1. Introduction
It is universally acknowledged that almost all dynamical systems in practice are unavoidably affected by unknown inputs and faults that resulted from actuators, components, or sensors. The design of the fault diagnosis filter aims at improving the robustness against unknown inputs and sensitivity to fault. To achieve such a goal, some practical criteria have been applied, such as
norm,
norm, and
index [
1,
2,
3,
4,
5,
6]. Based on these criteria, some multi-objective optimization problems such as
/
,
/
and
/
have attracted the attention of many scholars [
7,
8,
9].
norm measures the unknown input’s maximum influence on the residual signal. By comparison, the fault’s minimum influence on residual signal is measured by
index, which is first introduced in [
7]. In the past few years, the fault diagnosis filter design involving
/
has attracted the attention of many scholars [
10,
11,
12,
13,
14,
15,
16,
17].
The
index was first defined in the frequency domain. The
index at zero frequency was defined and investigated in [
2], which means the smallest singular value of non-zero. In [
5], the generalized famous KYP lemma was used to characterize
index problem of finite frequency. The
index of all frequency range was investigated by matrix inequality and equality in [
6], which denotes the minimum singular value. The
/
fault detection of finite frequency and infinite frequency were characterized in [
12,
13]. However, in practical applications, the frequency domain is very limited. Corresponding
index problems of time domain have attracted many scholars’ interest. In [
9], the optimal solutions to robust
/
problem in infinite and finite horizon were given for linear time-varying systems, which generalize corresponding resolutions to the time domain. In [
11], the
index was attributed to the existence of the solution to differential Riccati equation for time-varying systems, and corresponding results were extended to systems whose initial condition was unknown.
/
-optimization was used to design the fault detection filter for nonlinear systems in discrete-time in [
10]. For time-invariant systems, the method of matrix factorization was used to develop the
/
FD problem in [
14]. In [
15], the FD observer design was formulated as an
/
problem, which the solution was given via LMI formulation for the T-S fuzzy system. Fault detection for linear and nonlinear discrete-time systems were discussed in [
16,
17]. In [
18,
19], for discrete and continuous time-varying systems with Markov jump, corresponding finite horizon and infinite horizon
index were investigated via GDREs.
Considering that systems in the practical world are always affected by stochastic disturbances, many researchers have transferred their interest in stochastic control problems from determinate systems to stochastic systems [
20,
21,
22,
23,
24,
25]. Especially, interest in stochastic systems of mean-field type has been increasing. The mean-field theory is developed to study the collective behaviors resulting from individuals’ mutual interactions in various physical and sociological dynamical systems [
26,
27]. To date, many results on finite and infinite horizon linear quadratic optimal control of mean-field stochastic systems have been presented, we refer the reader to [
28,
29,
30,
31] and references therein. For mean-field stochastic differential and difference equations, the
control and finite horizon mixed
/
control of mean-field stochastic systems were characterized in [
32,
33,
34,
35]. Refs. [
36,
37] investigated finite and infinite horizon Pareto-based optimality of mean-field stochastic systems.
index of mean-field stochastic system was characterized in [
38].
In [
19], the
index of classic stochastic differential equation with Markov jump was studied. The corresponding results will be generalized to the mean-field stochastic differential equation with the Markov jump in this paper. Since the expectations
and
appear in equations, the problem is not a simple generalization. The main contributions of this note are as follows: Based on existing research results, the definition and attribute of
index are extended to mean-field stochastic systems. The appearance of
,
and Markov jumps in differential equations lead to more higher difficulty in mathematical deductions. This note illustrates the
index of mean-field stochastic continuous time-varying systems with Markovian jump in the finite horizon. The main result is about the necessary and sufficient condition of the
index greater than a given positive number, which is given by coupled generalized differential Riccati equation.
The remainder of the note is arranged as follows: The system considered in this article is formulated and some preliminary results are presented in
Section 2,
Section 3 gives the main results in terms of generalized differential Riccati equations. Numerical example is provided in
Section 4 to show the effectiveness of our obtained results.
Section 5 presents the conclusions of this article.
Notation. is the set of all dimensional real vectors. . denotes the transpose of W. () means that W is positive definite (positive semi-definite) symmetric matrix. : the set of all -dimensional real matrices. is the identity matrix. is the set of all real symmetric matrices . A wide (square or tall) system is the system whose inputs dimensions is more than (is equal to or less than) the outputs dimensions. is the space of nonanticipative stochastic processes with respect to an increasing algebras satisfying . is the class of -valued functions which are once continuously differentiable with respect to , and twice continuously differential with respect to , except possibly at the point .
