Synthesis of Linear Distributed Control and Coupled Oscillators with Multiple Limit Cycles
This dissertation presents a synthesis scheme of distributed controller for continuous time, finite dimensional, linear time-invariant systems. We show that under the assumption that the interconnection of the control units is characterized by a strongly connected directed graph, any centralized controller admits a distributed synthesis with an arbitrary accuracy. This idea applies to a wide variety of control problems such as observer design, stabilizing control, eigenstructure assignment, etc. We further specialize our theories for multi-agent systems with local observability and propose a lower order distributed controller with internal model to achieve eigenstructure assignment. Then by replacing the linear internal model by a nonlinear oscillator network, we show the potential of nonlinear distributed control to achieve pattern formation with the amplitude stabilized. Next, we consider the design of nonlinear oscillator network with linear coupling to have a stable limit cycle with prescribed oscillation profile. We give the suﬀicient conditions for the orbital stability of the limit cycle. Moreover, we demonstrate that the embedding of multiple limit cycles into the oscillator network is possible. Finally, we consider the design of nonlinear oscillator network with nonlinear coupling to overcome some limitations of the linearly coupled oscillators. Likewise, conditions for the orbital stability of multiple limit cycles are provided.