Showing papers in "Ire Transactions on Automatic Control in 1959"

Complex-curve fitting

Abstract: The mathematical analysis of linear dynamic systems, based on experimental test results, often requires that the frequency response of the system be fitted by an algebraic expression. The form in which this expression is usually desired is that of a ratio of two frequency-dependent polynomials. In this paper, a method of evaluation of the polynomial coefficients is presented. It is based on the minimization of the weighted sum of the squares of the errors between the absolute magnitudes of the actual function and the polynomial ratio, taken at various values of frequency (the independent variable). The problem of the evaluation of the unknown coefficients is reduced to that of the numerical solution of certain determinants. The elements of these determinants are functions of the amplitude ratio and phase shift, taken at various values of frequency. This form of solution is particularly adaptable to digital computing methods, be-because of the simplicity in the required programming. The treatment is restricted to systems which have no poles on the imaginary axis; i.e., to systems having a finite, steady-state (zero frequency) magnitude.

On the general theory of control systems

Abstract: This paper deals with further advances of the author's recent work on optimal design of control systems and Wiener filters. Specifically, we consider the problem of designing a system to control a plant when (1) not all state variables are measurable, (2) the measured state variables are contaminated with noise, and (3) there are random disturbances. An explicit design procedure (well adapted to digital computation) is presented. In addition, some fundamental new concepts (controllability, observability, etc.) are introduced. A general theory of control systems is outlined which answers many basic questions (what is controllable? why? how?) and gives a highly efficient method of computation. This paper is to be published in the Proceedings of the First IFAC Moscow Congress by Butterworth Scientific Publications in 1960.

On adaptive control processes

Abstract: One of the most challenging areas in the field of automatic control is the design of automatic control devices that 'learn' to improve their performamce based upon experience, i.e., that can adapt themselves to circumstances as they find them. The military and commercial implications of such devices are impressive, and interest in the two main areas of research in the field of control, the USA and the USSR, runs high. Unfortunately, though, both theory and construction of adaptive controllers are in their infancy, and some time may pass before they are commonplace. Nonetheless, development at this time of adequate theories of processes of this nature is essential. The purpose of our paper is to show how the functional equation technique of a new mathematical discipline, dynamic programming, can be used in the formulation and solution of a variety of optimization problems concerning the design of adaptive devices. Although, occasionally, a solution in closed form can be obtained, in general, numerical solution via the use of high-speed digital computers is contemplated. We discuss here the closely allied problems of formulating adaptive control processes in precise mathematical terms and of presenting feasible computational algoritbms for determining numerical solutioms. To illustrate the general concepts, consider a system which is governed by the inhomogeneous Van der Pol equation \ddot{x} + \mu(x^{2} - 1) \dot{x} + x = r(t), 0 \leq t \leq T , where r(t) is a random function whose statistical properties are only partially known to a feedback control device which seeks to keep the system near the unstable equilibrium state x = 0, \dot{x} = 0 . It proposes to do this by selecting the value of μ as a function of the state of the system at time t , and the time t itself. By observing the random process r(t) , the controller may, with the passage of time, infer more and more concerning the statistical properties of the function r(t) and thus may be expected to improve the quality of its control decisions. In this way the controller adapts itself to circumstances as it finds them. The process is thus an interesting example of adaptive control, and, conceivably, with some immediate applications. Lastly, some areas of this general domain requiring additional research are indicated.

Control system analysis and design via the second method of lyapunov: (I) continuous-time systems (II) discrete time systems

Abstract: I - Continuous-Time Systems - The "second method of Lyapunov is the most general approach currently in the theory of stability of dynamic systems. After a rigorous exposition of the fundamental concepts of this theory, applications are made to (a) stability of linear stationary, linear nonstationary, and nonlinear systems; (b) estimation of transient behavior; (c) control-system optimization; (d) design of relay servos. The discussion is essentially self-contained, with emphasis on the thorough development of the principal ideas and mathematical tools. Only systems governed by differential equations are treated here. Systems governed by difference equations are the subject of a companion paper. II - Discrete-Time Systems-The second method of Lyapunov is applied to the study of discrete-time (sampled-data) systems. With minor variations, the discussion parallels that of the companion paper on continuous-time systems. Theorems are stated in full but motivation, proofs, example, and so on, are given only when they differ materially from their counterparts in the continuous-time case. Part I has been published by the American Society of Mechanical Engineers as Paper No. 59-NAC-2. Part II has been published by the American Society of Mechanical Engineers as Paper No. 59-NAC-3.

