# Robust Local Regularity and Controllability of Uncertain TS Fuzzy Descriptor Systems

Authors: Shinn-Horng Chen; Wen-Hsien Ho; Jyh-Horng Chou
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The robust local regularity and controllability problem for the Takagi-Sugeno (TS) fuzzy descriptor systems is studied in this paper. Under the assumptions that the nominal TS fuzzy descriptor systems are locally regular and controllable, a sufficient criterion is proposed to preserve the assumed properties when the structured parameter uncertainties are added into the nominal TS fuzzy descriptor systems. The proposed sufficient criterion can provide the explicit relationship of the bounds on parameter uncertainties for preserving the assumed properties. An example is given to illustrate the application of the proposed sufficient condition.


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## Body

## 1. Introduction

Recently, it has been shown that the fuzzy-model-based representation proposed by Takagi and Sugeno [1], known as the TS fuzzy model, is a successful approach for dealing with the nonlinear control systems, and there are many successful applications of the TS-fuzzy-model-based approach to the nonlinear control systems (e.g., [2–19] and references therein). Descriptor systems represent a much wider class of systems than the standard systems [20]. In recent years, some researchers (e.g., [4–6, 8, 21–28] and references therein) have studied the design issue of the fuzzy parallel-distributed-compensation (PDC) controllers for each fuzzy rule of the TS fuzzy descriptor systems. Both regularity and controllability are actually two very important properties of descriptor systems with control inputs [29]. So, before the design of the fuzzy PDC controllers in the corresponding rule of the TS fuzzy descriptor systems, it is necessary to consider both properties of local regularity and controllability for each fuzzy rule [23]. However, both regularity and controllability of the TS fuzzy systems are not considered by those mentioned-above researchers before the fuzzy PDC controllers are designed. Therefore, it is meaningful to further study the criterion that the local regularity and controllability for each fuzzy rule of the TS fuzzy descriptor systems hold [30].On the other hand, in fact, in many cases it is very difficult, if not impossible, to obtain the accurate values of some system parameters. This is due to the inaccurate measurement, inaccessibility to the system parameters, or variation of the parameters. These parametric uncertainties may destroy the local regularity and controllability properties of the TS fuzzy descriptor systems. But, to the authors’ best knowledge, there is no literature to study the issue of robust local regularity and controllability for the uncertain TS fuzzy descriptor systems.The purpose of this paper is to present an approach for investigating the robust local regularity and controllability problem of the TS fuzzy descriptor systems with structured parameter uncertainties. Under the assumptions that the nominal TS fuzzy descriptor systems are locally regular and controllable, a sufficient criterion is proposed to preserve the assumed properties when the structured parameter uncertainties are added into the nominal TS fuzzy descriptor systems. The proposed sufficient criterion can provide the explicit relationship of the bounds on structured parameter uncertainties for preserving the assumed properties. A numerical example is given in this paper to illustrate the application of the proposed sufficient criterion.


