Elsevier

Automatica

Volume 62, December 2015, Pages 274-283
Automatica

Brief paper
Synchronization under space and time-dependent heterogeneities

https://doi.org/10.1016/j.automatica.2015.09.033Get rights and content

Abstract

We study output synchronization of nonlinear systems subject to a class of spatially and temporally-varying heterogeneities. We begin by addressing reaction–diffusion PDEs, and show that an incremental passivity property in the nominal reaction dynamics is fundamental in guaranteeing spatial homogenization of trajectories. Our distributed control achieves homogenization by defining an internal model subsystem for each output corresponding to the respective heterogeneity. We also design a similar distributed internal model control that achieves output synchronization of a network of incrementally passive nonlinear ODE systems in the presence of heterogeneities. We prove that our control laws effectively compensate for desynchronizing effects due to the differences between the heterogeneous inputs. We illustrate our algorithms on ring oscillators and bistable systems. The main contribution of the paper is a constructive and unified approach to guaranteeing homogenization under heterogeneities for both reaction–diffusion PDE and diffusively-coupled ODE systems.

Introduction

Spatially distributed models with diffusive coupling are crucial to understanding the dynamical behavior of a range of engineering and biological systems. This form of coupling encompasses, among others, feedback laws for coordination of multi-agent systems, electromechanical coupling of synchronous machines in power systems, and local update laws in distributed agreement algorithms. Synchronization of diffusively-coupled nonlinear systems is an active and rich research area (Hale, 1997), and conversely, developing conditions that rule out synchrony is also important, as they facilitate study of spatial pattern formation. One of the major ideas behind pattern formation in cells and organisms is based on diffusion-driven instability (Segel and Jackson, 1972, Turing, 1952), which occurs when higher-order spatial modes in a reaction–diffusion PDE are destabilized by diffusion (Cross and Hohenberg, 1993, Hsia et al., 2012, Jovanović et al., 2008, Olfati-Saber and Murray, 2004, Othmer et al., 2009).

Several works in the literature study the case of static interconnections between nodes in full state models (Arcak, 2011, Pecora and Carroll, 1998, Pogromsky and Nijmeijer, 2001, Russo and Di Bernardo, 2009, Scardovi et al., 2010, Stan and Sepulchre, 2007, Wang and Slotine, 2005) or phase variables in phase coupled oscillator models (Chopra and Spong, 2009, Dorfler and Bullo, 2012, Kuramoto, 1975, Strogatz, 2000). Common to much of the literature is the assumption that the agents to be synchronized are homogeneous with identical dynamics, and are furthermore not subject to disturbances.

However, recent work has considered synchronization and consensus in the presence of exogenous inputs. In  Bai, Freeman, and Lynch (2010), the authors addressed the problem of robust dynamic average consensus (DAC), in which the use of partial model information about a broad class of time-varying inputs enabled exact tracking of the average of the inputs through the use of the internal model principle (Francis & Wonham, 1976) and the structure of the proportional–integral average consensus estimator formulated in  Freeman, Yang, and Lynch (2006). The internal model principle has been useful in establishing necessary and sufficient conditions for output regulation (Pavlov & Marconi, 2008) and synchronization (De Persis and Jayawardhana, 2012, Wieland et al., 2011, Wieland et al., 2013). Bürger and De Persis (2013) proposed internal model control strategies in which controllers were placed on the edges of the interconnection graph to achieve output synchronization under time-varying disturbances. Recent work has also addressed robust synchronization in cyclic feedback systems (Hamadeh, Stan, & Gonçalves, 2008) and in the presence of structured uncertainties (Dhawan, Hamadeh, & Ingalls, 2012).

Spatial homogenization of reaction–diffusion PDEs has also been studied in the literature. In  Arcak (2011), the author gave a Lyapunov inequality for the Jacobian of the reaction term, significantly reducing conservatism of dominant approaches making use of global Lipschitz bounds (Conway, Hoff, & Smoller, 1978). However, the problem of spatial homogenization of reaction–diffusion PDEs subject to heterogeneities has not yet been addressed. Heterogeneities may be manifested by disturbances entering distributed systems: constant and sinusoidal disturbances are especially common in control systems, due to biases in outputs of sensors and actuators, vibrations, unexpected input loads, etc. Without compensating for the heterogeneities, a system may be unable to attain spatial uniformity.

