A low‐dimensional realization algorithm for periodic input/output behavioral systems

Transfer function estimation of turboalternator from practical operating data by spectral analysis

This study aims at the simultaneous solution of several quaternion linear systems with the same Hermitian and positive definite coefficient matrix by employing the conjugate gradient method. We consider the setting when the quaternion Hermitian positive definite coefficient matrix at hand is very large so that direct methods are not applicable. In the study, we first transform linear quaternion systems into real linear systems. The transformed real linear systems have special structure due to the fact that they are real representations of quaternion systems. Benefitting from the special structure, we further reduce the size of these linear systems. Then a block conjugate gradient method is applied to the resulting reduced real linear systems. The solution obtained after applying the conjugate gradient method is a real representation of the solution of the original quaternion problem. Thus, a conversion of this real solution to the quaternion setting is performed in the end.

A new canonical form for the minimal realizations of finite dimensional multi-input multi-output linear systems possessing cyclic state spaces has been developed. This minimal realization is called the ‘observable-pair realization’ and may be used as a direct summand of minimal realizations of general linear systems. An algorithm, suitable for computer implementation, is presented for calculating the observable-pair realizations of transfer function matrices of cyclic systems. It is shown that this algorithm is readily used as part of a second algorithm for calculating the minimal realizations of general transfer functions. Finally, it is proved that in a special sense the observable-pair realizations are unique. Notes † Communicated by Professor J. H. Westcott

The problems of realization and feedback stabilization of autonomous linear neutral systems are studied in the context of linear systems over commutative rings. Connections are made to the theory of retarded delay systems as well as to generalized linear systems over a field.

The relation between the classical G. Floquet characteristic exponents (1879) and entities called ED (essential D) eigenvalues and PED (primary essential D) eigenvalues for specified scalar periodic linear systems is established. In particular, some important results on the existence and uniqueness of periodic ED-eigenvalues and periodic PED-eigenvalues for these systems are obtained, and it is shown that the mean value of periodic PED-eigenvalues for these systems are their classical Floquet characteristic exponents. Using these results, the authors derive two necessary and sufficient stability criteria for F-nonderogatory and F-derogatory scalar periodic linear systems. The derived stability criteria not only serve to unify the stability theory for the general scalar time-varying linear system, but can also be used to develop novel techniques for evaluating and/or approximating the Floquet characteristic exponents for scalar periodic linear systems. >

The realization of linear time-varying systems specified by an analytic weighting pattern is approached in a novel manner using an algebraic framework defined over the ring of analytic functions. Realizations are given by a state representation consisting of a first-order vector differential equation and an output equation, both with analytic coefficients. Various new criteria for realizability are derived, including conditions given in terms of the finiteness of modules over the ring of analytic functions generated by the elementary rows or columns of a (generalized) Hankel matrix. These results are related to local criteria for realizability specified in terms of the rank of matrix functions, as developed in the work of Silverman and Meadows [5], [8], [9] and Kalman [7]. It is shown that the construction of minimal realizations reduces to the problem of computing a basis for a finite free module defined over the ring of analytic functions. A minimal realization algorithm is then derived using a constructive procedure for computing bases for finite free modules over a Bezout domain. The Silverman-Meadows realization algorithm [5] is a special case of the procedure given here. In the last part of the paper, the realization algorithm is applied to the problem of system reduction.

The field of coherent optics has stimulated much interest and indeed excitement over the past decade, and a number of engineering applications have been brought to light. The most significant of these are the subject of this senior- or graduate-level text, which was originally prepared--but is not limited to--electrical engineering students. It emphasizes the analogy between optical and electrical systems, both of which, for example, are capable of performing Fourier transform operations and signal filtering and processing.The book is designed for students without an intensive background in electromagnetic theory and classical optics. Its discussion of diffraction is based on scalar theory, and it approaches information processing and holography by means of the elementary point concept and linear system theory. This approach simplified the analysis so that solutions may be directly calculated, and it will appeal to engineering students because of their familiarity with the concepts of the impulse response of linear systems.After an opening presentation of the basic properties of linear systems and Fourier transformations, the book develops the theory of diffraction. The topics taken up include, among others, Fraunhofer and Fresnel diffraction, the reciprocity theorem, Huygens' principle, Kirchhoff's integral, the Fresnel zone plate, the Rayleigh criterion, and Abbe's sine condition. This part closes with a discussion of coherent theory and the mutual coherence function.The next part, on information processing, covers the Fourier transform properties of lenses and linear optical imaging systems, filtering, the basic properties of photographic film as a recording medium, film-grain noise and signal-to-noise ratio, the information channel capacity of photographic film, and optical resolving power and its relation to the uncertainty of information and physical realizability.The final part throws a clear light on the subject of holography. The presentation includes both linear and nonlinear holograms and takes up wavefront construction and reconstruction, magnifications, resolution limits and bandwidth requirements, finite-point analysis linear optimization techniques, syntheses of optimum nonlinear spatial filters, and applications. Among the latter are microscopic wavefront reconstruction, multiexposure holographic interferometry, time-average interferometry, and contour generation.

