Computation of nonlinear balanced realization and model reduction based on Taylor series expansion

Model Reduction by Moment Matching for Linear and Nonlinear Systems

“…Let H be the Hankel matrix associated to system (27), let H ν be the square matrix composed of the first ν rows and the first ν columns of H, and let σH ν be the square matrix composed of the first ν rows and the second to ν + 1 columns of H. By [30, Proposition 2], system (27) admits a positive diagonal realization if and only if H ν > 0 and the number of positive (negative, zero) eigenvalues of σH ν is equal to the number of positive (negative, zero) eigenvalues of S. Let ∆ = (CΠ) ′ and note that…”

“…Finally, approximation errors based on the Hankel operator of the system have been widely considered [14], [15], [16]. This approach leads to the so-called balancing realization problem [17], [18], which has been also studied in the nonlinear framework [19], [20], [21], [22], [23], [24], [25], [26], [27]. Secondly, the concept of simplicity.…”

“…Energy-based methods have been discussed, for example, in [21], [23], [32], [33]; model reduction of systems in special forms, such as differential-algebraic systems, bilinear systems or mechanical/Hamiltonian systems, have been discussed in [34], [35], [36], [37], [38], and reduction around a limit cycle or a manifold has been discussed in [39], [40]. Some computational issues have been addressed in [41], [42], [40], [27]. Finally, proper orthogonal decomposition has been proposed as a model reduction method applicable for linear and nonlinear systems, see, for example, [43], [44], [42], [45], [46], where theoretical issues, error bounds and applications have been discussed.…”

Balanced Realization and Model Order Reduction for Nonlinear Systems Based on Singular Value Analysis

“…In order to compute it explicitly, we employ the Taylor series approximation up to the 5th order in a similar way to [8]. First of all, it follows from Theorems 4 and 7 that the singular value functions σ i (·)'s are given by…”

“…Since then, many results on nonlinear state-space balancing, related minimality considerations, balancing near invariant manifolds, computational issues for model reduction, flow balancing, trajectory piecewise linear balancing, and empirical balancing for nonlinear systems have appeared in the literature; see, e.g., [7,8,10,11,12,14,16,21,23,25,27,28,30,31].…”