Second-order Sensitivity Research Articles

Second-Order Sensitivity Analysis of Multimodal Eigenvalues and Related Optimization Techniques*

ABSTRACT Making use of stationarity of the Rayleigh quotient, this paper presents a method for second-order sensitivity analysis of multimodal eigenvalues. Results are applied to formulate a sequential quadratic programming approach for solving problems of multimodal optimal design of structures.

Alternate Methods of Utilizing Cross-Section Sensitivity Coefficients in Radiation Shielding Problems

Attempts to devise techniques for rapidly calculating radiation transport in relatively simple shields has led to the development of two calculational models that are based on the use of cross-section sensitivity coefficients and are possible improvements over the traditional linear model. The two models, one an exponential model and the other a power model, were tested, along with the linear model, by applying them to 1- and 2-m-thick concrete slab problems in which the water content, reinforcing steel content, and composition of the concrete were varied. Comparing the results obtained with the three models with those obtained from an exact one-dimensional discrete ordinates transport calculation indicated that the exponential model, named the ''BEST model'' (for basic exponential shielding trend), is a particularly promising predictive tool for shielding problems dominated by exponential attenuation. When applied to a deep penetration sodium problem, the BEST model also yielded better results than did calculations based on second-order sensitivity theory.

Optimum sensitivity derivatives of objective functions in nonlinear programming

The feasibility of eliminating second derivatives from the input of optimum sensitivity analyses of optimization problems is demonstrated. This elimination restricts the sensitivity analysis to the first-order sensitivity derivatives of the objective function. It is also shown that when a complete first-order sensitivity analysis is performed, second-order sensitivity derivatives of the objective function are available at little additional cost. An expression is derived whose application to linear programming is presented.

Calculating transfer function and its first- and second-order sensitivities using one network analysis

The method calculates the transfer function and its sensitivities at a single frequency for both passive and active networks including constant delays. As the method requires only a single network analysis it is particularly useful for large networks. Although it uses nodal admittance representation it can be extended to other methods of network analysis.

Comment: Calculating transfer function and its first- and second-order sensitivities using one network analysis

Comment: Calculating transfer function and its first- and second-order sensitivities using one network analysis

Erratum: Calculating transfer function and its first- and second-order sensitivities using one network analysis

Erratum: Calculating transfer function and its first- and second-order sensitivities using one network analysis

Reply: Calculating transfer function and its first- and second-order sensitivities using one network analysis

The method calculates the transfer function and its sensitivities at a single frequency for both passive and active networks including constant delays. As the method requires only a single network analysis it is particularly useful for large networks. Although it uses nodal admittance representation it can be extended to other methods of network analysis.

Sensitivity analysis for premixed, laminar, steady state flames

We are involved in the modeling of premixed, laminar, one-dimensional, steady state flames. Such models require a large number of input parameters, the values of which may have large uncertainties. To determine the importance of a given input parameter in evaluating an output function, it is useful to be able to perform a sensitivity analysis. We have developed a computer program to do this for any given flames. The output functions are expressed as Taylor expansions. Both first- and second-order sensitivity coefficients can be computed. As a test case, a sensitivity analysis was carried out for a set of H 2O 2N 2 flames. The computed sensitivity coefficients show clearly which mechanisms are important for a given flame. By considering the second-order coefficients and by comparing expansions with exact solutions, some conclusions are reached about the range over which the linear expansions are reasonably accurate. Expansion in terms of the logarithm of the output functions is also considered, and is more accurate in many cases.

Second-order sensitivity derivatives in structural analysis

Haug's (1981) expressions for the calculation of static constraint function second derivatives have resulted in an order-of-magnitude saving over conventional methods for the calculation of such derivatives. A similar but simplified procedure is presented which results in a reduction of computational cost by an additional factor of two. The results are presented in a notation that is different from that of Haug, but more common in other work on structural synthesis and analysis.

Hydrophone nonlinearity measurements

Measurements of the second-order sensitivity, i.e., nonlinear response, are reported for several hydrophones in common use. The measurement technique utilizes a dual-frequency incident wave with closely spaced frequencies. The difference-frequency output of the hydrophone is then measured to yield the second-order sensitivity. The primary frequencies ranged from 15–300 kHz. Hydrophones of the same type and first-order sensitivity can have quite different second- order sensitivities. The nonlinearities are not electrostrictive in origin and are appreciably larger than pseudosound levels. They may be mechanical nonlinearities of the bond or boot materials.

Derivation and application of second-order uncertainty analysis for normally distributed parameters

A second-order uncertainty analysis methodology for problems involving generally correlated, but normally distributed parameters, is presented. All quantities required to apply this uncertainty analysis methodology are expressed analytically in terms of the customary covariance matrices and in terms of first-and second-order sensitivities. An application of this methodology to evaluate the variances of the initial conversion ratio and the infinite multiplication factor for a PWR unit cell is also presented.

