Interval State Estimation of Systems with Metzler Polytopic Models

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: - The paper deals with the design of interval observers for interval-deﬁned strictly Metzler polytopic

positive systems. The stability conditions for the proposed structure of the interval observer are formulated using

linear matrix inequalities to ensure a positive estimate of the system state. The proposed method makes it possible

to calculate time-varying lower and upper estimates of the state vector, assuming that the disturbance is bounded.

Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

Key-Words: - Metzler systems, parametric constraints, diagonal stabilisation, linear matrix inequalities, applied

interval analysis, interval observers.

Received: March 19, 2024. Revised: July 11, 2024. Accepted: August 26, 2024. Published: September 25, 2024.

1 Introduction

For linear systems with non-negative states, [1], [2],

the standards of their description, based on linear

equations, must be supplemented with additional

parametric constraints, [3], by which the positivity

of the evolution of the state variables is deﬁned us-

ing the properties of the Metzler matrices, [4], [5]. A

suitable uniﬁcation, considering the principle of di-

agonal stabilization of such deﬁned positive Metzler

systems is the design strategy based only on linear

matrix inequalities (LMIs), being proposed in [6].

Unlike systems with ﬁxed parameters, the ap-

proach outlined in [7] provides an estimate of the sys-

tem state for given bounds on the system matrices. A

modiﬁed approach, using a technique based only on

the parametric properties of Metzler matrices in the

synthesis of interval observers, is presented in [8]. It

is also worth noting that this task can be well for-

mulated for positive Metzler systems using the rep-

resentation of matrix bounds by the LMI structure,

which is implicitly veriﬁed in [9]. Analogously, the

synthesis of state observers for Takagi-Sugeno fuzzy

systems was formulated in [10], also considering the

guarantee of the positivity of the system state in poly-

topic linear systems, [11].

Unlike ordinary state observers, the outputs of the

interval observer are the time-varying lower and up-

per estimates of the state vector, and the states of the

system fall into the interval deﬁned in this way, even

in the presence of stationary input disturbances, [12],

[13]. Due to some system advantages, provided that

the description of uncertain Meztler systems has ma-

trix parameter constraints, the so-called principle of

cooperative observers is commonly used in the syn-

thesis, which assumes that the resulting matrices of

the dynamics of the interval observer will be Metzler

and Hurwitz, and in the optimization, as a tuning pa-

rameter, is used H∞norm of the disturbance transfer

function matrix, [14], [15], [16]. A survey on inter-

val observer design using positive system approach is

presented in [17].

Focusing on the above mentioned strategies, the

target adaptation of the authors’ results in strictly

Metzler positive systems to the synthesis of positive

interval state observers, as well as the interdepen-

dence of diagonal stabilization with the representa-

tion of interval constraints of this class of systems in

the synthesis task, constitute the main topic of this

paper. The presented new LMI formulation of the

deﬁnition of parametric and interval constraints re-

ﬁnes the design conditions so that the inequalities are

sharp and result in strictly positive observer gain ma-

trices. Since the LMI problem is formulated in this

way, the interval observer synthesis conditions in-

clude interval constraints and guarantee the H∞dis-

turbance input cost in the interval estimation error,

and a positive estimate of the lower observer state

vector. Since only a set of LMIs is used to deﬁne

the synthesis conditions of a positive interval state

observer, the methodology provides a standard envi-

ronment for implementation. The uncertainties are

modeled using a polytopic framework and the calcu-

lated gains of the interval observer are optimized for

this class of uncertainties. Although the basic proce-

dures are also applicable under more general assump-

tions in the design of interval observers for uncertain

positive continuous-time linear systems, the proposed

solution assumes that the polytopic uncertainties are

time invariant, which may be a conservative assump-

tion.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.60

Dušan Krokavec

E-ISSN: 2224-2880

571

Volume 23, 2024

In Section 2, the parametrization of the restric-

tions of Mezler matrix structures is analyzed with re-

gard to the synthesis of state observers of positive

systems, and Section 3, different from the existing

papers, presents the basis of the procedures deﬁning

the synthesis conditions for strictly Metzler positive

interval observers based on LMIs. Section 4 is the

extension of the results to the robust fault detection.

The the efﬁciency and validity of the proposed solu-

tion is illustrated in Section 5 on a numerical example

and, within the given concept, the analysis of the re-

sults is generalized in Section 6, providing a scope of

the authors further research work in the future.

