Tangential interpolatory projections for a class of second-order index-1 descriptor systems and application to Mechatron
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Tangential interpolatory projections for a class of second‑order index‑1 descriptor systems and application to Mechatronics Md. Motlubar Rahman1 · M. Monir Uddin2 · L. S. Andallah1 · Mahtab Uddin3 Received: 25 July 2020 / Accepted: 27 October 2020 © German Academic Society for Production Engineering (WGP) 2020
Abstract This paper studies the model order reduction of second-order index-1 descriptor systems using a tangential interpolation projection method based on the Iterative Rational Krylov Algorithm (IRKA). Our primary focus is to reduce the system into a second-order form so that the structure of the original system can be preserved. For this purpose, the IRKA based tangential interpolatory method is modified to deal with the second-order structure of the underlying descriptor system efficiently in an implicit way. The paper also shows that by exploiting the symmetric properties of the system the implementing computational costs can be reduced significantly. Theoretical results are verified for the model reduction of the piezo actuator based adaptive spindle support which is second-order index-1 differential-algebraic form. The efficiency and accuracy of the method are demonstrated by analyzing the numerical results. Keywords Interpolatory projections · Iterative Rational Krylov Algorithm · Structure-preserving model order reduction · Second-order index-1 systems · Piezo actuator based adaptive spindle support
1 Introduction We discuss the Iterative Rational Krylov Algorithm (IRKA) based tangential interpolation projection technique for the model reduction of second-order differential algebraic equations (DAEs) together with output equation which are given by
M11 v̈ (t) + L11 v(t) ̇ + K11 v(t) + K12 𝜂(t) = F1 u(t),
(1a)
K21 v(t) + K22 𝜂(t) = F2 u(t),
(1b)
H1 v(t) + H2 𝜂(t) + Da u(t) = y(t),
(1c)
where v(t) ∈ ℝn1 , 𝜂(t) ∈ ℝn2 are the states, u(t) ∈ ℝm are the inputs and y(t) ∈ ℝp are the outputs, and * M. Monir Uddin [email protected] 1
Department of Mathematics, Jahangirnagar University, Savar, Dhaka 1342, Bangladesh
2
Department of Mathematics and Physics, North South University, Dhaka 1229, Bangladesh
3
Institute of Natural Sciences, United International University, Dhaka 1212, Bangladesh
matrices M11 , L11 , K11 , K12 , K21 and K22 are sparse. The matrix Da ∈ ℝp×m represents the direct feed-through from the input to the output. We consider that number of inputs and outputs is greater than one i.e., the system is multiinputs and multi-outputs (MIMO). We also assume that the block matrix K22 is non-singular. In the previous literature see, e.g., [1] such system was defined as index-1 system. This system is called symmetric if the matrices M11 , L11 , T , H1 = F1T and K11 , K22 and Da are symmetric, and K21 = K12 T H2 = F2 . Such structure systems arise in many applications, for examples in the modeling of the mechanical and electrical networks (see e.g., [2–4]) where the constraints are imposed to control the dynamic behavior of the systems or
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