On adaptive control systems

Abstract: An attempt is made in the subject paper to evolve a basic philosophy for adaptive control systems. A method is described for determining the system impulse response from measurements of instantaneous system input and output. The impulse response is expanded in a Taylor series, to facilitate solution of the convolution integral. From a knowledge of the impulse response and the system error, the necessary correction to the system forcing function is determined, in a manner similar to that used for determination of the impulse response. The techniques developed are applied to two systems - one stable and one unstable. Curves of the results are presented.

Fundamental theory of automatic linear feedback control systems

Abstract: The reasons for using feedback are reviewed. The beneficial aspects of feedback are quantitatively expressed in sensitivity functions and noise transmission functions. The physical constraints on the controlled process (or plant) determine the maximum number of independent functions realizable. Any configuration with the same number of degrees of freedom may be used. With this approach the study of conditional feedback, model feedback, combined positive and negative feedback, etc. is of secondary interest. The benefits of feedback are paid for in gain-bandwidth of active elements over and above what is needed to physically do the job. The minimum price that must be paid is independent of configuration. The system with two degrees of freedom is studied in detail. Two methods are presented for the precise design of a system that will be as insensitive as may be desired to large parameter variations. One method uses root-locus techniques and is suitable for systems with a small number of dominant poles and zeros. The second method is based on frequency response and can be used for systems of any complexity. Numerical examples are given.

Signal stabilization of self-oscillating systems

Abstract: The hunt (self-oscillations) of a physical system may often be removed by the introduction of an appropriate stabilizing signal which changes the open loop gain of a closed loop system in a non-linear manner. The theory of stabilization developed in this paper explains experimental results reported by R. Oldenburger in 1957. With the aid of Fourier series the designer can determine the periodic signal to be inserted at one point in a loop to yield a desired stabilizing input to a nonlinear element in the loop. This is illustrated by sinusoidal and triangular inputs to a nonlinear element. An example where a limiter is the only nonlinearity, is employed to illustrate the theory. The input-output characteristics of non-linear elements are in practice always modified by the presence of extra signals such as "noise." Further, nonlinearities are always present in physical systems. The effect of extra signals on nonlinearities and system performance is thus of concern in the general study of physical systems, regardless of whether or not the problem of stability is involved. The results of this paper for the problem of stability extend readily to the problem of system performance for arbitrary disturbances. This paper is to be published in the Proceedings of the First IFAC Moscow Congress by Butterworth Scientific Publications, in 1960.

Extension of phase plane analysis to quantized systems

Abstract: The increasing applications of numerical control of processes have created a need for new methods of synthesis of control equipment. The method presented is applicable to systems commanded by discretely valued inputs, and processes whose outputs may be similarly quantized. Periodic sampling is not required. The most suitable sampling is by transmission of only significant data, as the new value obtained when the data are changed by a given increment. In certain cases, transmission of data by this means can be used to increase channel capacity. When the data are so quantized, the error signal is constrained to a finite number of discrete values, each of which may be associated with an area in the phase plane. Within each such area, the trajectories of any process subject to phase plane representation are a family of parallel curves. Thus, analytic synthesis may be simplified in the case of certain nonlinear processes. Graphical design is facilitated without requiring deduction of a mathematical representation of the process. The method is illustrated by synthesis of several systems involving a simple linear process.

