Transient Analysis and Control for Scalable Network Systems

The rapidly evolving domain of network systems poses complex challenges, especially when considering scalability and transient behaviors. This thesis aims to address these challenges by offering insights into the transient analysis and control design tailored for large-scale network systems. The thesis consists of three papers, each of which contributes to the overarching goal of this work.The first paper, A closed-loop design for scalable high-order consensus, studies the coordination of nth-order integrators in a networked setting. The paper introduces a novel closed-loop dynamic named serial consensus, which is designed to achieve consensus in a scalable manner and is shown to be implementable through localized relative feedback. In the paper, it is shown that the serial consensus system will be stable under a mild condition — that the underlying network contains a spanning tree — thereby mitigating a previously known scale fragility. Robustness against both model and feedback uncertainties is also discussed.The second paper, Closed-loop design for scalable performance of vehicular formations, expands on the theory on the serial consensus system for the special case when n=2, which is of special interest in the context of vehicular formations. Here, it is shown that the serial consensus system can also be used to give guarantees on the worst-case transient behavior of the closed-loop system. The potential of achieving string stability through the use of serial consensus is explored.The third paper, Input-output pseudospectral bounds for transient analysis of networked and high-order systems, presents a novel approach to transient analysis of networked systems. Bounds on the matrix exponential, coming from the theory on pseudospectra, are adapted to an input-output setting. The results are shown to be useful for high-order matrix differential equations, offering a new perspective on the transient behavior of high-order networked systems

Digraphs with distinguishable dynamics under the multi-agent agreement protocol

This work studies the ability to distinguish digraphs from the output response of some observing agents in a multi-agent network under the agreement protocol. Given a fixed observation point, it is desired to find sufficient graphical conditions under which the failure of a set of edges in the network information flow digraph is distinguishable from another set. When the latter is empty, this corresponds to the detectability of the former link set given the response of the observing agent. In developing the results, a powerful extension of the all-minors matrix tree theorem in algebraic graph theory is proved which relates the minors of the transformed Laplacian of a directed graph to the number and length of the shortest paths between its vertices. The results reveal an intricate relationship between the ability to distinguish the responses of a healthy and a faulty multi-agent network and the inter-nodal paths in their information flow digraphs. The results have direct implications for the operation and design of multi-agent systems subject to multiple link losses. Simulations and examples are presented to illustrate the analytic findings

