## Keywords

Synchronization; stability; distributed control; networked systems; networked control systems; pinning control; nonlinear systems; nonlinear control; complex networks; small worlds; random networks; chaotic oscillators; chaos

## Abstract

Stability analysis of networked dynamical systems has been of interest in many disciplines such as biology and physics and chemistry with applications such as LASER cooling and plasma stability. These large networks are often modeled to have a completely random (Erdös-Rényi) or semi-random (Small-World) topologies. The former model is often used due to mathematical tractability while the latter has been shown to be a better model for most real life networks. The recent emergence of cyber physical systems, and in particular the smart grid, has given rise to a number of engineering questions regarding the control and optimization of such networks. Some of the these questions are: How can the stability of a random network be characterized in probabilistic terms? Can the effects of network topology and system dynamics be separated? What does it take to control a large random network? Can decentralized (pinning) control be effective? If not, how large does the control network needs to be? How can decentralized or distributed controllers be designed? How the size of control network would scale with the size of networked system? Motivated by these questions, we began by studying the probability of stability of synchronization in random networks of oscillators. We developed a stability condition separating the effects of topology and node dynamics and evaluated bounds on the probability of stability for both Erdös-Rényi (ER) and Small-World (SW) network topology models. We then turned our attention to the more realistic scenario where the dynamics of the nodes and couplings are mismatched. Utilizing the concept of ε-synchronization, we have studied the probability of synchronization and showed that the synchronization error, ε, can be arbitrarily reduced using linear controllers. We have also considered the decentralized approach of pinning control to ensure stability in such complex networks. In the pinning method, decentralized controllers are used to control a fraction of the nodes in the network. This is different from traditional decentralized approaches where all the nodes have their own controllers. While the problem of selecting the minimum number of pinning nodes is known to be NP-hard and grows exponentially with the number of nodes in the network we have devised a suboptimal algorithm to select the pinning nodes which converges linearly with network size. We have also analyzed the effectiveness of the pinning approach for the synchronization of oscillators in the networks with fast switching, where the network links disconnect and reconnect quickly relative to the node dynamics. To address the scaling problem in the design of distributed control networks, we have employed a random control network to stabilize a random plant network. Our results show that for an ER plant network, the control network needs to grow linearly with the size of the plant network.

## Notes

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## Graduation Date

2015

## Semester

Summer

## Advisor

Vosoughi, Azadeh

## Degree

Doctor of Philosophy (Ph.D.)

## College

College of Engineering and Computer Science

## Department

Electrical Engineering and Computer Science

## Degree Program

Engineering and Computer Science

## Format

application/pdf

## Identifier

CFE0005834

## URL

http://purl.fcla.edu/fcla/etd/CFE0005834

## Language

English

## Release Date

August 2015

## Length of Campus-only Access

None

## Access Status

Doctoral Dissertation (Open Access)

## Subjects

Dissertations, Academic -- Engineering and Computer Science; Engineering and Computer Science -- Dissertations, Academic

## STARS Citation

Manaffam, Saeed, "Stability and Control in Complex Networks of Dynamical Systems" (2015). *Electronic Theses and Dissertations*. 692.

https://stars.library.ucf.edu/etd/692