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Automatic tracking systems are conventionally analysed and optimized on the basis of assumed linear operation, deterministic inputs, and the use of the minimum mean square error criterion in the presence of noise. This approach is limited by (a) the fact that the error detector is linear for only a small range of error signals, and (b) the mean square error criterion is not pertinent for a large class of applications. This paper reports an analysis of automatic tracking systems which overcomes in part these two limitations. The Fokker-Planck diffusion equation is used to find the probability distribution of the output of the tracking system and from this the probability of loss is found as a function of various system parameters. A digital computer was used to obtain numerical results for non-linear error detector characteristics.

This paper studies the optimum control of multi-variable systems by considering the behaviour of the two-variable on-off system described by the equationx¨+Ax˙=u

The purpose of this paper is to investigate the optimum design of predictor control systems subject to statistical or deterministic inputs, under appropriate constraints on the control signal. The control signal constraints treated in this paper are: (a) the magnitude of the control signal equal to a constant (this is the bang-bang control problem), and (b) the magnitude of the control signal less than or equal to some constant.

In this paper the concept of a functional Taylor series is applied directly to the functional relation between the solution and forcing function of a large class of differential equations, yielding the Volterra series. A simple, convenient method is presented for calculating the Volterra kernels in the Volterra series representation of responses with zero and non-zero initial conditions, for a broad class of time-varying non-linear physical systems.

This paper discusses the applicability of digital techniques to various optimization problems and not only demonstrates the utility of these techniques, but confirms that digital computers may be used for the study of plant behaviour.

Time-optimal control systems with random noise disturbances are discussed in the paper. It is shown that for a class of control systems the controller programme which provides minimum time of the transient for expected values of system coordinates, assures also the minimum expected value of the transient time. The construction method of switching surfaces to provide the minimum of expected transient time based upon the conventional equation of optimal switching surface, and characteristics of different noisy channels of the system, is given. The examples of the second-order time-optimal systems with noise disturbances are considered. Results of analogue computer simulation of the control systems with plant transfer function in the presence of random noise disturbances applied in various points of the system are given.

The article describes the general theory of the synthesis of hybrid, multi-parameter control systems acted upon by disturbances of an arbitrary form and with constant command input signals. The control loops contain controllers which operate in parallel and can be realized by means of an automatic digital computer and continuously acting controllers. A solution is presented according to which disturbances uk effect only the controlled variables xi, i = k, and the degree of this effect can be limited in accordance with the selected criterion of the quality of control, or according to other suitable conditions of control. Control loops of this kind are called quasi-invariant control loops. The paper describes the application of the criterion of the finite number of control steps and of the criterion of the least square of deviations. The validity of the solution of the Wiener-Hopf integral equation is extended to control loops containing digital computers.

The first part of the paper deals with the general problem of forcing a non-linear process to follow a specified (discrete-time) trajectory. The performance is measured by a functional of the vector error and input sequences and the inputs are assumed to be constrained. By the use of standard methods of calculus the optimum controller is shown to require the solution of a system of difference equations of twice the order of the process to be controlled with boundary conditions at the terminal instant dependent on the performance measure. A simple geometric interpretation of the problem is given.

This paper unifies and extends the ideas and techniques used by the authors in relation with the minimal time regulator problem. The system considered is a multiple input system described by a differential equation of the form $ = F(x) + Bu(t) where u is either a PAM or PWM bounded control vector. This paper consists of five sections. In the first, the problem is carefully stated in a suitable general setting, and the concept of optimal strategy is defined; in the second, an optimal strategy for a multiple input PAM system with a completely controllable linear plant is described fully, and a special case is discussed; in the third, the concept of equivalent optimal strategies is defined, and sufficient conditions for equivalence are indicated. In the fourth section equivalence is used to derive the optimal strategy of a PWM system whose linear plant has real and distinct eigenvalues, and finally in the last section, some extensions are mentioned.

Necessary and sufficient algebraic conditions for the roots of a real polynomial to lie inside the unit circle are given in table form. In this form, the constraints are obtained only by evaluation of second-order determinants. The connection and identity existing between the stability criterion established in this paper and that of a previously obtained criterion1 are discussed. The usefulness of the table may be found in those cases where the coefficients of the real polynomial are given in numbers. It is similar to Routh's table obtained for the continuous case.

