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. Recent generalizations of the Chebyshev inequality (1) provide upper bounds on the probability of a multivariate random vector ξ ∈ R n falling outside a prescribed confidence region Ξ ⊆ R n if only a few low-order moments of ξ
We present a new method to bound the performance of causal controllers for uncertain linear systems with mixed state and input constraints. The performance is measured by the expected value of a discounted linear quadratic cost function over an infinite horizon. Our method computes a lower bound on the lowest achievable cost of any causal control policy. We compare our lower performance bound with the best performance achievable using the restricted class of disturbance affine control policies, both of which can be computed by solving convex conic optimization problems that are closely connected. The feasible sets of both convex programs have a natural relationship with respect to the maximal robust control invariant (RCI) set of the control problem. We present two numerical examples to illustrate the utility of our method.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.