Showing papers in "Journal of The Franklin Institute-engineering and Applied Mathematics in 1983"
TL;DR: The Chebyshev series approach appears to have certain advantages over other orthogonal series, and they may therefore be more suitable for the study of the problems of identification, analysis and optimal control.
Abstract: The problems of identification, analysis and optimal control have been recently studied via orthogonal functions. The particular orthogonal functions used up to now are the Walsh, the block-pulse and the Laguerre functions. In this paper, the Chebyshev functions are introduced and solutions for the aforementioned problems are established. The algorithms proposed are analogous to those already derived for the Walsh, block-pulse and Laguerre functions. The Chebyshev series approach presented here appears to have certain advantages over other orthogonal series, and they may therefore be more suitable for the study of the problems of identification, analysis and optimal control.
166 citations
TL;DR: In this article, the authors extend the finite dimensional stochastic maximum principle to a large class of distributed parameter systems, which includes all actual distributed parameters and can be easily adapted to any particular situation.
Abstract: The paper extends the finite dimensional stochastic maximum principle to a large class of distributed parameter systems. Although this class of systems does not include all actual distributed parameter systems, the approach of this paper can be easily adapted to any particular situation.
155 citations
TL;DR: In this article, the generalized block pulse operational matrices of differentiation were derived and the inversions of rational and irrational transfer functions were proposed in a simple, accurate and efficient way.
Abstract: A more rigorous derivation for the generalized block pulse operational matrices is proposed in this paper. The Riemann-Liouville fractional integral for repeated fractional (and operational) integration is integrated exactly, then expanded in block pulse functions to yield the generalized block pulse operational matrices. The generalized block pulse operational matrices perform as s -α (α\s>;0,α∈ R ) in the Laplace domain and as fractional (and operational) integrators in the time domain. Also, the generalized block pulse operational matrices of differentiation which correspond to s α (α\s>;0,α∈ R ) in the Laplace domain are derived. Based on these results, the inversions of rational and irrational transfer functions are proposed in a simple, accurate and efficient way.
41 citations
TL;DR: In this article, the block transformation principle is used for the block implementation of two-dimensional (2D) digital filters. But this technique is restricted to one-dimensional block implementations.
Abstract: A novel principle known as block transformation is developed for the block implementation of two-dimensional (2-D) digital filters. It employs the one-dimensional (1-D) block implementation concept forwarded by Burrus ( 7 ) as a basic tool. Using this block transformation principle three distinct forms of 2-D block equations are derived. The generalized nature of the approach presented here facilitates the straightforward extension of this technique to higher dimensional digital filters.
36 citations
TL;DR: In this article, Pontryagin's maximum principles are applied to obtain the solution of minimum mass convective fin with variable heat transfer coefficients. But the results are given in a dimensionless form and they are not given in tables or graphs, in order to aid design procedures.
Abstract: Pontryagin's maximum principles are applied to obtain the solution of minimum mass convective fins with variable heat transfer coefficients. It is shown that the volume and the width of the fin are uniquely related to the specified heat transfer rate and the properties of the material. For a power law spatial variation of the heat transfer coefficient, the results are set forth in a dimensionless form and they are given in tables or graphs, in order to aid design procedures. Several properties relating to the optimum volume, base thickness and the width of the fins with the properties of the material are also discussed. The method can also be used to solve problems of minimum mass fins having restricted width, often encountered in practice.
31 citations
TL;DR: In this paper, a procedure is developed which uses bond graph modeling and a digital computer to determine if semi-active control can provide a suitable performance in an application where totally active control is considered.
Abstract: A procedure is developed which uses bond graph modeling and a digital computer to determine if semi-active control can provide a suitable performance in an application where totally active control is considered. The application areas principally addressed are those in which the disturbance inputs to the system are of zero mean, i.e. shock and vibration control. The procedure is developed through examples and then generalized to systems of high order and large complexity. The procedure consists basically of designing a control strategy suitable for totally active control and then enforcing a passivity constraint on the actuating device. Experience has shown that semi-active control approaches that of totally active control in most vibration isolation applications.
27 citations
TL;DR: A survey of the recent advances in the field of distributed parameter systems covers stability, controllability, optimal control, observability, state estimation, identification, sensitivity, computational methods, simulation, and applications.
Abstract: A survey of the recent advances in the field of distributed parameter systems is provided. The results are presented in ten sections, each one covering a distinct topic of the field. These topics are stability, controllability, optimal control, observability, state estimation, identification, sensitivity, computational methods, simulation, and applications.
