Hankel Norm Approximation Research Articles

State-Space and Distributed Lag Modelling of Dynamic Economic Processes Based on Singular Value Decompositions (With an Application to the Dutch Economy)

In this paper concepts and techniques from system theory will be used to obtain state-space (Markovian) models of dynamic economic processes instead of the usual ARMA-models. In this respect the concept of state will be reviewed and also Hankel factorisation, Hankel norm approximation and balanced realizations for stochastic models. The suggested procedures, which includes singular value decompositions, steady-state Kalman filtering and prediction-error estimation will be applied to some behavioral equations of a small macro-economic model representing the Dutch economy.

Weighted Hankel-norm approximation: Calculation of bounds

Frequency-domain bounds are computed for the approximation error involved in the optimum Hankel-norm approximation procedure. The bounds differ from those arising in the unweighted optimal Hankel-norm approximation procedure by a multiplicative factor depending solely on the weighting function. This factor becomes unity when the weighting function is constant.

Optimal Hankel-norm approximation of stable systems with first-order stable weighting functions

Given a stable rational transfer function G( s) and weighting function W( s), the problem of finding G ̂ (s) of MacMillan degree k so as to minimise ∥W(G - G ̂ ∥ H is considered. This problem is solved for W( s) =( s- β)/( s- α) with no assumptions on the signs of α and β. This gives rise to approximations where ∥W(G - G ̂ )∥ ∞ can be accurately bounded from above and below in terms of the Hankel singular values of WG (when α, β > 0).

Controller design for distributed systems based on Hankel-norm approximations

A procedure for obtaining a finite-dimensional reduced-order model for a class of distributed systems is developed. It is based on a combination of a modal approximation and an optimal Hankel-norm approximation and the approximation error can be accurately predicted. The reduced-order model can be used to design a precompensator for the original system as its performance and robust stability characteristics can be deduced from the reduced-order model and finitely many modal parameters. The procedure is applied to a parabolic system.

Power spectrum reduction by optimal Hankel norm approximation of the phase of the outer spectral factor

A new approach to stochastic processes power spectrum reduction, based on Akaike's canonical correlation between the process past and future, is developed. As a first result, the correlation between the canonical past and the canonical future is given its precise functional-analytic interpretation as the Hankel operator generated lay the all-pass function characterizing the phase of the process outer spectral factor. The canonical correlation coefficients turn out to he nothing else but the singular values of this Hankel operator. Next, the all-pass function characterizing the phase of the outer spectral factor is reduced using the optimal Hankel-norm procedure of Adamjan, Arov, and Krein. The reduction of the all-pass phase function is implemented with the safeguards necessary for the magnitude information to be carried all the way through. This involves Bode's relations between the gain and the phase of an outer function, Nehari's extension, and the winding number of the all-pass phase function which is itself shown to be related to the definition of the past and the future. Finally, the net result is a reduced-order, outer spectral factor that matches the high-order spectral factor, fairly well.

Spectral factor reduction by phase matching: the continuous-time single-input single-output case†

A procedure for reducing the order of a random process spectral factor, without altering its stable minimum-phase property, is presented. It is based upon the balanced or the Hankel-norm approximation of a certain all-pass function characterizing the phase of the process spectral factor. The motivation for this particular methodology for model order-reduction stems from the canonical correlation analysis of Akaike (1975). Further, Bode's gain-phase relation and Nehari's extension used in conjunction provide a functional analytic justification of the reduction procedure. The net result of this procedure is a stable, minimum-phase reduced-order spectral factor that matches the original one fairly well. The paper draws heavily on the early discrete-time paper of Jonckheere and Helton (1984). In the paper we treat the continuous-time case and extend the results of Jonckheere and Helton in several directions.

On the Hankel-norm approximation of linear stochastic systems

We derive ordering and interlocking properties for the singular values of the block-Hankel matrices corresponding to different spectral factors of a given spectral density matrix. The results are then applied to Hankel-norm approximation of SISO stochastic systems. In particular we show that the minimum phase model may be approximated in Hankel norm by systems of a certain prescribed dimension with better accuracy than any other model with the same output process. We also provide upper bounds on the gain in accuracy obtained by choosing the minimum phase model over some other model.

L<sup>∞</sup>optimization and Hankel approximation

An important L^{\infty} -optimization problem arising in feedback system design is shown to be equivalent to an optimal Hankel-norm approximation problem. This enables the problem to be solved using state-space methods based on balanced realizations.

Frequency-weighted optimal Hankel-norm approximation of stable transfer functions

This paper shows how to introduce arbitrary frequency weighting into the optimal Hankel-norm approximation problem for scalar, finite-dimensional, linear, time-invariant systems. The procedure has applications to the closed-form design of reduced-order controllers.

Canonical correlations of past inputs and future outputs for linear stochastic systems

The dependence of future outputs on past inputs in a stochastic system is investigated by means of the canonical correlation analysis. In particular a classification of minimal stochastic realizations of a given process with respect to the information transmitted from past inputs to future outputs is presented. The results are then rephrased in terms of certain block-Hankel matrices and their relevance for the Hankel-norm approximation of stochastic systems is discussed.

All optimal Hankel-norm approximations of linear multivariable systems and theirL,∞-error bounds†

The problem of approximating a multivariable transfer function G(s) of McMillan degree n, by Ĝ(s) of McMillan degree k is considered. A complete characterization of all approximations that minimize the Hankel-norm is derived. The solution involves a characterization of all rational functions Ĝ(s) + F(s) that minimize where Ĝ(s) has McMillan degree k, and F(s) is anticavisal. The solution to the latter problem is via results on balanced realizations, all-pass functions and the inertia of matrices, all in terms of the solutions to Lyapunov equations. It is then shown that where σ k+1(G(s)) is the (k+l)st Hankel singular value of G(s) and for one class of optimal Hankel-norm approximations. The method is not computationally demanding and is applied to a 12-state model.

Optimal Hankel-norm approximation of continuous-time linear systems

A problem on optimal approximation of continuous-time linear systems is studied. The performance measure (error) is chosen to be the spectral norm of the difference between the Hankel operators associated with the original system and the approximant. It is shown that the Hankel operators associated with continuous-time systems and the Hankel matrices associated with discrete-time systems are related by an interesting correspondence property via bilinear transforms. This fact is then used to derive the continuous-time results (theory and algorithms) from the established discrete-time ones. Some simple examples are presented.

A two-variable approach to the model reduction problem with Hankel norm criterion

A two-variable approach to the model reduction problem with Hankel norm criterion is discussed. The problem is proved to be reducible to obtaining a two-variable all-pass rational function, interpolating a set of parametric values at specified points inside the unit circle. A polynomial formulation and the properties of the optimal Hankel norm approximations are then shown to result directly from the general form of the solution of the interpolation problem considered.

A state-space formulation for optimal Hankel-norm approximations

The optimal Hankel-norm approximation problem studied in [1] is reformulated in a state-space setting. The crucial extension theorem is reestablished in this framework. The minimal degree optimal approximation is then derived in terms of state-space parameters