Towards Time-Limited $\mathcal H_2$-Optimal Model Order Reduction

In order to solve partial differential equations numerically and accurately, a high order spatial discretization is usually needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized systems and hence reduce computational complexity. A particular class of MOR techniques are $\mathcal H_2$-optimal methods such as the iterative rational Krylov subspace algorithm (IRKA) and related schemes. However, these methods are used to obtain good approximations on a infinite time-horizon. Thus, in this work, our main goal is to discuss MOR schemes for time-limited linear systems. For this, we propose an alternative time-limited $\mathcal H_2$-norm and show its connection with the time-limited Gramians. We then provide first-order optimality conditions for an optimal reduced order model (ROM) with respect to the time-limited $\mathcal H_2$-norm. Based on these optimality conditions, we propose an iterative scheme, which, upon convergence, aims at satisfying these conditions approximately. Then, we analyze how far away the obtained ROM due to the proposed algorithm is from satisfying the optimality conditions. We test the efficiency of the proposed iterative scheme using various numerical examples and illustrate that the newly proposed iterative method can lead to a better reduced-order compared to the unrestricted IRKA in the finite time interval of interest.

[1]  James Lam,et al.  On H2 model reduction of bilinear systems , 2002, Autom..

[2]  L. Meier,et al.  Approximation of linear constant systems , 1967, IEEE Transactions on Automatic Control.

[3]  Patrick Kürschner,et al.  Balanced truncation model order reduction in limited time intervals for large systems , 2017, Adv. Comput. Math..

[4]  D. Wilson.  Optimum solution of model-reduction problem , 1970 .

[5]  M. Freitag,et al.  Balanced truncation and singular perturbation approximation model order reduction for stochastically controlled linear systems , 2016, 1611.05494.

[6]  Jer-Nan Juang,et al.  Model reduction in limited time and frequency intervals , 1990 .

[7]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[8]  Peter Benner,et al.  Model reduction for stochastic systems , 2015 .

[9]  Serkan Gugercin,et al.  H2 Model Reduction for Large-Scale Linear Dynamical Systems , 2008, SIAM J. Matrix Anal. Appl..

[10]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[11]  Peter Benner,et al.  Interpolation-Based H2-Model Reduction of Bilinear Control Systems pdfsubject , 2011 .

[12]  Peter Benner,et al.  Dimension Reduction of Large-Scale Systems , 2005 .

[13]  H. V. D. Vorst,et al.  Model Order Reduction: Theory, Research Aspects and Applications , 2008 .

[14]  Peter Benner,et al.  An H2-type error bound for balancing-related model order reduction of linear systems with Lévy noise , 2017, Syst. Control. Lett..

[15]  Martin Redmann,et al.  An $\mathcal H_2$-Type Error Bound for Time-Limited Balanced Truncation , 2017, 1710.07572.