333
1 INTRODUCTION
In positive systems inputs, state variables and outputs
take only nonnegative values. 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 behavior can be found in engineering,
management science, economics, social sciences,
biology and medicine, etc. Positive linear systems are
defined on cones and not on linear spaces. Therefore,
the theory of positive systems is more complicated and
less advanced. An overview of state of the art in
positive systems theory is given in the monographs [1,
4, 8].
Positive linear systems with different fractional
orders have been addressed in [9, 10]. Stability of
standard and positive systems has been investigated in
[5, 15, 17, 20] and of fractional systems in [3, 6, 13, 14].
Descriptor positive systems have been analyzed in [11,
12]. Linear positive electrical circuits with state
feedbacks have been addressed in [2, 15].The global
stability of the nonlinear systems with positive linear
parts has been analyzed in [7 ].
In this paper the main results of [7] will be extended
to fractional nonlinear systems and the global stability
of fractional nonlinear systems with negative
feedbacks and positive not necessary asymptotically
stable linear parts will be addressed.
The paper is organized as follows. In section 2 some
preliminaries concerning positive linear systems are
given and it is shown that the coefficients of the
transfer matrices of positive asymptotically stable
linear systems are positive. The main result of the
paper is given in section 3 where the sufficient
conditions for the global stability of the fractional
nonlinear feedback systems with positive linear parts
are established. Concluding remarks are given in
section 4.
The following notation will be used:
- the set of
real numbers,
nm
- the set of nm real matrices,
mn
+
- the set of nm real matrices with nonnegative
entries and
1nn
++
=
, Mn - the set of nm Metzler
matrices (real matrices with nonnegative off-diagonal
entries), In - the nm identity matrix.
Global Stability of Fractional Feedback Systems
with Positive Linear Parts
T. Kaczorek
Bialystok University of Technology, Białystok, Poland
ABSTRACT: The global (absolute) stability of fractional nonlinear systems with negative feedbacks and positive
not necessary asymptotically stable linear parts is addressed. It is shown that the coefficients of the transfer matrix
of fractional positive asymptotically stable systems are positive. Sufficient conditions for the global stability of
the fractional nonlinear systems with positive linear parts are established.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 19
Number 1
March 2025
DOI: 10.12716/1001.19.01.39
334
2 PRELIMINARIES
Consider the 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
,
nm
B
,
pn
C
,
pm
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].
Definition 1. [7] 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. [7] 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. [6, 12] The positive fractional system
(1) (for u(t)=0) is called asymptotically stable if
lim ( ) 0
t
xt
→
=
for any
(0)
n
x
+
. (3)
Theorem 2. [6, 12] The positive linear system (1) (for
u(t)=0) 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)
The transfer matrix of the system (1) is given by
1
( ) [ ] , .
n
T s C I A B D s
−
= − + =
(6)
Theorem 3. If the matrix
n
AM
is Hurwitz and
nm
B
+
,
pn
C
+
,
pm
D
+
of the linear positive
system (1), then all coefficients of the transfer matrix (3)
are positive.
Proof is similar to the proof given in [7] for the
standard positive linear systems..
Example 1. Consider the fractional positive linear
system (1) with the matrices
2 1 1
, , 1 1 , 2 ,
2 3 2
A B C D
−
= = = =
−
(7)
Note that the matrix A given by (7) is Hurwitz since
its characteristic polynomial
2
2
21
det 5 4
23
IA
+−
− = = + +
−+
(8)
has positive coefficients (Theorem 2).
Using (7) and (6) we obtain
1
1
2
2
()
1
21
2 13 19
1 1 2
2
23
54
n
T C I A B D
−
−
= − + =
+−
++
+=
−+
++
(9)
The transfer function (9) has positive coefficients.
3 MAIN RESULT
Consider the nonlinear feedback system shown in
Figure 1 consisting of the fractional linear part
described by the equations
()d x t
Ax bu
dt
=+
, (10a)
y cx=
, (10b)
where
()
n
x x t=
,
()u u t=
,
()y y t=
,
nn
A
,
n
b
,
1 n
c
and of the nonlinear element with the
characteristic u=f(e) shown in Figure 2.
Figure 1. Nonlinear feedback system.
Figure 2. Characteristic of the nonlinear element.
The characteristic of the nonlinear element satisfies
the condition
(0) 0f =
and
()
0
fe
k
e
,
k +
. (12)
It is assumed that the linear part (10) is positive, i.e.
335
n
AM
,
n
b
+
,
1 n
c
+
, (13)
but not necessary asymptotically stable.
