1. Introduction
With the development of social industries, there are higher requirements for the establishment of mathematical models. Fractional order models are more accurate in describing the actual systems, which make it easier to study and control these systems. Fractional calculus and fractional-order system (FOS) theories have attracted extensive attention, and the related theories are becoming more and more mature [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11].
Singular systems include differential equations describing the dynamic characteristics of state variables and algebraic equations describing the static relationship of state variables [
12]. The studies of singular fractional-order systems have more practical significance, and the singular FOS theories have the prospect of a very broad application. Many FOS theories have been extended to singular fractional-order systems. Stability is the basis of control systems, including singular FOS. The authors of [
13,
14,
15] focus on the regularity, impulse-free, and admissibility properties and establish sufficient and necessary conditions for the admissibility for singular FOSs. Reference [
16] presents new admissibility conditions of singular FOS with order
expressed in a set of strict linear matrix inequalities (LMI). References [
17,
18] study the robust stabilization of uncertain singular FOSs. With the development of stability theory of singular FOSs, the study of stabilization has also received high attention. The static and dynamic output feedback stabilization problems of singular FOSs have been considered in [
19,
20,
21], and the observer-based stabilization for a class of singular FOSs have been considered in [
22].
Filtering is one of the most important basic problems in the field of control. The output signal is estimated by a certain filter for the unmeasurable signal inside the system. The classical Kalman filtering theory is based on an accurate mathematical model. It is assumed that the input noise is a strictly Gaussian process or a Gaussian sequence, but the two conditions cannot be met in practice. In the actual modeling, in addition to the stability, the system also needs to meet some performance indicators, for example,
,
performance index [
23,
24,
25,
26]. Compared with traditional Kalman filter, the advantage of
filtering is that one does not need to accurately know the statistical properties of external interference signals. One only needs to assume that the external interference is energy-bounded and that the
filter has better robustness to the uncertainty in the model. These advantages make
filtering more widely used, and many scholars apply it to various complex systems [
27,
28,
29,
30]. The reduced-order filtering, whose order is lower than that of the system, is a very important issue in many applications [
31,
32]. For singular FOSs, the reduced-order
filtering design problem has not been sufficiently solved [
33,
34]. Therefore, the research on these issues is more challenging, and has more theoretical and practical application value.
In this paper, we investigate the problem of reduced-order filter design for singular FOSs with the fractional commensurate order . Our purpose is to design appropriate filters such that the filtering error systems are admissible and the transfer functions from the disturbance to the filtering error output satisfy a prescribed -norm bound constraint. We provide sufficient and necessary conditions for filtering design for singular FOSs. Numerical examples illustrate the effectiveness of the methods.
The paper is organized as follows. In
Section 2, we provide the problem formulation and useful lemmas. In
Section 3, the main results are presented.
Section 4 gives numerical examples to illustrate our proposed results.
Section 5 is the conclusion.
Notation 1. Throughout this paper, and denote the inverse and the transpose of X, respectively. () denotes that X is positive (negative). Given , , the orthogonal complement satisfies and . The notation stands for . Furthermore, }. The symmetric term in a matrix is denoted by *. Matrices, if not explicitly stated, are assumed to have compatible dimensions.
2. Problem Formulation and Preliminaries
In this paper, we study the
filtering problem for singular FOS. Consider the following singular FOS described by
where
is the state vector;
is the measurement;
is the signal to be estimated, and
is the disturbance input that belongs to
. The matrix
may be singular,
.
, and
L are known real constant matrices with appropriate dimensions,
. We use the Caputo fractional derivative, which, with order
for a function
, is defined as:
where
h is an integer satisfying
and
is the Gamma function.
Definition 1 ([
14]).
The triple or the system is called regular if is not identically zero. It is called impulse-free if . It is called stable if all the roots of satisfy , where is the spectrum of . The triple is said to be admissible if is regular, impulse-free, and stable. The transfer function between
and
is
Consider the following filter
where
is the state vector,
,
is the estimator of
. Matrices
, and
are to be determined.
Let
,
. Then, the filtering error system is described as
where
and the transfer function between
and
is
The purpose of this work is to solve the following problem.
Problem 1. Given the singular FOS in (1) and a prescribed bound , find a reduced-order filter in (4), such that the following two conditions hold. (1) The filtering error system in (5) is admissible. (2) The transfer function satisfies .
If the problem has a solution, then we call the reduced-order filtering problem is solvable.
Especially when
, the reduced-order filter in (
4) is deformed to
where
is the estimator of
and
is to be determined.
For later development, we first introduce some lemmas.
Lemma 1 ([
19]).
The triple with is admissible, if and only if there exist and Q such thatwhere is an arbitrarily matrix of full column rank and satisfies . Lemma 2 ([
34]).
Given a scalar , the singular FOS in (1) is admissible and the transfer function satisfies if and only if there exist matrices such that:where According to the above lemmas, we can easily get the following Lemma 3.
Lemma 3. Given a scalar , the singular FOS in (1) is admissible and the transfer function satisfies if and only if there exist matrices and Q such thatwhere is an arbitrary matrix of full column rank and satisfies . Proof. By Lemma 2, letting
and left- and right-multiplying (
11) by
and
U, respectively, (
10) and (
11) are found to be equivalent to that there exist matrices
, such that
where
Using Lemma 1, we obtain the condition (
13). □
Lemma 4 ([
29]).
