Quadratic Lyapunov functions for cooperative control of networked systems

This paper addresses the problem of applying the Lyapunov direct method to stability analysis of cooperative systems. In particular, cooperative stability and its associated Lyapunov function are sought in terms of the topological properties of a cooperative system. Along this direction, the only available result is for the time-invariant case, it requires irreducibility, and the corresponding Lyapunov function is of form V = Sigma(i)pix(i)(T)x(i) but its time derivative is only negative semi-definite. In this paper, cooperative control Lyapunov function is defined to be a function which is positive definite with respect to the consensus set x = c1 and whose value monotonically decreases along the system trajectory. A necessary and sufficient condition in terms of the properties of a topology is found for the existence of cooperative control Lyapunov function and, through the introduction and development of an average system, the necessary and sufficient condition is extended to the case of time-varying topology. It is also shown that, under the condition, the cooperative system has the desired stability property and the corresponding Lyapunov function is always of form V = Sigma(i) pi (x(i) - x(k))(T) (x(i) - x(k)), where k can be determined using the proposed condition for any collection of time-varying topologies.