339
1 INTRODUCTION
In positive systems inputs, state variables and outputs
take only nonnegative values for any nonnegative
inputs and nonnegative initial conditions [4, 1, 9].
Examples of positive systems are industrial processes
involving chemical reactors, heat exchangers and
distillation columns, storage systems, compartmental
systems, water and atmospheric pollutions models. A
variety of models having positive behaviour can be
found in engineering, management science, economics,
social sciences, biology and medicine, etc. An overview
of state of the art in positive systems theory is given in
the monographs [1, 4, 9, 13, 17].
Mathematical fundamentals of the fractional
calculus are given in the monographs [21, 22, 13, 17].
The positive fractional linear systems have been
investigated in [3, 5, 7, 10-14, 17, 20, 23-25]. Positive
linear systems with different fractional orders have
been addressed in [10, 11, 27]. Descriptor positive
systems have been analyzed in [2, 12] and their
stabilization in [26, 27]. Linear positive electrical
circuits with state feedbacks have been addressed in [2,
17]. The superstabilization of positive linear electrical
circuits by state feedbacks have been analyzed in [15]
and the stability of nonlinear systems in [16, 18]. The
global stability of nonlinear systems with negative
feedbacks and positive not necessary asymptotically
stable linear parts has been investigated in [6, 8]. The
global stability of nonlinear standard and fractional
positive feedback systems has been considered in [15].
In this paper the global stability of nonlinear
fractional orders feedback multi-input multi-output
systems with interval matrices of positive linear parts
will be addressed.
The paper is organized as follows. In section 2 the
basic definitions and theorems concerning the positive
different fractional orders linear systems are recalled.
The stability of fractional interval positive linear
systems is analyzed in section 3. New sufficient
conditions for the global stability of these feedback
nonlinear systems with interval matrices of positive
linear parts are established in section 4. In section 5 a
procedure for calculation of gain matrix
characterizing the class of nonlinear element is
presented and illustrated by numerical examples.
Concluding remarks are given in section 6.
The following notation will be used:
- the set of
real numbers,
nm
- the set of nm real matrices,
Global Stability of Different Fractional Orders Multi
Input Multi Output Nonlinear Feedback Systems with
Interval Matrices of Positive Linear Parts
T. Kaczorek & Ł. Sajewski
Bialystok University of Technology, Białystok, Poland
ABSTRACT: The global stability of continuous-time fractional orders multi–input multi-output nonlinear
feedback systems with interval matrices of positive linear parts is investigated. New sufficient conditions for the
global stability of these class of positive nonlinear systems are established. A procedure for computation of gain
matrix characterizing the class of nonlinear elements is given and illustrated on simple example.
http://www.transnav.eu
the International Journal
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Volume 19
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March 2025
DOI: 10.12716/1001.19.01.40
340
nm
+
- the set of nm real matrices with nonnegative
entries and
1nn
++
=
, Mn - the set of nn Metzler
matrices (real matrices with nonnegative off-diagonal
entries), In - the nn identity matrix,
(:, )An
- sum of
all elements of nth column.
2 POSITIVE DIFFERENT FRACTIONAL ORDERS
LINEAR SYSTEMS
Consider the fractional continuous-time linear
system
()
( ) ( ), 0 1
d x t
Ax t Bu t
dt
= +
(1a)
( ) ( ) ( )y t Cx t Du t=+
, (1b)
where
()
n
xt
,
()
m
ut
,
()
p
yt
are the state,
input and output vectors and
nn
A
,
mn
B
,
np
C
,
mp
D
,
0
( ) 1 ( ) ( )
, ( )
(1 )
()
t
d x t x dx
dx
d
dt t
==
−
−
(1c)
is the Caputo fractional derivative and
1
0
( ) , Re( ) 0
zt
z t e dt z
−−
=
(1d)
is the gamma function [13, 18].
Definition 1. [13, 18] The fractional system (1) is
called (internally) positive if
()
n
xt
+
and
()
p
yt
+
,
0t
for any initial conditions
(0)
n
x
+
and all inputs
()
m
ut
+
,
0t
.
