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Model Reduction of Generalized State Space Systems
Published in Mohammad Monir Uddin, Computational Methods for Approximation of Large-Scale Dynamical Systems, 2019
The techniques to reduce the state space dimension of an LTI continuous-time system (4.1) LTI system are well established. See, e.g., [6,7,31,83] for an overview. In a broad sense, there are two techniques: the Gramian-based methods and the moment matching-based methods.moment matching The Gramian-based methods include optimal Hankel norm approximation [77], singular perturbation approximation [36,70,116], dominant subspaces projection [113,134], dominant subspaces projectionfrequency weighted balanced truncation [67,193],dominant subspaces projectiondominant pole algorithm and balanced truncation (BT) dominant pole algorithm [125,146,163]. On the other hand, moment matchingbalanced truncationBT can be implemented efficiently via rational Krylov methods discussed in rational Krylov method [5,66,69,72,74,180]. The concept of projection for rational interpolation of the transfer function was first proposed in [173]. In [81] Grimme showed how to obtain the required projection using the rational Krylov method of Ruhe [140]. Later on, the authors in [5,85] generalized Grimme’s idea to generate a reduced model which is an optimal H2 approximation to the original system in the sense that it minimizes the H2 norm. The implementing algorithmIRKA is called the Iterative Rational Krylov Algorithm (IRKA).rational Krylov algorithm
Frequency-weighted ℋ2-optimal model order reduction via oblique projection
Published in International Journal of Systems Science, 2022
Umair Zulfiqar, Victor Sreeram, Mian Ilyas Ahmad, Xin Du
Another important class of projection-based MOR techniques is the Krylov subspace-based methods wherein the full-order system is projected onto a low-dimensional subspace spanned by the columns of a matrix constructed so that the projected reduced system achieves moment matching, i.e. the ROM matches some coefficients of the series expansion of the original transfer function at some selected frequency points (Beattie & Gugercin, 2014). Among these methods is the famous iterative rational Krylov algorithm (IRKA) (Gugercin et al., 2008; Van Dooren et al., 2008), which constructs a local optimum for the -optimal MOR problem, i.e. the best among all the ROMs with the same modal configuration and size in minimising the -norm of the error transfer function. Unlike the BT method, IRKA (Gugercin et al., 2008; Van Dooren et al., 2008) does not require the solutions of large-scale Lyapunov equations. Thus it is computationally efficient and can handle large-scale systems. Some other projection-based algorithms for the -optimal MOR problem include but are not limited to Ahmad et al. (2010b), Ibrir (2018), and Yan and Lam (1999). IRKA is heuristically generalised to the frequency-weighted scenario in Anić et al. (2013) and Zulfiqar and Sreeram (2018). The algorithms proposed in Anić et al. (2013) and Zulfiqar and Sreeram (2018) ensure less -norm of the weighted error transfer function; however, they do not seek to construct local optimum for the frequency-weighted -optimal MORproblem.
HSH-norm optimal MOR for the MIMO linear time-invariant systems on the Stiefel manifold
Published in International Journal of Control, 2022
In the fields of engineering, a lot of physical models can be directly described by LTI systems (see e.g. Chahlaoui & Dooren, 2005; Freund, 2000; Z. D. Wang & Unbehauen, 1999). Based on the first-order necessary conditions for optimality, an iterative rational Krylov algorithm (IRKA) and a MIMO iterative rational interpolation method (MIRIAm) have been developed in Bunse-Gerstner et al. (2010) and Gugercin et al. (2008). Wolf and Panzer (2015) obtains an pseudo-optimal approximation by using the alternating direction implicit (ADI) iteration. Manifold optimisation is an efficient tool for solving the constrained optimisation problems, which has drawn a lot of attention. It aims at seeking a critical point of the cost function with manifold constraints. Absil et al. (2008) has deeply studied some commonly encountered manifolds, including the unit sphere, the Stiefel manifold, and the Grassmann manifold. Yan and Lam (1999) regards the optimal MOR problem of LTI systems as a constrained optimisation problem on the Stiefel manifold and constructs the gradient flow algorithm. The Riemannian trust-region method on the Stiefel manifold and the Grassmann manifold for the optimal MOR problem of LTI systems is established in Sato and Sato (2015). In addition, Sato and Sato (2016) shows the optimal MOR problem on the product manifold and proposes a Riemannian conjugate gradient method by using the scaled vector transport. Recently, a gradient descent optimisation on the Grassmann manifold for linear parameter-varying systems is proposed in Benner et al. (2019). An advantage of optimisation over Riemannian manifolds is that the convergence of related algorithms can be conveniently guaranteed.