Singular systems which are also referred to as descriptor systems,
semi-state systems, differential- algebraic systems or generalized
state-space systems have attracted much attention because of their
extensive applications in the Leontief dynamic model, electrical and
mechanical models, etc. This monograph presented up-to-date research
developments and references on stability analysis and design of
nonlinear singular systems. It investigated the problems of practical
stability, strongly absolute stability, input-state stability and
observer design for nonlinear singular systems and the problems of
absolute stability and multi-objective control for nonlinear singularly
perturbed systems by using Lyapunov stability theory, comparison
principle, S-procedure and linear matrix inequality (LMI), etc.
Practical stability, being quite different from stability in the sense
of Lyapunov, is a significant performance specification from an
engineering point of view. The basic concepts and results on practical
stability for standard state-space systems were generalized to singular
systems. For Lur'e type descriptor systems (LDS) which were the feedback
interconnection of a descriptor system with a static nonlinearity,
strongly absolute stability was defined and Circle criterion and Popov
criterion were derived. The notion of input-state stability (ISS) for
nonlinear singular systems was defined based on the concept of ISS for
standard state-space systems and the characteristics of singular
systems. LMI-based sufficient conditions for ISS of Lur'e singular
systems were proposed. Furthermore, observer design for nonlinear
singular systems was studied and some observer design methods were
proposed by the obtained stability results and convex optimization
algorithms. Finally, absolute stability and multi-objective control of
nonlinear singularly perturbed systems were considered. By Lyapunov
functions, absolute stability criteria of Lur'e singularly perturbed
systems were proposed and multi-objective control of T-S fuzzy
singularly perturbed systems was achieved. Compared with the existing
results, the obtained methods do not depend on the decomposition of the
original system and can produce a determinate upper bound for the
singular perturbation parameter.
