Local Lyapunov Functions for Consensus in Switching Nonlinear Systems

Abstract

Abstract-This note presents two theorems on asymptotic state consensus of continuous time nonlinear multi-agent systems. The agents reside in R m and have switching interconnection topologies. Both the first theorem, formulated in terms of the states of individual agents, and the second theorem, formulated in terms of the pairwise states for pairs of agents, can be interpreted as variants of Lyapunov's second method. The two theorems complement each other; the second provides stronger convergence results under weaker graph topology assumptions, whereas the first often can be applied in a wider context in terms of the structure of the right-hand sides of the systems. The second theorem also sheds some new light on well-known results for consensus of nonlinear systems where the right-hand sides of the agents' dynamics are convex combinations of directions to neighboring agents. For such systems, instead of proving consensus by using the theory of contracting convex sets, a local quadratic Lyapunov function can be used.

Stability Analysis of Parameter Varying Genetic Toggle Switches Using Koopman Operators

“…From here, one can work through a method of one's choice (i.e., Wu's algorithm [46], Lyapunov function [54], poles via Lyapunov transform [47], etc.) to verify if a particular choice of µ(t) and λ(t) provide an asymptotically stable system.…”

“…Also, note that the affine linear time-varying input u(t) permits us a limited fixed point selection. From here, one can work through a method of one's choice (i.e., Wu's algorithm [46], Lyapunov function [54], poles via Lyapunov transform [47], etc.) to verify if a particular choice of time-varying parameters provide an asymptotically stable system.…”

A lifting method for analyzing distributed synchronization on the unit sphere

“…We will use Theorem 1 in Thunberg, Hu & Goncalves (2017). Under the condition that the graph is uniformly strongly connected, if we can show that any closed disc (or ball) in the equatorial place with radius less than 1 is forward invariant for the y i 's and we can find a function V : R d−1 → R + such that V is 1) positive definite, 2) max i∈V V (y i (t)) is decreasing as a function of t, and 3)V (y i (t)) is strictly negative if i ∈ arg max j∈V {V (y j )} and there is j ∈ N i (t) such that y i = y j .…”

“…On this set the right-hand side of (13) is Lipschitz continuous in z and piece-wise continuous in t. Now the procedure in the rest of the proof is analogous to the one in Proposition 6. Since the right-hand side of (13) is Lipschitz continuous in z and piece-wise continuous in t, we can use Theorem 2 in Thunberg, Hu & Goncalves (2017) to find a continuously differentiable function W :…”

“…At any finite time the solution is the same as the original system. However, we can use the results in Thunberg, Hu & Goncalves (2017) to show that the solution to the modified system converges to A z and in particular all the z i 's converge to a fixed point that is nonzero. This means that after some finite time T , the point 0 is not contained in H(t, z 0 ) for the modified as well as for the original system.…”