Showing papers in "International Journal of Control in 1966"
TL;DR: In this article, a second-order method of successively improving a control sequence for a non-linear discrete-time system is derived, which is motivated by dynamic programming and facilitates the application of gradient methods to stochastic problems which will be the subject of a future paper.
Abstract: A second-order method of successively improving a control sequence for a non-linear discrete-time system is derived. One step convergence is obtained for linear systems with quadratic performance functions. Although the results are of interest in their own right, a second-order method for continuous-time systems is obtained by formally allowing the sampling interval to approach zero. The equations so obtained differ slightly, because of the method of derivation, from results already obtained using the calculus of variations approach. The difference, which is an advantage of the method described in this paper, is that one vector differential equation less has to be integrated. The approach used in the derivation is motivated by dynamic programming and facilitates the application of gradient methods to stochastic problems which will be the subject of a future paper.
336 citations
TL;DR: In this paper, it is shown that the stochastic optimal control problem can be viewed as a problem in the theory of control of distributed parameter systems, and the authors apply dynamic programming to this distributed parameter problem.
Abstract: The present paper is the sequel to a previous paper by the author. The main purpose of the present paper is to justify the claim made curlier that the approach to stochastic optimal control previously employed can be extended to the case of noisy observations of the state. The partial differential equation, for the conditional probability density of the present state given the past history of the noisy observations, is discussed. Using this equation, it is shown that the stochastic optimal control problem can be viewed as a problem in the theory of control of distributed parameter systems. Dynamic programming is then applied to this distributed parameter problem, to obtain a stochastic Hamilton-Jacobi equation in function space.
83 citations
TL;DR: In this paper, a series of experiments were performed investigating this type of sub-optimal control and it was found that although the proper choice of the control parameter leads to easily implemented closed loop controls, there are major drawbacks to its use.
Abstract: For the majority of all dynamic systems open loop optimal control generally results from optimization theory whereas a closed loop control is normally desired. By accepting a degradation in performance a closed loop control termed ‘ specific optimal control ’ can be obtained. A series of experiments were performed investigating this type of sub-optimal control. It was found that although the proper choice of the control parameter leads to easily implemented closed loop controls, there are major drawbacks to its use. The coefficients in the control law ore highly dependent upon the final time T, the system parameters, and the initial state of the system. Furthermore, the performance of the system was found to be a function of the algebraic form chosen for the control law. To determine the most optimum control law entails performing a series of computer experiments. This last difficulty may be circumvented in part by embedding the specific optimal control problem in an identification problem. Resul...
40 citations
TL;DR: In this paper, the problem of determining optimal control of a linear system affected by noise and with incomplete information about the actual state when the criterion is quadratic and the control signal is restricted to be a linear transformation of observations is considered.
Abstract: The problem considered is that of determining optimal control of a linear system affected by noise and with incomplete information about the actual state when the criterion is quadratic and the control signal is restricted to be a linear transformation of observations. An algorithm for optimization giving monotone convergence is presented and the existence of a solution is proved. A necessary condition is derived with the aid of the algorithm and the question of uniqueness is discussed.
39 citations
TL;DR: In this article, the Lyapunov stability method was applied to two problems in the dynamic behaviour of round-rotor synchronous machines. But the results were of a relatively weak sufficient nature and in a restricted region of the state-space.
Abstract: The Lyapunov stability method is applied to two problems in the dynamic behaviour of round-rotor synchronous machines. In one case analytical results of a sufficient nature are obtained from an approximate mathematical model which explain the dependence of the self-oscillation of a machine on the stator impedance and the rotor excitation, and good agreement is found with experimental results. The other case is the ‘two-machine’ problem of two synchronous machines connected by a transmission line subject to a fault. Using the complete set of Park's equations an analytical expression is obtained for the transient stability surface. In general this is of a fairly weak sufficient nature but in a restricted region of the state-Space it gives a close agreement with the computed region of stability. The dynamic equations of the machines involve trigonometrieal functions of the state variables, and in this study the variable gradient method of Schultz and Gibson has been found useful for generating Lyapu...
