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Model-Order Reduction Using Variational Balanced Truncation with Spectral Shaping Payam Heydari, Member, IEEE, and Massoud Pedram, Fellow, IEEE

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Model-Order Reduction Using Variational Balanced Truncation with Spectral Shaping Payam Heydari, Member, IEEE, and Massoud Pedram, Fellow, IEEE IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 53, NO. 4, APRIL 2006 879 Model-Order Reduction Using Variational Balanced Truncation With Spectral Shaping Payam Heydari, Member, IEEE, and Massoud Pedram, Fellow, IEEE Abstract—This paper presents a spectrally weighted balanced the reduced system. Extensions of explicit moment-matching truncation (SBT) technique for tightly coupled integrated circuit techniques have been proposed recently, to reduce the linear interconnects, when the interconnect circuit parameters change as time-varying (LTV) as well as nonlinear dynamic systems [7], a result of statistical variations in the manufacturing process. The salient features of this algorithm are the inclusion of the parameter [8]. variation in the RLCK interconnect, the guaranteed passivity of the An alternative model reduction technique is the balanced reduced transfer function, and the availability of provable spec- realization [9]–[11]. While being widely investigated in con- trally weighted error bounds for the reduced-order system. This trol system theory, balanced realization techniques have not paper shows that the variational balanced truncation technique received the same attention as Krylov-based and Pade-based produces reduced systems that accurately follow the time- and fre- quency-domain responses of the original system when variations in model reduction techniques in the context of interconnect the circuit parameters are taken into consideration. Experimental analysis, partly because these methods involve computation-in- results show that the new variational SBT attains, in average, 30% tensive algorithms [9], [11], [12]. Furthermore, it is well known more accuracy than the variational Krylov-subspace-based model- that the balancing transformation may be poorly conditioned order reduction techniques. when the system is nearly uncontrollable or unobservable [9]. Index Terms—Balanced truncation, interconnect, Lur’e equa- The most significant drawback of the conventional balanced re- tions, model-order reduction, passivity, process variation, spectral alization techniques lies in their inability to guarantee a passive shaping. reduced-order system. However, the balanced truncation tech- nique can achieve smaller reduced-order models with a better I. INTRODUCTION error control than those obtained using Krylov-subspace-based order reduction techniques. ODEL reduction techniques enable circuit designers Recently, in [13], the authors developed a guaranteed passive to capture the interconnect effects with a much shorter M balancing transformation for the model-order reduction of large computational time than that required for simulation of the full linear time-invarian (LTI) systems. circuit. On the other hand, as the minimum feature sizes shrink The balanced realization-based model reduction methods to the subquarter microns, geometrical variations in the line provide an error-bound between the transfer function of width, metal height, and dielectric thickness due to process the original and that of the reduced system.The bottleneck in variations have more pronounced effects on the reliability balanced truncation methods is the computational complexity and performance of VLSI circuits [1]. As a consequence, it for solving the Lyapunov equations. [10] uses the truncated is crucial to assess the impact of these process variations on balanced realization technique as well as the Schur decompo- model-order reduction techniques. sition to develop an efficient numerical method for the order Among various classes of model reduction techniques, reduction of a large LTI system. [11] and [12] propose efficient explicit moment-matching algorithms ( asymptotic waveform algorithms to solve the two Lyapunov equations in order to evaluation (AWE) [2], rapid interconnect circuit evaluation obtain the controllability and the observability grammians. The (RICE) [3]) and Krylov-subspace-based methods (Pact [4], algorithms are based on the alternated direction implicit (ADI) Pade approximation via the Lanczos proces (PVL) [5], pas- method that was first proposed in [14], [15]. sive reduced-order interconnect macromodeling algorithm A shortcoming of these model reduction techniques is that (PRIMA) [6]) have been most commonly employed for gen- they do not reshape the frequency spectrum to emphasize error erating the reduced-order models of the interconnects. The minimization in some frequency range of interest. Furthermore, computational complexity of these model-order reduction they do not address the numerical difficulties when the system is techniques