Mathematical analysis of k-path Laplacian operators on simple graphs

A set of links and nodes are the fundamental units or components used to represent complex networks. Over the last few decades, network studies have expanded and matured, increasingly making use of complex mathematical tools. Complex networks play a significant role in the propagation of processes, which include for example the case of epidemic spreading, the diffusion process, synchronisation or the consensus process. Such dynamic processes are critically important in achieving understanding of the behaviour of complex systems at different levels of complexity - examples might be the brain and modern man-made infrastructures. Although part of the study of the diffusion of information in the dynamic processes, it is generally supposed that interactions in networks originate only from a node, spreading to its nearest neighbours, there also exist long-range interactions (LRI), which can be transmitted from a node to others not directly connected. The focus of this study is on dynamic processes on networks where nodes interact with not only their nearest neighbours but also through certain LRIs. The generalised k-path Laplacian operators (LOs) Lk, which account for the hop of a diffusive particle to its non-nearest neighbours in a graph, control this diffusive process, describing hops of nodes vi at distance k; here the distance is measured as the length of the shortest path between two nodes. In this way the introduction of the k-path LOs can facilitate conducting more precise studies of network dynamics in different applications. This thesis aims to study a generalised diffusion equation employing the transformed generalised k-path LOs for a locally finite infinite graph. This generalised diffusion equation promotes both normal and super diffusive processes on infinite graphs. Furthermore, this thesis develops a new theoretical mathematical framework for describing superdiffusion processes that use a transform of the k-path LOs defined on infinite graphs. The choice of the transform appeared to be vitally important as the probability of a long jump should be great enough. As described by other researchers the fractional diffusion equation (FDE) formed the mathematical framework employed to describe this anomalous diffusion. In this regard,it is taken that the diffusive particle is not just hopping to its nearest node but also to any other node of the network with a probability that scales according to the distance between the two places. Initially, we extend the k-path LOs above to consider a connected and locally finite infinite network with a bounded degree and investigate a number of the properties of these operators, such as their self-adjointness and boundedness. Then, three different transformations of the k-path LOs, i.e. the Laplace, Factorial and Mellin transformations as well as their properties, are studied.In addition, in order to show a number of applications of these operators and the transformed ones, the transformed k-path LOs are used to obtain a generalised diffusion process for one-dimensional and two-dimensional infinite graphs.First, the infinite path graph is studied, where it is possible to prove that when the Laplacian- and factorial-transformed operators are used in the generalised diffusion equation, the diffusive processes observed are always normal, independent of the transform parameters. It is then proven analytically that when the k-path LOs are transformed via a Mellin transform and plugged into the diffusion equation, the result is a super diffusive process for certain values of the exponent in the transform. Secondly, we generalise the results on the superdiffusive behaviour generated by transforming k-path LOs from one-dimensional graphs to 2-dimensional ones. Our attention focuses on the Abstract Cauchy problem in an infinite square lattice. A generalised diffusion equation on a square lattice corresponding to Mellin transforms of the k-path Laplacian is investigated. Similar to the one-dimensional case also for the graph embedded in two-dimensional space,we could observe superdiffusive behaviour for the Mellin transformed k-path Laplacian. In comparison to the one-dimensional case, the conclusion reached is that the asymptotic behaviour of the solution of the Cauchy problem is much subtler.A set of links and nodes are the fundamental units or components used to represent complex networks. Over the last few decades, network studies have expanded and matured, increasingly making use of complex mathematical tools. Complex networks play a significant role in the propagation of processes, which include for example the case of epidemic spreading, the diffusion process, synchronisation or the consensus process. Such dynamic processes are critically important in achieving understanding of the behaviour of complex systems at different levels of complexity - examples might be the brain and modern man-made infrastructures. Although part of the study of the diffusion of information in the dynamic processes, it is generally supposed that interactions in networks originate only from a node, spreading to its nearest neighbours, there also exist long-range interactions (LRI), which can be transmitted from a node to others not directly connected. The focus of this study is on dynamic processes on networks where nodes interact with not only their nearest neighbours but also through certain LRIs. The generalised k-path Laplacian operators (LOs) Lk, which account for the hop of a diffusive particle to its non-nearest neighbours in a graph, control this diffusive process, describing hops of nodes vi at distance k; here the distance is measured as the length of the shortest path between two nodes. In this way the introduction of the k-path LOs can facilitate conducting more precise studies of network dynamics in different applications. This thesis aims to study a generalised diffusion equation employing the transformed generalised k-path LOs for a locally finite infinite graph. This generalised diffusion equation promotes both normal and super diffusive processes on infinite graphs. Furthermore, this thesis develops a new theoretical mathematical framework for describing superdiffusion processes that use a transform of the k-path LOs defined on infinite graphs. The choice of the transform appeared to be vitally important as the probability of a long jump should be great enough. As described by other researchers the fractional diffusion equation (FDE) formed the mathematical framework employed to describe this anomalous diffusion. In this regard,it is taken that the diffusive particle is not just hopping to its nearest node but also to any other node of the network with a probability that scales according to the distance between the two places. Initially, we extend the k-path LOs above to consider a connected and locally finite infinite network with a bounded degree and investigate a number of the properties of these operators, such as their self-adjointness and boundedness. Then, three different transformations of the k-path LOs, i.e. the Laplace, Factorial and Mellin transformations as well as their properties, are studied.In addition, in order to show a number of applications of these operators and the transformed ones, the transformed k-path LOs are used to obtain a generalised diffusion process for one-dimensional and two-dimensional infinite graphs.First, the infinite path graph is studied, where it is possible to prove that when the Laplacian- and factorial-transformed operators are used in the generalised diffusion equation, the diffusive processes observed are always normal, independent of the transform parameters. It is then proven analytically that when the k-path LOs are transformed via a Mellin transform and plugged into the diffusion equation, the result is a super diffusive process for certain values of the exponent in the transform. Secondly, we generalise the results on the superdiffusive behaviour generated by transforming k-path LOs from one-dimensional graphs to 2-dimensional ones. Our attention focuses on the Abstract Cauchy problem in an infinite square lattice. A generalised diffusion equation on a square lattice corresponding to Mellin transforms of the k-path Laplacian is investigated. Similar to the one-dimensional case also for the graph embedded in two-dimensional space,we could observe superdiffusive behaviour for the Mellin transformed k-path Laplacian. In comparison to the one-dimensional case, the conclusion reached is that the asymptotic behaviour of the solution of the Cauchy problem is much subtler

