Ostensible Metzler Linear Uncertain Systems: Goals, LMI Synthesis,

Constraints and Quadratic Stability

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 deals with the design problem for a class of linear continuous systems with dynamics

prescribed by the system matrix of an ostensible Metzler structure. The novelty of the proposed solution lies in

the diagonal stabilization of the system, which uses the idea of decomposition of the ostensible Metzler matrix,

preserving the incomplete positivity of the system during the synthesis. The proposed approach creates a uniﬁed

framework that covers compactness of interval system parameter representation, Metzler parametric constraints,

and quadratic stability. Combining these extensions, all of the conditions and constraints are expressed as linear

matrix inequalities. Implications of the results, both for design and for research directions that follow from the

proposed method, are discussed at the end of the paper. The efﬁciency of the method is illustrated by a numerical

example.

Key-Words: - positive and incomplete positive systems, strictly Metzler systems, ostensible Metzler matrices,

state feedback, interval state observers, linear matrix inequalities.

Received: November 9, 2022. Revised: July 11, 2023. Accepted: August 12, 2023. Published: September 7, 2023.

1 Introduction

Linear time invariant systems offer many properties

for their adaptation to speciﬁc control problems and

give potential conditions under which they will be-

have in a predetermined manner. Since in practice

there are often requirements for the positivity of sys-

tem states, [1], [2], the synthesis of their control must

take into account such state restrictions. However,

the speciﬁc nature of the problem results in a design

procedure tolerating the system positiveness by ad-

ditional constraints, [3], [4], when focusing on the

dynamical systems. Concerning on a class of linear

dynamical systems with positive states, the Metzler

matrix theory, [5], due to its particular structure, pro-

vides an alternative solution in the analysis and syn-

thesis of linear positive systems, [6]. Setting the Ja-

cobian matrix to a Metzler structure for cooperative

systems, this property stays a key candidate for the

use in interval observers, [7], [8].

Because of the strong nonnegative property, there

are remarkable impacts that are valid only for lin-

ear positive systems. Above all, unlike general linear

systems, the positive systems asymptotic or quadratic

stability have to be lossless reﬂected by considering

linear matrix inequality (LMI) principle, using pos-

itive deﬁnite diagonal matrices, [9]. These particu-

lar forms simplify stabilization analysis and allow in

the same vein the design of structured and decentral-

ized controllers and observers, [10]. A strictly LMI-

based approach for design under Metzler constraints,

reﬂecting the diagonal stabilization principle (DSP),

is given in [11]. Motivated by the problem of incom-

plete positive observation and control design, addi-

tional insights into the analysis with Metzler paramet-

ric constraints is provided in [12], [13].

The LMIs compatibility in design of ostensible

Metzler systems is presented in the paper, summa-

rizing an algorithmic platform with relationships to

system stability, incomplete internal positivity and

the ostensible Metzler parametric constraints. Instead

of using algebraic techniques, the approach is based

on the Lyapunov matrix inequality and diagonal ma-

trix variables, when constructing LMIs for an equiva-

lent ostensible Metzler system matrix representation.

Generalized duality in the controller and observer de-

sign task is proven to be modiﬁable for uncertain li-

near incomplete positive systems with interval osten-

sible Metzler parameters. To the best of the author’s

knowledge, this approach represents a new LMI syn-

thesis method for this class of systems.

Online: In Section 2 the parametrisation princi-

ples of positive systems is outlined and in Section

3 short characterization of systems with ostensible

Metzler system matrices is presented. Design con-

ditions to ostensible interval observer synthesis are

derived in Section 4, while the approach is illustrated

by usage to a model with interval ostensible Metzler

system matrices in Section 5. The presented approach

and the application example are ﬁnally discussed in

Section 6.

