Regularity and Controllability Robustness of TS Fuzzy Descriptor Systems with Structured Parametric Uncertainties

Regularity and controllability robustness of TS fuzzy descriptor

systems with structured parametric uncertainties

Shinn-Horng Chen

a

, Wen-Hsien Ho

b

, Jinn-Tsong Tsai

c

, Jyh-Horng Chou

b,d,e,⇑ a

Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, 415 Chien-Kung Road, Kaohsiung 807, Taiwan, Republic of China b

Department of Healthcare Administration and Medical Informatics, Kaohsiung Medical University, 100 Shin-Chuan 1st Road, Kaohsiung 807, Taiwan, Republic of China

c

Department of Computer Science, National Pingtung University of Education, 4-18 Min-Sheng Road, Pingtung 900, Taiwan, Republic of China d

Institute of Electrical Engineering, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchao, Kaohsiung 824, Taiwan, Republic of China

e_{Department of Electrical Engineering, National Kaohsiung University of Applied Sciences, 415 Chien-Kung Road, Kaohsiung 807, Taiwan, Republic of China}

a r t i c l e

i n f o

Article history: Received 22 June 2013

Received in revised form 30 November 2013 Accepted 25 January 2014

Available online 21 February 2014 Keywords:

Robust controllability

Takagi–Sugeno (TS) fuzzy descriptor system Structured parametric uncertainties

a b s t r a c t

The problem of robust global regularity and controllability is considered in Takagi–Sugeno (TS) fuzzy descriptor control systems with structured parametric uncertainties. Sufficient conditions are proposed to ensure both global regularity and controllability in these uncertain systems. The conditions also provide the explicit relationships of bounds on parametric uncertainties to achieve regularity and controllability. One numerical and two engineering examples are given to illustrate the applications of the proposed sufficient conditions.

Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction

The fuzzy model proposed by Takagi and Sugeno[1], known as the TS fuzzy model, has proven effective for dealing with nonlinear control systems, and the TS-fuzzy-model-based approach has been successfully applied in many nonlinear control systems (see[2–14]and references therein). Biglarbegian et al. considered the controllability problem in TS fuzzy control systems in[15], and we also analyzed the robust controllability and observability of uncertain TS-fuzzy-model-based control systems in[16,17].

However, since the class of descriptor systems is much wider than that of standard systems[18], fuzzy descriptor systems need further study. In recent years, some researchers (see, e.g.,[4–7,19–25]and references therein) have studied design issues in fuzzy parallel-distributed-compensation (PDC) controllers for TS fuzzy descriptor control systems. Here, it should be emphasized that the structures of descriptor systems are more complex than those of standard systems and contain not only finite dynamical modes, but also infinite nondynamical and dynamical modes[18]. Infinite dynamical modes can generate undesired impulse behavior. Two important properties of descriptor systems with control inputs are regularity and controllability [26]. Regularity ensures the existence and uniqueness of a solution for a descriptor system. The controllability property determines whether the descriptor system can be changed by control inputs. Three important properties of descriptor control systems that must be considered when characterizing globally structural controllability

http://dx.doi.org/10.1016/j.ins.2014.01.049

0020-0255/Ó 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author at: Institute of Electrical Engineering, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchao, Kaohsiung 824, Taiwan, Republic of China. Tel.: +886 7 6011000; fax: +886 7 6011066.

E-mail address:choujh@nkfust.edu.tw(J.-H. Chou).

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properties from various views are C-controllability (complete-controllability), R-controllability (reachable-controllability), and I-controllability (impulse-controllability)[26]. C-controllability/R-controllability determines whether a control input can derive the descriptor system in finite time in order to obtain the desired final values of system state variables. The I-con-trollability determines whether a controller can be designed such that the closed-loop descriptor system is impulse-free. Therefore, a controller design must consider both regularity and controllability properties (I-controllability, R-controllability and C-controllability) of the descriptor control system[29]. However, no studies reported in the literature have considered both regularity and controllability in TS fuzzy descriptor control systems when designing fuzzy PDC controllers ([4–7,19– 25]; and references therein). Thus, the criteria under which TS fuzzy descriptor control systems have both regularity and controllability need further clarification[27].

In practice, obtaining accurate values of some system parameters is often difficult, if not impossible, due to inaccurate measurements, inaccessible system parameters, or variation of parameters. These parametric uncertainties may destroy both the regularity and controllability properties of TS fuzzy descriptor control systems. Therefore, the authors have inves-tigated both regularity and controllability problems in TS fuzzy descriptor control systems since 2009[28–30]. Chen et al.