Consensus-Based Attitude Maneuver of Multi-spacecraft with Exclusion Constraints

Some space missions involve cooperative multi-vehicle teams, for such purposes as interferometry and optimal sensor coverage, for example, NASA Terrestrial Planet Finder Mission. Cooperative navigation introduces extra constraints of exclusion zones between the spacecraft to protect them from damaging each other. This is in addition to external exclusion constraints introduced by damaging or blinding celestial objects. This work presents a quaternion-based attitude consensus protocol, using the communication topology of the team of spacecraft. The resulting distributed Laplacians of their communication graph are applied by semidefinite programming (SDP), to synthesize a series of time-varying optimal stochastic matrices. The matrices are used to generate various cooperative attitude maneuvers from the initial attitudes of the spacecraft. Exclusion constraints are satisfied by quaternion-based quadratically constrained attitude control (Q-CAC), where both static and dynamic exclusion zones are identified every time step, expressed as time-varying linear matrix inequalities (LMI) and solved by semidefinite programming

A Dose Relationship Between Brain Functional Connectivity and Cumulative Head Impact Exposure in Collegiate Water Polo Players.

A growing body of evidence suggests that chronic, sport-related head impact exposure can impair brain functional integration and brain structure and function. Evidence of a robust inverse relationship between the frequency and magnitude of repeated head impacts and disturbed brain network function is needed to strengthen an argument for causality. In pursuing such a relationship, we used cap-worn inertial sensors to measure the frequency and magnitude of head impacts sustained by eighteen intercollegiate water polo athletes monitored over a single season of play. Participants were evaluated before and after the season using computerized cognitive tests of inhibitory control and resting electroencephalography. Greater head impact exposure was associated with increased phase synchrony [r (16) > 0.626, p < 0.03 corrected], global efficiency [r (16) > 0.601, p < 0.04 corrected], and mean clustering coefficient [r (16) > 0.625, p < 0.03 corrected] in the functional networks formed by slow-wave (delta, theta) oscillations. Head impact exposure was not associated with changes in performance on the inhibitory control tasks. However, those with the greatest impact exposure showed an association between changes in resting-state connectivity and a dissociation between performance on the tasks after the season [r (16) = 0.481, p = 0.043] that could also be attributed to increased slow-wave synchrony [F (4, 135) = 113.546, p < 0.001]. Collectively, our results suggest that athletes sustaining the greatest head impact exposure exhibited changes in whole-brain functional connectivity that were associated with altered information processing and inhibitory control

Pose consensus based on dual quaternion algebra with application to decentralized formation control of mobile manipulators

This paper presents a solution based on dual quaternion algebra to the general problem of pose (i.e., position and orientation) consensus for systems composed of multiple rigid-bodies. The dual quaternion algebra is used to model the agents' poses and also in the distributed control laws, making the proposed technique easily applicable to time-varying formation control of general robotic systems. The proposed pose consensus protocol has guaranteed convergence when the interaction among the agents is represented by directed graphs with directed spanning trees, which is a more general result when compared to the literature on formation control. In order to illustrate the proposed pose consensus protocol and its extension to the problem of formation control, we present a numerical simulation with a large number of free-flying agents and also an application of cooperative manipulation by using real mobile manipulators

Multi-Spacecraft Attitude Path Planning Using Consensus with LMI-Based Exclusion Constraints

Space missions involving multi-vehicle teams require the cooperative navigation and attitude slewing of the spacecraft or satellites, for such purposes as interferometry and optimal sensor coverage. This introduces extra constraints of exclusion zones between the spacecraft, in addition to the default exclusion constraints already introduced by damaging or blinding celestial objects. In this work, we present a quaternion-based attitude consensus protocol by using the communication topology of the spacecraft team. By using the Laplacian matrix of their communication graph and a semidefinite program, a synthesis of a time-varying optimal stochastic matrix P is done, which is used to generate various consensus and cooperative attitude trajectories from the initial attitudes of the spacecraft. The concept of quaternion-based quadratically constrained attitude control is then employed to satisfy cone avoidance constraints, where exclusion zones are identified, expressed as linear matrix inequalities (LMI), and solved by semidefinite programming (SDP)