Wavelet-Based Passivity Preserving Model Order Reduction for Wideband Interconnect Characterization

Transcription

1 Wavelet-Based Passivity Preserving Model Order Reduction for Wideband Interconnect Characterization Mehboob Alam, Arthur Nieuwoudt, and Yehia Massoud Rice Automated Nanoscale Design Group Rice University, Houston, TX rand.rice.edu, Abstract Model order reduction plays a key role in determining VLSI system performance and the optimization of interconnects. In this paper, we develop an accurate and provably passive method for model order reduction using adaptive wavelet-based frequency selective projection. The waveletbased approach provides an automated means to generate low order models that are accurate in a particular range of frequencies. The results indicate that our approach provides more accurate reduced order models than the spectral zero method with uniform interpolation points and the zero-shift and multi-shift Block Arnoldi-based techniques. 1 Introduction Motivated by the expanding complexity of nanoscale integrated circuits, the model order reduction (MOR) of RLC interconnect has been the focal point of substantial research efforts [1 5]. Model order reduction techniques can be divided into two categories, Singular Value Decomposition (SVD) methods and Krylov subspace projection methods. SVD methods preserve the stability and offer a minimum error bound. However, SVD methods suffer from high computational complexity of order n 3 and do not preserve the passivity of the system [6 1]. For large scale systems, moment-matching projection methods provide a tractable alternative to SVD methods since they are computationally less expensive [11 13]. However, there is no guarantee of stability, and passivity is not preserved in many cases. Recently proposed methods based on spectral zero interpolation have demonstrated the ability to preserve stability and passivity while achieving similar computational advantages as Krylov subspace projection techniques [14 16]. However, techniques for efficiently choosing the spectral zeros for accurate wideband approximation have yet to be developed. In this paper, we develop an adaptive, passivity preserving methodology for the selection of interpolation points in spectral zero-based model order reduction. We dynamically select expansion points by applying Haar wavelets to detect complex changes in the frequency points spanned by the spectral zeros of the system and select the dominant interpolation points. The adaptive scheme provides a low order realization with optimized matching of the system response for a given range of frequencies. The preservation of passivity is guaranteed by selecting interpolation points as a subset of the system s spectral zeros. The approximate low-order model can then be directly constructed from the projection matrices derived from these interpolants. In order to demonstrate the efficiency of the approach, we apply our technique to an RLC network representing an interconnect wire. The results indicate that the wavelet-based method provides higher accuracy approximate models than techniques based on moment matching and uniform interpolation point selection. 2 Model Order Reduction of Descriptor Systems 2.1 Model Order Reduction Background An RLC representation of a linear dynamic system can be written using the following state space representation: Cẋ(t) = Gx + Bu, (1) y = L T x + Du where C R nxn, G R nxn, B R nxm and L T R pxn are the matrices defining the linear maps between inputs, outputs, and internal state variables generated using common techniques such as modified nodal analysis, n is the number of state variables, m is the number of inputs, and p is the number of outputs. The variable x represents the internal state, and u and y are the input and output vectors, respectively. In the state space representation of RLC circuits,