For sake of convenience, throughout this paper

used notations reﬂect usual conventionality so that

xT,XTdenotes the transpose of the vector x, and the

matrix X, respectively, diag [·]marks a (block) di-

agonal matrix, for a square symmetric matrix X≺0

means its negative deﬁniteness, Inlabels the n-th or-

der unit matrix, X−1,ρ(X)signify the inverse and

the eigenvalue spectrum of a square matrix X, the

symbol ∗denotes a block-symmetric element in LMI

matrix variables, R(R+) marks the set of (nonnega-

tive) real numbers, Rn×m(Rn×m

+) refers to the set of

(nonnegative) real matrices and Mn×n

−+indicates the

set of strictly Metzler matrices.

2 Generalized Metzler Systems

The given task categorises that a set of matrices A∈

Mn×n

−+,B∈Rn×r

+,C∈Rm×n

+,D∈Rn×d

+belongs

to the polytopic uncertainty domain

O:=





a∈ Q,(A,B,C,D) (a) :

(A,B,C,D) (a)= s

i=1

ai(Ai,Bi,Ci,Di)





(1)

Q=(a1, ...as) :

i=1

ai=1; ai>0, i = 1, ...s(2)

where Qis the unit simplex, Ai∈Mn×n

−+,Bi∈

Rn×r

+,Ci∈Rm×n

+,Di∈Rn×d

+are constant ma-

trices and ai, i = 1,2, . . . , s are uncertainties.

Since ais restricted to the unit simplex as (2), the

matrices (A,B,C,D) (a)are afﬁne functions of the

uncertain parameter vector a∈Rn

+and the system is

described by a convex combination of vertex matrices

(Ai,Bi,Ci,Di,), i = 1, . . . , s.

The class of uncertain polytopic systems is cha-

racterized by multi-input and multi-output (MIMO)

dynamics, represented by the compact form

q(t) =

i=1

ai(Aiq(t) + Biu(t) + Did(t)) (3)

y(t) =

i=1

aiCiq(t)(4)

where q(t)∈Rn

+,u(t)∈Rr,y(t)∈Rmare vectors

of the state, input, and output variables and d(t)∈Rd

is the bounded.

A short overview of new trends and starting points

in this research area can be found in [18].

To the system (3), (4) can be designed an observer

with Luenberger structure, given by the formula

qe(t) =

i=1

ai(Aiqe(t) + Biu(t))+

i=1

aiJiC(q(t)−qe(t))

(5)

ye(t) =

i=1

aiCiqe(t)(6)

where Ji∈Rn×m

+,i= 1, . . . , s, are the observer

gain matrices, vector qe(t)∈Rn

+is the state vector of

the observer and ye(t)∈Rm

+is the estimated system

output vector.

Consequently, the observer (5), (6) can be rewrit-

ten as:

qe(t) =

i=1

ai(Aeiqe(t) + Biu(t) + JiCq(t)(7)

ye(t) =

i=1

aiCiqe(t)(8)

where

Aei =Ai−JiCi(9)

whilst Aei ∈Mn×n

−+has to be strictly Metzler and

Hurwitz.

Since positive linear systems are only diagonally

stabilizable, [19], it is appropriate to use positive deﬁ-

nite diagonal matrix variables in formulating stability

conditions and combine them with Metzler paramet-

ric constraints.

Since A∈Mn×n

−+is strictly Metzler, its descrip-

tion is characterized by negative diagonal elements

and strictly positive off diagonal elements, which

means formal, [5]

all <0, alj >0, l 6=j, ∀l, j ∈ h1, ni(10)

and then, if A={alj } ∈ Mn×n

−+is represented in the

equivalent rhombic structures, [20]

AΘ=







a11

a21 a22

a31 a32 a33

....

an1an2an3· · · ann

a12 a13 · · · a1n

a23 · · · a2n

....

an−1,n







(11)

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.60

Dušan Krokavec

E-ISSN: 2224-2880

572

Volume 23, 2024

the set of diagonal matrix inequalities for h=

0,1, . . . , n −1,











A(κ+h, κ)≺0, h = 0

A(κ+h, κ)0, h = 1, . . . , n −1

A(κ+h, κ) =

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

(12)

implying from diagonals of (11), can be used to rep-

resent (10).

