A Closed-Form Solution to Observer Design Problem for Ostensible

Metzler Takagi-Sugeno Systems

DUŠAN KROKAVEC

Department of Cybernetics and Artiﬁcial Intelligence

Faculty of Electrical Engineering and Informatics

Technical University of Košice

Letná 9, 042 00 Košice

SLOVAKIA

Abstract: - This paper addresses the state estimation problems related to the generalized fuzzy observer design

for ostensible Metzler Takagi-Sugeno (T-S) systems. Attention is focused on design constraints for the concept

of diagonal stabilization and positivity of observer gain matrices. On the basis of some new interpretations, the

parameterizations of ostensible Metzler T-S fuzzy systems is presented, which opens the way to the solution of

the design problem using only the principle of linear matrix inequalities. The same approach is intended to ensure

the stability of the dynamics of the estimation error. The presented method extends and generalizes the results

that have been presented in the literature so far.

Key-Words: - Metzler Takagi-Sugeno fuzzy system, ostensible Metzler matrix structures, diagonal stabilization,

parametric constraints, state-space methods, linear matrix inequalities.

Received: April 17, 2024. Revised: August 21, 2024. Accepted: September 13, 2024. Published: October 30, 2024.

1 Introduction

Many of the nonlinear systems can be represented

as an aggregation of a set of linear mathematical

models using the fuzzy implication of local dynamics

by the fuzzy Takagi-Sugeno (T-S) approach, [1].

Based on this idea, speciﬁc fuzzy design methods

have appeared especially in the ﬁeld of fuzzy

control, [2], [3], [4]. Since TS fuzzy models

are differential inclusions, these methods resort to

stability analysis using quadratic Lyapunov function

when using the representation of models in the state

space, although other fuzzy inference systems can

be used, [5], [6]. Currently, T-S fuzzy systems are

growing in popularity because they have powerful

modeling and control in ship control systems, [7], [8],

multi-agent systems, [9], data-oriented ﬂexible-joint

robot structures, [10], and T-S network control

systems, [11], [12]. A similar trend can be seen in the

use of T-S fuzzy observers in fault diagnosis, [13],

[14], and in fault-tolerant control, [15], of nonlinear

systems.

As for positive nonlinear systems, modern

methods of synthesis of their control are aimed at

less restrictiveness than offered by methods based on

linearization techniques. Inspired by linear positive

systems, [16], [17], the use of linear state space

theory in solving the control synthesis problem has

led to T-S structures of positive nonlinear systems

based on the matrices of the Metzler structure, [18].

Since the design conditions are in the form of

linear matrix inequalities (LMIs), the problem can

be numerically solved by the latest LMI techniques.

Already the ﬁrst technological applications have

shown that the structure of the ostensible Metzler

matrix suits reality, [19]. This class is represented

by ostensible Metzler T-S fuzzy systems, which

includes models of chaotic systems given by the

Lorenz equation with an input term, [20], [21],

[22]. New control algorithms based on fuzzy T-S

methods have been proposed for water turbine control

systems, [23], [24], actuator fault-tolerant control of

wind turbines, [25], and extended fuzzy control for

aircraft engines, [26], whose T-S models are based

on ostensible Metzler matrix structures.

In the mentioned papers, the authors focus only

on the synthesis of the control of T-S fuzzy positive

systems, and in [27], only design conditions for

T-S fuzzy positive observers with strictly Metzler

matrices can be found, which generalize the results

for linear positive systems, [28]. In general, a positive

observer is driven by the output of a positive system

such that the estimation error is asymptotically stable

and positive. Because ostensible Metzler T-S fuzzy

systems are not internally positive, for state observers

based on this class of models it is necessary to

modify the standard conditions of synthesis due to

incomplete positivity. Based on these facts, the paper

formulates a new approach to the design of fuzzy

observers for ostensible Metzler T-S fuzzy systems.