2 singularity can arise in the matrix C. However, it has been shown that a simple row and column permutation of C coupled with the partitioning and elimination of selected state variables will produce a non-singular system [9]. Assuming that we are interested in the admittance of the transfer function, we define A = G 1 C, K = G 1 B. (2) Taking the Laplace transform, the admittance matrix becomes H(s) =D + L T (G + sc) 1 B. (3) In terms of A and K, (3) can be expressed as H(s) =D + L T (si n A) 1 K. (4) For moment matching, the block moments are defined as H(s) =M + M 1 s + M 2 s 2 + M 3 s 3 +, (5) where M i R nxn. The block moments are computed using M i = L T A i K. The block Krylov space generated by these matrices implicitly forms the basis vectors and is defined as K(A, K) =[K, AK, A 2 K A k 1 K]. (6) The reduced system is of the form Ĝ = V GV, Ĉ = V CV, ˆB = V B, ˆL T = L T V, (7) where V is an orthogonal projector and the order of Ĝ<< G and Ĉ << C. One of the popular techniques for computing the orthogonal projectors is the Block Arnoldi method, which has been the basis for several previous techniques [12, 17]. The Block Arnoldi method iteratively produces basis vectors in the following form AV k = V k H k + r k e k, (8) where V k are the orthogonal basis vectors generated at iteration k of the algorithm. Several issues complicate the implementation of Krylov-based projection methods [2]. Numerical issues may cause the basis vectors (V k )tolose orthogonality, thereby corrupting the projectors and mitigating the effectiveness of the reduced system approximation. Additionally, the reduced system may also not be stable or passive unless certain methods are employed [12,17]. Since the Arnoldi iteration first approximates the eigenvalues associated with the high-frequency poles of the system, it may produce reduced order models that do not match the original system s response in the frequencies of interest. By applying shifted-arnoldi to provide a multi-point interpolation of the reduced system at a different frequency points, the function can be approximated across a wide-range of frequencies [16]. However, choosing these interpolation points to minimize the computational complexity may be problematic. 2.2 Passivity-Preserving Model Order Reduction It is known that the passivity of a linear time invariant system is preserved if its transfer function, H(s), is positive real. The positive realness of the system is achieved if H(s) is analytical for Re(s) > and H(s) + H(s) is nonnegative definite Hermitian in Re(s) > [14, 15]. The first condition is satisfied for real systems and the second condition implies the existence of a rational function with a stable inverse. Therefore, there exists a projector VW with VW obtained by interpolating the transfer function so that the projected system is both stable and passive. As a result, we seek interpolation points that are positive real. In the linear system defined in (4), (A, K) are reachable and (L T, A) are observable. The matrix A is also assumed to be stable with eigenvalues residing in the left-hand plane. Consequently, the system passivity is equivalent to the positive realness of the associated transfer function H(s) =D + L T (si n A) 1 K. (9) The spectral zeros of a system are defined as the zeros of the quantity H(s)+H T ( s). Furthermore, it is true that H(s)+H T ( s) = r(s)r( s) d(s)d( s), (1) where d(s) and n(s) are the denominator and numerator of the transfer function, respectively, and r(s)r( s) = n(s)d( s)+d(s)n( s). The roots of the polynomial r(s), which are defined as the stable spectral zeros, are in the left hand plane and the coefficients are real. This means that the spectral zeros cannot be purely imaginary. The critical result is that if the interpolation points are chosen to be the spectral zeros of the original system, the reduced system is both stable and passive. In our implementation, we have used Implicitly Restarted Arnoldi to solve the generalized eigenvalue problem to handle higher order systems. This makes the computational cost comparable to Krylov methods (O(kN 2 )) and significantly lower than balanced truncation methods (O(N 3 )). The eigenvalues are the spectral zeros and lie in the left half plane as H ( λ i )+H (λ i )=. In the computation of spectral zeros using our proposed method, if D + D is singular, we must solve the generalized eigenvalue problem as defined in [14]. In order to obtain an accurate approximation of the original system for the frequencies of interest, we need to adaptively select the finite spectral zeros of the system.

