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A note on the stochastic realization problem
 Hemisphere Publishing Corporation
, 1976
"... Abstract. Given a mean square continuous stochastic vector process y with stationary increments and a rational spectral density such that (oo) is finite and nonsingular, consider the problem of finding all minimal (wide sense) Markov representations (stochastic realizations) of y. All such realizati ..."
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Cited by 98 (23 self)
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Abstract. Given a mean square continuous stochastic vector process y with stationary increments and a rational spectral density such that (oo) is finite and nonsingular, consider the problem of finding all minimal (wide sense) Markov representations (stochastic realizations) of y. All such realizations are characterized and classified with respect to deterministic as well as probabilistic properties. It is shown that only certain realizations (internal stochastic realizations) can be determined from the given output process y. All others (external stochastic realizations)require that the probability space be extended with an exogeneous random component. A complete characterization of the sets of internal and external stochastic realizations is provided. It is shown that the state process of any internal stochastic realization can be expressed in terms of two steadystate KalmanBucy filters, one evolving forward in time over the infinite past and one backward over the infinite future. An algorithm is presented which generates families Of external realizations defined on the same probability space and totally ordered with respect to state covariances. 1. Introduction. One
ReducedOrder Modeling Techniques Based on Krylov Subspaces and Their Use in Circuit Simulation
 Applied and Computational Control, Signals, and Circuits
, 1998
"... In recent years, reducedorder modeling techniques based on Krylovsubspace iterations, especially the Lanczos algorithm and the Arnoldi process, have become popular tools to tackle the largescale timeinvariant linear dynamical systems that arise in the simulation of electronic circuits. This pape ..."
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Cited by 53 (10 self)
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In recent years, reducedorder modeling techniques based on Krylovsubspace iterations, especially the Lanczos algorithm and the Arnoldi process, have become popular tools to tackle the largescale timeinvariant linear dynamical systems that arise in the simulation of electronic circuits. This paper reviews the main ideas of reducedorder modeling techniques based on Krylov subspaces and describes the use of reducedorder modeling in circuit simulation. 1 Introduction Krylovsubspace methods, most notably the Lanczos algorithm [81, 82] and the Arnoldi process [5], have long been recognized as powerful tools for largescale matrix computations. Matrices that occur in largescale computations usually have some special structures that allow to compute matrixvector products with such a matrix (or its transpose) much more efficiently than for a dense, unstructured matrix. The most common structure is sparsity, i.e., only few of the matrix entries are nonzero. Computing a matrixvector pr...
Algorithms for Model Reduction of Large Dynamical Systems
, 1999
"... Three algorithms for the model reduction of largescale, continuoustime, timeinvariant, linear, dynamical systems with a sparse or structured transition matrix and a small number of inputs and outputs are described. They rely on low rank approximations to the controllability and observability Gram ..."
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Cited by 43 (1 self)
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Three algorithms for the model reduction of largescale, continuoustime, timeinvariant, linear, dynamical systems with a sparse or structured transition matrix and a small number of inputs and outputs are described. They rely on low rank approximations to the controllability and observability Gramians, which can eciently be computed by ADI based iterative low rank methods. The rst two model reduction methods are closely related to the wellknown square root method and Schur method, which are balanced truncation techniques. The third method is a heuristic, balancingfree technique. The performance of the model reduction algorithms is studied in numerical experiments.
Decay Bounds for Solutions of Lyapunov Equations: The Symmetric Case
, 1999
"... We present two new bounds for the eigenvalues of the solutions to a class of continuoustime and discretetime Lyapunov equations. These bounds hold for Lyapunov equations with symmetric coefficient matrices and righthand side matrices of low rank. They reflect the fast decay of the nonincreasingly ..."
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Cited by 36 (2 self)
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We present two new bounds for the eigenvalues of the solutions to a class of continuoustime and discretetime Lyapunov equations. These bounds hold for Lyapunov equations with symmetric coefficient matrices and righthand side matrices of low rank. They reflect the fast decay of the nonincreasingly ordered eigenvalues of the solution matrix.
Model Reduction Methods Based on Krylov Subspaces
 Acta Numerica
, 2003
"... This paper reviews the main ideas of reducedorder modeling techniques based on Krylov subspaces and describes some applications of reducedorder modeling in circuit simulation ..."
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Cited by 36 (6 self)
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This paper reviews the main ideas of reducedorder modeling techniques based on Krylov subspaces and describes some applications of reducedorder modeling in circuit simulation
Guaranteed Passive Balancing Transformations for Model Order Reduction
, 2002
"... The major concerns in stateoftheart model reduction algorithms are: achieving accurate models of sufficiently small size, numerically stable and efficient generation of the models, and preservation of system properties such as passivity. Algorithms such as PRIMA generate guaranteedpassive models ..."
