By Sheldon Tan, Lei He

Version order aid (MOR) recommendations lessen the complexity of VLSI designs, paving how one can larger working speeds and smaller function sizes. This ebook offers a scientific advent to, and remedy of, the major MOR equipment hired as a rule linear circuits, utilizing real-world examples to demonstrate the benefits and drawbacks of every set of rules. Following a evaluate of conventional projection-based innovations, insurance progresses to complex 'state-of-the-art' MOR tools for VLSI layout, together with HMOR, passive truncated balanced cognizance (TBR) tools, effective inductance modeling through the VPEC version, and structure-preserving MOR innovations. the place attainable, numerical tools are approached from the CAD engineer's viewpoint, averting complicated arithmetic and permitting the reader to tackle actual layout difficulties and enhance more beneficial instruments. With sensible examples and over a hundred illustrations, this publication is appropriate for researchers and graduate scholars of electric and desktop engineering, in addition to practitioners operating within the VLSI layout undefined.

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**Example text**

QED. 1 Introduction Model order reduction methods for linear and non-linear dynamic systems in general can be classiﬁed into two categories [6]: 1. Singular-value-decomposition (SVD) based approaches 2. Krylov-subspace-based approaches. Krylov-subspace-based methods have been reviewed in Chapter 2. In this chapter, we focus on the SVD-based reduction methods. Singular value decomposition is based on the lower rank approximation, which is optimal in the 2-norm sense. The quantities for deciding how a given system can be approximated by a lower-rank system are called singular values, which are the square roots of the eigenvalues of the product of the system matrix and its adjoint.

Proof: To prove the corollary, we only need to look at the expected highest-order 36 Projection-based model order reduction algorithms moment of the reduced system m2q−1 ; 2q−1 −1 mr2q−1 = lTc (A−1 Ar br r Er ) q−1 −1 q−1 −1 = lTc (A−1 (A−1 Ar br r Er ) r Er )(Ar Er ) = lT V (W T AV )−1 W T EV (W T AV )−1 W T EV q−1 × (W T AV )−1 W T EV T (W T AV ) (W T AV )−1 W T EV rq−1 = gq−1 T = gq−1 W T EV rq−1 = lT A−1 (EA−1 )q−1 E(A−1 E)q−1 A−1 b = lT (A−1 E)q−1 A−1 E(A−1 E)q−1 A−1 b = lT (A−1 E)2q−1 A−1 b = m2q−1 .

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 .