A convex optimization technique is used to

represent design constraints within the concept of

diagonal stabilization and positivity of observer gain

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.33

Dušan Krokavec

E-ISSN: 2224-2856

316

Volume 19, 2024

matrices. To solve the limitations generated by

ostensible Metzler matrix structures, a new efﬁcient

approach is presented to ﬁnd feasible solutions and

parameterize the gains of the ostensible Metzler T-S

fuzzy observer. Under LMI-based design conditions,

positive observer gains are determined such that all

modes of the observer subsystems are asymptotically

stable and ostensible Metzler. Mixed pair of

ostensible and strictly Metzler matrices are used as

an example to demonstrate the effectiveness of the

proposed method.

This correspondence is structured as follows. In

Sect. 2 essential limiting features are included with

reference to Metzler T-S fuzzy models and their

projection into the task of Metzler T-S fuzzy observer

design is outlined in Sect. 3. The formulation of

the fuzzy observer stability problem using the LMI

approach is the subject of Sect. 4. In Sect. 5

a numerical example is presented to illustrate the

synthesis procedure proposed in this paper and Sect.

6 draws conclusions regarding the effectiveness of the

presented approaches and potential future research

directions in this area.

The following notations are used throughout the

paper: xT,XTdenotes the transpose of the vector

x, respectively the matrix X,diag (·)denotes the

(block) diagonal matrix, the notation ≺0for the

square symmetric matrix Xmeans that Xis negative

deﬁnite, the symbol Indenotes the identity matrix of

the nth order, R+,Rn

+are sets of nonnegative real

numbers and n-dimensional real vectors, Rn×n

+refers

to the set of nonnegative real matrices and Rn×n

−+,

Rn×n

−⊕ covers the set strictly or ostensible Metzler

matrices.

2 Metzler T-S Fuzzy Models

Considering the T-S fuzzy approach to positive

nonlinear MIMO continuous-time dynamic systems

then the system state and output, q(t),y(t), are given

q(t) =



i=1

hi(ϑ(t))(Aiq(t) + Biu(t)) (1)

y(t) = Cq(t)(2)

where q(t)∈Rn

+,u(t)∈Rr,y(t)∈Rm

+,Ai∈

Mn×n

−+,Bi∈Rn×r

+,C∈Rm×n

+and hi(θ(t)) is

averaging weight for the i-th rule, where

0≤hi(ϑ(t)) ≤1,



i=1

hi(ϑ(t)) = 1 ∀i∈ ⟨1, s⟩(3)

while sis the number of linear sub-models) and

ϑ(t) = [ θ1(t)θ2(t)· · · θq(t)](4)

is q-dimensional vector of measured premise

variables. Further details can be found in [29].

To reﬂect the system positiveness, nonnegative

matrices Bi∈Rn×r

+and C∈Rn

+have to be

considered when matrices Ai∈Rn×n

−+are Metzler.

Deﬁnition 1 [30] A square matrix A∈Mn×n

−+is

strictly Metzler if its diagonal elements are negative

and its off-diagonal elements are positive and the

following constraints result

alh <0, l =h, alh >0, l =h, ∀l, h ∈ ⟨1, n⟩(5)

Since positive systems are only diagonally

stabilizable, [31], using the circulant permutation

matrix L∈Rn×n

+they matrix parameters have to be

diagonally parameterized.

Deﬁnition 2 [32] The square matrix L∈Rn×n

+is

circulant permutation matrix if

L=0T1

In−10.(6)

Deﬁnition 3 [33] If Metzler A={alj } ∈ Mn×n

−+is

represented in the equivalent rhombic structure

AΘ=







a11

a21 a22

a31 a32 a33

....

an1an2an3· · · ann

a12 a13 · · · a1n

a23 · · · a2n

....

an−1,n







(7)

then the set of diagonal matrix inequalities











AΘ(p+h, p)≺0, h = 0

AΘ(p+h, p)≻0, h = 1, . . . , n −1

AΘ(p+h, p) =

diag [a1+h,1· · · an,n−ha1,n−h+1· · · ah,n]

(8)

implying from diagonals of (7) is equivalent to (6).