Given a symmetric matrix and two matrices , there exists a matrix Θ to solve the following matrix inequalityif and only if the following two conditions are satisfiedIn this case, the matrix Θ can be expressed aswhere matrices and Δ are free parameters satisfyingand Λ and Σ are defined by 3. Main Results
In this part, we will introduce the design methods of reduced-order filter and zeroth-order filter for singular FOSs.
Theorem 1. The reduced-order filtering problem for the singular FOS in (1) is solvable with an -th order filter in the form (4) if and only if there exist matrices , and Q, satisfyingwhereWith the feasible solutions , and , the parameters in (4) are given bywhere and Σ satisfy (20)–(22) with parameters , and Π as follows.In addition, , , where , satisfying . Proof. The matrices in (
5) can be decomposed as
where
□
By Lemma 3, the filtering error system in (
5) is admissible and the transfer function
of the error system satisfies
if and only if there exist matrices
, such that
where
is an arbitrary matrix of full column rank and satisfies
. From (
30), we obtain that
which means the matrix
is nonsingular and
,
,
can be decomposed as
where the decomposition is compatible with
, and
. Due to
,
, we can obtain
,
,
, and
. By Lemma 1 in [
11],
P and
are invertible. Let
. By left- and right-multiplying
by
and
U, respectively, it is easy to deduce that
,
,
is invertible [
10]. Since
we have
where
.
On the other hand, with (
29), the LMI in (
30) can be expressed as
which is rewritten as
where
Based on Lemma 4, the LMI in (
36) has a solution
G if and only if the following two inequalities hold,
where
Then (
37) and (
38) are equivalent to
Set
,
; then from (
34),
,
, (
39) and (
40) are equivalent to (
23) and (
24),
. The proof is completed.
Remark 1. Theorem 1 gives a necessary and sufficient condition for solving the reduced-order filtering problem of singular FOS. The inequalities in (23) and (24) are LMIs, while the constraint has to resort to a numerical algorithm based on alternating projections in [28], which is used later for solving the same problems; see for instance [12] and [30]. Remark 2. When , Theorem 1 reduces to the reduced-order filtering design for singular integer-order systems. In this special case, the necessary and sufficient condition is modified as that there exist matrices and , satisfying (23) and (24), which is equivalent to Theorem 3 in [12]. This verifies that Theorem 1 includes the result in [12] as a special case. When
in (
1), we can obtain the following result for the reduced-order
filtering problem of FOS below:
Corollary 1. Consider the FOS in (41). The reduced-order filtering problem of FOS is solvable by an -th order filter in the form (4) if and only if there exist matrices and , satisfyingwhere . Proof. The proof is similar to that of Theorem 1, so we omit it. □
Remark 3. We can deduce an equivalent form of Corollary 1. Notice that (42) and (43) are equivalent towhere and , . Multiplying (42) by γ, left- and right-multiplying (43) by diag(), then applying Schur complement, we deduce the equivalent form by setting , . When , Corollary 1 reduces to the reduced-order filtering design for integer-order systems. In this special case, the necessary and sufficient condition is modified such that there exist matrices and , satisfying (44) and (45), which is equivalent to the condition of Theorem 1 in [29] with the case . Corollary 1 is an extension of Theorem 1 in [29]. Now, we study the zeroth-order filtering problem of singular FOS and give the following result.
Theorem 2. The zeroth-order filtering problem for the singular FOS in (1) is solvable with a filter in the form (8) if and only if there exist matrices and Q satisfyingwhereWith the solutions P and Q, the parameter in (8) is given bywhere , and Σ satisfy (20)–(22) with parameters as follows. Proof. Letting
, the filtering error system can be described as
By Lemma 3, the filtering error system in (
51) is admissible and the transfer function
of the error system satisfies
if and only if there exist matrices
such that
Furthermore, the above LMI can be separated as
where
Based on Lemma 4, (
53) has a solution
, if and only if the following two inequalities hold,
where
then (
55) and (
56) are equivalent to (
46) and (
47). □
Remark 4. Theorem 2 presents a necessary and sufficient condition for designing the zeroth-order filter of singular FOS with LMIs. When , Theorem 2 reduces to the zeroth-order filtering design for integer-order systems. In this special case, the necessary and sufficient condition is modified such that there exists a matrix satisfying (46) and (47), which is equivalent to Theorem 5 in [12]. From Theorem 2, the following Corollary 2 is obviously true.
Corollary 2. The zeroth-order filtering problem of FOS in (41) is solvable by a filter in the form (8) if and only if there exist matrix satisfyingwhere . Remark 5. Corollary 2 provides a necessary and sufficient condition for designing the zeroth-order filter of FOS. When , Corollary 2 reduces to the zeroth-order filtering design for integer-order systems. In this special case, the necessary and sufficient condition is modified as that there exists a matrix satisfying (58) and (59). Through a similar analysis of Remark 3, this result is equivalent to Theorem 3 in [29] with the case . Corollary 2 can be regarded as a generalization of Theorem 3 in [29]. Remark 6. In this paper, we study the problem of reduced-order filter design for commensurate fractional-order systems. We believe that the results are useful for further research on noncommensurate order systems.