Generalizations of adjoint networks techniques for RLC interconnects model-order reductions

Abstract

The adjoint network reduction technique has been shown to reduce 50% of the computational complexity of constructing the congruence transformation matrix. The method was suitable for analyzing the special multi-port driving-point impedance of RLC interconnect circuits. This paper extends this technique for the general circumstances of RLC interconnects. Comparative studies among the conventional methods and the proposed methods are also investigated. Experimental results will demonstrate the accuracy and the efficiency of the proposal method.

Description

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to generalizations of adjoint networks techniques, and more particularly to generalizations of adjoint networks techniques for RLC interconnects model-order reductions.

2. Description of Related Art

With the advances in deep submicron semiconductor techniques, the parasitic effects of interconnects has no longer been ignored [reference 2]. To completely characterize the signal integrity issues, interconnects are often modeled as large RLC networks. To this end, in order to reduce the computational time of the large-scale RLC interconnect networks, model-order reduction methods have been emerged recently [references 3, 6 and 9].

A consensus has emerged that of many model-order reduction techniques, the moment matching approach, or the so-called Krylov subspace projection method, seems to be the most viable one [references 3, 4, 5, 8 and 11]. In general, these methods can be divided into two categories: one-sided projection methods [references 8 and 11] and twosided projection methods [references 4 and 5]. The one-sided projection methods use the congruence transformation to generate passive reducedorder models while the two-sided ones can not be guaranteed. In recent works, we have proposed the adjoint network reduction technique to further reduce the cost about yielding the congruence transformation matrix [reference 7]. The method was suitable for analyzing the special multi-port driving-point impedance of RLC interconnect circuits.

The purpose of this paper is to extend the adjoint network technique for general RLC interconnect networks. First, relationships between an original MNA network and its corresponding adjoint MNA network are explored. Second, the congruence transformation matrix can be constructed by using the resultant biorthogonal bases from the Lanczos-type algorithms. Therefore, less storage and computational complexity are required in our proposed technique.

SUMMARY OF THE INVENTION

The main objective of the present invention is to provide improved generalizations of adjoint networks techniques for RLC interconnects model-order reductions.

The adjoint network reduction technique has been shown to reduce 50% of the computational complexity of constructing the congruence transformation matrix. The method was suitable for analyzing the special multi-port driving-point impedance of RLC interconnect circuits. This paper extends this technique for the general circumstances of RLC interconnects. Comparative studies among the conventional methods and the proposed methods are also investigated. Experimental results will demonstrate the accuracy and the efficiency of the proposal method.

Further benefits and advantages of the present invention will become apparent after a careful reading of the detailed description with appropriate reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a mech circuit diagram;

FIG. 2 shows the frequency responses of the voltage Vout in FIG. 1; and

FIG. 3 shows the relative errors in the frequency responses in FIG. 2.

DETAILED DESCRIPTION OF THE INVENTION

The dynamics of RLC interconnect networks can be represented by the following modified nodal analysis (MNA) formulae [reference 8 and 11]: $\begin{array}{cc}M\frac{dx\left(t\right)}{dt}=-\mathrm{Nx}\left(t\right)+\mathrm{Bu}\left(t\right),y\left(t\right)=D^{T}x\left(t\right)\mathrm{where}M=\left[\begin{array}{cc}C& 0\\ 0& L\end{array}\right],N=\left[\begin{array}{cc}G& E\\ -E^T& R\end{array}\right],x\left(t\right)=\left[\begin{array}{c}v\left(t\right)\\ i\left(t\right)\end{array}\right],& \left(1\right)\end{array}$ x(t)εRⁿ is the state vector, u(t)εR^m is the input vector, y(t)εR^p is the output vector, and M,NεR^nxn, BεR^nxm,and DεR^nxp are so-called the MNA matrices. M and N containing capacitances in C, inductances in L, conductances in G and resistances in R are positive definite, M is symmetric and N is non-symmetric. E presents the incident matrix for satifying Kirchhoff's current law. x(t) contains node voltages V(t)εR^nv and branch currents of inductors i(t)εRⁿⁱ, where n=n_v+n_i. If the m-port driving-point impedance is concerned, then p=m and D=B.

Let the signature matrix be defined as S=diag(I_nv,−I_ni) so that the symmetric properties of the MNA matrices are as follows [reference 11]: S⁻¹=S,SMS=M, and SNS=N^T   (2) Under this situation, if port driving-point impedance is concerned, that is, each port is connected with a current source, then B^T=└B_v^T0┘, where B_vεR^nvxm, and SB=B.

