Analysis of Interconnected Oscillators by Dissipativity Theory

Abstract

Abstract-This paper employs dissipativity theory for the global analysis of limit cycles in particular dynamical systems of possibly high dimension. Oscillators are regarded as open systems that satisfy a particular dissipation inequality. It is shown that this characterization has implications for the global stability analysis of limit cycle oscillations: i) in isolated oscillators, ii) in interconnections of oscillators, and iii) for the global synchrony analysis in interconnections of identical oscillators.Index Terms-Global limit cycle analysis, global synchronization, Hopf and pitchfork bifurcations, networks of oscillators.

Robustness of Aggregation in Networked Dynamical Systems

“…Relaxed co-coercivity and co-coercivity are the operator counterpart of the properties of output feedback incremental passivity and output strict incremental passivity defined for state-space systems in [7]. In general, to prove that an operator is (relaxed) co-coercive, it is possible to use a storage function approach assuming zero initial conditions (the interested reader is referred to the references [14] and [15]).…”

“…The operator describing the systems is assumed to satisfy an incremental condition called relaxed co-coercivity [6]. This notion is the operator counterpart of the incremental feedback passivity condition proposed in a state-space setting in [7] to analyze synchronization of coupled oscillators.…”

Robust synchronization in networks of cyclic feedback systems

“…As described in [12], [8], the operator Π measures the lack of consensus between the elements of a vector z ∈ R N in the following sense: the j th element of the vector Πz is the difference between the j th element of z and the average of all the elements of z. Note that Π T Π = Π.…”

“…The oscillator parameters are as follows b 1 = b 2 = 1 2 , p 1 = 17 and p 2 = 20. We take as the nominal system the first oscillator (where j = 1) and so the second oscillator's deviation from the first is such thatφ 12 Using the passivity approach of Theorem 2 however, we find that the upper bound on the total output synchronization error is given by (25):σ = 0.2182.…”

Contraction theory on Riemannian manifolds

“…Extensions and generalizations of this line of work are presented in [20,21,22,3,23]. Of particular interest are [3,23], which provide a coordinate-invariant formulation of incremental stability.…”