Optimal Hankel-norm Approximation Research Articles

Transformation of Two-Dimensional Roesser Model Into Two-Dimensional Decoupled Model: Application to Model Reduction via Balanced Framework

Two transformations are presented in this research; the first approach takes a generic two-dimensional (2-D) framework and transforms it into a diagonalized 2-D framework without resorting to minimal rank-decomposition criteria. Afterward, it converts the diagonalized system into two subsystems (i.e., one-dimensional (1-D) cascaded systems). Moreover, the proposed transformation retains the symmetrical structure of the two decomposed subsystems. Likewise, the second transformation converts 1-D submodels to the balanced structure used for model reduction operations. The proposed model reduction approach efficiently performs the model reduction task on a given 2-D diagonalized model by utilizing optimal Hankel norm approximation. The numerical results demonstrate that the proposed transformations efficiently convert the standard 2-D framework to a diagonalized model form and develop a stable reduced-order model.

Hankel-norm approximation of large-scale descriptor systems

Hankel-norm approximation is a model reduction method for linear time-invariant systems, which provides the best approximation in the Hankel semi-norm. In this paper, the computation of the optimal Hankel-norm approximation is generalized to the case of linear time-invariant continuous-time descriptor systems. A new algebraic characterization of all-pass descriptor systems is developed and used to construct an efficient algorithm by refining the generalized balanced truncation square root method. For a wide practical usage, adaptations of the introduced algorithm towards stable computations and sparse systems are suggested, as well as an approach for a projection-free algorithm. To show the approximation behavior of the introduced method, numerical examples are presented.

Model Reduction by Balanced Truncation for Systems with Nuclear Hankel Operators

We prove the H-infinity error bounds for Lyapunov balanced truncation and for optimal Hankel norm approximation under the assumption that the Hankel operator is nuclear. This is an improvement of the result from Glover, Curtain, and Partington [SIAM J. Control Optim., 26 (1998), pp. 863--898], where additional assumptions were made. The proof is based on convergence of the Schmidt pairs of the Hankel operator in a Sobolev space. We also give an application of this convergence theory to a numerical algorithm for model reduction by balanced truncation.

Model reduction by extended minimal degree optimal Hankel norm approximation

This paper presents an extended minimal degree optimal Hankel norm approximation (MDOHNA) based order reduction algorithm using a basis-free descriptor which circumvents the requirement of computing balanced realized model for order reduction. Conjunction of system decomposition algorithm (Singh and Nagar, 2004) [21], (Kumar et al., 2012) [30,31] with MDOHNA is used as extension for order reduction of unstable systems. The developed algorithm is applicable for stable/unstable, linear time invariant, minimal/non-minimal, continuous/discrete-time systems as well. Further, effectiveness of the algorithm over existing techniques is validated with the help of a numerical example.

Interpolatory [formula omitted] model reduction

We introduce an approach to H∞ model reduction that is founded on ideas originating in realization theory, interpolatory H2-optimal model reduction, and complex Chebyshev approximation. Within this new framework, we are able to formulate a method that remains effective in large-scale settings with the main cost dominated by sparse linear solves. By employing Loewner “data-driven” partial realizations within each optimization cycle, computationally demanding H∞ norm calculations can be completely avoided. Several numerical examples illustrate that our approach will produce high fidelity reduced models consistently exhibiting better H∞ performance than those produced by balanced truncation; these models often are as good as (and occasionally better than) those models produced by optimal Hankel norm approximation. In all cases, reduced models are produced at far lower cost than is possible either with balanced truncation or with optimal Hankel norm approximation.

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.

All optimal Hankel-norm approximations and their error bounds in discrete-time†

This paper investigates a parallel problem to Glover (1984): approximate a multivariable transfer function G(z) of McMillan degree n by of McMillan degree smaller than k in discrete-time. A state-space solution is derived to the optimal Hankel-norm approximation problem, together with characterization to all optimal Hankel-norm approximations. It is shown that the error bound derived in Glover (1984) holds for discrete-time systems as well.

Order Reduction for Large-Scale Finite Element Models: A Systems Perspective

Large scale finite element models are routinely used in design and optimization for complex engineering systems. However, the high model order prevents efficient exploration of the design space. Many model reduction methods have been proposed in the literature on approximating the high dimensional model with a lower order model. These methods typically replace a fine scale model with a coarser scale model in schemes such as coarse graining, macro-modeling, domain decomposition and homogenization. This paper takes a systems perspective by stating the model reduction objective in terms of the approximation of the mapping between specified input and output variables. Methods from linear systems theory, including balance truncation and optimal Hankel norm approximation, are reviewed and compared with the standard modal truncation. For high order systems, computational load, numerical stability, and memory storage become key considerations. We discuss several computationally more attractive iterative schemes that generate the approximate gramian matrices needed in the model reduction procedure. A numerical example is also included to illustrate the model reduction algorithms discussed in the paper. We envision that these systems oriented model reduction methods complementing the existing methods to produce low order models suitable for design, optimization, and control.

Optimal Hankel norm approximation for the Pritchard-Salamon class of infinite-dimensional systems

The optimal Hankel norm approximation problem is solved under the assumptions that the system Σ(A, B, C) is an exponentially stable, regular Pritchard-Salamon infinite-dimensional system. An explicit parameterization of all solutions is obtained in terms of the system parametersA, B, C.

Optimal Hankel Norm Model Reduction By Truncation Of Trajectories

We show how optimal Hankel-norm approximations of dynamical systems allow for a straightforward interpretation in terms of system trajectories. It is shown that for discrete time single-input systems optimal reductions are obtained by cutting 'balanced trajectories', i.e., by disconnecting the past and future in the input-output pairs relating to left- and right singular vectors of the system. A self-contained proof of optimality is given, and formulas are derived in terms of Faddeev sequences. Some parallels with the literature are briefly indicated.