Evidently, the equivalent conditions hold

(LhA(κ+h, κ)LhT≺0h= 0

LhA(κ+h, κ)LhT0h= 1, . . . , n −1

(13)

where L∈Rn×nof the structure

L=0T1

In−10(14)

is the circulant form of a permutation matrix, [21].

The Metzler matrix (9) can be parameterized as

follows:

Lemma 1 (see, [20]). Applying for observer dynam-

ics (9) with Ai∈Mn×n

−+,Ci∈Rm×n

+,Ji∈Rn×m

the conditions for diagonal parametrisation of Aei ∈

Mn×n

−+are given by using the deﬁned diagonal matri-

ces Ai(κ+h, κ)∈Rn×n

+,Cik ∈Rn×n

+,Jik ∈Rn×n

Jikh =LhTJik Lh∈Rn×n

+constructed such that

Ci=



im

,Cik =diag cT

ik=diag [cik1· · · cikn]

(15)

Ji=[ji1· · · jim],Jik =diag [jik]=diag [jik1· · · jikn]

(16)

and the observer system matrices Aei ∈Mn×n

−+,i=

1, . . . , s, are parameterizable by

Aei =

n−1

h=0

LhAi(κ+h, κ)−

k=0

JikhCik (17)

Theorem 1 Matrices Aei ∈Rn×n

−+for all i∈ h1, si

are strictly Metzler and Hurwitz if for by the system

deﬁned strictly Metzler matrices Ai∈Rn×n

−+and

nonnegative matrices Ci∈Rm×n

+,D∈Rn×d

+there

exist positive deﬁnite diagonal matrices P,Vik ∈

Rn×n

+and positive scalars ξ∈R+such that for

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

P0,Vik 0, ξ > 0(18)

P Ai(κ, κ)−

k=1

VikCik ≺0(19)

P LhAi(κ+h, κ)LhT−

k=1

VikLhCik LhT0

(20)





Ξi∗ ∗

DTP−ξId∗

Ci0−ξIm

≺0(21)

Ξi=P Ai+AT

iP−

k=1

VikllTCik −

k=1

CikllTVik

(22)

Conﬁrming feasibility for i= 1, . . . , s, then by

computing

Jik =P−1Vik,jik =Jik l,Ji= [ji1· · · jim]

(23)

the observer parameters are achieved.

Hereafter, ∗is the symmetric item in a symmetric

matrix.

Proof: Because the state observation produces

e(t) = q(t)−qe(t),ey(t) =

i=1

aiCie(t)(24)

whilst qe(0) = 0is freely assignable, it should be

noticed that using (24)

e(t) =

i=1

aiAeie(t) + Dd(t)(25)

and it is possible for stable (25) to deﬁne a positive

function v(e(t)>0

v(e(t)) = eT(t)P e(t)+

+ξ−1Zt

(eT

y(ν)ey(ν)−ξ2dT(ν)d(ν))dν

(26)

with a positive deﬁnite diagonal matrix (PDDM) P∈

Rn×n

+and a positive scalar ξ∈R+. In this way it is

necessary to request that the time derivative

˙v(e(t)) = ˙

eT(t)P e(t) + eT(t)P˙

e(t)+

+ξ−1eT

y(t)ey(t)−ξdT(t)d(t)(27)

is negative for all observer error trajectories or, by

substituting (25) that negative is the inequality

˙v(e(t)) =

i=1

aieT(t)(AT

eiP+P Aei)e(t)+

i=1

ai(eT(t)P Dd(t) + dT(t)DTP e(t))+

+ξ−1

i=1

j=1

aiajeT(t)CT

iCje(t)−

−ξdT(t)d(t)(28)

In order to derive LMIs with respect to

d(t) = heT(t)dT(t)i(29)

it follows that

˙v(ed(t)) =

i=1

j=1

aiajeT

d(t)Ωij ed(t)<0(30)

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.60

Dušan Krokavec

E-ISSN: 2224-2880

573

Volume 23, 2024

where it can be stated

Ωij ="AT

eiP+P Aei +ξ−1CT

iCj∗

DTP−ξId#≺0

(31)

Φi=



P Aei +AT

eiP∗ ∗

DTP−ξId∗

Ci0−ξIm



≺0(32)

respectively, when applying the Schur-complement

with respect to a block partitioning of the matrix.

Thus,

˙v(ed(t)) =

i=1

aieT

d(t)Φied(t)<0(33)

and the observer design requires ﬁnding for each ver-

tex i= 1, . . . , s such PDDMs P,Vik and positive

scalar ξthat conform to LMI (21).

