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The analysis of time series data can be significantly eased by the use of orthogonal polynomials. This technique is based on traditional regression analysis, but the necessity for solving the associated normal equations is eliminated. Also, the orthogonal polynomial technique includes a means to statistically test the level of fit of the independent terms in a forecast model (e.g., constant, linear, or quadratic). This paper shows the application of orthogonal polynomial technique to the analysis of time series data, but more importantly develops a mathematical means for updating the forecast model from one period to the next without completely recomputing the model from stored historical data. This result considerably improves the ease of application of the orthogonal polynomial technique and increases the value of the technique in practice.

The paper shows how the parameter plane technique can be applied to the characteristic equation of the optimal linear control system chosen to minimize a quadratic performance criterion. The technique, as applied to a general polynomial equation, is described paying particular attention to the plotting of constant relative damping, constant undamped natural frequency and constant real pole position characteristic curves in the parameter plane which has two variable parameters as co-ordinates. The method is then applied to an optimal linear control system characteristic equation which is written in a convenient form for determining the structure of the characteristic equation coefficients in terms of the variable parameters. Particular attention is given to two parameters contained within a diagonal control weighting matrix and by observing the characteristic curves in the parameter plane the effect of different values of these two parameters on the optimal transient performance can be investigated without...

By the introduction of the time integral of the output variable of a second-order system, the optimal feedback control law with respect to the quadratic performance criterion gives the conventional proportional-integral-derivative (PID) control. Some comparisons between the weighting factors of the quadratic criterion and the parameters of the PID controller are given. The values of the weighting factors corresponding to the classical methods of Cohen-Coon and Ziegler-Nichols are calculated ; negative values of the weighting factors are obtained.

The problem of obtaining output feedback gains to minimize a quadratic performance criterion which includes sensitivity variables is discussed. The equivalent stochastic problems are also studied and when a Kalman filter is allowed, the minimum variance estimate of the state is employed to implement the control. A set of non-linear matrix equations are obtained which constitute the necessary conditions that must be satisfied for an optimal solution. For the deterministic case it is shown that when no sensitivity is considered and the matrix which relates output and input is invertible, the usual Riccati formulation is obtained. The problem is also reformulated using a compensator and it is shown that it can be cast into an instantaneous feedback form.

Explicit expressions are derived for evaluation of the general time-weighted quadratic performance index associated "with a discrete-time constant linear system. The method used relies on formulating the problem in terms of Kronecker matrix products, which enable a formula for the sum of a scalar series to be applied. Similarities with the corresponding problem for continuous-time systems are demonstrated.

The capital budgeting problem is treated as a part of the general theory of choice, where utility is to be maximized subject to the opportunities and constraints. The utility function is assumed to be quadratic to take care of any risk-averse behavior of the shareholders and a quadratic mixed zero-one integer programming model is developed. Special solution techniques for the resulting mixed zero-one integer quadratic programming model are discussed.

Recently proposed discrete-model representation of turbo-charged diesel engines are recast in state-space form and used in an optimal control study. It is known that the turbo-charger has considerable influence on the dynamical behaviour of the engine and this is accounted for by incorporating a weighting of the air-fuel ratio in the performance index. This leads to a rather unusual form of the quadratic performance functional and the optimal controls are determined using both dynamic programming and discrete minimum principle techniques. This leads to extensions and some simplification of known theoretical results.

A Bayesian decision theoretic approach is employed to compare sampling schemes designed to estimate the reliability of series and parallel systems by testing individual components. Quadratic loss is assumed and schemes are found which minimize Bayes risk plus sampling cost. Several kinds of initial information concerning the reliability of the individual components, all of which assume the components function independently, are considered. The case for which the initial information is in terms of the system's reliability is briefly considered and related to the aforementioned case

The sensitivity of an observer to variations in the parameters of the system being observed is considered. An equation is developed for the error in the observer output caused by variations in the system parameters. An analogue computer design procedure is utilized to minimize a quadratic performance index. An example is provided to illustrate this procedure.

