Roots of Polynomials with DV(Transportation) Lag.

Cuthbert Nyack

The applet here can be used to find the roots of polynomials (order 1 to 5) with DV or Transportation lag e^-Ts.
For second order the eqn is s² + a(1)s + a(2)e^-a(3)s.
For third order the eqn is s³ + a(1)s² + a(2)s + a(3)e^-a(4)s. etc.

Newton's method is used by the applet so all the limitations of Newton's method may become apparent when using the applet.
Like the other applets on this page, an iterative method is used starting from the origin and the different roots are found by moving the origin to the vicinity of the roots which are located at the starting points of the red/blue boundaries.
Because of the sensitivity to the starting point which can occur 8 scrollbars(38 to 45) are used to locate the starting point. The different scrollbars set the starting point with different resolutions.

A fifth order system without DV lag is shown below. This poly has 1 real and 2 pairs of complex conjugate roots. Starting from (-1.5,0) the real root is -1.535916.

Changing the origin to (-1.0, 1.7) locates one pair of CC root at -0.9905795 ± j1.65365211.

Changing the origin to (0.7, 0.999) locates the other pair of CC root at 0.75853777 ± j1.00079057.

With DV lag, there are now an infinite number of roots because of the periodicity of e^-Tsi. The image below shows some of the roots when DV lag is added.

The following 3 images shows how the 3 pairs of CC poles are found.

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