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Nth-order Flat Approximation of the Signum Function by a Polynomial
In the interval studied, the signum function, sgn x, was demonstrated to be uniquely approximated by an odd polynomial f sub n (x) of order 2n-1, for which the approximation is nth order flat with respect to the points (1,1) and (-1, -1). A theorem was proved which states that for even integers n or = 2, the approximating polynomial has a pair of nonzero real roots or - x sub n such that the x sub n form a monotonically decreasing sequence which converges to the root of 2 as n approaches infinity. For odd n i, f sub n (x) represents a strictly increasing monotonic function for all real x. As n tends to infinity, f sub n (x) converges to sgn x uniformly in two interval ranges.
NASA Technical Note
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The Determination of a Stability Indicative Function for Linear Systems with Multiple Delays
A theoretical study is made of the stability of a class of linear differential-difference equations with multiple delays. A direct method for determining the exact stability boundaries for homogeneous, linear differential-difference equations with constant coefficients and constant delays is formulated. This formulation results in a stability indicative function, depending on a single parameter, which determines the number of roots of the transcendental characteristic equation that have positive real parts. It is proved that the system is stable if and only if this function has a value of zero. A second-order system with delays in the velocity and position feedback terms is considered as an example, and the stability regions for this system are determined for a range of delays and coefficients. It is observed that introduction of a delay has a definite destabilizing effect on the system, and introduction of a second delay has a compounding effect to further reduce stability. However, this example clearly illustrates that certain combinations of delays can stabilize an unstable system. This phenomenon is discussed from a theoretical point of view.
