Ming-Hong Lai

Model-order reductions for MIMO systems using global Krylov subspace methods

This paper presents theoretical foundations of global Krylov subspace methods for model order reductions. This method is an extension of the standard Krylov subspace method for multiple-inputs multiple-outputs (MIMO) systems. By employing the congruence transformation with global Krylov subspaces, both one-sided Arnoldi and two-sided Lanczos oblique projection methods are explored for both single expansion point and multiple expansion points. In order to further reduce the computational complexity for multiple expansion points, adaptive-order multiple points moment matching algorithms, or the so-called rational Krylov space method, are also studied. Two algorithms, including the adaptive-order rational global Arnoldi (AORGA) algorithm and the adaptive-order global Lanczos (AOGL) algorithm, are developed in detail. Simulations of practical dynamical systems will be conducted to illustrate the feasibility and the efficiency of proposed methods.

MIMO Interconnects Order Reductions by Using the Multiple Point Adaptive-Order Rational Global Arnoldi Algorithm

We extend the adaptive-order rational Arnoldi algorithm for multiple-inputs and multiple-outputs (MIMO) interconnect model order reductions. Instead of using the standard Arnoldi algorithm for the SISO adaptive-order reduction algorithm (AORA), we study the adaptive-order rational global Arnoldi (AORGA) algorithm for MIMO model reductions. In this new algorithm, the input matrix is treated as a vector form. A new matrix Krylov subspace, generated by the global Arnoldi algorithm, will be developed by a Frobenius-orthonormal basis. By employing congruence transformation with the matrix Krylov subspace, the one-sided projection method can be used to construct a reduced-order system. It will be shown that the system moment matching can be preserved. In addition, we also show that the transfer matrix residual error of the reduced system can be derived analytically. This error information will provide a guideline for the order selection scheme. The algorithm can also be applied to the classical multiple point MIMO Pade approximation by the rational Arnoldi algorithm for multiple expansion points. Experimental results demonstrate the feasibility and the effectiveness of the proposed method.

The Multiple Point Global Lanczos Method for Multiple-Inputs Multiple-Outputs Interconnect Order Reductions

The global Lanczos algorithm for solving the RLCG interconnect circuits is presented in this paper. This algorithm is an extension of the standard Lanczos algorithm for multiple-inputs multiple-outputs (MIMO) systems. A new matrix Krylov subspace will be developed first. By employing the congruence transformation with the matrix Krylov subspace, the two-side oblique projection-based method can be used to construct a reduced-order system. It will be shown that the system moments are still matched. The error of the 2q-th order system moment will be derived analytically. Furthermore, two novel model-order reduction techniques called the multiple point global Lanczos (MPGL) method and the adaptive-order global Lanczos (AOGL) method which are both based on the multiple point moment matching are proposed. The frequency responses using the multiple point moment matching method have higher coherence to the original system than those using the single point expansion method. Finally, simulation results on frequency domain will illustrate the feasibility and the efficiency of the proposed methods.

Applications of tree/link partitioning for moment computations of general lumped RLC networks with resistor loops

A new moment computation technique for general lumped circuits with resistor loops is proposed. Using the concept of tearing, a lumped network can be portioned into a spanning tree and several resistor links. The contributions of network moments from each tree and the corresponding links can be determined independently. By combining the conventional moment computation algorithms and the reduced ordered binary decision diagram (ROBDD), the proposed method can compute system moments efficiently.

The global Lanczos method for MIMO interconnect order reductions

We present the global Lanczos algorithm for solving the RLCG interconnect circuits in this paper. This algorithm is an extension of the standard Lanczos algorithm for systems with multi-input multi-output (MIMO). A new matrix Krylov subspace will be developed first. By employing the congruence transformation with the matrix Krylov subspace, the two-side oblique projection-based method can be used to construct a reduced-order system. It will be shown that the system moments is still matched. The error of the 2q-th order system moment will be derived analytically. Finally, simulation results on both frequency domain and time domain will illustrate the feasibility and the efficiency of the proposed method

Moment computations for R(L)C interconnects with multiple resistor loops using ROBDD techniques

A new moment computation technique for general lumped R(L)C interconnect circuits with multiple resistor loops is proposed. Using the concept of tearing, a lumped R(L)C network can be partitioned into a spanning tree and several resistor links. The contributions of network moments from each tree and the corresponding links can be determined independently. By combining the conventional moment computation algorithms and the reduced ordered binary decision diagram (ROBDD), the proposed method can compute system moments efficiently

Applications of AOGL Model-Order Reduction Techniques in Interconnect Analysis

We apply the adaptive-order global Lanczos (AOGL) to generate the MIMO macromodel of high-speed VLSI interconnects. Theoretical foundations of the AOGL method for multiple point moment matching are investigated. In addition, the order selection of the reduced system was discussed. Experimental result on practical large-scale interconnect designs demonstrate that the reduced system generated by the AOGL method has better coherence to the original system both in time domain and in frequency domain. It has illustrated the feasibility and the efficiency of the AOGL method.

Model-Order Reduction Algorithm with Structure Preserving Techniques

An approach for the model-order reduction technique of the VLSI interconnect systems is presented in this paper. The conventional model-order reduction techniques usually adopt the congruence transformation to generate the reduced system, but they often destroy the structure of the system mathematical formulation. The new structure-preserving approach calculates the system moments by the recursively and construct the corresponding projection matrix. The structure of the reduced system can be preserved and it achieves the 2q-order moment matching property, and the recursively moment calculation technique could great reduce the computational cost. In addition, the effective impedance model can be obtained from the 0-th model-order reduction method. The experimental results show the higher accuracy and the higher resource saving of our method

Interconnect model reductions by using the AORA algorithm with considering the adjoint network

This work proposes a hybrid method for interconnect model-order reductions. First, the adaptive-order rational Arnoldi (AORA) method is investigated. An extension of the classical multi-point Pade approximation by using the rational Arnoldi iteration approach are studied. The adaptive-order can be achieved by choosing the expansion frequency corresponding to the maximum output moment error. Secondly, the adjoint network technique is studied. By exploring symmetric properties of the MNA formulation, the computational cost of constructing the congruence transformation matrix can be reduced by 50% compared with the conventional methods.

Moment Computations of Distributed Coupled RLC Interconnects with Applications to Estimating Crosstalk Noise

A method is proposed to compute moments of distributed coupled RLC interconnects. Both uniform line models and non-uniform line models will be developed. Considering both self inductances and mutual inductances in multi-conductors, recursive moment computations formulae of lumped coupled RLC interconnects are extended to those of distributed coupled RLC interconnects. By using the moment computation technique in conjunction with the projection-based order reduction method, the inductive crosstalk noise waveform can be accurately and efficiently estimated. Fundamental developments of the proposed approach will be described. Simulation results demonstrate the improved accuracy of the proposed method over the traditional lumped methods.