For a linear Nth-order time-invariant system with scalar control variable and quadratic performance index, a procedure is given for computing the optimal value of the feedback vector k. Instead of determining, as a first, step, the N(N+1)/2 elements of a symmetric matrix, the K values of k are computed directly, leading to substantial simplifications, and making the method well suited for higher-order systems. In contrast to other methods starting from the same premise, no restrictions are imposed on the form of matrix A. In addition, and based on the previous considerations, a solution of the inverse optimal control problem is also presented. The form of the weighting matrix in the performance index can be chosen arbitrarily (e.g. as diagonal). In contrast to other known procedures, the computations are straightforward, and again no restrictions are imposed on the form of the system.

The state of a continuous system most naturally enters the performance index as a discrete sum if there is a finite set of observation times, or if there are only a few key times over which errors really matter. Such continuous-discrete problems are studied here for linear plants under a broad class of continuous metrics on control and summed metrics on trajectory error. The approach is through the dual optimization problem and alignment properties of the primal and dual solution. The dual problem, is often more attractive computationally than the primal, and is always finite-dimensional as opposed to the infinite-dimensional primal problem. Occasionally the dual problem has a straightforward analytical solution, as in the case of the quadratic continuous-discrete regulator. In any case, information on the structure of the primal control is available through the transformation even before the dual problem is solved. Three examples are detailed involving L1, L2 and L∞ measures on trajectory errors.

In a normal linear model necessary and suffieient for parameiers of she coveriance matrix to be estimable is that they are quadratic estimable. This does not coincide with identifiability.

A method is presented to synthesize a quadratic state regulator for analytic nonlinear discrete-time systems. The applicability of the method is tested in practice in computer control of a laboratory scaled process.

Wheeler [6] proposes simple formulas for quickly determining the sample sizes necessary to achieve a prescribed power. We find Wheeler's evaluation lacking in accuracy because of his application of a linear approximation to quadratic formulas. Therefore, “portable power” is potentially harmful to the experimenter who may not be knowledgeable in the theory.

Bayesian estimation and design are considered for a dichoto-mous response surveillance model for defectives. The probability of a defective item after storage time t is assumed to be given by F(t) = l-exp(-At), 0 < t, λ < ∞. Surveillance is carried out by removing k lots from storage and observing the size (ni), storage time (ti) and number of defectives (ri) for each lot. The ri are assumed to be independently and binomially distributed with respective expectations (ni).F(ti) A prior gamma distribution is assumed to be available for x. the posterior distribution of x is Hayes estimate assuming quadratic loss. A formula for the corresponding Bayes risk is derived as are some recursive relationships to aid in computation. A design procedure is given for selecting surveillance times and sample sizes successively one lot at a time. Tables of optimum surveillance times and Bayes risks at these times as functions of the prior parameters and sample size are provided to help with this selection procedure.

Techniques are developed for designing fixed-order time-invariant dynamic compensators for linear stochastic multivariable servomechanisms subject to plant disturbances and measurement noise. A quadratic performance criterion is employed and the resulting parameter optimization problem is treated using a state-variable formulation. Computable expressions are derived for the cost and its gradient with respect to the compensator parameters. The optimal parameters are then determined using a gradient-type minimization algorithm. An illustrative numerical example is presented.

The decision-theoretic approach to point estimation involves the choice of an estimate to minimize the expected loss associated with the estimate. The purpose of this paper is to indicate the influence of risk aversion on point estimates for classes of payoff functions including the piecewise linear and quadratic payoff functions. Increased risk aversion results in a point estimate closer to zero for a quadratic payoff function and a lower estimate with a piecewise linear payoff function, for example.

This paper presents a method of finding the optimal pulse-width-modulated input sequence from the equivalent pulse-amplitude-modulated input sequence. The optimal pulse-amplitude-modulated input is obtained using either the discrete maximum principle or dynamic programming. Both linear time-varying and time-invariant systems are considered for a class of quadratic performance index. A third-order numerical example is solved.

A quadratic performance index together with a set of prescribed closed-loop eigenvalues are considered as criteria for designing discrete multi-variable control systems. The design method presented in the paper is an extension of an earlier solution of the equivalent problem for continuous systems.

