This paper proposes a novel approach to construct data-driven online solutions to optimization problems (P) subject to a class of distributionally uncertain dynamical systems. The introduced framework allows for the simultaneous learning of distributional system uncertainty via a parameterized, control-dependent ambiguity set using a finite historical data set, and its use to make online decisions with probabilistic regret function bounds. Leveraging the merits of Machine Learning, the main technical approach relies on the theory of Distributional Robust Optimization (DRO), to hedge against uncertainty and provide less conservative results than standard Robust Optimization approaches. Starting from recent results that describe ambiguity sets via parameterized, and control-dependent empirical distributions as well as ambiguity radii, we first present a tractable reformulation of the corresponding optimization problem while maintaining the probabilistic guarantees. We then specialize these problems to the cases of 1) optimal one-stage control of distributionally uncertain nonlinear systems, and 2) resource allocation under distributional uncertainty. A novelty of this work is that it extends DRO to online optimization problems subject to a distributionally uncertain dynamical system constraint, handled via a control-dependent ambiguity set that leads to online-tractable optimization with probabilistic guarantees on regret bounds. Further, we introduce an online version of Nesterov's accelerated-gradient algorithm, and analyze its performance to solve this class of problems via dissipativity theory.
Established approaches to obtain generalization bounds in data-driven optimization and machine learning mostly build on solutions from empirical risk minimization (ERM), which depend crucially on the functional complexity of the hypothesis class. In this paper, we present an alternate route to obtain these bounds on the solution from distributionally robust optimization (DRO), a recent data-driven optimization framework based on worst-case analysis and the notion of ambiguity set to capture statistical uncertainty. In contrast to the hypothesis class complexity in ERM, our DRO bounds depend on…
We study stochastic optimization problems with chance and risk constraints, where in the latter, risk is quantified in terms of the conditional value-at-risk (CVaR). We consider the distributionally robust versions of these problems, where the constraints are required to hold for a family of distributions constructed from the observed realizations of the uncertainty via the Wasserstein distance. Our main results establish that if the samples are drawn independently from an underlying distribution and the problems satisfy suitable technical assumptions, then the optimal value and optimizers of …
This paper is motivated by addressing open questions in distributionally robust chance-constrained programs (DRCCP) using the popular Wasserstein ambiguity sets. Specifically, the computational techniques for those programs typically place restrictive assumptions on the constraint functions and the size of the Wasserstein ambiguity sets is often set using costly cross-validation (CV) procedures or conservative measure concentration bounds. In contrast, we propose a practical DRCCP algorithm using kernel maximum mean discrepancy (MMD) ambiguity sets, which we term MMD-DRCCP, to treat general no…
Distributionally Robust Optimization for Sequential Decision Making
Imperial College Business School, Imperial College London（帝国理工学院商学院）；Neuri Pte Ltd（Neuri私人有限公司）；Department of Industrial Systems Engineering and Management, National University of Singapore（新加坡国立大学工业系统工程与管理系）
Residual-Aware Distributionally Robust EKF: Absorbing Linearization Mismatch via Wasserstein Ambiguity
Department of Mechanical Science and Engineering, Grainger College of Engineering, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校格雷格工程学院机械科学与工程系）；Department of Electrical and Computer Engineering and Automation and Systems Research Institute, Seoul National University（首尔国立大学电气与计算机工程系及自动化与系统研究所）；Center for Scientific Innovation and Education and the National Polytechnic University of Armenia（亚美尼亚国立理工大学科学创新与教育中心）
This article extends the optimal covariance steering (CS) problem for discrete time linear stochastic systems modeled using moment-based ambiguity sets. To hedge against the uncertainty in the state distributions while performing covariance steering, distributionally robust risk constraints are employed during the optimal allocation of the risk. Specifically, a distributionally robust iterative risk allocation (DR-IRA) formalism is used to solve the optimal risk allocation problem for the CS problem using a two-stage approach. The upper-stage of DR-IRA is a convex problem that optimizes the ri…
This paper considers a security constrained dispatch problem involving generation and line contingencies in the presence of the renewable generation. The uncertainty due to renewables is modeled using joint chance-constraint and the mismatch caused by contingencies and renewables are handled using reserves. We consider a distributionally robust approach to solve the chance-constrained program. We assume that samples of the uncertainty are available. Using them, we construct a set of distributions, termed ambiguity set, containing all distributions that are close to the empirical distribution u…
Collaborative-Online-Learning-Enabled Distributionally Robust Motion Control for Multi-Robot Systems
School of Automation and Intelligent Sensing, Shanghai Jiao Tong University, Shanghai, China（自动化与智能感知学院，上海交通大学，上海，中国）；Department of Mechanical Engineering, University of Victoria, Victoria, Canada（机械工程系，维多利亚大学，维多利亚，加拿大）
The increasing share of renewables in the electricity generation mix comes along with an increasing uncertainty in power supply. In the recent years, distributionally robust optimization has gained significant interest due to its ability to make informed decisions under uncertainty, which are robust to misrepresentations of the distributional information (e.g., from probabilistic forecasts). This is achieved by introducing an ambiguity set that describes the potential deviations from an empirical distribution of all uncertain parameters. However, this set typically overlooks the inherent depen…