Lemma 1 [33] Using (6)-(8) the parameterization of

a Metzler matrix A∈Mn×n

−+means the relation

n−1



h=0

LhA(p+h, p)(9)

Applying for a Metzler Ae=A−JC ∈Mn×n

−+,

J∈Rn×m

+,C∈Rm×n

+address the following

parametrization

Ae=

n−1



h=0

LhA(p+h, p)−



k=0

JkhCk(10)

where diagonal Jkh,Ck∈Rn×n

+are given as

CT= [c1· · · cm],Cdk =diag cT

k(11)

J= [j1· · · jm],Jk=diag [jk](12)

where Jkh =LhTJkLh.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.33

Dušan Krokavec

E-ISSN: 2224-2856

317

Volume 19, 2024

The results of Lemma 1 show that all terms related

to the parametrization of the matrix Aeare diagonal

matrices.

3 Metzler T-S Fuzzy Observer

The system state estimation assumes the observer to

Metzler T-S fuzzy system (1), (2) in the form

qe(t) =



i=1

hi(ϑ(t))(Aiqe(t) + Biu(t))+



i=1

hi(ϑ(t))JiC(q(t)−qe(t))

(13)

ye(t) = Cqe(t)(14)

where qe(t)∈Rn

+is the estimation of the system

state vector, Ji∈Rn×m

+,i= 1, . . . , s is the set of the

observer gain matrices. Since (13), (14) implies

qe(t) =



i=1

hi(ϑ(t))(Ai−JiC)qe(t)+



i=1

hi(ϑ(t))(Biu(t) + JiCq(t))

(15)

and for every Metzler matrix Ai∈Mn×n

−+diagonal

constraints can be obtained analogously to (8)











AΘi(p+h, p)≺0, h = 0

AΘi(p+h, p)≻0, h = 1, . . . , n −1

AΘi(p+h, p) =

diag [ai,1+h,1· · · ai,n,n−hai,1,n−h+1· · · ai,h,n]

(16)

the parametrization of Aei =Ai−JiC∈Mn×n

−+is

Aei =

n−1



h=0

LhAΘi(p+h, p)−



k=0

JikhCk(17)

and the diagonal matrices Jikh,Ck∈Rn×n

+are

deﬁned as

CT= [c1· · · cm],Ck=diag cT

k(18)

Ji= [ji1· · · jim],Jik =diag [jik](19)

where Jikh =LhTJik Lh.

Having in mind constraints (19) it is not hard to

establish the following:

Lemma 2 [34] The Metzler T-S fuzzy observer (13),

(14) is stable if there exist positive deﬁnite diagonal

matrices P,Rik ∈Rn×n

+such that with lT=

[1 · · · 1] for i= 1,2, . . . , s,h= 1,2, . . . , n −1,

k= 1,2, . . . , m,

P≻0,Rik ≻0(20)

P AΘi(p, p)−



k=1

RikCk≺0(21)

P LhAΘi(p+h, p)LhT−



k=1

RikLhCkLhT≻0(22)

P Ai+AT

iP−



k=1

(RikllTCk+CkllTRik)≺0

(23)

When the above conditions hold, the set of strictly

positive Jiis given by (19), where

Jik =P−1Rik,jik =Jikl(24)

and Aei ∈Mn×n

−+are strictly Metzler and Hurwitz.

Remark 1 The conditions (21) guarantee that

elements on the main diagonals of Aei are strictly

negative and the set of LMIs (22) guarantee that

the off-diagonal elements of Aei are strictly positive

(Aei are strictly Metzler) if Aiare strictly Metzler,

Cis non-negative and Jiare strictly positive. The

Lyapunov matrix inequalities (23) force that all Aei

will be Hurwitz. The condition and solution for the

existence of observer gains Jiare clearly obtained

from Lemma 2.

4 Ostensible Metzler T-S Systems

In the considered case, the state space equation of a

system is also (1) but minimally one system matrix

from the set of Ai∈Mn×n

−⊙ ,i∈ ⟨1, s⟩, is ostensible

Metzler.

Deﬁnition 4 The matrix A∈Mn×n

−⊖ is ostensible

Metzler if in the structure Athere is at least one

negative off-diagonal element, while the number of

positive off-diagonal elements of Ais predominant

and all diagonal elements of Aare negative.