26 citations
TL;DR: In this article, the authors considered the non-linear generalization of an optimal control problem of Butkovskii and Sakawa, in which a system satisfying a diffusion-like nonlinear partial differential equation is controlled by regulating conditions in the medium in contact with the system at one of its spatial boundaries.
Abstract: The non-linear generalization of an optimal control problem of Butkovskii and Sakawa is considered, in which a system satisfying a diffusion-like non-linear partial differential equation is controlled by regulating conditions in the medium in contact with the system at one of its spatial boundaries. A necessary condition in the form of a minimum principle is developed, and a algorithm for steep descent calculation is obtained. A numerical example for a special case is included.
25 citations
TL;DR: In this paper, a method for determining control laws for distributed parameter systems is discussed, which leads to feedback control systems which are completely stable and have improved speeds of response over those of the corresponding uncontrolled systems.
Abstract: In this paper a method for determining control laws for distributed parameter systems is discussed. The method leads to feedback control systems which are completely stable and have improved speeds of response over those of the corresponding uncontrolled systems. Specific results are obtained for systems describable by parabolic and hyperbolic partial differential equations with controls introduced at the boundary and in the interior of the systems' spatial domains. Numerical examples are also given to substantiate the validity of certain theoretical results.
25 citations
TL;DR: In this article, it was shown that the optimal feedback law u = − Dx is insensitive to (i.e. is unaltered by) a wide variety of persistent changes in A and B.
Abstract: Previous work on the sensitivity of optimal linear control systems [xdot] = Ax + Bu with quadratic performance index has concentrated on the effects of small changes in parameters. It is shown in this paper that the optimal feedback law u = − Dx is insensitive to (i.e. is unaltered by) a wide variety of persistent changes in A and B. Some explicit expressions for these variations in A and B are presented, including the cases when B is fixed, and when the change in A is known. If A is fixed any change in B results in a change in D, so that the system is more sensitive to changes in B than in A.
24 citations
TL;DR: In this paper, the exact mode of oscillation of a time-invariant feedback system containing relays is determined based on a state space approach in the real time domain, and results in a set of transcendental algebraic equations which can easily be solved by digital computation.
Abstract: A method for determining the exact mode of oscillation of a time-invariant feedback system containing relays is presented. This method is based on tho state space approach in the real time domain, and results in a set of transcendental algebraic equations which can easily be solved by digital computation. General state transition and control matrices for plants with n poles and m zeros are derived. For complicated plant characteristics, the switching equations can be obtained easily by simple superposition of the switching equations of ‘elementary plants’. The general expression is given in discrete form, and may be reduced to a continuous form when no impulse control is present, otherwise a combined form is used. Switching equations for elementary plants up to third order are listed for reference, and illustrative examples are given with results compared by analogue simulation.
24 citations
TL;DR: In this article, the Volterra series is used to investigate aperiodic solutions of certain second-order differential equations with nonlinear damping and nonlinear restoring force, and the approximation involved is small over a wide range of values of linear damping.
Abstract: The Volterra series is used to investigate aperiodic solutions of certain second-order differential equations with non-linear damping and non-linear restoring force. Solutions are obtained in the phase plane for trajectories with zero initial conditions. These are obtained by approximating the Volterra series, and it is shown that the approximation involved is small over a wide range of values of linear damping.
23 citations
TL;DR: In this article, a modified version of the conventional π theorem is shown to be invalid when certain variables can change sign, and a modified pi theorem applicable to variables which can change signs is proved.
Abstract: For some simple control sytems, controllers are found which minimize integral-square-error- and similar performance criteria, subject to a constraint on control signal magnitude. The method used is that of setting up and solving Bellman's partial differential equation of continuous dynamic programming. Dimensional methods are used in the solution of Bellman's equation. The techniques are close to those of Wonham (1963), which are discussed and clarified. It is shown incidentally that the conventional pi theorem of dimensional theory is invalid when certain variables can change sign. A modified pi theorem, applicable to variables which can change sign, is proved.