[28,29]studied the robustness of local regularity and controllability in TS fuzzy descriptor control systems with structured uncertainties. Assuming that nominal TS fuzzy descriptor control systems are locally regular and controllable (i.e., each fuzzy rule of the nominal TS fuzzy descriptor control systems has a full row rank for its corresponding regularity and controllability matrix), Chen et al. [28,29]presented a sufficient criterion for preserving the assumed local properties when structured parameter uncertainties are included in nominal TS fuzzy descriptor control systems. However, as illustrated inRemark 6

in Section3below, the local regularity and controllability of TS fuzzy descriptor control systems cannot ensure global reg-ularity and controllability. Therefore, the issue of global regreg-ularity and controllability for TS fuzzy descriptor control systems is worthy of study. Earlier, we presented the results of a preliminary study of the robustness of global regularity and the I-controllability problem in TS fuzzy descriptor control systems[30]. However, that preliminary study did not discuss the robustness of global R-controllability and C-controllability. Here, the purpose is to present an approach for investigating the problem of globally robust regularity and controllability in TS fuzzy descriptor control systems with structured paramet-ric uncertainties and in which the considered controllability properties include C-controllability, R-controllability and I-con-trollability. Sufficient conditions are also proposed to ensure that the TS fuzzy descriptor control systems are robustly globally regular and controllable. The proposed sufficient conditions provide the explicit relationship of the bounds on struc-tured parametric uncertainties to achieve the properties of regularity and controllability. One numerical and two engineer-ing examples are given to illustrate the application of the proposed sufficient conditions.

2. Robust Regularity and controllability

Based on the approach of using sector nonlinearity in fuzzy model construction, the given nonlinear control model with parametric uncertainties can derive both the fuzzy set in the premise part and the linear dynamic model with parametric uncertainties in the consequent part in the exact TS fuzzy control model with parametric uncertainties[5]. The continu-ous-time TS fuzzy descriptor system with parametric uncertainties for nonlinear control systems with structured parametric uncertainties can be obtained in the following form:

eRi_{: IF z}

1ðtÞ is Mi1and . . . and zgðtÞ is Mig;

THEN E _xðtÞ ¼ ðAiþ

D

AiÞxðtÞ þ ðBiþ

D

BiÞuðtÞ; ð1Þ

or the uncertain discrete-time TS fuzzy descriptor system can be described by

eRi_{: IF z}

1ðkÞ is Mi1and . . . and zgðkÞ is Mig;

THEN Exðk þ 1Þ ¼ ðAiþ

D

AiÞxðkÞ þ ðBiþ

D

BiÞuðkÞ; ð2Þ

with the initial state vector x(0), where eRi_{ði ¼ 1; 2; . . . ; NÞ denotes the i-th 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) = [u1(t), u2(t), . . . , up(t)]T and u(k) = [u1(k), u2(k), . . . , up(k)]T denote the p-dimensional input vectors, zi(t) and zi(k)

(i = 1, 2, . . . , g) are the premise variables, E, Aiand Bi(i = 1, 2, . . . , N) are the n  n, n  n and n  p consequent constant

matri-ces, respectively,DAiandDBi(i = 1, 2, . . . , N) are the parametric uncertain matrices existing in the system matrices Aiand the

input matrices Bi, respectively, of the consequent part of the i-th rule due inaccurate measurements, inaccessible system

parameters, or variation in parameters, and Mij(i = 1, 2, . . . , N and j = 1, 2 , . . . , g) are the fuzzy sets. Here, matrix E may be

a singular matrix with rank(E) 6 n. In many applications, matrix E is the structure information matrix rather than a param-eter matrix, i.e., the elements of E contain only structural information for the considered problem.

Although many interesting problems have very few uncertain parameters, these uncertain parameters may be entered in many system and input matrices[31,32]. Therefore, structured parametric uncertain matricesDAiandDBiare in the forms

D

Ai¼ Xm k¼1

e

ikAik and

D

Bi¼ Xm k¼1

e

ikBik; ð3Þ

where

e

ik(i = 1, 2, . . . , N and k = 1, 2, . . . , m) are the elemental parametric uncertainties and Aikand Bik(i = 1, 2, . . . , N and

k = 1, 2, . . . , m) are the given n  n and n  p constant matrices, respectively, which are prescribed a priori to denote infor-mation that is linearly dependent on elemental parametric uncertainties

e

ik.

The resulting TS fuzzy descriptor control system with structured parametric uncertainties inferred from Eq.(1)or(2)is

E _xðtÞ ¼X N i¼1 hiðzðtÞÞ ðAð iþ

D

AiÞxðtÞ þ ðBiþ

D

BiÞuðtÞÞ ð4Þ or Exðk þ 1Þ ¼X N i¼1 hiðzðkÞÞ ðAð iþ

D

AiÞxðkÞ þ ðBiþ

D

BiÞuðkÞÞ; ð5Þ respectively, in which zðtÞ ¼ z1ðtÞ; z2ðtÞ; . . . ; zgðtÞ T

denotes the g-dimensional premise vector, hiðzðtÞÞ ¼ wiðzðtÞÞ=

PN

i¼1wiðzðtÞÞ; wiðzðtÞÞ ¼Pgj¼1MijðzjðtÞÞ, and Mij(zj(t)) are the membership grades of zj(t) in fuzzy sets Mij(i = 1, 2, . . . , N and

j = 1, 2, . . . , g). For all t, hi(z(t)) P 0 andPNi¼1hiðzðtÞÞ ¼ 1.

