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Senior managers when solving problems commonly use analogical reasoning, allowing a current ‘target problem’ situation to be compared to a valid previous experienced ‘source problem’ from which a potential set of ‘candidate solutions’ may be identified. We use a single-echelon of the often-quoted Forrester (1961) production-distribution system as a case ‘target model’ of a complex production and inventory control system that exhibits bullwhip. Initial analogical reasoning based on ‘surface similarity’ would presuppose a classic control engineering ‘source model’ consisting of a phase-lag feedback system for which it is difficult to derive the transfer function. Simulation alone would have to be relied on to mitigate the bullwhip effect. By using z-transform block diagram manipulation, the model for a single-echelon, consisting of 17 difference equations with five feedback loops is shown to have exact analogy to Burns and Sivazlian's (1978) second order system that has no feedback. Therefore, this more appropriate ‘source model’ is based on a deeper understanding of the ‘behavioral similarities’ which indicates that the bullwhip effect is not in the case of the ‘target model’ due to feedback control but due to a first-order derivative, ‘phase advance’, term in the feed forward numerator path. Hence a more appropriate ‘candidate solution’ can be found via the use of a ‘recovery’ filter. An interdisciplinary framework for exploiting control engineering block diagram manipulation, utilizing analogical reasoning, in a practical setting is presented, as is an example in a contemporary supply chain situation.

It is pointed out that by approximating the impulse response of a linear time-invariant stable system by a finite sum of given exponentials, the problem of estimating the transfer function is considerably simplified. The author shows how the complexity can be reduced further by using orthogonalized exponentials. The analysis is based on the result that the corresponding normal equations will then have a Toeplitz structure. The z-transform of orthogonalized exponentials corresponds to discrete Kautz functions, which generalize discrete Laguerre functions to the several, possibly complex, poles case. Hence, by appropriate choice of time constants Kautz models give low-order useful approximations of many systems of interest. In particular, resonant systems can be well approximated using Kautz models with complex poles. Several basic results on transfer function estimation are extended to discrete Kautz models, for example, frequency-domain quality properties and efficient numerical techniques., QC 20120104. Sponsors: IEEE Control Systems Soc

Papers presented at the meeting treat various Army statistical and design problems. Subject matter include ridge regression, response surface fitting, order statistics, radar signal improvement, digital filters, life time estimates, risks, forecasting, linear regression models, adaptive median smoothing, fitting an ellipse, approximating functions, selection criteria, armor combat models, experimental designs, probability densities, testing hypotheses, quantile estimation, rank transformation, design of field tests, crossover experiments, time series analysis, spectral limits, confidence limits, reliability, camouflage development, gravity anomalies, and allocation methods.

Advances (left shifting) and delays (right shifting) are shown to be inverses for the Laplace and z-transforms. This is true for both bilateral and unilateral Laplace and z-transforms. This result is not in agreement with some literature on these subjects, and the oversight made in that literature is pointed out. (Author)

The alpha-beta-gamma tracking equations are written in compact form and then Z transformed. The resulting expressions for estimated position, estimated velocity, estimated acceleration, and predicted position are in terms of the measured position and Z (the Z transform variable). These equations are first solved for the ratio of output (an estimated or predicted quantity) to input (the measured position) and then Z is replaced by exp(-j (omega) T) (omega is the radian frequency and T is the sample interval). After this change, the magnitude and phase for various output-input ratios is plotted as a function of omega T. Linear plots and Bode plots (especially useful for real-time control system design) are presented. Equations are given for the frequency at which the output is 0.707 of the input, for the minimum of the output to input ratio and for the overshoot (amount by which the output exceeds the input). A synopsis of previously available statistical properties of alpha-beta-gamma trackers is also included.

There are many situations where a great increase in the resolution of power spectral densities, PSD's, would solve important problems. The problem areas range from radar, speech compression, seismic signal processing, to spectral analysis of optical signals. The maximum entropy method (MEM) can, under certain circumstances, provide an order of magnitude increase in resolution over conventional techniques. The fundamental concepts and mathematical background are given and, most importantly, the peculiarities and new features introduced by the MEM technique are explained. Some signals are good candidates for this technique while others are not; thus, methods are described to distinguish between the two. After the techniques to obtain the MEM-PSD are explained, an application to Fourier Spectroscopy is given to illustrate the results (as compared to conventional means). This review employs certain features to make the MEM approach accessible to those without previous knowledge about it. These features include: a short history of the PSD which describes the evolution over time of the basic concepts; a discussion of the higher principle of maximum entropy in the context of probability theory; a simple introduction to the Z-transform and how it introduces the basic spectral line shape of the MEM-PSD as well as other peculiarities of the method which distinguish it from the conventional approach; the physics behind the reflection coefficients which are introduced by MEM; and an extensive annotated bibliography which should make the literature available.