The analysis of sampled-data control systems with a periodically time-varying sampling rate

Abstract: The z-transform is used to solve sampled-data systems which have a periodically time-varying sampling rate, i.e., systems which have a repetitive sampling pattern in which the time duration between the individual samples is not constant. Such systems are described by linear difference equations with periodic coefficients; however, the difference equation which describes the system at sampling instants corresponding to KN, where N is the period of the coefficients of the difference equation, and K = 0, 1, 2, …, is a linear difference equation with constant coefficients. Thus by forming a series of difference equations which individually describe the system at sampling instants corresponding to KN, KN+1, KN+2, …, (K+1)N−1, the time varying features of the system are in essence removed from the analysis and the z-transform can be used to solve the resulting constant coefficient difference equations. Also, the response between the sampling instants can be found using the solutions of these difference equations. The method presented is straightforward and can be used to analyze any linear sampled-data system with a periodic sampling pattern. Such a condition could occur, for example, when a computer is time shared by more than one system or in some telemetering devices which periodically give to control systems information on quantities being monitored but in which the desired information is not available at equally spaced intervals of time. This method can also be used to obtain an approximate solution for the output of any linear system which is excited by a periodic but nonsinusoidal forcing function and, because of the flexibility of the sampling pattern, should give more accurate results than an approximation which uses equally spaced samples. In this analysis, only periodicity of the sampling pattern is assumed, and no relationship between the individual sampling intervals is required. A few examples have been introduced to illustrate the analytical procedure and the features of the response of a system to sinusoidal inputs is indicated in one of the examples.

A parameter tracking servo for adaptive control systems

Abstract: This paper describes one very general approach to the design of adaptive control systems. The particular systems considered are process adaptive. The dynamic characteristics of the physical process are determined by the parameter tracking servo. The parameters thus determined are used to program the process' controller. The parameter tracking servo is a closed loop self-adjusting system. It consists of the following elements; 1) the physical process, 2) the learning model, 3) the adjusting mechanism. The learning model and the physical process are subjected to the same input signals. Their outputs are compared and the resultant error is fed to the adjusting mechanism where some function of this error is used to adjust the parameters of the learning model. The mechanism will continuously track the parameters of the physical process as they change with time in some unknown manner. The adjusting mechanism operates on an approximation to the method of steepest descent. These equations are derived for a first order process and the over-all systems is analyzed. The equations describing the tracking servo's operation are both non-linear and non-autonamous. System response as a function of input signal, gain, and error function are described analytically. Experimental results are included to demonstrate the validity of the analytic solutions.

Characteristics of the human operator in simple manual control systems

Abstract: The characteristics of simple pursuit and compensatory manual control systems were measured with, a family of gaussian input signals having power-density spectra that covered a range of bandwidths, center frequencies, and some variety of shapes. The experimental results, presented in the form of graphs, show the nature of the dependence of human operator characteristics upon input-signal characteristics. The superiority of pursuit systems over compensatory systems is clearly demonstrated. Simple analytic models that approximate these measured results are derived for both systems. The compensatory model is highly developed and relations among its parameters and those of the input have been obtained. The pursuit model is not so well developed and only approximate relations among its parameters and the input parameters have been found. The measured results and the analytic models together provide a description of manual control systems that should be useful in design of control systems.

Reaction wheel attitude control for space vehicles

Abstract: An attitude control system consisting of motor-driven inertial wheels in conjunction with an over-riding mass ejection system is proposed for use in space vehicles. Control by mass ejection is usedto compensate for initial distrubances during separation from the booster, and for removal of unwanted momentum stored in the wheels. The use of reaction wheels permits fine, damped attitude control. A laboratory model of a single-axis control system was constructed for experimentation and evaluation. The choice of a suitable platform configuration, selection of a prime mover for the inertial wheel, and the philosophy in the design of the electronics and pneumatics are discussed. Emphasis was placed upon minimization of weight and power consumption. System evaluation includes a discussion of efficiencies, reliability, and torque-speed-power relationships. Sources of disturbances, methods of sensing, and general equations of motion are presented in the Appendix.

Adaptive or self-optimizing control systems — A bibliography

Abstract: Adaptive, self-adjusting, or self-optimizing servos are designed for operation in a slowly-changing environment as opposed to servos intended for a fixed environment Optimalizer controls and similar devices which hunt for and adjust to a pre-set optimum condition are considered as adaptive servos The references which follow are a selective sampling of the latest material on this subject taken from the open literature and technical reports Servos which are designed to operate at some pre-set optimum based on pre-filtering of input signals evolved from Wiener's optimum filter theory, and, accordingly, references to the latter topic have also been included

D-decomposition analysis of automatic control systems

Abstract: Development of the basic theory of D-decomposition by a general polynomial expression, the characteristic function of a linear lumped parameter automatic control system, is followed by account of: the basic theory of parametric stability; unity-feedback variable-gain analysis, with subsequent generalization to numeric and frequency-dependent feedback; transient response determination; and sampled-data system analysis, as effected by D-decomposition theory. Application of theory in practice is illustrated by numerous illustrative examples.