Before investigating the global robustness of regularity and controllability in the uncertain TS fuzzy descriptor control system in Eq.(4)or Eq.(5), the following definitions and lemmas must be introduced.

Definition 1. [33]: The system fbE; bA; bBg, which denotes bE _xðtÞ ¼ bAxðtÞ þ bBuðtÞ or bExðk þ 1Þ ¼ bAxðkÞ þ bBuðkÞ with bE; bA and bB being, respectively, the n  n, n  n and n  p matrices, is called completely controllable (C-controllable) if, for any t1> 0 (or

k1> 0), x(0) 2 Rnand w 2 Rn, there exists a p-dimensional control input u(t) (or u(k)) such that x(t1) = w (or x(k1) = w), where

matrix bE may be singular matrix with rankðbEÞ 6 n.

Definition 2. [33]: The system fbE; bA; bBg is called R-controllable if it is controllable in the reachable set.

Definition 3. [18]: The system fbE; bA; bBg is called impulse controllable (I-controllable) if the state feedback uðtÞ ¼ bK xðtÞ (or uðkÞ ¼ bK xðkÞÞ is such that the closed-loop system ðbE; bA þ bB bK Þ is impulse-free.

Definition 4. [18]: The system fbE; bA; bBg is called strongly controllable (S-controllable) if it is both R-controllable and I-controllable.

Definition 5. [34]: The measure of a n  n complex matrix W is defined as

l

ðWÞ  lim

h!0

ðkI þ hWk  1Þ

h ;

where kk is the induced matrix norm on the n  n complex matrix.

Lemma 1 [33]. The system fbE; bA; bBg is regular if and only if rank bEnbEd

h i ¼ n2_{, where bE} n2 Rn 2_n and bEd2 Rn 2_n2 are given by bEn¼ bE 0    0 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 and bEd¼ bA bE bA       bE bA 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 : ð6Þ

Lemma 2 [35]. Suppose that the system fbE; bA; bBg is regular. The system fbE; bA; bBg is I-controllable if and only if

rank bAbSE bE bB

h i

¼ n; ð7Þ

where bSE2 RnðnrÞis the maximum right annihilator matrix of bE in which r ¼ rankðbEÞ.

Lemma 3 [26]. Suppose that the system fbE; bA; bBg is regular. The system fbE; bA; bBg is R-controllable if and only if rank bEd bEb

h i ¼ n2_,

where bEd2 Rn

2_n2

is given in Eq.(6)and bEb¼ diagfbB; bB; . . . ; bBg 2 Rn

2_np

.

Lemma 4 [36]. Suppose that the system fbE; bA; bBg is regular. The system fbE; bA; bBg is C-controllable if and only if it is R-controllable and rank bhE bBi¼ n.

Lemma 5 [34]. The matrix measures of matrices W and V,

l

(W) and

l

(V), respectively, are well defined for any norm and have the following properties:

(i)

l

(±I) = ± 1, for the identity matrix I,

(ii) kWk 6 

l

(W) 6 Re(k(W)) 6

l

(W) 6 kWk, for any norm kk and any matrix W 2 Cnn_;

(iii)

l

(W + V) 6

l

(W) +

l

(V), for any two matrices W, V 2 Cnn_;

(iv)

l

(

c

W) =

cl

(W), for any matrix W 2 Cnnand any non-negative real number

c

; where k(W) denotes any eigenvalue of W, and Re(k(W)) denotes the real part of k(W).

Where the induced matrix norms are 1-norm, 2-norm, and 1-norm and where k = 1, 2, 1, the corresponding matrix measures

l

k() are easily calculated as

ðiÞ

l

1ðWÞ ¼ max j Reð wjjÞþP n i ¼ 1 i–j j wijj 2 6 6 6 4 3 7 7 7 5; ðiiÞ

l

2ðWÞ ¼ max i ½kiðW þ W _Þ=2; ðiiiÞ

l

₁ðWÞ ¼ max i Reð wiiÞþP n j ¼ 1 j–i j wijj 2 6 6 6 4 3 7 7 7 5;

where wijis the ij-th element of matrix W and where ki() denotes the i-th eigenvalue.

Lemma 6. For any

c

< 0 and any matrix W 2 Cnn_,

_l

₍

_c

_{W) = }

_cl

_(W).