The analysis of non-linear oscillations in sampled-data control systems is treated. The system considered here has the non-linear element in discrete signal portions. In the first half of the paper the strict analytical method is proposed. By using it, the periodic equilibrium states and their stability can be strictly analysed, without any assumption on the waveform, if the period of the oscillation is an integer multiple of the sampling period.

One of the more important applications of the theory of sampled data control systems is that of the design of a compensator for a feedback system when the characteristics of the plant are specified. This problem is relatively simple when a discrete compensator is used. It is, however, much more difficult in the case of a continuous compensator because of the impossibility of separating in the overall operator or transfer function the contributions of the plant and compensator operators. Such solutions as have been proposed either use approximate methods, or introduce extra samplers or subsidiary feedback loops into the system. In this paper a graphical method is described for determining the operator of a simple continuous compensator for a given plant operator, assuming that the order of the system and the poles of the overall operator are specified. It is assumed, moreover, that the compensator must be physically realizable with a plant whose load has finite inertia.

In this paper the problems connected with the synthesis of optimum pulse systems for the control of dynamic media by means of linear models are considered. These problems are stated, using the mathematical tool of polynomial equations, which make possible the synthesis of efficient pulse systems. The analytical conditions for the efficiency of the pulse systems, which combine the usual conditions of stability of linear systems with the conditions of ‘approximation’ of modelling, are formulated. The inadequacy of the usual criteria of stability for the synthesis of efficient systems is demonstrated. An outline is given of the possible applications of the method of polynomial equations in the theory of automatic control.

The paper describes a new class of discrete data control systems in which a portion of the system makes use of pulse frequency modulation. This concept arose as an abstraction from the study of neural communication links in physiological control systems. An integral pulse frequency modulator is defined as one which emits a standard pulse whenever the integral of the input variable reaches a threshold value. The frequency of the output pulse train varies in a linear manner with the input magnitude. Modulators producing pulse trains of one or two signs are introduced and approximate frequency characteristics are obtained; they have discontinuous and discrete features, and depend upon signal amplitude, frequency, and initial phase angle. Feedback control systems making use of single-signed modulators in two parallel feedforward paths are described, and some of the unique stability considerations are studied. Continuous systems that approximate these stable discrete-data systems are constructed, and their transient properties considered.

The design of the digital controller frequently follows the aim to obtain the system with the fastest response. By the practical application of this criterion, cases are often encountered where the transient error, caused by an input of typical form, has a very short duration but an inadmissible magnitude. This shortcoming is removed by the numerical method of controller design presented in the paper.

This paper extends the theory of dual-input systems to the problem of limiting the activity of an intermediate variable in the system in order to avoid saturation or excessive power consumption. The derivation of the general dual-input Wiener filter which includes the constraint is first outlined and discussed and then three degenerate cases are completely solved. The Lagrange multiplier approach is used to handle the constraint mathematically. To demonstrate completely the concepts involved, a numerical example for a case is shown in detail. The numerical example demonstrates how one might ‘trade off’ or compromise in an actual system design in order to find the ‘best’ solution.

The paper shows that the classical method, due to Quine and others, of simplification of dipoles with contacts, is a consequence of axioms I—V and may therefore be applied not only to dipoles with series or parallel contacts and to circuits with diodes, but also to circuits with triodes, transistors or cryotrons.

The mathematical equations for optimal and adaptive control are now known. However, comprehensive methods for their numerical solution are not available. This paper gives a discussion of approximate computer methods of finding numerical solutions to suitable engineering accuracy. The importance of setting up the problem in a suitable form is stressed, and illustrated with an example. Three explicit approximation techniques are described, one a trajectory method, and the others function space methods. For each technique a worked example is given.

In this paper the optimal time problem is considered for multiple input systems when each input may be subject to a separate and possibly different type of constraint.

The paper considers: (1) two linear filters ϕ1 and ϕ2 connected in parallel, with an input signal Θ (t), and (2) closed-loop systems for adjusting the parameters Xi of filter ϕ2, which employ a search modulation Δxi (t) of these parameters. The self-adjustment network includes a detector (linear or square-law) for the output quantity of the filter ϕ2 – ϕ1, phase discriminators and averaging filters Wϕ.