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Notations: Throughout the paper xT,XTde-

notes the transpose of the vector x, and the matrix X,

respectively, diag [·]marks a (block) diagonal ma-

trix, for a square symmetric matrix X≺0means

that Xis negative deﬁnite matrix, Inlabels the n-

th order unit matrix, X−1,ρ(X)signiﬁes the inverse

and the eigenvalue spectrum of a square matrix X,

the maximum real part of the eigenvalues of Xis de-

noted by λ0:= maxλ∈ρ(X)ℜ(λ), indicator R(R+)

marks the set of (nonnegative) real numbers, Rn×n

(Rn×n

+) refers to the set of (nonnegative) real matrices

and Mn×n

−+indicates the set of matrices with Metzler

structure, for any matrix X={xij } ∈ Rn×n

+then

X>0,(X≥0) denotes that xij >0 (xij ≥0) for

all i, j.

2 Internal System Positivity

This section considers a class linear time-invariant

systems described by:

Ξ:˙

q(t) = Aq(t) + Bu +Dd(t)

y(t) = Cq(t)(1)

where q(t)∈Rn

+,y(t)∈Rm

+are positive, d(t)∈

+is positive and bounded, B∈Rn×r

+,C∈Rm×n

D∈Rn×d

+are nonnegative and A∈Mn×n

−+is Met-

zler. Then Ξis noted as internally positive.

A strictly Metzler matrix A∈Mn×n

−+is charac-

terized by its negative diagonal elements and by its

strictly positive off diagonal elements. Consequently,

a strictly Metzler Ais so limited by n2parametric

constraints

aii <0, aij >0, i =j, ∀i, j ∈ ⟨1, n⟩(2)

and, to guaranty (2) in design, DSPs have to be used.

Theorem 1 (Metzler matrix parametrisation, [11]).

Let A∈Mn×n

−+be strictly Metzler, then the following

two expressions for parametrization are equivalent.

(i)A=n−1



h=0

A(ν, ν +h)LhT

(ii)A=n−1



h=0

LhA(ν+h, ν)

(3)

where for A={aij } ∈ Mn×n

−+,h= 0, . . . , n −1,

AΘ(ν, ν +h) =

diag [a1,1+h· · · an−h,n an−h+1,1· · · an,h](4)

AΘ(ν+h, ν) =

diag [a1+h,1· · · an,n−ha1,n−h+1· · · ah,n](5)

L=0T1

In−10(6)

L∈Rn×nis the circulant permutation matrix, [14].

Remark 1 If a strictly Metzler A∈Mn×n

−+is repre-

sented in the following circulant rhombic forms

(i)AΘ=





−a11 a12 · · · a1n

−a22 · · · a2na21

....

−ann an1· · · an,n−1







(ii)AΘ=







a11

a21 a22

....

an1an2· · · ann

a12 · · · a1n

....

an−1,n





(7)

then diagonals of (7(i)), (7(ii)) imply the diagonal

matrix structure (4), (5), respectively.

Evidently, (2) can be interpreted as

(i)A(ν, ν +h)≺0, h = 0

A(ν, ν +h)≻0, h = 1, . . . , n −1

(ii)A(ν, ν +h)≺0, h = 0

A(ν+h, ν)≻0, h = 1, . . . , n −1

(8)

or, equivalently,

(i)LhA(ν, ν +h)LhT≺0, h = 0

LhA(ν, ν +h)LhT≻0, h = 1, . . . , n −1

(ii)LhA(ν, ν +h)LhT≺0, h = 0

LhA(ν+h, ν)LhT≻0, h = 1, . . . , n −1

(9)

since for any Z=diag [z1z2· · · zn],Z∈Rn×n

+, the

following circular shift operation

Ldiag [z1· · · zn]LT=diag [znz1· · · zn−1](10)

doesn’t change its deﬁniteness.

Theorem 2 ,[15]. Consider the autonomous closed

loop control structure

q(t) = (A−BK)q(t) = Acq(t)(11)

where A∈Mn×n

−+,B∈Rn×r

+,K∈Rr×n

Using the set of diagonal matrices AΘ(ν, ν +h)∈

Rn×n

+, as well as Bl∈Rn×n

+,Kl∈Rn×n

+,Khl =

LhTKlLh∈Rn×n

+constructed such that

K=









,B= [b1· · · br](12)

Kl=diag kT

l=diag [kl1· · · kln]

Bl=diag [bl] = diag [bl1· · · bln](13)

then the closed-loop system matrix Ac=A−BK ∈

Mn×n

−+can be parameterized as

Ac=

n−1



h=0 AΘ(ν, ν +h)−



l=0

BlKlhLhT(14)

