractable semidefinite representation. Moreover, we demonstrate that both the Chebyshev and Gauss bounds can be obtained within a unified framework using a generalized notion of unimodality. We also offer new perspectives on the computational solution of generalized moment problems, since we use concepts from Choquet theory instead of traditional duality arguments to derive semidefinite representations for worst-case probability bounds.

Distributionally Robust Control of Constrained Stochastic Systems

We investigate the control of constrained stochastic linear systems when faced with limited information regarding the disturbance process, i.e., when only the first two moments of the disturbance distribution are known. We consider two types of distributionally robust constraints. In the first case, we require that the constraints hold with a given probability for all disturbance distributions sharing the known moments. These constraints are commonly referred to as distributionally robust chance constraints. In the second case, we impose conditional value-at-risk (CVaR) constraints to bound the expected constraint violation for all disturbance distributions consistent with the given moment information. Such constraints are referred to as distributionally robust CVaR constraints with second-order moment specifications. We propose a method for designing linear controllers for systems with such constraints that is both computationally tractable and practically meaningful for both finite and infinite horizon problems. We prove in the infinite horizon case that our design procedure produces the globally optimal linear output feedback controller for distributionally robust CVaR and chance constrained problems. The proposed methods are illustrated for a wind blade control design case study for which distributionally robust constraints constitute sensible design objectives.

Infinite Horizon Performance Bounds for Uncertain Constrained Systems

We present a new method to bound the performance of controllers for uncertain linear systems with mixed state and input constraints and bounded disturbances. We take as a performance metric either an expected-value or minimax discounted cost over an infinite horizon, and provide a method for computing a lower bound on the achievable performance of any causal control policy in either case. Our lower bound is compared to an upper performance bound provided by restricting the choice of controller to one that is affine in the observed disturbances, and we show that the two bounds are closely related. In particular, the lower bounds have a natural interpretation in terms of affine control policies that are optimal for a problem with a restricted disturbance set. We show that our performance bounds can be computed via solution of a finite-dimensional convex optimization problem, and provide numerical examples to illustrate the efficacy of our method.

Optimal control for load alleviation in wind turbines

Nowadays, trailing edge flaps on wind turbine blades are considered to reduce loading stresses in wind turbine components. In this paper, an optimal control synthesis methodology for the design of gust load controllers for large wind turbine blades is proposed. We discuss a control synthesis approach that minimises the power expenditure of the actuated trailing edge flap, while at the same time guaranteeing that certain blade load measures remain bounded in a probabilistic sense. To illustrate our proposed control design methodology, a standard NREL 5-MW reference turbine was considered. The obtained numerical results indicate that through the use of optimal feedback considerable reductions in loading stresses could be achieved for moderate actuation power.

Multivariate Chebyshev Inequality With Estimated Mean and Variance

ABSTRACT A variant of the well-known Chebyshev inequality for scalar random variables can be formulated in the case where the mean and variance are estimated from samples. In this article, we present a generalization of this result to multiple dimensions where the only requirement is that the samples are independent and identically distributed. Furthermore, we show that as the number of samples tends to infinity our inequality converges to the theoretical multi-dimensional Chebyshev bound.

Distributionally robust expectation inequalities for structured distributions

Quantifying the risk of unfortunate events occurring, despite limited distributional information, is a basic problem underlying many practical questions. Indeed, quantifying constraint violation probabilities in distributionally robust programming or judging the risk of financial positions can both be seen to involve risk quantification under distributional ambiguity. In this work we discuss worst-case probability and conditional value-at-risk problems, where the distributional information is limited to second-order moment information in conjunction with structural information such as unimodality and monotonicity of the distributions involved. We indicate how exact and tractable convex reformulations can be obtained using standard tools from Choquet and duality theory. We make our theoretical results concrete with a stock portfolio pricing problem and an insurance risk aggregation example.