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The combination of Java applications as graphical user interfaces and the Matlab-Simulink environment as the simulation engine seems to be a good solution for taking a particular control lab to distant students. In this paper, it is presented a new kind of simulation environment for qualitative teaching on concepts of Control Engineering by means of a suitable configuration of experiments by the tutor or teacher.

Active power filter (APF) is one of the most important measures to deal with the power quality problem. Mathematically, APF model in a power grid is typical nonlinear. Since the passivity-based control is a powerful tool to study the stabilization of nonlinear system, it could be applied in current tracking controller design of APF. In this paper, the state space model of an n-phase APF is established first. Then, this model is applied to design a controller for current tracking from the viewpoint of passivity. Subsequently, the sufficient conditions to achieve gradual tracking and to keep the internal stability of the APF control system are given. Simulation results performed with PSCAD verify the validity of the approach.

This paper addresses stability and control issues of systems with uncertain and time-varying sampling period and time-delay. These arise e.g. in embedded or networked control applications where limited computation or communication resources have to be scheduled. The uncertain and time-varying sampling period and time delay are transformed into polytopic and additive norm-bounded uncertainties in the discretized system description. Control design and stability analysis methods are given in the form of LMIs applying switched parameter-dependent quadratic Lyapunov functions. A reduction algorithm is proposed in order to decrease the amount of LMIs necessary for stability analysis. The control design and stability analysis methods as well as the reduction algorithm are illustrated by an example.

The problem of real-time reliability analysis and prediction is considered in this paper based on degradation measurements. Given the nonlinear degradation path model with random parameters, the prior distribution of the random parameters is updated into a posterior distribution according to Bayesian formula based on the measurable degradation information. Then, the reliability during a certain period in the future can be predicted by Monte Carlo integration. An example by use of the fatigue crack growth data is given for illustration.

This paper is concerned with the output feedback stabilization problem of networked control systems (NCSs) with random communication delays. Mean-square stability is derived for the closed-loop NCSs. Based on this, the one-mode-dependent output feedback controller is designed, and the explicit expressions of the controller gains are obtained in terms of linear matrix inequalities (LMIs). A numerical example is provided to show the effectiveness and applicability of the proposed results.

This paper presents methods to analyze two-dimensional (2D) sheet variations obtained from a non-scanning full-sheet measuring system for control and diagnostics applications. A functional decomposition scheme is applied to extract and classify sheet variations in two dimensions. The 2D sheet variations are decomposed into controllable, diagnostic, and unclassifiable variation patterns. The disturbances that cause sheet variations can be identified with the separated patterns. The separated patterns give sheet-makers a new visibility to pinpoint the process problems and to improve sheet uniformity.

This paper presents a control algorithm that reduces the vibrations of a human-riding biped robot, HUBO FX-1 during walking. This kind of robot should handle the payload of at least 100 kg in order to carry a person easily. To secure a sufficient payload, it is important to reduce the weight of the robot. The weight of the robot is mainly composed of motors, reduction gears, and body frames. Reducing the weight of the body frame is more efficient use of power than reducing the capacities of motors and reduction gears. A disadvantage is that the leg structures become flexible, so serious structural vibrations are generated during walking. We solved this problem by using simple theoretical models and experiments based on sensory feedback. This control strategy is very effective for the weight reduction of biped robots.

The high penetration of renewable generation raises great concerns in grid operation on whether system will have sufficient flexibility for balancing the intermittent variation of renewable generation. With the development of smart grid, technologies like demand response, distribution generation, electric vehicle and energy storage become more popular. It is foreseen that with the increasing participation of these flexible load resources and high penetration of renewable generation, the system operations of the future interconnected power grid will become more complicated. This paper discusses the system operational challenges, analyzes the potentials of load flexibility, then proposes a multi-agent system control framework to support the interactive operations of “Source-Grid-Load”. An interactive leading control process is introduced to lead the flexible resources ramping with a sequence of advisory dispatches, reduce the balancing needs from the frequency control generators.

In this paper, an experimental vehicle with an active suspension system is regarded as a control object. First, a full-vehicle model representing this experimental vehicle was derived, then a control system was designed for this model using the H∞ control theory. The main purpose of this control system is to achieve robust performance. This control theory has been implemented in an experimental vehicle, and evaluated for robust performance in a four-wheel shaker and during actual driving. It has been confirmed that the desired robust performance is achieved in this closed-loop system.

This paper presents a speed planning method for mobile robots. The method is applied to a planned path and produces a speed profile which will be used for path execution. This goal is achieved by applying to stages: speed planning and speed profile generation. The first step computes a set of control speed points according to speed limitations of the vehicle, obtained from its kinematic and dynamic behaviour. Moreover, this stage considers the movile obstacles avoidance, which is included as a vehicle's motion constraint. The second step uses the information provided by the previous step to build a smooth speed profile.