Transportation lag — An annotated bibliography

Abstract: This bibliography is an attempt to survey the writings, in various fields of study, which deal with functions with retarded argument. This problem is characterized by a response to a stimulus which is identical to a normal response except that it is delayed in time. Some situations in which this transportation lag occurs include process control (distance-velocity lag), control of thermal systems (including control of nuclear reactors), rocket motor combustion (ignition and combustion lags), traveling waves, magnetic amplifiers, human link in control systems (reaction time), high-speed aerodynamic control, and economic systems (period of gestation or production lag). A bibliography is presented which lists and abstracts a number of the references dealing with this problem. Relevant references in two major categories have been omitted in this bibliography. Foreign language works and pure mathematical treatments have not been listed. A comprehensive list of these references may be found in an excellent bibliography on the subject.1 The format of the reference listings is an adaptation of standard bibliographical format convenient to the type of material presented. References are lettered according to the author's surname, rather than numbered, to facilitate future additions without breaking the continuity of the reference notation.

Adaptive flight control

Abstract: Need for adaptive control systems is explained, and results obtainable with two forms are discussed. Recognized approaches to achieving self-adaptive action are indicated; integrating and nonintegrating techniques are compared. One non-integrating form, the model-bistable system, is taken as an example, and analysis of its operation is developed. Its application to flight control is described, and its limitations in this and other applications are discussed. Effects of system performance requirements, including nature of input signals and disturbances, on the choice of adaptive control type are considered. Relations of mechanistic adaptive control systems to those found in the biological organism are discussed. This paper is to be published in the Proceedings of the First IFAC Moscow Congress by Butterworth Scientific Publications in 1960.

A dynamic programming approach to adaptive control processes

Abstract: In many multi-stage decision processes we face the problem of dealing with random variables whoe distributions are initially imperfectly known, but which become known with increasing accuracy as the process continues. In this paper we shall show how Dynamic Programming may be used to treat a class of such problems, which are currently called adaptive processes. After discussing the general theory, we shall illustrate the techniques by a specific example. For this example we derive simple computational algorithms, which are typical of those obtained for the whole class of problems under consideration.

On the analysis of bi-stable control systems

Abstract: The basic theory of bi-stable oscillating loop control is first sumaarized using standard describing function concepts. The effective transfer characteristic of the hi-stable switch is defined in terms of the ratio of both amplitude and frequency of the control error signal to amplitude and frequency of the loop oscillation signal. The effect of variation in loop oscillation amplitude with low frequency control error amplitude is considered. The relationship between switch gain at control frequencies and switch gain at oscillation frequency is noted. The inherent insensitivity of the bistable oscillating system to variations in linear loop parameters is discussed. The effect of variation in loop gain of the bi-stable system is compared with the effect of equal variations in a similar linear system. The possibility of bi-stable loop oscillation at more than one frequency is considered. The conditions are defined which insure oscillation at the intended oscillation frequency. Analog simulation tests used to verify these loop oscillation criteria are outlined.

Evaluating residues and coefficients of high order poles

Abstract: The powerful Laplace transform method for transient analysis by the partial fraction expansion technique has become quite popular in the fields of circuit and servo design. The theory of residues is usually used to find the coefficients of these fractions. The process is quite simple until second and higher order poles are included in the denominator. Previously, this has required a return to the calculus to find the additional coefficients required. This paper discloses a simple technique for finding these additional coefficients by algebraic processes. As a result both manual and machine computation can be performed more easily. The technique is described and its mathematical basis is rigorously proven.

The analysis of demodulating compensating networks

Abstract: The subject of this paper is the analysis of periodically switched electric networks. A method for determining a sampled-data system which is equivalent to a given member of a class of periodically switched electric networks is first presented. A technique for determining the response of the system to a given input function is then discussed in detail. Next it is shown that it is possible to identify a time-invariant electric network with the same response to a step-function input as the envelope of the response of the switched network exhibits when the input to the switched network is a suddenly applied constant-amplitude sinusoid at the switching frequency. This "equivalent" d-c network can be utilized to advantage in the design of carrier systems in which periodically switched electric networks are employed as compensators. Examples are then presented to illustrate the application of the theory, and the results obtained by this method are compared with those obtained by a different method, proposed earlier. Finally it is pointed out that the response calculated by the use of the method presented in this paper is theoretically exact and that the technique can be extended to encompass a wide class of periodically switched linear systems.

