mimo interconnects order reductions by using the global arnoldi algorithm

7/27/2019 MIMO Interconnects Order Reductions by Using the Global Arnoldi Algorithm

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MIMO Interconnects Order Reductions by Using

the Global Arnoldi Algorithm

Ming-Hong LaiGraduate Institute of Electronic Engineering

Chang Gung University

TaoYuan, Taiwan, R.O.C.

Email: [email protected]

Chia-Chi ChuDepartment of Electrical Engineering




Wu-Shiung FengDepartment of Electronic Engineering




Abstract We propose the global Arnoldi algorithm for MIMORLCG interconnect model order reductions. This algorithm isan extension of the standard Arnoldi algorithm for systemswith multi-inputs and multi-outputs (MIMO). By employing thecongruence transformation with the matrix Krylov subspace, theone-sided projection method can be used to construct a reduced-order system. Two kinds of reduced systems using the globalArnoldi algorithm will be proposed. The first

-th order of thesereduced systems moments will still be preserved. The transfermatrix residual error of the reduced system will also be derivedanalytically. Experimental results demonstrate the feasibility andthe effectiveness of the proposed method.

I. INTRODUCTION

Modern technological trends in interconnect modeling have

emphasized considerable attention in high-speed VLSI de-

signs. Traditionally, several projection-based methods, includ-

ing asymptotic waveform evaluation (AWE), Arnoldi algo-

rithm, Pade via Lanczos (PVL) have been adopted to solve

such tasks [4], [5]. Most of them only can handle single-input

single-out (SISO) systems. Extensions to the multi-input multi-

output (MIMO) system are still not completely solved. In liter-ature, MPVL, PRIMA, and the block Arnoldi (BA) algorithm,

have been proposed for MIMO interconnect reductions [1], [2],

[6], [10]. However, numerical ill-conditional problems, such as

breakdowns and deflations, will always arise in practical large-

scale interconnect examples when the order of the reduced

system is extremely high.

In this paper, we propose an alternative projection-based

method, called the global Arnoldi (GA) algorithm [8]. This

algorithm is an extension of the standard Arnoldi algorithm

for systems with multiple inputs and multiple outputs. It will

be shown that this new matrix Krylov subspace, generated

from the Frobenius orthonormalization process, indeed is the

union of system moments. By employing the congruencetransformation with this matrix Krylov subspace, the one-sided

projection method can be used to construct a reduced-order

system. Two reduced systems will be constructed and both of

them can achieve moment matching to the original system.

It can be proven that the transfer matrix of the first reduced

system is identical to those of the reduced system generated

by the BA algorithm. However, the computation complexity of

the GA algorithm seems to be cheaper [8]. The transfer matrix

residual error of the reduced system is derived analytically.

Error information in the reduced system will be a guideline

for the order selection scheme. In the second reduced-order

system, we can keep the simple formulation described in the

BA algorithm. In addition, the transfer matrix of the second

reduced system is identical to those of the original system with

additive perturbation matrix.

I I . PRELIMINARY

A linear, time-invariance, RLCG interconnect circuit can be

represented in the following modified nodal analysis (MNA)

formula:

and (1)

satisfy the Kirchhoffs voltage and current

laws.

indicates the nodes that supplied voltage

sources, in which that is the number of voltage source.

indicates the nodes that we measure the impulse

response. Both matrices and are allowed to be singular,

and we only assume that the pencil

is regular. Let

and

, where

is the selected expansion frequency and we assume that

is nonsingular. Eq. (1) can be rewritten as

and

(2)

and the reduced-order system MNA of Eq. (2) is described by

and

(3)

where

and

. The attributes of reduced-order

modeling of the linear dynamical system include replacing the

full-order system by a system of the same type but with a much

smaller state-space dimension. Furthermore, the reduced-ordermodel should also preserve essential properties of the full-

order system. Such a reduced-order model would let designers

efficiently analyze and synthesize the dynamical behavior of

the original system within a tight design cycle.

III. MODEL-ORDER R EDUCTIONS FOR MIMO SYSTEMS

A. The Block Arnoldi (BA) Algorithm

For a MIMO system, the BA algorithm is one of the well-

known methodologies to solve the MIMO system problem

1107 ISCAS 20060-7803-9390-2/06/$20.00 2006 IEEE


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TABLE I

THE GLOBALA RNOLDI ALGORITHM

Algorithm : (input: ; output:

) : /* Initialize */

: /* Iteration */for

for

end for

end for

[2]. The reduced system can be constructed by the following

congruence transformation:

and

(4)

where the projection matrix

generated from the Krylov

subspace

, and

is the initial matrix

extracted from the orthonormal matrix of . In the process of

iterations, we have the following recurrence relationship:

(5)

where

, and

is

constructed from the block Arnoldi algorithm.