TL;DR: In this paper, the problem of determining the optimal open loop control, when no observations at all are available, is treated by dynamic programming and a quasi-linearization type of algorithm for its solution is presented.
Abstract: This paper is presented, not so much as an end in itself concerning the solution of a definite engineering problem, but as an exploratory step toward the solution of the problem of optimal control of continuous time stochastic non-linear systems when only noisy observations of the state are available. In this paper, the problem of determining the optimal open loop control, when no observations at all are available, is treated by dynamic programming. The main results of the paper are to obtain the functional equation of dynamic programming and to present a quasi-linearization type of algorithm for its solution. The author's intention was to illustrate by these results that infinite dimensional function space is the most natural setting for stochastic non-linear problems. The results obtained give some insight into what can be expected in the more general case of noisy observations. In the Appendices an argument is presented to justify the proposed algorithm and an example is given for which an exa...
TL;DR: In this article, the authors discuss the nature of these difficulties and through several illustrative examples the various steps involved in solving differential games of varied degree of sophistication and difficulty are outlined.
Abstract: The theory of differential games is of recent origin. In attempting to formulate optimal control problems with uncertainty as differential games, several difficulties which are characteristic of differential game theory are encountered. The nature of these difficulties is discussed and through several illustrative examples the various steps involved in solving differential games of varied degree of sophistication and difficulty are outlined.
TL;DR: A sampling interval criterion is proposed and it is shown how this imposes an auxiliary contraint on a discrete-continuous system.
Abstract: The concept of signal dependent sampling is introduced into discrete-continuous control systems and a comparison has been made between periodic find signal dependent sampling systems. A sampling interval criterion is proposed and it is further shown how this imposes an auxiliary contraint on a discrete-continuous system. Performance surfaces in an object function-parameter space sire compared on the basis of the asymptotic equivalence of the non-linear signal dependent system and its linear periodic counterpart for a general second-order plant. A further comparison has been mode of the sampling requiremonts, necessary to minimize a particular object function, for the two different types of system.
TL;DR: In this article, the stability problem associated with a particular class of mixed distributed and lumped parameter feedback control systems is considered and sufficient conditions for stability and asymptotic stability of equilibrium are derived using a weak maximum principle for parabolic equations.
Abstract: In this paper the stability problem associated with a particular class of mixed distributed and lumped parameter feedback control systems is considered. These systems are in the form of a parabolic partial differential equation coupled with a non-linear ordinary differential equation. Sufficient conditions for stability and asymptotic stability of equilibrium are derived using a weak maximum principle for parabolic equations. The application of the main results is illustrated by an example.
TL;DR: In this paper, the sensitivity of linear systems with quadratic performance indexes is analyzed for sensitivity to small parameter variations and it is shown that the sensitivity coefficients depend on the solution of sets of linear equations similar in form to the well-known Lyapunov matrix equation.
Abstract: Optimal linear systems with quadratic performance indexes are analysed for sensitivity to small parameter variations In particular, if the parameters can be separated in a certain way, it is shown that calculation of the sensitivity coefficients depends on the solution of sets of linear equations similar in form to the well-known Lyapunov matrix equation An important example is when the elements of the matrices of the linear system are themselves regarded as the parameters Expressions arc obtained for the sensitivity of the performance index, the optimal control law and the state vector
TL;DR: The results of further analogue computer studies on discrete-continuous systems with signal dependent sampling constraints are presented and the concepts of signal dependent or time-varying thresholds and vector constraints are introduced.
Abstract: This paper presents the results of further analogue computer studies on discrete-continuous systems with signal dependent sampling constraints. Sampling interval criteria based on the modulus of integral of error and modulus of integral of output rate are both investigated and the concepts of signal dependent or time-varying thresholds and vector constraints are introduced.