## 2. Robust Local Regularity and Controllability Analysis

Based on the approach of using the sector nonlinearity in the fuzzy model construction, both the fuzzy set of premise part and the linear dynamic model with parametric uncertainties of consequent part in the exact TS fuzzy control model with parametric uncertainties can be derived from the given nonlinear control model with parametric uncertainties [5]. The TS continuous-time fuzzy descriptor system with parametric uncertainties for the nonlinear control system with structured parametric uncertainties can be obtained as the following form:
(2.1)R~i:IFz1isMi1and…andzgisMig,thenEix˙(t)=(Ai+ΔAi)x(t)+(Bi+ΔBi)u(t),
or the uncertain discrete-time TS fuzzy descriptor system can be described by
(2.2)R~i:IFz1isMi1and…andzgisMig,thenEix(k+1)=(Ai+ΔAi)x(k)+(Bi+ΔBi)u(k),
with the initial state vector x(0), where R~i(i=1,2,…,N) denotes the ith implication, N is the number of fuzzy rules, x(t)=[x1(t),x2(t),…,xn(t)]T and x(k)=[x1(k),x2(k),…,xn(k)]T denote the n-dimensional state vectors, u(t)=[u1(t),u2(t),…,up(t)]T and u(k)=[u1(k),u2(k),…,up(k)]T denote the p-dimensional input vectors, zi(i=1,2,…,g) are the premise variables, Ei, Ai, and Bi(i=1,2,…,N) are, respectively, the n×n, n×n and n×p consequent constant matrices, ΔAi and ΔBi(i=1,2,…,N) are, respectively, the parametric uncertain matrices existing in the system matrices Ai and the input matrices Bi of the consequent part of the ith rule due to the inaccurate measurement, inaccessibility to the system parameters, or variation of the parameters, and Mij(i=1,2,…,N and j=1,2,…,g) are the fuzzy sets. Here the matrices Ei(i=1,2,…,N) may be singular matrices with rank(Ei)≤n(i=1,2,…,N). In many applications, the matrices Ei(i=1,2,…,N) are the structure information matrices; rather than parameter matrices, that is, the elements of Ei(i=1,2,…,N) contain only structure information regarding the problem considered.In many interesting problems (e.g., plant uncertainties, constant output feedback with uncertainty in the gain matrix), we have only a small number of uncertain parameters, but these uncertain parameters may enter into many entries of the system and input matrices [31, 32]. Therefore, in this paper, we suppose that the parametric uncertain matrices ΔAi and ΔBi take the forms
(2.3)ΔAi=∑k=1mεikAik,ΔBi=∑k=1mεikBik,
where εik(i=1,2,…,N and k=1,2,…,m) are the elemental parametric uncertainties, and Aik and Bik(i=1,2,…,N and k=1,2,…,m) are, respectively, the given n×n and n×p constant matrices which are prescribed a priori to denote the linearly dependent information on the elemental parametric uncertainties εik.In this paper, for the uncertain TS fuzzy descriptor system in (2.1) (or (2.2)), each fuzzy-rule-nominal model Eix˙(t)=Aix(t)+Biu(t) or Eix(k+1)=Aix(k)+Biu(k), which is denoted by {Ei,Ai,Bi}, is assumed to be regular and controllable. Due to inevitable uncertainties, each fuzzy-rule-nominal model {Ei,Ai,Bi} is perturbed into the fuzzy-rule-uncertain model {Ei,Ai+ΔAi,Bi+ΔBi}. Our problem is to determine the conditions such that each fuzzy-uncertain model {Ei,Ai+ΔAi,Bi+ΔBi}for the uncertain TS fuzzy descriptor system (2.1) (or (2.2)) is robustly locally regular and controllable. Before we investigate the robust properties of regularity and controllability for the uncertain TS fuzzy descriptor system (2.1) (or (2.2)), the following definitions and lemmas need to be introduced first.Definition 2.1 (see [33]).