27 citations
TL;DR: For a certain class of systems including parabolic and delay systems it is possible to design finite-dimensional compensators and the system will be stabilized if the combined original system, together with the auxiliary system, is exponentially stable.
Abstract: The problem of stabilizing linear infinite dimensional systems by dynamic output feedback is discussed. The compensator is an auxiliary dynamic system driven by the current output (assumed finite dimensional) and the original system is controlled by a finite linear dimensional operation on this auxiliary state. The system will be stabilized if the combined original system, together with the auxiliary system, is exponentially stable. For a certain class of systems including parabolic and delay systems it is possible to design finite-dimensional compensators.
26 citations
TL;DR: In this paper, a single mass flexible rotor on damped flexible supports with active feedback control is considered and free vibrations and unbalance response of the rotor-control system are determined.
Abstract: This paper considers vibration control of a single mass flexible rotor on damped flexible supports with active feedback control. Both proportional and derivative feedback control are utilized. Free vibrations and unbalance response of the rotor-control system are determined. The effect of proportional control is to increase the critical speed of rotating machinery while the effect of derivative control is to reduce the amplitude of vibration. Generally larger ratios of support stiffness to shaft stiffness require larger values of control parameters to produce given amplitudes of vibration.
25 citations
TL;DR: In this article, the distributed parameter Linear Quadratic Gaussian (LQG) optimal sensor and actuator location problem is considered, and the optimal locations are chosen to minimize the performance criterion on the LQG problem.
Abstract: The distributed parameter Linear Quadratic Gaussian (LQG) optimal sensor and actuator location problem is considered. The sensor and actuator locations are chosen to minimize the performance criterion on the LQG problem. Necessary and sufficient conditions for the optimal locations are derived based on evolution operator theory. It is shown that the optimal sensor and actuator locations can be determined separately and that a duality holds between them. Numerical examples involving a one-dimensional heat conduction system are presented.
23 citations
TL;DR: Franklin invented lightning rods with the hope that they would dissipate thunderstorm electricity and thus prevent lightning from striking, but instead of preventing lightning, they were sometimes “struck” and became part of a lightning path to earth as mentioned in this paper.
Abstract: Franklin invented lightning rods with the hope that they would dissipate thunderstorm electricity and thus prevent lightning from striking. His invention was based on his findings that sharpened metal needles would allow electricity to flow silently through the air, away from highly charged objects. When his rods were used, however, instead of preventing lightning, they were sometimes “struck” and became part of a lightning path to earth. An analysis of the physics involved suggests that: 1. (a) The flow of electricity from sharpened conductors at the earth's surface does not dissipate thunderstorm electricity sufficiently to prevent lightning. 2. (b) The ionization and point discharges around the tip of a sharpened lightning rod limit the strength of the local electric field and reduce the probability of a lightning strike to the rod. The sharpened rod thus acts to protect itself against lightning discharges, but its protection does not extend to other objects in its vicinity. While a sharpened rod does not provide a preferred lightning path to earth, it can be used if no better paths are available. 3. (c) Elevated, blunt rods or horizontal conductors, suitably connected to earth, can provide better lightning paths to earth and therefore, better protection to structures in their vicinity than do sharpened rods. 4. (d) The connections from elevated conductors to earth need to be the most direct possible, with no abrupt changes in direction; impedance discontinuities created in down conductors at sharp bends cause reflections of lightning transients and may produce side flashes to other objects in their vicinity.
TL;DR: In this paper, a time-space boundary residual method with heat polynomials is proposed to solve one-dimensional heat conduction problems, which is based on an integral least squares criterion for the initial and boundary residuals.
Abstract: In this paper, we present a new method for solving unsteady heat conduction problems, which is based on a time–space boundary residual method with heat polynomials. More specifically, it employs an integral least squares criterion for the initial and boundary residuals so as to determine the unknown coefficients in a trial expansion of heat polynomials. Though it treats only one-dimensional cases, the present approach shows a good applicability for such heat conduction problems.
TL;DR: In this article, the effects of the external Biot numbers and axial conduction in fluid on the temperature distribution and the local Nusselt number are determined for both the upstream and downstream regions.
Abstract: Heat transfer in laminar tube flow with convective boundary conditions and axial heat conduction in the fluid is solved exactly. The effects of the external Biot numbers and the axial conduction in fluid on the temperature distribution and the local Nusselt number are determined for both the upstream and the downstream regions. In order to illustrate the effects of velocity profile, the results for the slug flow and the parabolic velocity profiles are plotted simultaneously. The heat transfer characteristics of the flow are found to be rather sensitive to the Peclet number, the external Biot numbers and the velocity profile in the thermal entrance region.