In this paper, some basic considerations concerning the design and use of hand-arm mechanical systems for dexterous manipulation tasks are presented. The UB robotic hand project is referred to as paradigm of this kind of manipulation systems.

A dynamic system is given with a transfer function of third order (one real pole, two complex poles, and two zeroes). It is desired to design a contactor control which yields a near-optimum follow-up of a given input. For zero-seeking systems and for step-inputs an "optimum" control problem can be defined easily; however, a general solution of the optimum problem is not yet available. The known particular cases yield switching surfaces in the phase space which depend on the particular pole and zero locations (parameter set). The realization of such surfaces is rather difficult and usually demands digital computer equipment. The author and her associates suggest in this paper switching functions which can be realized easily and which give good response for different parameter sets. Particular attention is given to the occurrence of discontinuities in linear switching functions and their advantages and disadvantages; to different types of chatter, which may or may not improve the system performance; to discontinuities in the switching function which can be avoided by feeding the output of the contactor back into the switching function. Analog computer tests show the performance of systems with linear switching functions. The choice of the coefficients in this linear switching function is discussed. The influence of imperfections of the control system on its performance is investigated. Comparison to linear feedback control systems are included. This paper is to be published in the Proceedings of the First IFAC Moscow Congress by Butterworth Scientific Publications in 1960.

Modern control systems are complex and must meet stringent performance requirements. There is an obvious need in such systems for development of design procedures for computer-controlled systems which make use of the highly developed computer art. This paper presents an attempt to apply computers in the solution of one class of such systems. The realization of the novel computer circuits required is outlined, and the overall design logic is presented. The designer versed in the art will have no difficulty integrating the individual circuits. It appears that analog circuits interconnected by means of switching devices will be useful in the realization of the required computing circuits. The required computer facility seems to be reasonable, and commensurate with the control problem. (Author)

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.

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.

In the paper an effective way of investigation of dynamical systems by means of their converted model is proposed. The reconstruction of a dynamical model and its total orientation for a task relies on expressing the relation between investigated characteristics of a system and its parameters in the form of a regression or approximating polynomial. Such a model is obtained on a data basis of few object or its dynamical model investigations, which are planned using the mathematical apparatus of the experimental design theory. It is most comfortable to carry over the active experiment on a simulation model of a system. The paper presents a mathematical apparatus and capabilities of the method. It is possible to use the offered way for the solution of selected problems at the stage of simulation model creation as well as of the system analysis and design. The method capabilities have been illustrated by examples of verification of permissible simplifications of simulation models, evaluation of model parameters, analysis of the influence of parameters on output characteristic and of optimum synthesis.

Facing the defect in CRI method in logical semantics and the problem in its application to longer reasoning chaining in expert system, this paper, based on triple I method in literature (Wang, 1999a), proposes a characteristic expansion algorithm of fuzzy reasoning. The algorithm is then used in the fuzzy reasoning modes of multidimension and multi-implication, from which the models' characteristic expansion algorithms are deducted. As a result, a new way-out is found in the application of fuzzy reasoning in the intelligent systems

Summary With the exception of a passing reference in an early note by Zadeh1, little or no attention appears to have been paid to the inverse describing function (IDF) problem. In essence this problem is concerned with extracting a description of a non-linear element given only its describing function. The problem is important as the final link in the nonlinear synthesis problem for automatic control systems, for various experimental determinations, and finally for itself. In this paper an analytic solution to the problem which results in expressions in the form of Volterra integral equations is provided. The conditions under which the representation for the non-linear element is unique are discussed and additional requirements for the unique definition of non-single-valued non-linear elements are given. As might be anticipated the describing function of a memory-type non-linear element is not sufficient to obtain a unique representation for the element. Practical computational techniques are mentioned for the machine solution of the IDF and several examples are given.

Summary This paper studies the optimum control of multi-variable systems by considering the behaviour of the two-variable on-off system described by the equation x ¨ + A x ˙ = u where u1, u2 = ± 1. By the application of Pontryagin's maximum principle, it is shown how the number of switchings and the drive ratio u1/u2, for an optimum trajectory (minimum settling time criterion), depend on the nature of the matrix A. For the case when the interaction terms a12 and a21 have the same sign, the maximum number of switchings for an optimum trajectory is three, and the drive ratio u1/u2 near the origin of the error phase space has the same sign as a12 and a21. A predictive controller described in this paper finds the three switching times by an iterative process which involves the application of logical rules to the predicted behaviour of the system, computed in a fast analogue model. This realizable controller is optimum when the two-by-two matrix A has a12 a21 ≥ 0; and will also control systems where a12 a21

Summary 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. The basic optimization problem is viewed, in very general terms, as a problem in decision theory. Based on available information, the decision must be as to which of the allowable control signals should be applied to the plant in order to achieve the optimum performance. A sum of squared error criterion is used to define the optimum in order to reduce the problem to a problem in linear decision theory. A piecewise-linear control boundary is found which can be used to generate the desired control signal. The realization of this control boundary is shown to be straightforward and inexpensive. This technique can be applied to linear time invariant plants of arbitrary order. Experimental results verifying the techniques which are developed are presented. Included in these is the control of the step response of a system with a saturation-limited unstable second order plant in the presence of noise.