The measure of a matrixW-∈Cn×n is defined as
(2.4)μ(W-)≡limθ→0(∥I+θW-∥-1)θ,
where ∥·∥ is the induced matrix norm on Cn×n.Definition 2.2 (see [34]).
The system{Ei,Ai,Bi} is called controllable, if for any t1>0 (or k1>0), x(0)∈Rn, and w∈Rn, there exists a control input u(t) (or u(k)) such that x(t1)=w (or x(k1)=w).Definition 2.3.
The uncertain TS fuzzy descriptor system in (2.1) (or (2.2)) is locally regular, if each fuzzy-rule-uncertain model {Ei,Ai+ΔAi,Bi+ΔBi}(i=1,2,…,N) is regular.Definition 2.4.
The uncertain TS fuzzy descriptor system in (2.1) (or (2.2)) is locally controllable, if each fuzzy-rule-uncertain model {Ei,Ai+ΔAi,Bi+ΔBi}(i=1,2,…,N) is controllable.Lemma 2.5 (see [34]).
The system{Ei,Ai,Bi} is regular if and only if rank[EniBdi]=n2, where Eni∈Rn2×n and Edi∈Rn2×n2 are given by
(2.5)Eni=[Ei0···0],Edi=[AiEiAi······EiAi].Lemma 2.6 (see [29, 35]).
Suppose that the system{Ei,Ai,Bi} is regular. The system {Ei,Ai,Bi} is controllable if and only if rank[EdiEbi]=n2 and rank[EiBi]=n, where Edi∈Rn2×n2 is given in (2.5) and Ebi=diag{Bi,Bi,…,Bi}∈Rn2×np.Lemma 2.7 (see [33]).
The matrix measures of the matricesW- and V-, namely, μ(W-) and μ(V-), are well defined for any norm and have the following properties:(i)
μ(±I)=±1, for the identity matrix I;(ii)
-∥W-∥≤-μ(-W-)≤Re(λ(W-))≤μ(W-)≤∥W-∥, for any norm∥·∥ and any matrix W-∈Cn×n;(iii)
μ(W-+V-)≤μ(W-)+μ(V-), for any two matrices W-,V-∈Cn×n;(iv)
μ(γW-)=γμ(W-), for any matrix W-∈Cn×n and any non-negative real number γ,where λ(W-) denotes any eigenvalue of  W-, and Re(λ(W-)) denotes the real part of λ(W-).Lemma 2.8.
For anyγ<0 and any matrix W-∈Cn×n,μ(γW-)=-γμ(-W-).Proof.
This lemma can be immediately obtained from the property (iv) in Lemma2.7.Lemma 2.9.
LetN-∈Cn×n. If μ(-N-)<1, then det(I+N-)≠0.Proof.
From the property (ii) in Lemma2.7 and since μ(-N-)<1, we can get that Re(λ(N-))≥-μ(-N-)>-1. This implies that λ(N-)≠-1. So, we have the stated result.Now, let the singular value decompositions ofRi=[EniEdi],Qi=[EdiEbi], and Pi=[EiBi] be, respectively,
(2.6)Ri=Ui[Si0n2×n]ViH,(2.7)Qi=Uri[Sri0n2×np]VriH,(2.8)Pi=Uci[Sci0n2×q]VciH,
where Ui∈Rn2×n2 and Vi∈R(n2+n)×(n2+n) are the unitary matrices, Si=diag{σi1,σi2,…,σin2}, and σi1≥σi2≥⋯≥σin2>0 are the singular values of Ri;Uri∈Rn2×n2 and Vri∈R(n2+np)×(n2+np) are the unitary matrices, Sri=diag{σri1,σri2,…,σrin2} and σri1≥σri2≥⋯≥σrin2>0 are the singular values of Qi;Uci∈Rn×n and Vci∈R(n+p)×(n+p) are the unitary matrices, Sci=diag{σci1,σci2,…,σcin} and σci1≥σci2≥⋯≥σcin>0 are the singular values of Pi;ViH, VriH, and VciH denote, respectively, the complex-conjugate transposes of the matrices Vi, Vri, and Vci.In what follows, with the preceding definitions and lemmas, we present a sufficient criterion for ensuring that the uncertain TS fuzzy descriptor system in (2.1) or (2.2) remains locally regular and controllable.Theorem 2.10.