Safety is a critical issue in learning-based robotic and autonomous systems as learned information about their environments is often unreliable and inaccurate. In this paper, we propose a risk-aware motion control tool that is robust against errors in learned distributional information about obstacles moving with unknown dynamics. The salient feature of our model predictive control (MPC) method is its capability of limiting the risk of unsafety even when the true distribution deviates from the distribution estimated by Gaussian process (GP) regression, within an ambiguity set. Unfortunately, t…
This paper proposes a distributionally robust unit commitment approach for microgrids under net load and electricity market price uncertainty. The key thrust of the proposed approach is to leverage the Kullback-Leibler divergence to construct an ambiguity set of probability distributions and formulate an optimization problem that minimizes the expected costs brought about by the worst-case distribution in the ambiguity set. The proposed approach effectively exploits historical data and capitalizes on the k-means clustering algorithm, in conjunction with the soft dynamic time warping score, to …
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 co…
This paper studies the problem of distributionally robust model predictive control (MPC) using total variation distance ambiguity sets. For a discrete-time linear system with additive disturbances, we provide a conditional value-at-risk reformulation of the MPC optimization problem that is distributionally robust in the expected cost and chance constraints. The distributionally robust chance constraint is over-approximated as a simpler, tightened chance constraint that reduces the computational burden. Numerical experiments support our results on probabilistic guarantees and computational effi…
Safely deploying robots in uncertain and dynamic environments requires a systematic accounting of various risks, both within and across layers in an autonomy stack from perception to motion planning and control. Many widely used motion planning algorithms do not adequately incorporate inherent perception and prediction uncertainties, often ignoring them altogether or making questionable assumptions of Gaussianity. We propose a distributionally robust incremental sampling-based motion planning framework that explicitly and coherently incorporates perception and prediction uncertainties. We desi…
Efficient Probabilistic Planning with Maximum-Coverage Distributionally Robust Backward Reachable Trees
Laboratory for Information and Decision Systems, Massachusetts Institute of Technology（信息与决策系统实验室，麻省理工学院）；Draper, Cambridge MA 02139（Draper，剑桥MA 02139）
Model predictive control (MPC) is an effective approach to control multivariable dynamic systems with constraints. Most real dynamic models are however affected by plant-model mismatch and process uncertainties, which can lead to closed-loop performance deterioration and constraint violations. Methods such as stochastic MPC (SMPC) have been proposed to alleviate these problems; however, the resulting closed-loop state trajectory might still significantly violate the prescribed constraints if the real system deviates from the assumed disturbance distributions made during the controller design. …
This paper considers a risk-constrained infinite-horizon optimal control problem and proposes to solve it in an iterative manner. Each iteration of the algorithm generates a trajectory from the starting point to the target equilibrium state by implementing a distributionally robust risk-constrained model predictive control (MPC) scheme. At each iteration, a set of safe states (that satisfy the risk-constraint with high probability) and a certain number of independent and identically distributed samples of the uncertainty governing the risk constraint are available. These states and samples are…
We present a novel data-driven distributionally robust Model Predictive Control formulation for unknown discrete-time linear time-invariant systems affected by unknown and possibly unbounded additive uncertainties. We use off-line collected data and an approximate model of the dynamics to formulate a finite-horizon optimization problem. To account for both the uncertainty related to the dynamics and the disturbance acting on the system, we resort to a distributionally robust formulation that optimizes the cost expectation while satisfying Conditional Value-at-Risk constraints with respect to t…
Distributionally Robust Planning with $\mathcal{L}_1$ Adaptive Control
National Polytechnic University of Armenia（亚美尼亚国立理工大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；American University of Armenia（亚美尼亚美国大学）；University of Michigan（密歇根大学）
This paper proposes a tractable distributionally robust chance-constrained conservation voltage reduction (DRCC-CVR) method with enriched data-based ambiguity set in unbalanced three-phase distribution systems. The increasing penetration of distributed renewable energy not only brings clean power but also challenges the voltage regulation and energy-saving performance of CVR by introducing high uncertainties to distribution systems. In most cases, the conventional robust optimization methods for CVR only provide conservative solutions. To better consider the impacts of load and PV generation u…
Distributionally Robust Control with End-to-End Statistically Guaranteed Metric Learning
School of Automation and Intelligent Sensing, Shanghai Jiao Tong University, PR China（自动化与智能感知学院，上海交通大学，中国）；Department of Mechanical Engineering, University of Victoria, Victoria, Canada（机械工程系，维多利亚大学，加拿大）
Seoul National University（首尔国立大学）；Automation and Systems Research Institute（自动化与系统研究院）
In this paper, a risk-aware motion control scheme is considered for mobile robots to avoid randomly moving obstacles when the true probability distribution of uncertainty is unknown. We propose a novel model predictive control (MPC) method for limiting the risk of unsafety even when the true distribution of the obstacles' movements deviates, within an ambiguity set, from the empirical distribution obtained using a limited amount of sample data. By choosing the ambiguity set as a statistical ball with its radius measured by the Wasserstein metric, we achieve a probabilistic guarantee of the out…
