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Chumachenko S.V.
Kostanai State
University A. Baitursynov, Kazakhstan
Some cases of finding determination
of Automatic Control System (ACS) at the stability boundary
Automation is one of the
main directions of scientific and technical progress and an important means for
improving of production efficiency. One of the most important characteristics
of control systems is stability, which is directly related to efficiency.
Unstable system does not perform control functions (is unusable) and, as a
consequence, it is ineffective. Great influence on the stability provides
feedback. Unstable control of systems occurs due to improper or very strong
action of main feedback. It occurs in the following cases:
- in case of doing
feedback positive instead of negative;
- in case of large
elements`s inertia of closed loop.
Theory of automotive
control has two ways of defining sustainability: experimental and analytical.
During using the
experimental method of determining the stability, we must have a valid system,
very accurate and sensitive equipment: for forming and fixing effects on the
system and registration system behavior after removing effects. This method of
determining the stability imposes the following restrictions on process: quite
slow process; protection for human health and environment. As a consequence,
the method of determining the stability can be applied during readjusting and
partial modernization of equipment.
During designing of ACS,
when there is a mathematical model, there are used analytical methods for
determining stability: finding stability control system on location of roots of
characteristic control and a various sustainability criterions.
In determining of
stability on roots location of the characteristic control, free movement of system
is described by uniform differential equations:
After transformation of
the differential equation we obtain its operational form (characteristic
equation):
Forced component of the
output value, depending on the type of external influence, is not affected on
the stability of the system.
Solving of equation (1):
where Ck – constants which depends
on the initial conditions;
pk – roots of characteristic equation (2).
Imaginary axis is the
boundary of stability in the root plane. If at least one root is
"right", then the system will be unstable. If there is a pair of
absolutely imaginary roots and all other roots are "left", then the
system is on the vibrational stability boundary. If there is a zero root, the
system is in the aperiodic boundary of
stability. If there are two roots, the system is unstable. Consequently, the
system in which the characteristic equation can be factorized p2 are unstable
[1].
If characteristic
equation of the system is higher than the third order (except for the
biquadratic equation), it is difficult to find roots, because there are not
general formulas of expressing roots of the characteristic equation through
equations coefficients. In this case, use different criteria for
sustainability: algebraic and frequency.
Algebraic application of
Raus and Hurwitz`s criterions and frequency criteria of Mikhailov don`t limited
by degree of characteristic equation. It is due to development of computer technology.
However, by the criterions of Hurwitz and Mikhailov, it is not possible to
determine the amount of the "right" roots of the characteristic
equation, unlike Raus.
Raus`s criteria is a
quite simple way to determine the stability of ACS of high order by using a
quite simple algorithm. However, using this criterion is difficult to determine
location of the system on stability boundary: aperiodic and vibrational.
Let`s consider some special cases of finding
definitions of automatic control system on stability boundary using Raus and
using Hurwitz stability criterion.
1). Hurwitz criterion
for the system is on aperiodic stability boundary, if an = 0. How
will the Raus table look like in this case? Consider equations for systems 3
and 4 order.
For a system with a
characteristic equation:
Raus table is:
|
à0 |
à2 |
0 |
|
à1 |
0 |
0 |
|
à2 |
0 |
0 |
|
0 |
0 |
0 |
For a system with a
characteristic equation:
Form:
|
à0 |
à2 |
0 |
|
à1 |
à3 |
0 |
|
|
0 |
0 |
|
à3 |
0 |
0 |
|
0 |
0 |
0 |
Comparing the tables for
two special cases, we can conclude that a similar results will be at higher
degrees of the characteristic equation. Therefore, if in the last row of the Raus
table (line number (n + 1)) all coefficients are zero, then the system is on
aperiodic boundary of stability [2].
2). To review of finding
of ACS on oscillatory stability boundary by Raus, consider the following
special case: the characteristic equation of the system is as follow:
Roots of this equation
are
For filling table of
Raus, we convert specified characteristic equation and obtain its next form:
By this expression we
can fill the table:
|
|
|
|
0 |
|
|
|
0 |
0 |
|
|
|
0 |
0 |
|
0 |
0 |
0 |
0 |
|
? |
? |
? |
? |
During finding of
coefficients of 5th line of the table, determining of the line coefficient
From that we can
conclude that if in line with the number n (in our case n = 4) are zeros, then
the system is on vibrational stability boundary [2].
Literature:
1. Senigov P.N. Automatic Control Theory: Lecture
notes. - Chelyabinsk: YUrGU 2001.
2. Automatic Control Theory: Manual book / Savin M.M.,
Elsukov V.S., Pyatina O.N.; edited by Professor Lachin V.I. - Rostov n / d:
Phoenix, 2007.