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Passivity enforcement using incomplete complex frequency hopping
Passivity enforcement methods entail the solution of optimization problems characterized by significant computational complexity. Optimal solutions require the use of positive-real lemma constraints which do not rely on linearization but scale rapidly in size owing to auxiliary slack variables. This paper presents how a chordal sparsity pattern can be used to solve this passivity enforcement problem more efficiently. The proposed algorithm uses graph theory and convex optimization to achieve a suboptimal solution that reduces the computation time required for the solution of the positive-real lemma passivity constraints by constraining the auxiliary Lyapunov variable to have a banded pattern. We provide numerical results using actual power system equipment to show that the proposed method gives faster suboptimal solutions.
This paper presents a novel approach to passivity enforcement of multiport frequency-dependent rational models. The strategy employs graph theory and convex optimization to extend the positive-definite matrix completion problem to passive system completions. The main distinguishing feature of the proposed technique relative to those in the literature is that passivity compensations are only applied to those parameters contributing to passivity violations, in a completion manner. Numerical experiments using actual transformer data reveals that the proposed method results in models with increased accuracy.
This paper addresses the problem of obtaining optimal black-box models of n-terminal power transformers for transient studies. We use actual transformer data to demonstrate that a transformer model endowed with both passivity and reciprocity can be optimally achieved only if the corresponding constraints are formulated and simultaneously included in the parameter estimation problem as opposed to enforcing them independently as two distinct post-processing model adjustments.
With the rapid developments in very large-scale integration (VLSI)
technology, design and computer-aided design (CAD) techniques, at both
the chip and package level, the operating frequencies are fast reaching
the vicinity of gigahertz and switching times are getting to the
subnanosecond levels. The ever increasing quest for high-speed
applications is placing higher demands on interconnect performance and
highlighted the previously negligible effects of interconnects such as
ringing, signal delay, distortion, reflections, and crosstalk. In this
review paper various high-speed interconnect effects are briefly
discussed. In addition, recent advances in transmission line
macromodeling techniques are presented. Also, simulation of high-speed
interconnects using model-reduction-based algorithms is discussed in
detail
In this paper, we present a methodology for generating guaranteed passive time-domain models of subsystems described by tabulated frequency-domain data obtained through measurement or through physical simulation. Such descriptions are commonly used to represent on- and off-chip interconnect effects, package parasitics, and passive devices common in high-frequency integrated circuit applications. The approach, which incorporates passivity constraints via convex optimization algorithms, is guaranteed to produce a passive-system model that is optimal in the sense of having minimum error in the frequency band of interest over all models with a prescribed set of system poles. We demonstrate that this algorithm is computationally practical for generating accurate high-order models of data sets representing realistic, complicated multiinput, multioutput systems.
Loewner Matrix interpolation based macromodeling methods can generate an accurate macromodel for large and complex structures. However, there is less flexibility in the choice of the macromodel order due to the need to ensure passivity. In this paper, an algorithm is proposed to obtain a reduced order model based on Loewner Matrix interpolation while still maintaining the passivity of the macromodel. This is achieved by first relaxing the passivity requirement during the order selection process, then a new passivity enforcement scheme with significantly improved speed and convergence properties is applied. The proposed approach results in a significantly reduced macromodel size while maintaining accuracy and passivity conditions. The
efficiency of the new approach is shown in the examples.
This paper presents a class of nonsmooth convex optimization methods for the passivity enforcement of reduced-order macromodels of electrical interconnects, packages, and linear passive devices. Model passivity can be lost during model extraction or identification from numerical field solutions or direct measurements. Nonpassive models may cause instabilities in transient system-level simulation, therefore a suitable postprocessing is necessary in order to eliminate any passivity violations. Different from leading numerical schemes on the subject, passivity enforcement is formulated here as a direct frequency-domain H∞ norm minimization through perturbation of the model state-space parameters. Since the dependence of this norm on the parameters is nonsmooth, but continuous and convex, we resort to the use of subdifferentials and subgradients, which are used to devise two different algorithms. We provide a theoretical proof of the global optimality for the solution computed via both schemes. Numerical results confirm that these algorithms achieve the global optimum in a finite number of iterations within a prescribed accuracy level.
This is a revised edition of a book which appeared close to two decades ago. Someone scrutinizing how the field has evolved in these two decades will make two interesting observations. On the one hand the observer will be struck by the staggering number of new developments in numerical linear algebra during this period. The field has evolved in all directions: theory, algorithms, software, and novel applications. Two decades ago there was essentially no publically available software for large eigenvalue problems. Today one has a flurry to choose from, and the activity in software development does not seem to be abating. A number of new algorithms appeared in this period as well. I can mention at the outset the Jacobi-Davidson algorithm and the idea of implicit restarts, both discussed in this book, but there are a few others. The most interesting development to the numerical analyst may be the expansion of the realm of eigenvalue techniques into newer and more challenging applications. Or perhaps, the more correct observation is that these applications were always there, but they were not as widely appreciated or understood by numerical analysts, or were not fully developed due to lack of software.