TL;DR: In this article, a matrix method is presented for the determination of the time response of time-invariant linear systems to a range of deterministic functions, where the factorized transfer function of the system is regarded as representing that of a chain of filters and the input function is a linear combination of certain basic functions.
Abstract: A matrix method is presented for the determination of the time response of time-invariant linear systems to a range of deterministic functions. The factorized transfer function of the system is regarded as representing that of a chain of filters and the input function is assumed to be a linear combination of certain basic functions. The coefficients of these basic functions form an input coefficient vector. The output of the system is shown to be obtainable by successive matrix multiplications of this coefficient vector by response matrices representing the individual filters. The explicit time response of the system is thus produced without resorting to Laplace Transform tables. The response matrices for standard filter elements are tabulated for a basic set of commonly occurring input functions. Extensions to non-zero initial conditions and discontinuities in the input functions are included.
TL;DR: In this article, the Popov criterion for the stability of linear finite-dimensional systems containing a single nonlinearity is extended to infinite-dimensional system with multiple nonlinearities.
Abstract: The Popov criterion for the stability of n. linear finite-dimensional system containing a single non-linearity is extended to infinite-dimensional systems with multiple non-linearities. Bather than examining stability of the states under zero input conditions the paper examines stability of the output.
TL;DR: In this paper, the authors considered a system with two non-linearities separated by a stabilizing network of the form S and (1 + ε)τS, where the input to the second nonlinear block is not a pure sinusoid but contains all higher harmonics.
Abstract: Systems containing two non-linearities separated by a stabilizing network of the form S and (1 + τS) are considered. The input to the second non-linear block is not a pure sinusoid but contains all higher harmonics. An equivalent non-linear characteristic is found which changes with frequency and amplitude, and a joint describing function is obtained. The method is believed to be novel and is applicable even if one non-linearity is of the backlash type.
TL;DR: Results show that this representation of a finite pulse width sampler by an impulse sampler in a sampled-data control system is only valid as long as the pulse width is small enough, otherwise the representation may result in appreciable errors.
Abstract: This paper describes the practical problem of determining errors in the State vector resulting from the representation of a finite pulse width sampler by an impulse sampler in a sampled-data control system. It is shown that the response of the impulse sampling system is the limit of the response of the corresponding finite pulse width system with on associated gain factor, as the pulse width tends to zero. Conditions for the convergence of the errors in the state vector and the asymptotic equivalence of the two systems are; considered. Results show that this representation is only valid as long as the pulse width is small enough, otherwise the representation may result in appreciable errors.
TL;DR: In this article, stability inequalities relating five system parameters have been derived using state apace analysis for n finite pulse sampled-data system and these inequalities have been utilized to compute absolute stability boundaries in n parameter space and in particular their variation with pulse width and sampling period.
Abstract: Stability inequalities relating five system parameters have been derived using state apace analysis for n finite pulse sampled-data system. Further, these inequalities have been utilized to compute absolute stability boundaries in n parameter space and in particular their variation with pulse width and sampling period is described. An expression for the control energy as a function of the state variables has been derived and computer results are presented which show the variation of the asymptotic value of control energy per interval with pulse width. It is further shown how points of inflection in the control energy per interval pulse width curve, may be utilized to minimize the sensitivity of control energy per interval to pulse width timing errors.
TL;DR: A method of employing a new transform technique in the design of discrete-time, time-varying, linear feedback systems is presented, which allows much of the insight already developed for time-invariant linear systems to be carried over to time-Varying linear systems.
Abstract: A method of employing a new transform technique in the design of discrete-time, time-varying, linear feedback systems is presented The chief advantage of this design method is that it allows much of the insight already developed for time-invariant linear systems to be carried over to time-varying linear systems The design method presented complements the one previously presented by Naylor and Waltz (1966)
TL;DR: Questions of optimum start-up and control therefore find mathematical expression as variational problems in two independent variables with those differential equations as aide conditions.
