|
Leonov, G. A., Reitmann, V., Smirnova, Vera B.;
Non-local methods for pendulum-like feedback systems (English)
Teubner-Texte zur Mathematik. Stuttgart: Teubner. vii, 242 p. (1992).
|
 |
 This
book is dedicated to the investigation of global behaviour of
dissipative pendulum-like equation. Chapter 1 presents basic definitions
and concepts for understanding global aspects of ODE with multiple equilibria:
the direct method of Lyapunov, the attractor concept, the theory of feedback
control equations and frequency-domain methods in absolute stability theory.\par
Chapter 2 introduces the global bifurcation theory of two-dimensional systems
with a periodic nonlinearity and the general concept of higher- dimensional
pendulum-like systems. In Chapter 3 (and the following ones) there are
demonstrate several new methods which extend the Lyapunov direct method
to systems with multiple equilibria.\par Chapter 4 shows that questions
concerning the global convergence of solutions of pendulum like systems
can be well formulated in terms of factor manifolds. Connections with differential
geometry approach to nonlinear systems are given.\par Chapter 5 presents
``the method of non-local reduction'' --- a method for stability investigation
of higher-dimensional systems of differential equations which employs the
stability results of differential equations in the plane, playing the role
of reduced equations.\par In Chapter 6, in order to get necessary conditions
for global convergence type, theorems which ensure the existence of circular
solutions and of cycles of various type are provided.\par In Chapter 7
some classes of pendulum-like systems describing the dynamics of synchronous
machines are considered. Chapter 8 extends some results of previous chapters
concerning boundedness, convergence and quasiconvergence to a class of
integro-differential equations with retarded argument which arrives from
phase synchronization problems.\par Chapter 9 is devoted to the cycle slipping
in phase-controlled systems and in the 10-th Chapter the authors show that
many results of the previous chapters also get through for discrete systems.
S.Anita
(Iasi)
Keywords: dissipative pendulum-like equation; global bifurcation
theory; global convergence; factor manifolds; method of non-local reduction;
phase-controlled systems
-
Classification:
-
93-02
Research monographs (systems and control)
-
93C15
Control systems governed by ODE
-
93D15
Stabilization of systems by feedback
-
70K10
Limit cycles (general mechanics)
93B29
Differential-geometric methods in systems theory
aus: MATH Database, Zentralblatt für Mathematik
/ Mathematics Abstracts:
Copyright
(c) 1997 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag.
The comparison principle has proven to be a useful tool in the
qualitative analysis of high-dimensional systems. A comparison technique, called the nonlocal
reduction principle, is presented in this book. The work is based on two ideas: Lyapunov functions,
which generate in the phase space a family of surfaces transversal to the vector field, and comparison systems,
which have the same qualitative properties as the original systems but have lower dimension than the former systems.
Initially, the nonlocal reduction method was developed for stability investigations of systems with angular coordinates.
Subsequently, it has been applied to general automatic control systems. Differential equations are interpreted,
in general, as feedback systems in this book. Sufficient conditions for global stability, existence of limit cycles,
boundedness of solutions, etc., are formulated in terms of transfer functions and have the form of frequency-domain
inequalities. Systems described by integro-differential equations and discrete-time systems are also included.
The distinguishing feature of the book is the combining of Lyapunov's second method with frequency domain techniques
in a natural way.
Anthony N. Michel (1-NDM-E; Notre Dame, IN)
aus: MATHEMATICAL REVIEWS; Clippings from Issue 94d.
|
|
