6
1 Introduction
modeling errors, are obtained. The energy norm indexes of the residual signals to the
external disturbance signals and fault signals are selected to illustrate robustness to the
external disturbance and sensitivity to the fault signals. The design of fault detection
observer or filter is described as an optimization problem. By using Lyapunov–
Krosovskii functional and LMIs technique, sufficient conditions for the existence
of fault detection observers or filters are proved and given, and an optimal design
method is also proposed. We also give simulation results to show the feasibility and
applicability of the designed fault detection methods.
1.3 Finite-Time and Finite-Frequency
It is known that the stability is always an important problem for control systems.
For the study of stability problems, we always mention Lyapunov stability theory,
whose key point is to construct a suitable Lyapunov function or functional. It should
be pointed out that Lyapunov stability theory is suitable for the study of asymptotic
stability. In general, asymptotic stability describes the steady-state performance of
a system, and represents the system characteristics in an infinite interval. It can
meet the requirements of practical engineering, but it can not reflect the transient
performance. In fact, an asymptotically stable system may have bad transient performance, which will cause bad effects and can not meet the requirements of industrial
production, such as excessive overshoot, severe oscillation, overshoot or saturation
nonlinearity, etc. Therefore, for some short-time working systems, such as missile
system, communication network system and robot control system, we not only study
the steady-state performance in Lyapunov sense, but also pay more attention to the
transient performance requirements of the dynamics, such as the deviation degree of
the system state trajectory from the equilibrium point at a certain time point. In order
to solve the problem of system transient performance, Dorato proposed a concept
of short-time stability [142] (i.e., finite short-time stability), and then analyzed the
finite short-time control problems, which have been widely used.
The so-called finite short-time stability means that the state trajectory of the system
does not deviate from the predetermined range in a given finite-time region. It should
be pointed out that the concept of finite short-time stability studied in this book is
different from the finite short-time stability studied in [143–147], in which it means
that the system state reaches an equilibrium point in a finite-time and stays in the
point. In fact, it examines the state behavior of the system in the “finite but infinite time
interval”. Therefore, the finite short-time control theory in this paper is applicable to
short time control systems, such as communication network [148], aircraft control
[149] and ATM network control system [150]. Considering that we study the state
behavior of the system in a short-time interval, we need to give the investigated
time-interval in advance.
In general, Lyapunov asymptotic stability theory studies the state behavior of the
system in infinite time interval. If we extend the finite-time interval to an infinite
time-domain, they have the same property. To some extent, Lyapunov asymptotic
1 Introduction
modeling errors, are obtained. The energy norm indexes of the residual signals to the
external disturbance signals and fault signals are selected to illustrate robustness to the
external disturbance and sensitivity to the fault signals. The design of fault detection
observer or filter is described as an optimization problem. By using Lyapunov–
Krosovskii functional and LMIs technique, sufficient conditions for the existence
of fault detection observers or filters are proved and given, and an optimal design
method is also proposed. We also give simulation results to show the feasibility and
applicability of the designed fault detection methods.
1.3 Finite-Time and Finite-Frequency
It is known that the stability is always an important problem for control systems.
For the study of stability problems, we always mention Lyapunov stability theory,
whose key point is to construct a suitable Lyapunov function or functional. It should
be pointed out that Lyapunov stability theory is suitable for the study of asymptotic
stability. In general, asymptotic stability describes the steady-state performance of
a system, and represents the system characteristics in an infinite interval. It can
meet the requirements of practical engineering, but it can not reflect the transient
performance. In fact, an asymptotically stable system may have bad transient performance, which will cause bad effects and can not meet the requirements of industrial
production, such as excessive overshoot, severe oscillation, overshoot or saturation
nonlinearity, etc. Therefore, for some short-time working systems, such as missile
system, communication network system and robot control system, we not only study
the steady-state performance in Lyapunov sense, but also pay more attention to the
transient performance requirements of the dynamics, such as the deviation degree of
the system state trajectory from the equilibrium point at a certain time point. In order
to solve the problem of system transient performance, Dorato proposed a concept
of short-time stability [142] (i.e., finite short-time stability), and then analyzed the
finite short-time control problems, which have been widely used.
The so-called finite short-time stability means that the state trajectory of the system
does not deviate from the predetermined range in a given finite-time region. It should
be pointed out that the concept of finite short-time stability studied in this book is
different from the finite short-time stability studied in [143–147], in which it means
that the system state reaches an equilibrium point in a finite-time and stays in the
point. In fact, it examines the state behavior of the system in the “finite but infinite time
interval”. Therefore, the finite short-time control theory in this paper is applicable to
short time control systems, such as communication network [148], aircraft control
[149] and ATM network control system [150]. Considering that we study the state
behavior of the system in a short-time interval, we need to give the investigated
time-interval in advance.
In general, Lyapunov asymptotic stability theory studies the state behavior of the
system in infinite time interval. If we extend the finite-time interval to an infinite
time-domain, they have the same property. To some extent, Lyapunov asymptotic