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RESEARCH PAPERS

J. Basic Eng. 1967;89(2):241-249. doi:10.1115/1.3609578.

The technique of signal stabilization is that of introducing an extra signal at the input to a nonlinearity to improve or otherwise alter its performance. In this paper this technique is extended to the introduction of both an extra signal and a nonlinear element to improve performance. Here a method is given for linearizing time-independent non-linearities in the sense of time averages by the introduction of a high frequency sawtooth wave and an ideal relay. The technique of pulse width modulation, employed in the communications field, is applied here. The method developed is applicable to feedback systems with a time-independent nonlinearity in the loop. The method is illustrated by applications to nonlinear systems with first and second-order lags. The concept of equivalent gain is used to simplify the mathematical approach so that the technique can be readily employed in engineering analysis and design. Good agreement is obtained between precise theory and approximations.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):251-262. doi:10.1115/1.3609586.

The optimal control of dynamical systems with conventional Mayer, Lagrange, and Bolza type performance indexes has been studied in some detail [1], [2], [3]. In the present paper, the optimal control of dynamical systems with a certain minimax type performance index, which cannot be expressed in the Mayer, Lagrange, or Bolza format, is studied. The form of the optimal control is described and certain geometric properties of the solution are discussed. Several examples are worked in detail to illustrate application of the proposed method of solution.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):263-266. doi:10.1115/1.3609589.

The problem of synthesizing a controller to yield a specified overall system phase margin requires specification of both the time constant (or constants) and the gain of the controller, but the choice of the two is not independent. A graphical technique is developed for plotting the relationship between the gain and the time constant in a straightforward manner. A logical criterion results upon which to base the controller design. The method is presented for a first order controller, but the approach may be extended to multiple time constant controllers. The method is applied in the frequency domain to linear elements. Phase shifting nonlinearities can be accommodated by judicious use of describing function results.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):269-272. doi:10.1115/1.3609592.

A new matrix formula for the inverse Laplace transformation is established. After substituting the eigenvalues and coefficients and performing some simple matrix operations, one can obtain the inverse Laplace transformation of the function in question. The regular Heaviside techniques involving partial fraction expansions, function differentiations, and so on, are avoided. Since the formula is general, it is particularly advantageous for use on high-order transfer functions; since the formula is exact, the results have no numerical errors. Hundreds of commonly used transform pairs can be replaced by this single matrix formula.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):273-282. doi:10.1115/1.3609595.

The problem of minimizing both the in-flight bending moments as well as the terminal drift of a flexible vehicle is considered. The performance criterion

V(T) = 12y2(T)+k2tT M2(ξ)dξ,
where y(T) is the drift, and M is the in-flight bending moment, is selected to achieve a compromise between excessive drift and excessive structural loading. The two-point boundary value problem for the design of the rigid-body control system is solved analytically for the linear, time-varying optimum control law. The flexibility of the vehicle is then accounted for, in a model consisting of two rigid sections hinged together with a torsional spring, by augmenting the rigid-body optimum control law with feedback terms proportional to the vehicle flexure and its rate. The results of a digital computer simulation indicate that the quasi-optimum control law obtained by this technique results in satisfactory performance, while the rigid-body control law is inadequate for a vehicle of moderate flexibility.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):283-286. doi:10.1115/1.3609596.

Exact analytical stability criteria are derived for the coupled-core kinetics equations. Four approximate models for describing the time distribution of the coupling neutrons are considered and a theorem proved by Pontryagin is used to establish the asymptotic stability of the systems. The criterion based on the Single Delta Function Model is compared with the one based on the Single Step Function Model.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):287-294. doi:10.1115/1.3609597.

The dynamic study of thermal processes is often restricted by the limitations of the temperature transducers. The thermocouple is one of the most employed transducers because of its small size, but it presents several problems such as: (a) The delivered output is very small and for precision measurements problems appear concerning signal to noise ratios, and (b) the transient response is often not well known and is also sometimes insufficient. For this reason thermocouples have been studied by varying the different parameters. To be able to get a reference, permitting accurate measurements of a temperature variation as a function of time, an optical method has been constructed called the “Schlieren” method which allows visualization of a temperature variation and thereby renders feasible the recording of these variations without any appreciable time delay. The thermocouple itself is installed in the water-channel in which the temperature variations are recorded by the foregoing principles. The comparison of the two recordings defines the time constant of the thermocouple. For these measurements a special device has been constructed permitting the temperature to vary without any influence upon the other parameters (flow, pressure, and so on). The experiments have been made for different flow rates. Using the experimental values, an electric model for the thermocouple’s behavior has been devised by which the equivalent thermal resistors and capacitors could be determined. Thus the time constants of the thermocouples corresponding to other parameter values can be calculated. Similar tests to determine time constants have been made for thermal resistors, hot wires, and hot films. All tests described have been made by measuring water temperatures, but the results could easily be applied to other media.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):295-299. doi:10.1115/1.3609598.