This report presents the approximation for Z(G(s)F(s)), which is based upon the second mean value theorem of the integral calculus. (Author)

This report covers the efforts undertaken to develop a mathematical model of the FAMMS I surface-to-surface missile suitable for preliminary analysis of the effects of digital computer sampling on system performance. Planar equations are developed and linearized to permit digital autopilot design to proceed. A design methodology is developed and described through application to a simplified version of the FAMMS I equations. (Author)

This report summarizes the results of research during the past two years, in understanding the resolution of the autoregressive (AR) spectral estimators and developing and evaluating computationally efficient autoregressive-moving average (ARMA) spectral estimators. The loss in the resolution of the AR spectral estimator in the presence of noise is related to the appearance of zeros in the z-plane. A parallel resonator model is developed to relate the loss in resolution (bandwidth expansion) to the signal-to-noise ratio and parameters of the noiseless signal model. A new technique for the identification of the order of an AR model was derived that shows substantial stability compared to the popular Akaike Information Criterion method. Order determination was emphasized, since increase in the order of the AR spectral estimator, to account for the presence of noise, is naturally accompanied by larger variance of the estimates and appearance of spurious peaks. Several sub-optimum (non-maximum-likelihood) ARMA spectral estimators were also developed. These methods are computationally efficient, but statistically not very stable for small data records. An evaluation of the statistical properties of the different sub-optimum ARMA techniques led to the evaluation of asymptotic bounds on the variances of the estimates of the parameters or the poles and zeros of the model through the evaluation of Fisher's information matrix.

An approximation for ZG(s)F(s) is developed using the second order Runge-Kutta integrator family. The residue for exact unit step response of a single real pole filter is then developed. The error for a complex exponential into a complex pole is also presented. (Author), Includes errata sheet dated 15 Jan 81.

A multi-dimensional sequence is not, in general, uniquely defined in terms of only the phase or magnitude of its Fourier transform. However, in this paper some conditions are developed under which a multi-dimensional sequence is uniquely defined by its phase. A similar set of conditions are then developed for the unique specification of a multi-dimensional sequence in terms of its Fourier transform magnitude. In both cases, it is initially assumed that either the phase or magnitude is known for all frequencies. The results are then generalized to the case in which the phase or magnitude is known only for a finite set of frequency values. (Author)

An eight-step procedure for the design of digital Butterworth low-pass filters is outlined which summarises the methods of several standard text-books on digital filter design. Examples of filters designed using the procedure are included. (Author)

An analog DDC reformatting memory has been designed and fabricated using an N-channel double level polysilicon gate process. This unique CCD structure employs two-dimensional charge transfer cells in a 32x32 element array which is accessed by means of integrated CCD demultiplexer and multiplexer structures resulting in greater dynamic range than observed in previous line-addressed designs. The design and operation of this structure are discussed, and examples of application in signal processor architectures are described. (Author)

This report describes the Cross-Multiplication technique for obtaining the response of a digital control system from its closed-loop transfer function. When the latter is expressed in the complex z-domain, the response is obtained at the sampling instants. If it is desired to know the response between the sampling instants, the submultiple method may be adapted to the cross-multiplication technique. When the transfer function is expressed in the modified z-transform domain, the response is obtained at any point within each sampling instant. The three approaches are described and applied to an example. (Author)

A detailed analytical and experimental study of the sensitivity of Wave Digital Filters has been conducted. The results indicate that the wave digital filter tends to achieve the same low sensitivity characteristic as the analogue circuit from which it was derived. Other results indicate relatively higher sensitivity to terminating resistance values compared with internal element values, lower sensitivity for algorithms derived from simple rather than multiple elements, and higher sensitivity at the critical frequencies. Finally the rms error due to the quantization in the number of bits in the multiplier coefficients has been measured at approximately 3 db per bit for the many examples tested. (Author)

