The implementation of advanced optimization-based control strategies in complex engineering systems promises significant improvements in efficiency and performance. However, the practical implementation of these strategies often faces substantial challenges, including distributed implementation requirements, nonlinear dynamics, system uncertainties, and insufficient robustness margins. This dissertation addresses these challenges through the lens of hybrid dynamical systems, developing robust and efficient control algorithms for a diverse range of applications. Our research explores four key areas, each offering distinct advantages for various scenarios: momentum-based methods for distributed optimization and real-time decision-making, nonsmooth approaches for fixed and finite-time stability, time-varying methods for prescribed-time convergence, and global optimization on manifolds.
In the domain of distributed optimization and real-time decision-making, we introduce novel hybrid momentum-based algorithms that overcome the limitations of related purely continuous-time approaches used in optimization, and extend their applicability to both potential and nonpotential game settings. These methods provide accelerated convergence rates over traditional gradient-based approaches, and are particularly effective in scenarios with low curvature.
Complementing this work, we investigate nonsmooth dynamics for fixed and finite-time stability. While momentum methods excel under certain conditions, nonsmooth approaches offer the potential for guaranteed convergence within a finite time, independent of initial conditions. Our approach applies both momentum-based and nonsmooth methods to practical problems such as traffic congestion management and accelerated learning, offering a comprehensive comparison of their respective strengths across different scenarios.
Our research extends to time-varying methods, utilizing suitable dynamic gains to shape the transient behavior of hybrid systems with preexisting uniform asymptotic stability properties. By interconnecting these dynamic gains with the original system, we obtain, in particular, prescribed-time stability results, guaranteeing convergence to a compact set within a user-defined finite time that is independent of initial conditions and problem parameters. We demonstrate the effectiveness of this method in complex scenarios, including systems with intermittent feedback, by applying it to switching systems with resets.
We address the challenge of global optimization on compact manifolds using gradient-free dynamics. Our approach overcomes topological obstacles that typically preclude the implementation of smooth, nonsmooth, and even time-varying techniques in these spaces. By harnessing the flexibility of hybrid systems, we develop mechanisms that not only render the manifold forward invariant but also have provable nonzero robustness margins for global optimization.
Finally, we address the challenge of robust global stabilization of Kapitza's pendulum's naturally unstable upright position. We propose a novel hybrid control approach that leverages multiple oscillating directions to achieve global asymptotic stability. This method overcomes the limitations of traditional smooth control techniques in addressing the pendulum's complex dynamics, demonstrating the power of hybrid systems in stabilizing counterintuitive nonlinear phenomena.
Throughout this dissertation, we employ tools from hybrid systems theory and Lyapunov stability analysis to provide rigorous convergence guarantees. Our theoretical results are substantiated by extensive numerical simulations, demonstrating the efficacy of the proposed methods across various engineering domains.
