Data-driven distributionally robust MPC for constrained stochastic systems

Abstract

In this paper we introduce a novel approach to distributionally robust optimal control that supports online learning of the ambiguity set, while guaranteeing recursive feasibility. We introduce conic representable risk, which is useful to derive tractable reformulations of distributionally robust optimization problems. Specifically, to illustrate the techniques introduced, we utilize risk measures constructed based on data-driven ambiguity sets, constraining the second moment of the random disturbance. In the optimal control setting, such moment-based risk measures lead to tractable optimal controllers when combined with affine disturbance feedback. Assumptions on the constraints are given that guarantee recursive feasibility. The resulting control scheme acts as a robust controller when little data is available and converges to the certainty equivalent controller when a large sample count implies high confidence in the estimated second moment. This is illustrated in a numerical experiment.

Safe, learning-based MPC for highway driving under lane-change uncertainty: A distributionally robust approach

“…This phenomenon, which is akin to overfitting, can be mitigated in a systematic manner by additionally constructing a set of probability vectors which account for potential misestimations. Such a set is commonly referred to as an ambiguity set and can be constructed in a variety of manners, each valid for different settings and underlying assumptions on the underlying system and data-generating distributions [50,49,51,52,16].…”

“…The ambiguity set (13) belongs to the general set of conic-representable risk measures, which allows us to tractably reformulate the OCP (24) as a standard nonlinear program, using duality-based techniques described in [12,28,52]. This section provides some exploratory and qualitative simulation results which motivate some of the decisions made for the configurations of further experiments.…”

Data-driven distributionally robust MPC using the Wasserstein metric

“…random process is a common assumption made in control literature, e.g. Arcari et al (2020); Coppens and Patrinos (2021). It assumes a priori that only the first two moments of the random process are acquired as partial distributional information, which can either be estimated or determined a priori Wan, Wang, Liu and Tong (2014).…”

“…The control policies for both finite horizon optimal control problem with expected quadratic objective and infinite horizon optimal control problem with an average cost can be determined concerning distributionally robust CVaR constraints modeled by moment-based ambiguity sets Van Parys, Kuhn, Goulart and Morari (2015). Recently, a data-driven distributionally robust MPC with a moment-based ambiguity set for quadratic objective function under multi-stage risk measures was proposed in Coppens and Patrinos (2021).…”

Distributionally Robust Decision Making Leveraging Conditional Distributions

“…Depending on the probability space, a DRO problem can be discrete (the random variable has discrete support), or continuous (the support is continuous). Computationally, the continuous problem is much more complex, and many methods rely on sampling to gain computation tractability [2], [3], [8]. Most existing data-driven methods assume i.i.d.…”