Abstract: Distributed systems such as chemical reactors, absorption columns and other similar apparatus used in the chemical industry have time-varying behaviour approximately describable by hyperbolic partial differential equations. Questions of optimum start-up and control therefore find mathematical expression as variational problems in two independent variables with those differential equations as aide conditions. Such problems have interesting mathematical features when the integral to be extremized is taken round a closed curve in the plane of the independent variables, and this curve includes finite straight segments parallel to the characteristics of the differential equations.
TL;DR: In this article, the problem of parameter estimation is formulated for the situation in which the measurement of every variable is subject to noise and the method of maximum likelihood is shown to produce estimators for which the amount of computation is prohibitive but these estimators can be used as a standard against which the accuracy of more easily computed methods may be judged.
Abstract: The parameter estimation problem is formulated for the situation in which the measurement of every variable is subject to noise. The method of maximum likelihood is shown to produce estimators for which the amount of computation is prohibitive but these estimators fire used as a standard against which the accuracy of more easily computed methods may be judged, A number of alternative methods are introduced, a form of weighted least squares being shown to require the least computation and to approximate the accuracy of the maximum likelihood estimators. An example of the appli-cation of this method is given.
TL;DR: In this paper, a study of feed back control systems with time-shared analogue elements is described, with particular emphasis on the practical problem of reducing interaction between loops and in particular the effect of varying the commutation duration has been determined.
Abstract: A study of feed back control systems with time-shared analogue elements is described. Continuous, ideal impulse commutated and finite pulse commutated systems have been compared for specific examples. Particular emphasis has been placed on the practical problem of reducing interaction between loops and in particular the effect of varying the commutation duration has been determined. A design procedure for time-sharing systems is suggested.
TL;DR: A method for testing and realization of threshold functions through classification of inequalities is suggested, showing that when the different inequalities involving the weights of the variables are represented in terms of their subscripts, the entire set of inequalities of any function can be classified into nine distinct types.
Abstract: A method for testing and realization of threshold functions through classification of inequalities is suggested in the present paper. The standard procedure of testing and realization of threshold functions consists in solving a system of linear inequalities in which the unknowns are the different weights to be assigned to the variables of the functions. In the paper it is first shown that when the different inequalities involving the weights of the variables are represented in terms of their subscripts only, thon, depending on the number and on the sum of the subscripts appearing on either side, the entire set of inequalities of any function can be classified into nine distinct types. The sot of inequalities is next expressed in terms of the least weight and the other different incremental weights, the knowledge of which along with that of the types of inequalities, furnishes information regarding I-realizability of the function and on the assignments of values to the different weights for integ...
TL;DR: In this article, a control system with backlash in the output gears and resolution in the error-sensing potentiometer is considered, and the effect of resolution is to destabilize the system.
Abstract: A control system with backlash in the output gears and resolution in the error-sensing potentiometer is considered. Certain peculiarities in the equivalent describing function are observed. Stability analysis discloses a low-amplitude and low-frequency limit cycle in the system with backlash and resolution while it is found that a no limit cycle exists in the same system without taking potentiometer resolution into consideration. Thus the effect of resolution is to destabilize the system. Analysis by a describing function method and phase-plane study give the same results.
TL;DR: It is shown that the primary information regarding such a decomposition into unate functions can be had from the cover table representation of the given function.
Abstract: The present paper is basically concerned with the problem of decomposition of general Boolean functions into unate sub-functions and a study of the associated properties. It is shown that the primary information regarding such a decomposition into unate functions can be had from the cover table representation of the given function. It is also shown how a knowledge of some of the geometrical properties of Boolean functions greatly facilitates recognition of threshold functions in certain special cases. A simplified procedure is finally suggested for realizing a given Boolean function with a fewer number of threshold logic elements. At every stage examples are worked out to show the decomposition and the recognition procedures and to illustrate the synthesis method.