Tadeusz Kaczorek
Computation of transition matrices of positive linear electrical circuits
JEL: L62 DOI: 10.24136/atest.2019.147
Data zgłoszenia:05.04.2019 Data akceptacji:26.06.2019
A method is proposed for calculation of transition matrices of posi-tive electrical circuits. It is shown that if the transition matrix is pre-sented as finite series of the Metzler matrix with real distinct eigen-values then the coefficients of the series are nonnegative function of time. The method is applied to positive linear electrical circuits.
Słowa kluczowe: positive, linear, electrical circuits, calculation, transition
matrix.
1.Introduction
A dynamical system is called positive if its trajectory starting from any nonnegative initial state remains forever in the positive orthant for all nonnegative inputs. An overview of state of the art in positive theory is given in the monographs [2, 9]. Variety of models having positive behavior can be found in engineering, especially in electrical circuits [17], economics, social sciences, biology and medicine, etc. [2, 9].
The positive electrical circuits have been analyzed in [4-8, 10, 17]. The constructability and observability of standard and positive electrical circuits has been addressed in [5], the decoupling zeros in [6] and minimal-phase positive electrical circuits in [7]. A new class of normal positive linear electrical circuits has been introduced in [8]. Positive fractional linear electrical circuits have been investigated in [11], positive linear systems with different fractional orders in [12, 13] and positive unstable electrical circuits in [14]. Zeroing of state variables in descriptor electrical circuits has been addressed in [16] and the realization problem of positive linear systems in [1].
In this paper a method for computation of transition matrices of positive linear electrical circuits will be presented and some specific properties of the coefficients of finite series presenting the transition matrices will be investigated.
The paper is organized as follows. In section 2 some basic defi-nitions and theorems concerning matrices and Cayley–Hamilton theorem are recalled. The computation of the matrix function and the transition matrix are addressed in section 3. The stability and computation of the transition matrices of positive linear electrical circuits are investigated in section 4. Concluding remarks are given in section 5.
The following notation will be used:
- the set of real num-bers,
nm - the set ofm
n
real matrices and
n
n1,m n
- the set ofn
m
matrices with nonnegative entries and1
n n,
M
n- the set ofn
n
Metzler matrices (real matri-ces with nonnegative off-diagonal entries),I
n- then
n
identity matrix, TA
denotes the transpose of the matrix A.2.Preliminaries
The characteristic polynomial of the matrix n n
A
0 1 1 1...
]
det[
)
(
I
n
A
n
a
n
n
a
a
(2.1)and its minimal polynomial
(
)
are related by [2, 15],
)
(
)
(
)
(
D
(2.2)where
D
(
)
is the greatest common divisor of entries of the adjoint matrix[
I
n
A
]
ad. If the eigenvalues
1,
2,
…,n
of the matrixA
are distinct, i.e.
i
j ifi
j
,
,
,...,
1
,
j
n
i
thenD
(
)
1
and
(
)
(
)
[3, 15] Consider a matrixA
nn with the minimal character-istic polynomial,
)
...(
)
(
)
(
)
(
1 1 1 2 1 r m m m
(2.3)where
1,
2,
…,
r are the eigenvalues of the matrixA
and
.
1n
m
m
r i i
It is assumed that the function
f
(
)
is well-defined on spectrum
A
{
1,
2,...
r}
of the matrixA
,
i.e.r k d df f d df f f k k k k k m m k m k k , 1,... ) ( ) ( ,..., ) ( ) ( ), ( 1 1 ) 1 ( ) 1 ( (2.4) are finite.
In this case the matrix
f
(A
)
is well-defined and it is giv-en by the Lagrange-Sylvester formula [3, 15]
r i i m im i i i i i if
Z
f
Z
f
Z
A
f
1 ) 1 ( ) 1 ( 2 1(
)
(
)
...
(
),
)
(
(2.5) where i i m j k i j k j k k n i i ijA
d
d
j
j
k
I
A
A
Z
1 1 1 1)
(
1
)!
1
(
)!
1
(
)
)(
(
(2.6) and.
,...,
1
,
)
(
)
(
)
(
A
A
i
r
i m i i
(2.7)Theorem 2.1. Let 0 1 1 1
...
)]
(
det[
)
(
A
I
f
A
a
na
a
n n n
(2.8) be the minimal characteristic polynomial of the matrix).
(A
f
Then the matrixf
(A
)
satisfies its characteristic equation, i.e..
0
)]
(
[
...
)]
(
[
)]
(
[
1 0 1 1
n n n nI
a
A
f
a
A
f
a
A
f
(2.9)Proof. Proof is given in [3, 15].
In particular case for
f
(
A
)
A
we obtain the classical Cayley–Hamilton theorem.3.Computation of the transition matrices
Definition 3.1. The matrix n n
ij
a
A
[
]
is call Metzlermatrix if
a
ij
0
fori
j
.
The set of
n
n
Metzler matrices will be denotedM
n.
