Showing papers in "International Journal of Control in 1965"
TL;DR: In this article, a practical and relatively simple method of measuring the Wiener kernels of a non-linear system is presented, which is based upon cross-correlation techniques and avoids orthogonal expansions such as those of the original Wiener method of measurement.
Abstract: A practical and relatively simple method of measuring the Wiener kernels of a non-linear system is presented. The method is based upon cross-correlation techniques and avoids orthogonal expansions such as those of the Wiener method of measurement. The application of this method to the experimental characterization of a non-linear system is discussed.
581 citations
TL;DR: In this article, a method of measuring the Volterra kernels of a finite-order non-linear system is presented, where the kernels are obtained individually as a multi-dimensional impulse response.
Abstract: A method of measuring the Volterra kernels of a finite-order non-linear system is presented. The kernels are obtained individually as a multi-dimensional impulse response. The multi-dimensional kernel transforms also can be obtained by the method described. As an extension, a technique of obtaining the Volterra kernels from a multi-dimensional step response is presented. This technique is useful for non-linear systems which can be considered to be of a given finite order for only a limited range of input amplitudes.
82 citations
TL;DR: In this paper, an approximate method of solution is developed for linear, stationary dynamical systems which is weakly coupled, where weak coupling is defined quantitatively in terms of a set of conditions which involve the characteristics of the sub-systems and the strengths of the interconnections.
Abstract: An approximate method of solution is developed for linear, stationary dynamical systems which is weakly coupled. A dynamical system which is built up by the interconnection of simpler sub-systems is termed weakly coupled when the degree of interaction between the constituent sub-systems is not large. For the purpose of the approximation weak coupling is defined quantitatively in terms of a set of conditions which involve the characteristics of the sub-systems and the strengths of the interconnections. The approximation allows the solution of the complete system to be expressed in terms of the solutions of modified sub-systems. Apart from the purely computational aspects, an application of the approximation helps towards an understanding of the way in which the sub-systems interact to produce the complete system behaviour. A number of examples are given of physical systems which exhibit weak coupling and one of these examples is developed in detail to demonstrate the application of the approximation.
45 citations
TL;DR: In this paper, a special problem which illustrates a new approach for discussing stability of non-linear systems under arbitrary input disturbances is discussed, where the system is assumed to be described by a nonlinear differential equation with forcing term.
Abstract: This report discusses a special problem which illustrates a new approach for discussing stability of non-linear systems under arbitrary input disturbances. The system is assumed to be described by a non-linear differential equation with forcing term. This equation is solved by 8 type of power series (Volterra series) which gives output explicitly in terms of input. Proof of convergence of this series gives criteria for stability of the system.
45 citations
TL;DR: In this paper, the analysis of general linear systems governed by n first-order state equations whose coefficients are subjected to either of two forms of random variation, namely Gaussian white or incremental Brownian type of process, is presented.
Abstract: The paper is an extension of a previous one (Ariaratnam and Graefe 1965), and presents the analysis of general linear systems governed by n first-Order state equations whose coefficients are subjected to either of two forms of random variation, namely Gaussian white or incremental Brownian type of process. The differences between the two cases are pointed out. The Fokker-Planck equation is set up for both cases and the equations governing the response moments are obtained. An example is given which incorporates both types of random variation of the coefficients.
41 citations
TL;DR: In this article, a control system with a plant consisting of three integrators with a saturable control input is considered, and the switching surface which minimizes integral-square-error is found, partly algebraically and partly numerically, by methods which start from Pontryagin's maximum principle.
Abstract: The paper deals principally with a control system which has a plant consisting of three integrators with a saturable control input. The command signal input is a step plus a ramp plus a parabola. The switching surface which minimizes integral-square-error is found, partly algebraically and partly numerically, by methods which start from Pontryagin's maximum principle. All optimum trajectories have an infinite number of switches before the origin is reached, except for two trajectories which have no switches. Some optimum trajectories have the property that the ratio of any two successive switching intervals is constant. All other optimum trajectories (apart from the two exceptional cases) converge rapidly towards these constant ratio trajectories. Thus, when finding optimum trajectories by backwards numerical computation from near the origin of the Hamiltonian system of equations, it is necessary to adjust the ‘initial’ values to be, not only small, but also close to a constant ratio trajectory. ...