Parametric structure design limits of an ostensible

Metzler matrix can be eliminated by using the matrix

eigenvalue principle. The essentials of the approach

are most easily understood in case of dealing with the

following lemma:

Lemma 3 [35] If for U,V∈Rn×nis

V=cU+dIn(25)

with c, d ∈R,c= 0 and In∈Rn×n, then the

eigenvectors of Uand Vare identical and

ηk=cλk+d(26)

where ηk, k = 1, . . . , n are eigenvalues of Vand λk

runs over the eigenvalues of U,

In such a way, the proposed approach is based

on separation of an ostensible Metzler matrix A∈

Mn×n

−⊖ to ensure the following is met:

A=Ap+Am(27)

where Ap∈Mn×n

−+is strictly Metzler and Am∈

Rn×n

−is constructed as element-wise negative and

Hurwitz using Lemma 3. Therefore, this concept

transforms the task of synthesis for ostensible

Metzler matrices into a task for strictly Metzler

ones, for which standard methods are available,

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.33

Dušan Krokavec

E-ISSN: 2224-2856

318

Volume 19, 2024

parameterizing only the matrix Ap∈Mn×n

−+with its

rhombic diagonals.

For such application the approach is reformulated

to the following:

Lemma 4 [36] For ostensible Metzler A∈Mn×n

−⊖

there exists a strictly Metzler Ap∈Mn×n

−+and an

element-wise negative and Hurwitz Am∈Rn×n

−

satisfying (27) with relation to positive scalars η, δ ∈

R+only if

Ap=Ad+A++ηΣ+pIn=A◦

p+pIn(28)

Am=A−−ηΣ−pIn=A◦

m−pIn(29)

Ad=diag [−a11 · · · − ann],Ad+pIn≺0(30)

A+=







0a+

12 · · · a+

1,n−1a+

21 0· · · a+

2,n−1a+

.....

n−1,1a+

n−1,2· · · 0a+

n−1,n

n1a+

n2· · · a+

n−1,n−10







(31)

A−=







0a−

12 · · · a−

1,n−1a−

a−

21 0· · · a−

2,n−1a−

.....

a−

n−1,1a−

n−1,2· · · 0a−

n−1,n

a−

n1a−

n2· · · a−

n−1,n−10







(32)

ij =aij if aij >0,

0if aij <0,

a−

ij =aij if aij <0,

0if aij >0,

i, j ∈ ⟨1, n⟩

i=j(33)

λk∈ρ(A◦

λo= max

k(λ+

k|λ+

k=real(λk)>0) (34)

p=λo+δ, Σ=





0 1 · · · 1 1

1 0 · · · 1 1

.....

1 1 · · · 1 0







(35)

Note, since any Aiis not strictly Metzler, the

system (1) in the related mode is not internally

positive.

Remark 2 If A◦

p,A◦

mare constructed for any η∈

R+as (28), (29), it yields

A= (Ad+A+

+ηΣ)+(A−

−ηΣ) = A◦

p+A◦

m(36)

Then, evidently, A◦

ptakes strictly Metzler structure

and A◦

mis a matrix which diagonal elements are

zeros and all its off-diagonal elements are negative.

Assuming (without loss of generality) that all

eigenvalues of A◦

mare distinct then

A◦

m−λoIn⪯0(37)

where λois deﬁned in (34). Since λo>0, to obtain a

stable D-stability region it can be set

Am=A◦

m−(λo+δ)In=A◦

m−pIn≺0(38)

where δ∈R+is a tuning parameter.

Introducing the error in the state observations as

e(t) = q(t)−qe(t)(39)

then, exploiting (1) and (13), it is obtained

e(t) =



i=1

hi(θ(t))Ai(q(t)−qe(t))−

−



i=1

hi(θ(t))Ji(y(t)−ye(t))

(40)

which can be written using (2), (17) as follows

e(t) =



i=1

hi(θ(t))Aeie(t)(41)

where, with the above modiﬁcations,

Aei =Ai−JiCstrictly M etzler mode

Aei =Api −JiCostensible M etzler mode

(42)

and the above presented approach can be modiﬁed

to derive the stable ostensible Metzler T-S fuzzy

observer.