In this paper, we consider synchronization of nonlinear systems satisfying an incremental passivity property and subject to a class of heterogeneities including constants and sinusoids. Motivated by the robust DAC estimator in  Bai et al. (2010), we provide a unified solution to output synchronization of incrementally passive spatially-distributed systems in the presence of heterogeneities. We show that incrementally passive system dynamics allow us to embed an incrementally passive internal model in the controller, thereby compensating for desynchronizing effects due to the differences between the heterogeneities. For reaction–diffusion PDEs, we present a distributed control law that includes a PDE-based internal model and achieves spatial homogenization in the presence of spatially and temporally-varying heterogeneities. We further show that the internal model control renders average spatial homogenization, where the system solutions asymptotically synchronize to a solution corresponding to the average of the heterogeneities. Our controller applies to systems with multiple input and output channels and allows non-identical heterogeneities to enter each channel.

For the ODE case, we consider a group of agents whose dynamics are incrementally passive. The coupling between the agents is described by bidirectional graphs. We propose a distributed control law that defines an internal model subsystem at each node corresponding to the heterogeneous inputs and achieves output synchronization. A key consequence is that local communication, computation and memory requirements are independent of the number of systems in the network and the network connectivity, which is of interest in dense networks under processing and communication constraints. The number of states in our model furthermore scales linearly in relation to the number of subsystems. Our approach additionally incorporates a proportional feedback term that is advantageous for systems that have a shortage of incremental passivity, allowing such systems to synchronize in the presence of disturbances if the coupling between agents is sufficiently strong.

We illustrate the effectiveness of our proposed controls using three numerical examples. In the examples, the dynamics of the nonlinear systems each have a shortage of incremental passivity. In particular, we demonstrate spatial homogenization of ring oscillators and systems with nominal bistable dynamics. We also illustrate output synchronization of compartmental ODE systems with bistable dynamics. The coupling of the compartmental systems is described by a clustered graph, which has three densely connected components with sparse external links between each pair of clusters. The simulation results verify that our internal model control indeed enables synchronization of the three clusters.

The rest of this paper is organized as follows. Section  2 presents preliminaries on incremental passivity. Our main result on spatial homogenization of PDEs under heterogeneities is presented Section  3. Section  4 presents output synchronization of diffusively coupled compartmental incrementally passive systems of ODEs under input disturbances. In Section  5, we illustrate the effectiveness of our control law on models with ring oscillator dynamics and bistable dynamics. Conclusions and future work are discussed in Section  6.

Notation

Let 1N be the N×1 vector with all entries 1. Let 0N be the N×1 vector with all entries 0. Let the transpose of a real matrix A be denoted by AT. Let IN denote the N×N identity matrix. Let the notation M=blkdiag{M1,,Mp} denote the block diagonal matrix M with matrices Mi, i=1,,p, along the diagonal. Let L2(Ω) denote the space of square integrable functions with time-independent domain Ω.

Section snippets

Preliminaries

Consider a collection of N dynamical systems Hi, i=1,,N, defined by: Hi:ẋi=f(xi)+g(xi)uii=1,,Nyi=h(xi), in which xiRn, uiRp, yiRp, and f():RnRn, g():RnRn×p, and h():RnRp are continuously differentiable maps. Hi is said to satisfy an incremental output-feedback passivity property (Scardovi et al., 2010, Stan and Sepulchre, 2007) if there exists a positive definite storage function S:RnR such that for any two solution trajectories xa(t) and xb(t) of Hi with input–output pairs ua(t),ya

Reaction–diffusion PDEs

In this section, we formulate the problem of spatial homogenization for systems of reaction–diffusion PDEs.

