Reference:
Zs. Lendek,
R. Babuska, and
B. De Schutter,
"Stability of cascaded fuzzy systems and observers," IEEE
Transactions on Fuzzy Systems, vol. 17, no. 3, pp. 641-653, June
2009.
Abstract:
A large class of nonlinear systems can be well approximated by
Takagi-Sugeno (TS) fuzzy models, with linear or affine consequents. It
is well-known that the stability of these consequent models does not
ensure the stability of the overall fuzzy system. Therefore, several
stability conditions have been developed for TS fuzzy systems. We
study a special class of nonlinear dynamic systems that can be
decomposed into cascaded subsystems, represented as TS fuzzy models.
We analyze the stability of the overall TS system based on the
stability of the subsystems, and prove that the stability of the
subsystems implies the stability of the overall system. The main
benefit of this approach is that it relaxes the conditions imposed
when the system is globally analyzed, thereby solving some of the
feasibility problems. Another benefit is that by using this approach,
the dimension of the associated linear matrix inequality (LMI) problem
can be reduced. For naturally distributed applications, such as
multi-agent systems, the construction and tuning of a centralized
observer may not be feasible. Therefore, we extend the cascaded
approach also to observer design, and use fuzzy observers to
individually estimate the states of these subsystems. A theoretical
proof of stability and simulation examples are presented. The results
show that the distributed observer achieves the same performance as
the centralized one, while leading to increased modularity, reduced
complexity, lower computational costs, and easier tuning. Applications
of such cascaded systems include multi-agent systems, distributed
process control, and hierarchical large-scale systems.