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Theorem 3 ,[16]. Consider the autonomous ob-

server dynamics

e(t) = (A−JC)e(t) = Aee(t)(15)

where A∈Mn×n

−+,C∈Rm×n

+,J∈Rn×m

Using the set of diagonal matrices AΘ(ν+h, ν)∈

Rn×n

+, as well as Cl∈Rn×n

+,Jl∈Rn×n

+,Jlh =

LhTJlLh∈Rn×n

+constructed such that

J= [j1· · · jm],C=





(16)

Jl=diag [jl] = diag [jl1· · · jln]

Cl=diag cT

l=diag [cl1· · · cln](17)

then the observer matrix Ae=A−JC ∈Mn×n

−+

can be parameterized as

Ae=

n−1



h=0

LhAi(ν+h, ν)−



l=0

JlhCl(18)

To generalize the above results, the parameteriza-

tions duality can be found.

Theorem 4 Used parameterisations (14) and (18)

are dual of each other.

Proof: The proof is by direct veriﬁcation. For

parametrisation (18), the matrix transpose implies



n−1



h=0

LhAΘ(ν+h, ν)−



l=0

JlhClT

n−1



h=0 AT

Θ(ν+h, ν)−



l=0

lJT

lhLhT

(19)

Constructing by (ii)AΘfrom (7) a rhombic ma-

trix (ii)AT

Θfor the transposed matrix AT, then

(ii)AT

Θ=







a11

a12 a22

....

a1na2n· · · ann

a21 · · · an1

....

an,n−1







(20)

and, evidently, for h= 0, . . . , n −1

Θ(ν+h, ν) = AΘ(ν, ν +h)(21)

Thus, with these changes of diagonal matrices

Θ(ν+h, ν)⇐AΘ(ν, ν +h)

l⇐Bl,JT

lh ⇐Klh

(22)

it can get the equivalent dual formulation, which is

the same as (14). This concludes the proof.

All presented theoretical coberings result from the

principle of diagonal stabilization of positive Metzler

systems. The interested readers can ﬁnd more details

on this topic in the papers, [11], [15], [16].

3 Incomplete System Positivity

The system is described by (1), where B∈Rn×r

C∈Rm×n

+,D∈Rn×d

+are nonnegative matrices

but A∈Mn×n

−⊙ is ostensible Metzler. Since Ais

not strictly Metzler, such a system Ξis not internally

positive, which means that only some of its state vari-

ables are non-negative.

Deﬁnition 1 The matrix A∈Mn×n

−⊖ is ostensible

Metzler if all diagonal elements of Aare negative,

at least one of the off-diagonal of elements is nega-

tive, while the number of non-negative off-diagonal

elements Ais predominant.

When working with the system Ξ(1), covering the

ostensible Metzler matrix A∈Mn×n

−⊖ , the proposed

idea of the parametrization is to separate Aso that

A=Ap+Am,A={aij }n

i,j=1 (23)

where Ap∈Mn×n

−+is strictly Metzler and Am∈

Rn×n

−is element-wise negative and Hurwitz. The ba-

sic implication is related to the next theorem.

Theorem 5 ,[17]. If for X,Y∈Rn×nit can be set

Y=cX+dInwith scalars c, d ∈R,c= 0 and

In∈Rn×n, then eigenvalues ηk, k = 1, . . . , n of Y

are

ηk=cλk+d(24)

where λkruns over the eigenvalues of Xand the

eigenvectors of Xand Yare identical.

Theorem 6 ,[12]. If there are positive scalars η, δ ∈

R+such that for ostensible Metzler A∈Mn×n

−⊖

there exists a strictly Metzler Ap∈Mn×n

−+satisfying

(23) as well as an element-wise negative and Hurwitz

Am∈Rn×n

−thus hold

Ap=Ad+A++ηΣ+pIn=A◦

p+pIn(25)

Am=A◦

m−pIn,A◦

m=A−−ηΣ(26)

λo= max

kλ+

k|λ+

k=real(λk)>0(27)

Ad=diag [−a11 −a22 · · · − ann](28)

Σ=





0 1 · · · 1 1

1 0 · · · 1 1

.....