TL;DR: In this paper, it was shown that, when dual structural models are used, elements can be assembled in arbitrary numbers to achieve any desired accuracy without problems of derivative causality. But the reason for the dual structure can be seen from physical reasoning.
Abstract: There is a strong analogy between mechanical structural elements and magnetic circuit components which can be extended to include hysteretic energy losses associated with yielding in mechanical systems and saturation in magnetic components. Procedures for developing finite state models of hysteretic components are presented using a small number of basic elements. Using bond graph techniques, it is shown that, when dual structural models are used, elements may be assembled in arbitrary numbers to achieve any desired accuracy without problems of derivative causality. The reason for the dual structure can be seen from physical reasoning. The shape of the hysteresis curve for magnetic materials generally requires more elements for a given degree of approximation than the shape of the typical mechanical hysteresis curve.
TL;DR: In this paper, a modified Cauer form of continued fraction (MCF) is used to match a set of both the time-moments and Markov parameters of the system and the model, giving a better approximation to the system response at all times.
Abstract: An algorithm, amenable for programming on a digital computer, has been presented for the modelling of linear discrete-time systems, as an alternative to the procedure of Shamash ( 1 ). The transformations inherent in the procedure are easily accomplished by the synthetic division technique. With the use of modified Cauer form of continued fraction (MCF), the new method matches a set of both the time-moments and Markov parameters of the system and of the model, as in the procedure of Parthasarathy and Singh ( 2 ), giving a better approximation to the system response at all times. A distinct feature of the proposed algorithm compared with the earlier methods of discrete system reduction ( 1 ),( 2 ), is that a number of reduced-order models are generated simultaneously; this allows scope for better selection in choosing the right model for system analysis and design.
TL;DR: In this paper, the authors provide an assessment of the contributions of active feedback control theory toward the general solution of problems in active control of flexible spacecraft, and elucidates open questions which still require further research.
Abstract: Large flexible spacecraft are required for several planned space missions. A proposal that active feedback control be employed to reduce the effects of structural bending and vibration in such spacecraft has provided a significant practical challenge to the theory of control of distributed parameter systems, and particularly hyperbolic systems. This limited survey provides an assessment of the contributions of this theory toward the general solution of problems in active control of flexible spacecraft, and elucidates open questions which still require further research. The development of a useful design methodology is found to require a close interplay between control theory and related areas such as modelling, aggregation, and sensor—actuator selection.
TL;DR: This paper introduces the use of piecewise continuous constraint bounds and extends the formal rigorous base of the constrained scheme by considering such items as convergence characteristics, the existence of safety zones and self-adaptive attributes.
Abstract: To improve the versatility, stability and efficiency of the Newton—Raphson technique, further extensions and formalisms are generated in the constrained family of algorithms. Specifically, this paper; (i) introduces the use of piecewise continuous constraint bounds and, (ii) extends the formal rigorous base of the constrained scheme by considering such items as convergence characteristics, the existence of safety zones and self-adaptive attributes. To benchmark the enhancements, the results of several numerical experiments involving large deformation elasticity problems are presented. These illustrate the inherent stability and efficiency of the constrained incremental scheme in handling problems with varying curvature definiteness.
TL;DR: In this paper, the classical method of harmonic balance was extended to difference equations, which can be used to obtain an approximation to the periodic solutions of a special class of second-order nonlinear difference equations containing a small parameter.
Abstract: We extend to difference equations the classical method of harmonic balance. We show that the method can be used to obtain an approximation to the periodic solutions of a special class of second-order nonlinear difference equations containing a small parameter. Two examples illustrating the method are presented.
TL;DR: In this article, an algorithm for numerical evaluation of zero order Hankel transform is developed, which involves one FFT evaluation and a summation of the interpolated FFT components, and compares favorably with other available techniques with regard to speed, accuracy, and ease of usage.
Abstract: An algorithm for numerical evaluation of zero order Hankel transform is developed. The method involves one FFT evaluation and a summation of the interpolated FFT components. It compares favorably with other available techniques with regard to speed, accuracy, and ease of usage.
TL;DR: In this article, a method for the design of 2D analog and recursive digital filters is presented, starting from a structure in the analog domain, suitable even or odd parts of two-variable Hurwitz polynomials are generated.