A dynamic system identification technique is described in which a digital computer model is constructed and adjusted until the model state transient follows the system state transient to a prescribed accuracy. This identification technique does not require testing signals and the observation and computation times are short relative to the longest system time constant. The principle of this identification technique is demonstrated by an example consisting of a combination jet pipe electrohydraulic servo and analog computer system. The system’s state-space dimension was larger than the model and the system had measurement noise. Favorable results of the experiment indicate that the method will be applicable to such problems as optimal control, adaptive control, learning control, and on-line component parameter identification.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):300-306. doi:10.1115/1.3609599.

The problem considered here is that of controlling the flow rate through a nuclear rocket such that temperature gradients in the fuel elements, and the corresponding thermal stresses produced, do not exceed specified values. The desired control program is that which takes the system from steady-state conditions at a given flow rate to a higher, specified flow rate in minimum time without violating the thermal stress constraints. The system equations here are a pair of coupled, first-order, bilinear, partial differential equations and the thermal stress constraint is proportional to a product of state and control variables. By analyzing both the solution for a step in control and the coupling between control level and time response in the bilinear system, the form of the optimal control is deduced. It is shown how the optimal control law can be generated using a digital computer. Numerical results are given.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):307-310. doi:10.1115/1.3609600.

This paper develops a technique for finding Lyapunov functions for the class of nonlinear partial differential equations arising from a reactor system which takes into account the coupling of heat transfer, hydrodynamics, and time-dependent neutron diffusion. As a first step, a generalized Lyapunov function was developed for the linearized reactor system. The result provides the sufficient conditions to system stability (and/or asymptotical stability) with respect to the distributed system parameters. A new Lyapunov function for the nonlinear reactor system was constructed by adding nonlinear terms to that of the linear system. The result enables one to determine the region of stability and indicates the proper feedback function which would insure the global stability of the system.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):311-314. doi:10.1115/1.3609601.

A technique based on Lyapunov’s second method is described for deriving bounds for the performance of nonlinear control systems. As an illustration, a performance bound is derived for systems associated with the so-called Lurie problem. As opposed to conventional sensitivity analysis, performance bounds are also determined for systems designed using grossly inaccurate models.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):315-320. doi:10.1115/1.3609602.

A method of simulating a class of hyperbolic distributed systems without using space discretization is presented. Complex distributed systems are realized by means of subsystems of basic distributed elements. The time-delay operation forms the basis of the technique which is ideally suited to hybrid or digital computation. Using this method the system is simulated independently of its boundary conditions which may be dynamic in nature. An example of a simple hydraulic system is presented.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):321-326. doi:10.1115/1.3609603.

The dual input describing function (DIDF) is derived here by statistical methods for the difficult case of a memory type nonlinearity. Special mathematical techniques are used to simplify the problem so that closed form solutions can be obtained. The DIDF is employed to determine the effect of an external sinusoidal signal on system limit cycles. The theoretical results are verified by tests on an analogue computer.

Topics: Signals , Computers , Cycles
Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):327-333. doi:10.1115/1.3609604.

The partial differential equations describing distributed parameter systems may often be reduced to transcendental transfer functions with the aid of appropriate boundary conditions. In the analysis and synthesis of closed loop systems, the transcendental transfer functions have to be approximated in a suitable manner. In this paper, discrete-time model of distributed parameter systems is obtained. The model employs a sample and hold circuit in the loop. The response of the model system is compared with the response obtained by approximating the transcendental transfer function by root factor and other approximations. The stability of linear and nonlinear systems with distributed parameters is investigated by employing the Mikhailov stability criterion.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):334-340. doi:10.1115/1.3609605.

The effects of geometry, Reynolds number, and wall temperature on the separation of a fluid jet from a curved surface are investigated. The results are used to develop an electropneumatic signal converter which has no moving parts. Resistive heating is used to control the wall temperature, which in turn controls the jet separation from the curved surface.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):341-348. doi:10.1115/1.3609606.