To simplify the notation it is assumed that
i
j fori
j
.
Theorem 3.1. LetA
nn,
f
(
A
,
t
)
be a matrix function well-defined on the spectrum
A
{
1,
2,...
r}
of the matrix,
A
t
- time and it can be presented in the series form
0)
(
)
,
(
k k kt
A
a
t
A
f
(3.1)where
a
k(t
)
are some coefficients. Then
1 0)
(
)
,
(
n k k kt
A
c
t
A
f
(3.2) where , ... ) ( ... ) ( ... ) ( ) ( 1 1 2 2 2 1 1 1 1 0 n n n n n n t f t f t f t , ... ) ( 1 ... ) ( 1 ... ) ( 1 ) ( 1 1 2 2 1 1 1 1 n n n n n t f t f t f t … , , ) ( ) ( ) ( ... 1 ... 1 ... 1 ) ( 2 1 2 2 2 2 2 1 1 1 t f t f t f t n n n n n n n . ... 1 ... 1 ... 1 1 1 2 2 1 1 1 n n n n n (3.3 )Proof. Using the classical Cayley–Hamilton theorem we can
elimi-nate in (3.1) the matrix k
A
fork
n
n
,
1
,...
and obtain (3.2). It is well-known [3, 15] that (3.2) is also satisfied on the spectrum of the matrixA
1 0)
(
)
,
(
n j k j kt
c
t
f
for
k
1
,...,
n
.
(3.4) The relations (3.4) can be written in the form
)
(
)
(
)
(
)
)
(
)
(
...
1
...
1
...
1
2 1 1 1 0 1 1 2 2 1 1 1t
f
t
f
t
f
t
c
t
c
t
c
n n n n n n n
(3.5)Using the well-known the Cramer formula to (3.5) we obtain (3.3).
The matrix At
e
is called transition matrix of the linear systems.Theorem 3.2. If
A
M
n has distinct eigenvalues
1,
2,
…,n
than the coefficientsc
k(t
),
k
0
,
1
,...
n
1
in the formula
1 0)
(
n k k k AtA
t
c
e
(3.6) satisfy the following conditions:
1
,...,
1
for
0
0
for
1
)
0
(
n
k
k
c
k (3.7)
0
for
0
,
2
,...,
1
0
for
1
)
(
0k
n
k
dt
t
dc
t k (3.8) and0
)
(
t
c
k fort
0
andk
0
,
1
,...,
n
1
.
(3.9)Proof. The equality (3.6) follows from (3.2) for and are defined by
(3.3). From (3.3) we have
1
)
0
(
)
(
0 0 0
tt
c
and(
)
0(
0
)
0
0
k t kt
c
fork
0
,
1
,...
n
1
(3.10) since forf
(
A
,
t
)
e
At,
1
,
0
t t ke
k
1
,...,
n
,
0(
0
)
and
k(
0
)
fork
1
,...,
n
1
has two identicalcolumns. Note that k t t k
e
dt
d
0 fork
1
,...,
n
,
and (0) 0 for 0,2,..., 1 1 1 ) 0 ( ) ( ) ( 0 0 k n k dt t d dt t dc k k t t k for (3.11)since
k(
0
)
fork
0
,
2
,...,
n
1
has two identical columns. The condition (3.9) is satisfied since0
)
(
)
(
)
(
)
(
1 2 2 2 1
t
c
t
c
t
c
t
c
k k k k fork
0
,
1
,...,
n
1
andt
1,
t
2
[
0
,
].
(3.12)Example 3.1. Consider the matrix At
e
for 22
1
0
1
M
(3.13)with the eigenvalues
1
1
,
2
2
.
By Theorem 3.2 we have
A
t
c
I
t
c
e
At)
(
)
(
2 1 0
(3.14) It will be shown that0
)
(
0t
c
for
t
0
,
c
0(
0
)
1
,
lim
c
0(
t
)
0
t and0
)
(
1t
c
for
t
0
,
c
1(
0
)
1
,
lim
c
1(
t
)
0
.
tUsing (3.3) for (3.13) we obtain
0 2 ) ( 1 1 1 ) ( 2 1 2 1 2 2 1 2 1 0 2 1 2 1 t t t t t t e e e e e e t c for
t
0
(3.15) and0
)
(
1
1
1
1
1
)
(
2 1 2 2 1 1 1 2 2 1
t t t t t te
e
e
e
e
e
t
c
(3.16) Note that0
2
2
)
(
)
(
2 1 2 2 1 0 1
2
t t t te
e
e
e
dt
t
dc
for0
t
(3.17) and0
2
)
(
1
)
(
2 1 2 1 2 1
2
1
t t t te
e
e
e
dt
t
dc
for0
t
(3.18) from (3.18) it follows that0
)
(
1
dt
t
dc
for 1 2 2 1ln
1
0
t
and 1(
)
0
dt
t
dc
for 1 2 2 1ln
1
0
t
. (3.19)Therefore, the conditions of Theorem 3.2 are satisfied (see Fig. 3.1).
