By Sheldon Tan, Lei He

Version order relief (MOR) recommendations decrease the complexity of VLSI designs, paving how to greater working speeds and smaller characteristic sizes. This booklet offers a scientific creation to, and remedy of, the foremost MOR tools hired mostly linear circuits, utilizing real-world examples to demonstrate the benefits and drawbacks of every set of rules. Following a evaluate of conventional projection-based ideas, insurance progresses to complex 'state-of-the-art' MOR tools for VLSI layout, together with HMOR, passive truncated balanced awareness (TBR) tools, effective inductance modeling through the VPEC version, and structure-preserving MOR recommendations. the place attainable, numerical equipment are approached from the CAD engineer's standpoint, heading off complicated arithmetic and permitting the reader to tackle genuine layout difficulties and strengthen better instruments. With sensible examples and over a hundred illustrations, this ebook is appropriate for researchers and graduate scholars of electric and computing device engineering, in addition to practitioners operating within the VLSI layout undefined.

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**Example text**

This can be easily obtained by ˆ Once all the poles are computed, we ˆ −1 C. performing the eigendecomposition of G then compute the residues at any node using aforementioned residue computation methods. We want to stress that the major diﬀerence of the projection methods used for pole computations here and the full-blown subspace projection-based model order reduction methods like PRIMA [85] is that we only need to compute the dominant poles here instead of the reduced system. Since the dominant poles are shared by all the transfer functions, we only need one transfer function to compute them.

24) may not be positive deﬁnite. 1 and the algorithm simply breaks. 9 Practical implementation and numerical issues 49 process, which can make a given system unstable even though the original system is stable and passive. In this section, we present a method to deal with this problem [10]. The basic idea is to decompose the system into two parts, the stable part and unstable part, and then perform the reduction on the stable part only. 59) such that G− is stable and G+ is unstable. Then any of the absolute or relative error state-space truncation methods for model reduction can be applied to G− in ˜ − , and the reduced order system order to obtain a reduced-order transfer function G ˜ ˜ − (s) + G+ (s).

Next we perform SVD on X . 7) where Σ and Σk are diagonal matrices with singular values in a decreasing order along the diagonal and Σk is a k × k diagonal matrix. As a result, we have the eigendecomposition of the response Gramian as X X T = U ΣΣT U T ≈ Uk Σk ΣTk UkT . 8) Let x(t) ≈ Uk η(t), η(t) ∈ Rk ; then we have the following reduced system in terms of η(t), η(t) ˙ = Uk T f (Uk η(t), u(t)), y(t) = h(Uk η(t), u(t)). 10) Thus the approximation of the state x(t) uses a low-dimensional space that is spanned by the k leading columns of U , which essentially are the dominant eigen vectors of the response Gramian X X T .