Robust Control of Variable Flow Processes

Models of continuous flow processes with variable parameters are studied and methods developed for their robust control. Such processes have proven invariant under variable flow and volume, if they are presented as functions of composite variables which have been introduced recently. Their finite impulse response (FIR) models are obtained e.g. by physical testing or identification, and expressed in terms of discrete, equidistant values of the appropriate variable. In the present study, such a model of a time-variable process with a long time delay is considered a high-order model; this is subsequently reduced to a low-order state-space model by means of the optimal Hankel-norm approximation. The robust feedback control is developed in terms of the same variable on the basis of the low-order model. The application of the control algorithm to the simulated process proves the approach superior to such conventional and robust control methods which have the regular time as their independent variable under unsteady conditions.

Model reduction with balanced realizations

This paper considers the use of balanced realizations in the context of plant and controller order reduction. The emphasis is placed on reduced-order models obtained by residualizing the less controllable and observable states of a balanced realization. The residualized system, unlike a truncated system or an optimal Hankel-norm approximation, maintains the steady-state gain of the system. In addition, balanced residualization is seen to give good matching at low and medium frequencies. This is favourable with regards to both plant and controller reduction. The method is demonstrated on examples and detailed comparisons made with balanced truncation and optimal Hankel-norm approximation. The importance of DC-gain matching and scaling in different set-ups is highlighted. Ways of reducing the 2-DOF (degree of freedom) H ∞ controller of Hoyle el al. (1991) are discussed and important conclusions drawn. A new proof of the error bound for balanced residualization is given in an appendix.

Formula omitted] approximation of a parabolic system regarded as a delay system

We present an approach to the problem of finding an L ∞ approximant of the infinite-dimensional system describing the diffusion of heat in a wall. We show that this system can be regarded as a delay system with Laplace variable √s. We are able to get results, established in Zwart et al. (1988), about partial fraction expansion for delay systems, achieved by some adjustment to the specificities of our particular case. The determination of an L ∞ approximant is realised in two steps, using the optimal Hankel-norm approximation.

Discrete Time H∞ Model Reduction Problem

In this paper the H∞ model reduction problem of discrete-time systems is studied. A characterization to the solution is obtained by converting the H∞ approximation problem into an optimal Hankel norm approximation problem via an imbedding process developed by the authors.

One-step extension approach to optimal Hankel-norm approximation and H/sup infinity /-optimization problems

A methodology is presented for Hankel approximation and H/sup infinity /-optimization problems that is based on a new formulation of a one-step extension problem which is solved by the Sarason interpolation theorem. The parameterization of all optimal Hankel approximants for multivariable systems is given in terms of the eigenvalue decomposition of an Hermitian matrix composed directly from the coefficients of a given transfer function matrix phi . Rather than starting with the state-space realization of phi , the authors use polynomial coefficients of phi as input data. In terms of these data, a natural basis is given for the finite dimensional Sarason model space and all computations involve only manipulations with finite matrices.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Optimal Hankel-norm approximation approach to model reduction of large-scale Markov chains

A model reduction problem of certain large-scale Markov chains under an optimal criterion for Hankel-norm approximation is discussed. The multi-dimensional Markov chain under investigation is assumed to have a finite-dimensional stationary state-transition matrix, which is first reformulated as a multi-input/multi-output (MIMO) linear time-invariant (LT1) stochastic system. Consequently, the resulting large-scale MIMO LTI stochastic system has a closed-form best approximant in the Hankel-norm from a specified class of stable lower-dimensional MIMO LTI systems.

An efficient algorithm on optimal Hankel-norm approximation for multivariable systems

This note provides a novel methodology for Hankel approximation and H/sup infinity /-optimization problems, based on a new formulation of the one-step extension problem, which is solved by the Sarason interpolation theorem. The present method does not require an initial balanced realization, and the parameterization of all optimal solutions is given in terms of the eigenvalue decomposition of a Hermitian matrix composed directly from the coefficients of the given transfer function matrix. >

IIR filter design via optimal Hankel–norm approximation

A new technique for digital filter design is presented. Based on the singular value decomposition of the Hankel matrix, balanced realisation and all-pass functions, an IIR filter is obtained via an optimal Hankel–norm approximation. The error between the optimal filter with order r and the desired filter is found to be equal to the (r + l)th singular value of the Hankel matrices. The designed low-pass filter and the differentiator are given to illustrate the proposed design algorithm.

Realisation and Approximation of Linear Infinite-Dimensional Systems with Error Bounds

The class of linear infinite-dimensional systems with finite-dimensional inputs and outputs whose impulse response h satisfies $h \in L_1 \cap L_2 (0,\infty ;\mathbb{C}^{p \times m} )$ and induces a nuclear Hankel operator is said to be of nuclear type. For this class of systems it is shown that balanced or output normal realisations always exist and their truncations converge to the original system in various topologies. Furthermore, explicit $L_\infty$ bounds on the transfer function errors, $L_1$ and $L_2$ bounds on the impulse response errors, and Hilbert-Schmidt and nuclear bounds on the Hankel operator errors are obtained. These truncations also generate an approximating sequence to the optimal Hankel-norm approximations to the original system, and various error bounds of these approximants are deduced.

Remarks on stochastic model reduction

In this note, we give a solution to the stochastic system model reduction problem in terms of the approximation of the output stochaslic process power spectrum by applying balanced realization or optimal Hankel norm approximation to the asymptotically stable and minimum phase spectral factor. We also give explicit bounds for the approximation error. The effectiveness of this method is compared to the one by Jonckheere and Helton.