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Optimal H 2 Moment Matching-Based Model Reduction for Linear Systems through (Non)convex Optimization

Author

Listed:
  • Ion Necoara

    (Department of Automatic Control and Systems Engineering, Politehnica University of Bucharest, Splaiul Independentei 313, 060042 Bucharest, Romania
    Gheorghe Mihoc—Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, Romania
    These authors contributed equally to this work.)

  • Tudor-Corneliu Ionescu

    (Department of Automatic Control and Systems Engineering, Politehnica University of Bucharest, Splaiul Independentei 313, 060042 Bucharest, Romania
    Gheorghe Mihoc—Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, Romania
    These authors contributed equally to this work.)

Abstract

In this paper, we compute a (local) optimal reduced order model that matches a prescribed set of moments of a stable linear time-invariant system of high dimension. We fix the interpolation points and parametrize the models achieving moment-matching in a set of free parameters. Based on the parametrization and using the H 2 -norm of the approximation error as the objective function, we derive a nonconvex optimization problem, i.e., we search for the optimal free parameters to determine the model yielding the minimal H 2 -norm of the approximation error. Furthermore, we provide the necessary first-order optimality conditions in terms of the controllability and the observability Gramians of a minimal realization of the error system. We then propose two gradient-type algorithms to compute the (local) optimal models, with mathematical guarantees on the convergence. We also derive convex semidefinite programming relaxations for the nonconvex Problem, under the assumption that the error system admits block-diagonal Gramians, and derive sufficient conditions to guarantee the block diagonalization. The solutions resulting at each step of the proposed algorithms guarantee the achievement of the imposed moment matching conditions. The second gradient-based algorithm exhibits the additional property that, when stopped, yields a stable approximation with a reduced H 2 -error norm. We illustrate the theory on a CD-player and on a discretized heat equation.

Suggested Citation

  • Ion Necoara & Tudor-Corneliu Ionescu, 2022. "Optimal H 2 Moment Matching-Based Model Reduction for Linear Systems through (Non)convex Optimization," Mathematics, MDPI, vol. 10(10), pages 1-19, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:10:p:1765-:d:821073
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    References listed on IDEAS

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    1. Peter Benner & Enrique S. Quintana-Ortí & Gregorio Quintana-Ortí, 2000. "Balanced Truncation Model Reduction of Large-Scale Dense Systems on Parallel Computers," Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis Journals, vol. 6(4), pages 383-405, December.
    2. Michal Kocvara & Michael Stingl, 2012. "PENNON: Software for Linear and Nonlinear Matrix Inequalities," International Series in Operations Research & Management Science, in: Miguel F. Anjos & Jean B. Lasserre (ed.), Handbook on Semidefinite, Conic and Polynomial Optimization, chapter 0, pages 755-791, Springer.
    3. Wei-Gang Wang & Yao-Lin Jiang, 2018. "optimal model order reduction on the Stiefel manifold for the MIMO discrete system by the cross Gramian," Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis Journals, vol. 24(6), pages 610-625, November.
    Full references (including those not matched with items on IDEAS)

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