Theorem 1 The Metzler T-S fuzzy observer (13), (14)

with ostensible Metzler modes is stable if there exist

positive deﬁnite diagonal matrices P,Rik ∈Rn×n

such that with lT= [1 · · · 1] for i= 1,2, . . . , s,h=

1,2, . . . , n −1,k= 1,2, . . . , m,

P≻0,Rik ≻0(43)

P AΘi(p, p)−



k=1

RikCk≺0(44)

P LhAΘi(p+h, p)LhT−



k=1

RikLhCkLhT≻0(45)

P Ai+AT

iP−



k=1

(RikllTCk+CkllTRik)≺0

(46)

When the above conditions hold, the set of strictly

positive Jiis given by (19), where

Jik =P−1Rik,jik =Jikl(47)

Proof: Deﬁning the Lyapunov function

v(e(t)) = eT(t)P e(t)>0(48)

where P∈Rn×n

+is positive deﬁnite diagonal

matrix, then (48) implies for strictly negative of time

derivative of Lyapunov function

˙v(e(t)) = ˙

e(t)P e(t) + eT(t)P˙

e(t)(49)

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.33

Dušan Krokavec

E-ISSN: 2224-2856

319

Volume 19, 2024

Substituting (41) into (49) gives

˙v(e(t))

=eT(t)



i=1

hi(θ(t))(P Aei +AT

eiP)e(t)

(50)

P Aei +AT

eiP≺0∀i(51)

respectively. Since the diagonal matrix representation

is necessary, the expression (50) implies the

following result:

P(Ai−



k=1

JikllTCk)+

+(Ai−



k=1

JikllTCk)TP≺0

(52)

and with the notation

Rik =P Jik (53)

(52) implies (46).

Applying the circular shifted structures to cover

sets of algebraic constraints then (17) for h= 0

implies

AΘi(p, p)−



k=1

JikCdk ≺0(54)

and multiplying the left side by a positive deﬁnite

diagonal matrix P∈Rn×n

+then (54) implies

P Ai(p, p)∆−



k=1

P JikCk≺0(55)

and with the notation (53) then (55) implies (44).

Since Jikh =LhTJik Lh,LhLhT=In,

pre-multiplying the left side by Pand

post-multiplying the right side by ThTthen (17)

results

P LhAΘi(p+h, p)LhT−

−



k=1

P LhLhTJikLhCkLhT≻0(56)

and with the notation (53) then (56) implies (45).

This concludes the proof.

Note, the proposed design conditions have no

tuning parameters, tuning parameters are occurred

only in separation of a ostensible Metzler matrices.

The invertibility of Pis guaranteed by the fact that

P∈Rn×n

+is positive deﬁnite diagonal matrix.

5 Illustrative Example

The example reﬂects the ostensible Metzler T-S

equations (1), (2), where for s= 2,n= 3,m= 2

A1=



−0.272 1.94 1.45

0.058 −3.96 0.10

0.100 0.08 −2.91



, ρ(A1)=





−0.1863

−2.9634

−3.9923







A2=



−0.272 1.94 −0.45

0.058 −3.14 0.10

0.100 −0.04 −2.91



, ρ(A2)=





−0.2474

−2.9454

−3.1293







B1=B2=



0.50 1.00

1.00 0.90

0.70 1.10



,C=1.1 0 0

0 1.5 0

Since A2is ostensible Metzler

A2d=diag [−0.272 −3.14 −2.91]

A+

2=



0 1.94 0

0.058 0 0.10

0.100 0 0



,L=



0 0 1

1 0 0

0 1 0





A−

2=



0 0 −0.45

0 0 0

0−0.04 0



,Σ=



0 1 1

1 0 1

1 1 0



,l=







Setting µ= 0.005 then

A◦

2m=



0−0.005 −0.455

−0.005 0 −0.005

−0.005 −0.045 0





ρ(A◦

2m)={−0.0642 0.0321 ±0.0238 i}

which implies λ0= 0.0321 and with δ= 0.003 gives

A2m=



−0.0351 −0.0050 −0.4550

−0.0050 −0.0351 −0.0050

−0.0050 −0.0450 −0.0351





ρ(A2m)={−0.0993 −0.0030 ±0.0238 i}

A2p=



−0.2369 1.9450 0.0050

0.0630 −3.1049 0.1050

0.1050 0.0050 −2.8749





ρ(A2p)={−0.1919 −2.9057 −3.1191}

To reﬂect diagonal LMIs structures, the

representations of Care given as

C1=diag [1.100],C2=diag [0 0.5 0]

and the representations of A1,A2pare

AΘ 1(p, p) = diag [−0.272 −3.96 −2.91]

