Balanced truncation model reduction for linear time-varying systems

References

• P. Benner, P. Kürschner, and J. Saak, An improved numerical method for balanced truncation for symmetric second order systems, Math. Comput. Model. Dyn. Syst. 19 (2013), pp. 593–615.
   Google Scholar
• T. Reis and T. Stykel, Balanced truncation model reduction of second-order systems, Math, Comput. Model. Dyn. Syst. 14 (2008), pp. 391–406.
   Google Scholar
• H. Sandberg and A. Rantzer, Balanced truncation of linear time-varying systems, IEEE Trans. Automat. Control. 49 (2004), pp. 217–229.
   Web of Science ®Google Scholar
• S. Shokoohi, L. Silverman, and P. Van Dooren, Linear time-variable systems: balancing and model reduction, IEEE Trans. Automat. Control. 28 (1983), pp. 810–822.
   Web of Science ®Google Scholar
• E.I. Verriest and T. Kailath, On generalized balanced realizations, IEEE Trans. Automat. Control. 28 (1983), pp. 833–844.
   Web of Science ®Google Scholar
• T. Kailath, Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980.
   Google Scholar
• A.J. Laub, M.T. Heath, C.C. Paige, and R.C. Ward, Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms, IEEE Trans. Automat. Control. 32 (1987), pp. 115–122.
   Web of Science ®Google Scholar
• M.S. Tombs and I. Postlethwaite, Truncated balanced realization of a stable nonminimal state-space system, Internat. J. Control. 46 (1987), pp. 1319–1330.
   Web of Science ®Google Scholar
• P. Benner and H. Mena, Rosenbrock methods for solving Riccati differential equations, IEEE Trans. Automat. Control. 58 (2013), pp. 2950–2957.
   PubMed Web of Science ®Google Scholar
• C. Choi and A.J. Laub, Efficient matrix-valued algorithms for solving stiff Riccati differential equations, IEEE Trans. Automat. Control. 35 (1990), pp. 770–776.
   Web of Science ®Google Scholar
• L. Dieci, Numerical integration of the differential Riccati equation and some related issues, SIAM J. Numer. Anal. 29 (1992), pp. 781–815.
   Web of Science ®Google Scholar
• H. Mena, Numerical solution of differential Riccati equations arising in optimal control problems for parabolic partial differential equations, Ph.D. thesis, Escuela Politecnica Nacional, 2007.
   Google Scholar
• P. Benner, P. Kürschner, and J. Saak, Efficient handling of complex shift parameters in the low-rank Cholesky factor ADI method, Numer. Algorithms. 62 (2013), pp. 225–251.
   Web of Science ®Google Scholar
• P. Benner, P. Kürschner, and J. Saak, A reformulated low-rank ADI iteration with explicit residual factors, Proc. Appl. Math. Mech. 13 (2013), pp. 585–586.
   Google Scholar
• J.-R. Li and J. White, Low rank solution of lyapunov equations, SIAM J. Matrix Anal. Appl. 24 (2002), pp. 260–280.
   Web of Science ®Google Scholar
• T. Penzl, Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case, Systems Control Lett. 40 (2000), pp. 139–144.
   Web of Science ®Google Scholar
• P. Benner, R.C. Li, and N. Truhar, On the ADI Method for Sylvester equations, J. Comput. Appl. Math. 233 (2009), pp. 1035–1045.
   Web of Science ®Google Scholar
• N. Lang, H. Mena, and J. Saak, On the benefits of the factorization for large-scale differential matrix equation solvers, Linear Algebra Appl. 480 (2015), pp. 44–71.
   Web of Science ®Google Scholar
• M. Bollhöfer and A.K. Eppler, Low-Rank Cholesky Factor Krylov subspace methods for generalized projected Lyapunov equations, in System Reduction for Nanoscale IC Design, P. Benner, ed., Mathematics in Industry, Springer International Publishing, Berlin, Heidelberg, 2016.
   Google Scholar
• K. Dekker and J.G. Verwer, Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, Elsevier, Amsterdam, 1984.
   Google Scholar
• E. Hairer and G. Wanner, Solving Ordinary Differential Equations II – Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics, Vol. 14, Springer-Verlag, Berlin, Heidelberg, 2002.
   Google Scholar
• V. Druskin and V. Simoncini, Adaptive rational Krylov subspaces for large-scale dynamical systems, Systems Control Lett. 60 (2011), pp. 546–560.
   PubMed Web of Science ®Google Scholar
• J.G. Verwer, E.J. Spee, J.G. Blom, and W. Hundsdorfer, A second Order Rosenbrock method applied to Photochemical dispersion problems, SIAM J. Sci. Comput. 20 (1999), pp. 1456–1480.
   Web of Science ®Google Scholar
• P. Benner and J. Saak, Linear-quadratic regulator design for optimal cooling of steel profiles, Technical Report SFB393/05-05, Sonderforschungsbereich 393 Parallele Numerische Simulation für Physik und Kontinuumsmechanik, TU Chemnitz, D-09107 Chemnitz, 2005 Available from http://www.tu-chemnitz.de/sfb393/sfb05pr.html.
   Google Scholar
• E. Hairer, S.P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I – Nonstiff Problems, second, Springer Series in Computational Mathematics, 2nd ed., Vol. 8, Springer-Verlag, Berlin Heidelberg, 1993.
   Google Scholar
• J.M. Burgers, Application of a model system to illustrate some points of the statistical theory of free turbulence, Proc. Roy. Netherl. Acad. Sci. (Am- sterdam). 43 (1940), pp. 2–12.
   Google Scholar
• J.M. Burgers, A Mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1 (1948), pp. 171–199.
   Google Scholar
• J.M. Burgers, Statistical problems connected with asymptotic solutions of the one-dimensional nonlinear diffusion equation, in Statistical Models and Turbulence, M. Rosenblatt and C. Van Atta, eds., Lecture Notes in Physics, Vol. 12, Springer, Berlin Heidelberg, 1972, pp. 41–60.
   Google Scholar
• S. Hein, MPC-LQG-based optimal control of parabolic PDEs, Ph.D. thesis, TU Chemnitz, 2009.
   Google Scholar
• P. Benner, S. Görner, and J. Saak, Numerical solution of optimal control problems for parabolic systems, in Parallel Algorithms and Cluster Computing. Implementations, Algorithms, and Applications, K.H. Hoffmann and A. Meyer, eds., Lect. Notes Comput. Sci. Eng., Vol. 52, Springer-Verlag, Berlin/Heidelberg, 2006, pp. 151–169.
   Google Scholar
• S.K. Godunov, Ordinary Differential Equations with Constant Coefficient, Translations of Mathematical Monographs, Vol. 169, AMS, Providence, RI, 1997.
   Google Scholar