MS16.1: Control and Synchronization in Nonlinear Systems

Self-balancing robots have gained prominence in domestic and industrial applications due to their notably smaller footprint than their naturally stable counterparts. However, the dynamics and control of such robots pose significant engineering challenges. In this study, we develop a planar dynamics model to represent a self-balancing robot and employ the Lagrange method to derive its equations of motion. Subsequently, we design a sliding mode controller to regulate the robot's pitch angle. Our analysis also highlights the impact of actuation delay, which we explore using numerical methods.

We discuss synchronization in networks of dynamical systems which interact via nonlinear coupling functions. First, we characterize nonlinear (potential) instabilities that drive systems apart in absence of coupling. Next, we constructively design nonlinear coupling functions that target these (potential) instabilities, therewith guaranteeing network synchronization. As the designed nonlinear coupling functions are active only when needed, these couplings are more energy efficient and less sensitive to output noise compared to linear coupling functions, which are commonly used to establish network synchronization. We demonstrate the performance advantages of our synchronizing nonlinear coupling by means of numerical simulations of a network with FitzHugh-Nagumo oscillators.

This study introduces a damping control scheme for the mitigation of the rotation of a cylinder through electromagnetic interaction between two magnets. Leveraging the nonlinear dynamics of the interaction, the results showcase effective control performance and efficiency. Consequently, the proposed method lays a promising foundation for a non-contact rotation control technique, particularly applicable for offshore wind turbine installations.

We present observer designs for hybrid systems in two scenarios: one where the jump times (i.e., the times when discontinuities in solutions occur) are known, and the other where these are unknown. For the first case, we first consider systems with linear maps by proposing a Kalman-like observer, then those with nonlinear maps by coupling a high-gain observer with a jump-based one under Lyapunov analysis. While the high-gain observer reconstructs the instantaneously observable part of the state from the flow output, the other one estimates the rest from the jump output and a fictitious one that models its interaction with the first part at jumps. When the jump times are unknown, we propose an observer by gluing the time domain, relying on transforming the hybrid system into some continuous-time dynamics without jumps. Applications to a walking robot and the stick-slip friction phenomenon are presented.