Reduction of Second-Order Network Systems With Structure Preservation

Cheng, Xiaodong; Kawano, Yu; Scherpen, Jacquelien M. A.

doi:10.1109/tac.2017.2679479

Cited by 52 publications

(77 citation statements)

References 34 publications

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“…In particular, it would be interesting to apply our result to the reduction of large scale networks [19], [20] which may have degenerate dynamics. Moreover, the development of controllers for slowfast systems with non-hyperbolic dynamics needs to be further investigated, particularly when more than two timescales are involved.…”

Section: Discussionmentioning

confidence: 99%

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Model Order Reduction and Composite Control for a Class of Slow-Fast Systems Around a Non-Hyperbolic Point

Jardón-Kojakhmetov

Scherpen

2017

IEEE Control Syst. Lett.

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Model order reduction and composite control for a class of slow-fast systems around a nonhyperbolic point Jardón Kojakhmetov, H.; Scherpen, Jacquelien M.A. Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LCSYS.2017.2703983, IEEE Control Systems LettersModel order reduction and composite control for a class of slow-fast systems around a non-hyperbolic point H. Jardón-Kojakhmetov 1 and Jacquelien M.A. Scherpen 1Abstract-In this letter we investigate a class of slow-fast systems for which the classical model order reduction technique based on singular perturbations does not apply due to the lack of a Normally Hyperbolic critical manifold. We show, however, that there exists a class of slow-fast systems that after a well-defined change of coordinates have a Normally Hyperbolic critical manifold. This allows the use of model order reduction techniques and to qualitatively describe the dynamics from auxiliary reduced models even in the neighborhood of a non-hyperbolic point. As an important consequence of the model order reduction step, we show that it is possible to design composite controllers that stabilize the (non-hyperbolic) origin.

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Section: Discussionmentioning

confidence: 99%

“…Thus, it is evident that it is quite difficult to conclude anything about the dynamics of (19) from the "reduced systems" (20). So we proceed by following Section III.…”

Section: A Example 1 (Model Order Reduction)mentioning

confidence: 99%

Model Order Reduction and Composite Control for a Class of Slow-Fast Systems Around a Non-Hyperbolic Point

Jardón-Kojakhmetov

Scherpen

2017

IEEE Control Syst. Lett.

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“…, where the symmetric matrixL is given by (7). Thus, a state space representation for the error system is given byẋ…”

Section: The Single Integrator Casementioning

confidence: 99%

“…with the symmetric matrixL given by (7). Analogous to the proof of Theorem 3, we first apply Lemma 1 to the error systeṁ…”

Section: Proof Of Theorem 4 First Note That the Transfer Functionŝ Ofmentioning

confidence: 99%

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Model reduction of linear multi-agent systems by clustering with $$\varvec{\mathcal {H}_2}$$ H 2 and $$\varvec{\mathcal {H}_\infty }$$ H ∞ error bounds

Jongsma

Mlinarić

Grundel

et al. 2018

Math. Control Signals Syst.

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Firstly, we will establish an a priori upper bound for the H 2 model reduction error in case that the agent dynamics is an arbitrary multivariable input-state-output system. Secondly, for the single integrator case, we will derive an explicit formula for the H ∞ model reduction error. Thirdly, we will prove an a priori upper bound for the H ∞ model reduction error in case that the agent dynamics is a symmetric multivariable input-state-output system. Finally, we will consider the problem of obtaining a priori upper bounds if we cluster using arbitrary, possibly non almost equitable, partitions.

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Reduction of a district heating model using network decomposition

Rein

Möhring

Damm

et al. 2019

Proc Appl Math and Mech

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In this contribution we study model order reduction of nonlinear transport networks for district heating systems. In these, heating energy injected at a centralized power plant is transported to consumers. They are modeled by a hyperbolic differential algebraic system with large state space dimension. The network structure introduces sparse system dynamics, which transform to a dense reduced system leading to unacceptable computational costs [1]. To exploit the benefits of sparsity, sub-parts of the network are reduced separately in a structure preserving way using Galerkin projection [2]. After introducing the dynamical model, we discuss a decomposition strategy which aims at minimizing the number of observables for each subnetwork. We demonstrate its numerical benefits at an existing large scale heating network.The dynamics of heating networks are described by one dimensional incompressible Euler equations. The latter incorporate the advection of the energy density ϕ by a purely time dependent volume flow q(t) in each pipeline, as well as the conservation of momentum. In this, the pressure difference along a pipeline is dominated by frictional forces, allowing to neglect the contribution of acceleration in a quasi-stationary limit. After spatial discretization of the advection equation using the upwind scheme, a state space decomposition is performed allowing to formulate the differential algebraic equationThe algebraic constraints (3) result from the coupling of pipelines at nodes of the network and gather three contributions: the conservation of volume, Kirchhoff's second law claiming that pressure differences along network loops sum up to zero, and the volumeflow balance at consumers. It forces the product of volume flow q at consumer stations and observed energy density y to equal the power consumption u H . Hence, (3) collects quadratic constraints in the state variables ϕ, q. The system description is decomposed to a main network ϕ 0 to which the thermal control u T is applied, and c ∈ N subnetworks forming ϕ which receive observablesỹ as artificial inputs. Observables y are measured by both main-and subnetworks, whileỹ is the system state of the main network at attachment points to subnetworks.Ã,B,C represent continued block diagonal matrices. Their blocks A s , B s , C s , s ∈ {1, .., c} describe the dynamics of the subnetworks. Both A s (q), and B s (q) depend on the volume flow q, and can be represented in an affine decompositionWe approximate the input-output characteristics of each system s ∈ {0, .., c} by moment matching of the volume flow dependent transfer function H(q, ω) = C(ω1 − A(q)) −1 B(q) in frequency domain. Considering the volume flow q as a timevarying parameter to the advection of the energy density, local Galerkin projections V s e (q e ) are performed for representative volume flow vectors q e . These are picked using a greedy strategy and ultimately combined to a global projection V s using a singular value decomposition for each s ∈ {0, .., c}. Application of V s to (1-3) defines t...

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Reduction of Second-Order Network Systems With Structure Preservation

Cited by 52 publications

References 34 publications

Model Order Reduction and Composite Control for a Class of Slow-Fast Systems Around a Non-Hyperbolic Point

Model Order Reduction and Composite Control for a Class of Slow-Fast Systems Around a Non-Hyperbolic Point

Model reduction of linear multi-agent systems by clustering with $$\varvec{\mathcal {H}_2}$$ H 2 and $$\varvec{\mathcal {H}_\infty }$$ H ∞ error bounds

Reduction of a district heating model using network decomposition

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