Proof. From property (iv) inLemma 5, this lemma can be immediately obtained. h

Lemma 7. Let W 2 Cnn_{. If}

_l

_{(W) < 1, then det(I + W) – 0.}

Proof. Since

l

(W) < 1, then, according to property (ii) inLemma 5, Re(k(W)) P 

l

(W) > 1. This implies that k (W) – 1. Thus, det (I + W) – 0. h

The resulting uncertain TS fuzzy descriptor control system in Eq.(4)or(5)can be rewritten as

E _xðtÞ ¼ AxðtÞ þ BuðtÞ þX

N

i¼1

hiðzðtÞÞððeAiþ

D

AiÞxðtÞ þ ðeBiþ

D

BiÞuðtÞÞ ð8Þ

or

Exðk þ 1Þ ¼ AxðkÞ þ BuðkÞ þX

N

i¼1

hiðzðkÞÞððeAiþ

D

AiÞxðkÞ þ ðeBiþ

D

BiÞuðkÞÞ; ð9Þ

where eAi¼ Ai A, eBi¼ Bi B ði ¼ 1; 2; . . . ; NÞ, and where A and B are the given n  n and n  p constant matrices,

respec-tively, for ensuring the regularity and controllability of E _xðtÞ ¼ AxðtÞ þ BuðtÞ or Exðk þ 1Þ ¼ AxðkÞ þ BuðkÞ denoted by fE; A; Bg. That is, fE; A; Bg is regular, I-controllable, R-controllable and C-controllable. Thus, all necessary and sufficient con-ditions inLammas 1–4hold true for the descriptor system fE; A; Bg.

The problem is identifying the conditions under which the uncertain TS fuzzy descriptor control system fE; A þDA; B þDBg in Eq.(8)or(9)(i.e., in Eq.(4)or(5)) is globally and robustly regular and controllable where uncertainty matricesDA andDB areDA ¼PNi¼1hiðzðtÞÞðeAiþDAiÞ andDB ¼PNi¼1hiðzðtÞÞðeBiþDBiÞ., respectively.

Now, let the singular value decompositions of R0¼ E½ nEd, P0¼ ASE E B

h i ; Q0¼ E½ d Eb and M0¼ E B   be R0¼ U S O½ n2_nVH; ð10Þ P0¼ UI SI OnðnrþpÞ   VH I; ð11Þ Q0¼ URSR On2_npVH_R; ð12Þ

and M0¼ UC SC Onq   VH C; ð13Þ respectively, where En¼ E 0    0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ; Ed¼ A E A       E A 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 ;

SE2 Rn(nr) is the maximum right annihilator matrix of E in which r ¼ rankðEÞ; Eb¼ diagfB; B; . . . ; Bg; U 2 Rn

2_n2

and V 2 Rðn2_þnÞðn2_þnÞ

are the unitary matrices, S ¼ diagf

r

1;

r

2; . . . ;

r

n2g, and

r

1P

r

2P   P

r

n2>0 are the singular values of

R0; UI2 Rnn and VI2 R(2nr+p)(2nr+p) are the unitary matrices, SI= diag{

r

1,

r

2, . . . ,

r

n}, and

r

1P

r

2P   P

r

n> 0 are

the singular values of P0; UR2 Rn

2_n2

and VR2 Rðn

2_þnpÞðn2_þnpÞ

are the unitary matrices, SR¼ diagf

r

1;

r

2; . . . ;

r

n2g,

and

r

1P

r

2P   P

r

n2>0 are the singular values of Q0; UC2 Rnnand VC2 R(n+p)(n+p) are the unitary matrices, SC=

diag{

r

1,

r

2, . . . ,

r

n}, and

r

1P

r

2P   P

r

n> 0 are the singular values of M0; VH; VHI; V H R and V

H

C denote the

complex-conjugate transposes of matricesV, VI, VRand VC., respectively.

Assuming that the nominal TS fuzzy descriptor control system fE; A; Bg is regular and I-controllable, the following

Theorem 1gives the conditions that ensure that the uncertain TS fuzzy descriptor control system fE; A þDA; B þDBg in Eq.(8)or(9)(i.e., in Eq.(4)or(5)) is robustly and globally regular and I-controllable.

