Модели сетевых динамических систем, методы обеспечения устойчивости и достижения консенсуса тема диссертации и автореферата по ВАК РФ 00.00.00, доктор наук Парсегов Сергей Эрнестович

• Парсегов Сергей Эрнестович
• доктор наук
• 2024

Оглавление диссертации доктор наук Парсегов Сергей Эрнестович

Contents

1 Introduction

2 Overview of the obtained results

2.1 Opinion dynamics with interdependent issues

2.1.1 A multidimensional extension

2.1.2 Convergence and the steady opinions

2.1.3 Design of the MiDS matrix

2.1.4 A randomized gossip-based model

2.2 Uniform deployment on a segment

2.2.1 Stability and convergence of second-order dynamics

2.2.2 Fixed-time uniform deployment

2.3 Consensus

2.3.1 Fixed-time consensus in undirected networks

2.3.2 Hierarchical cyclic pursuit

3 Conclusion

4 Bibliography

5 Appendix

Article 1: A new model of opinion dynamics for social actors with multiple

interdependent attitudes and prejudices

Article 2: Network science on belief system dynamics under logic constraints

Article 3: Novel multidimensional models of opinion dynamics in social

networks

Article 4: Equidistant arrangement of agents on line: analysis of the algorithm and its generalization

Article 5: Problem of uniform deployment on a line segment for second-order

agents

Article 6: Uniform deployment of second-order agents on a line segment . . 95 Article 7: Nonlinear fixed-time control protocol for uniform allocation of

agents on a segment

Article 8: Fixed-time consensus algorithm for multiagent systems with

integrator dynamics

Article 9: Second-order agents on ring digraphs

Article 10: Hierarchical cyclic pursuit: Algebraic curves containing the

Laplacian spectra

Article 11: Laplacian spectra of two-layer hierarchical cyclic pursuit schemes

Введение диссертации (часть автореферата) на тему «Модели сетевых динамических систем, методы обеспечения устойчивости и достижения консенсуса»

Introduction

Simple averaging control laws based on local interactions have paved the way to a new class of models in modern control theory and, more widely, in the interconnected dynamical systems theory. Such systems (also known as multi-agent systems and network systems) consist of a large (as usual) number of identical subsystems and are supposed to achieve certain global goals. The subsystems, or agents, are coupled in some way and therefore share an amount of common information. The interest to such systems is motivated by numerous real-life applications; it turns out that use of simple rules of local communication in the absence of centralized control may lead to many attractive features. The motivation behind the research in this field is to identify advantages over conventional single-agent systems. First, it is the reduction in cost and complexity from hardware platform to software and algorithms, i.e., one large and expensive robot or unit can be replaced by several smaller and cheaper units while realizing a task with lower cost and complexity. Second, multi-agent systems are capable of solving many tasks that cannot be efficiently performed by a single-agent system, such as the surveillance task. In addition, multi-agent systems with decentralized control have preferential flexibility and robustness, and reduce the communication and computational load on the signal by using local neighbor-to-neighbor interaction. The consensus problem is a fundamental problem in the cooperative control of multi-agent systems, since many applications are based on the development of consensus algorithms. In multi-agent systems, the consensus problem is to develop a strategy for managing a group of agents to reach consensus (or agreement) on their

states of interest. The basic idea is that each agent updates its information state based on the information states of the states of its local neighbors such that the final information state of each agent converges to a common value. One of the significant contributions to consensus control is the introduction of graph theory into traditional control theory. As new problems and algorithms of cooperative control appeared, it became clear that the increasing complexity of both the agents themselves and the properties of the network through which they interact (i.e., from undirected graphs to directed graphs, from static topology to switching topology, from simple linear dependencies of the agents' states on the states of their neighbors to nonlinear couplings) significantly complicate the analysis of stability or the possibility of achieving consensus in such systems, while complicating the problems of design too. During the last 20 years, the complexity of models of network systems and related problems increased significantly starting from a simple continuous-time consensus model with first-order agents and undirected communication graph.

It is convenient to study the evolution of these models/problems and classify them considering separately three main entities comprising a network system (three independent dimensions of complexity): agent complexity, interaction graph complexity, and coupling function complexity.

This summary describes new models of network dynamical systems and network control algorithms related to areas such as opinion dynamics in a social network, formation control, and the classical consensus problem. The results obtained advance the science of network systems in each of the directions of the mentioned complexity space. Thus, a new model with multidimensional interdependent opinions, its randomized version, was proposed in the framework of the problem of modeling social network dynamics [1], [2], [3]. The problems of stability and convergence were investigated, and results were obtained for different types of the matrix of multi-issues dependence structure and its identification. More adequate high-order agent models as well as new control protocols were proposed for the formation control problems of uniform deployment on a fixed segment and cyclic pursuit [4], [5], [6], [7], [8], [9]. In addition, for both the deployment on a segment

problem and the general consensus problem, a distributed nonlinear control protocol that guarantees stabilization in a finite globally bounded time was proposed [10], [11].

The results obtained form the basis for a comprehensive analysis, modeling and optimization of network systems. The research contributes to nonlinear control theory, algebraic graph theory and stability/consensusability analysis of network systems. Thus, the results obtained in the dissertation develop and enrich the network systems science along all three dimensions of the complexity space.

Object and goals of the dissertation.

The object of the dissertation is network dynamical systems or multi-agent systems. The goals of the thesis are to develop new models of network systems, corresponding control laws, which guarantee the desired properties of the closed-loop system. This includes both the development of new agent models, new control protocols, the introduction and research of new interaction topologies, as well as analyzing their impact on the operation of the entire network system.

The obtained results:

1. We propose novel stabilizing control protocols for equidistant deployment of second-order agents on a segment. These results are obtained in papers [4], [5], [6].

