Derivation of The Routh Array - The Cauchy Index

The Cauchy Index

Given the system:


\begin{align} f(x) & {} = a_0x^n+a_1x^{n-1}+\cdots+a_n & {} \quad (1) \\ & {} = (x-r_1)(x-r_2)\cdots(x-r_n) & {} \quad (2) \\ \end{align}


Assuming no roots of lie on the imaginary axis, and letting


= The number of roots of with negative real parts, and
= The number of roots of with positive real parts


then we have



Expressing in polar form, we have



where



and



from (2) note that



where



Now if the ith root of has a positive real part, then (using the notation y=(RE,IM))


\begin{align} \theta_{r_i}(x)\big|_{x=j\infty} & = \angle(x-r_i)\big|_{x=j\infty} \\ & = \angle(0-\mathfrak{Re},\infty-\mathfrak{Im}) \\ & = \angle(-\mathfrak{Re},\infty) \\ & = \lim_{\phi \to -\infty}\tan^{-1}\phi=-\frac{\pi}{2} \quad (9)\\ \end{align}


and



Similarly, if the ith root of has a negative real part,



and



Therefore, when the ith root of has a positive real part, and when the ith root of has a negative real part. Alternatively,



and



So, if we define



then we have the relationship



and combining (3) and (16) gives us


and


Therefore, given an equation of of degree we need only evaluate this function to determine, the number of roots with negative real parts and, the number of roots with positive real parts.


Figure 1
versus


Equations (13) and (14) show that at, is an integer multiple of . Note now, in accordance with (6) and Figure 1, the graph of vs, that varying over an interval (a,b) where and are integer multiples of, this variation causing the function to have increased by, indicates that in the course of travelling from point a to point b, has "jumped" from to one more time than it has jumped from to . Similarly, if we vary over an interval (a,b) this variation causing to have decreased by, where again is a multiple of at both and, implies that has jumped from to one more time than it has jumped from to as was varied over the said interval.


Thus, is times the difference between the number of points at which jumps from to and the number of points at which jumps from to as ranges over the interval provided that at, is defined.


Figure 2
versus


In the case where the starting point is on an incongruity (i.e., i = 0, 1, 2, ...) the ending point will be on an incongruity as well, by equation (16) (since is an integer and is an integer, will be an integer). In this case, we can achieve this same index (difference in positive and negative jumps) by shifting the axes of the tangent function by, through adding to . Thus, our index is now fully defined for any combination of coefficients in by evaluating over the interval (a,b) = when our starting (and thus ending) point is not an incongruity, and by evaluating



over said interval when our starting point is at an incongruity.


This difference, of negative and positive jumping incongruities encountered while traversing from to is called the Cauchy Index of the tangent of the phase angle, the phase angle being or, depending as is an integer multiple of or not.

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