General approach to control theory based on the methods of Lyapunov

Abstract: This discussion is an introduction to the second method of Lyapunov in determining the stability of linear and nonlinear control systems. Although almost universally referred to in Russia and although it is becoming an increasingly important concept, this method has not been used in English-speaking countries because of the lack of suitable translations. This introduction to the method will not be published because it is essentially a condensation and introduction of the authors' paper given in Session IV.

On the optimum synthesis of multipole control systems in the weiner sense

Abstract: This paper is concerned with obtaining the optimum system in the Weiner sense for the multipole system. Earlier literature has shown how to obtain the mean-square value of the error when the multipole system transfer function has been specified, but thus far no published work has shown how to solve the synthesis problem, in general, for this case. The principal reason that this problem has appeared to be impossible of analytic solution thus far for cross correlation between the inputs is based on the fact that the usual variational approach results in a set of untractable simultaneous integral equations involving many complicated cross products of the desired weighting functions and the variational functions. The synthesis problem for the system is first solved for the case in which there is no correlation between the inputs to the various terminals. The result for the optimum weighting functions in this case is presented in equation (24), and the resultant mean-squared value of the error is shown in equation (25). Following this, the far more complicated case of the synthesis problem when the inputs to all the various terminals are correlated is considered. In this case, a rather unique technique is utilized to avoid the difficulties inherent in the use of the usual variational techniques. Through the technique utilized in this paper, the usual set of untractable simultaneous integral equations is completely avoided, and instead a set of ordinary algebraic equations results. The set of equations for this case is shown in equation (64), and in matrix form in equation (65). The resultant solution for the optimum physically realizable transfer functions is shown in equation (77). It is also shown, as a check, that the solution for the case of correlated inputs reduces to the solution obtained for the case of uncorrelated inputs. The paper then concludes with an illustrative example for the more complicated case of correlated inputs. The possibilities of applications of the results of this paper to such fields as the guidance and control of astronautical vehicles, military fire control systems, bombing navigation systems, process control systems, automatic milling machines, air traffic control, nuclear reactor control, etc., are fairly evident.

The American Automatic Control Council

Abstract: An international meeting of automatic control was held in Heidelberg, Germany, September 25–29, 1956, under the auspices of the joint control Committee of the German electrical and mechanical engineering societies. This committee is known as the VDI/VDE — Fachgruppe Regelungstechnik. Other international meetings on automatic control had been held previously. On July 16–21, 1951, one took place in Cranfield, England. This was called the “Conference on Automatic Control.” It was followed by the “Frequency Response Symposium” in New York City, December 1–2, 1953. The Heidelberg meetings were such a success that a demand arose for an international federation which could assist the host country in the matter of publicity, securing papers, and other aspects of the organization of future international control meetings.

On optimal computer control

Abstract: This paper is concerned with the optimal computer control of a linear dynamic element of plant. It is required that the outputs of the plant exactly equal a set of desired outputs at the uniformly spaced time instants, t, t_{N}, t_{2N}, ... . The system is optimal in the sense that a quadratic form of the control effort, which is proportional to the control energy, is a minimum. The optimal system is shown to be a linear, periodically time-varying feedback system, in which the generation of the control input is related to the solution of the adjoint equations of the plant. The paper concludes with a discussion of the effects of uncontrolled inputs and certain problems encountered in making the system adapt to slow parameter variations. This paper is to be published in the Proceedings of the First IFAC Moscow Congress by Butterworth Scientific Publications in 1960.