Abstract: In this paper, a method for the design of 2-D analog and recursive digital filters is presented. Starting from a structure in the analog domain, suitable even or odd parts of two-variable Hurwitz polynomials are generated. This enables 2-variable very strictly Hurwitz polynomials (VSHP) to be obtained, † thus avoiding non-essential singularities of the second kind. Thus it will ensure a stable 2-D recursive digital filter obtained by the use of bilinear transformations. Examples are given to illustrate the method.
TL;DR: In this paper, the authors present a review of recent analyses of spurious phenomena near interfaces and boundaries of numerical approximations of hyperbolic equations using Fourier transforms and the concept of sinusoidal wave propagation.
Abstract: Recent analyses of spurious phenomena near interfaces and boundaries of numerical approximations of hyperbolic equations have produced a host of interesting, concrete results which are reviewed in this paper What characterizes these analyses is that they rely on tools of mathematical physics, in particular in the systematic use of Fourier transforms and the concept of sinusoidal wave propagation This leads to a description of numerical inaccuracy in terms of dispersion, and a description of spurious numerical solutions in terms of wave packets and group velocities It is in providing the mathematics needed to describe spurious reflection at numerical boundaries and at interfaces in mesh refinement that the most applied results of this theory are obtained
TL;DR: In this article, a topology on Hilbert spaces is defined and the continuity of a map with respect to such a topological is equivalent to the uniform S-continuity around the origin, according to the Gross definition.
Abstract: In this paper a topology on Hilbert spaces is defined. The continuity of a map with respect to such a topology is equivalent to the uniform S-continuity around the origin, according to the Gross definition. An example of an application, in which the achieved gives a simple tool for providing the existence of stochastic solutions, is reported for a quite general class of non- linear differential equations driven by white noise.
TL;DR: A simple and flexible algorithm is presented to find stable reduced models, provided the original system has a set of dominant poles and a parameter to control the approximation for small and large t .
Abstract: A simple and flexible algorithm is presented to find stable reduced models, provided the original system has a set of dominant poles. The proposed technique applies to the multi- variable case as well and provides a parameter to control the approximation for small and large t .
TL;DR: In this paper, a series of nonperiodic Lame functions were investigated for the even and odd Dirichlet and Neumann problems. Butler et al. presented an application of the Lame function in an analytic study of electromagnetic diffraction by a perfectly conducting elliptic cone.
Abstract: An investigation into nonperiodic Lame functions is presented. Those functions are given in series form for the even and odd Dirichlet and Neumann problems. Results are shown for the eigenfunctions and eigenvalues. An application of the Lame functions in an analytic study of electromagnetic diffraction by a perfectly conducting elliptic cone is presented.
TL;DR: In this article, a two-step iterative method (1,2 ) for a reduction in the order of linear continuous time systems, given in the state equation or the transfer function, is extended to reduce discrete-time systems.
Abstract: A two-step iterative method ( 1,2 ) for a reduction in the order of linear continuous-time systems, given in the state equation or the transfer function, is extended to reduce discrete-time systems. The method requires the optimization of the residues and eigenvalues (or poles) belonging to an objective function. The objective function to be minimized is chosen as the finite sum of the squares of the error between the step responses of the reduced model and the original system. This scheme is continued cyclically until the objective function is satisfactorily minimized. By investigating the initial selection of the eigenvalues in the reduced-order model, it is found that the dominant eigenvalues of the original system give a good approximation. Further, the resulting model is always stable, assuming the original system is stable. As shown in a numerical example, the proposed method is superior to the other methods of model reduction in both steady-state and transient responses, and in the value of the sum of the squares of the error.
TL;DR: In this article, the determinant of the admittance matrix of the network can be made to yield VSHP (a two-variable Hurwitz polynomial without nonessential singularities of the second kind) under certain conditions involving the sub-determinants of the gyrator matrix.
Abstract: An n -port gyrator, terminated by s 1 -type capacitors in m 1 -ports, by s 2 -type capacitors in the next m 2 -ports and by resistors in the remaining (n-m 1 -m 2 ) -ports is considered. The determinant of the admittance matrix of the network can be made to yield VSHP (a two-variable Hurwitz polynomial without non-essential singularities of the second kind) under certain conditions involving the sub-determinants of the gyrator matrix. With the gyrator constants as variables and the above conditions as constraints, some 2-D stable low-pass filters have been designed using a suitable optimization procedure. The method is illustrated by examples.
TL;DR: In this article, a state-space formulation for two-terminal higher-order elements is provided, and properties such as local controllability, input-observability, passivity and losslessness are investigated.