Theorem 1. [13, 18] The fractional system (1) is
positive if and only if
, , ,
n m p n p m
n
A M B C D
+ + +
(2)
Definition 2. [13, 18] The positive fractional system
(1) (for u(t)=0) is called asymptotically stable (the
matrix A is Hurwitz) if
lim ( ) 0
t
xt
→
=
for any
(0)
n
x
+
. (3)
Theorem 2. [13, 18] The fractional positive linear
system (1) is asymptotically stable if and only if one of
the following equivalent conditions is satisfied:
1. All coefficient of the characteristic polynomial
1
1 1 0
( ) det[ ] ...
nn
n n n
p s I s A s a s a s a
−
−
= − = + + + +
(4)
are positive, i.e.
0
i
a
for
0,1,..., 1in=−
.
2. There exists strictly positive vector
1
[]
TT
n
=
,
0
k
,
1,...,kn=
such that
0A
or
0
T
A
. (5)
Theorem 3. The fractional positive system (1) is
asymptotically stable if the sum of entries of each
column (row) of the matrix A is negative.
Proof. Using (5) we obtain
11 1 1 11 1
1
11
... 0
... ...
... 0
nn
n
n nn n n nn
a a a a
A
a a a a
= = + +
(6)
and the sum of entries of each column of the matrix A
is negative since
0
k
,
1,...,kn=
. The proof for rows
is similar. □
Now consider the fractional linear system with two
different fractional orders
1
11 12 1 1
21 22 2 2
2
()
()
()
()
()
d x t
A A x t B
dt
ut
A A x t B
d x t
dt
=+
, (7a)
1
12
2
()
( ) [ ]
()
xt
y t C C
xt
=
, (7b)
where
0 , 1
,
1
1
()
n
xt
and
2
2
()
n
xt
are the
state vectors,
ij
nn
ij
A
,
i
nm
i
B
,
i
pn
i
C
; i, j =
1,2;
()
m
ut
is the input vector and
()
p
yt
is the
output vector. Initial conditions for (7a) have the form
10
1 10 2 20 0
20
(0) , (0) and
x
x x x x x
x
= = =
. (7c)
Remark 1. The state equation (7a) of fractional
continuous-time linear systems with two different
fractional orders has similar structure as the 2D
Roeesser type models.
Definition 3. The fractional system (7) is called
positive if
1
1
()
n
xt
and
2
2
()
n
xt
and
()
p
yt
+
,
0t
for any initial conditions
1
10
n
x
+
,
2
20
n
x
+
and all input vectors
m
u
+
,
0t
.
Theorem 4. [10, 11] The fractional system (7) for
10;10
is positive if and only if
11 12 1
21 22 2
1 2 1 2
,,
[ ] ( )
nm
n
pn
A A B
A M B
A A B
C C C n n n
+
+
= =
= = +
(8)
Theorem 5. [13] The fractional positive system (7) is
asymptotically stable if and only if one of the following
equivalent conditions is satisfied:
1. All coefficients of the characteristic polynomial
1
2
11 12
21 22
0
( , ) det
0
n
n
n
Is
AA
p s s
AA
Is
=−
(9)
are positive.
2. There exists strictly positive vector
1
[]
n
=
,
0
k
,
1,...,kn=
such that
0A
or
0
T
A
. (10)
Theorem 6. The fractional positive system (7) is
asymptotically stable if the sum of entries of each
column (row) if the matrix
A
is negative.
341
Proof is similar to the proof of Theorem 3.
