8
1 Introduction
jumping system and how to give a reasonable finite-time definition is the key points.
For this purpose, He and Liu first gave the definitions of finite-time stability, finitetime boundedness and finite-time stabilization of continuous-time and discrete-time
linear multi-model jumping systems, and designed the relevant stabilization controllers in finite-time sense [176]. Then the uncertain parameters are introduced into
the state variables and control variables, and the finite-time state feedback stabilization is studied for multi-model jumping system [177]. On this basis, Luan et
al. studied the finite-time stability of multi-model jumping systems with partially
unknown switching probability [178], and the relevant conclusion was extended to
discrete-time multi-model jumping system by Zuo et al. [179, 180]. Furthermore,
Ma and Jia studied the input-output finite-time stability of multi-model jumping
system [181]. Wen et al. studied the finite-time stability and stabilization of multimodel jumping system with time-varying delays [182]. The relevant conclusions
are extended to generalized multi-model jumping system [183] and Ito multi-model
jumping system [184].
In order to further study the controller synthesis problem of multi-model jumping
system, He and Liu introduced robust H ∞ control performance index into the finitetime control performance index, and designed the finite-time delayed H ∞ controller
for time-delay multi-model jumping system [185]. Then, the observer-based state
feedback controller design problem for the time-delay multi-model jumping system
is studied [186]. On this basis, the conclusions about finite-time control are applied
to discrete-time multi-model jumping systems [187, 188], time-varying delay multimodel jumping systems [189–191], gain scheduling multi-model jumping systems
[192–194], nonlinear multi-model jumping systems [195–197], etc.
State estimation and filtering is an important branches of information fusion
and signal processing. In many industrial applications, it always contains uncertain
parameters, and it is difficult to obtain the accurate system model. Therefore, in the
research of control theory, we often use state estimation and robust filtering methods
to estimate the states. As mentioned in the “Roust filtering and fault detection” parts,
the main robust filtering methods include H ∞ estimation and L 2 − L ∞ estimation.
Based on finite-time stability, finite-time stabilization and finite-time control analysis,
the finite-time H ∞ filtering and L 2 − L ∞ filtering problems of time-delay Markov
jump systems are studied in [198–200]. For some unknown cases of stochastic switching, finite-time H ∞ filters are respectively designed in [201] and [202]. Furthermore,
the relevant conclusions are extended to the design of finite-time observer [203] and
discrete time-delay multi-model jumping systems [204]. For other conclusions on
finite-time filtering of multi-model jumping system, we can refer to [205–224].
We study the finite-time control of dynamical systems in time domain; in other
aspects, we can study the system performance in frequency domain, which represents the system information with a different dimension. In time domain, we design
controllers and filters such that the system can meet the performance requirements of
practical engineering, such as system stability, robustness, etc. In frequency domain,
we make the frequency response characteristics of the closed-loop system, for example, amplitude frequency characteristics, pole distribution, meeting the requirements
such that the controlled system has corresponding performance indicators [225]. In
1 Introduction
jumping system and how to give a reasonable finite-time definition is the key points.
For this purpose, He and Liu first gave the definitions of finite-time stability, finitetime boundedness and finite-time stabilization of continuous-time and discrete-time
linear multi-model jumping systems, and designed the relevant stabilization controllers in finite-time sense [176]. Then the uncertain parameters are introduced into
the state variables and control variables, and the finite-time state feedback stabilization is studied for multi-model jumping system [177]. On this basis, Luan et
al. studied the finite-time stability of multi-model jumping systems with partially
unknown switching probability [178], and the relevant conclusion was extended to
discrete-time multi-model jumping system by Zuo et al. [179, 180]. Furthermore,
Ma and Jia studied the input-output finite-time stability of multi-model jumping
system [181]. Wen et al. studied the finite-time stability and stabilization of multimodel jumping system with time-varying delays [182]. The relevant conclusions
are extended to generalized multi-model jumping system [183] and Ito multi-model
jumping system [184].
In order to further study the controller synthesis problem of multi-model jumping
system, He and Liu introduced robust H ∞ control performance index into the finitetime control performance index, and designed the finite-time delayed H ∞ controller
for time-delay multi-model jumping system [185]. Then, the observer-based state
feedback controller design problem for the time-delay multi-model jumping system
is studied [186]. On this basis, the conclusions about finite-time control are applied
to discrete-time multi-model jumping systems [187, 188], time-varying delay multimodel jumping systems [189–191], gain scheduling multi-model jumping systems
[192–194], nonlinear multi-model jumping systems [195–197], etc.
State estimation and filtering is an important branches of information fusion
and signal processing. In many industrial applications, it always contains uncertain
parameters, and it is difficult to obtain the accurate system model. Therefore, in the
research of control theory, we often use state estimation and robust filtering methods
to estimate the states. As mentioned in the “Roust filtering and fault detection” parts,
the main robust filtering methods include H ∞ estimation and L 2 − L ∞ estimation.
Based on finite-time stability, finite-time stabilization and finite-time control analysis,
the finite-time H ∞ filtering and L 2 − L ∞ filtering problems of time-delay Markov
jump systems are studied in [198–200]. For some unknown cases of stochastic switching, finite-time H ∞ filters are respectively designed in [201] and [202]. Furthermore,
the relevant conclusions are extended to the design of finite-time observer [203] and
discrete time-delay multi-model jumping systems [204]. For other conclusions on
finite-time filtering of multi-model jumping system, we can refer to [205–224].
We study the finite-time control of dynamical systems in time domain; in other
aspects, we can study the system performance in frequency domain, which represents the system information with a different dimension. In time domain, we design
controllers and filters such that the system can meet the performance requirements of
practical engineering, such as system stability, robustness, etc. In frequency domain,
we make the frequency response characteristics of the closed-loop system, for example, amplitude frequency characteristics, pole distribution, meeting the requirements
such that the controlled system has corresponding performance indicators [225]. In