1 1 · · · 1 0







λk∈ρ(A◦

p=λo+δ

Ad+pIn≺0

(29)

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







(30)

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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







(31)

ij =aij if aij >0

0if aij <0a−

ij =aij if aij <0

0if aij >0(32)

applying for all i, j ∈ ⟨1, n⟩, i =j, where ρ(A◦

m)is

the spectrum of eigenvalues of A◦

It can be noted that the positive system rep-

resented by Ap∈Mn×n

−+,B∈Rn×r

+,C∈

Rm×n

+,D∈Rn×d

+, can be used in a design with

Metzler parametric constraints, parametrizing analo-

gously Ap∈Mn×n

−+by its rhombic representation as

presented above and using the principle of duality in

relation to speciﬁc design tasks.

If q(0) and an ostensible Metzler A∈Mn×n

−⊖ are

intervally given as, [18],

0≤q(0) ≤q(0) ≤q(0) ,A≤A≤A(33)

decoupling of A,A∈Mn×n

−⊖ ,A=aij n

i,j=1,A=

{aij }n

i,j=1 has to be done that

A=Ap+Am,A=Ap+Am(34)

where Ap,Ap∈Mn×n

−+,Am,Am∈Rn×n

−.

The parametrization for ostensible interval Met-

zler systems can be generalized as follows:

Corollary 1 If there are positive scalars η, η, δ, δ ∈

R+such that for the ostensible Metzler A,A∈

Mn×n

−⊖ there exist strictly Metzler Ap,Ap∈Mn×n

−+

satisfying (34) as well as element-wise negative and

Hurwitz Am,Am∈Rn×n

−thus hold

Ap=Ad+A++ηΣ+pIn=A◦

p+pIn

Ap=Ad+A++ηΣ+pIn=A◦

p+pIn

(35)

Am=A−−ηΣ−pIn=A◦

m−pIn

Am=A−−ηΣ−pIn=A◦

m−pIn

(36)

λo= maxk(λ+

k|λ+

k=real(λk)>0)

λo= maxk(λ+

k|λ+

k=real(λk)>0) (37)

Ad=diag [−a11 −a22 · · · − ann]

Ad=diag [−a11 −a22 · · · − ann](38)

λk∈ρ(A◦

m)λk∈ρ(A◦

p=λo+δ p =λo+δ

Ad+pIn≺0Ad+pIn≺0

(39)

where Σis from (29) and A+,A+,A−,A−are con-

structed as in (30)–(32).

4 Ostensible Interval Observer

Using all the interval system parameters listed above,

in relation to the system input and output related data,

the interval observer equations are

qe(t) = Aeqe(t) + Bu(t) + Jy(t)

qe(t) = Aeqe(t) + Bu(t) + Jy(t)(40)

where qe(t)∈Rn,qe(t)∈Rnare respectively

the lower and the upper system state vector esti-

mates. Thus, using the observer parameters in (40)

J∈Rn×n

+,Ae,Ae∈Mn×n

+⊙with connection to sys-

tem (1) it evident that

Ae=A−JC,Ae=A−JC (41)

y(t) = Cq(t),ye(t) = Cqe(t)

y(t) = Cq(t),ye(t) = Cqe(t)(42)

Using the observation errors, [21],

e(t) = q(t)−qe(t),e(t) = q(t)−qe(t)(43)

and substituting the system and observer equations

into (43) it follows that

e(t) = Apee(t) + Ame(t) + Dd(t)

e(t) = Apee(t) + Ame(t) + Dd(t)(44)

when constructing

Ape =Ap−JC,Ape =Ap−JC

Ae=Ape +Am,Ae=Ape +Am

(45)

which predeﬁne the conditions for interval observer

quadratic stability.

It is need to impose Ape,Ape ∈Mn×n

−+to be

strictly Metzler and Hurwitz as well as to impose

Am,Am∈Rn×n

−to be element-wise negative and

Hurwitz when implementing for ostensible Metzler

lower and upper matrices A=Ap+Am∈Mn×n

−⊖ ,

A=Ap+Am∈Mn×n

−⊖ , whilst the matrices

Ae,Ae∈Mn×n

−⊖ need to be ostensible Metzler and

Hurwitz.

