Reinforcement learning has been successfully used to solve difficult tasks in complex unknown environments. However, these methods typically do not provide any safety guarantees during the learning process. This is particularly problematic, since reinforcement learning agents actively explore their environment. This prevents their use in safety-critical, real-world applications. In this paper, we present a learning-based model predictive control scheme that provides high-probability safety guarantees throughout the learning process. Based on a reliable statistical model, we construct provably accurate confidence intervals on predicted trajectories. Unlike previous approaches, we allow for input-dependent uncertainties. Based on these reliable predictions, we guarantee that trajectories satisfy safety constraints. Moreover, we use a terminal set constraint to recursively guarantee the existence of safe control actions at every iteration. We evaluate the resulting algorithm to safely explore the dynamics of an inverted pendulum and to solve a reinforcement learning task on a cart-pole system with safety constraints.

Abstract-One of the most fundamental problems when designing controllers for dynamic systems is the tuning of the controller parameters. Typically, a model of the system is used to obtain an initial controller, but ultimately the controller parameters must be tuned manually on the real system to achieve the best performance. To avoid this manual tuning step, methods from machine learning, such as Bayesian optimization, have been used. However, as these methods evaluate different controller parameters on the real system, safety-critical system failures may happen. In this paper, we overcome this problem by applying, for the first time, a recently developed safe optimization algorithm, SAFEOPT, to the problem of automatic controller parameter tuning. Given an initial, low-performance controller, SAFEOPT automatically optimizes the parameters of a control law while guaranteeing safety. It models the underlying performance measure as a Gaussian process and only explores new controller parameters whose performance lies above a safe performance threshold with high probability. Experimental results on a quadrotor vehicle indicate that the proposed method enables fast, automatic, and safe optimization of controller parameters without human intervention. SUPPLEMENTARY MATERIALA video demonstrating the proposed safe, automatic controller optimization on a quadrotor vehicle can be found at http://tiny.cc/icra16 video. A Python implementation of the algorithm is available in [1].

Abstract-Control theory can provide useful insights into the properties of controlled, dynamic systems. One important property of nonlinear systems is the region of attraction (ROA), a safe subset of the state space in which a given controller renders an equilibrium point asymptotically stable. The ROA is typically estimated based on a model of the system. However, since models are only an approximation of the real world, the resulting estimated safe region can contain states outside the ROA of the real system. This is not acceptable in safety-critical applications. In this paper, we consider an approach that learns the ROA from experiments on a real system, without ever leaving the true ROA and, thus, without risking safety-critical failures. Based on regularity assumptions on the model errors in terms of a Gaussian process prior, we use an underlying Lyapunov function in order to determine a region in which an equilibrium point is asymptotically stable with high probability. Moreover, we provide an algorithm to actively and safely explore the state space in order to expand the ROA estimate. We demonstrate the effectiveness of this method in simulation.

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Selecting the right tuning parameters for algorithms is a pravelent problem in machine learning that can significantly affect the performance of algorithms. Data-efficient optimization algorithms, such as Bayesian optimization, have been used to automate this process. During experiments on real-world systems such as robotic platforms these methods can evaluate unsafe parameters that lead to safety-critical system failures and can destroy the system. Recently, a safe Bayesian optimization algorithm, called SafeOpt, has been developed, which guarantees that the performance of the system never falls below a critical value; that is, safety is defined based on the performance function. However, coupling performance and safety is often not desirable in practice, since they are often opposing objectives. In this paper, we present a generalized algorithm that allows for multiple safety constraints separate from the objective. Given an initial set of safe parameters, the algorithm maximizes performance but only evaluates parameters that satisfy safety for all constraints with high probability. To this end, it carefully explores the parameter space by exploiting regularity assumptions in terms of a Gaussian process prior. Moreover, we show how context variables can be used to safely transfer knowledge to new situations and tasks. We provide a theoretical analysis and demonstrate that the proposed algorithm enables fast, automatic, and safe optimization of tuning parameters in experiments on a quadrotor vehicle.