Theorem 7. The solution of the equation (7a) for
0 1; 0 1
with initial conditions (7c) has the
form
1
00
2
0
()
( ) ( ) ( ) ( )
()
t
xt
x t t x M t u d
xt
= = + −
, (11)
where
1 10 2 01
1 1 2 2
11 12 11 12
1
1 1 2 2
2
21 22 21 22
1 2 1 2
11 1 12 2 11 12
1 2 1 2
21 1 22 2 21 22
( ) ( ) ( )
( ) ( ) ( ) ( ) 0
0
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
M t t B t B
t t t t
B
B
t t t t
t B t B t t
t B t B t t
= + =
= + =
+
==
+
1
2
B
B
(12a)
and
0
00
()
( 1)
kl
kl
kl
t
tT
kl
+
==
=
+ +
, (12b)
( 1) 1
1
00
()
( 1)
kl
kl
kl
t
tT
kl
+ + −
==
=
+ +
, (12c)
( 1) 1
2
00
()
( 1)
kl
kl
kl
t
tT
kl
+ + −
==
=
+ +
, (12d)
11 12
21 22
10 1, 01 , 1
for 0
for 1, 0
00
00
for 0, 1
for 1
n
kl
k l k l
I k l
AA
kl
T
kl
AA
T T T T k l
−−
==
==
=
==
+ +
(12e)
Proof is given in [11].
Note that if
=
then from (12b) we have
0
0
()
( 1)
kk
k
At
t
k
=
=
=
+
. (13)
3 STABILITY OF FRACTIONAL INTERVAL
POSITIVE LINEAR SYSTEMS
Consider the fractional interval positive linear
continuous-time system
, 0 1,
dx
Ax
dt
=
(14)
where
n
txx = )(
is the state vector and the interval
matrix
n
MA
is defined by
AAA
or equivalently
[ , ]A A A
. (15)
Definition 4. The fractional interval positive system
(14) is called asymptotically stable if the system is
asymptotically stable for all matrices
n
MA
satisfying the condition (15).
By condition (5) of Theorem 2 the positive system
(14) is asymptotically stable if there exists strictly
positive vector
0
such that the condition (5) is
satisfied.
For two fractional positive linear systems
,
n
dx
Ax A M
dt
=
(16a)
and
,
n
dx
Ax A M
dt
=
(16b)
there exists a strictly positive vector
n
+
such that
0A
and
0A
(17)
if and only if the systems (16) are asymptotically stable.
Remark 2. As according to condition 2 of Theorem
2 (Theorem 5), in general case, it is not necessary to find
the same
n
+
vector for two different systems.
However, for interval systems, it is reasonable for
choose one
n
+
vector for (16a) and (16b).
Theorem 8. If the matrices
A
and
A
of fractional
positive systems (16) are asymptotically stable then
their convex linear combination
(1 )A k A kA= − +
for
01k
(18)
is also asymptotically stable.
Proof. By condition (5) of Theorem 2 if the fractional
positive linear systems (16) are asymptotically stable
then there exists strictly positive vector
n
+
such
that (17) holds. Using (5) and (17) we obtain
[(1 ) ] (1 ) 0A k A kA k A kA
= − + = − +
(19)
for
01k
. Therefore, if the positive linear systems
(16) are asymptotically stable then their convex linear
combination (18) is also asymptotically stable. □
Theorem 9. The interval positive system (14) is
asymptotically stable if and only if the positive linear
systems (16) are asymptotically stable.
Proof. By condition (5) of Theorem 2, if the matrices
n
MA
,
n
MA
are asymptotically stable, then there
exists a strictly positive vector
n
+
such that (5)
holds. The convex linear combination (18) satisfies the
condition
0
A
if and only if (19) holds. Therefore,
the interval system (14) is asymptotically stable if and
only if the positive linear system is asymptotically
stable. □
Example 1. Consider the fractional interval positive
linear continuous-time system (14) with the matrices
3 2 2 1
,
2 4 1 3
AA
−−
==
−−
. (20)
342
Using the condition (5) of Theorem 2 we choose
T
]11[=
and we obtain
3 2 1 1
0
2 4 1 2
A
−−
= =
−−
(21a)
and
2 1 1 1
0.
1 3 1 2
A
−−
= =
−−
(21b)
Therefore, the matrices (20) are Hurwitz.
These considerations can be easily extended to
positive different fractional orders linear systems (7).
4 GLOBAL STABILITY OF FRACTIONAL
NONLINEAR POSITIVE FEEDBACK SYSTEMS
Consider the m-input p-output (MIMO) nonlinear
feedback system shown in Figure 1 which consists of
the positive fractional linear part, the nonlinear
element with the matrix characteristic u=f(e) and the
feedback with positive gain matrix H. The positive
fractional linear part is described by the equations (7)
with the interval matrices
[ , ] , [ , ] , [ , ] .
n m p n
n
A A A M B B B C C C
++
(22)
It is assumed that the interval matrix
n
MA
~
is
Hurwitz.