Methods of using fluidic devices for sensing and controlling temperature were developed for gas turbine engine application. Frequency modulation and phase discrimination techniques were used both in the sensing of the temperature and in the comparison of the sensed temperature with the reference value. These techniques proved to be practical for measurement at a point and can be adapted for average temperature measurements. Temperature information was transmitted 5 ft by a pneumatic carrier frequency signal. Key fluidic circuitry was built and tested at temperatures ranging from 70 F to 2000 F. The temperature limitations were found to be those inherent in the material from which the devices were fabricated. No degradation of performance was noted at high temperatures.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):349-356. doi:10.1115/1.3609607.

The design of a sampled-data nonlinear system for three-axes acceleration measurement is discussed. The basic specific force equation is given, and the equation for the indicated velocity (integral of indicated acceleration) is derived to show the dependence of the error on the shape of the limit cycle trajectory. Emphasis is on how certain physical parameters, such as damping and dead zone width, may be chosen to fulfill the design requirement of minimum error in the indicated velocity output of the system. Expressions for design purposes relating physical parameters, such as damping, and phase plane switch line spacing to limit cycle period are derived. High limit cycle frequencies, or low periods, are shown to be less objectionable. An expression is given for the approximate dead zone width for minimum error from dynamic average and static offsets of the limit cycle trajectories. A multiscale-normalized phase plane technique is described and used to determine both the transient and final limit cycle trajectories for assumed step acceleration inputs at arbitrary points during an existing limit cycle.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):357-364. doi:10.1115/1.3609608.

A derivative of the output of a nonlinear element is taken with respect to the input. For commonly occurring nonlinearities which are piecewise linear, this derivative is sectionally constant with respect to the input. This property of the derivative is used to reduce the computational work required for deriving the describing function. The concept of this derivative is applied to the study of the effect of a high-frequency signal on the input-output relationship of a system containing a limiter with hysteresis. This signal may be regarded as an extra signal introduced into a system to improve the performance of the nonlinear component. The mathematical analysis of the effect is simplified if the extra signal is a triangular wave instead of sinusoidal. The extra triangular signal is applied to removing the jump phenomenon which exists in a feedback control system with a limiter with hysteresis.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):365-370. doi:10.1115/1.3609609.

The effects of nonuniform error quantization on feedback control system stability from the viewpoint of the existence of limit cycles are discussed. The existence of limit cycle is investigated for both step and ramp inputs and results are compared to actual system simulation. Guidelines are given for the selection of a quantizer to allow maximum velocity for a given number of error quantization levels with no loss of position accuracy.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):371-378. doi:10.1115/1.3609610.

The amplitude frequency response (transfer gain curve) of 0.170-in-ID blocked pneumatic lines of the type used in fluidic systems was experimentally determined. Several lengths (20 ft and less) of tubing at several mean pressures (10 to 40 psig) were studied over the frequency range of 1-1000 cps. The electric-pneumatic analogy was used to develop theoretical predictions of the gain curves. Correlation with experiment showed that a frequency-dependent resistance and a frequency-dependent conductance were required in the analogy when the signal frequency was somewhat greater than a characteristic frequency of the line. A high-frequency model, based on the work of Nichols, was developed; it predicted the resonant gains within 2 db and the resonant frequencies within 10 percent.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):379-383. doi:10.1115/1.3609611.

The optimum feedback control of controllable linear distributed stationary systems is discussed. A linear closed-loop system is assured by restricting the criterion to be the integral of quadratics in the state and control. Feedback is obtained by expansion of the linear closed-loop equation in terms of uncoupled modes. By incorporating symbolic functions into the formulation, one can treat boundary condition control and point observable systems that are null-delta controllable.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):385-390. doi:10.1115/1.3609615.

A necessary condition for the optimum control of a linear system with input delay or transportation lag is derived in terms of the system weighting function. For a general quadratic integrand in an integral-type index of performance and a linear time invariant system with input delay, an expression for the optimum controller is given in the Laplace domain in terms of the system transfer function. This expression simplifies the calculation of the optimum controller. The results are applicable to systems with or without input delay. An example is given to illustrate the procedure.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):393-398. doi:10.1115/1.3609619.

