Generalization of Phase-Plain Methods

In case the singular formation is a point, the following hold true:

1 Let
w be a vector corresponding with a process variable shown as a local coordinate; u, the time change vector of this; w0 and u0, their value in the singular point.

Also, let
r = w – w0
v = u
(as the change in the singular point is zero, u0 = 0), where r is the vector pointing from the singular point towards the examined point, and v is it change-velocity vector.

The r – v right-angle coordinate system created by linear transformation is a plain even in the case “w” variable is dependent on many other variables which could be shown only in a multi-dimensional phase space.

As a result of the transformation, the processes can always be examined in one phase plain.

2 Let us break variable “r” down to the following components:
r = rd + rs + ri
where d is abducing, that is, destabilizing, s is approaching, that is, stabilizing, while i is neither abducing nor approaching, therefore it is inert.

The same in the case of vector “v”:
v = vd + vs
as due to the definition, the third component is a zero vector.

Thus, the examination of asymptotic stability based on the scalar product of the two vectors is in each case simplified to an examination in this plain.

Dr. Endre Simonyi