Model Reduction by Moment Matching for Linear and Nonlinear Systems

Astolfi, Alessandro

doi:10.1109/tac.2010.2046044

Cited by 288 publications

(198 citation statements)

References 84 publications

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“…We are currently working on possible ways to numerically solve the underdetermined systems (20), (23) and (26). * In Astolfi [2010], the signal generator is assumed to be Poisson stable 1 , so that the steady-state of the nonlinear system is well-defined. In linear MOR, however, shifts σ i ∈ C \ λ 2 (M , D, K) can be used.…”

Section: Analysis Discussion and Limitationsmentioning

confidence: 99%

“…A few years ago, the concept of moment matching and Krylov subspaces known from linear MOR has been carried over to nonlinear first-order systems by Astolfi [2010].…”

Section: Introductionmentioning

confidence: 99%

“…steady-state) interpretation of moment matching for linear second-order systems. Based on Astolfi [2010], we then transfer this reduction method to the nonlinear second-order case and provide the corresponding nonlinear, Sylvester-like PDE. In Section 4, simplifications are proposed to avoid the PDE, and achieve a simulationfree second-order nonlinear moment matching algorithm, which relies on the solution of nonlinear systems of equations.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nonlinear Moment Matching for the Simulation-Free Reduction of Structural Systems

2019

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This paper transfers the concept of moment matching to nonlinear structural systems and further provides a simulation-free reduction scheme for such nonlinear second-order models. After first presenting the steady-state interpretation of linear moment matching, we then extend this reduction concept to the nonlinear second-order case based on Astolfi [2010]. Then, similar simplifications as in Cruz Varona et al. [2019] are proposed to achieve a simulation-free nonlinear moment matching algorithm. A discussion on the simplifications and their limitations is presented, as well as a numerical example which illustrates the efficiency of the algorithm.

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Section: Analysis Discussion and Limitationsmentioning

confidence: 99%

“…A few years ago, the concept of moment matching and Krylov subspaces known from linear MOR has been carried over to nonlinear first-order systems by Astolfi [2010].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonlinear Moment Matching for the Simulation-Free Reduction of Structural Systems

2019

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“…If the number of subsystems is large, we first reduce their number using clustering techniques and then perform subsystem approximation. To the best of our knowledge, in the time-domain moment matching framework [2], [11], finding a reduced model of the network optimally w.r.t. the H 2 -norm of the approximation error in the family of ν order models that matches a set of ν moments, while preserving the network topology and stability, is an open question.…”

Section: Motivationmentioning

confidence: 99%

$H_2$ Model Reduction of Linear Network Systems by Moment Matching and Optimization

Necoara

Ionescu

2020

IEEE Trans. Automat. Contr.

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In this paper we study the problem of model reduction of linear network systems. We aim at computing a reduced order stable approximation of the network with the same topology and optimal w.r.t. H2-norm error approximation. Our approach is based on time-domain moment matching framework, where we optimize over families of parameterized reduced order models matching a set of moments at arbitrary interpolation points. The parameterization of the low order models is in terms of the free parameters and of the interpolation points. For this family of parameterized models we formulate an optimizationbased model reduction problem with the H2-norm of error approximation as objective function while the preservation of some structural and physical properties yields the constraints. This problem is nonconvex and we write it in terms of the Gramians of a minimal realization of the error system. We propose two solutions for this problem. The first solution assumes that the error system admits a block diagonal observability Gramian, allowing for a simple convex reformulation as semidefinite programming, but at the cost of some performance loss. We also derive sufficient conditions to guarantee block diagonalization of the Gramian. The second solution employs a gradient projection method for a smooth reformulation yielding (locally) optimal interpolation points and free parameters. The potential of the methods is illustrated on several network examples.

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“…Let us apply the obtained reduction rules on the following example with the network of 30 nodes, see Figure 7. We assume that all the subsystems are ISS with the following gains ij : 3,1 .t / D 5 6 t 2 , 1,2 .t / D 2t, 2,13 .t / D 1 5 p t , 13,30 .t / D t , 30,29 .t / D 4t 2 , 29,28 .t / D p t, 29,23 .t / D 3 10 t 2 , 15,11 .t / D t 2 , 11,16 .t / D 2 3 t , 16,11 .t / D t , 16,15 This subgraph can be aggregated in several steps:…”

Section: Proofmentioning

confidence: 99%

Reduction of the small gain condition for large‐scale interconnections

Kosmykov

2013

Intl J Robust & Nonlinear

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SUMMARYThe small gain condition is sufficient for input-to-state stability (ISS) of interconnected systems. However, verification of the small gain condition requires large amount of computations in the case of a large size of the system. To facilitate this procedure we aggregate the subsystems and the gains between the subsystems that belong to certain interconnection patterns (motifs) using three heuristic rules. These rules are based on three motifs: sequentially connected nodes, nodes connected in parallel and almost disconnected subgraphs. Aggregation of these motifs keeps the main structure of the mutual influences between the subsystems in the network. Furthermore, fulfillment of the reduced small gain condition implies ISS of the large network. Thus such reduction allows to decrease the number of computations needed to verify the small gain condition. Finally, an ISS-Lyapunov function for the large network can be constructed using the reduced small gain condition. Applications of these rules is illustrated on an example.

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Model Reduction by Moment Matching for Linear and Nonlinear Systems

Abstract: Abstract-

Cited by 288 publications

References 84 publications

Nonlinear Moment Matching for the Simulation-Free Reduction of Structural Systems

Nonlinear Moment Matching for the Simulation-Free Reduction of Structural Systems

$H_2$ Model Reduction of Linear Network Systems by Moment Matching and Optimization

Reduction of the small gain condition for large‐scale interconnections

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