1.3 Finite-Time and Finite-Frequency
7
stability is a special case of finite-time stability. Secondly, to study the finite shorttime stability, it is necessary to give the boundedness of the state trajectory of the
system according to the practical application requirements. In general, although the
state trajectory of the system is also required to be bounded, there is no requirement
to set a fixed value in advance. Therefore, comparing with Lyapunov asymptotic
stability, finite-time stability concerns the position of the system state deviating from
the equilibrium point, that is, the overshoot. Moreover, the finite-time theory is different from the invariant sets principle. If a set is called as an invariant set, it requires
not only the initial system states belonging to this set, but also the state at any time
belonging to this set. It obviously shows that finite-time stability concerns with the
stability of an infinite interval, which belongs to a concept in space.
After putting forward the concept of “short-time stability”, Dorato extended the
results of finite-time to linear time-varying systems [151, 152]. Weiss and Infante
introduced the concept of finite-time convergence stability [153, 154], and extended
the conclusion to nonlinear systems to propose the so-called finite-time boundedness [155]. Then, Michel and Wu [156] extended the above results to discrete-time
systems.
As we all know, the design of controllers is also a very important problem in control
theory. In 1969, Garrard et al. studied the design problem of finite-time stability
control [157]. In the subsequent research, he gave a further finite-time comprehensive
conclusion of system control [158]. In 1974, Filippo and Dorato studied the controller
design of linear system and designed a controller to make the system be stable in a
finite-time interval and to meet the linear quadratic performance index. The results
were applied to the control system of the space shuttle [149]. Then, Grlljie applied
the concept of finite-time stability to the controller design of adaptive systems [159].
It is a pity that although some valuable results are studied for finite-time stability and
controller design, these methods are hardly to be deployed in practice.
With the development of LMIs techniques, the results of finite-time stability and
controller design are easily to calculate and realize. The first paper to study the finitetime stability analysis and controller design by LMIs techniques was published in
proceedings of the 36th IEEE Conference on Decision and Control in 1997. In this
paper, Dorato et al. [160] used LMIs to design the state feedback control law of
linear systems via finite-time stabilization. Then, the conclusion was applied to ATM
network [150]. In 2005, Orino et al. proposed the concept of finite-time convergence
stability of nonlinear systems with a fixed dwell time by using feedback linearization
method [161]. In proceedings of the 46th IEEE Conference on Decision and Control,
Amate et al. published a paper on the finite-time stability of linear systems [162].
In this paper, the authors adopted the polyhedral Lyapunov function, that is, the
initial state of the system and the offset region of the state trajectory related to the
equilibrium point are described by ellipsoid. For other conclusions on finite-time
stability analysis and controller design of linear and nonlinear systems, we can refer
to [163–175].
Different from conventional non-stochastic system, multi-model jumping system
needs to consider the stochastic switching between subsystems and the influence of
stochastic factors. How to analyze the finite-time control problem of multi-model
7
stability is a special case of finite-time stability. Secondly, to study the finite shorttime stability, it is necessary to give the boundedness of the state trajectory of the
system according to the practical application requirements. In general, although the
state trajectory of the system is also required to be bounded, there is no requirement
to set a fixed value in advance. Therefore, comparing with Lyapunov asymptotic
stability, finite-time stability concerns the position of the system state deviating from
the equilibrium point, that is, the overshoot. Moreover, the finite-time theory is different from the invariant sets principle. If a set is called as an invariant set, it requires
not only the initial system states belonging to this set, but also the state at any time
belonging to this set. It obviously shows that finite-time stability concerns with the
stability of an infinite interval, which belongs to a concept in space.
After putting forward the concept of “short-time stability”, Dorato extended the
results of finite-time to linear time-varying systems [151, 152]. Weiss and Infante
introduced the concept of finite-time convergence stability [153, 154], and extended
the conclusion to nonlinear systems to propose the so-called finite-time boundedness [155]. Then, Michel and Wu [156] extended the above results to discrete-time
systems.
As we all know, the design of controllers is also a very important problem in control
theory. In 1969, Garrard et al. studied the design problem of finite-time stability
control [157]. In the subsequent research, he gave a further finite-time comprehensive
conclusion of system control [158]. In 1974, Filippo and Dorato studied the controller
design of linear system and designed a controller to make the system be stable in a
finite-time interval and to meet the linear quadratic performance index. The results
were applied to the control system of the space shuttle [149]. Then, Grlljie applied
the concept of finite-time stability to the controller design of adaptive systems [159].
It is a pity that although some valuable results are studied for finite-time stability and
controller design, these methods are hardly to be deployed in practice.
With the development of LMIs techniques, the results of finite-time stability and
controller design are easily to calculate and realize. The first paper to study the finitetime stability analysis and controller design by LMIs techniques was published in
proceedings of the 36th IEEE Conference on Decision and Control in 1997. In this
paper, Dorato et al. [160] used LMIs to design the state feedback control law of
linear systems via finite-time stabilization. Then, the conclusion was applied to ATM
network [150]. In 2005, Orino et al. proposed the concept of finite-time convergence
stability of nonlinear systems with a fixed dwell time by using feedback linearization
method [161]. In proceedings of the 46th IEEE Conference on Decision and Control,
Amate et al. published a paper on the finite-time stability of linear systems [162].
In this paper, the authors adopted the polyhedral Lyapunov function, that is, the
initial state of the system and the offset region of the state trajectory related to the
equilibrium point are described by ellipsoid. For other conclusions on finite-time
stability analysis and controller design of linear and nonlinear systems, we can refer
to [163–175].
Different from conventional non-stochastic system, multi-model jumping system
needs to consider the stochastic switching between subsystems and the influence of
stochastic factors. How to analyze the finite-time control problem of multi-model