This chapter presents a survey on realization theory for linear hybrid systems. Recall that for linear systems, realization theory addresses the problem of existence and minimality of a linear time-invariant state-space representation of an input–output map (transfer function). The results of linear realization theory have turned out to be useful for control synthesis, model reduction, and system identification. In this chapter, we address a similar problem for linear hybrid systems and linear switched systems. We will also discuss the implications of realization theory for estimation and control of hybrid systems.

Poincaré half-maps can be used to characterize the behavior of recurrent dynamical systems. Their usefulness is demonstrated for a linear three-dimensional single-loop feedback system. In this example everything can be calculated analytically. The resulting half-maps are "benign" endomorphic maps with a complicated topological structure. This is surprising since the combination of two such half-maps (yielding an ordinary Poincaré map) always implies simple behavior in a linear system. The method has a direct bearing on the theory of piecewise linear systems -like the well-known Danziger-Elmergreen system of hormonal regulation.

For the further development we use the state-variable description of linear time-invariant systems. This is the standard form used in linear system theory. Introducing this notation brings some advantages. The state-variable description is the most general form and is equally applicable for scalar and for multivariable systems. Matrix calculus is employed instead of working with polynomials. The whole theory of linear systems is available.

On Padé-type model order reduction of J-Hermitian linear dynamical systems

A method is presented for predicting the total response, in both frequency and time, of an unknown linear system when only the measured continuous wave (cw) magnitude is available. The approach is based on approximating the square of the measured magnitude by a rational function, from which various system transfer functions in terms of complex frequency are deduced. These transfer functions may or may not be at minimum phase. The corresponding impulse response is then obtained by taking the inverse Laplace transform of the transfer function. The impulse response of the minimum-phase case rises faster initially to its first maximum than the nonminimum-phase counterparts. This result confirms that, for the same cw magnitude response, the accumulative energy contained in the impulse response is the greatest when the transfer function is at minimum phase. Physical meaning of the energy content is also discussed.

There are many useful computations such as correlations and convolutions that can be implemented using FTs. In fact, taking advantage of computationally efficient DFT techniques such as the FFT often executes much faster than more straightforward implementations. Subsequent chapters reuse these tools in an optical context. For example, convolution is used in Ch. 5 to simulate the effects of diffraction and aberrations on image quality, and structure functions are used in Ch. 9 to validate the statistics of turbulent phase screens. Three of these tools, namely convolution, correlation, and structure functions, are closely related and have similar mathematical definitions. Furthermore, they are all written in terms of FTs in this chapter. However, their uses are quite different, and each common use is explained in the upcoming sections. These different uses cause the implementations of each to be quite different. For example, correlations and structure functions are usually performed on data that pass through an aperture. Consequently, their computations are modified to remove the effects of the aperture. The last computation discussed in this chapter is the derivative. Like the other computations in this chapter, the method presented is based on FTs to allow for efficient computation. The method is then generalized to computing gradients of two-dimensional functions. While derivatives and gradients are not used again in later chapters, derivatives are discussed because some readers might want to compute derivatives for topics related to optical turbulence, like simulating the operation of wavefront sensors. <strong>3.1 Convolution</strong> We begin this discussion of FT-based computations with convolution for a couple of reasons. First, convolution plays a central role in linear-systems theory.¹⁴ The output of a linear system is the convolution of the input signal with the system's impulse response. In the context of simulating optical wave propagation, the linear-systems formalism applies to coherent and incoherent imaging, analog optical image processing, and free-space propagation.

The two-image depth from defocus problem is considered as a system identification problem. The underlying phenomenon of defocusing is modelled as a linear system with an appropriate transfer function. Using a general measure for the spread, a relation between the spread and the system transfer function is obtained. A parametric transfer function is utilised in deriving an analytical expression for the spread. The two images taken with different camera parameters then become the input-output of such a system. An algorithm for estimating the coefficients of the transfer function is derived. The method is tested on synthetic images as well as images obtained from a camera.

Stability of Periodic Linear Systems and the Geometry of Lie Groups