To have the diagonal constraints it yields for (9)

P(Ai−JiCi)+(Ai−JiCi)TP

=PAi−

k=1

jikcT

ik+Ai−

k=1

jikcT

ikTP

=PAi−

k=1

JikllT

Cik+Ai−

k=1

JikllT

CikT

(34)

Respecting the basic rule (9) when including system

parameters in the LMI structure means that

PAei =PAi−P

k=1

JikllT

Cik =P Ai

−

k=1

VikllT

Cik

(35)

where

Vik =P Jik (36)

and applying the last given, then (34) implies (22),

whilst (32) gives (21).

Since the observer system matrix Aei ∈Mn×n

−+is

parameterizable as (17) where Jikh =LhTJik Lh,

pre-multiplying the left side by Pand post-

multiplying the right side by LhTthen (17), (36) im-

plies

P LhAi(κ+h, κ)LhT−P

k=0

JikhCik LhT

=P LhAi(κ+h, κ)LhT−

k=0

VikLhCik LhT

(37)

Thus, one can conclude that (37) implies the Met-

zler parametric constraints as (18) for h= 0 and (18)

for h > 0as LMIs, which is the key to close the proof.

3 Interval Observer Design

It can be considered that in (3), (4) the parameters

(Ai,Ci) and the initial state of the system q(0) are

unknown, but their upper and lower matrix and vector

bounds are known, and element-wise holds for all i∈

h1, siand t≥0

Ai≤Ai≤Ai,Ci≤Ci≤Ci(38)

0≤q(0) ≤q(0) ≤q(0) (39)

d≤d(t)≤d,d=−d(40)

Ai,Ai∈Mn×n

−+,Ci,Ci∈Rm×n

+,d∈Rd. Con-

straint (40) is basic in the literature of interval ob-

servers supposing that the disturbances are assumed

to be bounded with known bounds.

Considering the interval-given parameters of the

system, it is possible to deﬁne an interval observer

qe(t) =

i=1

aiAiqe(t)+Biu(t)+Ji(y(t)−ye(t))

i=1

aiAeiqe(t) + Biu(t) + JiCiq(t)(41)

qe(t) =

i=1

aiAiqe(t)+Bu(t)+Ji(y(t)−ye(t))

i=1

ai(Aeiqe(t) + Bu(t) + JiCiq(t)

(42)

ye(t) =

i=1

aiCiqe(t),ye(t) =

i=1

aiCiqe(t)

(43)

where for t≥0, if qe(0) = q(0),qe(0) = q(0), it is

expected that

0≤qe(t)≤q(t)≤qe(t)(44)

Aei =Ai−JiCi,Aei =Ai−JiCi(45)

when considering priori nonnegative matrices Bi∈

Rn×r

+,Ji∈Rn×m

+,Di∈Rn×d

+,Ci,Ci∈Rm×n

and strictly Metzler Ai,Ai∈Rn×n

−+.

The diagonal parametrisation constraints from

Lemma 1 can be generalized as follows:

Lemma 2 Applying for observer dynamics (45) with

Ai,Ai∈Rn×n

−+,Ci,Ci∈Rm×n

+,Ji∈

Rn×m

+, the conditions for diagonal parametrisation

of Aei,Aei ∈Mn×n

−+are given by using the rhombic

diagonal matrices Ai(κ+h, κ),Ai(κ+h, κ)∈Rn×n

and diagonal matrices Cik,Cik ∈Rn×n

+,Jik ∈

Rn×n

+,Jikh =LhTJik Lh∈Rn×n

+constructed such

that

Ci=



im

,Cik =diag cT

ik=diag [cik1· · · cikn]

(46)

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.60

Dušan Krokavec

E-ISSN: 2224-2880

574

Volume 23, 2024

Ci=



im

,Cik =diag cT

ik=diag [cik1· · · cikn]

(47)

Ji=[ji1· · · jim],Jik =diag [jik]=diag [jik1· · · jikn]

(48)

and the observer system matrices Aei,Aei ∈Mn×n

−+

are parameterizable as

Aei =

n−1

h=0

LhAi(κ+h, κ)−

k=0

JikhCik (49)

Aei =

n−1

h=0

LhAi(κ+h, κ)−

k=0

JikhCik (50)

It is no restriction to assume in the diagonal con-

straints that d∈Rdacts through the gain D∈Rn×d.

