Ph. D. Project
Title:
Analysis and design of networked k-contractive systems
Dates:
2024/10/01 - 2027/09/30
Supervisor(s):
Other supervisor(s):
Airlie Chapman
Description:
Networked systems are increasingly present in modern society. The study of these systems becomes essential for many fields of application. Without being exhaustive, the analysis of networked systems is motivated by applications such as: the spread of an epidemic following interactions between different individuals; the propagation of opinions/rumors in a social network; the evolution of the climate following interactions between the environment and economic and social agents; the dynamics of robots cooperating to collectively carry out a task. We can add technological applications such as the control of autonomous vehicle platoons, or the collaborative control of robots on an assembly line. In this context, a scientific priority of the CID department concerns the development of mathematical/theoretical tools for the analysis of interconnected systems.
The asymptotic behavior of nonlinear dynamic systems cannot be reduced to attraction towards a fixed point or a unique and stationary solution. This is particularly true when non-local behaviors are considered and/or when we are not dealing with stabilization issues. For this, the notion of incremental stability [LS98] began to play a fundamental role in control theory. In short, a system is incrementally stable (or contractive) if the distance between trajectories with different initial conditions decreases asymptotically to zero, without necessarily converging to an equilibrium point. Contraction properties have been shown to be effective in many control problems, such as output regulation [GAAM22], multi-agent synchronization [JCVB21], observer design [GATA23], etc. Therefore, it has become natural to search for generalizations of such a tool.
In this project, we focus on the recent notion of k-contraction [WKM22]. It generalizes the concept of contraction of distances to geometric objects of dimension k. Thus, the 1-contraction corresponds to the contraction of lengths, that is to say, to contraction in the usual sense. Therefore, the 2-contraction corresponds to the contraction of surfaces, the 3-contraction to the contraction of volumes, and so on. For k>1, such a notion can be used to study the asymptotic behavior of nonlinear systems, such as the presence of (possibly multiple) equilibrium points, limit cycles, and chaotic behavior without relying on tools of local stability.The objective of this thesis is to study the k-contractive properties of networked systems. Indeed, oscillations, limit cycles and similar behaviors are typical of many nonlinear systems that can be described as multi-agent systems. This type of framework is found in many applications, such as decision-making processes, training control of mobile robots, aerial vehicles, biological interaction networks [AARS21], neural networks, and many others.
The interest of the project is twofold. On the one hand, it would make it possible to study and justify the asymptotic behaviors of the solutions of network systems, by examining their k-contractive properties. On the other hand, it would enable the development of new tools to control networked systems by appropriately imposing k-contractive properties on the network, either through feedback design or by modifying the communication protocol .
The thesis will be developed in two phases. First, we will focus on creating new analysis tools. Indeed, existing approaches are either based on matrix measurements [WKM22], which require intensive calculations and are not suitable for feedback design, or on the use of flat metric structures [FS18], which are however not not suitable for large classes of systems. In particular, we will first develop a Riemannian type (and/or Lyapunov type) approach to study the k-contraction (similar to [LS98] for 1-contraction). This will allow us to consider more general classes of systems, for which sufficient and necessary conditions for k-contraction will be developed. Next, we will study how k-contractive systems behave under the effect of external signals (and therefore input-output stability properties). The second phase of the thesis will focus on the distributed aspect, that is to say, the interconnection of k-contractive systems. In particular, we will study the network interconnection of k-contractive systems, the role of the communication protocol (the graph) in the synchronization process and how to modify it to obtain certain asymptotic properties. To conclude, we will study how to design feedback controllers achieving k-contractive properties for the closed loop.
[AARS21] D. Angeli, M.A. Al-Radhawi, and E.D. Sontag. A robust Lyapunov criterion for nonoscillatory behaviors in biological
interaction networks. IEEE Transactions on Automatic Control, 67(7):3305-3320, 2021.
[FS18] F. Forni and R. Sepulchre. Differential dissipativity theory for dominance analysis. IEEE Transactions on Automatic Control, 64(6):2340-2351, 2018.
[GAAM22] M. Giaccagli, D. Astolfi, V. Andrieu, and L. Marconi. Sufficient conditions for global integral action via incremental
forwarding for input-affine nonlinear systems. IEEE Transactions on Automatic Control, 67(12):6537-6551, 2022.
[GATA23] M. Giaccagli, V. Andrieu, S. Tarbouriech, and D. Astolfi. LMI conditions for contraction, integral action, and output
feedback stabilization for a class of nonlinear systems. Automatica, 154:111106, 2023.
[JCVB21] S. Jafarpour, P. Cisneros-Velarde, and F. Bullo. Weak and semi-contraction for network systems and diffusively coupled
oscillators. IEEE Transactions on Automatic Control, 67(3):1285-1300, 2021.
[LS98] W. Lohmiller and J.J. E. Slotine. On contraction analysis for non-linear systems. Automatica, 34(6):683-696, 1998.
[WKM22] C. Wu, I. Kanevskiy, and M. Margaliot. k-contraction: Theory and applications. Automatica, 136:110048, 2022.
