Subsection POLES and ZEROS
Poles and Zeros
The main window of this subsection resumes the frequency domain structure of the LTI current model; the graphic (pzmap) shows the positions of its poles and transmission zeros in the complex plane (the poles are plotted as x's and the zeros are plotted as o's), while the buttons POLES and TZEROS allow the visualization of their values in the text-fields on the right of the window (a yellow background points out the instability of a pole or zero).
Note: the button POLES
is disabled if the total number of poles is more than 16, while the button TZEROS
is disabled if the total number of transmission zeros is more than 8 or equal
to 0 (in the last case, also the button Zeros analysis is inactive).
Eigenstructure
(or poles analysis)
Pressing the button Eigenstructure, it's possible to open the following window in which is performed the analysis of the poles and of the corresponding eigenvectors.
For each eigenvalue, which can be selected with the buttons << and >>, the corresponding eigenvector is computed, then the absolute values of its components along the state space are displayed in a bar graphic; at the top of this diagram are also pointed out the value of the pole in the complex plane and its values of controllability and observability (gramians) that are computed as the magnitude of the intersection point between the eigenvector corresponding to the pole and the controllability/observability ellipsoid.
The button CTRB-OBSV allows to visualize the values of ctrb/obsv gramian of every poles in a 3-dimensional graphic, treating them as heights above the complex plane: the values of the considered pole are pointed out with circles.
The function associated with the Zeros Analysis button starts the examination of the model transmission zeros (if they exist).
As usual, the user can select a transmission zero with the pushbuttons << and >>: the index and the value of the considered zero are indicated in the first two text-fields above these buttons. Then, for every zero (zo), the transfer matrix G(zo) of the current model and the parameter xo are evaluated in the following way:
G(zo) = C pinv( zo I - A ) B +D
[U S V] = svd( G(zo) )
Un = U( : , #outputs ) Vn
= V( : , #inputs )
xo = pinv( zo I - A ) B Vn
where the parameter xo represents the initial condition that, together with the input signal
u(t) = uo exp(zo t) ¹ 0,
causes y(t)=0 [1].
The bar graphics represent respectively: the absolute value of xo along the states, the absolute value of Un along the outputs and the absolute value of Vn along the inputs.
Note: all the graphics are associated with
a popup menu that can be activated pressing the right mouse button
over them.