cross-Gramian Matrix Research Articles

Fast Low‐Rank Empirical Cross Gramians

AbstractThe cross Gramian matrix encodes the input‐output coherence of linear control systems and is used in projection‐based model reduction. The empirical cross Gramian is a data‐driven variant of the cross Gramian which also extends to nonlinear systems. A drawback of the empirical cross Gramian for large‐scale systems is its full order and dense structure; yet, it may be computed column‐wise. Using the hierarchical approximate proper orthogonal decomposition (HAPOD), this partial computability can be exploited to obtain a truncated projection for model order reduction without assembling a full cross Gramian. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Note on Compressed Sensing of Structured Sparse Wavelet Coefficients From Subsampled Fourier Measurements

We consider signal recovery from Fourier measurements using compressed sensing (CS) with wavelets. For discrete signals with structured sparse Haar wavelet coefficients, we give the first proof of near-optimal recovery from discrete Fourier samples taken according to an appropriate variable density sampling scheme. Crucially, in taking into account such structured sparsity—known as sparsity in levels—as opposed to just sparsity, this result yields recovery guarantees that agree with the empirically observed recovery properties of CS in this setting. This result complements a recent theorem in Adcock et al. [Breaking the coherence barrier: A new theory for compressed sensing, arXiv preprint arXiv: 1302.0561, 2014.], which addressed the case of continuous time signals. Moreover, we provide a significantly shorter and more expositional argument, which clearly illustrates the key factors governing recovery in this setting: namely the division of frequency space into dyadic bands corresponding to wavelet scales, the near-block diagonality of the Fourier/wavelet cross-Gramian matrix, and the structured sparsity of wavelet coefficients.

A note on the cross Gramian for non-symmetric systems

ABSTRACTThe cross Gramian matrix is a tool for model reduction and system identification, but it is only applicable to square control systems. For symmetric systems, the cross Gramian possesses a useful relation to the system's associated Hankel singular values. Yet, many real-life models are neither square nor symmetric. In this work, concepts from decentralized control are used to approximate a cross Gramian for non-symmetric and non-square systems. To illustrate this new non-symmetric cross Gramian, it is applied in the context of model order reduction.

The Empirical Cross Gramian for Parametrized Nonlinear Systems

The cross gramian matrix can be used for model order reduction as well as system identification of linear control systems, which are frequently used in the sciences. The empirical cross gramian is solely computed from trajectories and hence extends beyond linear state-space systems to nonlinear systems. In this work the applicability of the empirical cross gramian also for parametrized systems is demonstrated and assessed using a nonlinear benchmark problem.

A New Approach to Compute the Cross-Gramian Matrix and its Application in Input-Output Pairing of Linear Multivariable Plants

A New Approach to Compute the Cross-Gramian Matrix and its Application in Input-Output Pairing of Linear Multivariable Plants

Frequency-domain model reduction approach to design IIR digital filters using orthonormal bases

A new and numerically efficient technique for the design of reduced-order IIR digital filters is presented. This technique is based on computing the cross-gramian matrix from the state–space representation of the full-order filter in an arbitrary frequency range of operation. The modified cross-gramian is derived in the frequency domain and computed by solving only one Lyapunov equation. This technique does not require a minimal system to start with, and the reduced order can be obtained without calculating the conventional balancing transformation. Instead, orthonormal bases are computed to find the left and right eigenspaces associated with the large eigenvalues of the modified cross-gramian matrix. These eigenspaces are used to convert the high-order to a reduced-order models. The phase linearity is also discussed and it is shown that the proposed method can be used to transform a linear-phase FIR filter to a reduced-order IIR filter and the phase linearity is preserved over the interested frequency band, which is commonly be the passband. A comparison between the conventional time-domain methods such as balanced truncation (BT) and optimal Hankel-norm approximation (OHA); and the proposed frequency-domain model reduction technique is presented through several filters design examples to illustrate the effectiveness of the proposed technique.