34 citations
TL;DR: In this paper, the sensitivity of the index of performance to parameter variations in optimal control systems is examined, and it is shown that the value of the performance index, after the system parameters have deviated from a nominal set of values, is still given by a symmetric positive definite quadratic form of the initial state.
Abstract: The sensitivity of the index of performance to parameter variations in optimal control systems is examined in this paper. It is shown, in the case of linear optimal systems with quadratic performance criteria, that the value of the performance index, after the system parameters have deviated from a nominal set of values, is still given by a symmetric positive definite quadratic form of the initial state. The matrix of this quadratic form is governed by a special case of the matrix Riccati equation. It is shown also, that similar results hold for the performance index sensitivity function. Because the sensitivity problem closely parallels the original optimization problem, the computational techniques used in the design of the optimal system may be reapplied in the sensitivity analysis.
24 citations
TL;DR: In this paper, the authors present the concepts of controllability and observability of discrete-time systems in terms of the output as well as the state and discuss some of the differences with and similarities to the continuous-time case.
Abstract: The purpose of this paper is to present the concepts of controllability and observability (due to Kalman) of discrete-time systems in terms of the output as well as the state and to discuss some of the differences with and similarities to the continuous-time case.
24 citations
IBM1
TL;DR: In this paper, a first-order system with random parameters and random forcing is studied, where the analysis is concentrated on the probability distributions, and it is shown that considerable qualitative information can be obtained from Feller's classification of the singular points of the forward and backward Kolmogorov equations.
Abstract: A first-order system with random parameters and random forcing is studied. The analysis is concentrated on the probability distributions. It is shown that considerable qualitative information can be obtained from Feller's classification of the singular points of the forward and backward Kolmogorov equations. It is found that there is a drastic difference between the cases of uncorrelated and strongly correlated disturbances. The existence of stationary distributions is shown and their structure is analysed; it is found that the steady-state distributions are of the Pearson type. Some examples exhibit in detail the differences between uncorrelated and strongly correlated disturbances, giving the rather surprising effect that by making the fluctuations of the parameters sufficiently large, the probability of finding the state of the system in tin interval around the origin can be made arbitrarily close to one ‘peaking’. The results of some numerical computations are presented. A case where the energy of the fluctuation in parameters is limited within a certain frequency band shows that this situation is different from the case of ‘ white noise ’. For example, ‘ peaking ’ of the distributions does not occur. It is found that, for the purpose of analysing probability distributions the system obtained can be approximated by a different system with white noise coefficients, These results are also illustrated by numerical computations. (Less)
22 citations
TL;DR: In this paper, a method for determining whether a given Volterra kernel can be synthesized exactly by using only a finite number of multipliers is presented, where the use of the techniques discussed here is illustrated by examples.
Abstract: A problem in the practical application of the Volterra series to nonlinear theory is the synthesis of systems with a given set of Volterra kernels. In this article we present a method of determining whether a given Volterra kernel can be synthesized exactly by using only a finite number of multipliers. A method for the synthesis of such kernels is also presented. The use of the techniques discussed here is illustrated by examples.
21 citations
TL;DR: In this article, an explicit steady-state solution to the Fokker-Planck equation for an nth order, ordinary linear differential equation with constant coefficients excited by a single white noise source by making use of a Liapunov function was found.
Abstract: An explicit, steady–state solution is found to the Fokker–Planck equation for an nth order, ordinary linear differential equation with constant coefficients excited by a single white noise source by making use of a Liapunov function. Liapunov functions are also used to derive a more general result of Wang and Uhlenbeck.