Note that in both cases, the necessity of the system

matrix separation approach has to be preserved.

Using the equivalent procedure for the system

parametrization, then

Ape =n−1



h=0

LhAp(ν+h, ν)−r



j=0

JjhCj

Ape =n−1



h=0

LhAp(ν+h, ν)−r



j=0

JjhCj

(46)

when applying appropriately the rhombic diagonal

principle. These representations are captured by gen-

eralization of inequalities (9).

A statement of ostensible Metzler interval ob-

server design procedure is provided by the following

theorem.

Theorem 7 The matrices Aep ,Aep ∈Rn×n

−+are

strictly Metzler and Hurwitz and the matrices Ae,

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Ae∈Rn×n

−⊕ are ostensible Metzler and Hurwitz if

for ostensible Metzler A=Ap+Am∈Rn×n

−⊕ ,

A=Ap+Am∈Rn×n

−⊕ and non-negative C∈

Rm×n

+there exist positive deﬁnite diagonal matrices

P,Rl∈Rn×n

+and positive scalars µ, µ ∈R+such

that for h= 1, . . . , n −1,lT= [1· · · 1]∈Rn

P≻0,Rk≻0(47)





Π∗ ∗

P−µId∗

C0−µIm

≺0,



Π∗ ∗

P−µId∗

C0−µIm

≺0

(48)

P Ap(ν, ν)−m



l=1

RlCl≺0

P Ap(ν, ν)−m



l=1

RlCl≺0

(49)

P LhAp(l+h, l)LhT−m



l=1

RlLhClLhT≻0

P LhAp(l+h, l)LhT−m



l=1

RlLhClLhT≻0

(50)

Π=P Ap+AT

pP+P Am+AT

mP−

−



l=1

(RlllTCl+ClllTRl)

Π=P Ap+AT

pP+P Am+AT

mP−

−



k=1

(RkllTCk+CkllTRk)

(51)

A feasible task for J∈Rn×m

+implies

Jl=P−1Rl,jl=Jll,J= [j1· · · jm](52)

Hereafter, ∗is the symmetric item in a symmetric

matrix.

Proof: Choosing Lyapunov function in the following

form

v(e(t)) = eT(t)P e(t)−η

∫

dT(τ)d(τ)dτ+

+η−1t

∫

y(τ)ey(τ)dτ

(53)

where P∈Rn×m

+is a diagonal positive deﬁnite ma-

trix and η∈R+is a positive scalar, then along all

stable lover errors

˙v(e(t)) = ˙

eT(t)P e(t) + eT(t)P˙

e(t)+

+η−1eT

y(t)ey(t)−ηdT(t)d(t)

(54)

and applying in (54) the observer error dynamics it

gets to

˙v(e(t))

=eT(t)(AT

eP+P Ae)e(t)+

+eT(t)P Dd(t) + dT(t)DT

P e(t))+

+µ−1eT(t)CTCe(t)−µdT(t)d(t)

(55)

The equality (55) can be compactly written con-

structing a common notation ed(t)as follows

ed(t) = eT(t)dT(t)(56)

then there is a reasonable ground to conclude that the

following have to yield

˙v(eed(t)) = eT

ed(t)Π◦eed(t)<0(57)

where, evidently,

Π◦=AT

eP+P Ae+µ−1CTC P D

DTP−µIrd≺0(58)

After applying the property of Schur complement

with relation to the element µ−1CTCthen





P Ae+AT

eP∗ ∗

DTP−µId∗

C0−µIm



≺0(59)

and it can be set

P Ape =P(Ap−JC)

=P Ap−



k=1

P jkcT

=P Ap−



k=1

P JkllTCk

(60)

where vector l∈Rn

+is used to uncover the diagonal

matrix structures.

