The Mathematics of Networks of Linear Systems

Fuhrmann, Paul A.; Helmke, Uwe

doi:10.1007/978-3-319-16646-9

Cited by 56 publications

(54 citation statements)

References 201 publications

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“…We do not claim that this lemma is new, and similar results can be found, for instance, in [24]. Still, we give the proof for the completeness of the discussion.…”

Section: Basic Algorithm By Problem Decompositionmentioning

confidence: 74%

Convergence Rate Analysis of Multi-Agent Positive Systems under Formation Control: An Efficient Algorithm by Exploiting Positivity

Ebihara

Peaucelle

Arzelier

2016

SICE Journal of Control, Measurement, and System Integration

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: This paper is concerned with convergence rate analysis of multi-agent positive systems under formation control. Recently, we have shown that very basic multi-agent systems under formation control can be modeled as interconnected positive systems, and desired formation can be achieved by designing interconnection matrices appropriately. In such formation control, the resulting convergence performance (i.e., convergence rate) varies according to the interconnection matrices and this fact motivates us to develop an efficient algorithm for the analysis of the convergence rate. In this paper, assuming that the dynamics of agents are positive and homogeneous, we conceive such an algorithm by problem decomposition. We show that the decomposition to smaller size problems and drastic reduction of computational burden become possible by making full use of the positivity of the agents.

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“…We do not claim that this lemma is new, and similar results can be found, for instance, in [24]. Still, we give the proof for the completeness of the discussion.…”

Section: Basic Algorithm By Problem Decompositionmentioning

confidence: 74%

Convergence Rate Analysis of Multi-Agent Positive Systems under Formation Control: An Efficient Algorithm by Exploiting Positivity

Ebihara

Peaucelle

Arzelier

2016

SICE Journal of Control, Measurement, and System Integration

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“…We do not to intend to reproduce in the paper the results established there, but rather our contribution related to ensemble controllability is to demonstrate that if the set of control vector fields { f i } m i=1 is distinguished, then the ensemble version of Rachevsky-Chow criterion can be easily verified in analysis and fulfilled in system design. For ensemble control of linear systems, we refer the reader to [12,13], [14,Ch. 12] and references therein.…”

Section: Literature Reviewmentioning

confidence: 99%

“…Observability of a continuum ensemble system has been mostly addressed within the class of linear systems. We first refer the reader to [14,Ch. 12] where the following ensemble of linear systems is investigated:…”

Section: Literature Reviewmentioning

confidence: 99%

Structure theory for ensemble controllability, observability, and duality

Chen

2019

Math. Control Signals Syst.

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Ensemble control deals with the problem of using a finite number of control inputs to simultaneously steer a large population (in the limit, a continuum) of control systems. Dual to the ensemble control problem, ensemble estimation deals with the problem of using a finite number of measurement outputs to estimate the initial state of every individual system in the ensemble. We introduce in the paper a novel class of ensemble systems, termed distinguished ensemble systems, and establish sufficient conditions for controllability and observability of such systems. Every distinguished ensemble system has two key components, namely a set of distinguished control vector fields and a set of codistinguished observation functions. Roughly speaking, a set of vector fields is distinguished if it is closed (up to scaling) under Lie bracket, and moreover, every vector field in the set can be obtained by a Lie bracket of two vector fields in the same set. Similarly, a set of functions is codistinguished to a set of vector fields if the Lie derivatives of the functions along the given vector fields yield (up to scaling) the same set of functions. We demonstrate in the paper that the structure of a distinguished ensemble system can significantly simplify the analysis of ensemble controllability and observability. Moreover, such a structure can be used as a guiding principle for ensemble system design.We further address in the paper the problem about existence of a distinguished ensemble system for a given manifold. We provide an affirmative answer for the case where the manifold is a connected semi-simple Lie group. Specifically, we show that every such Lie group admits a set of distinguished vector fields, together with a set of codistinguished functions. The proof is constructive, leveraging the structure theory of semi-simple real Lie algebras and representation theory. Examples will be provided along the presentation of the paper illustrating key definitions and main results.

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“…We next address ourselves to the tensor product 2 of F l (z l ) and F r (z r ), resulting in F l ⊗ F r , a p l p r × m l m r -valued rational function. Tensor product of rational functions is discussed in [5,Section 5.2].…”

Section: Introductionmentioning

confidence: 99%

Realization of tensor product and of tensor factorization of rational functions

Alpay

Lewkowicz

2019

Quantum Stud.: Math. Found.

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We here first study the state space realization of a tensor-product of a pair of rational functions. At the expense of "inflating" the dimensions, we recover the classical expressions for realization of a regular product of rational functions. Then, under an additional assumption that the limit at infinity of a given rational function exists and is equal to identity, an explicit formula for a tensor-factorization of this function, is introduced.

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The Mathematics of Networks of Linear Systems

Cited by 56 publications

References 201 publications

Convergence Rate Analysis of Multi-Agent Positive Systems under Formation Control: An Efficient Algorithm by Exploiting Positivity

Convergence Rate Analysis of Multi-Agent Positive Systems under Formation Control: An Efficient Algorithm by Exploiting Positivity

Structure theory for ensemble controllability, observability, and duality

Realization of tensor product and of tensor factorization of rational functions

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