TL;DR: In this paper, a discrete form of Pontryagin's maximum principle was shown to be fallacious and a corresponding correct but weaker result was derived, and certain classes of problem were identified for which the original strongor result is valid.
Abstract: A discrete form of Pontryagin's Maximum Principle recently proposed by a number of authors, is shown to be fallacious and a corresponding correct but weaker result is derived. Certain classes of problem are identified for which the original strongor result is valid.
TL;DR: In this article, the output autocorrelation function of a half-wave smooth limiter is derived using two different procedures: (1) the direct expansion method of Amiantov and Tikhonov, and (2) application of Price's theorem to obtain a differential equation relating ψ(τ) to the normalized input autocorerelation function, ρ(τ).
Abstract: Under the assumption of a stationary Gaussian input, the output autocorrelation function, ψ(τ), of a half-wave smooth limiter is derived using two different procedures: (1) the ‘direct’ expansion method of Amiantov and Tikhonov, and (2) application of Price's theorem to obtain a differential equation relating ψ(τ) to the normalized input autocorrelation function, ρ(τ). In both cases, the resulting expressions are more tractable for numerical calculations than the previously derived double series expansion due to Kaufman and Roberts.
TL;DR: In this paper, it is shown that the minimizing function (Boolean representation of the prime implicant table) of a switching function can suitably be split up into a number of sub-functions such that the sum terms of each sub-function can be arranged in any of the four possible distinct matrices called connected cover term matrices.
Abstract: The central idea developed in the present paper involves the decomposition of the prime irnplicant covering problems of switching functions into the number of readily tractable sub-problems. It is shown that the minimizing function (Boolean representation of the prime implicant table) of a switching function can suitably be split up into a number of sub-functions such that the sum terms of each of these sub-functions can be arranged in any of the four possible distinct matrices called connected cover term matrices. The irredundant solutions of the sub-problem corresponding to each of the connected cover term matrices can be readily obtained, without the generation of a single redundant or duplicate solution, by following a systematic procedure suggested in the paper. The paper is concerned with the detailed study of the properties of the connected cover term matrices.
TL;DR: In this article, it is shown that a hard direction pulse field is necessary before creep can occur, and the significant features of the pulse field are found to be the amplitude, the rates of rise and fall of pulse edge, and total number of pulses involved.
Abstract: ‘ Creep ’ is the name given to the apparent lowering of the magnetic coorcivity of thin magnetic films in certain pulsed fiold conditions. The theory of creep has been discussod elsewhcre (Beeforth and Hulyer 1963, Green et al. 1964, Middlehoek 1962); the work reported here aims at examining the practical implications for thin film computer stores. It is shown that a hard direction pulse field is necessary before creep can occur, and the significant features of the pulse field are found to be the amplitude, the rates of rise and fall of the pulse edge, and the total number of pulses involved. So far as the films themselves are concerned, creep is affected by differences in film thickness and also film composition. One application of thin magnetic films is in high speed computer storage. It is shown that creep limits the maximum storage density, but that correct operation of a carefully designed store should always be possible.
TL;DR: In this article, the authors generalized sampling to include the sampling of a function and its partial derivatives up to order K-means-1, and the reconstruction of the sampled function f(t) has the form of a truncated Taylor series expansions about each of the sample points which lie on a periodic lattice.
Abstract: The work of II previous paper, utilizing the gradient in sampling theory, is generalized further to include the sampling of a function and its partial derivatives up to order K ≥ 1. The reconstruction of the sampled function f(t) has the form of a. sum of truncated Taylor series expansions about each of the sample points which lie on a periodic lattice. The single function g(t), of the vector from the sample point to a generic one, which multiplies each series, hag a J+1ourier transform G(r) which must satisfy sets of partial differential equations on the ’ lattice sets ’ of the support of F(r). The necessary and sufficient conditions which insure the validity of the reconstruction demand that the ‘ associated set ’ of each lattice set be among the zeros of n certain kind of polynomial function which in turn can act as u solution for G(r) on the lattice set.