Thus, (59) implies (48), (51) when using the sub-

stitutions

P Jk=Rk,Ae=Ape +Am(61)

According to the parametrization, pre-multiplying

the left side by Pand post-multiplying the right side

by LhTthen, with Jlh =LhTJlLh∈Rn×n

+, (9),

(46) gives

n−1



h=0 P LhAp(ν+h, ν)LhT−



l=0

P JlLhClLhT

(62)

and using (61) then (62) implies for h= 0 the lower

part of (49) and for h > 0(62) implies the lower part

of (50).

Since analogously can be set the LMIs working on

e(t), this closes the proof.

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5 Illustrative Example

The considered system (1), (2) is built on the param-

eters of linearized dynamic model of F-404 engine,

[19], with A≤A

A=



−1.4600 0 2.4280

−0.8357 −2.4−0.3788

0.3107 0 −2.1300

,D=



1



A=



−1.4600 0 2.4280

−0.3357 −1.4−0.3788

0.3107 0 −2.1300

,CT=



1 0

0 0

0 1



B=



0.4182 5.2030

0.3901 −0.1245

0.5186 0.0236



and the derived design parameters are

Ad=diag [−1.46 −2.4−2.13]

Ad=diag [−1.46 −1.4−2.13]

A+=A+=



0 0 2.4280

0 0 0

0.3107 0 0 

,l=



1



A−=



0 0 0

−0.8357 0 −0.3788

0 0 0 

,Σ=



0 1 1

1 0 1

1 1 0



A−=



0 0 0

−0.3357 0 −0.3788

0 0 0 

,L=



0 0 1

1 0 0

0 1 0



With A◦

m=A−−ηΣ,A◦

m=A−−ηΣ, where

η= 0.005 then

A◦

m=



0−0.0050 −0.0050

−0.8407 0 −0.3838

−0.0050 −0.0050 0 



A◦

m=



0−0.0050 −0.0050

−0.3407 0 −0.3838

−0.0050 −0.0050 0 



ρ(A◦

m) = {−0.0808 0.0150 0.0758}

ρ(A◦

m) = {−0.0627 0.0050 0.0577}

and λ0= 0.0758,λ0= 0.0577.

To deﬁne D-stability regions it is set δ= 0.003,

δ= 0.03, which deﬁne p=λ0+δ,p=λ0+δthe

Hurwitz matrices Am=A◦

m−pIn,Am=A◦

m−pIn

and the strictly Metzler Ap=Ad+A++ηΣ+pIn,

Ap=Ad+A++ηΣ+pInso that Ap≤Ap,

Am=



−0.0788 −0.0050 −0.0050

−0.8407 −0.0788 −0.3838

−0.0050 −0.0050 −0.0788



Am=



−0.0807 −0.0050 −0.0050

−0.3407 −0.0807 −0.3838

−0.0050 −0.0050 −0.0777



ρ(Am) = {−0.1596 −0.0738 −0.0030}

ρ(Am) = {−0.1504 −0.0827 −0.0300}

Ap=



−1.3812 0.0050 2.4330

0.0050 −2.3212 0.0050

0.3157 0.0050 −2.1512



Ap=



−1.3723 0.0050 2.4330

0.0050 −1.3123 0.0050

0.3157 0.0050 −2.1423



Using Ap,Apit can be found that the matrix vari-

ables, which provide a solution by SeDuMi, [20], are

P=diag [2.2680 3.3453 2.4098],

R1=diag [1.4454 0.0062 0.2474], γ = 4.2335

R2=diag [2.8826 0.0060 1.1582], γ = 4.4557

J=



0.6373 1.2710

0.0018 0.0018

0.1027 0.4806



These infuse the strictly Metzler and Hurwitz ma-

trices Ape =Ap−JC,Ape =Ap−JC and

the ostensible Metzler and Hurwitz matrices Ae=

A−JC,Ae=A−JC

Ape=



−2.0185 0.0050 1.1620

0.0032 −2.3212 0.0032

0.2130 0.0050 −2.6318



Ape=



−2.0096 0.0050 1.1620

0.0032 −1.3123 0.0032

0.2130 0.0050 −2.6229



ρ(Ape) = {−1.7406 −2.3213 −2.9096}

ρ(Ape) = {−1.3122 −1.7319 −2.9007}

Ae=



−2.0973 0 1.1570

−0.8375 −2.4−0.3806

0.2080 0 −2.6106



Ae=



−2.0973 0 1.1570

−0.3375 −1.4−0.3806

0.2080 0 −2.6106



ρ(Ae) = {−1.8003 −2.4000 −2.9076}

ρ(Ae) = {−1.4000 −1.8003 −2.9076}

Note, the positions of negative off-diagonal ele-

ments in (A,Ae),(A,Ae)are preserved.

