## Structure preserving model order reduction of large sparse second-order index-1 systems and application to a mechatronics model.(English)Zbl 1348.93025

Summary: Nowadays, mechanical engineers heavily depend on mathematical models for simulation, optimization and controller design. In either of these tasks, reduced dimensional formulations are obligatory in order to achieve fast and accurate results. Usually, the structural mechanical systems of machine tools are described by systems of second-order differential equations. However, they become descriptor systems when extra constraints are imposed on the systems. This article discusses efficient techniques of Gramian-based model-order reduction for second-order index-1 descriptor systems. Unlike, our previous work, here we mainly focus on a second-order to second-order reduction technique for such systems, where the stability of the system is guaranteed to be preserved in contrast to the previous approaches. We show that a special choice of the first-order reformulation of the system allows us to solve only one Lyapuov equation instead of two. We also discuss improvements of the technique to solve the Lyapunov equation using low-rank alternating direction implicit methods, which further reduces the computational cost as well as memory requirement. The proposed technique is applied to a structural finite element method model of a micro-mechanical piezo-actuators-based adaptive spindle support. Numerical results illustrate the increased efficiency of the adapted method.

### MSC:

 93A15 Large-scale systems 93B11 System structure simplification 93C15 Control/observation systems governed by ordinary differential equations 93C95 Application models in control theory
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### References:

 [1] DOI: 10.1007/s11740-012-0410-x · doi:10.1007/s11740-012-0410-x [2] M.M. Uddin,Computational methods for model reduction of large-scale sparse structured descriptor systems, Ph.D. thesis, Otto-von-Guericke-Universität, Magdeburg, 2015. Available at http://nbn-resolving.de/urn:nbn:de:gbv:ma9:1-6535 [3] DOI: 10.1080/13873950701844170 · Zbl 1151.93010 · doi:10.1080/13873950701844170 [4] DOI: 10.1080/13873954.2010.540822 · Zbl 1221.93039 · doi:10.1080/13873954.2010.540822 [5] DOI: 10.1137/040605552 · Zbl 1078.65058 · doi:10.1137/040605552 [6] DOI: 10.1016/j.laa.2004.12.013 · Zbl 1103.93017 · doi:10.1016/j.laa.2004.12.013 [7] DOI: 10.1080/00207178408933239 · Zbl 0543.93036 · doi:10.1080/00207178408933239 [8] DOI: 10.1080/00207178708933971 · Zbl 0642.93015 · doi:10.1080/00207178708933971 [9] DOI: 10.1002/nla.v15:9 · doi:10.1002/nla.v15:9 [10] DOI: 10.1137/S1064827598347666 · Zbl 0958.65052 · doi:10.1137/S1064827598347666 [11] DOI: 10.1007/s11075-012-9569-7 · Zbl 1267.65047 · doi:10.1007/s11075-012-9569-7 [12] DOI: 10.1080/13873954.2013.794363 · Zbl 1305.93043 · doi:10.1080/13873954.2013.794363 [13] DOI: 10.1016/j.cirp.2008.03.051 · doi:10.1016/j.cirp.2008.03.051 [14] DOI: 10.1016/j.cirp.2010.03.029 · doi:10.1016/j.cirp.2010.03.029 [15] DOI: 10.1109/9.544000 · Zbl 0859.93015 · doi:10.1109/9.544000 [16] DOI: 10.1016/j.laa.2004.03.032 · Zbl 1102.93008 · doi:10.1016/j.laa.2004.03.032 [17] DOI: 10.1002/gamm.v36.1 · doi:10.1002/gamm.v36.1 [18] Benner P., Electron. Trans. Numer. Anal. 43 pp 142– (2014) [19] Marcus M., A Survey of Matrix Theory and Matrix Inequalities (1964) · Zbl 0126.02404 [20] DOI: 10.1137/1.9780898718881 · Zbl 1119.65021 · doi:10.1137/1.9780898718881 [21] DOI: 10.1137/1.9780898718003 · doi:10.1137/1.9780898718003 [22] DOI: 10.1109/TPWRS.2008.926693 · doi:10.1109/TPWRS.2008.926693 [23] DOI: 10.1137/070681910 · Zbl 1216.76015 · doi:10.1137/070681910
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