Abstract
This paper focuses on the model reduction problem for a special class of linear parameter-varying systems. This kind of systems can be reformulated as bilinear dynamical systems. Based on the bilinear system theory, we give a definition of the \(\mathcal {H}_{2}\) norm in the generalized frequency domain. Then, a model reduction method is proposed based on the gradient descent on the Grassmann manifold. The merit of the method is that by utilizing the gradient flow analysis, the algorithm is guaranteed to converge, and further speedup of the convergence rate can be achieved as well. Two numerical examples are tested to demonstrate the proposed method.
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Acknowledgments
The authors would like to thank the European Cost Action: TD1307-European Model Reduction Network (EU-MORNET) for the funding of a short-term scientific mission, which leads to this work.
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Communicated by: Anthony Nouy
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Appendices
Appendix A: Proof of Theorem 5
Recall the Euclidean gradient given by (3.3). Rewrite it into two parts.
Applying Theorem 2 and Corollary 1, we can derive the upper bound of the reduced-order Gramians Rr, Qr and the solutions of the Sylvester equations X and Y, which are
Applying the bounds in (A.1) and (A.2), the upper bound of JV 1 is derived as
The upper bound of JV 2 is computed as follows:
As a result, the upper bound of JV is
In order to derive \(\| \dot {J}_{V} \|_{F}\), we need to obtain the upper bound on \(\| \dot {X} \|_{F}\), \(\| \dot {Y} \|_{F}\), \(\| \dot {R}_{r} \|_{F}\), and \(\| \dot {Q}_{r} \|_{F}\). Consider (B.4) with Ψ1 given by (B.8). Applying Theorem 2, it is not difficult to derive that
In the same way, it can be computed that
Differentiating JV 1 and computing its Frobenius norm, we obtain
Now differentiate JV 2 and compute its Frobenius norm to obtain
Summing them up, we can derive
Note that here, \(\dot {V}\) stands for \(\dot {\mathfrak {V}}(\alpha )\).
Appendix B: Proof of Corollary 4
The inequality in (3.13) requires up to the third-order derivatives of the objective function \(J(\mathfrak {V}(\alpha ))\) with respect to α. Denote the matrix Φ(α) as
The derivatives of \(J(\mathfrak {V}(\alpha ))\) are computed as
Note that here, we abbreviate \(\mathfrak {V}(\alpha )\) as V for convenience and the matrix Γk given by (3.7) is skew-symmetric. Let \( W(\alpha ) = \begin {pmatrix} X(\alpha )^{\top } & R_{r}(\alpha ) \end {pmatrix} \) denote the second row of Φ(α). Since \(\| \mathfrak {V}(\alpha ) \| = 1\), it is not difficult to derive the following:
To derive the bound on |J(3)(α)|, we need to compute the bound on ∥W(α)∥F and its first three derivatives. Since W(α) depends on the step size, it is reasonable to consider \( \max _{0 \leq \alpha \leq \tau _{k}} \| W(\alpha ) \|_{F} \) and its derivatives for some positive scalar τk, which varies over the iteration. To do so, we consider the generalized Lyapunov equation associated with Φ(α) and the derivatives of the generalized Lyapunov equation. Let,
denote the system matrices of the error system in the k th iteration. The generalized Lyapunov equations associated with \({\Phi }(\alpha ),\ \dot {\Phi }(\alpha ), \ \ddot {\Phi }(\alpha )\), and Φ(3)(α) are
where,
Denote the second row of Ψi, i = 0, 1, 2, 3, as \(Z_{{\Phi }_{i}}, \ i = 0,1,2,3\). According to Theorem 2, the bound of ∥W(3)(α)∥F satisfies
Now apply LemmaA.3 in [27] and let Ωk denote \(Z_{{\Phi }_{3}}(0)\), then we obtain the following:
where βi, i = 0, 1, 2, 3 are given by (3.16) to (3.20). Again, repetitively applying LemmaA.3 in [27], we derive that
Substituting the above inequalities to (B.11), it can be obtained that
To make sure that \(\max _{0 \leq \alpha \leq \tau _{k}} \| W^{(3)}(\alpha ) \|_{F}\) is bounded, it must hold that
It is also not difficult to show that the upper bound of W(α) and its first three derivatives satisfy a system of linear inequalities as follows:
By following the proofs of Lemma 4.2 and 4.3 in [27], the proof can be completed.
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Benner, P., Cao, X. & Schilders, W. A bilinear \(\mathcal {H}_{2}\) model order reduction approach to linear parameter-varying systems. Adv Comput Math 45, 2241–2271 (2019). https://doi.org/10.1007/s10444-019-09695-9
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DOI: https://doi.org/10.1007/s10444-019-09695-9
Keywords
- Model order reduction
- Linear parameter-varying systems
- Bilinear dynamical systems
- Gradient descent
- Grassmann manifold