Optimal sampled-data controls for linear processes with quadratic criteria are determined through application of the discrete minimum principle. The effect of sampling on the closed-loop system's performance is investigated and the asymptotic behaviour of the optimal cost for large sampling periods is determined. The resulting design method is applicable to continuous, sampled-data and discrete regulators.

A Bayesian treatment is given of a two-phase sampling problem when sampling is from a double dichotomy, losses are quadratic, and sampling costs are linear. The motivating question is: how does one determine an optimal first phase sample from a dichotomous population given that an optimal follow-up sample of nonrespondents in the first phase is to be taken?

Necessary and sufficient conditions are developed for the existence and calculation of well-defined Riccati differential equation solutions associated with quadratic loss minimization problems. Of particular interest is the fact that a. covariance condition is involved. The disclosure of this condition not only extends the range of optimal control problems for which a solution, guaranteed to be well defined, may be calculated, but also introduces an approach for establishing the existence of well-defined solutions in other problems involving covariance conditions, as for example, in a time-varying spectral factorization procedure. This paper is concerned with finite time results, whilo a companion paper considers tho infinite time case.

A linear time and state dependent approximation to a non-linear and non-stationary system is optimized with respect to a quadratic performance index by treating it as an instantaneously linear stationary system. By allowing for the variations of the model a realizable adaptive typo of controller can be devised.

The general problem of determining the relative energies of a system of four weakly-interacting transition-metal ions is considered when there is spin but no spatial degeneracy. Interaction hamiltonians that are quadratic and biquadratic in the spins are employed. The irreducible tensor method is used.

In this paper two computational techniques are presented, to solve non-linear differential games which have a value. It is assumed that there may be interaction in the dynamics of the players and that the conflicting index of performance of the game is a scalar function of the final state of the players of the miss-distance type. The first of these techniques is based on the Newton-Raphson method in function space in association with one of the two following non-iterative procedures to solve linear differential games with quadratic pay-off function : the generalized Riccati transformation or the transition matrix method. The second computational technique presented is based on the gradient of the Hamiltonian function with respect to the strategies. Some numerical examples are given to illustrate the methods.

A Study is mode of Weiner's least-square optimization with quadratic constraints. The control system input consists of a random message plus noise. The message has a general form of power spectrum and the noise is white. Both message and noise are uncorrelated and stationary in the statistical sense. A digital computer is used to find the values of the Lagrange multipliers for given mean square constraints simultaneously imposed on two separate plant variables. The computation is performed by means of a specially devised goal-seeking programme.

(1939). A Certain Quadratic Diophantine Invariant. The American Mathematical Monthly: Vol. 46, No. 4, pp. 203-208.

The choice of an experimental design suitable for fitting a graduating polynomial can be made according to a number of criteria, depending on the problem involved. Difficulties arise when, although the factors are continuous in nature, the number of levels is specified by some external considerations. For example, if some factors can be examined at only two levels, the graduating function cannot include quadratic terms in those variables, but all second order terms for variables to be examined at three or more levels can be permitted. For such cases, a restricted model and special experimental designs are needed. This paper considers the problem when there are some factors at two levels and some factors at three levels, and when there are some factors at two levels and some factors at four levels.

A general procedure for estimating the population mean in random effects models of the nested and/of classification type is considered. The suggested estimator is unbiased and consistent (with respect to the structure of the experimental design). It is also optimal with respect to a particular quadratic, location-sensitive criterion. Finally, for experimental designs which contain a certain degree of structural balance, the suggested estimator coincides with the sample mean.

Optical experimental designs are developed for the inverse estitator in the Linear Calibration problem using the crieterion of minimum intetrated or average mean squared error. Designs are developed for a linear approximation when the true model is Linear and when it is quadratic. In both cases, the optimal designs depend on unknown model parameters and are not realistically usable. Eowever, designs are shown to exist which are near optimal and do not depend on unknown model parameters

The equilibrium of a dislocation distribution is considered for the case where the applied stress field is a rational function. A function of a complex variable is introduced and it is shown that for equilibrium this function satisfies a quadratic equation. The stress field of the distribution and the distribution itself are different aspects of this complex function and are determined together. The known prevalence of square roots in distributions is a consequence of finding the roots of the quadratic equation. The connection between this complex representation of equilibrium and that used in Part IV for dislocation dynamics is discussed.

