By Sheldon Tan, Lei He
Version order aid (MOR) ideas decrease the complexity of VLSI designs, paving how you can larger working speeds and smaller function sizes. This e-book offers a scientific advent to, and therapy of, the major MOR equipment hired quite often linear circuits, utilizing real-world examples to demonstrate the benefits and drawbacks of every set of rules. Following a assessment of conventional projection-based options, assurance progresses to complex 'state-of-the-art' MOR equipment for VLSI layout, together with HMOR, passive truncated balanced awareness (TBR) tools, effective inductance modeling through the VPEC version, and structure-preserving MOR suggestions. the place attainable, numerical equipment are approached from the CAD engineer's point of view, fending off advanced arithmetic and permitting the reader to tackle actual layout difficulties and improve more beneficial instruments. With functional examples and over a hundred illustrations, this publication is acceptable for researchers and graduate scholars of electric and machine engineering, in addition to practitioners operating within the VLSI layout undefined.
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Extra resources for Advanced Model Order Reduction Techniques in VLSI Design
We will introduce positive-real TBR (PR-TBR) in this section. 3 Lur’e equations We assume D + D T > 0. For RLC models in a modiﬁed nodal analysis (MNA) format, we have A + AT ≤ 0, B = C T , and D = 0. The MNA system can then be transformed into an equivalent form with D + D T > 0. We will discuss this case in the following sections. Using the Schur complement, the solution of the positive-real lemma is equivalent to the following Lur’e equations. 41) and Xc ≥ 0 (Xc is positive semideﬁnite), Xc is analogous to the controllability Gramians.
2 (Moment matching connection for symmetric systems). 66). Proof: This corollary can easily be proved by noticing that, in this case, the two subspaces colspV = Kq (A−1 E, A−1 b) and colspW = Kq (A−T E T , A−T l) are the same because of the symmetric nature of the system. 1. QED. 1 is that for many practical RC circuits, whose MNA matrices are symmetric, using the one-side method such as the Arnoldi method can lead to more accurate results and the reduced models are also passive. For general RLC circuits, their matrices are not symmetric, but by splitting the projection matrices, which makes the matrices subblock-level symmetric, the reduction process can still lead to double accuracy (matching 2q instead of q moments) as shown in  and in Chapter 8 for general structure-preserved model order reduction methods.
These eigenvalues, λi (Wc Wo ), are called the Hankel singular values, and contain useful information about the input-output behavior of the system. In particular, small eigenvalues of Wc Wo correspond to internal sub-systems that have a weak eﬀect on the input-output behavior of the system and are almost non-observable or non-controllable or both. These are fundamental invariants that determine how well a model can be approximated by a reduced-order model. They play the same role for dynamical systems that the singular values play for ﬁnitedimensional matrices.
Advanced Model Order Reduction Techniques in VLSI Design by Sheldon Tan, Lei He