Let Ω be a bounded and connected domain in Rr with smooth boundary Ω, and consider the PDE: x(t,χ)t=f(x(t,χ))+=1pg(x(t,χ))u(t,χ)y(t,χ)=h(x(t,χ)),=1,,p, where χΩ is the spatial variable, x(t,χ)Rn is the state variable with initial condition x(0,χ)=x0(χ), f():RnRn, g()=[g1gp]:RnRn×p, and h()=[h1ThpT]T:RnRp are continuously differentiable maps, and the inputs u(t,χ)

Compartmental systems of ODEs

In relation to homogenization of each output y(t,χ) across the spatial domain Ω, we now derive a control law that guarantees synchronization of each like output yi,k of a compartmental system of ODEs across compartments i=1,,N.

Consider the case where the input ui for each Hi in (1) is subject to a class of unknown heterogeneities ϕi(t)Rp, i.e.,  ui=ui+ϕi, in which each heterogeneity ϕi can be characterized by ξ̇i=Aξi,ξi(0)Rdϕi=Cξi, with A=ATRd×d, CRp×d, and the pair (A,C) observable.

Numerical examples

In this section, we consider several numerical examples, demonstrating the effectiveness of the internal model approach in guaranteeing spatial homogenization in the presence of heterogeneities and disturbances.

Conclusions and future work

We have studied spatial homogenization of incrementally passive nonlinear systems, and designed distributed control laws that guarantee spatial homogeneity in the presence of a class of spatially and temporally-varying heterogeneities. Our controller has the advantage of not requiring knowledge of the initial conditions of the heterogeneities, and is amenable to heterogeneities that vary over multiple input and output channels. Future work includes designing adaptation laws to reduce time to

S. Yusef Shafi received his M.S. and Ph.D. degrees in Electrical Engineering and Computer Sciences from the University of California, Berkeley (2011 and 2014), and his B.S. degree in Applied Mathematics from the University of California, Los Angeles (2008). From 2008 to 2009, he was a Software Engineer at Northrop Grumman. His research is in distributed control and optimization, with applications to diffusively-coupled networks, multi-agent systems, and machine learning for the smart grid. He

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    S. Yusef Shafi received his M.S. and Ph.D. degrees in Electrical Engineering and Computer Sciences from the University of California, Berkeley (2011 and 2014), and his B.S. degree in Applied Mathematics from the University of California, Los Angeles (2008). From 2008 to 2009, he was a Software Engineer at Northrop Grumman. His research is in distributed control and optimization, with applications to diffusively-coupled networks, multi-agent systems, and machine learning for the smart grid. He is currently an Engineer at Nest, a Google company.

    He Bai is an Assistant Professor in Mechanical and Aerospace Engineering at Oklahoma State University. He received the B.Eng degree from the Department of Automation at the University of Science and Technology of China, Hefei, China, in 2005, and the M.S. and Ph.D. degrees in Electrical Engineering from Rensselaer Polytechnic Institute in 2007 and 2009, respectively. From 2009 to 2010, he was a Post-doctoral Researcher at the Northwestern University, Evanston, IL. Before joining Oklahoma State University, he was a Senior Research and Development Scientist at UtopiaCompression Corporation, working with the Air Force and DARPA on Unmanned Aircraft Systems (UAS) research and technologies. He was Principal Investigator (PI) or co-PI for a number of SBIR/STTR projects on UAS sense-and-avoid, cooperative target tracking, and target handoff in GPS-denied environments. He has also published a research monograph “Cooperative control design: a systematic passivity-based approach” in Springer in 2011. He was listed in “Who’s Who in America” in 2009 and 2012. His research interests include nonlinear control and estimation, multi-agent systems, sensor fusion, path planning, intelligent control, and robotics.

    The material in this paper was partially presented at the 53rd IEEE Conference on Decision and Control, 2014. This paper was recommended for publication in revised form by Associate Editor Nicolas Petit under the direction of Editor Miroslav Krstic.

    1

    Tel.: +1 510 642 3214; fax: +1 510 643 7846.

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    Now at: Nest Labs (Google Inc.), Palo Alto, CA, United States.

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