ISSN 1068-3712, Russian Electrical Engineering, 2017, Vol. 88, No. 12, pp. 827–831. © Allerton Press, Inc., 2017.
Original Russian Text © A.M. Kamachkin, S.A. Sogonov, V.N. Shamberov, 2017, published in Elektrotekhnika, 2017, No. 12, pp. 57–62.
The Decomposition Method in Issues of Researching Electric-Power
Systems with Dry Friction
A. M. Kamachkin
*, S. A. Sogonov
, and V. N. Shamberov
St. Petersburg State University, St. Petersburg, Russia
St. Petersburg State Marine Technical University, St. Petersburg, 190121 Russia
Received November 14, 2017
Abstract⎯Electrical engines are widespread in all areas of industry and technology. Electrical actuators are widely
used as an actuating motor regulating the interconnected controls of different machines. Increased demands for
precision in interconnected controls make it very important to consider dry friction in the executive mechanism.
The consideration of dry friction in a mathematical model of an automatic system considerably complicates its
analysis. In all known practical cases of research into such issues, the research is carried out by means of computing
or analytically, but using a considerably simplified model of the dry-friction law. This simplified presentation does
not allow one to uncover and understand the true (or apparent) reasons for loss of stability in a system accompanied
by friction self-oscillations of different kinds (periodical, chaotic etc.). The decomposition method of parameter
space presented in the current work allows investigation of nonautonomous nonlinear multivariable systems via
basic subsystems (both linear and nonlinear) that have a lower-order state space and are amenable to strict analysis.
Keywords: many-dimensional nonlinear dynamical system, state space, state variables, nonsingular linear
transformation, decomposition of the system, basic subsystems
Electric machines are widely used as executive
electric motors that convert an input electrical signal
into the speed of angular rotation (or into the rotation
angle) of the shaft. The working element, which
together with the electric motor forms the actuator of
the automatic system, is connected with the motor
shaft. The presence of dry friction in the mechanism
can cause an auto-oscillatory mode in the system,
which often leads to accidents of differing severity.
To study this phenomenon, first of all, it is neces-
sary to correctly take into account dry friction in the
mathematical model of the automatic system under
study. The investigation of automatic systems with dry
friction makes the process of research nonlinear. In
most practical cases, a simplified description of dry
friction has been used for rigorous analytical studies—
the increased level of frictional forces at rest over fric-
tional forces of motion and the presence of a falling
section in the friction characteristic were ignored,
passed over by the mass of the moving part of the
mechanism, etc. A simplified representation of dry
friction (which is not always sufficiently substantiated)
did not allow the reasons to be identified and under-
stood for the loss of stability of the system accompa-
nied by frictional auto-oscillations of various kinds.
The mathematical models presented in this article
take into account the majority of physically significant
features of dry friction [1, 2], which has allowed solv-
ing a number of fundamental problems concerning the
dynamic behavior of automatic systems with drive
electric motors and electric drives [3–5].
Almost all known problems that have been solved
by rigorous analytical investigation of automatic sys-
tems with dry friction concerned one-dimensional
systems, while the technique raises urgent questions
regarding the study of multidimensional systems,
which have recently become quite widespread.
In the framework of the proposed method,
dynamic systems of the following types are considered:
In Eqs. (1), A, B, and C are real matrices of dimen-
sions (n × n), (n × m), and (m × n), respectively; D is
a real diagonal matrix of dimensions (m × m); ,
x(t), and y(t) are the column vectors of the variables of
dimensions (n × 1), (n × 1), and (m × 1), respectively
(here, n ≥ m); and Ψ(t) is a column vector of dimen-
sions (m × 1) of external disturbing influences
() () () ();
() () () ();
(0) ; (0) ; (0) ;
() sin ,
( ) sin ,...,
() sin .