Summary There are known controlled systems such as chemical plants or airplanes whose differential equations are not known completely to the controller because of environmental changes, ageing, etc. In many systems of this kind the best which can be accomplished is to construct a multistage optimalizing process which converges to the optimum control. In the case of zero-memory, non-linear plant processes of this kind can be constructed if the gradient of the performance measure is known or can be determined experimentally. In this paper an extension of this method is considered for the case of non-linear plants having memory and changing randomly in time. In the two introductory parts of the paper the assumptions and the necessary and sufficient conditions of optimality are formulated. It is shown that at any stage of optimalization the best change of the input signal should be adjoint to the mean value of the gradient of the performance measure. Then an optimalizing process, based on the so-called fixed point theorem, is constructed. It is shown that for certain classes of non-linear random plants all the necessary information about the generalized gradient can be obtained experimentally. As an example an optimalizing process which minimizes the cost of input energy and maximizes the output gain of a non-linear plant has been constructed and is discussed.

Summary Non-linear programming problems may be met with in the selection of optimal automatic control systems. This paper presents a method of solving a problem in non-linear programming. The essence of the method consists in reducing the set problem to a repeated search for a solution of a linear programming problem and the choice of values for certain additional parameters that are introduced.

Summary 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.

Summary 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. Results discussed in the early part of the paper show that analytical methods for the classes of problems dealt with involve mathematically intractable expressions; the analogue computational procedures also do not yield satisfactory results. The explicit digital relationships and the corresponding systems and structures developed in the paper overcome these difficulties. The digital methods are expected to prove most useful because of the availability of high-speed digital computers capable of handling increasingly large amounts of data. Ideas for further investigations are included.

Summary 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.

Summary 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.

Summary 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. The second part of the paper is concerned with a more specific class of problem in which the process is linear, and the performance objective is one of the following: (1) To force the process from an arbitrary initial state as close as possible to a specified terminal state in a fixed number of steps. (2) To force the process from the initial state to a specified terminal state in a fixed number of steps with minimum ‘effort’. (3) To force the process from the initial state to a specified terminal state in a minimum number of steps. A special-purpose computer which performs the required computation in real time is proposed as the on-line optimum controller.

Summary 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.

Summary 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. Conditions on the numbers of the roots inside, outside, or on the unit circle are also discussed, within the cases when the determinants are zero or non-zero. Also, necessary and sufficient conditions are formulated for all the roots to be inside a circle of radius σ less than unity, and also the conditions when the roots are to lie between plus and minus unity in the z plane. Various examples of discrete systems are presented which illustrate the applications of the new stability criterion as well as the other conditions formulated in this paper. In concluding the paper, various analytical stability criteria applied to linear discrete systems are enumerated and compared, with emphasis on the advantageous applications of each.

Summary 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. In the second half, two different applications of describing function methods to the sampled-data systems are proposed. One is defined in the same way as in continuous systems, and it is accurately applicable to the sinusoidal oscillation of which the period is sufficiently larger than the sampling period. The other is defined by considering the phase relation of oscillations at sampling instants, and it is applicable when both frequencies are closely approached. Moreover, in some examples illustrated, the existence of ‘hard self-excitations’ in sampled-data systems is suggested.

Summary 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.

Summary 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.

Summary 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.

Summary 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. In this method the finite settling time is combined with the minimum of integral squared error. In this integral either the same importance is allotted to all errors during the control process, or only the errors occurring after the first sampling period are being considered. The physical meaning of the second case is the requirement of the computer liquidating the error, as far as possible, during one sampling period, and not instantaneously. The paper shows that good results can be obtained particularly by the second method. The method of calculating is arranged in such a way that both alternatives can be investigated simultaneously. The author succeeded in arranging the calculating procedure into a simple scheme. A numerical example is given.

Summary 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.

Summary 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. An example is given in section VII. Upon replacing axiom IV by IV* the method may be applied to circuits with multivalued elements, e. g. relays with real time functioning, polarized relays, relays with multipositional armatures and rotary switches.

Summary 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.