Temperature-stable surface-acoustic-wave delay lines on gallium arsenide:second-order effects and sensitivities

Temperature-compensated surface-acoustic-wave delay lines on GaAs have been realised. Temperature coefficient of frequency (TCF) variation against temperature exhibited a 2nd-order coefficient comparable to that of the quartz ST-X. Investigations in sensitivities have shown that the compensating layer thicknesses can be easily controlled to achieve the temperature compensation.

First and second-order eigenvalue sensitivities of aggregated models

Analytical expressions are derived for first- and second-order eigenvalue sensitivities of aggregated models with respect to the parameters of the original high-order system. It is shown that these sensitivities are identical with the corresponding sensitivities of the original system and independent of the choice of the aggregation matrix. A numerical example is included to illustrate the results.

On sampling period sensitivities of the optimal stationary sampled-data linear regulator

In this paper the stationary sampled-data linear regulator is considered. The third and fourth-order sensitivities, about the zero sampling period, of the cost matrix and the third-order sensitivity of the feedback gain matrix are computed. Also the first and second-order sensitivities, about an arbitrary sampling period, of the cost and feedback gain matrices are computed. Moreover, the relationships between the cost matrix sensitivities and the feedback gain matrix sensitivities are made clear.

Recursive digital filter synthesis via gradient based algorithms

The three gradient-based algorithms of 1) steepest descent, 2) Newton's method, and 3) the linearization algorithm are applied to the problem of synthesizing linear recursive filters in the time domain. It is shown that each of these algorithms requires knowledge of the associated recursive filter's first-order sensitivity vectors, and, in the case of the Newton method, second-order sensitivity vectors as well. Systematic procedures for generating these sensitivity vectors by computing the response of a companion filter structure are then presented. Using the ideal low-pass filter as a design objective, it is then demonstrated that the linearization algorithm is particularly well suited for recursive filter design. On the other hand, the steepest descent and Newton methods are found to work rather poorly for this class of problems. Reasons for these empirical observed results are postulated.

Sensitivity analysis of ordinary differential equation systems—A direct method

Given a system of time dependent ordinary differential equations, dot y i = f i(c 1, c 2, …, y 1, y 2, …, t) , where c k are rate parameters, we simultaneously solve for both y i and a set of sensitivity functions, ϱy i ϱc k , over all times t. These partial derivatives measure the sensitivity of the solution with respect to changes in the parameters c k . Often these parameters are not accurately known. An example is given from atmospheric chemical kinetics using constant as well as time varying (diurnal) rate parameters. For the purposes of this paper, our calculations considered both first- and second-order contributions to Δ y with respect to Δc. It is found that second-order sensitivity terms can be highly significant, but tend to be too costly for present widespread application.

An active filter with a distributed RGC line†

A new positive feedback active filter using a RGC line is proposed. The filter has zero Q-sensitivity of the dominant pole with respect to gain variation. In comparison with other positive feedback filters it is characterized by small second-order sensitivity and having gain less than 1.

A simple bookkeeping scheme for computing sensitivities of symbolic transfer functions

A method of computing the normalized sensitivities of symbolic transfer functions is presented here for utilization in design applications which can take advantage of the stored symbolic transferfunction coefficients that are generated by recently developed symbolic transfer-function generation algorithms. The normalized sensitivities are computed in terms of quantities that are generated by the transferfunction generation algorithm, so that no additional differentiation or analysis is needed. Matrix formulation of the sensitivities further simplifies the acquisition. In addition, second-order sensitivities are shown to be obtainable in terms of first-order sensitivities.

Computer-Aided Analysis of Microwave Circuits

The most relevant techniques that have either found or should find useful application in analyzing microwave circuit performances in the frequency domain are surveyed. The particular needs of the microwave engineer are briefly discussed. Circuit equation formulations in terms of voltages and currents and wave variables are presented and the solution of the set of circuit equations by sparse-matrix techniques is illustrated. Methods based on multiport connection are also reviewed. The techniques for computing the first- and second-order sensitivity are illustrated and a comparison is made between the direct method and the transpose-matrix method, which is in certain cases similar to the method based on the adjoint circuit.

Sub-optimal design of the linear regulator with incomplete state feedback via second-order sensitivity†

Abstract Sensitivity analysis is applied to the design of the linear regulator with quadratic performance index. A Taylor expansion is used to develop expressions for sensitivity coefficients to second order for variations in the state feedback coefficients. The expression for second-order sensitivity at the optimum is particularly simple, and is applied to tho problem of designing a sub-optimal controller with incomplete state feedback.