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A Review on Model Reduction by Moment Matching for Nonlinear Systems

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Feedback Stabilization of Controlled Dynamical Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 473))

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Abstract

The model reduction problem for nonlinear systems and nonlinear time-delay systems based on the steady-state notion of moment is reviewed. We show how this nonlinear description of moment is used to pose and solve the model reduction problem by moment matching for nonlinear systems, to develop a notion of frequency response for nonlinear systems, and to solve model reduction problems in the presence of constraints on the reduced order model. Model reduction of nonlinear time-delay systems is then discussed. Finally, the problem of approximating the moment of nonlinear, possibly time-delay, systems from input/output data is briefly illustrated.

Dedicated to Laurent: a pioneer in the land of control

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Notes

  1. 1.

    The matrices A, B, C, and the zeros of (2.9) fix the moments. Then, given any observable pair (L, S) with S a non-derogatory matrix with characteristic polynomial (2.9), there exists an invertible matrix \(T\in \mathbb {R}^{\nu \times \nu }\) such that the elements of the vector \(C\varPi T^{-1}\) are equal to the moments.

  2. 2.

    A matrix is non-derogatory if its characteristic and minimal polynomials coincide.

  3. 3.

    Note that the results of this section are local.

  4. 4.

    See [53, Chapter 8] for the definition of Poisson stability.

  5. 5.

    \(V_\xi \) and \(V_{\xi \xi }\) denote, respectively, the gradient and the Hessian matrix of the scalar function \(V:\ \xi \mapsto V(\xi )\).

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Authors and Affiliations

  1. Imperial College London, London, SW7 2AZ, UK

    Giordano Scarciotti & Alessandro Astolfi

  2. University of Rome “Tor Vergata”, Via Del Politecnico 1, 00133, Rome, Italy

    Alessandro Astolfi

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  1. Giordano Scarciotti
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  1. Centre Automatique et Systèmes CAS, MINES ParisTech, PSL Research University, Centre Automatique et Systèmes CAS, Paris Cedex 06, France

    Nicolas Petit

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Scarciotti, G., Astolfi, A. (2017). A Review on Model Reduction by Moment Matching for Nonlinear Systems. In: Petit, N. (eds) Feedback Stabilization of Controlled Dynamical Systems. Lecture Notes in Control and Information Sciences, vol 473. Springer, Cham. https://doi.org/10.1007/978-3-319-51298-3_2

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