2. Preliminaries
In this section, a useful lemma will be given for the following stochastic Markov jump systems of the mean-field type in continuous-time:
where,
,
and
are the system state, control input and regulated output, respectively.
is the one-dimensional standard Brownian motion,
denotes the expectation.
is a continuous-time discrete-state Markov process, whose values is taken in
and has the transition probability described by
where
and
for
determine the transition rate from mode
i to mode
j, and
with
for all
.
,
,
,
,
,
,
,
,
,
,
and
are corresponding weighted coefficient matrices. In different practical problems, their meanings are different.
,
,
,
,
,
,
,
,
,
,
and
when
, are assumed to be continuous matrix-valued functions of proper dimensions. The process
and
are defined on filtered probability space
with the natural filter
, and
is independent of
. For any given
and
, the unique solution of (
1) is denoted by
with deterministic initial condition
.
Definition 1. For system (1), the index is defined as Remark 1. When and denote fault signal and residual signal, respectively, the minimum sensitivity of system (1) from input to output is depicted as . For system, , so it is supposed that system (1) is a system or system. For
,
and
, we want to investigate the condition of the smallest sensitivity greater than
, i.e.,
. Define
and
then
yields
.
Lemma 1. [39] (Generalized It formula): Let be -adapted process, , . Then for given , we havewhere For system (
1), by taking mathematical expectations, we can express
and
as following:
Lemma 2. For system (1), assume and are differentiable, where with . Letandwhere
and Then for , we have Proof. Applying Lemma 1 for
and
, respectively, we obtain
where
In addition,
with
and
According to (
9) and (
10), we can obtain (
8), the proof is end. □
3. Finite Horizon Mean-Field Type Stochastic Index
The mean-field type stochastic index will be investigated in this section, the sufficient and necessary condition of , which means , .
Theorem 1. If for a given there exist and, such that the following GDREs are fulfilledthen . Proof. In view of
, by Lemma 2, for any
with
,
, it can be obtained that
By (
11) and the technique of completing squares, it follows that
where
,
Since
and
, (
12) provides
, which means
.
To prove
, we define the operator
with its realization
and define the operator
with its realization
Then
and
exist, which are determined by
and
respectively, where
Since
and
are continuous functions on
, there exist
and
for
, such that
and
on
. Let
, it follows that
and
for
. So, there exist constants
and
, such that
for
, which yields
So,
it is concluded that
which ends the proof. □
In what follows, the necessary condition of
will be given for the following square and time-invariant system:
Theorem 2. For system (13) and some given , which satisfies and , if , there exist unique and , satisfying the following GDRE Moreover,
and
are minimized by
with
and
where
is the state trajectory of system (
13) when
, and
where
Proof. It will be proved that
can imply that there is a unique solution
of (
14) on
. Otherwise, according to standard theory of differential equations, there is a finite escape time for (
14), i.e., (
14) has a unique solution
on a maximal interval
with
, and
becomes unbounded when
. Next, a contradiction will be derived.
For
,
, similar to the method of Theorem 3.1, it can be shown that
where
In addition, it is obviously that there exists
such that
Combining (
16) and (
17), it yields immediately
Let
be the solution of (
13) with initial state
and
, by linearity
Suppose
and
satisfy the following equation
one has
where
,
According to (
16), there exists
such that
Combining (
20) and (
21), it follows that
Obviously, there exists
such that
Therefore, there is some constant
satistying the following inequality
According to Lemma 1,
Let
and
respectively, there is
such that
From (
22), (
24), and (
25), it can be obtained that
Combining (
18) and (
26), it yields
. So
can not be unbounded when
, which lead to a contradiction. Therefore, there is unique
and
satisfying (
14) on
.
Moreover, similar to Equation (
18), we have
□
For system (
13), Theorem 1 and Theorem 2 can yield the following equivalence relationships immediately:
Theorem 3. For system (13) and some given , which satisfies and , the following are equivalent: (1) ,
(2) there exist unique and , satisfying GDRE (14) Theorem 4. For system (13) and some given , which satisfies and , , then the satisfying (14) decreases as T increases for . Proof. Suppose
,
is optimal when
, set
The time-invariance of
yields
. Therefore, for
, we can obtain
which indicates that
decreases as
T increases for
. □