Theorem 1. The uncertain TS fuzzy descriptor control system fE; A þDA; B þDBg in Eq.(8)or(9)(i.e., in Eq.(4)or(5)) is robustly and globally regular and I-controllable if the following inequalities simultaneously hold true

XN i¼1

l

ðRiÞ þ XN i¼1 Xm k¼1

e

ik

u

ik<1 ð14aÞ and XN i¼1

l

ðPiÞ þ XN i¼1 Xm k¼1

e

ik/ik<1; ð14bÞ

where i = 1, 2, . . . , N, and k ¼ 1; 2; . . . ; m; eRi¼ diagfeAi; . . . ; eAig 2 Rn

2_n2 ; Ri¼ On2_neRi h i 2 Rn2ðn2_þnÞ , Ri¼ S1UHRiV½In2;O_n2_nT, eRik¼ diagfAik; . . . ;Aikg 2 Rn 2_n2 , Rik¼ On2_neRik h i 2 Rn2ðn2_þnÞ , Rik¼ S1UHRikV½In2;O_n2_nT, Pi¼ eAiSE Onn eBi h i 2 Rn2ð2nrþpÞ_, Pi¼ S1I U H IPiVI½In;OnðnrþpÞT, Pik¼ A½ ikSE OnnBik 2 Rn 2_ð2nrþpÞ , Pik¼ S1I U H IPikVI½In; OnðnrþpÞT,

u

ik¼

l

ðRikÞ; for

e

ikP0; 

l

ðRikÞ; for

e

ik<0; ( /ik¼

l

ðPikÞ; for

e

ikP0; 

l

ðPikÞ; for

e

ik<0; (

where matrices S, U, and V, are defined by Eq.(10), matrices SI, UIand VIare defined by Eq.(11), and In2denotes the n2 n2identity

matrix.

Proof. See Appendix A. h

Theorem 2 given below presents the conditions that ensure the uncertain TS fuzzy descriptor control system fE; A þDA; B þDBg in Eq.(8)or (9)(i.e., in Eq.(4) or(5)) is robustly and globally regular and R-controllable under the assumption that the nominal TS fuzzy descriptor control system fE; A; Bg is regular and R-controllable.

Theorem 2. The uncertain TS fuzzy descriptor control system fE; A þDA; B þDBg in Eq.(8)or(9)(i.e., in Eq.(4)or(5)) is robustly and globally regular and R-controllable if the following inequalities simultaneously hold

XN i¼1

l

ðRiÞ þ XN i¼1 Xm k¼1

e

ik

u

ik<1 ð15aÞ and

XN i¼1

l

ðQiÞ þ XN i¼1 Xm k¼1

e

ikhik<1; ð15bÞ where i = 1, 2, . . . , N, and k = 1, 2, . . . , m; Qi¼ eAi eBi eAi eBi       eAi eBi 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 2 Rn2ðn2þnpÞ; ð16aÞ Qik¼ Aik Bik Aik Bik       Aik Bik 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 2 Rn2ðn2_þnpÞ ; ð16bÞ hik¼

l

ðQikÞ; for

e

ikP0; 

l

ðQikÞ; for

e

ik<0; ( ð16cÞ

and

u

ikare given inTheorem 1; Qi¼ S1R U H

RQiVR½In2; O_n2_npT, Qik¼ S1R U H

RQikVR½In2;O_n2_npT, matrix SR, URand VRare defined in

Eq.(12).

Proof. See Appendix B. h

Under the assumption that the uncertain TS fuzzy descriptor control system fE; A þDA; B þDBg is robustly and globally regular and R-controllable, the followingTheorem 3gives the condition under which the uncertain TS fuzzy descriptor con-trol system fE; A þDA; B þDBg in Eq.(8)or(9)(i.e., in Eq.(4)or(5)) is also robustly and globally C-controllable.

Theorem 3. The uncertain TS fuzzy descriptor control system fE; A þDA; B þDBg in Eq.(8)or(9)(i.e., in Eq.(4)or(5)) is robustly and globally regular and C-controllable if the inequalities in(15a) and (15b)and the following inequality simultaneously hold true

XN i¼1

l

ðMiÞ þ XN i¼1 Xm k¼1

e

ikdik<1; ð17Þ where i = 1, 2, . . . , N, and k ¼ 1; 2; . . . ; m; Mi¼ OneBi h i 2 RnðnþpÞ_{, M} ik¼ O½ nBik 2 RnðnþpÞ, Mi¼ S1C U H CMiVC½In; OnpT, Mik¼ S1C U H CMikVC½In; OnpT, dik¼

l

ðMikÞ; for

e

ikP0; 

l

ðMikÞ; for

e

ik<0; (

matrices SC, UCand VCare defined in Eq.(13).

Proof. See Appendix C. h

Remark 1. For a matrix W 2 Rab_with

_a

_{6b, if the matrix W has a full row rank, the singular value decomposition of W is[37]}

W ¼ U

R

VH_{; ð18Þ}

where U 2 Raa_{and V 2 R}bb_{are the unitary matrices,R}

¼ ½S; OaðbaÞ is an

a

 b matrix whose off-diagonal entries are all

zeros, S ¼ diagf

r

1;

r

2; . . . ;

ra

g, and

r

1P

r

2P   P

r

a> 0 are the singular values of W. Since the rank of the matrix W is

equal to the number of its nonzero singular values[37], and sinceR¼ UH_WV,

rankðWÞ ¼ rankð

R

Þ ¼ rankðUH_{WVÞ: ð19Þ}

Since rankðS1_{RÞ ¼}

_a

_,

rankðS1

_R

_{Þ ¼ rankðS}1_UH_{WVÞ ¼}

_a

_{¼ rankðWÞ: ð20Þ}

Remark 2. According toDefinition 4, the sufficient conditions that ensure the S-controllability of the uncertain TS fuzzy descriptor control system fE; A þDA; B þDBg can be immediately obtained fromTheorems 1 and 2.