The transfer functions of the state variables and of the outputs are defined as X(s)=(n+sM)⁻¹ and Y(s)=D^TX(s). Given a frequency s₀εC, let A=−(N+s₀M)⁻¹M and R=(N+s₀M)⁻¹B, where N+s₀M is assumed nonsingular. The Taylor series expansion of X(s) about s₀ is given by $X\left(s\right)=\sum _{i=0}^{\infty }{X}^{\left(i\right)}\left(s_0\right)\left(s-s_0\right),\mathrm{where}$$X^{\left(i\right)}\left(s_0\right)=A^{i}R$ is the ith-order system moment about s₀. Similarly, the ith-order output moment about s₀ is calculated asY⁽ⁱ⁾(s₀)=D^TX⁽ⁱ⁾(s₀).

Suppose that the above system is large-scale and sparse. The model-order reduction problem is to seek a q-order system, where q<<n, such that $\begin{array}{cc}\hat{M}\frac{d\hat{x}\left(t\right)}{dt}=-\hat{N}\hat{x}\left(t\right)+\hat{B}u\left(t\right),\hat{y}\left(t\right)={\hat{D}}^T\hat{x}\left(t\right)& \left(3\right)\end{array}$ where {circumflex over (x)}(t)εR^q,{circumflex over (M)},{circumflex over (N)}εR^qxm,{circumflex over (B)}εR^qxm, and {circumflex over (D)}εR^qxp. The corresponding ith-order system moment and output moment about s₀ is {circumflex over (X)}⁽ⁱ⁾(s₀)=(−({circumflex over (N)}+s₀{circumflex over (M)})⁻¹{circumflex over (M)})ⁱ({circumflex over (N)}+s₀{circumflex over (M)})⁻¹{circumflex over (B)} and Ŷ⁽ⁱ⁾(s₀)={circumflex over (D)}^T{circumflex over (X)}⁽ⁱ⁾(s₀). The purpose of the moment matching is to construct a reduced-order system such that Ŷ⁽ⁱ⁾(s₀)=Y⁽ⁱ⁾(s₀) for 0≦i≦k−1, where k is the order of moment matching.

One conventional solution for moment matching is using the one-sided Krylov subspace projection method [references 3, 8 and 11]. The kth-order block Krylov subspace is defined as K(A,R,k)=colsp([RAR . . . A^k−1R]). K(A,R,k) is indeed the subspace spanned by X⁽ⁱ⁾(s₀) for 0≦i≦k−1. The projection can be achieved by constructing V_qεR^nxq,q≦km, from the Krylov subspace K(A,R,k). Under this framework, we have x(t)=V_q{circumflex over (x)}_q(t) and the reduced-order model can be expressed as {circumflex over (M)}=V_q^TMV_q,{circumflex over (N)}=V_q^TNV_q,{circumflex over (B)}=B_q^TB,{circumflex over (D)}=V_q^TD   (4) It has been shown that X⁽ⁱ⁾(s₀)=V_q{circumflex over (X)}⁽ⁱ⁾(s₀) and Ŷ⁽ⁱ⁾(s₀)=Y⁽ⁱ⁾(s₀) for 0≦i≦k−1. The reduced-order model is guaranteed stable {circumflex over (M)} and {circumflex over (N)} are positive definite. Furthermore, it will be passive if the multi-port driving-point impedance is concerned.

Two types of algorithms can be employed to generate V_q from the Krylov subspace: the Arnoldi type [references 8 and 11] and the Lanczos type [references 4 and 5]. We use the notation V_q(A) to denote the orthonormal basis generated from the block Arnoldi algorithm from the Krylov subspace K(A,R,k). Similarly, we use the notation V_q(L) and W_q(L) to represent the biorthogonal bases yielded from the block Lanczos algorithm from the Krylov subspaces K(A,R,k) and K(A^T,D,k), respectively. In this case, W_q(L)^TV_q(L)=Δ_q, where Δ_q is a full rank diagonal matrix. In the past, either V_q(A) or V_q(L) has been used to generate the reduced-order model (4).

Traditionally, W_q(L) and V_q(L) are used to perform oblique projection {circumflex over (M)}=−W_q(L)^TAV_q(L),{circumflex over (N)}=W_q(L)^T(I+s₀A)V_q(L),{circumflex over (B)}=W_qT(L)^TR,{circumflex over (D)}=V_q(L)^TD Although the reduced-order system can match up to 2k-order moments, this model can not be guaranteed to be stable and passive. Variations of the Lanczos-type algorithms have also been proposed. For example, if the m-port driving-point impedance is concerned (D=B), the symmetric block Lanczos algorithm has been investigated. In this case, W_q=D_qV_q(L), where D_q is a diagonal matrix, only a half of the computational cost and storage are required [references 3 and 5].

Using both W_q(L) and V_q(L) in the one-sided projection method is still possible. It can be achieved by the adjoint network reduction technique. The details can be developed in the following section.

