Content uploaded by Mehran Mesbahi
Author content
All content in this area was uploaded by Mehran Mesbahi
Content may be subject to copyright.
A preview of the PDF is not available
Controllability of Multi-Agent Systems from a Graph-Theoretic Perspective
Abstract and Figures
In this work, we consider the controlled agreement problem for multi-agent networks, where a collection of agents take on leader roles while the remaining agents execute local, consensus-like protocols. Our aim is to identify reflections of graph-theoretic notions on system-theoretic properties of such systems. In particular, we show how the symmetry structure of the network, characterized in terms of its automorphism group, directly relates to the controllability of the corresponding multi-agent system. Moreover, we introduce network equitable partitions as a means by which such controllability characterizations can be extended to the multileader setting. (c) 2009 Society for Industrial and Applied Mathematics
Figures - uploaded by Mehran Mesbahi
Author content
All figure content in this area was uploaded by Mehran Mesbahi
Content may be subject to copyright.
... The latter is usually discussed within systemic risk research, which originally focused on the collapse or at least serious disruption of a system and was modeled primarily through network dynamics methods (Kharrazi et al., 2013;Kreis et al., 2019). The former can be related to the question of the controllability of complex networks which is usually determined to a great extent by the underlying networks degree distribution (see Liu et al., 2011) and analyzed through Graph-Theory, see for example Rahmani et al. (2009). We refer for an overview of disciplines and methodologies applied for complex systems analysis to Castellani and Hafferty (2009) and more specifically on quantitative approaches to Thurner et al. (2018). ...
Disasters associated with natural hazards as well as climate change are happening within complex socio-economic systems and desired system states, including sustainable development and resource management, are formulated on the global as well as regional and national levels. However, complex system approaches are yet only rudimentarily incorporated in related applications, and we discuss modeling as well as policy challenges focusing on fiscal risk. As an intermediate step we suggest a gap approach which we relate to fiscal stress levels a complex system may experience due to natural hazard events. We argue that in case of no gaps one can assume a no stress situation and therefore modeling of disruptions including cascading effects is less necessary. However, at the same time we also acknowledge that there is an urgent need to address corresponding challenges with complex system methods. Policy-wise our paper responds to concerns for real-world applications and can provide insights to support current discussions within the UNFCCC and Paris Agreement around both adaptation finance and the new funding arrangements for loss and damage from climate impacts established at COP27.
... Without completeness, we mention some applications of L(S). The value of the smallest eigenvalue λ(S) of matrix L(S) determines the convergence rate of a leader-follower networked dynamical system [13], as well as the effectiveness of pinning scheme of pinning control of complex dynamical networks [10], with large λ(S) corresponding to fast convergence speed and good pinning control performance. Finding L(S) with the maximum possible λ(S) has been shown to be an NPhard problem [14]. ...
Maximizing the smallest eigenvalue of the grounded Laplacian matrix is an NP-hard problem that involves identifying the Laplacian matrix's (n − k) × (n − k) principal submatrix obtained after removing k rows and corresponding columns. The challenge is to determine optimally the rows and columns to be deleted. Our proposed approach, motivated by the Gershgorin circle theorem, is used together with the degree centrality of the corresponding graph. Moreover, integer linear programming for the vertex cover problem has been employed as an additional method of solving the problem. The efficiency of the methods is demonstrated on real-world graphs.
This paper investigates optimal control for two kinds of multi-agent systems with the sampled-data based communication protocols. Necessary and sufficient conditions on data-sampling controllability are obtained for sampled-data based multi-agent systems, which indicate that sampling periods affect the controllability of double-integrator systems, whereas the controllability of single-integrator systems is decided by the interconnection topology. To obtain the optimal control inputs under different objectives, two concepts of 2-norm-optimal control and infinity-norm-optimal control are proposed. For both kinds of systems, generalised inverse matrices of controllability test matrices are utilised to derive the 2-norm-optimal controls, and the infinity-norm-optimal controls are equivalently transformed to be the optimal solutions of some specific linear programming problems via matrix vectorization. Some illustrative examples are provided for the main results in this paper.
The association between pathological states and aberrant brain rhythms underscores the potential of brain rhythm modulation to facilitate the transition from pathological states back to physiological norms. It is important to ensure the feasibility of the brain rhythm modulation strategy. The concept of system controllability, a cornerstone of control science, serves as a crucial prerequisite for the creation of state feedback mechanisms designed to achieve desired system performance. Investigating the controllability of brain networks from a control standpoint lays a theoretical foundation for the practicality and strategic planning of neuromodulation. This study delves into the controllability of brain networks composed of neural mass models with four distinct rhythms RcR-
$\alpha$
, RcR-
$\beta$
, RcR-
$\theta$
and RcR-
$\gamma$
. Through the examination of how various inputs, model parameters, and the interplay between neural mass models influence controllability, we can observe the intricate relationship between brain networks controllability and rhythmic activity. Our findings suggest that the controllability of brain networks is particularly sensitive to changes in the external inputs or the strength of internal connections of RcR-
$\alpha$
and RcR-
$\gamma$
, in contrast to RcR-
$\beta$
and RcR-
$\theta$
. We hope that this research not only advances the understanding of neural regulation's feasibility but also informs the optimization of network dynamics and connectivity in neuromodulation strategy development.
