In this dissertation, the notion of symmetry in dynamical control systems is studied. Large-scale dynamical systems present significant challenges to control design and analysis. The energy-based analysis methods, namely dissipativity and passivity theory, are studied with the presence of symmetry in system dynamics and interconnection structures. It is shown that symmetry is a powerful tool in preserving system properties and performances such as dissipativity, passivity, stability, robustness and controllability.
In order to recognize the important role of symmetry in dynamical systems, background is first given on general definitions of symmetry for geometric control systems in Chapter 2. In addition, symmetry examples in nonlinear affine control systems and in multi-agent systems are also given. Chapter 3 discusses symmetry in multi-agent dissipative systems. Stability conditions are derived for large-scale systems by categorizing agents into symmetry groups and applying local control laws under limited interconnections with neighbors. Extensions on passivity results are presented in Chapter 4. Also the passivity indices are characterized for large-scale passive systems with symmetric interconnections. Approximate symmetry in terms of network effects is discussed in Chapter 5. Chapter 6 explores the relationship between symmetry and nonlinear controllability. Chapter 7 discusses experimental testing to determine the passivity indices using numerical optimization.
Overall, this dissertation provides new approaches in analyzing and synthesizing large-scale dynamical control systems with regard to energy and input-output relationships, where symmetry property provides interesting results for system analysis and control design. Numerical simulations and illustrative examples are presented to demonstrate the practical usage and to motivate the theory.