AΘ 1(p+ 1, p) = diag [0.058 0.08 1.45]

AΘ 1(p+ 2, p) = diag [0.10 1.94 0.10]

AΘ 2p(p, p) = diag [−0.2369 −3.1049 −2.8749]

AΘ 2p(p+ 1, p) = diag [0.063 0.005 0.005]

AΘ 2p(p+ 2, p) = diag [0.105 1.945 0.105]

The LMIs taken into account are those of Theorem

1 nad SeDuMi tolbox, [37], determines the feasible

solution

P=diag [0.8101 0.4452 0.6971]≻0

R11 =diag [0.6951 0.0133 0.0342]≻0

R12 =diag [0.7080 0.3038 0.0224]≻0

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.33

Dušan Krokavec

E-ISSN: 2224-2856

320

Volume 19, 2024

R21 =diag [0.6308 0.0147 0.0410]≻0

R22 =diag [0.7060 0.3200 0.0014]≻0

J1=



0.8580 0.8740

0.0299 0.6823

0.0490 0.0322



,J2=



0.7787 0.8715

0.0329 0.7187

0.0587 0.0021





which, with the positive gains, guarantees strictly

Metzler and Hurwitz matrices

Ae1=



−1.0442 0.8912 1.45

0.0311 −4.7788 0.10

0.0559 0.0414 −2.91





ρ(Ae1)={−0.9936 −2.9518 c−4.7875}

Ae2p=



−0.9377 0.8992 0.0050

0.0334 −3.9673 0.1050

0.0521 0.0025 −2.8749





ρ(Ae2p)={−0.9269 −2.8771 −3.9760}

and ostensible Metzler and Hurwitz matrix

Ae2=



−0.9728 0.8942 −0.4500

0.0284 −4.0024 0.1000

0.0471 −0.0425 −2.9100





ρ(Ae2)={−0.9746 −2.9052 −4.0055}

The ostnesible Metzler T-S fuzzy observer design

task is transformed into an equivalent form and leads

to a standard design task. Based on the diagonalized

matrix representations it is veriﬁed that the design

conditions allow existence of positive Jifor i∈

⟨1,2⟩such that Lyapunov principle establishes

asymptotic stability of the observer equilibrium.

Due to the diagonal principle of stabilization,

these results are relevant when the fuzzy observer

uses the measured premise variables.

6 Concluding Remarks

Observing the properties of the state space model

suggests that it is possible to obtain a ﬁnite number

of LMIs in the design for a strictly Metzler a

Hurwitz fuzzy observer dynamics matrices with

positive gains. To obtain a solution for the ostensible

structures of Metzler matrices, the design method is

modiﬁed to provide the feasibility of a set of LMIs,

where the conditions take into account the separation

of the ostensible structures of Metzler matrices

and the resulting modiﬁed parametric constraints.

The approach favors standard LMI representation to

handle the stability of the fuzzy observer, where the

proposed conditions guarantee asymptotic stability.

The separation takes into account those performance

requirements that are associated with the positivity of

the system’s trajectories. Since ostensible Metzler TS

fuzzy systems are not internally positive, the initial

states of the observer should be chosen appropriately

to allow for non-negative observer error.

The proposed approach provides a platform for

the development of ostensible Metzler T-S fuzzy

observers that are resilient to parameter uncertainties

and perturbations. These topics are currently under

investigation and will be addressed in the future. It

is possible to envisage in the future control structures

of T-S fuzzy networks and cooperative control of T-S

fuzzy multi-agent systems for agents whose system

matrices are ostensible Metzler.

References:

[1] T. Takagi and M. Sugeno, "Fuzzy identiﬁcation

of systems and its applications to modeling and

control," IEEE Trans. Syst. Man. Cybern., vol.

15, no. 1, pp. 116132, 1985.

[2] T.M. Guerra and L. Vermeiren, "Control laws

for TakagiSugeno fuzzy models, Fuzzy Sets

Syst., vol. 120, no. 1, pp. 95-108, 2001.

