Chebyshev Technique Research Articles

A Chebyshev technique for solving nonlinear optimal control problems

A numerical technique for solving nonlinear optimal control problems is introduced. The state and control variables are expanded in the Chebyshev series, and an algorithm is provided for approximating the system dynamics, boundary conditions, and performance index. Application of this method results in the transformation of differential and integral expressions into systems of algebraic or transcendental expressions in the Chebyshev coefficients. The optimum condition is obtained by applying the method of constrained extremum. For linear-quadratic optimal control problems, the state and control variables are determined by solving a set of linear equations in the Chebyshev coefficients. Applicability is illustrated with the minimum-time and maximum-radius orbit transfer problems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Positivity and unimodality as stabilizers of the analytic extrapolation of a function known with errors

Positivity and unimodality hypotheses on an unknown function χ1(x) confer Stieltjes character to another function H1(z), known in a discrete set of real points and affected by errors caused by experimental measurements, and impose constraints on the coefficients of its formal expansion which limit the universe of approximant functions, so acting as stabilizers of the analytic extrapolation. The type I Padé approximants, built with the coefficients of the formal expansion, provide rigorous bounds on the function in the cut complex plane. The application of a Stieltjes–Chebyshev technique allows approximations to the function, even on the cut, to be obtained. The physical problem of K±p forward elastic scattering is approached by the previous method, and bounds on the coupling constant and real part of the amplitude are found.

Design of transducer arrays with tapered side-lobe heights

Transducer arrays for many underwater sound applications have customarily been designed for equal sidelobe levels. This is, perhaps, as much due to the convenience of the analytical Dolph—Chebyshev technique as to any specific requirement. Using a simple adaptation of a previously reported numerical technique for symmetrical arrays [G. L. Wilson, J. Acoust. Soc. Am. 59, 195 (1976)], designs can be obtained whose directional response has a taper on the envelope enclosing the side lobes. The method is equally applicable to sum and to difference patterns. Examples are given. [This work was sponsored by the Naval Sea Systems Command, Code SEA-034.]

Squared amplitude frequency response and Chebyshev techniques applied to continued fraction transfer-function approximates

A continued fraction (Pade) approximant is discussed, based on the ‘ amplitude-squared frequency response ’ of the original system. A study is also made of modifications to the Pade approximant, due to Maehly, based on Chebyshev rather than Maclaurin series. By introducing the concept of ‘ shifting ’ allied to the amplitude-squared technique, the reduced model can be made an approximant over a specified range of frequencies, including band-pass systems. It is concluded that these techniques offer some interesting extensions to the continued fraction model reduction methods but fail to provide the all-purpose solution sought after ; the Chebyshev techniques, already established in numerical analysis, are not so successful in model reduction.

Impulse Noise Performance Bounds for Binary Correlation Receivers

A general performance equation is derived for digital receivers in the presence of impulse noise. Two bounding techniques are employed to simplify the results. First, the Schwarz inequality removes the dependence on both the impulse noise waveforms and the signal-set basis functions. Second, the Chebyshev inequality is applied to the Schwarz results to additionally eliminate the statistical dependence on the probability density function of the noise waveform weighting factor. The bound curves are compared to specific theoretical results where the noise waveforms and probability density functions have been assumed. The tightness of the bound curves is a function of the signal-energy-to-noise-weighting-factor variance ratio and converges for the Schwarz bound as the ratio increases. Considerably looser bounds are obtained by the Chebyshev technique, but rough estimates of system performance can be obtained when the signalenergy-to-signal-energy-to-noise-variance ratio is the only known parameter.