Networked control of coupled subsystems: Spectral decomposition and low-dimensional solutions

Gao, Shuang; Mahajan, Aditya

doi:10.1109/cdc40024.2019.9029266

Cited by 5 publications

(4 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…See [26], [38] for the details of the solution method, which provides solutions to a more general class of graphon control problems with network couplings in states, controls and costs.…”

Section: B Minimum Energy Graphon State-to-state Controlmentioning

confidence: 99%

Graphon Control of Large-Scale Networks of Linear Systems

Gao

Caines

2020

IEEE Trans. Automat. Contr.

Self Cite

View full text Add to dashboard Cite

To achieve control objectives for extremely largescale complex networks using standard methods is essentially intractable. In this work, a theory of the approximate control of complex network systems is proposed and developed by the use of graphon theory and the theory of infinite dimensional systems. First, the graphon dynamical system models are formulated in an appropriate infinite dimensional space in order to represent arbitrary-size networks of linear dynamical systems, and to define the convergence of sequences of network systems with limits in the space. The exact controllability and the approximate controllability of graphon dynamical systems are then investigated. Second, the minimum energy state-to-state control problem and the linear quadratic regulator problem for systems on complex networks are considered. The control problem for the graphon limit systems is solved in each case and an approximation is defined which yields control laws for the original control problems. Furthermore, the convergence properties of the approximation schemes are established. A systematic control design methodology is developed within this framework. Finally, numerical examples of networks with randomly sampled weightings are presented to illustrate the effectiveness of the graphon control methodology.

show abstract

“…See [26], [38] for the details of the solution method, which provides solutions to a more general class of graphon control problems with network couplings in states, controls and costs.…”

Section: B Minimum Energy Graphon State-to-state Controlmentioning

confidence: 99%

Graphon Control of Large-Scale Networks of Linear Systems

Gao

Caines

2020

IEEE Trans. Automat. Contr.

Self Cite

View full text Add to dashboard Cite

show abstract

“…We want to reduce the infection level, limit the actions cost and make sure that subpopulations on the network receive almost the same amount of resources as their weighted neighbors for equity purposes. This forms a good example to illustrate the polynomial structures appearing in the cost functions in [14], [38].…”

Section: Controlling Epidemic Processes On Network Based On Specmentioning

confidence: 99%

“…The number L of non-zero eigenvalues can be much smaller than N . The optimal solution [38] to be executed at community i is given by…”

Section: Controlling Epidemic Processes On Network Based On Specmentioning

confidence: 99%

Spectral Representations of Graphons in Very Large Network Systems Control

Gao

Caines

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

Self Cite

View full text Add to dashboard Cite

Graphon-based control has recently been proposed and developed to solve control problems for dynamical systems on networks which are very large or growing without bound (see Gao and Caines, CDC 2017, CDC 2018). In this paper, spectral representations, eigenfunctions and approximations of graphons, and their applications to graphon-based control are studied. First, spectral properties of graphons are presented and then approximations based on Fourier approximated eigenfunctions are analyzed. Within this framework, two classes of graphons with simple spectral representations are given. Applications to graphon-based control analysis are next presented; in particular, the controllability of systems distributed over very large networks is expressed in terms of the properties of the corresponding graphon dynamical systems. Moreover, spectral analysis based upon real-world network data is presented, which demonstrates that low-dimensional spectral approximations of networks are possible. Finally, an initial, exploratory investigation of the utility of the spectral analysis methodology in graphon systems control to study the control of epidemic spread is presented.

show abstract

“…The solution method was first developed for linear quadratic mean-field control problems in [12], [20] (where couplings are of rank one). This idea is further generalized and applied to the control of graphon and network coupled systems in [4], [21], [22]. A closely related recent paper [23] discusses decoupling linear quadratic control problems based on state transformations (or coordinate transformation) to formulate equivalent problems and applies it to generate low-complexity solutions to costsensitive linear quadratic control problems with mean-field couplings.…”

Section: Introductionmentioning

confidence: 99%

Subspace Decomposition for Graphon LQR: Applications to VLSNs of Harmonic Oscillators

Gao,

Caines

2020

Preprint

Self Cite

View full text Add to dashboard Cite

Graphon control has been proposed and developed in [1]-[4] to approximately solve control problems for very largescale networks of linear dynamical systems. In this paper, linear quadratic regulation (LQR) problems for graphon dynamical systems are studied. Graphon couplings appear in states, controls and cost, and these couplings may be represented by different graphons. Based on invariant subspace decompositions, this work provides a solution method for a class of such problems where the local dynamics is homogeneous but the network couplings are heterogeneous among the coupled subsystems. By exploring a common invariant subspace of the couplings, the original problem is decomposed into a network coupled LQR problem of finite dimension and a decoupled infinite dimensional LQR problem. A centralized optimal solution and a nodal collaborative optimal control solution are established. The complexity of these solutions involves solving one nd × nd dimensional Riccati equation and one n × n Riccati equation, where n is the dimension of each nodal agent state and d is the dimension of the (nontrivial) invariant subspace shared by the coupling operators. For situations where the graphon couplings do not admit exact lowrank representations, approximate control is developed based on low-rank approximations. Finally, an application to the regulation of harmonic oscillators coupled over large networks with uncertainties is demonstrated.

show abstract

Networked control of coupled subsystems: Spectral decomposition and low-dimensional solutions

Cited by 5 publications

References 38 publications

Graphon Control of Large-Scale Networks of Linear Systems

Graphon Control of Large-Scale Networks of Linear Systems

Spectral Representations of Graphons in Very Large Network Systems Control

Subspace Decomposition for Graphon LQR: Applications to VLSNs of Harmonic Oscillators

Contact Info

Product

Resources

About