Simulating the deﬁned uncertain system within

constraints A≤A≤A,qe(0) ≤q(0) ≤qe(0),

σ2

d= 0.012, where

A=



−1.4600 0 2.4280

−0.5857 −1.9−0.3788

0.3107 0 −2.1300

,q(0) = 



0.250

3.750

0.025



and utilizing the system forced mode

u(t) = W w(t),W=2 0

0 1,w(t) = 0.352

0.076

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0 1 2 3 4 5 6 7 8

t [s]

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.1

q1(t)

q1L(t)

q1U(t)

Figure 1: Convergence of the ﬁrst state variable

simulation results with the initial conditions

q(0) = 



0.25

3.75

0.02

,qe(0) = 



0.15

0.00

0.00

,qe(0) = 



0.4

0.0

0.1



are given in Figure 1. These simulation results show

the performance of the proposed interval observer,

where the black color curved line denotes the ﬁrst

state variable trajectory and the blue and red curved

lines denote its upper and lower estimations. Since

the ﬁrst state variable is positive, it can be seen that

its behavior is correctly intervally estimated, guaran-

teing exponential convergence of the state variable

estimation error.

6 Concluding Remarks

The main objectives in this paper are parameterisa-

tions approaches in design for ostensible Metzler sys-

tems with interval-speciﬁed dynamics and bounded

system disturbances. The diagonal matrix variables

and the proposed LMI structures reﬂect the key idea

to obtain auxiliary Metzler and Hurwitz matrix struc-

tures of Ap,Apwhile Lyapunov function and the re-

lated LMIs form the base of the quadratic stability.

Despite the design conditions complexity, state es-

timation using ostensible Metzler interval observers

is robust to the changes covered in plant dynamics

by given interval bounds, taking into account that the

positivity of the lower positive state estimation need

to be keep.

In the synthesis it is simple to deﬁne sequentially

different D-regions of stability for Am,Amby using

the parameters p, p > 0and so to guaranty conse-

quently that Ap≤Ap, when forcing interval bounds

on positive state variables, as well as to ﬁnd via LMIs

the acceptable rate of convergence of estimation er-

rors. Although these tasks are parametrical depen-

dent, their interactive predeﬁnition is possible as a

rule.

Since scalar variables µ,µare related to the sys-

tem dependency on the parameters and can be tuned

in LMIs, they can be used for attenuation when guar-

anteing interval observer quadratic stability under un-

known disturbance.

The approach certainly requires further investiga-

tion the ostensible Metzler system matrices in depen-

dence on the un-structural set of negative off-diagonal

elements. It is worth highlighting that there are also

many unexplored theoretical and applied aspects of

problems in systems with non-strictly Metzler matrix

structures and ostensible Metzler matrix representa-

tion in control, e.g., of drones and unmanned aerial

vehicles, [22], [23], or constructing Metzler matrix

representations to match the properties of the interval

observers, [24].

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] K.J. Arrow, "The genesis of dynamic systems

governed by Metzler matrices," in Mathematical

Economics and Game Theory. Springer, Berlin,

1977.

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

Systems with Positive Characteristics. Singa-

pore: Springer Nature, 2017.

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

negative Matrices in Dynamic Systems. New

York: John Wiley & Sons, 1989.

[6] C. Briat, "Robust stability and stabilization of

uncertain linear positive systems via integral lin-

ear constraints: L1-gain and L∞-gain character-

ization," Int. J. Robust Nonlinear Control, vol.

23, no. 17, pp. 1932–1954, 2013.

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

"Interval observers for uncertain biological sys-

tems," Ecological modelling, vol. 133, no. 1,

pp. 45–56, 2000.