B. The Global Arnoldi (GA) Algorithm

We will introduce an alternative technique, called the Global

Arnoldi algorithm, to solve the reduced-order system for

MIMO systems.The Kronecker product is used to replace the

multiplication in the Arnoldi process. We define a vector-

valued function associated with a matrix and closely re-

lated to the Kronecker product. The system moment matrix

can also be associated with a vector-function,

we have

. Under this framework,

the input matrix is treated as a stacked vector form

and the GA algorithm is the standard Arnoldi algorithm

applied to a new matrix pair

. Since the

matrix is treated as a stacked vector, the inner product will also

be modified by the equation ,

accordingly. The GA algorithm will recursively generate the

Frobenius orthonormal basis

from the matrix

Krylov subspace

span

with the following properties [9]

for

(6)

for (7)

where

represents the trace of inner product

trace

The associated

norm, called the Frobenius norm, is defined as

trace

. The pseudo code of the GA algorithm is

outlined in Table I [8].

As can be seen from the GA algorithm, linear dependence

between the vector-columns of the generated matrix

has no effect on the algorithm. In fact as we

are working with a matrix Krylov subspace, the GA algorithm

allows us to generate the Frobenius-orthonormal basis.This is

a major difference between the GA and the BA algorithms.

Let

denotes the matrix,

denotes the

upper Hessenberg matrix from the GA algorithm, the

following relation is satisfied:

(8)

It is worthy of mentioning that the GA algorithm breaks down

at step if and only if

and in this case an

invariant subspace is obtained. This corresponds to a lucky

breakdown. On the other hand, for the BA algorithm, a

serious breakdown may occur and deflation techniques are

required. Various techniques have been proposed to solve this

task [3].

Lemma 1: [8] Let

be the matrix defined by

, where the matrices

are constructed by the GA algorithm, Then, we have

(9)

Having developed the relationships between system mo-

ments and Krylov subspace, now we are in the position to

construct the reduced-order system.

C. The Proposed Reduced-Order System - Type I

In this paper, the reduced system is chosen as

and

(10)

where

is the pseudo inverse of

.

For the special case

,

and

. This

is the standard Arnoldi algorithm for SISO system and

.

The notation of the reduced-order system can be further

simplified

where

is defined as

. The

can

be simplified as

By similar techniques developed in the BA algorithm,

moment matching can be achieved. Here we omit the details.

Theorem 1: (Moment Matching) For ,

the output moments

of the reduced system (10)

generated from the global Arnoldi algorithm will be the samewith those of the original MNA

in Eq. (2).

Lemma 2: [7] The reduced transfer matrix is defined as

, where

. The projected transfer matrix will be unchanged if the

projectors and are replaced by other matrices

and

which span the same respective spaces, where

and are invertible.

Since the

and

span the same respective space, by

Lemma 2, both reduced systems (4) and (10) can achieve the

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moment matching up to orders. In fact, transfer matrices

of both reduced systems are identical. The next theorem will

illustrate this result.

Theorem 2: Both transfer matrices of the reduced systems

generated by the BA algorithm and that of the GA algorithm

are identical.

Proof: In the block Arnoldi algorithm, we can rewrite

as

, then the projection matrix can be rep-

resented as

, where

is an

upper triangular matrix. On the other hand, according to the

aid of Lemma 1, the corresponding projection matrix generated

from the global Arnoldi algorithm can also be represented

as

, where

is an upper

triangular matrix, too.

Since

,

and

is an upper triangular matrix and is nonsingular.

By Lemma 2, it can be shown that the transfer functions are

identical. This completes the proof.

D. Residual Error

Since the exact transfer matrix error between the original

MNA and the reduced system is not easily derived analytically.

Here, we also use the notion residual error to describe their

difference [4]. Let the residual error

be defined as

(11)

where

is an approximate solution of

. It can be eas-

ily seen that if , then

. When either the

BA algorithm or the GA algorithm is applied, the approximate

state variable must belong to the Krylov subspace. That

is,

or

. The following theorem

describes analytical expressions of this residual error.

Theorem 3: Suppose that steps of the GA algorithm havebeen performed, Frobenius orthonormal matrix

and thecorresponding upper Hessenberg matrix

are obtained.

Let

be any approximate solution of

and

bean approximate solution after performing steps of the GA

algorithm, i.e.,

and

be the residualerror. The residual error

can be expressed as

(12)The error bound estimation can be proceeded as follows.

Suppose that all eigenvalues of

are simple and

be the eigenvalue

decomposition of

. Eq. (12) can be simplified as

(13)

where

Since is a high-pass

matrix,

min

. By taking the

norm from

both sides of Eq. (13), we have

where is the condition number of a matrix. The above

error estimation only involves

,

and

. Com-

paring with previous error expressions [11], less computational

CL

CL

CL

CL

CL

CL

CL

CL

CL

CL

CL

CL

CL

CL

CL

CL

CL

CL

CL

CL

CL

CL

CL

CL

CL

Vs1

Rs

Vs2

Rs

Scope2

Scope1

Fig. 1. The mesh40line circuit structure.

cost will be involved in the proposed formulation. Since the

computational cost of

is very time-consuming, we can

only consider

in our order selection scheme. Instead

of using the absolute value directly, we will use the relative

value

as a stopping criterion to terminate the

iteration process. If

is sufficiently small, the original system

and the reduced system in nearly identical.

E. The Proposed Reduced-Order System - Type II

Another possible reduced system can be defined as

and

(14)

where

.

can be further simplified

as

The above reduced system can still achieve the moment

matching property.