3 3 Wavelet Based Adaptive Interpolation Point Selection and System Reduction A transform is used to highlight certain characteristics of a given system that may not be clear in the original representation. Typical transforms include the Fourier and Wavelet transforms, which are used for frequency representation and multi-resolution analysis, respectively. In signal processing, the transform s energy compaction and decorrelation properties are used to compact the original signal s energy and to represent it using fewer coefficients. Therefore, the discrete wavelet transform (DWT) of a signal represents the correlation between the signal and the wavelet function. To obtain wavelet coefficients, the signal is processed using a filter bank, whose response is given by ψ [19]. 1D DWT computation using the direct approach is defined as ψj h (n) =ΣL 1 i= h iψj 1 h (2n i), ψ g j (n) =ΣM 1 i= g iψ g j 1 (2n i), (11) where ψ j (n) is the input N point sequence that is zero outside n N 1, h(i) and g(i) are the low and high pass filter coefficients respectively, and j is the current decomposition level. L and M are the order of the low pass and high pass filters. The 1D decomposition leads to non-uniform subband coding and representation of the system response [18]. In this paper, thresholding is used to determine the dominant spectral zero in each subband to dynamically select the dominant subset of spectral zeros for projection. Adaptive approximation uses nonlinear selection and produces a close match to the original system. This can be used to automate the process of interpolation point selection for model order reduction. In the following section, we derive and implement a new method for adaptively choosing the interpolation points in the spectral zero method using the wavelet transform. 3.1 Adaptive Interpolation Point Selection using Wavelet Transform The wavelet transform provides a fast method for multiresolution analysis and adaptation. At each level of decomposition, the signal is represented by compressed and decorrelated coefficients. The motivation is to derive and implement a method to adaptively select spectral zeros to allow efficient projection of the system in the Krylov subspace. Therefore, we develop an adaptive method for choosing the spectral zeros that accurately captures the original system s behavior across a given range of frequencies. The approximation and thresholding technique adapted in our method is based on a 1-D wavelet using the inexpensive Haar wavelet transform. The spectral zeros are selected using hard thresholding where spectral zeros less than a certain value are equated to zero. Hard thresholding allows us to choose between different wavelet bases and coefficients to facilitate more efficient selection and matching. In the implementation, the Haar discrete wavelet transform is used to decompose the given data into an orthogonal basis set. The basis functions dynamically detect sharp changes in the data. In this implementation, we apply a fast in-place discrete Haar function, which is defined on a set of N equally spaced points with values equal to the magnitude of the imaginary part of the spectral zeros. The wavelet around a point v is defined as { 1 if u r<v, ψ(i, j) (12) 1 if v r<w, where i and j are integers. The Haar function satisfies the orthonormality condition N 1 1 { ψ(i, j)ψ 1 i = j (i, j) (13) N i j n= Therefore, the sequence on the interval n<ncan be represented as sum of Haar functions F n = i,j where the coefficients are a i,j = 1 N N 1 n= a i,j ψ(i, j) (14) F n ψ(i, j). (15) The transform given by i and j is applied to the spectral zeros for adaptive selection. The procedure for computing the Haar transform consists of two steps: (1) initialize a 1D array with the spectral zeros equal to N = 2 j and (2) sweep the array at each level of wavelet decomposition by applying the Haar basis function. After level decomposition, the transformed coefficients are decomposed into subbands. The decimation by decomposing the array into subbands will facilitate the selection of the projected spectral zeros. The goal is to capture the complex behavior of the dynamic system. The detailed components of the wavelet transform contain the sharp variations and highlight the complex behavior of the signal. Depending on the order of the reduced model, the parameters of the wavelet transform as well as the threshold are selected to give the desired number of spectral zeros. The output of the adaptive wavelet method is the set of spectral zeros selected to match the critical areas of the system s response in the reduced order system. Figure 1 shows the spectral zeros (O) of an RLC model and selected spectral zeros by the proposed method (*). We

4 Imaginary 4 x Spectral Zero Selected Spectral Zero A. Assume that H ( s)+h(s) =D T + D + K T ( si A T ) 1 L T (17) +L T (si A) 1 K. Where W T O W O = D+D T and PH = H LW O. Equation (17) can be reduced to H ( s)+h(s) =W T O W O + K T [( si A T ) 1 P (18) Real Figure 1. Spectral zeros (O) of an RLC model with adaptively selected spectral zeros ( )using Haar wavelet transform. select the spectral zeros with imaginary part near the frequency of interest to match the response close to the selected frequency. Once we have chosen the spectral zeros, we use the eigenvectors corresponding to the selected spectral zeros to construct the projectors. It is important to note that the reduced order model generated by our proposed method is passive and stable. The discrete wavelet transform is only used to adaptively select spectral zeros for the projection of the system and does not transform the original system. Since the original system is reduced with spectral zeros selected using the wavelet transform, the reduced system is both passive and stable. 3.2 Preservation of Stability and Passivity In this section, we consider the positive real transfer function H(s). First, we show the necessary and sufficient conditions for the preservation of passivity and then demonstrate that the projected system is both stable and passive. H(s) is a nxn matrix that is a real rational function of the complex variable s. The positive real lemma under the assumption H( ) < can be proved with sufficient conditions for the minimal realization. Let (A, K, L T,D) be the realization of H(s) =D + L T (si n A) 1 K.ThenH(s) is positive real if and only if there exist real matrices P, M, and W O with P positive symmetric such that PF + F T P = MM T (16) PH = H LW O WO T W O = D + D T The sufficient proof for positive real lemma can be derived for spectral factorization of H ( s)+h(s). The poles of H(s) have non-positive real part and are the eigenvalues of +P (si A) 1 ]K + K T ( si A T ) 1 LW O +W T O M T (si A) 1 K. With further manipulation, and since PF + F T P = MM T, equation (18) will be reduced to following final expression: H ( s)+h(s) =[WO T + K T ( si A T ) 1 L] (19) [W O + M T (si A) 1 ]K = W T ( s)w (s). In (19), H ( s) +H(s) is the spectral factorization. A generalization of the scalar result confirms the nonnegative character of H ( s)+h(s). Since any matrix of the form W W is nonnegative, it follows that H ( s) +H(s), which proves the sufficient condition for passivity. This proves positive real lemma for the spectral factorization of the system transfer function. The passivity of the projected system can be proven in the following manner. Given a passive system (3) described by A, K, L and D, if the original system is passive, this means the transfer function is positive real. If interpolation points s j are chosen to satisfy the following property of the transfer function, H ( λ i )+H( λ i )=, (2) then the projected system is both stable and passive [15]. In our method, the adaptively selected spectral zeros are zeros of the spectral factor and they satisfy (2). Therefore, the reduced order system is both stable and passive. The spectral factors are the eigenvalues of A L T. (21) A K L T K D + D The stability and passivity can be proved by taking partial real Schur decomposition (AQ = ɛqr) whereq T = [X T,Y T,Z T ]. Therefore, the system can be block partitioned into A K A L T L T K D + D X Y Z = X Y R. (22) Suppose that R in (22) satisfies Re(λ) > for all λ σ(r), thenx T Y = Y T X is symmetric. By solving (22), we obtain following expression: R T (Y T X X T Y )+(Y T X X T Y )R =. (23)

