Approximation Of Linear Systems Research Articles

Space–bandwidth-efficient realizations of linear systems

One can obtain either exact realizations or useful approximations of linear systems or matrix-vector products that arise in many different applications by implementing them in the form of multistage or multichannel fractional Fourier-domain filters, resulting in space-bandwidth-efficient systems with acceptable decreases in accuracy. Varying the number and the configuration of filters enables one to trade off between accuracy and efficiency in a flexible manner. The proposed scheme constitutes a systematic way of exploiting the regularity or structure of a given linear system or matrix, even when that structure is not readily apparent.

Parametric and nonparametric identification of linear systems in the presence of nonlinear distortions-a frequency domain approach

This paper studies the asymptotic behavior of nonparametric and parametric frequency domain identification methods to model linear dynamic systems in the presence of nonlinear distortions under some general conditions for random multisine excitations. In the first part, a related linear dynamic system (RLDS) approximation to the nonlinear system (NLS) is defined, and it is shown that the differences between the NLS and the RLDS can be modeled as stochastic variables with known properties. In the second part a parametric model for the RLDS is identified. Convergence in probability of this model to the RLDS is proven. A function of dependency is defined to detect and separate the presence of unmodeled dynamics and nonlinear distortions and to bound the bias error on the transfer function estimate.

High-order approximations of linear control systems via Runge-Kutta schemes

It is well known that classical Runge-Kutta approximations for dynamical systems do not converge with high order when the control is not smooth with respect to time. We consider here a generalization of RK schemes for linear systems which preserves its order with measurable controls, and obtain as consequence a result of high-order approximation for the reachable set.

On application of the describing function method for optimization of feedback control systems

Two H∞ optimization problems of a nonlinear tracking control system: the problem of a nonlinear controller and the problem of a nonlinear plant are considered in the paper. The describing function method is used for linearization of a feedback control system. Theorems, which enable one to replace the optimization of a nonlinear system by the optimization of an approximate linear system are proven in the paper. Methods of H∞ optimization are used to find the structure of an optimal controller of the approximate system. © 1997 by John Wiley & Sons, Ltd.

Passive Network Approximation of Linear Systems: Application to Fractional Power Pole Models

Continued fraction expansions of Padé approximants and their RC ladder realisations are investigated, in the case. of general linear time invariant operator with fading memory, represented with convolution operators. Stleltjes integrals and power series are then introduced to study the passivity of the network approximations. The case of fractional power pole model is then analysed to illustrate the approach.

Consumer demand in Japan: An analysis using the Deaton-Muellbauer system

The Linear Approximate Deaton-Muellbauer system is applied to Japanese time series family budget data to analyze the consumer demand for 16 commodity categories. A good deal of similarities in the results between the present and previous studies are observed. Changes in demands per annum are attributed to the effects of changes in real income, relative prices and a time trend, revealing that the income effect is substantially greater in terms of an annual rate of change in quantity purchased than the substitution effect for many commodities. Economic growth policy is very important and statistically significant for the expansion of domestic demand and imports. Further commodity-specific research is required.

Model Reduction of Linear Discrete-Time Periodic Systems Using Hankel-Norm Approximations

In this paper we study model reduction of discrete-time periodic systems using Hankel-norm approximation. We review some ancillary results from the linear discrete time-invariant and discrete-time periodic systems. Then, we introduce a direct approach of Hankel-norm approximations for linear discrete time-invariant systems. Further, we extend the results to the linear discrete-time periodic systems. Simulation results show that the Hankel-norm approximation for discrete-time periodic systems has the desired error bounds. The balanced periodic system is obtained by solving the Lyapunov equations using recently proposed methods.