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Cited by 28 (4 self)
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The major concerns in stateoftheart model reduction algorithms are: achieving accurate models of sufficiently small size, numerically stable and efficient generation of the models, and preservation of system properties such as passivity. Algorithms such as PRIMA generate guaranteedpassive models, for systems with special internal structure, using numerically stable and efficient Krylovsubspace iterations. Truncated Balanced Realization (TBR) algorithms, as used to date in the design automation community, can achieve smaller models with better error control, but do not necessarily preserve passivity. In this paper we show how to construct TBRlike methods that guarantee passive reduced models and in addition are applicable to statespace systems with arbitrary internal structure.
How to Make Theoretically Passive ReducedOrder Models Passive in Practice
 In Proc. IEEE 1998 Custom Integrated Circuits Conference
, 1998
"... This paper demonstrates that, in general, implementations of circuit reduction methods can produce unstable and nonpassive models even when such outcomes are theoretically proven to be impossible. The reason for this apparent contradiction is the numeric roundoff inherent in any finiteprecision co ..."
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Cited by 21 (11 self)
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This paper demonstrates that, in general, implementations of circuit reduction methods can produce unstable and nonpassive models even when such outcomes are theoretically proven to be impossible. The reason for this apparent contradiction is the numeric roundoff inherent in any finiteprecision computer implementation. This paper introduces a new variant of the symmetric, multiport, Pad'e via Lanczos algorithm (SyMPVL) that, even in practice, is guaranteed to produce stable and passive models for all the circuits characterized by pairs of symmetric, positive semidefinite matrices. The algorithm is based by a new band Lanczos process with coupled recurrences. A number of circuit examples are used to illustrate the results. Introduction In recent years, model reduction for extracted RC(L) circuits has become an important part of the VLSI design methodology. Parasitic extraction programs typically produce large lumped RC (or even RLC) circuits as models of the structures that link the ...
Passive ReducedOrder Models for Interconnect Simulation and their Computation via KrylovSubspace Algorithms
 In Proc. 36th ACM/IEEE Design Automation Conference
, 1998
"... This paper studies a general projection technique based on block Krylov subspaces for the computation of reducedorder models of multiport RLC circuits. We show that the resulting reducedorder models are always passive, yet they still match at least half as many moments as the corresponding reduce ..."
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Cited by 19 (9 self)
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This paper studies a general projection technique based on block Krylov subspaces for the computation of reducedorder models of multiport RLC circuits. We show that the resulting reducedorder models are always passive, yet they still match at least half as many moments as the corresponding reducedorder models based on matrixPad 'e approximation. Moreover, for the special cases of RC, RL, and LC circuits, the reducedorder models obtained by projection and by matrixPad'e approximation are identical. For general RLC circuits, we show how the projection technique can easily be incorporated into the SyMPVL algorithm to obtain passive reducedorder models, in addition to the highaccuracy matrixPad'e approximation that characterizes SyMPVL, at essentially no extra computational costs. Connections between SyMPVL and the recently proposed reducedorder modeling algorithm PRIMA are also discussed. Numerical results for interconnect simulation problems are reported. 1 Introduction Electr...
A convex programming approach for generating guaranteed passive approximations to tabulated frequencydata
 IEEE Trans. on ComputerAided Design of Integrated Circuits and Systems
, 2004
"... Abstract—In this paper,we present a methodology for generating guaranteed passive timedomain models of subsystems described by tabulated frequencydomain data obtained through measurement or through physical simulation. Such descriptions are commonly used to represent on and offchip interconnect ..."
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Cited by 16 (1 self)
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Abstract—In this paper,we present a methodology for generating guaranteed passive timedomain models of subsystems described by tabulated frequencydomain data obtained through measurement or through physical simulation. Such descriptions are commonly used to represent on and offchip interconnect effects,package parasitics,and passive devices common in highfrequency integrated circuit applications. The approach,which incorporates passivity constraints via convex optimization algorithms,is guaranteed to produce a passivesystem 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 highorder models of data sets representing realistic, complicated multiinput,multioutput systems. Index Terms—Behavior modeling,convex optimization,convex programming,interconnect modeling,rational fitting,system identification. I.
A partial PadéviaLanczos method for reducedorder modeling
 Linear Algebra Appl
, 1999
"... The classical Lanczos process can be used to efficiently generate Pad'e approximants of the transfer function of a given singleinput singleoutput timeinvariant linear dynamical system. Unfortunately, in general, the resulting reducedorder models based on Pad'e approximation do not preserve the s ..."
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Cited by 16 (7 self)
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The classical Lanczos process can be used to efficiently generate Pad'e approximants of the transfer function of a given singleinput singleoutput timeinvariant linear dynamical system. Unfortunately, in general, the resulting reducedorder models based on Pad'e approximation do not preserve the stability, and possibly passivity, of the original linear dynamical system. In this paper, we describe the use of partial Pad'e approximation for reducedorder modeling. Partial Pad'e approximants have a number of prescribed poles and zeros, while the remaining degrees of freedom are used to match the Taylor expansion of the original transfer function in as many leading coefficients as possible. We present an algorithm for computing partial Pad'e approximants via suitable rank1 updates of the tridiagonal matrices generated by the Lanczos process. Numerical results for two circuit examples are reported. Key words: Lanczos algorithm; Linear dynamical system; Transfer function; Stability; Passi...