2
1 Introduction
in discrete-time case), and the switching between different modes is determined by
Markov switching probability. Because of the different modes of the system, we call
it as multi-model jumping system.
In fact, we can regard the subsystem of each mode as the sub-model of the whole
multi-model jumping system, and these sub-models are integrated by certain switching laws or jump probabilities. From this point of view, multi-model jumping system is essentially different from other multi-model control systems or distributed
systems, which are linearized by Takagi–Sugeno (T–S) fuzzy rules and linear differential inclusion (LDI) rules. In this sense, multi-model jumping system is a kind
of stochastic systems, a kind of hybrid systems and a kind of multi-model control
systems. With the deepening of research, it is found that the common characteristic
of this kind of multi-model jumping systems is that it is often affected by the external
environment, internal structures and other random mutation factors in the operation
process, which makes the system parameters jump stochastically. It shows that many
natural systems can be abstracted as multi-model jumping system. Therefore, the
study of multi-model jumping system has very important theoretical and practical
significance.
In the past few decades, there have been a lot of research results on multi-model
jumping system, which mainly focus on stochastic stability, stochastic stabilization,
controller design and comprehensive analysis, robust filtering and fault detection.
For example, Swarder discussed the optimal control problem of linear multi-model
jumping system with the theory of maximum minimum principle [2] in 1969. Then,
in 1971, Wonham applied dynamic programming to deal with the optimal control of
linear multi-model jumping system [3].
Since the 1980s and 1990s, multi-model jumping system has aroused great interest
of scholars. In 1990, Ji and chizeck explicitly proposed the concepts of stochastic stabilization and stochastic controllability of multi-model jumping system, and proved
the necessary and sufficient conditions for stochastic stabilization and controllability
of systems by using stochastic Lyapunov functionals and so-called weak infinitesimal operators [5]. In 1992, Feng et al. discussed a series of problems related to
stochastic stability of multi-model jumping system, and pointed out that stochastic
stability, mean square stability and mean square exponential stability are sufficient
conditions of almost asymptotically stability of multi-model jumping system [6], and
these stochastic stability conditions are equivalent. Then, Boukas et al. respectively
studied the stochastic stabilization, H ∞ control and guaranteed cost control of multimodel jumping system [7–9]. Park et al. considered the predictive control problem
[10, 11] of multi-model jumping system respectively in 1997 and 2002. Costa et
al. analyzed the H 2 control problem [12] of continuous-time multi-model jumping
system in 1999, and successfully established the relationship between the H 2 norm
index and the controllable/observable Gram matrix for discrete-time multi-model
jumping system to extended the research results to the robust H 2 control problem of
multi-model jumping system with uncertain parameters [13] by using convex optimization tools in 2000. De Farias et al. used dynamic output feedback strategy to
study the H 2 and H ∞ control [14]. For other conclusions on stochastic stability and
stochastic control of multi-model jumping system, we can refer to [15–47].
1 Introduction
in discrete-time case), and the switching between different modes is determined by
Markov switching probability. Because of the different modes of the system, we call
it as multi-model jumping system.
In fact, we can regard the subsystem of each mode as the sub-model of the whole
multi-model jumping system, and these sub-models are integrated by certain switching laws or jump probabilities. From this point of view, multi-model jumping system is essentially different from other multi-model control systems or distributed
systems, which are linearized by Takagi–Sugeno (T–S) fuzzy rules and linear differential inclusion (LDI) rules. In this sense, multi-model jumping system is a kind
of stochastic systems, a kind of hybrid systems and a kind of multi-model control
systems. With the deepening of research, it is found that the common characteristic
of this kind of multi-model jumping systems is that it is often affected by the external
environment, internal structures and other random mutation factors in the operation
process, which makes the system parameters jump stochastically. It shows that many
natural systems can be abstracted as multi-model jumping system. Therefore, the
study of multi-model jumping system has very important theoretical and practical
significance.
In the past few decades, there have been a lot of research results on multi-model
jumping system, which mainly focus on stochastic stability, stochastic stabilization,
controller design and comprehensive analysis, robust filtering and fault detection.
For example, Swarder discussed the optimal control problem of linear multi-model
jumping system with the theory of maximum minimum principle [2] in 1969. Then,
in 1971, Wonham applied dynamic programming to deal with the optimal control of
linear multi-model jumping system [3].
Since the 1980s and 1990s, multi-model jumping system has aroused great interest
of scholars. In 1990, Ji and chizeck explicitly proposed the concepts of stochastic stabilization and stochastic controllability of multi-model jumping system, and proved
the necessary and sufficient conditions for stochastic stabilization and controllability
of systems by using stochastic Lyapunov functionals and so-called weak infinitesimal operators [5]. In 1992, Feng et al. discussed a series of problems related to
stochastic stability of multi-model jumping system, and pointed out that stochastic
stability, mean square stability and mean square exponential stability are sufficient
conditions of almost asymptotically stability of multi-model jumping system [6], and
these stochastic stability conditions are equivalent. Then, Boukas et al. respectively
studied the stochastic stabilization, H ∞ control and guaranteed cost control of multimodel jumping system [7–9]. Park et al. considered the predictive control problem
[10, 11] of multi-model jumping system respectively in 1997 and 2002. Costa et
al. analyzed the H 2 control problem [12] of continuous-time multi-model jumping
system in 1999, and successfully established the relationship between the H 2 norm
index and the controllable/observable Gram matrix for discrete-time multi-model
jumping system to extended the research results to the robust H 2 control problem of
multi-model jumping system with uncertain parameters [13] by using convex optimization tools in 2000. De Farias et al. used dynamic output feedback strategy to
study the H 2 and H ∞ control [14]. For other conclusions on stochastic stability and
stochastic control of multi-model jumping system, we can refer to [15–47].