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Video tags
Subtitles
Russian
00:00:01
stable automatic00:00:03
control system stability criteria00:00:07
line of drawn sars calculation of the roots00:00:11
of characteristic equations is simple for00:00:15
equation 1 2 steve not a general expression00:00:18
for the roots of equations of the third fourth00:00:20
degree cumbersome and of00:00:22
little use in general the expression for the00:00:25
roots of equations of higher degrees00:00:27
there is no numerical determination of the roots of00:00:31
equations and does not cause difficulties At00:00:33
present, however, there are well-00:00:36
developed methods for assessing stability00:00:38
and rules for analyzing stability without00:00:41
calculating roots, and this is very important00:00:44
at the stage when the system does not yet exist and when it is00:00:48
just being created, these rules are00:00:51
called stability criteria,00:00:53
they allow in some cases not only00:00:56
to determine whether the system is stable or not, but00:00:59
also to determine the influence of various00:01:01
parameters and structural changes on00:01:03
stability, there are various forms of00:01:06
stability criteria;00:01:08
however, all of them are mathematically00:01:11
equivalent, since the conditions under which the00:01:13
roots of the characteristic equation00:01:15
are located on the left side of the complex00:01:17
plane, the stability criterion00:01:19
is divided into algebraic and00:01:22
frequency algebraic criteria; these are00:01:25
algebraic procedures over the00:01:28
coefficients of the characteristic00:01:29
equation to These include the00:01:31
Routh Hurwitz stability criterion, the00:01:33
mayor, and the code; other algebraic00:01:38
criteria for a system described by00:01:41
equations above the 4th degree make it00:01:43
possible to determine only the stability of00:01:45
the system for given numerical values of the00:01:48
coefficients of the equation,00:01:50
but it is difficult to answer the question of how00:01:53
to change the parameters of the system to make00:01:55
it stable; the frequency00:01:59
stability criterion was first formulated00:02:01
and test and modernized by Mikhailov, the00:02:04
composition of the frequency criteria is00:02:08
clarity and also the possibility of00:02:10
using experimental00:02:12
frequency characteristics.00:02:15
Mikhailov’s stability00:02:17
criterion is based on the principle of00:02:20
the argument is essentially and its00:02:22
geometric interpretation, consider the00:02:24
characteristic polynomial of a linear00:02:26
system of a different order with positive00:02:28
coefficients, which is a necessary00:02:30
condition for stability00:02:32
appearance characteristic and went00:02:34
to the sight and00:02:35
and the solvent slide we assume that lambda is00:02:38
equal to jim mega we get the00:02:40
characteristic polynomial in the form00:02:42
in which it is presented on the third line of00:02:44
the slide where the real part x contains00:02:48
only an even degree of omega understands00:02:51
only an odd degree very system is00:02:55
presented on the fourth line of glory00:02:59
we will depict d&g amides on the complex00:03:01
plane at x frame grams of00:03:08
infinity00:03:09
x equals plus minus infinity y equals00:03:12
plus minus infinity the sign00:03:15
is determined by the exponent00:03:17
characteristic form of the hodograph MDF and omega00:03:20
are shown in the figure send the third00:03:22
drumnom because in this case it goes through 500:03:25
quadrants the graph constructed in the year00:03:28
is called the Mikhailov curve also00:03:30
characteristic curve to the delegate graph of00:03:32
the vector d Hajime00:03:34
formulation of the Mikhailov stability criterion00:03:36
video stability of a linear00:03:39
system of etheric order necessary00:03:41
sufficient for changes in the arguments of the00:03:43
function d&g omega when the frequency changes00:03:46
from zero to infinity would be equal to00:03:50
n.p. in half, that is, since it is00:03:54
depicted on the slide,00:03:58
the formulation is simple and00:04:00
pleasant, but not entirely accurate, so that the00:04:02
stability conditions are sufficient; it is00:04:04
only necessary that all n00:04:07
roots of the characteristic equation00:04:09
be left-handed, that is, among them there should00:04:11
be no roots lying on the imaginary axis and00:04:15
turning into complex polynomial Jane00:04:18