2. We develop fixed-time control law for equidistant allocation on a segment in finite globally bounded time. This result is obtained in paper [10].

3. We design a fixed-time control protocol that ensures consensus in networks of first-order agents in finite globally bounded time. This result is obtained in paper [11].

4. We propose a model of network dynamics with interconnected opinions of agents and its generalization to the case of asynchronous interaction. These results are obtained in papers [1], [2], [3].

5. We propose an algorithm for identifying the matrix of multi-issues dependence structure. These results are obtained in papers [1], [3].

6. We propose hierarchical models of cyclic pursuit and methods for investigating consensusability. These results are obtained in papers [8], [9].

7. We develop a method for localization of the spectrum of the graph Laplacian matrices of hierarchical cyclic pursuit problems using high-order algebraic curves. These results are obtained in papers [7], [9].

Author's contribution includes the mathematical problem formulations, the development of theoretical statements, mathematical models and methods, analysis and generalization of the results.

The novelty of the proposed research lies in the development of new models, methods, control laws and analysis of them. In particular, in the dissertation the author proposes:

• A model of opinion dynamics with interdependent issues;

• A control law for stabilization of a formation of second-order agents on a segment;

• A distributed fixed-time control for stabilization and consensus of first-order agents;

• A method for exact localization of the Laplacian spectra for the family of ring digraphs.

The scope of dissertation is covered in 17 publications, among those we specifically mention papers [2], [3], [9] in Q1-journals; papers [4], [6] in Q2-journals; and papers [10], [1] in CORE A conference proceedings.

According to regulations of the Dissertation Council in Computer Sciences of Higher School of Economics 12 papers are listed below. The defense is performed

based on 11 of them (namely, first 7 from the list of first-tier publication, two second-tier publications, and two other publications).

Список литературы диссертационного исследования доктор наук Парсегов Сергей Эрнестович, 2024 год

REFERENCES

Behroozi, F. and Gagnon, R. (1979). Cyclic pursuit in a plane. Journal of Mathematical Physics, 20(11), 22122216.

Bruckstein, A.M., Cohen, N., and Efrat, A. (1991). Ants,

crickets and frogs in cyclic pursuit. Citeseer. Darboux, G. (1878). Sur un probleme de geometrie elementaire. Bulletin des Sciences Mathématiques et

Astronomiques, 2(1), 298-304.

Davis, P.J. (2013). Circulant matrices. American Mathematical Soc.

Hara, S., Tanaka, H., and Iwasaki, T. (2013). Stability analysis of systems with generalized frequency variables. IEEE Transactions on Automatic Control, 59(2), 313326.

Hu, J., Bhowmick, P., Arvin, F., Lanzon, A., and Lennox, B. (2020). Cooperative control of heterogeneous connected vehicle platoons: An adaptive leader-following approach. IEEE Robotics and Automation Letters, 5(2), 977-984.

Iqbal, M., Leth, J., and Ngo, T.D. (2017). Generalized hierarchical block circulant structure of multi-agent systems. International Journal of Control, Automation and Systems, 15(3), 1275-1286.

Iqbal, M., Leth, J., and Ngo, T.D. (2018). Cartesian product-based hierarchical scheme for multi-agent systems. Automatica, 88, 70-75.

Iyengar, S.S. and Brooks, R.R. (2016). Distributed sensor •networks: sensor networking and applications (Volume Two). CRC press.

Johnsen, E.C. (1973). Essentially doubly stochastic matrices. Linear & Multilinear Algebra, 1, 33-45.

Klamkin, M. and Newman, D. (1971). Cyclic pursuit or "the three bugs problem". The American Mathematical Monthly, 78(6), 631-639.

Li, Z. and Duan, Z. (2017). Cooperative control of multiagent systems: a consensus region approach. CRC press.

Marshall, J.A., Broucke, M.E., and Francis, B.A. (2004). Formations of vehicles in cyclic pursuit. IEEE Transactions on automatic control, 49(11), 1963-1974.

Mukherjee, D. and Ghose, D. (2016). Generalized hierarchical cyclic pursuit. Automatica, 71, 318-323.

Olfati-Saber, R., Fax, J.A., and Murray, R.M. (2007). Consensus and cooperation in networked multi-agent systems. Proceedings of the IEEE, 95(1), 215-233.

Parsegov, S. and Chebotarev, P. (2018). Second-order agents on ring digraphs. In Proc. 22nd International Conference on System Theory, Control and Computing (ICSTCC), 609-614. IEEE.

Polyak, B.T. and Tsypkin, Y.Z. (1996). Stability and robust stability of uniform systems. Automation and Re-mote Control, 57(11), 1606-1617.

Ren, W. and Beard, R.W. (2008). Distributed consensus in multi-vehicle cooperative control. Springer.

Segall, I. and Bruckstein, A. (2020). Guidance of agents in cyclic pursuit. arXiv preprint arXiv:2007.00949.

Sharma, B.R., Ramakrishnan, S., and Kumar, M. (2013). Cyclic pursuit in a multi-agent robotic system with double-integrator dynamics under linear interactions. Robotica, 31(7), 1037-1050.

Smith, S.L., Broucke, M.E., and Francis, B.A. (2005). A hierarchical cyclic pursuit scheme for vehicle networks. Automatica, 41(6), 1045-1053.

Tsubakino, D. and Hara, S. (2012). Eigenvector-based intergroup connection of low rank for hierarchical multiagent dynamical systems. Systems & Control Letters, 61(2), 354-361.

Tsubakino, D., Yoshioka, T., and Hara, S. (2013). An algebraic approach to hierarchical lqr synthesis for large-scale dynamical systems. In Proc. 9th Asian Control Conference, 1-6. IEEE.