Remark 3. If the parametric uncertainties are not considered (i.e.,

e

ik= 0), then the proposed sufficient conditions in Eqs.

(14), (15) and (17)can be used to check the global regularity and controllability of nominal TS fuzzy descriptor control systems.

Remark 4. The proposed sufficient conditions in Eqs.(14), (15) and (17)give the explicit relationship of the bounds on

e

ik

(i = 1, 2, . . . , N and k = 1, 2, . . . , m) required for both regularity and controllability. Additionally, the bound obtained under the proposed sufficient conditions, on

e

ik(i = 1, 2, . . . , N and k = 1, 2, . . . , m) are not necessarily symmetric with respect to the

origin of the parameter space regarding

e

ik(i = 1, 2, . . . , N and k = 1, 2, . . . , m).

Remark 5. To obtain less conservative results, various methods based on Lyapunov function and on linear matrix inequality (LMI) have been proposed for analyzing stability and design controllers for TS fuzzy descriptor control systems. Here, how-ever, it should be noted that not all techniques based on Lyapunov function or on LMI can solve the regularity and control-lability problems of TS fuzzy descriptor control systems. That is, for TS fuzzy descriptor control systems, mathematical methods and tools for solving regularity and controllability problems substantially differ from tools for analyzing stability and design controllers. Additionally, a literature review shows that no studies have solved the global regularity and control-lability (simultaneously considering I-controlcontrol-lability, R-controlcontrol-lability and C-controlcontrol-lability) problems in TS fuzzy descriptor control systems. Although the sufficient conditions proposed here are conservative, this study is apparently the first to solve the global regularity and controllability problems (simultaneously considering I-controllability, R-controllability and C-trollability) in TS fuzzy descriptor control systems. Another advantage is that the freedom to choose A and B can reduce con-servatism. In Eqs.(14), (15) and (17), the left-hand-sides of the inequalities are considered the objective functions to be minimized, and the proposed criteria of Eqs.(14), (15) and (17)can be considered constraints. The problem of analyzing glo-bal regularity and controllability then becomes a constrained-optimization problem. Thus, by using the fitness function J = 1/ (1 +X), whereXdenotes the left-hand-side of the inequality, in Eqs.(14), (15) or (17), evolutionary optimization methods such as the hybrid Taguchi-genetic algorithm (HTGA)[38]can be used to determine the most suitable matrices A and B for systematically reducing the conservatism of the proposed sufficient conditions. The detailed steps for using HTGA to solve global regularity and controllability problems are as follows.

(I) Generate Initial Population

In the real coding representation, each chromosome is encoded as a vector of floating-point numbers with the same length as the vector of the design variables. The real coding representation is accurate and efficient because it is closest to the real design space. Moreover, the string length is the number of design variables. The real coding technique is used to solve the global regularity and controllability. For convenience and simplicity, let us denoteH= (x1, x2, . . . , xi, . . . , xq),

where xi(i = 1, 2, . . . ,

q

) denote the respective coefficients fa11; a12; . . . ; a1n; . . . ann; b11; . . . b1m; . . . bnpg and

q

is the total

number of designed coefficients. In the initialization procedure, the following algorithm produces pschromosomes,

where psdenotes the population size:

Step 1: Generate a uniformly distributed random number b, where b 2 [0, 1].

Step 2: Let xi= li+ b(ui li), where xidenotes the ith element ofHand where liand uiare the lower and the upper

bounds of xi, respectively. Repeat

q

times to obtain a vector (x1, x2, . . . , xq).

Step 3: Repeat the above two steps pstimes to obtain psinitial feasible solutions.

(II) Crossover Operation for Generating Diverse Offspring

The crossover operators used here, which are one-cut-point operators integrated with arithmetical operators derived from convex set theory[39], generate new offspring by randomly selecting one cut-point, exchanging the right parts of two parents, and then calculating the linear combinations at the cut-point genes. Since the cut-point is selected randomly, each gene has a chance to be one cut-point. For example, if two parents, H1= (x1, x2, . . . , xk, . . . , xq) and H2= (y1, y2, . . . ,

yk, . . . , yq), are crossed after the k-th position, the resulting offspring are H01¼ ðx1;x2; . . . ;x0k;ykþ1;ykþ2; . . . ;yqÞ and

H0

2¼ ðy1;y2; . . . ;y0k;xkþ1;xkþ2; . . . ;xqÞ, respectively, where xk0 ¼ xkþ bðyk xkÞ; y0k¼ lkþ bðuk lkÞ; lkand ukare the domain

of yk, and b is a uniformly distributed random number, in which b 2 [0, 1].