The asymptotic behavior of nonlinear dynamic systems cannot be reduced to attraction towards a fixed point or a unique and stationary solution. This is particularly true when non-local behaviors are considered and/or when we are not dealing with stabilization issues. For this, the notion of incremental stability [LS98] began to play a fundamental role in control theory. In short, a system is incrementally stable (or contractive) if the distance between trajectories with different initial conditions decreases asymptotically to zero, without necessarily converging to an equilibrium point. Contraction properties have been shown to be effective in many control problems, such as output regulation [GAAM22], multi-agent synchronization [JCVB21], observer design [GATA23], etc. Therefore, it has become natural to search for generalizations of such a tool.
In this project, we focus on the recent notion of k-contraction [WKM22]. It generalizes the concept of contraction of distances to geometric objects of dimension k. Thus, the 1-contraction corresponds to the contraction of lengths, that is to say, to contraction in the usual sense. Therefore, the 2-contraction corresponds to the contraction of surfaces, the 3-contraction to the contraction of volumes, and so on. For k>1, such a notion can be used to study the asymptotic behavior of nonlinear systems, such as the presence of (possibly multiple) equilibrium points, limit cycles, and chaotic behavior without relying on tools of local stability.The objective of this thesis is to study the k-contractive properties of networked systems. Indeed, oscillations, limit cycles and similar behaviors are typical of many nonlinear systems that can be described as multi-agent systems. This type of framework is found in many applications, such as decision-making processes, training control of mobile robots, aerial vehicles, biological interaction networks [AARS21], neural networks, and many others.
The interest of the project is twofold. On the one hand, it would make it possible to study and justify the asymptotic behaviors of the solutions of network systems, by examining their k-contractive properties. On the other hand, it would enable the development of new tools to control networked systems by appropriately imposing k-contractive properties on the network, either through feedback design or by modifying the communication protocol .
The thesis will be developed in two phases. First, we will focus on creating new analysis tools. Indeed, existing approaches are either based on matrix measurements [WKM22], which require intensive calculations and are not suitable for feedback design, or on the use of flat metric structures [FS18], which are however not not suitable for large classes of systems. In particular, we will first develop a Riemannian type (and/or Lyapunov type) approach to study the k-contraction (similar to [LS98] for 1-contraction). This will allow us to consider more general classes of systems, for which sufficient and necessary conditions for k-contraction will be developed. Next, we will study how k-contractive systems behave under the effect of external signals (and therefore input-output stability properties). The second phase of the thesis will focus on the distributed aspect, that is to say, the interconnection of k-contractive systems. In particular, we will study the network interconnection of k-contractive systems, the role of the communication protocol (the graph) in the synchronization process and how to modify it to obtain certain asymptotic properties. To conclude, we will study how to design feedback controllers achieving k-contractive properties for the closed loop.
[AARS21] D. Angeli, M.A. Al-Radhawi, and E.D. Sontag. A robust Lyapunov criterion for nonoscillatory behaviors in biological
interaction networks. IEEE Transactions on Automatic Control, 67(7):3305-3320, 2021.
[FS18] F. Forni and R. Sepulchre. Differential dissipativity theory for dominance analysis. IEEE Transactions on Automatic Control, 64(6):2340-2351, 2018.
[GAAM22] M. Giaccagli, D. Astolfi, V. Andrieu, and L. Marconi. Sufficient conditions for global integral action via incremental
forwarding for input-affine nonlinear systems. IEEE Transactions on Automatic Control, 67(12):6537-6551, 2022.
[GATA23] M. Giaccagli, V. Andrieu, S. Tarbouriech, and D. Astolfi. LMI conditions for contraction, integral action, and output
feedback stabilization for a class of nonlinear systems. Automatica, 154:111106, 2023.
[JCVB21] S. Jafarpour, P. Cisneros-Velarde, and F. Bullo. Weak and semi-contraction for network systems and diffusively coupled
oscillators. IEEE Transactions on Automatic Control, 67(3):1285-1300, 2021.
[LS98] W. Lohmiller and J.J. E. Slotine. On contraction analysis for non-linear systems. Automatica, 34(6):683-696, 1998.
[WKM22] C. Wu, I. Kanevskiy, and M. Margaliot. k-contraction: Theory and applications. Automatica, 136:110048, 2022.
Keywords:
Nonlinear systems, incremental stability, k-contraction, networked systems
Conditions:
Duration: 3 ans
Place: CRAN Nancy, University of Lorraine
Salary: around 1700 net income
Profile: Strong background in mathematics and nonlinear control, good skills in Matlab and facility to communicate in English. Motivation and interest towards the academic and applied research
Cooperation with University of Melbourne, visit of few months to plan
Place: CRAN Nancy, University of Lorraine
Salary: around 1700 net income
Profile: Strong background in mathematics and nonlinear control, good skills in Matlab and facility to communicate in English. Motivation and interest towards the academic and applied research
Cooperation with University of Melbourne, visit of few months to plan
Department(s):
| Control Identification Diagnosis |
Funds:
Ph.D. contract of University of Lorraine
Publications: