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We propose an approach to design a Model Predictive Controller (MPC) for constrained Linear Time Invariant systems performing an iterative task. The system is subject to an additive disturbance, and the goal is to learn to satisfy state and input constraints robustly. Using disturbance measurements after each iteration, we construct Confidence Support sets, which contain the true support of the disturbance distribution with a given probability. As more data is collected, the Confidence Supports converge to the true support of the disturbance. This enables design of an MPC controller that avoids conservative estimate of the disturbance support, while simultaneously bounding the probability of constraint violation. The efficacy of the proposed approach is then demonstrated with a detailed numerical example. 

Playing the cup-and-ball game is an intriguing task for robotics research since it abstracts important problem characteristics including system nonlinearity, contact forces and precise positioning as terminal goal. In this paper, we present a learning model based control strategy for the cup-and-ball game, where a Universal Robots UR5e manipulator arm learns to catch a ball in one of the cups on a Kendama. Our control problem is divided into two sub-tasks, namely (i) swinging the ball up in a constrained motion, and (ii) catching the free-falling ball. The swing-up trajectory is computed offline, and applied in open-loop to the arm. Subsequently, a convex optimization problem is solved online during the ball’s free-fall to control the manipulator and catch the ball. The controller utilizes noisy position feedback of the ball from an Intel RealSense D435 depth camera. We propose a novel iterative framework, where data is used to learn the support of the camera noise distribution iteratively in order to update the control policy. The probability of a catch with a fixed policy is computed empirically with a user specified number of roll-outs. Our design guarantees that probability of the catch increases in the limit, as the learned support nears the true support of the camera noise distribution. High-fidelity Mujoco simulations and preliminary experimental results support our theoretical analysis 

Frequency regulation is crucial for balancing the supply and demand of modern electricity grids. To provide regulation services, it is important to understand the capability of flexible resources to track regulation signals. This paper studies the problem of submitting capacity bids to a forward regulation market based on historical regulation signals. We consider an aggregator who manages a group of flexible resources with linear dynamic constraints. He seeks to find the optimal capacity bid, so that real-time regulation signals can be followed with an arbitrary guaranteed probability. We formulate this problem as a chance-constrained program with unknown regulation signal distributions. A sampling and discarding algorithm is proposed. It provably provides near-optimal solutions at a guaranteed probability of success without knowing the distribution of the regulation signals. This result holds for resources with arbitrary linear dynamics and allows arbitrary intra-hour data correlations. We validate the proposed algorithm with real data via numerical simulations. Two cases are studied: (1) CAISO market, where providers separately submit capacity estimates for regulation up and regulation down signals, (2) PJM market, where regulation up and down capacities are the same. Simulation results show that the proposed algorithm provides near-optimal capacity estimates for both cases.