Summary 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. Specifically the control of a plant described by an nth order, linear, time varying differential equation and which has r inputs and m outputs is considered. Each input ui(t) is constrained by a condition of the form [ ∫ 0 T | u i ( t ) | pi dt ] 1 p i ≤ L i , for i = 1 , 2 , ... , r where pi ≥ 1. Note that the constraints are applied to each input separately so that this problem includes those practical cases when some inputs may be limited in magnitude, some in energy, and others in still different ways. The problem is to find the set of inputs ui(t) which satisfy the constraints and which will make the plant outputs equal to a set of desired outputs in minimum time T. The technique presented in this paper for solving the problem is an extension of the methods of functional analysis first used by Krasovskii and Kulikowski to solve optimal time problems with a single constrained input. A useful feature of the solution presented in this paper is that a very general norm is defined for the input vector u(t) = [u1(t), u2(t), …, ur(t)] which incorporates all the separate constraints into a single constraint condition. This permits a solution of the optimal time problem for this case, which may be obtained with no more formal difficulty than in the single input case.

Summary 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. On the assumption that the amplitudes of the signals Δxi (t) are small, approximate (and, within the limits of the small-parameter method, entirely correct) expressions are found for the control actions in the self-adjustment networks in terms of the frequency characteristics of the filters ϕ1 and ϕ2 and the present spectra of the signals Δxi (t) and Θ (t). The differential equations derived for the selfadjustment processes take account of the limited memory and passband of the filters ϕl5 ϕ2 and Wϕ, and cover the case where the frequency characteristic of the filter is a functional of the signals Δxi (t). These equations lead readily to a number of necessary conditions for the stability of the self-adjustment process, and also to the desirability, with a high-frequency sinusoidal search signal Δxi (t), of using two phase discriminators with reference voltages in quadrature, so as to make use also of the phase modulation on the carrier signal. As a simplified mathematical abstraction, detailed consideration is given to the case of an almost periodic signal Θ (t) and a test signal similar in nature to white noise. Some attention is devoted also to quasistationary self-adjustment modes of operation. Illustrative examples are given of the calculation of actual systems with several adjustable parameters.

Summary 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. The paper also considers approximate methods of solving optimal control problems on systems with distributed parameters, based on approximating the equations for the distribution functions by ordinary differential equations.

Summary 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.

Summary 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.

Summary 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. A practical result of the introduction of sufficient coordinates is hat the optimal control breaks down in two units, one producing he sufficient coordinates, the other providing optimal control.

Summary 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.

Summary In this paper consideration is given to the system of differential equations d x d t = f ( x , η , t ) + u ( c , y , t ) where x (t) is an n-dimensional vector, y (t) an m-dimensional vector, n) (t) a certain (in general random) vector function, and c a constant vector. It is assumed that a certain trajectory x = Ψ (t) in phase space is given for 0 ≤ t ≤ T(0 The paper shows the methods of programme control used.

Summary 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. A new design principle for the systems invariant to any continuous control function is developed.Usually error signal is independent of control action only if the right-hand side of the system's non-homo-genious differential equation vanishes. To meet this requirement one must use control action derivatives in the control algorithm. In this case any variation of system parameters would make these invariancy conditions invalid. It is shown that with the aid of open-loop variable structure systems one can perform a wide range of control functions without static errors. In this case there is no need to meet this classical invariancy condition. Note that combined servo systems with variable open-loop structure are insensitive to certain variations of the system parameters.

Summary 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. These systems are based on the structures which admit an unlimited increase of gains without disturbing the stability. It is shown that the presence of the disturbance at the input of the compensating network (when the disturbance can be measured) or the introduction of the model of the plant to the system provides, for the chosen structure of the whole system, the removal of the influence of external disturbances. It is concluded that the sensitivity, according to Bode, for the mentioned structures does not depend on the parameters of the plant, nor does its value tend to zero under the corresponding increase of the gain.

Summary 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. Instead of general derivations the comparison is demonstrated, using a very simple control system as an example. In each case the variance of the measuring unit output is estimated, this varies statistically because of the finite averaging time. From the variance the necessary time constants of the smoothing filter are determined. Results: From the three devices the ‘stepwise’ searching method requires long measuring intervals. The method with a ‘periodic searching signal’ is better because the search by trial and error is replaced by an adjustment directed uniquely to the minimum, but the measuring interval is in the same order of magnitude. The method with a ‘reference model’, however, has also the advantage of a shorter measuring time.

Summary 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.

Summary 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 principle of operation of the control programme is as follows. In each cycle of operation of the system is performed an extrapolation of the input and search requirements (by use of ‘fictitious’ cycles within the machine) of the control signal which minimizes the system error. The paper gives logical schemes of the control programme, as well as the results of an experimental investigation.