is primarily due to the matrix-vector products. How- nearly uncontrollable or unobservable. In [16]–[18], a new nu- ever, these methods do not provide a provable error bound for merically stable, frequency-weighted balanced truncation tech- nique was presented. The proposed method gives a definite a Manuscript received June 22, 2004; revised July 24, 2005. This paper was priori bound on the weighted error, and is guaranteed to be recommended by Associate Editor M. Stan. stable even when both input and output weightings are utilized P. Heydari is with the Department of Electrical Engineering and Computer Science, University of California, Irvine, CA 92697-2625 USA (e-mail: at the same time. [email protected]). Analyzing the interconnect without taking into account the M. Pedram is with the Department of Electrical Engineering Systems, rather large variations of the interconnect geometries is not University of Southern California, Los Angeles, CA 90089 USA (e-mail: [email protected]). useful in practice. These variations are especially large in the Digital Object Identifier 10.1109/TCSI.2005.859772 inter-layer dielectric (ILD) thickness and the metal line width 1057-7122/$20.00 © 2006 IEEE 880 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 53, NO. 4, APRIL 2006 and height. These process-dependent geometrical variations the state vector constitutes node voltages across circuit capaci- have a definite impact on the total line and inter-wire coupling tances, voltage sources, currents flowing through inductors, and parasitics, which in turn results in variations in the signal delay current sources. The system matrix is readily obtained from and the coupling noise. In [19], Liu et al. studied the effect the conductance and susceptance matrices of the RLC circuit. of interconnect parameter variations on the Krylov-subspace Similar to [5], the representation of with respect to the sus- model-order reduction techniques. The paper adds the vari- ceptance, , and conductance, , matrices ability to the Krylov-subspace-based model reduction method is carried out around an expansion point , such [4], [6] by borrowing some ideas from the matrix perturbation that becomes nonsingular. The goal theory [20]. The authors allow two-dimensional variations on of the model-order reduction is to obtain a similar lower order the projection matrices. To compute the corresponding sensi- system tivities of the susceptance and conductance matrices to each dimensional variation, some sample points were picked up and (3) the dominant eigenvalues/eigenvectors were calculated. (4) The goal of the present paper is to study the effects of process where , , , . variations and spectral shaping on model-order reduction using . , the order of the reduced system, is much smaller than an extended balanced truncation technique and to propose an that of the original system. The balanced truncation technique efficient order reduction technique that includes these effects. carries out the order reduction by providing an error-bound The main contribution of this paper is the use of the spec- between the transfer functions of the original and the reduced trally weighted balanced truncation (SBT) method proposed in systems. [16]–[18] combined with a new variational balanced truncation Central to the state-space description and balanced truncation approach proposed in that accounts for process variations of dynamic systems are the controllability and observability resulting in a new variational SBT method [21]. The works of grammians. The key functions in formulating the passivity-pre- [18], [19], and [21] have laid the groundwork for interesting serving balanced truncation of linear systems are the posi- research on various extensions of balanced truncation method tive-definite controllability and observability grammians (e.g., [13]) and model-order reduction of parameterized inter- and of the system, which are obtained by solving Lur’e connects (e.g., [22]). To preserve the passivity, the proposed equations [13], i.e., algorithm employs results of [13]. As the future work, we will investigate the systematic way of determining the input/output weighting functions for the SBT method. This paper is organized as follows. Section II gives a brief overview of the balanced realization technique. Section III de- (5) scribes the frequency-weighted model reduction proposed in [16]–[18]. A discussion about the effect of process variations on interconnect modeling is provided in Section IV. Also, in Sec- (6) tion IV,the variational balanced truncation with spectral shaping is illustrated, and a theoretical comparison between the pro- The controllability grammian is a measure of how much the posed algorithm and the work presented in [19] is provided. input energy is coupled to the states of the system. The observ- Section V discusses the experimental results including the com- ability grammian is a measure of how the states
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