1.1 Background and Research Status of Multi-model Jumping System
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In recent years, multi-model jumping system has been widely used in various
fields, such as communication system [48, 49], economic system [50], power system [51], robot mechanical system [52, 53], network system [54], etc. Comparing
with the linear multi-model jumping system, the research of nonlinear multi-model
jumping system is not well-considered, which is mainly due to the complexity of
nonlinearities. In earlier research of linear dynamical systems, we always refer to
solve the corresponding Ricatti equation. It is more convenient and feasible to transform Ricatti equation to linear matrix inequalities (LMIs) [55]. However, it is difficult
to obtain a general controller for nonlinear multi-model jumping system. In 1998,
Aliyu and Boukas [56] tried to use Hamilton–Jacobi equation to derive sufficient
conditions for stochastic stability and H ∞ performance index of multi-model jumping system. Unfortunately, a global solution to the Hamilton–Jacobi equation can
not be obtained by numerical or analytical methods.
With the development of T–S fuzzy control technology and LDI-based neural
network technology, the control problem of nonlinear multi-model jumping system
becomes relatively easy. The corresponding nonlinear system can be linearized by
using T–S fuzzy rules [51–54] or LDI-based neural network technology [57–59] to
obtain a feasible controller. Till now, there are a lot of research results about nonlinear multi-model jumping system using these methods. For example, for uncertain
continuous-time fuzzy systems with Markov jump parameters, Nguang et al. proposed the H ∞ controller and filter [60] design methods for fuzzy systems with output
feedback, and extended the results to fuzzy singularly perturbed systems [61]. By
introducing the method of free weight matrix, Wu and Cai [52] gave a sufficient
condition of robust stability for a class of uncertain nonlinear multi-model jumping
system, and designed a mode-independent controller. On this basis, by introducing
multiple free weight matrices, Dong and Yang [62] discussed the design method of
state feedback controller for multi-model jumping system with less conservatism. For
other conclusions on stochastic stability and stochastic control of nonlinear multimodel jumping system, we can refer to [63–80].
Up to now, the theory of multi-model jumping system has been developed for
fifty or sixty years. Although a lot of theoretical research results have been obtained
and applied to production practice, the research on this kind of stochastic systems
is still under development. Both the systematicness and perfection in theory aspects
and the maturity and standardization in technology aspects have not formed an intact
research system, which needs further study and discussion.
1.2 Roust Filtering and Fault Detection
State estimation and filtering are important branches of system control theory, signal
processing and information fusion. We always use Kalman filtering and Luenberger
observer theory to study estimation and filtering problems. However, these two methods are for deterministic systems, which require accurately known models and white
3
In recent years, multi-model jumping system has been widely used in various
fields, such as communication system [48, 49], economic system [50], power system [51], robot mechanical system [52, 53], network system [54], etc. Comparing
with the linear multi-model jumping system, the research of nonlinear multi-model
jumping system is not well-considered, which is mainly due to the complexity of
nonlinearities. In earlier research of linear dynamical systems, we always refer to
solve the corresponding Ricatti equation. It is more convenient and feasible to transform Ricatti equation to linear matrix inequalities (LMIs) [55]. However, it is difficult
to obtain a general controller for nonlinear multi-model jumping system. In 1998,
Aliyu and Boukas [56] tried to use Hamilton–Jacobi equation to derive sufficient
conditions for stochastic stability and H ∞ performance index of multi-model jumping system. Unfortunately, a global solution to the Hamilton–Jacobi equation can
not be obtained by numerical or analytical methods.
With the development of T–S fuzzy control technology and LDI-based neural
network technology, the control problem of nonlinear multi-model jumping system
becomes relatively easy. The corresponding nonlinear system can be linearized by
using T–S fuzzy rules [51–54] or LDI-based neural network technology [57–59] to
obtain a feasible controller. Till now, there are a lot of research results about nonlinear multi-model jumping system using these methods. For example, for uncertain
continuous-time fuzzy systems with Markov jump parameters, Nguang et al. proposed the H ∞ controller and filter [60] design methods for fuzzy systems with output
feedback, and extended the results to fuzzy singularly perturbed systems [61]. By
introducing the method of free weight matrix, Wu and Cai [52] gave a sufficient
condition of robust stability for a class of uncertain nonlinear multi-model jumping
system, and designed a mode-independent controller. On this basis, by introducing
multiple free weight matrices, Dong and Yang [62] discussed the design method of
state feedback controller for multi-model jumping system with less conservatism. For
other conclusions on stochastic stability and stochastic control of nonlinear multimodel jumping system, we can refer to [63–80].
Up to now, the theory of multi-model jumping system has been developed for
fifty or sixty years. Although a lot of theoretical research results have been obtained
and applied to production practice, the research on this kind of stochastic systems
is still under development. Both the systematicness and perfection in theory aspects
and the maturity and standardization in technology aspects have not formed an intact
research system, which needs further study and discussion.
1.2 Roust Filtering and Fault Detection
State estimation and filtering are important branches of system control theory, signal
processing and information fusion. We always use Kalman filtering and Luenberger
observer theory to study estimation and filtering problems. However, these two methods are for deterministic systems, which require accurately known models and white