In this paper, a backstepping control approach is developed and analyzed for a setting where the system model is partially unknown and is modeled using Gaussian processes. The proposed approach encompasses the classical backstepping and command filtered approaches as special cases. The tracking error is globally uniformly ultimately bounded, and the performance is shown to be improved by adding new training data. The stability analysis is carried out by employing a quadratic Lyapunov function and Tikhonov's theorem. The proposed method outperforms an established adaptive backstepping approach given sufficient training data.

When first principle models cannot be derived due to the complexity of the real system, data-driven methods allow us to build models from system observations. As these models are employed in learning-based control, the quality of the data plays a crucial role for the performance of the resulting control law. Nevertheless, there hardly exist measures for assessing training data sets, and the impact of the distribution of the data on the closed-loop system properties is largely unknown. This paper derives -based on Gaussian process models -an analytical relationship between the density of the training data and the control performance. We formulate a quality measure for the data set, which we refer to as ρ-gap, and derive the ultimate bound for the tracking error under consideration of the model uncertainty. We show how the ρ-gap can be applied to a feedback linearizing control law and provide numerical illustrations for our approach.

Combined harnessing of electrical and thermal energies could leverage their complementary nature, inspiring the integration of power grids and centralized heating systems in future smart cities. This paper considers interconnected power distribution network (PDN) and district heating network (DHN) infrastructures through combined heat and power units and heat pumps. In the envisioned market framework, the DHN operator solves an optimal thermal flow problem given the nodal electricity prices and determines the best heat production strategy. Variate coefficients of performance of heat pumps with respect to different load levels are considered and modeled in a disciplined convex optimization format. A two-step hydraulic-thermal decomposition method is suggested to approximately solve the optimal thermal flow problem via a second-order cone program. Simultaneously, the PDN operator clears the distribution power market via an optimal power flow problem given the demands from the DHN. Electricity prices are revealed by dual variables at the optimal solution. The whole problem gives rise to a Nash-type game between the two systems. A best-response decentralized algorithm is proposed to identify the optimal operation schedule of the coupled infrastructure, which interprets a market equilibrium as neither system has an incentive to alter their strategies. Numeric results demonstrate the potential benefits of the proposed framework in terms of reducing wind curtailment and system operation cost.

Control tasks with high levels of uncertainty and safety requirements are increasingly common. Typically, techniques that guarantee safety during learning and control utilize constraint-based safety certificates, which can be leveraged to compute safe control inputs. However, if model uncertainty is very high, the corresponding certificates are potentially invalid, meaning no control input satisfies the constraints imposed by the safety certificate. This paper considers a learning-based setting with a safety certificate based on a control barrier function second-order cone program. If the control barrier function certificate is valid, our approach leverages the control barrier function to guarantee safety. Otherwise, our method explores the system to recover the feasibility of the control barrier function constraint as fast as possible. To this end, we employ a method inspired by well-established tools from Bayesian optimization. We show that if the sampling frequency is high enough, we recover the feasibility of a control barrier function second-order cone program, guaranteeing safety. To the best of our knowledge, this corresponds to the first algorithm that guarantees safety through online learning without requiring a prior model or backup safe non-learning-based controller.

Learning-based techniques are increasingly effective at controlling complex systems. However, most work done so far has focused on learning control laws for individual tasks. Simultaneously learning multiple tasks on the same system is still a largely unaddressed research question. In particular, no efficient state space exploration schemes have been designed for multi-task control settings. Using this research gap as our main motivation, we present an algorithm that approximates the smallest data set that needs to be collected in order to achieve high performance across multiple control tasks. By describing system uncertainty using a probabilistic Gaussian process model, we are able to quantify the impact of potentially collected data on each learning-based control law. We then determine the optimal measurement locations by solving a stochastic optimization problem approximately. We show that, under reasonable assumptions, the approximate solution converges towards the exact one. Additionally, we provide a numerical illustration of the proposed algorithm.