This design task of interval observer can be so for-

mulated as a generalization of Theorem 1.

Theorem 2 The matrices Aei,Aei ∈Rn×n

−+for all

i∈ h1, siare strictly Metzler and Hurwitz if for by

the system deﬁned strictly Metzler matrices Ai,Ai∈

Rn×n

−+and non-negative matrices Ci,Ci∈Rm×n

D∈Rn×d

+there exist positive deﬁnite diagonal ma-

trices P,Vik ∈Rn×n

+and a positive scalar ξ∈R+

such that for all i= 1, . . . , s,h= 1, . . . , n −1,

lT= [1 · · · 1]

P0,Vik 0, ξ > 0(51)











P Ai(κ, κ)−m

k=1

VikCdk ≺0

P Ai(κ, κ)−m

k=1

VikCdk ≺0

(52)











P LhAi(κ+h, κ)LhT−m

k=1

VikLhCik LhT0

P LhAi(κ+h, κ)LhT−m

k=1

VikLhCik LhT0

(53)





Ξi∗ ∗

P−ξId∗

Ci0−ξIm



≺0,



Ξi∗ ∗

P−ξId∗

Ci0−ξIm

≺0

(54)











Ξi=

P Ai+AT

iP−m

k=1

VikllTCik −m

k=1

CikllT

Vik

Ξi=

P Ai+AT

iP−m

k=1

VikllTCik −m

k=1

CikllT

Vik

(55)

and according to (23), Ji∈Rn×m

+can be deter-

mined.

The proof is omitted for clarity of notation when

extending Theorem 1.

Assuming that each sub-model dynamics of inter-

val observer is strictly Metzler, nontrivial interval ob-

server matrices can be designed so that each is stable

using the proposed LMI-based algorithm. The choice

ensures that the observer error sets are described by

stable dynamics, satisfying stability and nonnegativ-

ity properties of the observation errors.

Starting from the initial state g(0) which veriﬁes

(39) and taking into account the system uncertainties,

then the positive gains of the interval observer are op-

timized by H∞approach to guarantee the attenuation

of disturbances effect. That means, the optimal de-

sign results are only tuned by the user-speciﬁed dis-

turbance attenuation. Since

e(t) = q(t)−q(t),e(t) = q(t)−q(t)(56)

e(t)−e(t) = q(t)−q(t)+ q(t)−q(t) = q(t)−q(t)

(57)

then

kq(t)−q(t)k ≤ ke(t)k+ke(t)k(58)

and the stability of the interval observer (41)-(43) can

be simply deduced.

4 Robust Fault Detection

The main application ﬁeld of interval observers is the

fault residual generation in the model-based fault di-

agnosis. The standard principle means to generate

fault residuals in dependency on the system output

y(t)and their estimated values ye(t), which can be

generated as

r(t) = Ce(t)(59)

where

e(t) = q(t)−qe(t)(60)

is the state variable estimation error. It is obvious that

the estimation error in the occurrence of an additive

fault cannot be zero while, in a fault-free operation,

the residuals are around zero. Nevertheless, when

considering a system affected by perturbations and

parameter uncertainties, the residuals deviate from

zero even in the fault-free scenario.

Considering the interval error dynamics described

by (56), the following robust fault residuals are given

r(t) = Ce(t),r(t) = Ce(t)(61)

and the fault detection test can be formulated as

0/∈[r(t),r(t)] (62)

which give the possibility to apply adaptive thresh-

olds on the residuals, [22].

To enable all followers tracking the trajectory

formed by the leader in the multi-agent system in the

presence of agent model interval parameters, external

disturbances and single actuator faults, the fault has to

be detected to determine the faulty agent in the team.

Based on the distributed strategy of fault detection

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.60

Dušan Krokavec

E-ISSN: 2224-2880

575

Volume 23, 2024

and isolation, the residual ﬂags can be generated cor-

responding to the residual signals of the team agents,

assuming that each agent constructs the deﬁned fault

pattern in the ﬂag, [23], [24]. Adaptation of the pre-

sented intervally deﬁned parametric topic for network

agent systems is moved for future research.