Theorem 4: (Moment Matching) For ,

the output moments

of the reduced system (14)

generated from the global Arnoldi algorithm will be the same

with output moments

of the original MNA in Eq.

(2).

If we define a perturbed system with the following descrip-tion:

and

(15)

Let the transfer matrix of perturbed system (15) is

.

The following theorem illustrates that the transfer matrix of

the reduced system defined by Eq. (14) is indeed equal to that

of the original MNA with additive perturbation matrix (15).

That is,

. This is an extension of

the SISO system developed in [4].

Theorem 5: The transfer matrix

of the per-

turbed system (15) is indeed equal to the transfer matrix

of the reduced system (14).

1 2 3 4 5 6 7 8 9 100

1

2

3

4

Iteration Number q

q

Fig. 2. Comparison of

for different .

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IV. NUMERICALE XAMPLES

In order to verify the proposed reduction techniques, an

RLC interconnect circuit is investigated. As shown in Fig.

1, an RLC mesh circuit with forty lines is investigated.

The line parameters are resistance: ; capacitance:

; inductance: ; driver resistance: , and

load capacitance: . Each line is long and is

divided into sections, the MNA matrices have dimension

. In our experiment, there are two input voltage

sources and two output sinks, i.e., and . The

experiment tries within the frequency range

and

the expand frequency point of reduced system at

.

As the GA algorithm proceeds, the value of

and

are recorded. From the simulation results shown in Fig. 2,

it is recommended that the order of the reduced system is

set to . Fig. 3 shows the flop comparisons between

two algorithms, it can be observed that the cost of the global

Arnoldi algorithm has great improvement to the block Arnoldi

algorithm. The FLOPS number in the global Arnoldi algorithm

are almost half of those in the block Arnoldi. Fig. 4 shows the

relative errors between the original system and three reducedsystems. Additionally, the CPU time to construct a -order

reduced system by using the global Arnoldi algorithm and the

block Arnoldi algorithm is seconds and seconds,

respectively. The global Arnoldi algorithm saves about

in CPU time. Fig. 5 shows the transfer matrix of the original

system and two reduced systems

,

. It can

be observed that the transfer matrix of the reduced system is

well matched the original system nearby the expansion point,

and the frequency response of reduced systems

and

are identical. Meanwhile, Fig. 5 also shows that the

transfer matrices

and that of the original system with

additive perturbation system

are identical.

V. CONCLUSIONSIn this paper, we proposed a novel model-order reduction

technique for general multi-input multi-output large scale

interconnect systems by using the GA algorithm. Extending

from the classical Arnoldi algorithm of the single-input single-

output system, this technique is more suitable for the high-

speed VLSI interconnect analysis. Moment matching of the

original system and the reduced system of the first -order

has also been proven. According to the residual error analysis,

it will provide a guideline in order selection scheme. The

perturbation system of the original system is also derived.

Numerical experiment also shows the high coherency nearby

the expansion point in the frequency response.

10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

Reduced System Order (q)

FlopsDifference(%)

Flops

Fig. 3. Flops comparison between the block Arnoldi and global Arnoldialgorithm.

0 5 1010

15

1010

105

Order (j)

RelativeError(%)

Y(j)

11(s

0)

0 5 1010

15

1010

105

Order (j)

RelativeError(%)

Y(j)

12(s

0)

0 5 1010

15

1010

105

100

Order (j)

RelativeError(%)

Y(j)21

(s0)

0 5 1010

15

1010

105

100

Order (j)

RelativeEr

ror(%)

Y(j)

22(s

0)

Global Arnoldi I

Global Arnoldi II

Block Arnoldi

Fig. 4. The relative error of the system moments of the mesh40line originalsystem and those of the two kinds of the reduced systems.

0 5 10 150

0.5

1

1.5

2

2.5

Freq. (GHz)

Mag.

H11

(s)

0 5 10 150

0.2

0.4

0.6

0.8

1

Freq. (GHz)

Mag.

H12

(s)

0 5 10 150

1

2

3

4

Freq. (GHz)

Mag.

H21

(s)

0 5 10 150

0.5

1

1.5

Freq. (GHz)

Mag.

H22

(s)

Original

Block Arnoldi

Global ArnoldiI

Global ArnoldiII

H

Fig. 5. The transfer matrices ,

,

and

ofmesh40line circuit of all source / sink pairs.

ACKNOWLEDGMENTThe authors would also like to thank the National Science Council, R.O.C.,

for financially supporting this research under Contract No. NSC92-2213-E-

182-001 and NSC93-2213-E-182-001 .

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[8] K. Jbilou, A. J. Riquet, Projection Methods for Large Lyapunov MatrixEquations, Linear Algebra and its Applications, to appear, 2005.

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[10] A. Odabasioglu, M. Celik and L. T. Pileggi, PRIMA: PassiveReduced-Order Interconnect Macromodeling Algorithm, IEEE Trans.on Computer-Aided Design of Integrated Circuits and Systems, Vol. 17,No. 8, pp. 645-654, 1998.

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mimo interconnects order reductions by using the global arnoldi algorithm

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