A technique for the use of semi-infinite planes to approximate the log magnitude versus log frequency characteristics of two-dimensional filters is presented (Extended Bode approach). This technique is applied to quarter plane filters and works well for separable transfer functions. Other non-separable canonic (basic) transfer functions are also studied in terms of their planar approximation. The general technique is shown to be useful for the insight it provides as well as being a simple approach to design. The double bilinear z-transform is studied for use with the two-dimensional analog transfer functions to convert them to the digital domain. (Author)

Tunable integration and trapezoidal convolution are reviewed and tunable trapezoidal convolution derived. The effects of tuning for a sine wave input to a tunable integrator are shown analytically. Also there are some examples as to how recurrence relations may be developed using z-transforms and trapezoidal convolution. (Author)

A Method is presented which yields the autocovariance function of a stationary discrete-time stochastic process in closed form. Special reference is made to the Box Jenkins forecasting methodology in which the underlying process is generated by passing white noise through a linear filter. The impulse response of the filter and its Z-transform, the transfer function, are obtained from the equation which defines the filter. The bilateral Z-transform of the autocovariance function is then derived from the transfer function, and is inverted following a partially fraction expansion. Several examples of this procedure are worked out in detail, and a summary of solutions for a number of cases is given. (Author), Supersedes Rept. no. RR-76-20 dated Aug 76, AD-A032 103.

Sampling theorem concepts are applied to certain classes of linear shift variant systems. Various sampling theorem characterizations arise from different bandlimiting assumptions on the system input and/or impulse response. These characterizations are also expressed in discrete form and in all cases considered, reduce to an identical computational form which can be evaluated with a generalized Z transform treatment. The fourier duals of the sampling theorems, wherein the system is characterized by its frequency response, are also presented. (Author), Proceedings of the Midwest Symposium on Circuits and Systems (20th), Part 2 held on Aug 77 at N. Hollywood, CA.

Formulas for the general term of the Taylor series expansion (impulse response) of a multidimensional recursive digital filter are developed. Of particular interest is a non-recursive combinatorial formulation involving the filter coefficients. These results are applied to develop several new specific stability criteria for low order two-dimensional filters, as well as for the development of some general filter design procedures to achieve a desired impulse response. Also included are some new filter structures and stability conditions for N-dimensional filters. (Author)

This report describes an approach that may be used to develop a simulation which employs some digital hardware in the guidance and control loop. The approach is applied to portions of the T-6 digital autopilot data processor. (Author)

An object of the invention is to provide a system for performing the discrete Fourier transform via convolution using a transversal filter. Another object of the invention is to provide a system for the more accurate implementation of the discrete Fourier transform. Yet another object of the invention is to provide a system wherein the block length is even, thereby permitting the use of the same transversal filters which would be used for the ordinary chirp-Z transform., Supersedes PAT-APPL-086 967-79, AD-D006 742.

This report describes the implementation of a Third Order Butterworth filter in the ACSL. Observations and comparisons are made with inputs and outputs. Both plots of the S and Z transfer functions were made with an ACSL program. The program and Bode plots are both described in this report. (Author)

A Z-transform algorithm, developed for the spectral analysis of signals, allows one to get closer to the poles of a signal and effectively reduces the signal's bandwidth and sharpens its peak point. It can give a high resolution, narrow-band frequency analysis with frequency spacing delta f or = 1/T, where T = total length of the analysis interval. This algorithm also enhances the signal poles that lie on circular or spiral contours that begin at almost any point in the Z-plane and the angular spacing of points in an arbitrary constant. Since this algorithm takes advantage of high-speed convolution, it is almost as fast and more flexible than the Fast Fourier Transform (FFT).

In the analysis of sampled-data or digital control systems it is convenient to use z-transform analytics. However, the computation of these transforms may be tedious and time consuming as well as being inaccurate for any but the most simple systems. This program computes the modified z-transform of a system whose Laplace transform, F(s); sample period, T; and delay parameter, m, are given. (Author)

An efficient method of inverting Z-transforms is given by a shortcut recursion formula that is herein programmed for the digital computer. The program is basic FORTRAN and its use is illustrated by means of a numerical example. (Author)