The problem of designing an optimal discrete-time feedback controller is discussed for a linear system governed by coupled partial and ordinary differential equations. For the case of spatially concentrated controls and a quadratic performance index, the minimum cost and the optimal discrete-time controls are determined. Evaluation of computational requirements demonstrate the applicability to on-line computer control.

The search for a linear Λ-minimax estimate (see, e.g., [19]) of a multivariate location parameter (with quadratic loss) is reduced to a problem of maximizing a continuous function on a compact, convex set—a problem which can be solved numerically with little difficulty. This reduction is accomplished by showing that a minimax theorem applies, thus changing a difficult minimax calculation to a simpler maximin problem. The technique is observed to be applicable to other, related problems. Also considered is the choice of experiment under the assumption that the decision maker can, at some cost, improve the specification of his prior distribution.

When the substrate of a 2 n–q factorial or fractional factorial experiment may be expected to show trends representable by linear and quadratic terms in time, then certain orderings spaced at equal time-intervals permit better estimation of the effects than do others. Some of these ordered plans are given for p – q = 2, 3, 4, 5. Simple methods are given for computing effects, trends, and efficiencies.

This paper discusses a problem in the spacing of observations in quadratic regression for most precise extrapolation. Let E(yi ) = α + βxi + γxi 2. The yi are uncorrelated observations with variance Vo; the xi are controlled. The problem answered is: at which x-values in a specified interval [xL, xH ] should N observations yi, i = 1 (1) N, be taken so that for a ξ > xH (or ξ < xL ) the least squares estimate of α + βξ + γξ2 has minimum variance? It is shown that the observations are located at x L, (xL + xH )/2, and xH. The distribution at these locations as a function of ξ is given. These results are also useful in designing experiments for testing the hypothesis of a quadratic relation versus that of a linear relation. * Work done under AEC Contract No. W-7405-eng.-26.

A sample of size N is taken from an equiprobable t-category multinomial distribution. The precise values of Pearson's X 2 and the likelihood ratio statistic A were computed for t = 3(1)6, N = 3(1)12; t = 6(1)14, N = 6(1)2t; t=15(1)18, N = 6(1)28. The logarithms of the tail areas were smoothed and fitted by means of quadratic expressions. Much information is given concerning the accuracy of the chi-squared fits to the distributions of X 2 and A. A table of −log10 P(χ2 > a) is given, for a = 1(1)50, and d.f. = 1(1)20. The distribution of the total number of repeats within cells is also discussed.

The thermal conductivity data of the Ne-Ar, Ne-Kr, Ar-Xe and Kr-Xe systems are reported as a function of composition and at temperatures 40°, 65° and 90°c. Similar results are also presented for the ternary systems Kr-Ar-Ne, Xe-Kr-Ar and a quaternary system Xe-Kr-Ar-Ne. A conductivity cell of the thick hot-wire type is used. The thermal conductivity values for the binary mixtures are compared with those obtained from rigorous theory and such procedures as approximate, semi-theoretical, empirical, linear mixing, reciprocal mixing, combination of linear and reciprocal mixing and quadratic equations. A similar study is also made for ternary and quaternary mixtures and some interesting conclusions are drawn concerning the relative appropriateness of different methods for computing thermal conductivity of multi-component gas mixtures.

This paper establishes conditions of optimality for linear systems whoso control functions are quantized either in magnitude or in switching instants, or both. Optimization of such systems hitherto has been limited primarily to dynamic programming techniques and direct enumeration. The results are derived from the total variations of the cost functional. The main theorem provides sufficient conditions of optimality for linear systems with quadratic and positive semi-definite performance index. The results are significant for digital control systems, pulse-code-modulated control systems and some types of pulse-frequency-modulated systems which are subject to quantized variations only. For the special case where the control functions are not quantized, the optimality conditions provided by the extended maximum principle are shown to be necessary and sufficient.

