The Evaluation section allows the user to evaluate in every detail the behavior of the closed loop system built with the controller that has been previously computed at the end of a synthesis or optimization section. Let us consider the following closed loop scheme,
where G(s) is the transfer
matrix of the current model that has to be controlled, and K(s) is the transfer
matrix of the controller, we have that:
|
Open Loop |
Fo = G K |
Fi = K G |
|
Sensitivity |
So = ( I + G K )-1 |
Si = ( I + K G )-1 |
|
Control Sensitivity |
Mo = K So |
Pi = Mi = G Si |
|
Complementary Sensitivity |
Po = To = So G K |
Ti = Si K G |
where Po(s) is the transfer matrix of the closed loop system from reference input R(s) to plant output Y(s) and Pi(s) is the transfer matrix of the closed loop system from plant input U(s) to plant output Y(s) (o\i denotes that the relative matrix is measured at the plant output\input).
All the functionalities of this section are accessible only through the commands of the evaluation menu and can be subdivided in the following groups:
· Det( I + GK ) = Det( I + KG)
· Open Loop Response, Sensitivity, Control Sensitivity, Complementary Sensitivity:
They plot the frequency responses of the corresponding
transfer matrix measured at the plant input and output; if the upper limit of
the maximum singular value of a matrix is defined (in case of H-2, H-Infinity,
H-Mix and Mu control) it's visualized into the same graphic.
· Gain and Phase Margins
· Pzmaps: Outputs & Inputs
They plot in the complex plane the poles and
transmission zeros of the Open Loop, Sensitivity, Control Sensitivity and
Complementary Sensitivity transfer matrices, measured either at the plant input
or output: the poles are plotted as blue x's and the zeros as green o's.
· Poles Drifting: Graphic & Values
They allow the user to visualize the way in which the
feedback loop acts on the poles of the open loop system (plant + controller);
the section relative to the command "Values" is limited to 20 poles
They allow visualizing respectively the robust
performance condition and the ultimate robust performance condition
through the frequency response of the maximum structured singular values ( m ) of
the matrices
[ T , T ; S , S ] , [ M , M ; S , S]
diag{ Si , Mi , Mo , So} , diag{ Si , Ti ,
To , So }
Since the computation time of these frequency responses is rather long,
the two commands are disabled (default condition). In order to activate (or
deactivate) these functionalities, it's necessary to set (or reset) the
"advanced-flag", stored in the global variable "stack", by
typing the command
stack.general.adv_flag = 1 (or 0)
at the Marlab prompt. Each time the user exits
from MIMOtool, the value of this flag is saved and it will be reloaded at the
next activation of the program.
· Graphic Main Results, Step Responses, Text Main Results:
These commands consider either Po(s)
or Pi(s) and provide a set of information about
their structure in the frequency domain (poles / zeros map, controllability,
observability, maximum eigenvalue, etc) and their time domain behavior (step
response, settling time, overshoot, etc).
Since the command relative to the step responses of the selected
transfer matrix plots the responses of all the channels contemporaneously, the
visualization could be not very clear: in this case the commands Axes Grouping … and Select I\O's … of the popup menu, which can
be activated by pressing the right mouse button over a graphic of the window,
allow the user to examine in details either a single channel or a group of
them.
Both the buttons BACK and CLOSE causes the closing of the evaluation section, but while the first one comes back to the previous window, the second one closes also the synthesis or optimization section in which the controller has been computed and comes back to the main synthesis window.
Note: all the
graphics in MIMOtool have its own popup menu, which can be activated by pressing the right mouse
button over them (see the relative help page for more details).
Note: the data structure help page explains how the controller can
be accessed directly from the Matlab command line.