One technique for model-order reductions is tot apply the corresponding adjoint MNA formulae. The adjoint network (or the dual system [reference 11]) of the system (1) is represented as [reference 10] $\begin{array}{cc}M\frac{d{x}_a\left(t\right)}{dt}=-N^T{x}_a\left(t\right)+\mathrm{Du}\left(t\right),y_a\left(t\right)=B^T{x}_a\left(t\right)& \left(5\right)\end{array}$ The system transfer function and its ith-order system moment about s₀ are defined as X_a(s)=(N^T+sM)⁻¹D and X_a⁽ⁱ⁾(s₀), respectively. By using the information of X_a⁽ⁱ⁾(s₀), the following theorem has been shown in [reference 11]. Theorem 1: If a matrix U is chosen as the congruence transformation matrix such that X⁽ⁱ⁾(s₀),X_a^(j)(s₀)}εcolsp(U) for 0≦i≦k−1 and 0≦j≦l −1,   (6) then Ŷ⁽ⁱ⁾(s₀)=Y⁽ⁱ⁾(s₀) for 0≦i≦k+l −1.

The technique can overcome the numerical instability problem when generating the basis matrix U if order k+l is extremely high. In this section, we summarize some properties of the adjoint network reduction technique. Relationships between X_a⁽ⁱ⁾(s₀) and X⁽ⁱ⁾(s₀) are derived explicitly. It can be contributed to reduce the computational cost of constructing U.

In this subsection, we will show that the congruence transformation matrix U can be constructed with the resultant biorthogonal bases V_q(L) and W_q(L) from the Lanczos-type algorithms. The theorems will be disclosed as below.

By changing the state variables: x_a(t)=(N^T+s₀M)⁻¹z_a(t). Equation (5) can be rewritten as ${M\left(N^T+s_{0}M\right)}^{-1}\frac{d{z}_a\left(t\right)}{dt}=-{N^T\left(N^T+s_{0}M\right)}^{-1}{z}_a\left(t\right)+\mathrm{Du},y_a\left(t\right)={B^T\left(N^T+s_{0}M\right)}^{-1}{z}_a\left(t\right).$ The corresponding system transfer function is Z_a(s)=(N^T+s₀M)(N^T+sM)⁻¹D=(N^T+s₀M)X_a(s) . Thus the ith-order system moments of Z_a(s) and X_a(s) about s₀ can be derived as follows: Z_a⁽ⁱ⁾(s₀)=(N^T+S₀M)X_a⁽ⁱ⁾(s₀)   (7) Through the introduction of Z_a⁽ⁱ⁾(s₀), the relationships between X_a⁽ⁱ⁾(s₀) and W_q(L) are observed in the following lemma. Lemma 1: Suppose that K(A^T,D,k)=colsp(W_q(L)). We have Z_a⁽ⁱ⁾(s₀)εcolsp(W_q(L)),   (8) X_a⁽ⁱ⁾(s₀)εcolsp((N^T+S₀M)⁻¹W_q(L),   (9) for Osisk-1. Proof: First, from (7), we have $\begin{array}{c}{Z}_a^{\left(i\right)}\left(s_0\right)={\left(N^T+s_{0}M\right)\left[-{\left(N^T+s_{0}M\right)}^{-1}M\right]}^i\left(N^T+s_{0}M\right)D\\ ={\left[-{M\left(N^T+s_{0}M\right)}^{-1}\right]}^{i}D\\ ={\left(A^T\right)}^{i}D.\end{array}$$\mathrm{Thus}$$\mathrm{colsp}\left(\left[Z_a^{\left(0\right)}\left(s_0\right)Z_a^{\left(1\right)}\left(s_0\right)\dots Z_a^{\left(k-1\right)}\left(s_0\right)\right]\right)=K\left(A^T,D,k\right)=\mathrm{colsp}\left(W_{q\left(L\right)}\right).$ Second, the above result implies that Z_a⁽ⁱ⁾(S₀)=W_q(L){circumflex over (Z)}_a⁽ⁱ⁾(s₀) for 0≦i≦k−1. Thus (9) can be proven as below: X_a⁽ⁱ⁾(s₀)=(N^T+s₀M)⁻¹W_q(L){circumflex over (Z)}_a⁽ⁱ⁾(S₀), for 0≦i≦k−1. Each X_a⁽ⁱ⁾(s₀) exists in the subspace spanned by columns of (N^T+S₀M)⁻¹W_q(L). This completes the proof. Theorem 2: Suppose that X⁽ⁱ⁾(s₀)εcolsp(V_q(L)), for 0≦i≦k−1, is a set of moments of x(s) about s₀. X_a⁽ⁱ⁾(s₀)εcolsp((N^T+s₀M)⁻¹W_q(L)) for 0≦i≦k−1. Suppose that V_q(L) and W_q(L) are biorthogonal matrices generated by the block Lanczos algorithm. Let U=[V_q(N^T+s₀M)⁻¹W_q] be the congruence transformation matrix for model-order reductions in (4), then Ŷ(s₀)=Y⁽ⁱ⁾(s₀), for 0≦i≦2k−1,   (10) Proof: From the projection theory in Section 2 and Lemma 1, we have X⁽ⁱ⁾(s₀)=U{circumflex over (X)}⁽ⁱ⁾(s₀), for 0i≦k−1, and X_a⁽ⁱ⁾(s₀)=U{circumflex over (X)}_a⁽ⁱ⁾(s₀), for 0≦i≦k−1. Then the result can be shown by Theorem 1.

