Gradient and Passive Circuit Structure in a Class of Non-linear Dynamics on a Graph

We consider a class of non-linear dynamics on a graph that contains and generalizes various models from network systems and control and study convergence to uniform agreement states using gradient methods. In particular, under the assumption of detailed balance, we provide a method to formulate the governing ODE system in gradient descent form of sum-separable energy functions, which thus represent a class of Lyapunov functions; this class coincides with Csisz\'{a}r's information divergences. Our approach bases on a transformation of the original problem to a mass-preserving transport problem and it reflects a little-noticed general structure result for passive network synthesis obtained by B.D.O. Anderson and P.J. Moylan in 1975. The proposed gradient formulation extends known gradient results in dynamical systems obtained recently by M. Erbar and J. Maas in the context of porous medium equations. Furthermore, we exhibit a novel relationship between inhomogeneous Markov chains and passive non-linear circuits through gradient systems, and show that passivity of resistor elements is equivalent to strict convexity of sum-separable stored energy. Eventually, we discuss our results at the intersection of Markov chains and network systems under sinusoidal coupling

Synchronization of Nonlinear Circuits in Dynamic Electrical Networks with General Topologies

Sufficient conditions are derived for global asymptotic synchronization in a system of identical nonlinear electrical circuits coupled through linear time-invariant (LTI) electrical networks. In particular, the conditions we derive apply to settings where: i) the nonlinear circuits are composed of a parallel combination of passive LTI circuit elements and a nonlinear voltage-dependent current source with finite gain; and ii) a collection of these circuits are coupled through either uniform or homogeneous LTI electrical networks. Uniform electrical networks have identical per-unit-length impedances. Homogeneous electrical networks are characterized by having the same effective impedance between any two terminals with the others open circuited. Synchronization in these networks is guaranteed by ensuring the stability of an equivalent coordinate-transformed differential system that emphasizes signal differences. The applicability of the synchronization conditions to this broad class of networks follows from leveraging recent results on structural and spectral properties of Kron reduction---a model-reduction procedure that isolates the interactions of the nonlinear circuits in the network. The validity of the analytical results is demonstrated with simulations in networks of coupled Chua's circuits

Control law synthesis for distributed multi-agent systems: Application to active clock distribution networks

International audienceIn this paper, the problem of active clock distribution network synchronization is considered. The network is made of identical oscillators interconnected through a distributed array of phase-locked-loops (PLLs). The problem of the PLL network design is reformulated, from a control theory point of view, as a control law design for a distributed multi-agent system. Inspired by the decentralized control law design methodology using the dissipativity input-output approach, the particular topology of interconnected subsystems is exploited to solve the problem by applying a convex optimization approach involving simple Linear Matrix Inequality (LMI) constraints. After choosing the dissipativity properties which is satisfied by the interconnection matrix, the constraints are transformed into an H ∞ norm constraint on a particular transfer function that must be fulfilled for global stability. Additional constraints on inputs and outputs are introduced in order to ensure the desired performance specifications during the H ∞ control design procedure

Distributed Stabilization of Nonlinear Multi-Agent Systems

The study of multi-agent systems (MASs) is focused on systems in which many autonomous agents interact and operate within a limited communication environment. The general goal of the MAS research is to design interconnection control laws such that all the dynamic agents in the group are synchronized to a desired common trajectory by exchanging information with adjacent agents over certain constrained communication networks. Based on the review and modification of existing results concerning the consensus control of linear heterogeneous MASs in Moreau (2004) [21], Scardovi and Sepulchre (2009) [25], Wieland et al (2011) [30], and Alvergue et al. (2013) [1], this thesis investigates the distributed stabilization of the heterogeneous MAS, consisting of N different continuous-time nonlinear dynamic systems, under connected communication graphs. The conditions for a nonlinear dynamic agent to be feedback equivalent to a strictly passive system are derived along with the feedback law. A distributed stabilization control protocol using state feedback is then proposed under the idea of feedback connection of two passive systems. It proves to be sufficient for only one or a few agents to have access to the reference signal for the MAS to achieve stability, which lowers the communication overhead from the reference to different agents. The result can be interpreted as an extension of the stabilizing law for linear MASs introduced in [1], and considered as a fundamental preliminary for the consensus research for nonlinear MASs in the future

Network Internal Signal Feedback and Injection: Interconnection Matrix Design

International audienceThe design of systems defined as networks (interconnections) of identical subsystems emerges as an interesting engineering problem, with some open issues. One of these issuesis how to "retune" the interconnection in order to ensure the stability and the performance of the system. Based on the LFT representation and on the input-output framework, we propose in this paper some efficient "retuning" methods using convex optimization involving LMI constraints. The proposed approach can be interpreted as an extension of usual state space methods. Its application is investigated for the design of a network of PLLs

Performance Control for Interconnection of Identical Systems: Application to PLL network design

International audienceIn this paper, the problem of the control law design for interconnected identical systems ensuring the global stability and the global performance properties is under consideration. Inspired by the decentralized control law design methodology using the dissipativity input–output approach, the problem is reduced to the problem of satisfying two conditions: (i) the condition on the interconnection and (ii) the condition on the local subsystem dynamics. Both problems are efficiently solved applying a (quasi‐) convex LMI optimization and standard H∞ synthesis. The proposed design methodology is applied to the control law design of a synchronous PLL network