Design of 2-D Linear Phase Digital Filters Using Schur Decomposition and Symmetries

In this paper we present a new and numerically efficient technique for designing 2-D linear phase octagonally symmetric digital filters using Schur decomposition method (SDM) and the diagonal symmetry of the 2-D impulse response specifications. This technique is based on two steps. First, the 2-D impulse response matrix is decomposed into a parallel realization of k sections, each comprising two cascaded linear phase SISO 1-D FIR digital filters. It is shown that using the symmetry property of the 2-D impulse response matrix and the fact that the left and right eigenspaces obtained by SDM are transpose of each other, the design problem of two 1-D digital filters is reduced to the design problem of only one 1-D digital filter in each section. In the second step, the 2-D linear phase FIR filter is converted to 2-D IIR filter. This is done by converting the constituent N-order 1-D FIR filters to an n-order IIR filters, where n < N using the given model reduction algorithm. The reduced order IIR filters are obtained without computing the balancing transformation, but by finding the orthonormal eigenspaces associated with the largest eigenvalues of the cross-Gramian matrix W CO. Two design examples are given to illustrate the advantages of the proposed technique.

Design of linear-phase IIR digital filters using singular perturbational model reduction

A new technique for the design of a passband linear-phase infinite impulse response (IIR) digital filter is presented. This technique is based on computing orthonormal bases, which span the left and right invariant subspaces associated with the large and small eigenvalues of the cross-Gramian matrix W/sub c0/, which tends to be symmetric and easily constructed. The orthonormal bases are computed by using the ordered real Schur form decomposition, which has been proved to be robust and numerically stable. The proposed technique is illustrated by several examples.

Model order reduction via real Schur-form decomposition

Abstract A new algorithm is presented for computing Moore's reduced-order transfer-function matrix without calculating the balancing transformation, which tends to be ill-conditioned, especially when the original system is non-minimal or when it has very nearly uncontrollable or unobservable modes. The algorithm is based on finding the eigenspaces associated with large eigenvalues of the cross-gramian matrix Wco using the real Schur-form decomposition. The algorithm does not require a minimal model to start with. The state-space realization obtained by this method is related to the balanced realization by a non-singular matrix. An example is presented to illustrate the proposed algorithm

On the cross-Gramian for symmetric MIMO systems

Recently, Laub et al. [1] and Fernando and Nicholson [2] extended the definition of the cross-Gramian matrix W_{co} to include symmetric MIMO systems where the matrix W_{co} is defined as an integral. In this correspondence, we define this matrix as a solution of a Lyapunov equation and prove the fundamental identity W_{co}^{2} = W_{c}W_{o} where W_{c} and W_{o} are the controllability and observability Gramians defined via Lyapunov equations. This definition of W_{co} complements the integral definition and has some advantages from theoretical and numerical points of view.

On a fundamental property of the cross- Gramian matrix

The cross-Gramian matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W_{c0} (T)</tex> for linear single-input single-output (SISO) dynamical systems has been related to the controllability and observability Gramians via <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W^{2}_{c0}(T)= W_{c}(T)W_{0}(T)</tex> for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T= \infty</tex> . The result is now proved for any arbitrary time interval <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</tex> , and is extended to symmetric multivariable systems which have symmetric transfer function matrices.

Reachability, observability, and minimality of MIMO systems

Reachability and observability of time-variant systems with equal number of inputs and outputs are investigated using the cross-Gramian matrix. A sufficient condition for minimality of time-invariant systems is also proposed.

On the structure of balanced and other principal representations of SISO systems

A new matrix W <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">co</inf> which can be considered as a cross-Gramian matrix which contains information about both controllability and observability is defined for single-input, single-output, linear systems. Using this matrix, the structural properties of linear systems are studied in the context of principal component analysis. The matrix W <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">co</inf> can be used in obtaining balanced and other principal representations without computation of the controllability and the observability Gramians. The importance of this matrix in model-order reduction is highlighted.

On the Cauchy index of linear systems

The Cauchy index for linear SISO systems is given by the signature of the cross-Gramian matrix W <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">co</inf> defined by Fernando and Nicholson [1]. This index is useful in the qualitative understanding of systems, and the importance of this index in internally balanced model reduction is illustrated.