Suppose that the each fuzzy-rule-nominal descriptor system{Ei,Ai,Bi} is regular and controllable. The uncertain TS fuzzy descriptor system in (2.1) (or (2.2)) is still locally regular and controllable (i.e., each fuzzy-rule-uncertain descriptor system {Ei,Ai+ΔAi,Bi+ΔBi} remains regular and controllable), if the following conditions simultaneously hold(2.9a)∑k=1mεikφik<1,(2.9b)∑k=1mεikθik<1,(2.9c)∑k=1mεikϕik<1,where i=1,2,…,N, and k=1,2,…,m:
(2.10)φik={μ(-Si-1UiHRikVi[In2,0n2×n]T),forεik≥0,-μ(Si-1UiHRikVi[In2,0n2×n]T),forεik<0,Rik=[0n2×nR~ik]∈Rn2×(n2+n),R~ik=diag{Aik,…,Aik}∈Rn2×n2,θik={μ(-Sri-1UriHQikVri[In2,0n2×np]T),forεik≥0,-μ(Sri-1UriHQikVri[In2,0n2×np]T),forεik<0,Qik=[AikAik···AikBikBik···Bik]∈Rn2×(n2+np),ϕik={μ(-Sci-1UciHPikVci[In,0n×p]T),forεik≥0,-μ(Sci-1UciHPikVci[In,0n×p]T),forεik<0,Pik=[0n×nBik]∈Rn×(n+p),
the matrices Si, Ui,Vi, Sri, Uri,Vri, Sci, Uci, and Vci(i=1,2,…,N) are, respectively, defined in (2.6)–(2.8), and In2 denotes the n2×n2 identity matrix.Proof.
Firstly, we show the regularity. Since each fuzzy-rule-nominal descriptor system{Ei,Ai,Bi}  (i=1,2,…,N) is regular, then, from Lemma 2.5, we can get that the matrix Ri=[EniEdi]∈Rn2×(n2+n) has full row rank (i.e., rank(Ri)=n2). With the uncertain matrices Ai+ΔAi and Bi+ΔBi, each fuzzy-rule-uncertain descriptor system {Ei,Ai+ΔAi,Bi+ΔBi} is regular if and only if
(2.11)R~i=Ri+∑k=1mεikRik
has full row rank, where Rik=[0n2×nR~ik]∈Rn2×(n2+n) and R~ik=diag{Aik,…,Aik}∈Rn2×n2. 							(2.12)It is known thatrank(R~i)=rank(Si-1UiHR~iVi).
Thus, instead of rank(R~i), we can discuss the rank of
(2.13)[In2,0n2×n]+∑k=1mεikR^ik,
where R^ik=Si-1UiHRikVi, for i=1,2,…,N and k=1,2,…,m. Since a matrix has at least rank n2 if it has at least one nonsingular n2×n2 submatrix, a sufficient condition for the matrix in (2.13) to have rank n2 is the nonsingularity of
(2.14)Li=In2+∑k=1mεikR-ik,
where R-ik=Si-1UiHRikVi[In2,0n2×n]T (for i=1,2,…,N and k=1,2,…,m).
Using the properties in Lemmas2.7 and 2.8 and from (2.9a), we get
(2.15)μ(-∑k=1mεikR-ik)=μ(-∑k=1mεikSi-1UiHRikVi[In2,0n2×n]T)≤∑k=1mμ(-εikSi-1UiHRikVi[In2,0n2×n]T)=∑k=1mεikφik<1.
From Lemma 2.9, we have that
(2.16)det(Li)=det(In2+∑k=1mεikR-ik)≠0.
Hence, the matrix Li in (2.14) is nonsingular. That is, the matrix R~i in (2.11) has full row rank n2. Thus, from the Lemma 2.5, the regularity of each fuzzy-rule-uncertain descriptor system {Ei,Ai+ΔAi,Bi+ΔBi} is ensured.
Next, we show the controllability. Since each fuzzy-rule-nominal descriptor system{Ei,Ai,Bi}(i=1,2,…,N) is controllable, then from Lemma 2.6, we have that the matrix Qi=[EdiEbi] has full row rank (i.e., rank(Qi)=n2) and Pi=[EiBi] has full row rank (i.e., rank(Pi)=n). With the uncertain matrices Ai+ΔAi and Bi+ΔBi, each fuzzy-rule-uncertain descriptor system {Ei,Ai+ΔAi,Bi+ΔBi} is controllable if and only if
(2.17)Q~i=Qi+∑k=1mεikQik,(2.18)P~i=Pi+∑k=1mεikPik
have full row rank, where
(2.19)Qik=[AikAik···AikBikBik···Bik]∈Rn2×(n2+np),
and Pik=[0n×nBik]∈Rn×(n+p).
It is known that(2.20)rank(Q~i)=rank(Sri-1UriHQ~iVri).
Thus, instead of rank(Q~i), we can discuss the rank of
(2.21)[In2,0n2×np]+∑k=1mεikQ^ik,
where Q^ik=Sri-1UriHQikVri, for i=1,2,…,N and k=1,2,…,m. Since a matrix has at least rank n2 if it has at least one nonsingular n2×n2 submatrix, a sufficient condition for the matrix in (2.21) to have rank n2 is the nonsingularity of
