The contributions of this thesis lie in the area of global finite-time observer design for a class of
nonlinear systems with bounded rational and mixed rational powers imposed on the incremental rate
of the nonlinear terms whose solutions exist and are unique for all positive time. In the thesis, two
different kinds of nonlinear global finite-time observers are designed by employing of finite-time
theory and homogeneity properties with different methods. The global finite-time stability of both
proposed observers is derived on the basis of Lyapunov theory.
For a class of nonlinear systems with rational and mixed rational powers imposed on the nonlinearities,
the first global finite-time observers are designed, where the global finite-time stability of the
observation systems is achieved from two parts by combining asymptotic stability and local finitetime
stability. The proposed observers can only be designed for the class of nonlinear systems with
dimensions greater than 3. The observers have a dynamic high gain and two homogenous terms, one
homogeneous of degree greater than 1 and the other of degree less than 1. In order to prove the global
finite-time stability of the proposed results, two homogeneous Lyapunov functions are provided, corresponding
with the two homogeneous items. One is homogeneous of degree greater than 1, which
makes the observation error systems converging into a spherical area around the origin, and the other
is of degree less than 1, which ensures local finite-time stability.
The second global finite-time observers are also proposed based on the high-gain technique, which
does not place any limitation on the dimension of the nonlinear systems. Compared with the first
global finite-time observers, the newly designed observers have only one homogeneous term and a
new gain update law where two new terms are introduced to dominate some terms in the nonlinearities
and ensure global finite-time stability as well. The global finite-time stability is obtained directly
based on a sufficient condition of finite-time stability and only one Lyapunov function is employed in
The validity of the two kinds of global finite-time observers that have been designed is illustrated through some simulation results. Both of them can make the observation error systems converge to
the origin in finite-time. The parameters, initial conditions as well as the high gain do have some
impact on the convergence time, where the high gain plays a stronger role. The bigger the high gain
is, the shorter the time it needs to converge. In order to show the performance of the two kinds of
observers more clearly, two examples are provided and some comparisons are made between them.
Through these, it can be seen that under the same parameters and initial conditions, although the
amplitude of the observation error curve is slightly greater, the global finite-time observers with a
new gain update law can make the observation error systems converge much more quickly than the
global finite-time observers with two homogeneous terms. In the simulation results, one can see that,
as a common drawback of high gain observers, they are noise-sensitive. Finding methods to improve
their robustness and adaptiveness will be quite interesting, useful and challenging.