(III) Taguchi Method for Generating Better Offspring

The orthogonal arrays of the Taguchi method are used to study a large number of design variables with a small number of experiments. The better combinations of designed variables are determined by the orthogonal arrays and the signal-to-noise ratios. The underlying concept of the Taguchi method is maximizing performance measures (signal-to-signal-to-noise ratios) by using orthogonal arrays to run a partial set of experiments. The two-level orthogonal array used in this study has Q factors, where Q is the number of design factors (variables) and each factor has two levels. To establish an orthogonal array of Q factors with two levels, let Ln(2n1) represent n  1 columns and n individual experiments corresponding to

the n rows, where n = 2k_{, k = a positive integer (k > 1), and Q 6 n  1. If Q < n  1, only the first Q columns are used while}

the other n  1  Q columns are ignored. The signal-to-noise ratio (

g

) refers to the mean-square-deviation in the objec-tive function. Let yidenote the output value of experiment i, where i = 1, 2, . . . , n and where n is the number of

experi-ments.The equation

g

i= (yi)2or (1/yi)2is used to describe the [degree of value of

g

each other what does this mean?]

in the orthogonal array experiments if the objective function is to be maximized (larger-the-better) or minimized (smal-ler-the-better), respectively. The effects of the various factors (variables) can be defined as Efl= sum of

g

ifor factor f at

level l, where i is the experiment number, f is the factor number, and l is the level number. The main objective of the matrix experiments is to choose the gene at the locus (factor) of each of the two chromosomes to be a new chromosome. At each locus (factor), a gene is chosen if the Eflhas the highest value in the experimental region. That is, the main

objec-tive is to determine the best or optimal level for each factor. The optimal level for a factor is the level that gives the high-est value of Eflin the experimental region. For the two-level problem, if Ei1> Ei2, the optimal level is 1 for factor i 2 [1, Q].

Otherwise, level 2 is optimal. After the optimal level for each factor is selected, the new chromosome is also obtained by combining the levels for each factor. Therefore, the systematic reasoning capability of the Taguchi method obtains a new chromosome with the best or close-to-best evaluation value of the objective function among those of 2Q_{combinations of}

factor levels, where 2Q_{is the total number of experiments needed for all combinations of factor levels. When Taguchi}

method is used to generate better offspring[40], two chromosomes at a time are randomly chosen from the population pool generated by the crossover operations to execute matrix experiments of an orthogonal array. The detailed steps for each experiment of the matrix experiment are as follows.

Step 1: Set j = 1. Generate two sets, U1and U2, each of which has Q design factors (variables). Allocate the Q design

factors in the first Q columns of the orthogonal array Ln(2n1), where n P Q + 1.

Step 2: Designate sets U1and U2as level 1 and level 2, respectively, by using a uniformly distributed random method

to select two chromosomes from the population pool generated by the crossover operations.

Step 3: Assign the values of level 1 from U1and level 2 from U2to level cells of the j experiment in the orthogonal

array.

Step 4: Calculate the fitness value and signal-to-noise ratio for the new chromosome.

Step 5: If j > n, then go to Step 6. Otherwise, j = j + 1, return to Step 3, and continue through Step 5. Step 6: Calculate the effects of the various factors (Ef1and Ef2), where f = 1, 2, . . . , Q.

Step 7: The gene of locus i of the new chromosome comes from U1, if Ei1> Ei2. Otherwise, it comes from U2, where

i = 1, 2, . . . , Q. Implementing the process for each gene at each locus obtains the new chromosome. (IV) Mutation Operation

The basic concept of mutation operation is also derived from convex set theory. Depending on mutation rate pm, each

gene in a single chromosome executes the mutation of the convex combination. For a given chromosomeH= (x1, x2,

-= (x1, x2, . . . , xi, xj, xk, . . . , xq), if element xiis selected to execute the mutation operation, and if the other xkis randomly

selected to join in, the resulting offspring becomesH0¼ ðx1;x2; . . . ;x0i;xj;xk; . . . ;xqÞ. The new gene x0iis x0i¼ ð1  bÞxiþ bxk,

and b is a uniformly distributed random number in which b 2 [0, 1].

Detailed Steps of the HTGA:

Step 1: Set parameters.

Input: population size G, crossover rate pc, mutation rate pm, and number of generations.

Output: values for J = 1/(1 +X), and matrices A and B.

