By Sheldon Tan, Lei He
Version order aid (MOR) innovations decrease the complexity of VLSI designs, paving the best way to greater working speeds and smaller characteristic sizes. This publication provides a scientific advent to, and therapy of, the major MOR tools hired quite often linear circuits, utilizing real-world examples to demonstrate the benefits and downsides of every set of rules. Following a assessment of conventional projection-based recommendations, insurance progresses to complex 'state-of-the-art' MOR equipment for VLSI layout, together with HMOR, passive truncated balanced consciousness (TBR) tools, effective inductance modeling through the VPEC version, and structure-preserving MOR concepts. the place attainable, numerical tools are approached from the CAD engineer's viewpoint, fending off advanced arithmetic and permitting the reader to tackle genuine layout difficulties and improve more beneficial instruments. With sensible examples and over a hundred illustrations, this ebook is acceptable for researchers and graduate scholars of electric and desktop engineering, in addition to practitioners operating within the VLSI layout undefined.
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Extra info for Advanced Model Order Reduction Techniques in VLSI Design
The basic idea of Poor Man’s TBR method is to recognize that the Gramian in the frequencydomain form can be approximated by using the state responses under impulse responses in frequency domain. 49) where A ∈ Rn×n , B ∈ Rn×p , C ∈ Rp×n , y(t), and u(t) ∈ Rp . , C = B T . In this case, both Gramians are equal. As a result, we only need to look at the controllability Gramian, X, ∞ X= T eAt BB T eA t dt. 51) where superscript H denotes the Hermitian transpose. We now consider evaluating X by applying numerical quadrature to the above equation, given a quadrature scheme with nodes ωk and weights ωk , and deﬁning zk = (jωk I − A)−1 B.
In this book, we call V the output Krylov subspace and W the input Krylov subspace. With this choice, the reduction method is called a two-side method as both input and output relations of the systems are involved. If only one of the two matrices V or W is chosen and the other one is chosen arbitrarily such that W T AV is non-singular, then the method is called a one-side MOR method. 65), we will ﬁnd that the numerical calculation of the matrix-vector products involved in the subspaces turns out to be unstable.
41) and Xc ≥ 0 (Xc is positive semideﬁnite), Xc is analogous to the controllability Gramians. In fact, it is the controllability Gramian for a system with the input-tostate mapping given by the matrix Kc (Kc is treated as input matrix). Similarly there is a dual set of Lur’e equations for Xo = XoT > 0: Jo , Ko are obtained from above equations by the substitutions A → AT , B → C T , C T → B. 44) have a corresponding observability quantity Xo ≥ 0 for a positive-real H(s). It can be veriﬁed that Xc Xo transform under similarity just as Wc Wo , so that their eigenvalues are invariant, and in fact they behave as the Gramians Wc and Wo .
Advanced Model Order Reduction Techniques in VLSI Design by Sheldon Tan, Lei He