Plants with distributed parameters are widespread in many branches of engineering. The problem of designing control systems which are the best in a given sense for such plants requires the development of new techniques of optimal control. Control systems with distributed parameters have a number of specific properties that make investigation into them very complex compared with lumped-constant systems. This paper gives a statement of optimal control problems on systems with distributed parameters for a fairly broad class of systems described by non-linear integral equations under arbitrary constraints. It formulates a maximum principle for this case, which gives the necessary conditions for determining optimal controls.

This paper deals with the solution of optimum control problems, using Pontryagin's maximum principle. Since an optimum control system which completes the transient response in minimum time behaves like a relay servo, the important task of the controller is to reverse the sign of the manipulated variable at the proper instants according to the operating condition of the system. A set of points at which the sign of the manipulated variable is to be reversed forms a switching surface in the phase space. In this paper the optimum switching surface of a linear third-order system is investigated. Also dealt with is the optimum switching curve of a non-linear control system governed by van der Pol's equation with an additional forcing term.

In this paper some new results concerning analysis and synthesis of time-optimal control systems are given. We assume that the motion of the controlled plant is described by linear ordinary differential equations with variable or constant coefficients, and the final states of the system form a convex region of phase space. Geometric properties of the isochronous region of these systems in phase space and the facts connected with these which have fundamental meaning for the solution of the synthesis problem are discussed. Moreover the facts demonstrated which are formulated in theorems have also their independent interests. At the end of this paper is given the method of synthesis of the optimal control function for constant as well as with variable coefficients, and illustrative examples.

Dynamic programming theory will solve, in principle, many of the problems connected with optimal systems synthesis, but the number of complex problems actually solved with this method is not large, because the calculations become more and more difficult the more complicated the problem becomes. Introduction of ‘sufficient coordinates’, on which the risk function depends, eases the situation, the ‘sufficient coordinates’ forming the space in which the Bellman equation is considered. In the paper a non-trivial example illustrates the effectiveness of introducing sufficient coordinates. In the example the sufficient coordinates are a combination of a posteriori probabilities and the dynamic variable of the controlled plant.

In Pontryagin's maximum principle and in ‘bang-bang’ control, the manipulated variable is assumed to have no inertia, so that its position changes instantaneously from one value to another. In practice this never happens, for instance in aircraft controls the elevators and ailerons are limited in speed and displacement. The problem is a special case of the more general problem of optimal control in bounded phase space. In this paper a method is given whereby the problem is treated by a limiting process. In place of the rigid bound, a cost function with a multiplied K is introduced for regions beyond the boundaries in phase space. It is shown that in the limit of K approaching infinity, both the added cost and the maximum excursion of the optimal path beyond the boundaries approach zero, thus deriving the optimal control condition. The main results of the paper are stated in the form of two theorems. Also considered is the practical significance of the results for systems with multiple saturation.

In this paper consideration is given to the system of differential equationsdxdt=f(x,η,t)+u(c,y,t)

The paper considers invariancy in the automatic control systems subject to external disturbances with known statistics. Invariancy conditions established by means of K(D) transform theory are generalized for systems with known disturbance statistics.

The paper describes the methods of designing automatic control systems with constant structures and constant parameters, the properties of which are equivalent to one of the adaptive systems.

It seems impossible to select the ‘best’ self-adjusting system out of the many proposals, but it is possible to compare them from a certain point of view. In this paper the point of view is the time required in measuring the parameters to be adjusted, considering random input signals. Three devices are compared which adjust one parameter of a control system in such a way that the R. M. S. error becomes a minimum.

The paper deals with the question of the use of self-learning processes in adaptive systems with least initial information. The mathematical expression, based on the phenomenon of entropy by which the dynamics of the learning and self-learning processes are experimentally determined, is given in the paper. Also introduced is the phenomenon of rate of learning which can give the value of the processes. Some concrete examples of the learning of automata and some living beings are discussed. In the paper are given some experimental data of the results of the self-learning system making the automatic synthesis of the corrective device. According to the experimental data the processes of searching can be significantly lessened by using the methods of learning of the adaptive system. The proposed mathematical expression can be used for the description of different learning processes and makes it possible to achieve some analogies between them. Experiments connected with the self-learning of the adaptive system were carried out on the special electronic machine of the Institute of Electronics, Automatics and Telemechanics of the Academy of Sciences of Georgia.