[8] E. Chambon, P. Apkarian, L. Burlion, "Metzler

matrix transform determination using a nons-

mooth optimization technique with an applica-

tion to interval observers," Proc. Conference

on Control and its Applications CT’15, Paris,

France, 205–211, 2015.

[9] G.P. Barker, A. Berman, R.J. Plemmons, "Pos-

itive diagonal solutions to the Lyapunov equa-

tions," Linear Multilinear Algebra, vol. 5, no. 4,

pp. 249–256, 1978.

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

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

vol. 61, no. 1, pp. 6-10, 2012.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

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Volume 18, 2023

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

ciples in strictly Metzlerian systems control de-

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

2018.

[12] D. Krokavec and A. Filasová, "State con-

trol of linear systems with potentially Met-

zler dynamics," in CONTROLO 2022, Lec-

ture Notes in Electrical Engineering, vol. 930,

Cham: Springer Nature) 689–701, 2022.

[13] D. Krokavec and A. Filasová, "Observer-ba-

sed control design for systems with potentially

Metzler dynamics," Prep. 22nd IFAC World

Congress, Yokohama, Japan, pp. 5550–5555,

2023.

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

New York: Cambridge University Press, 2013.

[15] D. Krokavec and A. Filasová, "Control de-

sign for linear strictly Metzlerian descriptor sys-

tems," Proc. 18th EUCA European Control

Conference ECC ’20, Saint-Petersburg, Russia,

pp. 2092–2097, 2020.

[16] D. Krokavec and A. Filasová, "Interval obser-

ver design for uncertain linear continuous-time

Metzlerian systems," Proc. 28th Mediterranean

Conference on Control and Automation MED

’20, Saint-Raphaël, France, pp. 1051–1056,

2020.

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

Analysis. New York: Springer, 2007.

[18] L. Jaulin, M. Kieffer, O. Didrit, K. Walter, Ap-

plied Interval Analysis with Examples in Pa-

rameter and State Estimation, Robust Control

and Robotics. London: Springer-Verlag, 2001.

[19] W.H. Kwon, P.S. Kim, P.G. Park, "A receding

horizon Kalman FIR ﬁlter for linear continuous-

time systems," IEEE Trans. Autom. Control,

vol. 44, mno. 11, pp. 2115-2120, 1999.

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

User’s Guide for SeDuMi Interface, Toulouse:

LAAS-CNRS, 2002.

[21] F. Mazenc and O. Bernard, "Interval obser-

vers for linear time-invariant systems with dis-

turbances," Automatica, vol. 47, no. 1, pp. 140–

147, 2011.

[22] K. Yong, "Disturbance interval observer-based

carrier landing control of unmanned aerial ve-

hicles using prescribed performance," Scientia

Sinica Informationis, vol.52, no. 9, pp. 1711–

1726, 2022.

[23] Y. Song, K. Yong, X. Wang, "Disturbance in-

terval observer-based robust constrained control

for unmanned aerial vehicle path following,"

Drones, vol. 7, no. 2, pp. 1–21, 2023.

[24] Y. Tian, K. Zhang, B. Jiang, X.G. Yan, "Inter-

val observer and unknown input observer-based

sensor fault estimation for high-speed railway

traction motor," J. Frankl. Inst., vol. 357, no 2,

pp. 1137–1154, 2020.

Acknowledgement

The work presented in this paper was supported by

VEGA, the Grant Agency of the Ministry of Educa-

tion and Academy of Science of Slovak Republic, un-

der Grant No. 1/0483/21. This support is very grate-

fully acknowledged.

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

Conflict of Interest

The author has no conflict 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

Dušan Krokavec newly addressed the duality prin-

ciple and the incidence of ostensible Metzler ma-

trix separation as well as LMIs for interval observer

quadratic stability and converted these tasks to an

LMI problem. The author has read and agreed to the

proposed version of the manuscript.

The work presented in this paper was supported by

VEGA, the Grant Agency of the Ministry of Educa-

tion and Academy of Science of Slovak Republic, un-

der Grant No. 1/0483/21. This support is very grate-

fully acknowledged.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.25

Dušan Krokavec

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262

Volume 18, 2023