(2.22)Gi=In2+∑k=1mεikQ-ik,
where Q-ik=Sri-1UriHQikVri[In2,0n2×np]T (for i=1,2,…,N and k=1,2,…,m).
Applying the properties in Lemmas2.7 and 2.8 and from (2.9b), we get
(2.23)μ(-∑k=1mεikQ-ik)=μ(-∑k=1mεikSri-1UriHQikVri[In2,0n2×np]T)≤∑k=1mμ(-εikSri-1UriHQikVri[In2,0n2×np]T)=∑k=1mεikθik<1.
From Lemma 2.9, we have that
(2.24)det(Gi)=det(In2+∑k=1mεikQ-ik)≠0.
Hence, the matrix Gi in (2.22) is nonsingular. That is, the matrix Q~i in (2.17) has full row rank n2.
And then, it is also known that(2.25)rank(P~i)=rank(Sci-1UciHP~iVci).
Thus, instead of rank(P~i), we can discuss the rank of
(2.26)[In,0n×p]+∑k=1mεikP^ik,
where P^ik=Sci-1UciHPikVci, for i=1,2,…,N and k=1,2,…,m. Since a matrix has at least rank n if it has at least one nonsingular n×n submatrix, a sufficient condition for the matrix in (2.26) to have rank n is the nonsingularity of
(2.27)Hi=In+∑k=1mεikP-ik,
where P-ik=Sci-1UciHPikVci[In,0n×p]T (for i=1,2,…,N and k=1,2,…,m).
Adopting the properties in Lemmas2.7 and 2.8 and from (2.9c), we obtain
(2.28)μ(-∑k=1mεikP-ik)=μ(-∑k=1mεikSci-1UciHPikVci[In,0n×p]T)≤∑k=1mμ(-εikSci-1UciHPikVci[In,0n×p]T)=∑k=1mεikϕik<1.
From Lemma 2.9, we get that
(2.29)det(Hi)=det(In+∑k=1mεikP-ik)≠0.
Hence, the matrix Hi in (2.27) is nonsingular. That is, the matrix P~i in (2.18) has full row rank n. Thus, from the Lemma 2.6 and the results mentioned above, the controllability of each fuzzy-rule-uncertain descriptor system {Ei,Ai+ΔAi,Bi+ΔBi} is ensured. Therefore, we can conclude that the uncertain TS fuzzy descriptor system in (2.1) (or (2.2)) is locally regular and controllable, if the inequalities (2.9a), (2.9b), and (2.9c) are simultaneously satisfied. Thus, the proof is completed.Remark 2.11.
The proposed sufficient conditions in (2.9a)–(2.9c) can give the explicit relationship of the bounds on εik(i=1,2,…,N and k=1,2,…,m) for preserving both regularity and controllability. In addition, the bounds, that are obtained by using the proposed sufficient conditions, on εik are not necessarily symmetric with respect to the origin of the parameter space regarding εik(i=1,2,…,N and k=1,2,…,m).Remark 2.12.
This paper studies the problem of robust local regularity and controllability analysis. If the proposed conditions in (2.9a)–(2.9c) are satisfied, each rule of the uncertain TS fuzzy descriptor system {Ei,Ai+ΔAi,Bi+ΔBi}is guaranteed to be robustly locally regular and controllable. This implies that, in the fuzzy PDC controller design, if the proposed conditions in (2.9a)–(2.9c) are satisfied, the PDC controller of each fuzzy rule can control every state variable in the corresponding rule of the uncertain TS fuzzy descriptor system {Ei,Ai+ΔAi,Bi+ΔBi}. However, here, it should be noticed that although the PDC controller of each control rule can control every state variable in the corresponding rule under the presented conditions being held, the PDC controller gains should be determined using global design criteria that are needed to guarantee the global stability and control performance [5], where many useful global design criteria have been proposed by some researchers (e.g., [4-6, 8, and 21-28] and references therein).