This paper considers an approach to the synthesis of an adaptive control system that provides for rapid adaptations. It leads to a model adaptive control system, but not to a model comparison system. A model is used as a dominant operator, and then the dynamics of process may be neglected. Time-domain synthesis of dominant operator is introduced. In the time-domain synthesis, performance index approach and model comparison approach may be transferred to the desired form of differential equation of error signal. Advantages of this type of adaptive control system are studied. An example and some experimental results are given.

The paper gives the results of a study of a system comprising a digital computer and a plant whose transfer function varies in time and is previously unknown. The plant and the digital computer are interlinked via transforming units. The system uses an adaptive control programme by which the coefficients of the plant transfer function are determined (an analogue of the plant is obtained). Control of the plant is effected in such a way that a criterion—the mean modulus of the error—is minimized. Determination of the plant transfer function coefficients is effected during the operation of the system.

The paper is devoted to the problem of teaching automatic systems pattern recognition. At the basis of the experiments that were conducted with regard to teaching, lies a profound and original hypothesis relating to the teaching process—a so-called ‘compactness hypothesis’. This hypothesis is interesting in that it opens a path to the mathematical description of the essence involved in the teaching process, which description is not attached to any concrete technical realization of the learning system nor to any concrete pattern type. At the same time, it prompted the experiments described in the present paper, and also made it possible to comprehend already-known studies on teaching.

The paper considers the method of auxiliary operator for the determination of the gradient of index of the quality of control with respect to tuned parameters for continuous self-tuning. A theoretical investigation is given of the dynamics and precision of the extremal tuning of parameters of an automatic control system with the utilization of a method of auxiliary operator. The effectiveness of the considered system of self-tuning with random disturbances is demonstrated.

This paper deals with the structure of a special purpose computer for the automatic search of the optimum operating condition of a process consisting of one time constant and a delay. The structure proposed is independent of the dynamics of the system and it is not necessary to identify the process dynamics in order to apply this method.

Control in closed systems, with incomplete a priori information about the plant, should, in the general case, include two types of action on the plant: (a) tentative actions, ‘experiments’, with the controlled plant, whose problem consists in studying it; (b) directing actions, whose task consists in adjusting the plant to the required operating conditions. The controlling action in optimum control, should, generally speaking, realize both of the above-mentioned tasks simultaneously, and because of this, has a dual character. Dual control problems are considered in this paper. The problem is set up, and its general solution given, for discrete-time systems, when a change in the plant's characteristics represents a discrete Markov process. Various examples are considered.

The methods treating the stability of a closed loop with single nonlinearity lead to stability conditions that assure stability regardless of the shape of the non-linearity within a section in the non-linear plane. Thus, the stability criteria are always more than sufficient conditions for non-linearities that accept some restrictions on its shape. These restrictions must cause relaxation of the stability conditions. In this paper, this approach is used for both inherently stable and unstable control systems to get stability criteria that take into consideration the restrictions on the shape of the non-linearity.

In this paper Liapunov's second method is applied to the stability for certain functions taking on given values, and some formulae, to estimate the decaying time in the general case, are obtained. The linear system with quasi-constant coefficients, and the quasi-reducible system, are studied in detail.

The purpose of this paper is to show how an axiomatic development of the theory of control systems can systematize important concepts. Starting with the idea of attainable set (the set of points which can be reached from a given one in a certain time) a set of 6 axioms is given to describe the behaviour of a general control system. Barbashin already gave such an axiomatic approach, and Zubov also used an axiomatic theory for proving stability theorems. But besides the strong form of stability and invariance of a set (given by Zubov) a weak form of these properties can be defined for general control systems. In general, almost every property of the classical results can be translated in a strong and a weak form for control systems. Accordingly, the powerful second method of Liapunov can also be applied in a strong and a weak form, in order to prove strong and weak properties, respectively.

The inverse problem of integral square estimation may be treated as a problem of inversion of the transient performance of linear stationary systems. The present paper discusses a method of solving this problem by means of spectra of square integral estimation introduced for this purpose. The analytical expressions obtained allow one to determine in a unique manner the transfer functions corresponding to any integral square estimation given by means of these spectra. They are therefore particularly useful for the synthesis of simple and multiple stationary linear systems; at the same time they enable one to select in a simple way the values of the parameters to satisfy the required static and dynamic characteristics of the systems.