5 Illustrative Example

The system in the example is described by the model

(3), (4) where

A1=



−0.1904 1.3580 1.0150

0.0406 −2.7720 0.0500

0.0700 0.0350 −2.0370

,C1=0.9 0 0

0 1.2 0

A2=



−0.2176 1.5520 1.1600

0.0464 −3.1680 0.0575

0.0800 0.0640 −2.3280

,C2=0.9 0 0

0 1.2 0

A1=



−0.1806 1.4420 1.0850

0.0994 −2.5480 0.0540

0.1400 0.0460 −1.7850

,C1=1.1 0 0

0 1.5 0

A2=



−0.2064 1.6480 1.2400

0.1136 −2.9120 0.0665

0.1600 0.0640 −2.0400

,C2=1.1 0 0

0 1.5 0

For all i= 1,2, it is not difﬁcult to conﬁrm that

Ai,Aiare strictly Metzler and Hurwitz, Ci,Ciare

nonnegative matrices and Ai≤Ai,Ci≤Ci.

In order to take into account the principle of diag-

onal stabilization, the diagonal representations from

Ci,Ciare

C11=C21=|1diag [0.9 0 0],C12=C22=diag [0 0.2 0]

C11=C21=diag [1.1 0 0],C12=C22=diag [0 0.5 0]

while the set of circular diagonal representations of

the matrices Ai,Aiis

A1(κ, κ) = diag [−0.1904 −2.7720 −2.0370]

A1(κ+1, κ) = diag [0.0406 0.0350 1.0150]

A1(κ+2, κ) = diag [0.0700 1.3580 0.0500]

A2(κ, κ) = diag [−0.2176 −3.1680 −2.3280]

A2(κ+1, κ) = diag [0.0464 0.0640 1.1600]

A2(κ+2, κ) = diag [0.0800 1.5520 0.0575]

A1(κ, κ) = diag [−0.1806 −2.5480 −1.7850]

A1(κ+1, κ) = diag [0.0994 0.0460 1.0850]

A1(κ+2, κ) = diag [0.1400 1.4420 0.0540]

A2(κ, κ) = diag [−0.2064 −2.9120 −2.0400]

A2(κ+1, κ) = diag [0.1136 0.0640 1.2400]

A2(κ+2, κ) = diag [0.1600 1.6480 0.0665]

The condition for inserting the task into the Se-

DuMi toolbox, [25], means the construction of a total

of N= 22 LMIs, of which Nmv = 8 are used to de-

clare the positivity of the matrix variables, Nst = 4

are needed to enter stability conditions and Npb = 10

is necessary to deﬁne the parametric constraints for

the structures of the Metzler matrices. Then, a feasi-

ble solution results

P=diag [4.5394 2.9283 4.8781], ξ = 5.6530

V11 =diag [3.9176 0.0493 0.1192]

V12 =diag [2.3752 1.1860 0.0461]

V21 =diag [3.9063 0.0554 0.1301]

V22 =diag [2.8408 1.1311 0.0721]

which implies the positive gains

J1=



0.8630 0.5232

0.0168 0.4050

0.0244 0.0094

,J2=



0.8605 0.5232

0.0189 0.4050

0.0267 0.0094



to construct Metzler and Hurwitz stable matrices

Ae1=



−0.9671 0.7301 1.0150

0.0255 −3.2580 0.0500

0.0480 0.0237 −2.0370



ρ(Ae1) = {−0.9148 −2.0809 −3.2664}

Ae2=



−0.9921 0.9241 1.1600

0.0294 −3.6540 0.0575

0.0560 0.0527 −2.3280



ρ(Ae2) = {−0.9342 −2.3746 −3.6653}

Ae1=



−1.1299 0.6571 1.0850

0.0809 −3.1555 0.0540

0.1131 0.0318 −1.7850



ρ(Ae1) = {−0.9542 −1.9349 −3.1814}

Ae2=



−1.1530 0.8631 1.2400

0.0928 −3.5195 0.0665

0.1307 0.0498 −2.0400



ρ(Ae1) = {−0.9658 −2.1938 −3.5529}

Note that programming the task using the LMI

plexity.

Using, for example, the method presented in [26],

the algorithm can easily be adapted by structured ma-

trix variables to design interval observers for purely

Metzler matrices in which some off-diagonal matrix

elements may be zero. In that case, the solution leads

to non-negative matrix structures Ji, which guaran-

tees that the stable matrices Aσ

ei,Aσ

ei are also purely

Metzlerian. Since in general Aσ

ei ∈Rn×n

−+for all

i∈ h1, siare Metzler and Hurwitz, the asymptotic

stability of the interval observer is guaranteed.