We establish a correspondence between the singular values of a transfer matrix evaluated along the imaginary axis and the
imaginary eigenvalues of a related Hamiltonian matrix. We give a simple linear algebraic proof, and also a more intuitive
explanation based on a certain indefinite quadratic optimal control problem. This result yields a simple bisection algorithm
to compute the H∞ norm of a transfer matrix. The bisection method is far more efficient than algorithms which involve a search over frequencies,
and the usual problems associated with such methods (such as determining how fine the search should be) do not arise. The
method is readily extended to compute other quantities of system-theoretic interest, for instance, the minimum dissipation
of a transfer matrix. A variation of the method can be used to solve the H∞ Armijo line-search problem with no more computation than is required to compute a single H∞ norm.
Rational macromodels must be passive in order to guarantee a stable simulation. This paper introduces a fast approach for enforcing passivity for S -parameter based pole-residue models, using a similar method previously introduced for Y -parameter models. The approach is based on perturbing the elements of the residue matrices with the perturbed residue matrix eigenvalues as free variables. This gives large savings for the CPU time and memory requirements. The implementation does not require sparse computations. Combining the passivity enforcement step with iterations and fast passivity assessment via a half-size test matrix gives a globally passive model. Error control strategies are implemented via least squares weighting. The approach is demonstrated for a two-port microwave filter, a four-port interconnect, a 48-port low-order package model, and a 28-port high-order package model.
This paper presents a new technique for the passivity enforcement of linear time-invariant multiport systems in state-space form. This technique is based on a study of the spectral properties of related Hamiltonian matrices. The formulation is applicable in case the system input-output transfer function is in admittance, impedance, hybrid, or scattering form. A standard test for passivity is first performed by checking the existence of imaginary eigenvalues of the associated Hamiltonian matrix. In the presence of imaginary eigenvalues the system is not passive. In such a case, a new result based on first-order perturbation theory is presented for the precise characterization of the frequency bands where passivity violations occur. This characterization is then used for the design of an iterative perturbation scheme of the state matrices, aimed at the displacement of the imaginary eigenvalues of the Hamiltonian matrix. The result is an effective algorithm leading to the compensation of the passivity violations. This procedure is very efficient when the passivity violations are small, so that first-order perturbation is applicable. Several examples illustrate and validate the procedure.
This paper deals with the characterization and enforcement of passivity for linear lumped interconnect macromodels. An adaptive accuracy-controlled frequency sampling process is employed to identify a set of frequency bands where the macromodel is locally passive. These results are employed as a preliminary step, enabling the fast computation of imaginary eigenvalues of the Hamiltonian matrix associated to the macromodel. Then, iterative perturbation is employed to remove these eigenvalues from the imaginary axis and to achieve global passivity. The resulting scheme is highly optimized for macromodels having large dynamic order and with a sparse structure. Significant speedup factors up to two orders of magnitude are achieved with respect to a standard implementation of the same passivity compensation scheme based on a full eigensolver
This paper addresses some issues related to the passivity of interconnect macromodels computed from measured or simulated port responses. The generation of such macromodels is usually performed via suitable least squares fitting algorithms. When the number of ports and the dynamic order of the macromodel is large, the inclusion of passivity constraints in the fitting process is cumbersome and results in excessive computational and storage requirements. Therefore, we consider in this work a post-processing approach for passivity enforcement, aimed at the detection and compensation of passivity violations without compromising the model accuracy. Two complementary issues are addressed. First, we consider the enforcement of asymptotic passivity at high frequencies based on the perturbation of the direct coupling term in the transfer matrix. We show how potential problems may arise when off-band poles are present in the model. Second, the enforcement of uniform passivity throughout the entire frequency axis is performed via an iterative perturbation scheme on the purely imaginary eigenvalues of associated Hamiltonian matrices. A special formulation of this spectral perturbation using possibly large but sparse matrices allows the passivity compensation to be performed at a cost which scales only linearly with the order of the system. This formulation involves a restarted Arnoldi iteration combined with a complex frequency hopping algorithm for the selective computation of the imaginary eigenvalues to be perturbed. Some examples of interconnect models are used to illustrate the performance of the proposed techniques.
Numerical methods for large eigenvalue problems. SIAM
- Y Saad