[3] K. Kiriakidis, "Robust stabilization of the

Takagi-Sugeno fuzzy model via bilinear matrix

inequalities," IEEE Trans. Fuzzy Syst., vol. 9,

no. 2, pp. 269-277, 2001.

[4] Y. Wang, H.R. Karimi, H.K. Lam, and

H. Shen, "An improved result on exponential

stabilization of sampled-data fuzzy systems,"

IEEE Trans. Fuzzy Syst., vol. 26, no. 6, pp.

3875–3883, 2018.

[5] L. Rutkowski. Flexible Neuro-Fuzzy Systems.

Structures, Learning and Performance

Evaluation. Boston: Kluwer Academic

Publishers, 2004.

[6] M. Mazandarani and X. Li, "Fractional fuzzy

inference system. The new generation of fuzzy

inference systems," IEEE Acces, vol. 8, pp.

126066-126082, 2020.

[7] W.J. Chang, Y.H. Lin, J. Du, and C.M. Chang,

"Fuzzy control with pole assignment and

variance constraints for continuous-time

perturbed Takagi-Sugeno fuzzy models:

Application to ship steering systems," Int. J.

Control Autom. Syst., vol. 17, pp. 2677-2692,

2019.

[8] W.J. Chang, Y.H. Lin, C.L. Pen, M.H. Tsai,

"Actuator saturated fuzzy controller design for

interval type-2 Takagi-Sugeno fuzzy models

with multiplicative noises, Processes, vol. 9, pp.

1-24, 2021.

[9] Z. Zhang, Y. Shi, Z. Zhang, and W. Yan,

"New results on sliding-mode control for

TakagiSugeno fuzzy multiagent systems," IEEE

Trans. Cybern., vol. 49, no. 5, pp. 1592-1604,

2019.

[10] B. Zhang and Y. Shin, "A data-driven approach

of Takagi-Sugeno fuzzy control of unknown

nonlinear systems," Appl. Sci., vol. 11, pp.

1-15, 2020.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.33

Dušan Krokavec

E-ISSN: 2224-2856

321

Volume 19, 2024

[11] J. Qiu, H. Gao, and S.X. Ding, "Recent

advances on fuzzy-model-based nonlinear

networked control systems. A survey," IEEE

Trans. Ind. Electron., vol. 63, no. 2, pp.

1207-1217, 2016.

[12] M. Aslam and Z. Ma, "Output regulation for

timedelayed TakagiSugeno fuzzy model with

networked control system," Hacet. J. Math.

Stat., vol. 52, pp. 1-21, 2023.

[13] I. Elleuch, A. Khedher, and K.B. Othman,

"State and faults estimation based on

proportional integral sliding mode observer for

uncertain TakagiSugeno fuzzy systems and its

application to a turbo-reactor," Int. J. Fuzzy

Syst., vol. 19, pp. 1768-1781, 2017.

[14] D. Zhao, H.K. Lam, Y. Li, S.X. Ding, and

S. Liu, "A novel approach to state and unknown

input estimation for TakagiSugeno fuzzy

models with applications to fault detection,"

IEEE Trans. Circuits Syst. I: Regul. Pap., vol.

67, no. 6, pp. 2053-2063, 2020.

[15] D. Kharrat, H. Gassara, A. El Hajjaji, and

M. Chaabane, "Adaptive observer-based H∞

FTC for T-S fuzzy systems. Application to cart

motion model," J. Frank. Inst., vol. 357, no. 17,

pp. 12062-12084, 2020.

[16] H. Nikaido, Convex Structures and Economic

Theory. New York: Academic Press, 1968.

[17] H.L. Smith, Monotone Dynamical Systems. An

Introduction to the Theory of Competitive and

Cooperative Systems. Providence: American

Mathematical Society, 1995.

[18] A. Berman, M. Neumann, and R. Stern,

Nonnegative Matrices in Dynamic Systems.

New York: John Wiley & Sons, 1989.

[19] K. Tanaka, T. Hori, and H.O. Wang, "New

parallel distributed compensation using time

derivative of membership functions. A fuzzy

Lyapunov approach," Proc. 40th IEEE

Conference on Decision and Control, Orlando,

USA, pp. 3942-3947, 2001.

[20] K. Tanaka, T. Ikeda and H.O. Wang, "An

LMI approach to fuzzy controller designs based

on relaxed stability conditions," Proc. 6th Int.