It is also assumed that if the linear part is unstable
then by suitable choice of the gain k1 we may obtain
(Figure 3) asymptotically stable positive linear part
with the transfer function
1
1
()
()
1 ( )
T
T
kT
=
+
(14)
and the nonlinear element with the characteristic
11
( ) ( )f e f e k e=−
satisfies the condition (Figure 4)
1
1 1 2 1
()
(0),
fe
f k k k k
e
= −
. (15)
Figure 3. Nonlinear feedback system with the gain k1.
Figure 4. Characteristic of the nonlinear element with the
gain k1.
Definition 3. The nonlinear system is called globally
(or absolutely) asymptotically stable if
lim ( ) 0
t
xt
→
=
for
any
(0)
n
x
+
.
Definition 4. The circle in the plane
( ( ), ( ))PQ
with center in the point
12
12
,0
2
kk
kk
+
−
and radius
21
12
2
kk
kk
−
will be called the
12
11
,
kk
−−
circle.
Theorem 4. The fractional nonlinear feedback
system (Figure 3) consisting of positive linear
asymptotically stable part with the transfer function
(14) and of nonlinear element with characteristic
satisfying the condition (15) is globally asymptotically
stable if the Nyquist plot of
1
( ) ( ) ( )T j P jQ
=+
of the
linear part is located on the right-hand side of the circle
12
11
,
kk
−−
.
Proof. Proof is based on the application of the
Lyapunov method to the positive nonlinear system [12,
15, 17]. As the Lyapunov function we choose the time
function
1
( ) 0, [0, )
At
T
V t e b t
= +
, (16)
where
1
[]
TT
n
=
is strictly positive vector, i.e.
0
k
,
1,...,kn=
.
The function
( ) 0Vt
for
[0, )t +
since
1 n
AM
is asymptotically stable and
n
b
+
.
From (16) we have
1
1
()
( ) 0
At
T
dV t
V t Ae b
dt
= =
for
[0, )t +
(17)
since
1
0
T
A
for the Hurwitz matrix
1 n
AM
(Theorem 2).
Therefore, by the Lyapunov theorem the fractional
positive nonlinear system is asymptotically stable if
1
0
At
ce b
for
[0, )t +
. (18)
Note that
1
1
11
( ) [ ] [ ]
At
n
T c e b c I A b
−
= = −L
, (19)
where
L
is the Laplace transform operator. From (18)
we obtain
1
1
Re ( ) 0Tj
k
+
for
0
and
21
0k k k= −
. (20)
Taking into account that
1
2 1 1 2 1
2
2 1 1
1 ( ) 1
Re ( ) Re
1 ( )
1 ( )
1
Re
1 ( )
Tj
Tj
k k k T j k k
k T j
k k k T j
+ = + =
− + −
+
−+
(21)
and that the border of asymptotic stability is the
j
axis we obtain
2
1
1 [ ( ) ( )]
1 [ ( ) ( )]
k P jQ
j
k P jQ
++
=
++
(22a)
or
12
{1 [ ( ) ( )]} 1 [ ( ) ( )]j k P jQ k P jQ
+ + = + +
. (22b)
From (22b) we have
12
( ) 1 ( )k Q k P
− = +
and
12
[1 ( )] ( )k P k Q
+=
(23)
and after elimination of
2
1 2 1 2
[1 ( )][1 ( )] ( ) 0k P k P k k Q
+ + + =
(24a)
or
22
12
1 2 1 2
1
( ) ( ) ( ) 0
kk
P P Q
k k k k
+
+ + + =
. (24b)
336
Note that (24b) can be rewritten in the form of the
equation
22
2
1 2 2 1
1 2 1 2
( ) ( )
22
k k k k
PQ
k k k k
+−
+ + =
(25)
which describes the circle
12
11
,
kk
−−
(see Figure 5).
This completes the proof. □
Figure 5. Nyquist plot with the circle
12
11
,
kk
−−
.
This theorem can be considered as an extension for
the fractional nonlinear systems with positive linear
parts of the Kudrewicz theorem presented in [18] for
nonlinear systems with standard linear parts.
Example 2. Consider the fractional nonlinear
system with unstable linear part with
2
( ) 2 3
()
()
1.8 0.1
L
T
M
+
==
+−
(26)
and nonlinear element with the characteristic u=f(e)
shown in Figure 6.
Figure 6. Characteristic of the nonlinear element of Example
2.
To obtain the fractional nonlinear system with
asymptotically stable linear part we choose k1=0.2 and
we obtain
1
11
22
( ) ( )
()
1 ( ) ( ) ( )
2 3 2 3
1.8 0.1 0.2(2 3) 2.2 0.5
TL
T
k T M k L
= = =
++
++
=
+ − + + + +
. (27)
Note that the characteristic of the nonlinear element
u=f(e) satisfies the condition (Figure 6)
()
0.2 2
fe
e
. (28)
In this case
1
2
23
( ) ( ) ( )
0.5 2.2
j
T j P jQ
j
+
= = +
−+
, (29)
where
23
2 2 2 2 2 2
1.4 1.5 2 5.6
( ) , ( )
(0.5 ) (2.2 ) (0.5 ) (2.2 )
PQ
++
= = −
− + − +
(30)
The Nyquist plot and the circle are shown on the
Figure 7. By Theorem 4 the nonlinear system is globally
stable.