Step 2: Initialize. Use J = 1/(1 +X), which is the fitness function defined for the HTGA, to calculate the fitness values of the initial population feasible for the constraint of Eq.(14), (15) or (17)where the initial population with the chromosomes of the form V ¼ fa11; a12; . . . ; a1n; . . . ann; b11; . . . b1m; . . . bnpg is randomly generated.

Step 3: Use roulette wheel approach to perform a selection operation.

Step 4: Perform crossover operation. The probability of the crossover is determined by crossover rate pc.

Step 5: Select a suitable two-level orthogonal array Lc(2c1) for the matrix experiments, where

c

denotes the number

of experimental runs and where

c

 1 is the number of columns in the orthogonal array. Step 6: Randomly choose two chromosomes at a time to execute the matrix experiments.

Step 7: Calculate the fitness values of the

c

experiments in the orthogonal array Lc(2c1) by using J = 1/(1 +X), and

penalize the fitness value of chromosomes that violate the constraint. Step 8: Calculate the effects of the various factors.

Step 9: Generate the optimal chromosome based on the results of Step 8. Step 10: Repeat Steps 6 through 9 until the expected number G  pcis obtained.

Step 11: Generate the population via Taguchi method.

Step 12: Perform a mutation operation in which the mutation probability is determined by mutation rate pm.

Step 13: Generate offspring population.

Step 14: Sort the fitness values in increasing order among the parents and offspring populations.

Step 15: Select the better G chromosomes feasible for the constraint as the parents of the next generation. Step 16: If the stopping criterion has been met, go to Step 17. Otherwise, repeat Steps 3 through 16. Step 17: Display the optimal chromosome and the optimal fitness value.

Hence, matrix G in Eq.(B4)is nonsingular. That is, matrix Q in Eq.(B1)has a full row rank n2_{. Thus, according toLemma 3, the}

global R-controllability of the uncertain TS fuzzy descriptor control system in Eq.(8)or(9)(i.e., in Eq.(4)or(5)) is ensured. Thus, the proof is completed.

Appendix C. Proof ofTheorem 3

According toLemma 4, matrix M0¼ E B has full row rank (i.e., rank(M0) = n) since the TS fuzzy descriptor control system

fE; A; Bg is C-controllable. According to uncertainty matricesDA andDB, the uncertain TS fuzzy descriptor control system fE; A þDA; B þDBg is C-controllable if and only if

M ¼ M0þ XN i¼1 hiðzðtÞÞMiþ XN i¼1 Xm k¼1 hiðzðtÞÞ

e

ikMik ðC1Þ

have full row rank, where Mi¼ OneBi

h i 2 RnðnþpÞ, and Mik¼ O½ nBik 2 RnðnþpÞ. It is known that rankðMÞ ¼ rank S1 C U H CMVC   : ðC2Þ

Thus, instead of rank(M), we discuss the rank of

½In; Onp þ XN i¼1 hiðzðtÞÞ bMiþ XN i¼1 Xm k¼1

e

ikhiðzðtÞÞ bMik; ðC3Þ where bMi¼ S1C U H CMiVC, and bMik¼ S1C U H

CMikVC, 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 under which the matrix in Eq.(C3)has rank n is the non-sin-gularity of F ¼ Inþ XN i¼1 hiðzðtÞÞMiþ XN i¼1 Xm k¼1

e

ikhiðzðtÞÞMik; ðC4Þ where Mi¼ S1C U H CMiVC½In;OnpT, and Mik¼ S1C U H

CMikVC½In;OnpT, ðfor i ¼ 1; 2; . . . ; N and k ¼ 1; 2; . . . ; mÞ.

Based on Eq.(17)and the properties inLemmas 5 and 6,

l

X N i¼1 hiðzðtÞÞMi XN i¼1 Xm k¼1 hiðzðtÞÞ

e

ikMik ! 6X N i¼1 hiðzðtÞÞ

l

ðMiÞ þ XN i¼1 Xm k¼1 hiðzðtÞÞ

l

ð

e

ikMikÞ 6X N i¼1

l

ðMiÞ þ XN i¼1 Xm k¼1

l

ð

e

ikMikÞ ¼ XN i¼1

l

ðMiÞ þ XN i¼1 Xm k¼1

e

ikdik<1: ðC5Þ

According to Eq.(C4)andLemma 7,

detðFÞ ¼ det Inþ XN i¼1 hiðzðtÞÞMiþ XN i¼1 Xm k¼1

e

ikhiðzðtÞÞMik ! –0: ðC6Þ

Hence, matrix F in Eq.(C4)is nonsingular. That is, matrix M in Eq.(C1)has the full row rank n. Thus, according toLemma 4, the global C-controllability of the uncertain TS fuzzy descriptor control system in Eq.(8)or(9)(i.e., in Eq.(4)or(5)) is en-sured. Thus, the proof is completed.

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