TL;DR: In this paper, a graphical procedure for determining the stability of a linear control system whose characteristic equation is the sum of two frequency dependent functions is described, using a dual-locus diagram.
Abstract: A graphical procedure is described for determining the stability of a linear control system whose characteristic equation is the sum of two frequency dependent functions. This procedure uses a dual-locus diagram. One locus is the polar plot of one of the frequency dependent functions; the second locus is the negative of the polar plot of the other function. The frequency distribution that exists along these two loci and their intersection determines the stability of the system. In comparison to the regular Nyquist method of stability analysis a number of specific advantages are associated with this procedure, A geometric justification and an illustrative example are included.
TL;DR: In this paper, the geometrical properties of root-loci for higher-order systems are studied and conditions for existence of such features are examined and procedures for their identification and location established.
Abstract: The paper is a study of the geometrical properties of root-loci for higher-order systems. Properties of general higher-plane algebraic curves are invoked to provide the broad framework of reference for systematic investigation of root-locus properties. Such an approach affords better integrated and more comprehensive understanding of the nature of the root-paths for higher-order systems. The discussion is a composite one, encompassing all T(N,M) systems of the order N + M Δ h = 5 and 6, whose equations involve fourth-degree powers in ω: N and M denote the number of poles and zeros, respectively, of the open-loop function G(S) where, S = σ + jω. The examples cited are illustrative of all significant features of higher-plane algebraic curves as manifest in root-locus shapes. Conditions for existence of such features in root-loci are examined and procedures for their identification and location established.
TL;DR: An approach to the design of adaptive and learning systems for achieving optimal control makes use of decision theory and dynamic programming.
Abstract: This paper presents an approach to the design of adaptive and learning systems for achieving optimal control. The design procedure makes use of decision theory and dynamic programming. The system up-dates the knowledge of the probability density function on the basis of the observed information pattern, The fresh information which is learned during each stage is used in performing the optimization process.
TL;DR: In this paper, a distributed parameter system governed by systems of linear partial differential equations has been treated, where the admissibly control functions are assumed to belong to a closed ball in the appropriate Banach space.
Abstract: In this paper distributed parameter systems governed by systems of linear partial differential equations have been treated. The admissibly control functions are assumed to belong to a closed ball in the appropriate Banach space. Formulae have been derived, such that the norm of the control function in the Banach space in which they are assumed to belong is minimized, such that the control becomes simultaneously time optimal.
TL;DR: The actual relationship between B and H in a magnetic material is compared with the B, H relationship of the model commonly used to represent hysteresis in this article, showing that the two are entirely different phenomena and may be treated as one only under special circumstances.
Abstract: The actual relationship between B and H in a magnetic material is compared with the B, H relationship of the model commonly used to represent hysteresis The effect of backlash is investigated and then compared with hysteresis The comparison shows that the two are entirely different phenomena and may be treated as one only under special circumstances
TL;DR: A theoretical study is made of optimizing an nth order system which has a random variation of gain of the plant of Gaussian white noise, or generalized Poisson process.
Abstract: A theoretical study is made of optimizing an nth order system which has a random variation of gain of the plant of (a) Gaussian white noise, or (b) generalized Poisson process. The problem is formulated by dynamic programming.
TL;DR: In this paper, the stability problem associated with linear time-lag systems with stochastic parameters is considered and a sufficient condition for ensuring almost sure asymptotic stability in the largo is derived.
Abstract: The stability problem associated with linear time-lag systems with stochastic parameters is considered. A sufficient condition for ensuring almost sure asymptotic stability in the largo is derived. Its application is illustrated by examples.
TL;DR: In this paper, the flowgraph representation of the z-transforms of various time functions and basic properties is developed, which enables a pictorial representation of all the operations and enables a quick and systematic solution.
