Analysis of Discrete and Hybrid Stochastic Systems by Nonlinear Contraction Theory

We investigate the stability properties of discrete and hybrid stochastic nonlinear dynamical systems. More precisely, we extend the stochastic contraction theorems (which were formulated for continuous systems) to the case of discrete and hybrid resetting systems. In particular, we show that the mean square distance between any two trajectories of a discrete (or hybrid resetting) contracting stochastic system is upper-bounded by a constant after exponential transients. Using these results, we study the synchronization of noisy nonlinear oscillators coupled by discrete noisy interactions.Comment: 6 pages, 1 figur

Nonlinear Model Reduction and Decentralized Control of Tethered Formation Flight by Oscillation Synchronization

This paper describes a fully decentralized nonlinear control law for spinning tethered formation flight, based on exploiting geometric symmetries to reduce the original nonlinear dynamics into simpler stable dynamics. Motivated by oscillation synchronization in biological systems, we use contraction theory to prove that a control law stabilizing a single-tethered spacecraft can also stabilize arbitrary large circular arrays of spacecraft, as well as the three inline configuration. The convergence result is global and exponential. Numerical simulations and experimental results using the SPHERES testbed validate the exponential stability of the tethered formation arrays by implementing a tracking control law derived from the reduced dynamics

Contraction and partial contraction : a study of synchronization in nonlinear networks

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2005.Includes bibliographical references (p. 121-128).This thesis focuses on the study of collective dynamic behaviors, especially the spontaneous synchronization behavior, of nonlinear networked systems. We derives a body of new results, based on contraction and partial contraction analysis. Contraction is a property regarding the convergence between two arbitrary system trajectories. A nonlinear dynamic system is called contracting if initial conditions or temporary disturbances are forgotten exponentially fast. Partial contraction, introduced in this thesis, is a straightforward but more general application of contraction. It extends contraction analysis to include convergence to behaviors or to specific properties (such as equality of state components, or convergence to a manifold). Contraction and partial contraction provide powerful analysis tools to investigate the stability of large-scale complex systems. For diffusively coupled nonlinear systems, for instance, a general synchronization condition can be derived which connects synchronization rate to net- work structure explicitly. The results are applied to construct flocking or schooling models by extending to coupled networks with switching topology. We further study the networked systems with different kinds of group leaders, one specifying global orientation (power leader), another holding target dynamics (knowledge leader). In a knowledge-based leader-followers network, the followers obtain dynamics information from the leader through adaptive learning. We also study distributed networks with non-negligible time-delays by using simplified wave variables and other contraction-oriented analysis. Conditions for contraction to be preserved regardless of the explicit values of the time-delays are derived.(cont.) Synchronization behavior is shown to be robust if the protocol is linear. Finally, we study the construction of spike-based neural network models, and the development of simple mechanisms for fast inhibition and de-synchronization.by Wei Wang.Ph.D