## 3.  Illustrative Example

Consider a two-rule fuzzy descriptor system as that considered by Wang et al. [21]. The TS fuzzy descriptor system with the elemental parametric uncertainties is described by(3.1a)R~1:IFz1isM11,thenE1x(k+1)=(A1+ΔA1)x(k)+(B1+ΔB1)u(k);(3.1b)R~2:IFz1isM21,thenE2x(k+1)=(A2+ΔA2)x(k)+(B2+ΔB2)u(k),where
(3.2)x(k)=[x1(k)x2(k)]T,E1=E2=[1000],A1=[0.84800-0.315],A2=[-0.236000.113],B1=B2=[11],ΔAi=∑k=12εikAik,ΔBi=∑k=12εikBik,A11=[00-0.10],A21=[000.20],A12=A22=[0000],Bi1=[00],Bi2=[0.10.1],M11=11+exp(-0.5(z1-0.3)),M21=exp(-0.5(z1-0.3))1+exp(-0.5(z1-0.3)),εi1∈[-11.1],εi2∈[-1.210],in whichi=1,2.Now, applying the sufficient conditions in (2.9a)–(2.9c) with the two-norm-based matrix measure, we can get the following:(I)
for the fuzzy rule 1:(3.3a)(i)∑k-12εikφik≤0.17460<1,forεi1∈[01.1],εi2∈[-1.210],(3.3b)(ii)∑k-12εikφik≤0.15873<1,forεi1∈[-10],εi2∈[-1.210],(3.3c)(iii)∑k-12εikθik≤0.03297<1,forεi1∈[01.1],εi2∈[010],(3.3d)(iv)∑k-12εikθik≤0.13861<1,forεi1∈[-10],εi2∈[010],(3.3e)(v)∑k-12εikθik≤0.25078<1,forεi1∈[-10],εi2∈[-1.20],(3.3f)(vi)∑k-12εikθik≤0.14514<1,forεi1∈[01.1],εi2∈[-1.20],(3.3g)(vii)∑k-12εikϕik=0<1,forεi1∈[-11.1],εi2∈[010],(3.3h)(viii)∑k-12εikϕik≤0.1200<1,forεi1∈[-11.1],εi2∈[-1.20];(3.4a)
(
i
)
∑
k
-
1
2
ε
i
k
φ
i
k
≤
0.97345
<
1
,
for
ε
i
1
∈
[
0
1.1
]
,
ε
i
2
∈
[
-
1.2
10
]
,
(3.4b)
(
ii
)
∑
k
-
1
2
ε
i
k
φ
i
k
≤
0.88496
<
1
,
for
ε
i
1
∈
[
-
1
0
]
,
ε
i
2
∈
[
-
1.2
10
]
,
(3.4c)
(
iii
)
∑
k
-
1
2
ε
i
k
θ
i
k
≤
0.56719
<
1
,
for
ε
i
1
∈
[
0
1.1
]
,
ε
i
2
∈
[
0
10
]
,
(3.4d)
(
iv
)
∑
k
-
1
2
ε
i
k
θ
i
k
≤
0.87768
<
1
,
for
ε
i
1
∈
[
-
1
0
]
,
ε
i
2
∈
[
0
10
]
,
(3.4e)
(
v
)
∑
k
-
1
2
ε
i
k
θ
i
k
≤
0.99740
<
1
,
for
ε
i
1
∈
[
-
1
0
]
,
ε
i
2
∈
[
-
1.2
0
]
,
(3.4f)
(
vi
)
∑
k
-
1
2
ε
i
k
θ
i
k
≤
0.168691
<
1
,
for
ε
i
1
∈
[
0
1.1
]
,
ε
i
2
∈
[
-
1.2
0
]
,
(3.4g)
(
vii
)
∑
k
-
1
2
ε
i
k
ϕ
i
k
=
0
<
1
,
for
ε
i
1
∈
[
-
1
1.1
]
,
ε
i
2
∈
[
0
10
]
,
(3.4h)
(
viii
)
∑
k
-
1
2
ε
i
k
ϕ
i
k
≤
0.1200
<
1
,
for
ε
i
1
∈
[
-
1
1.1
]
,
ε
i
2
∈
[
-
1.2
0
]
.From the results in (3.3a)–(3.3h) and (3.4a)–(3.4h), we can conclude that the uncertain TS fuzzy descriptor system (3.1a) and (3.1b) is locally robustly regular and controllable.


## 4. Conclusions

The robust local regularity and controllability problem for the uncertain TS fuzzy descriptor systems has been investigated. The rank preservation problem for robust local regularity and controllability of the uncertain TS fuzzy descriptor systems is converted to the nonsingularity analysis problem. Under the assumption that each fuzzy rule of the nominal TS fuzzy descriptor system has the full row rank for its related regularity and controllability matrices, a sufficient criterion has been proposed to preserve the assumed properties when the elemental parameter uncertainties are added into the nominal TS fuzzy descriptor systems. The proposed sufficient conditions in (2.9a)–(2.9c) can provide the explicit relationship of the bounds on elemental parameter uncertainties for preserving the assumed properties. One example has been given to illustrate the application of the proposed sufficient conditions. On the other hand, the issue of robust global regularity and controllability with evolutionary computation [36] for the uncertain TS fuzzy descriptor systems will be an interesting and important topic for further research.



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