Figure 1. The nonlinear feedback system
The characteristic f(e) of the nonlinear element
satisfies the condition
0 ( )f e Ke
or
,u Ke
(23a)
where
11 1
11
1
,,
p
m m mp p
kk
ue
u K e
u k k e
= = =
(23b)
and
1 1 1
(0) 0, ( ,..., ) , 1, ,
i p i ip p
f u f e e k e k e i m= = + + =
(23c)
Remark 3. The matrix K forms m, p-dimensional
convex surfaces which constricts possible dynamics of
nonlinear elements, i.e. for m = p = 1 the possible
trajectory f(e) of nonlinear element should fit in grey
cones presented in the Figure 2.
Figure 2. Example of possible trajectory f(e) of nonlinear
element constricted by K.
Definition 5. The fractional nonlinear positive
system is called globally stable if it is asymptotically
stable for all nonnegative initial conditions
(0) .
n
x
+
The following theorem gives sufficient conditions
for the global stability of the positive nonlinear system.
Theorem 10. The fractional nonlinear system
consisting of the positive asymptotically stable linear
part described by (7) with interval matrices (22), the
nonlinear element satisfying the condition (23) and the
feedback with positive gain matrix
pm
H
+
is
globally stable if, the sum of entries of each column
(row) of the matrix
for 0
(1 )
for 1
n
n
A BKHC M q
q A qA BKHC
A BKHC M q
+ =
− + + =
+ =
(24)
are negative.
Proof. The proof will be accomplished by the use of
the Lyapunov method [19, 20]. As the Lyapunov
function V(x) we choose
0)()(
22112211
+=+ xxxVxV
TT
for
11
22
,,
nn
x
x
x
++
= =
(25)
where
is strictly positive vector, i.e.
0
ij
,
;2,1=i
.,...,1 nj =
Using (25) and (22) we obtain
1
1 1 2 2
12
2
( ) ( )
[]
( ) ( ( )) ( )
TT
T T T
dx
d V x d V x
dt
dt dt
dx
dt
Ax Bu Ax Bf e A BKHC x
+ = =
+ = + +
(26)
since
.
~
)( xCKHKeefu ==
From (26) it follows that
1 1 2 2
( ) ( )
0
d V x d V x
dt dt
+
if
the sum of entries of each column (row) of the matrix
(24) is negative (Theorem 3) and the nonlinear positive
system is globally stable. □
5 PROCEDURE AND EXAMPLE
To find the
pm
K
+
satisfying the condition (24) for
the nonlinear positive system the following procedure
can be used.
343
Procedure 1.
Step 1. Using the matrices
,,A B C
of the positive
linear system and the matrix H compute the maximum
value of the matrix
K
such that the sum of all entries
of each column (row) of the matrix
A A BKHC=+
(27)
is negative.
Entries of the matrix K can be compute as the
solution of the linear matrix equation
,Gk h=
(28)
where the matrix G and the column vector h are
defined by the sum of entries of each column (row) of
the matrix (27) and vector k contains components of
matrix K. If
mp n
or/and
rank ,Gn
then we choose
arbitrarily
rank mp G−
nonnegative entries of the
matrix K.
Step 2. Using the matrices
,,A B C
of the positive
linear system and the matrix H compute the maximum
value of the matrix
K
such that the sum of all entries
of each column (row) of the matrix
A A BKHC=+
(29)
is negative.
Step 3. Taking into considerations
K
computed in
Step 1 and
K
from Step 2, find the desired
pm
K
+
for which, the matrices
A
and
A
are Hurwitz, i.e.
the characteristic f(e) satisfy the condition (23a).
Remark 4. The conditions of Theorem 2 can be also
used to compute the entries of the matrix K. Usually in
this case the computations are more complicated.