On the use of growing harmonic exponentials to identify static nonlinear operators

Abstract: The following paper describes a method of obtaining a polynomial characteristic function for a nonlinear static system. This function, F(x) = hx + mx^{2} + dx^{3} , is obtained by the application of a growing exponential x = \exp(t) to the input of the system and the filtering of the output h \exp(t) + m \exp(2t) + d \exp(3t) , into its separate components h \exp(t), m \exp(2t) , and d \exp(3t) . The values of these three components at t = 0 are the polynomial coefficients h, m , and d respectively. The identification of systems not exactly describable by a cubic gives rise to an error minimization problem; the technique described in this paper minimizes the weighted mean-square error, with a weighting function 1/x . This method is compared with the more widely known sinusoidal analysis of nonlinear systems. Experimental results are given.

Coherent optical data processing

Abstract: Coherent optical systems, which utilize the wave nature of light and the consequent diffraction phenomena, may often be used to supplement or even replace complex electronic equipment. Such systems are particularly adapted to the performance of certain linear mathematical operations, particularly those of an integral transform nature such as spectral analysis, convolution, auto- and cross-correlation, and matched filtering. The two-dimensional nature of optical systems, contrasted with the inherent one-dimensional nature of an electronic channel, allows a great reduction in equipment complexity for certain classes of operations. This paper discusses the theory behind optical channels and filters as outlined above, and also illustrates simple multi-channel optical systems which can carry out representative operations.

Random noise with bias signals in nonlinear devices

Abstract: A number of investigations have been made in recent years about the transmission of Gaussian noise through nonlinear devices. In many cases, simplification or approximations were needed to make analytical solutions possible, and only zero-average Gaussian input signals were used when the results were applied to feedback control systems. This paper presents a different approach to the problem of noise transmission through non-linear single-valued elements. Basically, amplitudes removed by nonlinear saturation or deadzones are replaced by impulses in the amplitude distribution functions of the output signals, and the resulting first and second moments of the output distribution are computed to yield the average and rms value of the output signal. The solution may be found by graphical or mathematical integration, a visual representation of the phenomenon is obtained, and input signals with any distributions having non-zero average values may be considered. It is shown that there is an equivalent transmission function or describing function for the average value of the noise, another for the rms value, and that one is a function of the other. Examples of the functions are given and the simpler functions with zero-average values are compared to the results obtained by other methods. Finally, the application of the noise describing functions to feedback control systems is discussed. Theoretical results are compared with those obtained from analog simulations.

Techniques for the optimum synthesis of multipole control systems with random processes as inputs

Abstract: This report considers the general problem of obtaining the optimum multipole system when the inputs to the system are stationary random processes. The system under investigation is linear and time invariant. The input to each terminal shall consist of signal and noise. The synthesis procedure is carried out under the basis that a fixed plant must be compensated in order to perform certain desired tasks. The design criterion employed is the minimum mean square error between actual outputs and ideal outputs of the system. A set of integral equations shall be obtained, which can be converted into algebraic equations through transformation. By solving these equations and using the method of undetermined coefficients, the transfer functions of the compensation can be uniquely determined.

Attitude control of space vehicles

Abstract: The problem of sensing and controlling the angular orientation of a space vehicle is discussed. Optical, radio, and inertial methods of angle-sensing are described, and application of control torques by reaction jets, flywheels, radiation pressure, and gravitational gradient are compared. Requirements for angular accuracy likely to be imposed by maneuvering, performing measurements for navigation, making scientific measurements, radio communication, and solar cell orientation are described. Attitude control requirements and techniques for earth satellites and deep-space probes are compared. This discussion will not be published.

Multi-loop automatic temperature control system design for fluid dynamics facility having several long transport delays

Abstract: This paper describes the systems analysis and design of an automatic temperature control system in the world's largest fluid dynamics testing complex consuming 470 megawatts of drive power. The plant treated in this paper can be described mathematically as a 110th order system having 130 feedback loops. Facility component characteristics are mostly non-linear, with prominent distributed parameters and ten long, variable fluid transport delay times. Important simplifications introduced to make the analysis and design work tractable include: (a) Application of the mathematical analogy between the theory of electric transmission lines and gas flow to validate the separation of temperature and pressure variations, (b) Selection of control frequency to permit lumped-parameter representation, (c) Design and construction of special computer devices and circuitry to simulate transport delays from 0.2 to 141 seconds in length. These valid simplifications reduce the plant to a 14th order system having about 50 feedback loops. The simplified system is simulated on an analog computer for economical synthesis and design of the controllers.