As can be seen from the example, the proposed

procedure transforms the synthesis problem into ac-

ceptable structural LMIs that are easily accessible by

calculation.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.60

Dušan Krokavec

E-ISSN: 2224-2880

576

Volume 23, 2024

6 Concluding Remarks

A key result is that it is possible to construct a set of

LMIs respecting given constraints to prescribe posi-

tive (non-negative) gains in the design of the system

Metzler and Hurwitz matrices of the interval state ob-

server of positive systems. The novelty of the given

LMI structure links interval bounds, parameters of

Metzler matrices and stability conditions directly in

the formulation of the problem and deﬁnes the crite-

ria to capture the boundedness and positiveness of the

interval estimation error dynamics.

Although there are many studies on interval ob-

server design approaches for estimating continuous-

time systems, discrete-time linear time-invariant sys-

tems, and bounded time-delay systems, [27], [28], the

research on interval observers has not been compre-

hensive, and there are many unsolved problems in the

design of interval observers for systems with input by

saturation and linear time-varying systems. The pro-

posed methodology could be extended to classes of

systems such as positive switching systems and inter-

connected positive systems, [29], [30]. Moreover, T-

S fuzzy systems with non-measurable premise vari-

ables is also a challenging task to become one of the

future research points, and also the design of a closed

loop interval observer supporting the stabilization of

possibly unstable plants by state feedback is an inter-

esting perspective. Special cases of these problems

may arise in various contexts associated with fault

detection a class of distributed multi-agent systems

based on ostensible Metzler agents [31].

References:

[1] H. Nikaido, Convex Structures and Economic

Theory, New York: Academic Press, 1968.

[2] H.L. Smith, Monotone Dynamical Systems.

An Introduction to the Theory of Competitive

and Cooperative Systems. Providence: Ameri-

can Mathematical Society, 1995.

[3] J. Shen,. Analysis and Synthesis of Dynamic

Systems with Positive Characteristics. Singa-

pore: Springer Nature, 2017.

[4] A. Berman and D. Hershkowitz, "Matrix diago-

nal stability and its implications," SIAM J. Dis-

crete Math., vol. 4, no. 3, pp. 377–382, 1983.

[5] A. Berman, M. Neumann, and R. Stern, Non-

negative Matrices in Dynamic Systems. New

York: John Wiley & Sons, 1989.

[6] D. Krokavec and A. Filasová, "LMI based prin-

ciples in strictly Metzlerian systems control de-

sign," Math. Probl. Eng., vol. 2018, p. 1–14,

2018.

[7] J.L. Gouzé, A. Rapaport, and M.Z. Hadj-Sadok,

"Interval observers for uncertain biological sys-

tems," Ecol. Modell., vol. 133, no. 1, pp. 45–56,

2000.

[8] F. Mazenc and O. Bernard, "Interval ob-

servers for linear time-invariant systems with

disturbances," Automatica, vol. 47, no. 1, pp.

140–1477, 2011.

[9] D. Krokavec and A. Filasová, "Interval ob-

server design for uncertain linear continuous-

time Metzlerian systems," Proc. 28th Mediter-

ranean Conference on Control and Automa-

tion MED’20, Saint-Raphaël, France, pp. 1051-

1056. 2020

[10] H. Ito and T.N. Dinh, "An approach to inter-

val observers for Takagi-Sugeno systems with

attractiveness guarantees," Proc. 58th Annual

Conference of the Society of Instrument and

Control Engineers of Japan, Hiroshima, Japan,

pp. 1268–1273, 2019.

[11] B. Marx, D. Ichalal, and J. Ragot, "Interval state

estimation for uncertain polytopic systems," Int.

J. Control, vol. 93, no. 11, pp. 2564-2576, 2020.

[12] J. Blesa, D. Rotondo, V. Puig, and F. Nejjari,

"FDI and FTC of wind turbines using the inter-

val observer approach and virtual actuators/sen-

sors," Control Eng. Pract., vol. 24, pp. 138-155,

2014.

[13] M. Bolajraf, M. Ait Rami, and U. Helmke, "Ro-

bust positive interval observers for uncertain

positive systems," IFAC Proceedings Volumes,

vol. 44, no. 1, pp. 14330-14334, 2011.

[14] N. Ellero, D. Gucik-Derigny, and D. Henry, "In-

terval observer for linear time invariant (LTI)

uncertain systems with state and unknown input

estimations," Journal of Physics: Conference

Series, vol. 659, pp, 1-12, 2015.

