Balanced truncation of linear time-invariant systems over finite-frequency ranges
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Balanced truncation of linear time-invariant systems over finite-frequency ranges Peter Benner1,2
· Xin Du3 · Guanghong Yang4 · Dan Ye4
Received: 1 October 2018 / Accepted: 27 October 2020 / Published online: 16 November 2020 © The Author(s) 2020
Abstract This paper discusses model order reduction of linear time-invariant (LTI) systems over limited frequency intervals within the framework of balanced truncation. Two new frequency-dependent balanced truncation methods are developed, one is singlefrequency (SF)-type frequency-dependent balanced truncation to cope with the cases that only a single dominating point of the operating frequency interval is pre-known, and the other is interval-type frequency-dependent balanced truncation to deal with the case that both the upper and lower bounds of the relevant frequency interval are known a priori. Error bounds for both approaches are derived to estimate the approximation error over a pre-specified frequency interval. In contrast to other error bounds for frequency-weighted or frequency-limited balanced truncation, these bounds are given specifically for the interval under consideration and are thus often sharper than the global bounds for previous methods. We show that the new methods generally lead to good in-band approximation performance, and at the same time provide accurate error bounds under certain conditions. Examples are included for illustration. Keywords Model order reduction · Balanced truncation · Linear time-invariant systems · Kalman-Yakubovich-Popov lemma Mathematics Subject Classification (2010) 93B40 · 93A15 · 93B11
Communicated by: Anthony Nouy This article belongs to the Topical Collection: Model reduction of parametrized Systems Guest Editors: Anthony Nouy, Peter Benner, Mario Ohlberger, Gianluigi Rozza, Karsten Urban and Karen Willcox Peter Benner [email protected]
Extended author information available on the last page of the article.
82 Page 2 of 34
Adv Comput Math (2020) 46: 82
1 Introduction and problem formulations We study model order reduction for linear time-invariant continuous-time systems, where, by abuse of notation, we will denote the system and its transfer function by G, and we will use the following representations in state-space, block matrix, and frequency domain form: G:
x(t) ˙ = Ax(t) + Bu(t) ⇐⇒ y(t) = Cx(t) + Du(t) ⇐⇒
A B (1) C D G(j ω) := C(j ωI − A)−1 B + D, G∼ =
where A ∈ Cn×n , B ∈ Cn×m , C ∈ Cp×n , D ∈ Cp×m , and at time t ∈ [0, ∞], m p x(t) ∈ Cn is the state vector, u(t) √ ∈ C is the input signal, and y(t) ∈ C is the output signal. Furthermore, j := −1 is the imaginary unit and ω ∈ R is the angular frequency in radians per second. Modeling of complex physical processes frequently leads to large order n. The corresponding high storage requirements and expensive computations make it often difficult to simulate or optimize such systems, and may be prohibitive to design a controller for such plant models. In these situations, model order reduction (MOR) becomes an expedient tool; see [1–8] for overviews on M
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