On approximation of stable linear dynamical systems using Laguerre and Kautz functions

Approximation of stable linear dynamical systems by means of so-called Laguerre and Kautz functions, which are the Laplace transforms of a class of orthonormal exponentials, is studied. Since the impulse response of a stable finite dimensional linear dynamical system can be represented by a sum of exponentials (times polynomials), it seems reasonable to use basis functions of the same type. Assuming that the transfer function of a system is bounded and analytic outside a given disc, it is shown that Laguerre basis functions are optimal in a mini-max sense. This result is extended to the “two-parameter” Kautz functions which can have complex poles, while the poles of Laguerre functions are restricted to the real axis. By conformai mapping techniques the “two-parameter” Kautz approximation problem is recast as two Laguerre approximation problems. Thus, the well-developed theory of Laguerre functions can be applied to analyze Kautz approximations. Unilateral shifts are used to further develop the connection between Laguerre functions and Kautz functions. Results on þ 2 and þ ∞ approximation using Kautz models are given. Furthermore, the weighted L 2 Kautz approximation problem is shown to be equivalent to solving a block Toeplitz matrix equation.

On the bilinear transformation of LQG-balanced realizations

This contribution shows that some useful properties of LQG-balanced realizations of continuous-time systems are preserved by applying bilinear transformations. This fact suggests that LQG-balancing of discrete systems and its approximation can be performed by using the typical schemes successfully adopted for continuous-time systems. In the paper the approximation of discrete linear time-invariant systems by using the singular perturbation approximation approach for LQG-balanced realizations is analyzed, showing its usefulness in approximating discrete models, including unstable ones, for closed-loop aims.

The local bifurcation of Ramsey equilibrium

The elasticity of substitution has been proposed as one factor in the generation of aggregate fluctuations in dynamic models with incomplete markets. We study the existence of periodic solutions in a one-sector neoclassical capital accumulation model under borrowing constraints with infinitely-lived heterogeneous agents. A dynamical system representing an equilibrium profile with only the most patient agent holding capital is analyzed when capital income is not an increasing function of total capital. Conditions for the linear approximation system at a steady state to have an eigenvalue of — 1 are found. A one-parameter family of maps based on a perturbation of the production function is introduced and the dynamical system is reduced to 1 dimension via an application of a center manifold theorem. Conditions for a stable flip bifurcation are shown to hold at the steady state.

A theoretical analysis of the performance of code division multiple access communications over multimode optical fiber channels-part I: transmission and detection

This paper presents results of a theoretical investigation to evaluate the performance of code division multiple access communications over multimode optical fiber channels in an asynchronous, multiuser communication network environment. The system is evaluated using Gold sequences for spectral spreading of the baseband signal from each user employing direct-sequence biphase shift keying and intensity modulation techniques. The transmission channel model employed is a lossless linear system approximation of the field transfer function for the /spl alpha/-profile multimode optical fiber. Due to channel model complexity, a correlation receiver model employing a suboptimal receive filter was used in calculating the peak output signal at the ith receiver. The performance measures for the system, i.e., signal-to-noise ratio and bit error probability for the ith receiver, are derived as functions of channel characteristics, spectral spreading, number of active users, and the hit energy to noise (white) spectral density ratio. >

Reduction of linear continuous-time multivariable systems by matching first- and second-order information

This paper considers the approximation of stable continuous-time multivariable linear systems from a finite number of Markov parameters and second-order information indexes. It is shown that, by properly choosing these indexes, it is possible to uniquely identify an input-output model of given order from an equal number of first- and second-order data.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Multiplicative Hankel norm approximation of linear multivariable systems

Abstract In control system design, specifications are commonly given in terms of log-magnitude quantities (i.e. decibels). This description induces a multiplicative or relative error criterion for plant model reduction. Techniques from the theory of additive Hankel norm approximation and spectral factorization are synthesized to produce a multiplicative approximant of a possibly unstable or non-square plant. It is shown that the reduced-order model preserves the unstable poles and right half-plane zeros of the original system. Explicit state-space formulae are provided for the construction of the reduced-order model and its error properties are discussed. The method is illustrated by a numerical example.