Figure 7. Nyquist plot with the circle (-5,-0.5).
4 CONCLUDING REMARKS
The global stability of fractional nonlinear systems
with negative feedbacks and positive linear parts has
been analyzed. The characteristics u=f(e) of the
nonlinear element satisfy the assumption (12) and the
linear parts described by the equations (11) are not
necessary asymptotically stable. The gain k1 of the
positive linear part has been chosen so that the transfer
function (14) is asymptotically stable and the
characteristic u=f(e) satisfies the condition (15).
It has been shown that the nonlinear systems are
globally asymptotically stable if the Nyquist plots of
the linear parts are located on the right-hand side of the
circles
−−
21
1
,
1
kk
. This theorem is an extension of
the Kudrewicz theorem presented in [18] for nonlinear
systems with standard linear parts.
The considerations have been illustrated by
numerical examples. The considerations can be
extended to the fractional nonlinear systems with
positive linear parts and with positive descriptor linear
parts.
-6 -4 -2 0 2 4 6
-5
-4
-3
-2
-1
0
1
2
3
4
5
P()
Q(
)
337
ACKNOWLEDGMENT
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.
REFERENCES
[1] Berman A. and Plemmons R.J.: Nonnegative Matrices in
the Mathematical Sciences, SIAM, 1994.
[2] Borawski K.: Modification of the stability and positivity of
standard and descriptor linear electrical circuits by state
feedbacks, Electrical Review, vol. 93, no. 11, 2017, 176-180.
[3] Busłowicz M., Kaczorek T.: Simple conditions for practical
stability of positive fractional discrete-time linear
systems, Int. J. Appl. Math. Comput. Sci., vol. 19, no. 2,
2009, 263-169.
[4] Farina L., Rinaldi S.: Positive Linear Systems; Theory and
Applications, J. Wiley, New York, 2000.
[5] Kaczorek T.: Analysis of positivity and stability of
discrete-time and continuous-time nonlinear systems,
Computational Problems of Electrical Engineering, vol. 5,
no. 1, 2015, 11-16.
[6] Kaczorek T.: Analysis of positivity and stability of
fractional discrete-time nonlinear systems, Bull. Pol.
Acad. Sci. Techn., vol. 64, no. 3, 2016, 491-494.
[7] Kaczorek T.: Global stability of nonlinear feedback
systems with positive linear parts, International Journal
of Nonlinear Sciences and Numerical Simulation. 2019 (in
Press).
[8] Kaczorek T.: Positive 1D and 2D Systems, Springer-
Verlag, London, 2002.
[9] Kaczorek T.: Positive linear systems with different
fractional orders, Bull. Pol. Acad. Sci. Techn., vol. 58, no.
3, 2010, 453-458.
[10] Kaczorek T.: Positive linear systems consisting of n
subsystems with different fractional orders, IEEE Trans.
on Circuits and Systems, vol. 58, no. 7, 2011, 1203-1210.
[11] Kaczorek T.: Positive fractional continuous-time linear
systems with singular pencils, Bull. Pol. Acad. Sci. Techn.,
vol. 60, no. 1, 2012, 9-12.
[12] Kaczorek T.: Descriptor positive nonlinear systems,
Advances in Intelligent Systems and Computing, vol. 550,
2017, Springer, 34-44.
[13] Kaczorek T.: Selected Problems of Fractional Systems
Theory, Springer, Berlin 2011.
[14] Kaczorek T.: Stability of fractional positive nonlinear
systems, Archives of Control Sciences, vol. 25, no. 4, 2015,
491-496.
[15] Kaczorek T.: Superstabilization of positive linear
electrical circuit by state-feedbacks, Bull. Pol. Acad. Sci.
Techn., vol. 65, no. 5, 2017, 703-708.
[16] Kaczorek T.: Theory of Control, vol. 2, PWN, Warsaw,
1981 (in Polish).
[17] Kaczorek T., Borawski K.: Stability of Positive Nonlinear
Systems, Proc. MMAR 2017, Międzyzdroje, 28-31.08.2017.
[18] Kudrewicz J.: Ustoicivost nieliniejnych sistem z obratnoj
swjazju, Avtomatika i Telemechanika, vol. 25, no. 8, 1964.
[19] Lyapunov A.M.: Obscaja zadaca ob ustoicivosti
dvizenija, Gostechizdat, Moskwa, 1963.
[20] Leipholz H.: Stability Theory, New York Academic
Press, 1970.