Theorem 2 demonstrates that a stable reduced-order model for general RLC circuits can be generated by the Lanczos-type algorithms. Moreover, 2k-order output moments are matched by performing k iterations of the algorithm.

Theorem 3: Suppose that X⁽ⁱ⁾(s₀)εcolsp(V_q(A)) for 0≦i—k−1 is a set of moments of X_(s) about s₀. V_q(A) is theorthonormal matrix generated by the block Arnoldi algorithm. Let U=[V_q(A)SV_q(A)] be the congruence transformation matrix for model-order reductions in (4), then Ŷ⁽ⁱ⁾(s₀)=Y⁽ⁱ⁾(s₀), for 0≦i≦2k−1   (11) Proof: Let P be a q×n diagonal matrix. Since W_q(L)=P(N^T+s₀M)V_q(L) and V_q(L)=V_q(A) [references 3 and 5], it can be shown that the subspace spanned by U=[V_q(A)SV_q(A)] and U=[V_q(L)(N^T+s₀M)⁻¹W_q(L)] are the same.

A mesh twelve-line circuit, presented in FIG. 1, is studied to show the efficiency of the proposed method. The line parameters are resistance: 1.0 O/cm, capacitance: 5.0 pF/cm, inductance: 1.5 nH/cm, driver resistance 3 O, and load capacitance: 1.0 pF. Each line is divided into 50 sections. Therefore, the dimension of the MNA matrices is n=1198, m=1, and p=1, and D≠B. We set the expansion frequency s₀=1 GHz, the iteration number k=15, and use a total of 1001 frequency points distributed uniformly between the frequency range {0,15 GHz} for simulations. The frequency responses of the original model and the reduced-order model generated by the following projections: (1) W_q(L) and V_q(L); (2) V_q(A); (3) U=V_2q(A); (4) U=[V_q(L)(N^T+s₀M)⁻¹W_q(L))]I are compared in FIG. 2. Their corresponding relative error, |Y(s)−Ŷ(s)|/|Y(s)|, are illustrated in FIG. 3.

The program was implemented in Matlab 6.1 with Pentium IV 2.8 GHz CPU and 1024 MB DRAM. The time to generate each reduced-order model and the corresponding average 1-norm relative error are summarized. The average 1-norm relative error is approximated by $\left(\sum _{i=1}^{1001}Y\left(s_i\right)-\hat{Y}\left(s_i\right)/Y\left(s_i\right)\right)/1001.$ Note that only 60% work is needed to generate the similar frequency response by using the proposed method.

Although the invention has been explained in relation to its preferred embodiment, it is to be understood that many other possible modifications and variations can be made without departing from the spirit and scope of the invention as hereinafter claimed.

REFERENCE

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Claims

1. Generalizations of adjoint networks techniques for RLC interconnects model-order reductions, the congruence transformation matrix U for the one-sided projection being defined as

2. The generalizations of adjoint networks techniques for RLC interconnects model-order reductions as claimed in claim 1, the purpose of the moment matching being to construct a reduced-order system such that Ŷ(i)(s0)=Y(i)(s0), wherein A. the ith-order output moment about so is calculated as Y(i)(s0)=DTX(i)(s0); and B. the model-order reduction problem is to seek a q-order system, where q<<n, such that M^⁢ⅆx^⁡(t)ⅆt=-N^⁢x^⁡(t)+B^⁢u⁡(t),y^⁡(t)=D^T⁢x^⁡(t)where {circumflex over (x)}(t)εRq,{circumflex over (M)},{circumflex over (N)}εRqxm, and {circumflex over (D)}εRqxp, and the corresponding ith-order system moment and output moment about s0 is {circumflex over (X)}(i)(s0)=(−({circumflex over (N)}+s0{circumflex over (M)})−1{circumflex over (M)})i({circumflex over (N)}+s0{circumflex over (M)})−1{circumflex over (B)} and Ŷ(i)(s0)={circumflex over (D)}T{circumflex over (X)}(i)(s0).