Omega that is, one more condition must be satisfied00:04:22
that I am not Gandhi is equal to zero both formulas00:04:25
are a mathematical expression of the00:04:27
Mikhailov stability criterion The00:04:30
figure shows Mikhailov curves of00:04:32
stable linear systems of various00:04:35
orders n equals 3 4 5 600:04:39
we can say and so for the stability of a00:04:42
linear system it is necessary and it is enough00:04:44
for Mikhailov to peck and pass00:04:46
sequentially to the00:04:48
quadrant counterclockwise and all the time surround the00:04:50
origin of coordinates and is also clearly visible in the00:04:53
figure if changes in the arguments of the00:04:56
Dr. J Omega function do not meet the00:05:00
Finn and dome criterion when the00:05:04
frequency changes from zero to infinity and are not00:05:07
fulfilled conditions d&g megane is equal to00:05:10
zero, then the system is unstable; the figure at the00:05:13
top of the slide shows an example of00:05:15
Mikhailov curves for its stable00:05:18
systems; it can be seen that it violates the rule of00:05:20
sequential park hosting00:05:21
quadrants; the figure at the bottom of the slide00:05:24
shows examples of Mikhailov clinics00:05:26
for systems located on the border of00:05:28
stability; that in the mole year the graph00:05:31
passes through the point practically the00:05:35
Mikhailov curve is constructed according to . with the complex00:05:37
shape of this curve, in practice this00:05:39
happens very often,00:05:40
there is never a guarantee that the entire Mikhailov curve has been constructed,00:05:43
for example, in the figure00:05:45
there is a very small part of it and00:05:47
Mikhailov, which makes this system00:05:50
unstable, this is a section indicated by00:05:54
dots, and00:05:55
when calculating the Mikhailov curve, this00:05:58
section is easily overlooked by fighters00:05:59
calculate the system is stable when00:06:01
using a complex technique, this00:06:04
option is always available00:06:06
therefore, in contrast to the one00:06:09
Mikhailov criterion method discussed above,00:06:11
more convenient, the main thing is more00:06:14
accurate method 2, the two Mikhailov criterion method00:06:17
is to use the00:06:19
property of displacement of the roots of the00:06:21
polynomial fixer omega 3 and omega00:06:24
functions d&g atomic00:06:26
on The figure on the left shows that going along the00:06:30
Mikhailov curve, the omega point is equal to zero, the00:06:33
direction of increasing omega leaves the00:06:36
x axis, then Alex intersects the y axis again,00:06:40
and so on, this means that the roots00:06:42
of the equation x amides are equal to zero y atom the00:06:46
game black must be explored one00:06:48
after another in a stable system00:06:51
apply x from amigo y from omega and00:06:54
the main location of the roots of omega this00:06:58
has the form approximately shown in the figure on the00:07:00
right in this example and is equal to 400:07:05
thus the conditions for the stability of00:07:07
the system is the mobility of the roots of00:07:11
the equation x the remaining grams y from Egor00:07:14
consider the example let as in the previous00:07:17
topic the characteristic polynomial the00:07:20
systems themselves have the form presented in the first00:07:22
line s line and then for dna equal x00:07:28
from omega plus g y equations00:07:33
are presented in the second line of the slide the00:07:35
roots of the equation are presented on the slide00:07:38
only non-negative00:07:40
frequency values are considered and no less than these00:07:44
roots are put down in advance in order of00:07:46
increasing frequency values00:07:48
the value of these roots must alternate,00:07:51
that is, in this case there must be00:07:53
omega 0 less they stroked less00:07:56
indignant the stability condition00:07:58
obtained from the previous topic remains the00:08:03
same to less than one from 1 plus 100:08:06
divided by t200:08:08
for this example the cool X's and Y's00:08:12
comedy looks like shown in In the figure,00:08:14
it is believed that Mikhailov’s clitoris is simpler00:08:17
and more clear for determining the stability of a00:08:19
system than the Hurwitz criterion for00:08:23
high-order systemsDescription:
Критерий устойчивости Михайлова для исследования устойчивости систем автоматического регулирования. Скринкаст лекции Николая Германовича Грибанова.
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