A contribution to matrix equations arising in system theory

AbstractA useful relationship between solutions is derived for a class of matrix equations arising in the system theory, particularly in the context of the orthogonal approximation of linear dynamic systems. The computational aspects of the result are discussed and illustrated on an example of practical importance.

Approximation in Control of Thermoelastic Systems

This paper develops an abstract framework for analysis and approximation of linear thermoelastic control systems, and for design of finite-dimensional compensators. The thermoelastic systems in this paper consist of abstract wave and diffusion equations coupled in a skew self-adjoint fashion. Linear semigroup theory is used to establish that the abstract thermoelastic models are well posed and to prove convergence of generic approximation schemes. Open-loop uniform exponential stability for a subclass of thermoelastic systems is proved via a Lyapunov function. An example involving the design of an optimal linear-quadratic-Gaussian (LQG) compensator for a thermoelastic rod illustrates the application of the abstract theory. Results of an extensive numerical study, including a comparison of the closed-loop performance of different compensator designs, are presented and discussed.

Remarks on approximations for linear dynamical systems

Two techniques for the approximate analysis of damped linear dynamical systems are compared.

Verification of partially coherent light diffraction models in x-ray lithography

The image formed in proximity x-ray lithography is normally computed using a physical optics model that takes in explicit account diffraction processes. It is important to understand the assumption implicit in the various models and the effects that they have on the image formation. We concentrate, in particular, on the effect of the type of illumination on the quality of the image that can be obtained in proximity x-ray lithography. In this paper, different methods of calculating the diffraction of partially coherent light are compared and the approximations implicit in each discussed. In a partially coherent system, the properties of the source play an intrinsic role in image formation, so they must be treated together with the imaging process. The full detailed calculations are too complex for implementation on even a supercomputer but, for some types of sources and optical systems it is possible to simplify the problem. Particularly important is the issue of the validity of the linear shift-invariant system approximation, since it allows great improvements in computational time. We prove the validity of the approximation and verify it using a full 4D numerical integration, based on the Rayleigh–Sommerfeld theory and realized on a Cray YMP8 supercomputer. Application of the results to some physical cases of 0.25 and 0.1 μm features is presented and discussed, illustrating the wide process latitude and resolution of x-ray lithography.

Approximation of discrete linear systems based on a dual criterion

A new method for approximating scalar discrete linear systems is described. The method can be classified as a time domain/frequency domain technique. Two reduced order models are obtained. One is obtained when the time domain specifications are in terms of the impulse response of the error. The other is derived when the time domain specifications are in terms of the step response of the error. The technique utilizes the input normal system representation to determine the order and the denominator polynomial of the reduced order model.

On the minimal partial realization of 2D discrete linear shift-invariant systems

The partial realization problem for 2D discrete linear shift-invariant systems is discussed. The authors attempt to establish a reasonable definition of minimality and propose a method for finding the degree and the system function of a minimal realization of a 2D linear system characterized by a given 2D impulse response array. The method used is based on the 2D Berlekamp-Massey algorithm which has a close connection with the 2D Hankel matrix derived from the 2D array. While it compares in efficiency with other methods for identification and approximation of 2D linear systems, it gives a novel approach to the problem in the sense that one does not need any knowledge or assumptions concerning the system degree. >

Description of a parametric maximum likelihood estimator in the frequency domain for multi-input, multi-output systems and its application to flight flutter analysis

The basic formula of a parametric maximum likelihood estimator (MLE) in the frequency domain for multi-input, multi-output (MIMO) systems is given. The proposed method takes into account the perturbation noise on all the measured input and output signals. It is an extension to MIMO systems of the single-input, single-output (SISO) estimation method called ELiS (Estimation of Linear Systems). The MIMO estimator will be shown to determine more accurately the natural frequencies and damping ratios of mechanical structures than the SISO estimator. The estimators are applied to very noisy flight flutter data.