By Sheldon Tan, Lei He
Version order relief (MOR) concepts lessen the complexity of VLSI designs, paving tips to greater working speeds and smaller characteristic sizes. This e-book provides a scientific advent to, and remedy of, the major MOR equipment hired usually 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 complicated 'state-of-the-art' MOR tools for VLSI layout, together with HMOR, passive truncated balanced consciousness (TBR) tools, effective inductance modeling through the VPEC version, and structure-preserving MOR thoughts. the place attainable, numerical equipment are approached from the CAD engineer's point of view, averting advanced arithmetic and permitting the reader to tackle actual layout difficulties and boost more advantageous instruments. With functional examples and over a hundred illustrations, this publication 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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If G + GT ≥ 0 and C = C T ≥ 0 and G + sC is invertible at least at one point s with Re(s) > 0, then the following matrix transfer function is positive real H(s) = B T (G + sC)−1 B. 85) The proof of the theorem can be found in . , W (s)H = G + GT + 2δC at s = δ + jω is positive semideﬁnite. The inverse of the matrix and congruence transformation also do not change the property of positive semideﬁnite. After we perform the model order reduction on both G and C respectively, the reduced system is still passive.
The reduced model matches the original systems in terms of block moments as deﬁned below, Y (s) = M0 + M1 s + M2 s2 + ... 94) where Mi ∈ RN ×N . The block moments can be computed as follows Mi = B T Ai R. , Ak−1 R}, where k = q/N . 97) where ri is the ith column vector of R, and l = q − kN . 7 Block Arnoldi method for MIMO systems. 1 The . operator is the truncation to the nearest integer towards zero. 98) 32 Projection-based model order reduction algorithms and V T V = I. 99) Hq = . , .. ..
Compute SVD of Cholesky factors U ΣV T = LTo Lc where Σ is diagonal positive and U ,V have orthonormal columns. Compute the balancing transformation matrices T = Lc V Σ−1/2 , T −1 = Σ−1/2 U T LTo . ˜ = T −1 B, C˜ = Form the balanced realization transformations as A˜ = T −1 AT , B CT . ˜ B, ˜ C˜ conformally. Select reduced model order and partition realization A, ˜ ˜ ˜ Truncate A, B, C to form the reduced realization. Error bounds One of the attractive aspects of TBR methods is that computable error bounds are available.
Advanced Model Order Reduction Techniques in VLSI Design by Sheldon Tan, Lei He