Example 2. Consider the nonlinear feedback system
shown in Figure 1 with the interval matrices of the
positive linear part
15.8 0.2 0.9 7.9 0.1 0.7
0.4 14.1 0.5 , 0.2 7.9 0.3 ,
0.5 0.4 20.1 0.1 0.2 12.1
0.5 0.4 0.4 0.3
0.3 0.5 , 0.2 0.4 ,
0.8 0.6 0.6 0.5
0.4 0.6 0.8 0.3 0.5 0.6
,
0.6 0.4 0.6 0.4 0.3 0.5
AA
BB
CC
−−
= − = −
−−
==
==
(30)
and the gain matrix
21
15
H
=
. (31)
From (30) it follows that m = p = 2 and
1 1 2 11 1 12 2
( , ) ,u f e e k e k e= +
2 1 2 21 1 22 2
( , ) ,u f e e k e k e= +
then we are looking for
11 12
22
21 22
kk
K
kk
+
=
, for which
the nonlinear feedback system is globally stable.
Using Procedure 1 we obtain:
Step 1. Using (27), (30) and (31) we obtain that the
sum of columns are the following:
11 12 21 22
11 12 21 22
11 12 21 22
(:,1) 2.24 5.44 2.1 5.1 14.9,
(:,2) 2.56 4.16 2.4 3.9 13.5,
(:,3) 3.52 6.08 3.3 5.7 18.7
A k k k k
A k k k k
A k k k k
= + + + −
= + + + −
= + + + −
(32)
and taking into consideration (28), we have
11
12
21
22
2.24 5.44 2.1 5.1 14.9
2.56 4.15 2.4 3.9 , , 13.5
3.52 6.08 3.3 5.7 18.7
k
k
G k h
k
k
= = =
. (33)
Since G=2, then we choose
2rank =− Gmp
elements of the vector k (entries of the matrix K) as
k21=1, k22=1 . Solution for (28) with (33) is the following
11
12
21
22
1.309
0.869
1
1
k
k
k
k
=
, (34)
thus for matrix
1.309 0.869
11
K
=
(35)
the system with
,,A B C
is stable.
Step 2. Similarly as in Step 1, using (29), (30) and
(31) we can write the linear matrix equation (28) in the
form
=
1.11
6.7
6.7
72.304.241.387.1
4.256.12.243.1
76.22.153.21.1
22
21
12
11
k
k
k
k
. (36)
Since
1.1 2.53 1.2 2.76
rank 1.43 2.2 1.56 2.4 2
1.87 3.41 2.04 3.72
=
, then we
choose two elements of the vector k as k11=1, k12=1.
Solution for (36) is the following
11
12
21
22
1
1
1.013
1.002
k
k
k
k
=
, (37)
thus for matrix
11
1.013 1.002
K
=
(38)
the system with
,,A B C
is stable.
344
Step 3. Taking under considerations (35) and (38),
we have that for
11
11
K
=
, the matrices
A
and
A
are Hurwitz since
(:,1) 0.01,
(:,2) 0.01,
(:,3) 0.06,
A
A
A
=−
=−
=−
…
(:,1) 0.02,
(:,2) 0.48,
(:,3) 0.1.
A
A
A
=−
=−
=−
(39)
Therefore, for nonlinear element satisfying the
condition
2
1
21
11
11
),(
0
0
e
e
eef
, (40)
the nonlinear feedback system with interval matrices
(30) of positive linear parts and the gain matrix (31) is
globally stable.
6 CONCLUDING REMARKS
The global stability of continuous-time different
fractional orders nonlinear feedback systems with
interval matrices of positive linear parts has been
investigated. New sufficient conditions for the global
stability of this class of positive nonlinear systems are
established (Theorem 10). The procedure for
calculation of gain matrix characterizing the class of
nonlinear element is presented and illustrated by
numerical example. The considerations can be
extended to discrete-time fractional different orders
nonlinear systems with interval matrices of positive
linear parts and scalar feedbacks. An open problem is
an extension of the considerations to nonlinear
different orders fractional systems with interval
matrices of their positive linear parts.
ACKNOWLEDGMENTS
The studies have been carried out in the framework of work
No. WZ/WE-IA/5/2023 and financed from the funds for
science by the Polish Ministry of Science and Higher
Education.
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