| dc.description.abstract | In this thesis, we consider the problem of absolute stability of continuous time
feedback systems with a single, time-varying nonlinearity. Necessary and sufficient
conditions for absolute stability of second-order systems in terms of system parameters
are developed, which are characterized by eigenvalue locations on the complex plane.
More specifically, our results are presented in terms of the associated matrix-pencil
{A+ bvc*, v E [11,1, /L2]}, where /Li, /J,2 E ffi., A is n x n-matrix, b and c are n-vectors.
The stability conditions require that the eigenvalues of all matrices A +bvc*, p1 ~ v ~
μ2, lie in the interior of a specific region of the complex plane ( a cone to be specific).
Thus, we have the following reformulation of the problem. Find the maximal cone
satisfying the following condition: If all eigenvalues of corresponding linear systems
belong to this cone, then system is absolutely stable. Known results show that this
cone is not smaller than { z E <C : 3; ~ arg z ~ 5;} ( called the 45°-Region). The result
is proven using Lyapunov functions of two different types. It is known that usually
the approach based on Lyapunov functions provides essentially sufficient conditions
for absolute stability. We will use a different technique which provides necessary
and sufficient conditions for absolute stability. The problem setting, the approach,
and methods to solve the problem will be presented in Chapter 3. The contents
of Chapters 1 and 2 include preliminary concepts, definitions, and facts basic to
the theory of feedback control systems. In Sections 3.1 and 3.2, we introduce basic
results of the theory of stability for feedback control systems (i.e., for systems of
arbitrary order n E z+). In particular, we will introduce the notion of absolute
stability for feedback control systems, linear differential inclusions, dual inclusions,
and asymptotic stability of linear inclusions. Sections 3.3, 3.4, and 3.5 are devoted
to the core of this thesis: the analysis of absolute stability of systems of order two
(i.e., n = 2). In Section 3.4, we present the proof of a variant of sufficient conditions
for absolute stability that was first introduced in [2], and in Section 3.5, we prove the
new result that shows the necessity of the condition given in Section 3.4. Chapter 4
is a summary of the results obtained in this thesis and highlight some possible future
investigations. | en_US |