A method is presented for the determination of ordinary differential equations to describe the performance of existing lumped-parameter, time-invariant, nonlinear physical systems. It is assumed initially that the nonlinear elements can be described by products of continuous functions of system variables and these system variables themselves, which consist of the input and output of the system and their time derivatives. It is also assumed that the system input may be specified and that the output can be measured. The method yields graphical representations of unknown nonlinear functions in an assumed system differential equation. Examples illustrating the accuracy of the procedure are presented, and results obtained in the identification of two physical systems are given.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):399-405. doi:10.1115/1.3609620.

A method is introduced for finding a functional polynomial fit of a continuous functional. This method radically reduces the number of measured records required by straightforward techniques for the second-degree approximation.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):406-414. doi:10.1115/1.3609621.

A new type of pneumatic oscillator is presented which produces triangular waveforms of pressure versus time. The device can be used for both high power and low power applications. It utilizes a freely floating disk which translates back and forth a short distance inside a housing between two dynamically unstable flapper nozzle valves. A theoretical analysis of the performance and stability of the device is given. A simplified analysis and design procedure which accounts for load flow is shown. Several prototype units were successfully operated and the experimental results obtained with them are described and compared with the predicted results.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):415-422. doi:10.1115/1.3609622.

The purpose of this paper is to investigate the phenomena of the decay and dispersion of a disturbance pulse as it travels in a fluid along a pipe. A disturbance has associated with it deviations in the mean velocity and pressure. The disturbance pulse, therefore, may be considered as a velocity or a pressure pulse. By decay we mean here a decrease in the intensity, and by dispersion an increase in the width of the pulse with time. In an ideal nonviscous fluid contained in a rigid pipe, decay and dispersion do not occur, but are observed in the viscous fluids. A reflection of the pulse from a closed free end of the pipe also causes decay and dispersion. It is observed that for sufficiently low values of the coefficient of friction the viscous effects may be approximated by a pure decay without dispersion.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):423-432. doi:10.1115/1.3609623.

Proper selection and ordering of the variables of uniform, linear, one-dimensional distributed, dynamic models are shown to simplify their analysis, particularly when several simultaneous energy flows are coupled. Symmetric and asymmetric product variables are identified in pairs, leading toward criteria for system symmetry and reciprocity and formulas for the desired transmission matrices. Standard operational matrix techniques allow the identification of generalized wave-scattering variables, leading to decoupled equations. Application of the technique is demonstrated for simple systems, counterflow heal exchangers, the Bernoulli-Euler beam, and flexible fluid-carrying tubes.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):433-439. doi:10.1115/1.3609624.

A method is developed for generating Liapunov functions, V, which have the property that, along the trajectories of a system under consideration, the scalar functions V̇ or (V̇ − Vt ) take on a preassigned or desired form. The method applies to both autonomous and nonautonomous systems and complements and extends other known methods and techniques for generating Liapunov functions. The method is called a format method since it is based upon a fundamental vector-matrix equation or format, v = [D + P]f , which mathematically represents every vector v which satisfies the scalar product v ·f = (V̇ − Vt ). The method is readily applied to very general classes of systems as well as to special and particular systems. The format method is illustrated by generating Liapunov functions for autonomous and nonautonomous systems. Three examples are given. An explicit expression for V and V̇ for second-order systems is given in terms of the components of a system ẋ = f and any, arbitrary real function p(x ; t). A Liapunov function is generated for a more general class of third-order systems than any which has been given heretofore. Also, it is shown how the basic vector format may be applied to inverse Liapunov problems.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):440-443. doi:10.1115/1.3609628.

This paper presents a method to determine the transfer function and input impedance of a pressurized fluid piping system. Distributed parameters are used to arrive at a transfer function of a single line, and then block-diagram feedback methods are used to model the system. The input impedance is derived from the feedback model, and methods are presented for finding the flow and pressure at any point in the system.

Commentary by Dr. Valentin Fuster
J. Basic Eng. 1967;89(2):445-449. doi:10.1115/1.3609632.

Pulse testing of systems, where the input is a controlled function of time, is a well-established procedure. One data reduction technique numerically calculates the Fourier transforms of the input and the response. The complex ratio is the FT of the system function. It is shown (a) that the correction function usually employed to improve the respective FTs cancels out and is a useless operation; (b) that the numerical system function FT fails at high frequencies because the approximation to the integral fails to converge; (c) the upper limit in frequency is ωδ ≤ 0.1π, not π as predicted by the sampled data theorem; (d) generally it is necessary to use ten times as many data points as has been the practice to attain a given bandwidth.

Commentary by Dr. Valentin Fuster

DISCUSSIONS

Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster

ERRATA

Commentary by Dr. Valentin Fuster

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