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Cuts and cycles in relative sensing and control of spatially distributed systems
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... We refer to such a feedback system setup, and the resulting host of control issues, as the problem of control over RSNs. Such systems have recently been considered by Smith and Hadaegh [14]; this reference has in fact motivated our studies on RSNs in [13]. Other related works include [1], [4], [9], [10], [16], and [17]. ...
... Suppose however that the actual relative states available to the controller is y j (t) := D(G j ) T x(t). As it was shown in [13], for a system without measurement noise, there exists a linear transformation T dj such that for all t, y d (t) = T dj y j (t). Equivalently one can view these transformations as a mechanism for control reconfiguration. ...
... Equivalently one can view these transformations as a mechanism for control reconfiguration. Accordingly, the controller K d constructed to operate on information geometry G d can be updated as K j = K d @BULLET T dj , so that it can operate on the information abstracted by the graph G j [13]. The central idea in deriving these transformations is the realization that the RSNs contain algebraic redundancies; for example, for any i, j, k ∈ [n] one has x ik (t) + x kj (t) − x ij (t) = 0. ...
In this paper we consider various structural aspects of estimation and control over relative sensing networks. We first expand on our earlier characterization of equivalent sensing and control topologies that incorporates the presence of measurement noise. Next, we consider the interplay between system minimality, network observability, and estimation problems. We conclude our presentation by pointing out a reciprocity between the relative sensing geometry on one hand, and the control structure for distributed dynamic systems on the other. Simulations and examples are incorporated throughout the paper to further complement the theoretical analysis.
... The matrix E F has full column rank, and N F is the orthogonal complement of E F ( [29,30]). 2) E C can be re-expressed as [40] ...
... By graph theory [41], the above expression represents the effective conductance between bus i and bus j in the resistive network whose admittance matrix is interpreted by L G (θ), so that the physical meaning of equivalent weight keeps unchanged. With the expression (40), the equivalent weight between any bus pair (i, j) and the relevant stability issues can be evaluated. An example of using equivalent weight to guide generation dispatch for stability improvement will be given in Section V. ...
This paper investigates small-disturbance angle stability of power systems with emphasis on the role of power network topology, which sheds new light on the instability mechanism. We introduce the concepts of active power flow graph and critical lines. It is shown that the inertia of the Laplacian matrix of this graph provides information on the stability and type of an equilibrium point. Then, the instability mechanism is elaborated from the impact of critical lines on the inertia of the Laplacian matrix. A stability criterion in terms of a critical line-based matrix is established. This criterion is a necessary and sufficient condition to judge the stability and type of an equilibrium point. It includes the existing results in the literature and applies to the unsolved cases where the critical lines exist but do not form cutsets. Moreover, we introduce the concept of equivalent weight between a pair of buses. Another stability criterion in terms of the equivalent weight is developed, from which the small-disturbance instability can be interpreted as the “electrical antagonism” between some buses in the power network resulting from the critical lines. The equivalent weight can also be used as a stability index and provides guidance for system operation. The obtained results are illustrated by numerical simulations.
... We know that for connected graphs, x c can always be written as x c = Z x τ as shown in Sandhu et al. (2005) ...
We present consensus analysis of systems with single integrator dynamics interacting via time-varying graphs under the event-triggered control paradigm. Event-triggered control sparsifies the control applied, thus reducing the control effort expended. Initially, we consider a multi-agent system with persistently exciting interactions and study the behaviour under the application of event-triggered control with two types of trigger functions- static and dynamic trigger.We show that while in the case of static trigger, the edge-states converge to a ball around the origin, the dynamic trigger function forces the states to reach consensus exponentially. Finally, we extend these results to a more general setting where we consider switching topologies. We show that similar results can be obtained for agents interacting via switching topologies and validate our results by means of simulations.
... Lemma 3: [44] E C can be re-expressed as E C = E T T C , where T C is the transformation matrix defined below ...