Abstract: This paper develops the flowgraph representation of the z-transforms of various time functions and basic properties. This enables a pictorial representation of all the operations and enables a quick and systematic solution. A sequence table of the properties is also developed so that the whole operation can be performed without recourse to memory. These two tables are used to develop solutions of difference equations and discreta systems which can be represented by difference equations.
TL;DR: In this article, the problem of obtaining the optimal control of two independently controlled systems having a common goal may be solved using the maximum principle of Pontryagin, provided certain special boundary conditions are obtained for the Hamiltonian system of differential equations.
Abstract: In the discussion of the theory of optimal cooperative rendezvous, it is shown that the problem of obtaining the optimal control of two independently controlled systems having a common goal may be solved using the maximum principle of Pontryagin This will be accomplished provided certain special boundary conditions are obtained for the Hamiltonian system of differential equations It is further demonstrated that the requirement that two state variables be equal at the rendezvous time implies that the adjoint variables associated with them are negatives of each other at that time Using this result, a first-order time-optimal rendezvous problem is solved and the results discussed Indications for further work are also shown
TL;DR: The paper is interested in determining the conditions for which there is no correlation between signal and noise, for a statistically stationary process the limitation is on the form of the input signal.
Abstract: The output from a non-linear element can be considered as composed of two portions, one proportional to the input signal and the other a distortion noise. The paper is interested in determining the conditions for which there is no correlation between signal and noise, For a statistically stationary process the limitation is on the form of the input signal. The time dependent mean density function, defined in the paper must be separable into the product of time and amplitude functions. Several examples are included.
TL;DR: In this article, the internal feedback of a generalized active two-port network is studied in terms of the z-, y-, h- and g-matrix representations, and it is shown that the return difference is non-unique in that it depends on the matrix representation adopted to describe the external behaviour of the network.
Abstract: The internal feedback existing within a generalized active two-port network is studied in terms of the z-, y-, h- and g-matrix representations. It is shown that the return difference is non-unique in that it depends on the matrix representation adopted to describe the external behaviour of the two-port network.
TL;DR: In this article, a generalized operational amplifier with two-port networks is considered and a minimum phase transfer function having poles and zeros anywhere in the left half of the complex frequency plane is realized.
Abstract: In a generalized operational amplifier, two-port networks are used for the input and feedback network components. By considering each one of these networks as a number of sub-networks composed of resistors and capacitors, and connected in parallel at their input as well as output ports, it is shown that it is possible to realize a minimum phase transfer function having poles and zeros anywhere in the left half of the complex frequency plane. The synthesis procedure is illustrated with an example involving a Butterworth low pass filter of order two. The stability performance of the frequency selective operational amplifier is also considered.
TL;DR: In this paper, the authors deal with the problem of using an analogue computer to determine the optimum control for systems in which there is a bounded constraint on the control signal and the difficulties arising out of this method are also discussed.
Abstract: The paper deals with the method of using an analogue computer to determine the optimum control for systems in which there is a bounded constraint on the control signal. The difficulties arising out of this method are also discussed. Two problems are considered. In each case the criterion of optimality is minimization of integral square error. The first problem concerns the control of a plant consisting of two integrators ; the control to take place in a given finite time, the command signal input being a step or a step plus a ramp. Ail analytic/numerical solution of this problem is given and compared with the analogue solution. The switching boundary is calculated. The second problem concerns the control of a plant consisting of three integrators with a command signal input of a step plns a ramp plus a parabola. The switching surface obtained is only optimal if the initial conditions are specially chosen, otherwise a result is obtained which is similar to Feldbnum's (1955) switching enrve for the...
TL;DR: In this paper, a method of directly testing whether the AND-OR or OR-AND form of a switching function is more economic for some simple cases is presented, and simplified expressions leading to economic three-level synthesis have also been derived.
Abstract: A method of directly testing whether the AND-OR or the OR-AND form of a switching function is more economic for some simple cases is presented. Simplified expressions leading to economic three-level synthesis have also been derived.