PhD Defence Suad Krilašević

Abstract:

Both societal and engineering systems are growing in complexity and interconnectivity, making it increasingly challenging, and sometimes impossible, to model their dynamics and behaviors. Moreover, individuals or entities within these systems, often refereed to as agents, have their own objectives that may conflict with one another. Examples include various economic systems where agents compete for profit, wind farms where upwind turbines reduce the energy extraction of downwind turbines, unwanted perturbation minimization in extremum seeking control, and cooperative source-seeking robotic vehicles. Despite having access to only limited observable information, it is crucial to ensure that all participants are content with the outcomes of these interactions. In this thesis, we choose to examine these problems within the framework of \emph{games}, where each agent has their own cost function and constraints, and all costs and constraints are interconnected. Since the notion of optimum in multi-agent problems is difficult to define, we often seek to find a Nash equilibrium, i.e., a set of decisions from which no agent has an incentive to deviate.

This thesis primarily explores the development of Nash equilibrium seeking algorithms for scenarios where agents' cost functions are unknown and can only be assessed through measurements of a dynamical system's output, referred to as the zeroth-order (derivative-free) information case. We specifically concentrate on scenarios where partial derivatives can be estimated from these measurements and subsequently integrated into a full-information algorithm. Existing approaches exhibit significant drawbacks, such as the inability to handle shared constraints, stringent assumptions on the cost functions, and applicability limited to agents with continuous dynamics.

The thesis is divided into three parts: equilibrium seeking via output derivative estimators, equilibrium seeking without projections, and hybrid output feedback for equilibrium seeking. In Part 1, we explore the development of zeroth-order (generalized) Nash equilibrium-seeking methods for dynamical systems using output derivative-based estimators. We identify an issue with an existing extremum seeking method and leverage the gained insights to design several algorithms. The first is a single-timescale NE seeking algorithm for a restricted class of linear systems, while the second is a multi-time scale algorithm for a broader range of nonlinear systems, which can solve GNEPs in the zeroth-order information case for the first time.

Part 2 aims to relax the strong monotonicity assumption on the pseudogradient mapping required in the zeroth-order GNE seeking. The main challenge arises from the fact that full-information continuous-time GNE seeking algorithms, which require only the monotonicity of the pseudogradient, also necessitate multiple pseudogradient evaluations that cannot be provided by estimation schemes. We propose a new projectionless algorithm to address these challenges and demonstrate its effectiveness in various practical scenarios, such as distortion reduction in photovoltaic current.

Part 3 focuses on extending existing averaging theory for discrete systems and singular perturbation theory for hybrid systems. By demonstrating practical stability of a multi-timescale discrete-time system with a practically stable averaged system, we can establish new results in discrete-time NE seeking. Moreover, by incorporating jumps from the boundary layer system of the restricted system into the singular perturbation theory, we can demonstrate stability for various systems where this was previously not the case. This allows us to show that discrete-time NE seeking algorithms can be applied in cases where agents have hybrid dynamics, significantly improving the real-life applicability of such algorithms.

To conclude, this thesis has established the stability of several zeroth-order game-theoretic control algorithms. In our final chapter, we evaluate how effectively we addressed our initial research questions. We also lay out potential directions for future research and acknowledge areas of interest or potential weaknesses that our research did not fully explore. This self-reflection allows us to set a clear path for further inquiries and improvements.