In the field of Systems and Control, optimal control problem-solving for complex systems is a core task. The development of accurate mathematical models to represent these systems’ dynamics is often difficult. This complexity comes from potential uncertainties, complex non-linearities, or unknown factors that might affect the system. Because of these challenges, there is a need for methods that can understand the dynamics using available data and control strategies that can work with such models without relying too much on expert knowledge or task-specific insights. These methods are essential for creating efficient and reliable solutions in a wide variety of applications within the discipline. The need for models that do not require expert knowledge has spurred the interest in applying machine learning methods to control problems. Probabilistic Inference for Learning COntrol (PILCO) is a model-based Reinforcement Learning (RL) algorithm known for its probabilistic approach to model-based RL. By employing Gaussian Process (GP) dynamics models, PILCO integrates uncertainties into its learning process, allowing it to derive control policies from limited data. PILCO’s use of the Squared Exponential (SE) kernel in its GP can restrict the learning capacity. Especially in higher-dimensional spaces, due to the SE kernel’s inherent smoothness assumption that might not capture complex or non-smooth dynamics effectively. The algorithm’s reliance on moment matching for approximating posterior distributions introduces another weakness, which can lead to inaccuracies in non-Gaussian or multi-modal contexts. These shortcomings may limit PILCO’s efficiency and scalability in more complex, higher-dimensional tasks or in situations where the underlying dynamics are not well-captured by the chosen kernel and approximation methods. This thesis introduces Deep Kernel PILCO (DKL PILCO), a novel framework that uses Deep Kernel Learning (DKL) for learning the dynamics, and the Unscented Transform (UT) to propagate the uncertainty. The effectiveness of this approach is demonstrated across various tasks, highlighting the potential of DKL and UT to enhance the scalability and efficiency of model-based RL methods such as PILCO, making it a promising candidate for real-world control applications.

In decision making problems, the ability to compute the optimal solution can pose a serious challenge. Dynamic Programming (DP) aims to provide a framework to deal with a category of such problems, namely ones that involve sequential decision making. By dividing the original control problem into sub-problems and solving it backwards in time, from the end of the time horizon to the start, the method can compute a map of optimal solutions with respect to the initial condition. In order to divide the original problem into subproblems the DP method takes advantage of the principle of optimality, which states that a sub-solution of the optimal solution should be the optimal solution for the equivalent subproblem. In control systems, where the state and decision spaces are continuous, the original DP framework can be intractable due to the size of the discretization needed to simulate the continuous space. Therefore, efficient approximations are needed to solve such problems. One promising method is called Conjugate Dynamic Programming (CDP). The CDP algorithm is able to transform the original framework and solve problems in the conjugate domain providing a computational advantage over the standard method. In this work, we aim to improve and extend the setting under which the CDP algorithm operates, thus providing a more concrete advantage over standard method . In that regard, we will extract the optimal control actions from within the CDP algorithm, eliminating the need for solving an extra optimization problem for their computation. In addition, we will introduce a different interpolation technique that can outperform the current one, in certain scenarios, thus granting the user more choice when solving a decision making problem.

The rising number of electricity consumers poses a challenge to power generators and grid operators in maintaining a balanced grid. Peak shaving is a technique that consists of shifting electricity consumption from hours of high demand to times of low demand, and has been gaining popularity in recent years. It allows to reduce peak loads which must be met at all times, which in turn reduces the need to resort to less efficient and more expensive power plants to see that all peaks are met. This paper proposes two methods to operate a battery energy storage system (BESS) for peak shaving of daily load profiles, based on the dynamic programming (DP) algorithm. The first approach treats historical profiles as deterministic paths and solves the DP algorithm for each profile. Then, an operating plan is obtained in real-time by combining the policies calculated for each path. The second approach treats data as stochastic paths, from which it derives a probabilistic model for the target path. Given this model, the DP algorithm obtains an operating plan for the entire day ahead. We also propose three modifications that can be applied to the two methods which aim at the improving the reliability of the methods. The performance of the proposed methods is examined through numerical experiments on both artificial and real data.

A well-established method for finding the optimal control policy for a given dynamical system is to solve the problem iteratively going from its terminal state "backwards" in time, known as Dynamic Programming Algorithm. For a generic problem with discrete state/action space, the algorithm has computational complexity of O(NM) for N states and M actions. In this thesis, we propose a novel numerical algorithm that approaches this problem in the conjugate domain, using the so-called Legendre-Fenchel Transform. In essence, the proposed approach is analogous to, and was inspired by Fast Fourier Transform, and how it can be beneficial to do computations/analysis in the frequency domain. In particular, this approach allows us to exploit the structure of the problem (e.g., in LQ control) to drastically reduce the computational complexity to O(N+M). Of course, this computational gain comes with a cost of introducing error.