SOLVING TERMINAL PROBLEMS FOR MULTIDIMENSIONAL
AFFINE SYSTEMS BASED ON TRANSFORMATION
TO A QUASICANONICAL FORM
D.A. Fetisov
Bauman Moscow State Technical University, Moscow, Russian Federation
e-mail:
dfetisov@yandex.ruThe paper considers a terminal problem for multidimensional affine systems, which
are not linearizable by a feedback. The affine system is transformed to a regular
quasicanonical form using a smooth nondegenerate change of variables within the
range of states. In addition, the terminal problem for the initial system is transformed
to the equivalent terminal problem for the system of a quasicanonical form. A method
of solving the terminal problems is proposed for the quasicanonical systems, which
is based on a concept of dynamics inverse problems generalization. The sufficient
condition for applying the proposed method is proved. The numerical procedure of
solving the terminal problems for the systems of a quasicanonical form is proposed.
There is an example of solution development of a terminal problem for a sixth-order
system using the above-mentioned method. The obtained results may be used for
solving problems of terminal control over technical systems.
Keywords
:
affine system, control, quasicanonical form, terminal problem.
Introduction.
Equivalent transformations of the systems with a control
provide many opportunities for solving various control theory problems.
Papers [1–3] present methods of controllability research, construction of
reachability sets, solving the problems of stabilization, and the terminal
problems based on the system transformation to certain canonical forms.
In this paper, the issue of terminal problems solution to the affine systems
is considered. Different approaches to this issue can be found in [1, 4, 5–9].
Papers [1, 4] describe methods of terminal problems solutions of the affine
systems, which are linearizable by a feedback, i.e. the systems that are
converted to linear controlled systems by a smooth nondegenerate change
of variables and a reversible change of controls. The methods for solving the
terminal problems of the linear controlled systems are well known and are
based on the application of the concept of dynamics inverse problems [10].
Nowadays, one of the most important challenges is the development of
methods for solving the terminal problems of the systems, which are not
linearizable by a feedback. Papers [5–8] set out the methods for solving
the terminal problems for such systems. However, these methods cover a
relatively small class of systems; the range of applicability of such methods
imposes severe restrictions on system dimensions. A special kind of the
system vector fields is often used. Thus, the issue of solving the terminal
problems for the affine systems, which are not linearizable by a feedback
is relevant. The present paper is dedicated to this problem.
Let us consider the following problem. For an affine system
˙
x
=
F
(
x
) +
m
X
j
=1
G
j
(
x
)
u
j
;
(1)
16
ISSN 1812-3368. Herald of the BMSTU. Series “Natural Sciences”. 2014. No. 5