5 2 Spectral zero uniform shift Original system Our method Spectral zero uniform shift 2 Original system Our method Amplitude (db) Original system 8 Our Method Amplitude (db) Original System Our Method Multi shifted Arnoldi PRIMA Zero shifted 1 12 Multi shifted Figure 2. Reduced order frequency response for an RLC network using uniform and wavelet selection of spectral zeros. Figure 4. Reduced order frequency response for an RLC network using our method, multishifted Arnoldi and PRIMA. 1 Our method Spectral zero uniform shift 1 Our Method Multi shifted Arnoldi PRIMA Error 1 5 Amplitude (db) Figure 3. Comparison of reduced order system absolute error response for the RLC network example Figure 5. Absolute error response for the RLC network example. The only unique solution to (23) is Y T X X T Y =. Therefore, our initial assumption and the positive realness proves that the system is both passive and stable. 4 Results In order to demonstrate the accuracy and efficiency of the dynamic wavelet based passivity preserving model order reduction method, we approximate the system response of a simple RLC network representing an interconnect wire. The results are compared for the following two cases: (1) uniform versus adaptive wavelet selection of spectral zeros and (2) adaptive wavelet selection versus other MOR methods. The circuit used to model the structure is an RLC network. The model takes into account the resistance, induc- tance, self-capacitance, coupling-capacitance and mutualinductance between segments in the interconnect wire. The complexity of the initial system is n =79. The results of this comparison can be seen in Figure 2. As can be seen, an order 16 approximation using the uniform spectral zero slection method does not accurately model the system. The approximate system response misses the resonant peaks of the original system. In contrast, the proposed technique with a low order of 14 closely matches the frequency response. The absolute error plot in Figure 3 shows that our method has significantly less error as compared to the uniform selection approach. Next, we compare accuracy of proposed method is compared with PRIMA (zero-shifted) Arnoldi [12] and multishifted Arnoldi [21]. The results of this comparison can be seen in Figure 4. An order 14 approximation gener-

6 ated using multi-shifted Arnoldi and PRIMA does not accurately approximate the system. The two methods significantly miss the last two peaks of the system response. However, the proposed technique almost perfectly matches the response over the frequencies of interest. The absolute error plot shown by Figure 5 further highlights that our method has significantly less error as compared to multi-shifted and PRIMA. 5 Conclusion In this paper, we developed an accurate and provably passive method for model order reduction using adaptive wavelet-based frequency selective projection. We dynamically select interpolation points among the spectral-zeros of a system by applying an inexpensive Haar wavelet and performing efficient spectral zero selection. The wavelet-based approach provides an automated means for generating low order models that are accurate in a particular range of frequencies. 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