The partial component consensus in mean‐square for delayed nonlinear multi‐agent systems (NMASs) with uncertain nonhomogeneous Markov switching (UNMS) topologies subjected to aperiodic denial‐of‐service (DoS) cyber‐attacks is investigated. Firstly, the partial component ( ω$$ \omega $$‐dimensional) consensus of the system ( n$$ n $$‐dimensional, 1≤ω≤n$$ 1\le \omega \le n $$) is considered in this paper. When ω=n$$ \omega =n $$, the partial component consensus degrades into identical consensus. Secondly, the communication topologies governed by uncertain nonhomogeneous Markov chains are random. Thirdly, the communication topologies suffer from aperiodic DoS cyber‐attacks. Then, in view of stochastic analysis technology, distributed control theory and Lyapunov stability theory, the partial component consensus conditions are obtained using permutation matrix and inequality scaling skills. Eventually, the correctness of the results is verified through an example.
The tensor vector product-based dynamical system (TVPDS) representation was pioneered in 2021 by Chen et al. [IEEE Trans. Netw. Sci. Eng.] as a novel framework for characterizing the multidimensional state dynamics of hypergraphs. Hypergraphs are generalizations of graphs where hyperedges can connect more than two nodes. The evolution of a TVPDS is characterized by the tensor vector product between a dynamic tensor and a state vector, which in fact can be equivalently reformulated in the form of homogeneous polynomial dynamical systems. By utilizing polynomial systems theory and tensor algebra, the system-theoretic properties of TVPDSs, such as stability, controllability, and observability, are investigated. In particular, the controllability and observability of hypergraphs are explored through the lens of the TVPDS representation.
The problem of coordination and control of multiple microspacecraft (MS) moving in formation is considered. Here, each MS is modelled by a rigid body with fixed center of mass. First, various schemes for generating the desired formation patterns are discussed. Then, explicit control laws for formation-keeping and relative attitude alignment based on nearest neighbor-tracking are derived. The necessary data which must be communicated between the MS to achieve effective control are examined. The time-domain behavior of the control system with a proposed inter-spacecraft communication/computing network.
Graphs.- Groups.- Transitive Graphs.- Arc-Transitive Graphs.- Generalized Polygons and Moore Graphs.- Homomorphisms.- Kneser Graphs.- Matrix Theory.- Interlacing.- Strongly Regular Graphs.- Two-Graphs.- Line Graphs and Eigenvalues.- The Laplacian of a Graph.- Cuts and Flows.- The Rank Polynomial.- Knots.- Knots and Eulerian Cycles.- Glossary of Symbols.- Index.
We consider constrained multiple-spacecraft reconfigurations outside a gravity well in deep space. As opposed to the single-spacecraft scenario, such reconfigurations involve collision avoidance constraints that can be embedded in a nonconvex, state-constrained optimal control problem. Due to the difficulties in solving this general class of optimal control problems, we adopt a heuristically motivated approach to multiple-spacecraft reconfigurations. Then, we proceed to prove the convergence properties of the proposed approach for reconfigurations involving an arbitrary number of spacecraft.
In this paper we introduce a novel control network protocol, Try-Once-Discard (TOD), for networked control systems (NCS), and provide, for the first time, an analytic proof of global exponential stability for both the new protocol and the commonly used statically scheduled access methods. Controllers are designed without regarding the presence of the network in the feedback loop, so consequently many controller design techniques may be employed. The performance of the new network protocol and the statically scheduled protocols are compared in simulations.
In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results.
We consider distributed dynamic systems operating over a graph or a network. The geometry of the network is assumed to be a function of the underling system's states-giving it a unique dynamic character. Certain aspects of the resulting abstract structure, having a mixture of combinatorial and system theoretic features, are then studied. In this venue, we will explore an interplay between notions from extremal graph theory and system theory by considering a controllability framework for such state-dependent dynamic graphs.
From the Publisher:This comprehensive treatment of the analysis and design of continuous-time control systems provides a gradual development of control theoryand shows how to solve all computational problems with MATLAB. It avoids highly mathematical arguments, and features an abundance of examples and worked problems throughout the book. Chapter topics include the Laplace transform; mathematical modeling of mechanical systems, electrical systems, fluid systems, and thermal systems; transient and steady-state-response analyses, root-locus analysis and control systems design by the root-locus method; frequency-response analysis and control systems design by the frequency-response; two-degrees-of-freedom control; state space analysis of control systems and design of control systems in state space.
This paper considers linear dynamic systems with an interconnection structure specified by a directed graph. We formulate linear matrix inequalities for computation of performance of the system based on dissipation inequalities, and define a notion of local dissipation. We derive a computational test for this which may be solved via semidefinite program. The structure of the resulting matrix inequalities is related to recent approaches to distributed control, and the results of this paper reduce to recent results in the case where the underlying graph structure is a rectangular array. Based on the analytical result, a state feedback synthesis approach is presented and it also can be cast as a semidefinite program.