Abstract: Higher-order elements are introduced to provide a logically complete formulation for nonlinear circuit theory. A distinctive feature of higher-order elements is that they possess internal dynamics that are more complicated than those of conventional circuit elements (namely, the resistor, inductor, capacitor and memristor). In this paper, we provide a state space formulation for studying two-terminal higher-order elements. State-space properties such as local controllability, input-observability, passivity and losslessness are investigated in detail.
TL;DR: In this paper, the authors used energy dissipation per cycle to determine an approximate frequency response of semi-active systems requiring only a calculator for numerical analysis, where the frequency response is defined as the closed-loop response of the total system.
Abstract: Semi-active systems are those in which a passive, dissipative element (damper, friction clutch, fluid orifice) is modulated, via low power inputs, to enhance the performance of its associated dynamic system. This type of control is inherently nonlinear, thus forcing the closed-loop response of the total system to be determined numerically using a digital computer. Although, strictly speaking, the frequency response for nonlinear systems does not exist, it is typical to simulate semi-active systems with sinusoidal input and present results as frequency transmissibilities. This paper uses energy dissipation per cycle to determine an approximate frequency response of semi-active systems requiring only a calculator for numerical analysis.
TL;DR: The Taylor-Cauchy transform method has been applied in the analysis of simultaneous nonlinear systems as discussed by the authors, and it is shown that the linear system to the nonlinear system, the input-output crosscorrelation φxy can be shown to be equal to the convolution of system impulse response h1 with the autocorrelation Άxx.
Abstract: This paper gives a general review of the Theory of Nonlinear Systems. In 1960, the author presented a paper “Theory of Nonlinear Control” at the First IFAC Congress at Moscow. Professor Norbert Wiener, who attended this Congress, drew attention to his work on the synthesis and analysis of nonlinear systems in terms of Hermitian polynomials in the Laguerre coefficients of the past of the input. Wiener's original idea was to use white noise as a probe on any nonlinear system. Applying this input to a Laguerre network gives u1, u2,…, us, and then to a Hermite polynomial generator gives V(α)'s. Applying the same input to the actual nonlinear system gives output c(t). Putting c(t) and V(α)'s through a product averaging device, we get c(t)V(α) = A α 2л s 2 , where the upper bar denotes time average and Aα's can be considered as characteristic coefficients of the nonlinear system. A desired output z(itt) may replace c(itt) to get a new set of Aα's. The Volterra functional method suggested by Wiener in 1942 has been greatlydeveloped from 1955 to the present. The method involves a multi-dimensional convolution integral with multi- dimensional kernels. The associated multi-dimensional transforms are given by Y.H. Ku and A.A. Wolf (J. Franklin Inst., Vol. 281, pp. 9–26, 1966). Wiener extended the Volterra functionals by forming an orthogonal set of functionals known as G-functionals, using Gaussian white noise as input. Volterra kernels and Wiener kernels can be correlated and form the characteristic functions of nonlinear systems. From an extension of the linear system to the nonlinear system, the input-output crosscorrelation φxy can be shown to be equal to the convolution of system impulse response h1 with the autocorrelation φxx. Using the white noise as input, where its power density spectrum is a constant, say, A, the crosscorrelation is given by φxy(σ) = Ah1(σ), while the autocorrelation is φxx(τ) = Au(τ). This extension forms the basis of an optimum method for nonlinear system identification. Measurement of kernels can be made through proper circuitry. Parallel to the Volterra series and the Wiener series, another series based on Taylor-Cauchy transforms developed since 1959 are given for comparison. The Taylor-Cauchy transform method can be applied in the analysis of simultaneous nonlinear systems. It is noted that the Volterra functional method and the Taylor-Cauchy transform method give identical final results. A selected Bibliography is appended not only to include other aspects of nonlinear system theory but also to show the wide application of nonlinear system characterization and identification to problems in biology, ecology, physiology, cybernetics, control theory, socio- economic systems, etc.
Abstract: The problem factorizing (separating) the transfer function of a given SISO 3-D discrete system, ie of a system depending on three independent variables, is considered. The 3-D system is assumed to be available in its transfer function representation, which is converted to a canonical state-space model by a simple inspection procedure. Then applying state-feedback to this canonical model we choose the feedback matrix gain (under certain conditions) such that the transfer function of the closed-loop system has the desireed factorized form, ie a product of three 1-D transfer functions each one being dependent on a single variable. The method is illustrated by a nontrivial numerical example.