A cutset is a concept of importance in both graph theory and many engineering problems. In this paper, cutset properties are studied and applied to transient stability problems in power systems. First, we reveal that cutsets are intrinsically linked to cycles, another important concept in graph theory. The main theoretical results include that any pair of edges in a cutset are concyclic (i.e., contained in the same cycle), and any edge together with some of its concyclic edges can form a cutset. Then, with the cycle constraint for the bus angles in power networks, we give a theoretical explanation of a common phenomenon in transient stability that the loss of synchronism initiates from the angle separation across the critical cutset in the underlying power network. This phenomenon is frequently observed but no proof has been presented yet. Moreover, an improved cutset index (ICI) is proposed based on these cutset properties. This index can better identify the vulnerable cutset and help to estimate the cutset-relevant unstable equilibrium points and critical energy for stability region determination. Numerical studies on two IEEE test systems show that the formation of critical cutset coincides with the explanation, and the ICI has better performance than the conventional cutset index especially in heavy load cases.
... Implementing this control law results in a closed-loop error dynamics identical to the earlier model (19). When there are cycles present in the network, we can again use the results from [14] to obtain a reduced order representation of the error dynamics. ...
Formation keeping strategies for groups of interconnected agents have recently been of great research interest in the systems community. In this work, we explore the utility of the edge variant of the graph Laplacian in the synthesis of formation keeping control laws. Along the way, we show a general duality relation between networked dynamic systems with measurement restrictions and those with control constraints. In both cases, it is shown that the closed loop error dynamics for the formation reduces to the edge agreement problem- which in turn-can be fully characterized via the spanning trees of the underlying interconnection topology.
This study deals with cooperative source localisation with a group of agents where the source is static/dynamic. In stead of the traditionally defined persistent excitation (PE) condition, it is shown that a looser condition - cooperative PE condition - is sufficient for the source localisation provided the topology graph is connected. An estimator is designed to estimate the relative coordinate among agents; a local cooperative estimator based on cooperative PE condition is designed to estimate the relative location of the source; based on the first two steps, a consensus-based estimation fusion algorithm is given to guarantee that estimation errors asymptotically converge to 0. The cooperative localisation method is then extended to the case of a dynamic source and is analysed for the disturbance case. Finally, several simulation results are provided to show the correctness of the conclusion and the effectiveness of the proposed method.
In this note, we consider certain structural aspects of estimation and control over relative sensing networks (RSNs). In this venue, using tools from basic algebraic graph theory-namely, the incidence matrix and cut and cycle spaces of a connect graph- we examine transformations among relative sensing topologies for a networked system. These transformations are parameterized for noise-free as well as noisy networks and their utility in the context of network-centric robust control and control reconfigurations is explored.
In real-life, many practical applications involve agents that are nonlinear and nonholonomic. This paper investigates output synchronization of multiple mobile agents in a plane. The output synchronization of mobile agent's aims are two aspects: making the speed of each agent synchronized and tracking desired trajectories in a group. First, each agent dynamics is described by an affine control system and the communication topology among the agents is represented by a digraph. Using adjacent information and incidence matrix, a decentralized control law is acquired, which is composed of alignment control and cohesion control. Second, the stability of the group is proved by the decentralized control law and the synchronization control objective is realized. Finally, ten mobile agents are simulated to show the effectiveness of the present control law.
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 distributed dynamic systems operating over state-dependent dynamic graphs or networks. Various aspects of the resulting abstract structure, having a mixture of combinatorial and control theoretic features, are then studied. Along the way, we explore an interplay between notions from the theory of graphs (elementary, extremal, and random), and system theory (controllability, invariance, etc.).
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.
1. The field of values 2. Stable matrices and inertia 3. Singular value inequalities 4. Matrix equations and Kronecker products 5. Hadamard products 6. Matrices and functions 7. Totally positive matrices.
The problemof formation and reconéguration of multiple microsatellite systems is addressed. A control scheme called the perceptive frame is adopted that integrates the decentralized feedback of each satellite with online sensor information to achieve the goal of formation keeping and intersatellite coordination. The proposed design algorithm has the advantage of relative position keeping and easy formation reconé guration.
In this paper system models arising from process-plant installations are derived in the form of an interconnected series of mixed, distributed and lumped-parameter realisations. The terminal characteristics of these models is shown to be a rational, multidimensional function of a finite set of transformed discrete time-delay variables which may be used in feedback-control studies. By employing a linear mapping of the support of the model it is established that stability can be assessed using a corresponding algebraic function to that of the denominator of the system model. Simple worked examples are used to illustrate the procedure.
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 disa