[15] Z. Shu, J. Lam, H. Gao, B. Du, and L. Wu, "Pos-

itive observers and dynamic output-feedback

controllers for interval positive linear systems,"

IEEE Trans. Circuits Syst., vol. 55, no. 10, pp.

3209-3222, 2008.

[16] T. Raissi and D. Eﬁmov, "Some recent re-

sults on the design and implementation of inter-

val observers for uncertain systems," Automa-

tisierungstechnik, vol. 66, no. 3, pp. 213-224,

2018.

[17] Z. Zhang and J. Shen, "A survey on interval ob-

server design using positive system approach,"

Franklin Open, vol. 4, pp. 1-10, 2023.

[18] H. Ito and T.N. Dinh, "Asymptotic and track-

ing guarantees in interval observer design for

systems with unmeasured polytopic nonlinear-

ities," IFAC-PapersOnLine, vol. 53, no 2, pp.

5010-5015, 2020.

[19] O. Mason, "Diagonal Riccati stability and pos-

itive time-delay systems," Syst. Control Lett.,

vol. 61, no. 1, pp. 6–10, 2012.

[20] D. Krokavec, "Ostensible Metzler linear uncer-

tain systems. Goals, LMI synthesis, constraints

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.60

Dušan Krokavec

E-ISSN: 2224-2880

577

Volume 23, 2024

and quadratic stability," WSEAS Trans. Syst.

Control, vol. 18, pp. 255-2062, 2023.

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

New York: Cambridge University Press, 2013.

[22] Y. Garbouj, T. N. Dinh, T. Raissi, T. Zouari,

and M. Ksouri, "Optimal interval observer for

switched Takagi-Sugeno systems. An applica-

tion to interval fault estimation," IEEE Trans.

Fuzzy Syst., vol. 29, no. 8, pp. 2296–2309,

2021.

[23] M.R. Davoodi, N. Meskin, and K. Khorasani,

"Simultaneous fault detection and control de-

sign for a network of multi-agent systems,"

Proc. 13th European Control Conference ECC

2014, Strasbourg, France:, pp. 575–581, 2014.

[24] T.V. Pham and Q.T.T. Nguyen, "H2/H_dis-

tributed fault detection and isolation for het-

erogeneous multi-agent systems," Proc. Int.

Conf. Advanced Technologies for Communica-

tions ATC 2021, Ho Chi Minh City, Vietnam,

pp. 83–88, 2021.

[25] D. Peaucelle, D. Henrion, Y. Labit, and K. Taitz,

User’s Guide for SeDuMi Interface. Toulouse:

LAAS-CNRS, 2002.

[26] A. Khan, W. Xie, L. Zhang, and L.W. Liu, "De-

sign and applications of interval observers for

uncertain dynamical systems," IET Circuits De-

vices Syst., vol. 14, pp. 721-740, 2020.

[27] T. Wang, Y. Li, W. Xiang, "Design of inter-

val observer for continuous linear large-scale

systems with disturbance attenuation," J. Frank.

Inst., vol 359, no 8, pp. 3910-3929, 2022.

[28] T. Pati, M. Khajenejad, S.P. Daddala, R.G. San-

felice, and S.Z. Yong, "Interval observers for

hybrid dynamical systems with known jump

times," Proc. 62nd IEEE Conf. Decision and

Control CDC 2023, Singapore, Singapore, pp.

7507–7513, 2023.

[29] N. Otsuka, D. Kakehi, "Interval switched pos-

itive observers for continuous-time switched

positive systems under arbitrary switching,"

IFAC-PapersOnLine, vol. 52, no. 11, pp.

250–255, 2019.

[30] N. Otsuka, D. Kakehi, and P. Ignaciuk, "Inter-

val reduced-order switched positive observers

for uncertain switched positive linear systems,"

Int. J. Control Autom. Syst., vol. 24, no. 4, pp.

1105–1115, 2024.

[31] X. Wang and G.H. Yang, "Distributed reliable

H∞consensus control for a class of multi-agent

systems under switching networks. A topology-

based average dwell time approach," Int. J.

Robust Nonlinear Control, vol. 26, no. 13, pp.

2767–2787, 2016.

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 MATHEMATICS

DOI: 10.37394/23206.2024.23.60

Dušan Krokavec

E-ISSN: 2224-2880

578

Volume 23, 2024