Fuzzy Systems Conference, Barcelona, Spain,

pp. 171-176, 1997.

[21] Y.W. Wang, Z.H. Guan, and O. Wang,

"Impulsive synchronization for TakagiSugeno

fuzzy model and its application to continuous

chaotic system," Phys. Lett. A, vol. 339, pp.

325-332, 2005.

[22] A. Senouci and A. Boukabou, "Predictive

control and synchronization of chaotic and

hyperchaotic systems based on a T-S fuzzy

model," Math. Comput. Simul., vol. 105, pp.

62-78, 2014.

[23] B. Wang, J. Xue, F. Wu, and D. Zhu,

"Finite time Takagi-Sugeno fuzzy control for

hydro-turbine governing system," J. Vib.

Control, vol. 24, no. 5, pp. 1001-1010, 2018.

[24] Y. Tian, B. Wang, P. Chen, and Y. Yang,

"Finite-time TakagiSugeno fuzzy controller

design for hydraulic turbine governing systems

with mechanical time delays," Renew. Energy,

vol. 173, pp. 614-624, 2021.

[25] A. Abdelmalek, A.T. Azar, and D. Dib,

"A novel actuator fault-tolerant control

strategy of DFIG-based wind turbines using

Takagi-Sugeno multiple models," Int. J. Control

Autom. Syst., vol. 16, pp. 1415-1424, 2018.

[26] W. Wang, J. Peng, and Y. Zhang, "Modeling

and control for an aero-engine based on the

Takagi-Sugeno fuzzy model," Aerospace, vol.

10, pp. 1-18, 2023.

[27] Y. Li a H. Zhang, "Positive observer design

for switched positive T-S fuzzy delayed systems

with dwell time constraints," ISA Trans., vol.

96, pp. 37-50, 2020.

[28] L.J. Liu, X. Zhao, X.M. Sun, and W. Wang,

"New approaches to positive observer design

of linear positive systems," Proc. 2017

Chinese Automation Congress, Jinan, China,

pp. 7195-7198, 2017.

[29] K. Tanaka and H.O. Wang, Fuzzy Control

Systems Design and Analysis. A Linear Matrix

Inequality Approach. New York: John Wiley &

Sons, 2001.

[30] J.M. Carnicer, J.M. Pena, R.A. Zalik, "Strictly

totally positive systems," J. Approx. Theory,

vol. 92, pp. 411–441, 1998.

[31] O. Mason, "Diagonal Riccati stability and

positive time-delay systems," Syst. Control

Lett., vol. 61, no. 1, pp. 6–10, 2012.

[32] R.A. Horn and C.R. Johnson, Matrix Analysis.

New York: Cambridge University Press, 1985.

[33] D. Krokavec and A. Filasová, "Observer-based

control design for systems with potentially

Metzler dynamics," IFAC PapersOnLine, vol.

56, no. 2, pp. 5065-5070, 2023

[34] D. Krokavec and A. Filasová, "State control

design of ostensible Metzler linear systems with

unsigned input parameters," Designs, vol. 8, no.

3, pp. 1-13, 2024.

[35] T.S. Shores, Applied Linear Algebra and Matrix

Analysis. Springer, New York, 2007.

[36] D. Krokavec, "Ostensible Metzler linear

uncertain systems. Goals, LMI synthesis,

constraints and quadratic stability," WSEAS

Trans. Syst. Control, vol. 18, pp. 255-2062,

2023.

[37] D. Peaucelle, D. Henrion, Y. Labit, and

K. Taitz, User’s Guide for SeDuMi Interface

1.04. Toulouse: LAAS-CNRS, 2002.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.33

Dušan Krokavec

E-ISSN: 2224-2856

322

Volume 19, 2024

Contribution of the Author to the Creation of the

Scientiﬁc Article

Dusan Krokavec carried out completely this paper.

Sources of Funding for Research Presented in the

Scientiﬁc Article

No funding was received for conducting this study.

Conﬂicts of Interest

The author has no conﬂicts of interest to

declare that is relevant to the content of this

article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.33

Dušan Krokavec

E-ISSN: 2224-2856

323

Volume 19, 2024