This paper deals with the optimization of sampled data systems subjected to several constraints and operating on noisy inputs; while signa and noise are supposed to be a random stationary, ergodic and stochastically independent time series, the optimization criterion is the R.M.S. error and saturating signals minimization.

Certain processes which require to be controlled by means of feedback loops have small perturbation static or dynamic characteristics whose parameters are functions of the unperturbed output level. The form of the response of such systems to large disturbances may be a significant design requirement. The paper describes the first results of a study of the response of controlled systems of this type, and their relation to characteristics which can be deduced from an extension of the small-perturbation roots-locus method of analysis for such processes.

Experiments being undertaken by Culham Laboratory, U.K.A.E.A. are carried in the attitude controlled heads of Skylark sounding rockets which point the whole payload at the sun to better than 1 degree. This paper reports on the fine alignment system within the experiment which maintains the position of a solar image to better than 1 arc minute. The system comprises a platform carrying a small imaging mirror which directs sunlight on to a split field sensor whose outputs, suitably processed and amplified energise the platform actuators to maintain the required position. Mounted on the same platform is the main imaging component of the experiment which thus maintains a solar image in fine alignment on the equipment.

The paper describes the evolution of attitude control techniques within the framework of the TIROS weather satellite program, starting in about 1958, which led to a particularly simple stabilization system used in the operational weather satellite system now being implemented for the United States Weather Bureau by NASA and RCA. The stabilization system employs a controlled form of simple spin stabilization, exploiting control torques generated by interaction with the earth’s magnetic field, to achieve in effect the same results which would be provided by a completely stabilized earth-oriented spacecraft. Possible future evolution of the basic ideas is also discussed.

In this paper we consider the rotational motion of a satellite whose center of mass is following a circular orbit. The satellite shape is a figure of revolution with A = B, and the axial moment of inertia C. Figure 1 shows the orbital coordinate system Oxyz determined by the radius vector and the transversal and binormal to the orbit and also the coordinate system Qxyz given by the principal central axes of inertia of the satellite. The orientation of the frame Oxyz with respect to the orbital coordinate system is described by the Euler angles ψ, θ, ϕ (ψ is the precession angle, iß is the nutation angle, and ϕ is the proper-motion angle). The satellite’s axis of symmetry coincides with the Oz axis, while the Oz axis is directed along the radius vector.

This paper deals with fuel and energy minimization problems in the attitude control of a satellite which is in a steady orbit around the earth. The earth-centered triad is taken as a reference frame. The torque producing elements considered include a reaction gas jet and a reaction wheel driven by a d-c motor. The optimal switching policy of the control functions is investigated, and the minimum fuel or energy consumption is determined for various initial states. Influence of a transition time on the amount of fuel or energy consumption is also discussed.

Description of main principles of antenna steering system utilizing a simple specialpurpose digital computer and a precision analog power servo drive. Input to system is 8-channel paper tape with predicted satellite positions in special format; typically 3 feet of tape required for one hour steering. Over-all pointing error ± 2 minutes of arc excluding prediction errors.

The digital guidance computer is the central control element in the Apollo Guidance and, Navigation System. The difficulties in preparing computer programs for the complex Apollo missions are aggravated by the need to keep the computer hardware to a minimum. This paper demonstrates how an effective solution to these problems is achieved with a combination of appropriate programming techniques and computer design.

Computer programming for control of space vehicles differs from other programming due to the necessity of achieving error -free performance without the opportunity of testing the program in its operational situation. This paper discusses the program content, the procedures for proving the correctness of the content and code of the program and the simuation tools which have been developed and used for that purpose. The primary simulation tool is a general vehicle simulation program that runs on a large scientific computer designed to allow for extensive exercising of the vehicle computer program in all mission phases. The particular features of the simulation program which enhances its utility for this purpose are described.

The overall Apollo spacecraft configuration includes two separate manned spacecraft. They are the command module designed for earth launch and reentry and cislunar flight including lunar orbit, and the lunar excursion module designed for the excursion from lunar orbit to the moon’s surface and return. Each spacecraft has its own independent guidance and control system. This paper decribes the Apollo spacecraft control systems including engine control and rocket engine configurations for attitude control. The Apollo guidance systems are not discussed.