Results of application of the perturbation approach along with stochastic approximations to the problem of optimizing non-linear stochastic systems subject to quadratic index are discussed. The perturbation approach is first introduced with the help of linear regulator systems. This is combined subsequently with statistical linearization in order to derive sub-optimal solutions for the problem of optimizing a non-linear system. Finally, a new stochastic approximation technique, that permits more accuracy than the linearization, is suggested for the non-linear problem. Numerical examples are presented to compare the results of the various approximation methods and also to illustrate the possible advantages of non-linearization.

The problem of the optimal feedback control of a class of linear stochastic systems with quadratic performance measure is studied. Employing an analogue of the Pontryagin maximum principle, a set of ordinary differential equations are derived for the gain parameters of the controller. It is shown that these equations exhibit an elemental form of adaptivity. By means of an example, the applicability of the certainty equivalence principle to problems of this type is investigated.

A new method of evaluating quadratic form performance indices, time and time-squared-weighted quadratic form performance indices for a linear time-invariant regulator system described by state variable equations [Xdot] = AX is presented in this paper. The performance indices are evaluated completely in time domain, and hence the method requires no knowledge of either the Laplace transform or the complex variable theory. It is brought out that this method is much simpler than the methods available in. the literature. A numerical example for a third-order system is worked out to illustrate the method. The performance indices ISE, ITSE and ISTSE (for type-1 unity feedback systems) that are extensively used and reported in the literature are obtained in this paper as a special case of the method used here for evaluating the more general quadratic form performance indices.

(1951). A Simple Iterative Solution for Certain Simultaneous Quadratic Equations. The American Mathematical Monthly: Vol. 58, No. 9, pp. 609-613.

This article investigates some theoretical and computational properties dealing with the selection of an “optimum” experiment in a constrained environment. The squared Euclidean norm of the information matrix is developed as a measure of optimality for any experimental factorial arrangement. The problem of experimental layout is structured as a mathematical optimization which can be transformed into a quadratic integer program. Computational results are given.

The parameter variation problem of optimal linear control systems with quadratic performance indices is considered. The change in performance due to persistent changes in parameters is easily evaluated by considering the performance index as the Lyapunov function. Assuming the control law to be unchanged, parameter variations that cause system performance deterioration or improvement, are studied on the parameter plane. The bounds on performance are also given. The last example deals with a piecewise linear system.

The paper explores the engineering potential of incorporating quadratic sensitivity terms in linear optimization problems as a means of sensitivity reduction to plant-parameter deviations in linear feedback control systems.

We consider arbit,rary regression models y = Xβ + e, where the covariance matrix from e can also be singulary. First we determine optimal homogeneous and inhomogeneous linear predictions for y ∗where the true equation for y ∗ be y ∗ = Xβ + e ∗ As the risk we use a mean quadratic (not necessary) distance. Then we study the possibilities for improving the best homogeneous unbiased prediction. We show, that this is possible, if there are additional informations about X β if for instance ∣X β∣ is bounded. After this me consider the improvement of a homogeneous prediction by admitting inhomogeneous predictions. In a further section we determine for (may be even singulary) models y = X β + e a GAUSS-MARKOV estimation for β and we get also statements of the question in which cases this estimation can be improved. Thereby the strict connection between unbiased predictions and G.-M. estimations is used.

(1968). Quadratic Polynomials with the Same Residues. The American Mathematical Monthly: Vol. 75, No. 9, pp. 969-973.

The problem of optimal control is considered for the system ˙x=Ax + Bu; x(0)= c, where A is subject to random change, u is restricted to a closed, bounded region U, and a quadratic performance index is used. A general formulation of the problem is given and results are obtained for a particular, simple behaviour of A The optimal system shows adaptive behaviour of an elementary kind, and uses the whole of the available control effort at each instant. It is not known whether the last property would persist for less simple behaviour of A, or for systems disturbed by noise.