0 5 10 0 0.5 1 t c0 (t ) 0 5 10 0 0.1 0.2 0.3 0.4 t c1 (t ) 0 5 10 -0.8 -0.6 -0.4 -0.2 0 t dc 0 (t )/ d t 0 5 10 -0.5 0 0.5 1 t dc 1 (t )/ d t
Fig. 3.1. Plots of coefficients ck(t)and their derivatives
Example 3.2. Consider the matrix function At
e
forA
M
3 with the eigenvalues
1
1
,
2
2
,
3
3
.
Note that the matrix
A
M
3 is given by the set of matrices(3.20)
for 3x3
P
monomial matrix. Using(3.3) for (3.10) we obtain(3.21) t t t t t t t t t e e e e e e e e e t c 3 2 2 3 1 3 2 1 3 2 1 2 3 1 3 1 1 2 3 2 2 3 3 2 2 2 2 1 1 2 3 2 2 2 1 1 2 3 4 2 5 ) )( ( ) ( ) )( ( ) ( ) )( ( ) ( 1 1 1 1 1 1 ) ( 3 2 1 3 2 1 (3.22) t t t t t t t t t e e e e e e e e e t c 3 2 2 3 1 3 3 2 1 2 3 1 2 1 2 3 3 2 2 2 2 1 1 3 2 1 2 2 1 2 1 ) )( ( ) )( ( ) )( ( 1 1 1 1 1 1 ) ( 3 2 1 3 2 1 (3.23)
From (3.21) - (3.23) is follows that
c
0(
0
)
1
,
c
1(
0
)
0
,
0
)
0
(
2
c
andc
k(
t
)
0
fort
0
.
Therefore the conditions of Theorem 3.2 are satisfied (see Fig. 3.2).
0 5 10 0 0.2 0.4 0.6 0.8 1 t c0 (t ) 0 5 10 0 0.1 0.2 0.3 0.4 0.5 t c1 (t ) 0 5 10 0 0.1 0.2 0.3 0.4 0.5 t c2 (t ) 0 5 10 -0.5 -0.4 -0.3 -0.2 -0.1 0 t dc 0 (t )/ d t 0 5 10 -0.5 0 0.5 1 t dc 1 (t )/ d t 0 5 10 -0.05 0 0.05 0.1 0.15 t dc 2 (t )/ d t
Fig. 3.2. Plots of coefficients ck(t)and their derivatives
4.The stability of positive electrical circuits and computation of their transition matrices
Consider the linear continuous-time electrical circuit described by the state equations
)
(
)
(
)
(
t
Ax
t
Bu
t
x
, (4.1a))
(
)
(
)
(
t
Cx
t
Du
t
y
, (4.1b) where nt
x
(
)
, mt
u
(
)
, pt
y
(
)
are the state, input and output vectors and nnA
, nmB
, pnC
, m pD
.Definition 4.1. [2, 10, 17] The electrical circuit (4.1) is called
(inter-nally) positive if n
t
x
(
)
and pt
y
y
(
)
,]
,
0
[
t
for any nx
x
0
(
0
)
and every mt
u
(
)
,]
,
0
[
t
.Theorem 4.1. [2, 10, 17] The electrical circuit (2.1) is positive if and
only if n
M
A
, n mB
, pnC
, p mD
. (4.2) It is well-known [2, 10, 17] that any linear electrical circuit com-posed of resistors, coils, capacitors and voltage (current) sources can be described by the state equations (4.1). Usually as the state variablesx
1(
t
)
,…,x
n(t
)
(the components of the state vector)
(t
x
) the currents in the coils and voltages on the capacitors are chosen.Theorem 4.2. The linear electrical circuit composed of resistors,
coils and voltage sources is positive for any values of the resistanc-es, inductances and source voltages if the number of coils is less or equal to the number of its linearly independent meshes and the direction of the mesh currents are consistent with the directions of the mesh source voltages.
Proof. Proof is given in [17].
Theorem 4.3. The R, L, C, e electrical circuits are not positive for
any values of their resistances, inductances, capacitances and source voltages if at least one their branch contains coil and capaci-tor.
Proof. Proof is given in [17].
Definition 4.2. [17] The positive electrical circuit (4.1) is called
asymptotically stable if
0
)
(
lim
x
t
t for all nx
0
. (4.3)Theorem 4.4 [17] The positive electrical circuit (4.1) is
asymptotical-ly stable if and onasymptotical-ly if
0
Re
k
fork
1
,...,
n
, (4.4)where
k is the eigenvalue of the matrixA
M
n and)
)...(
)(
(
]
det[
I
n
A
1
2
n . (4.5)Theorem 4.5. [3, 15] A symmetric matrix nn
A
(A
T
A
)has only real eigenvalues
k,k
1
,...,
n
.Theorem 4.6. Let
k,k
1
,...,
n
be real eigenvalues of the symmetric Hurwitz matrix nnA
. Then the matrixn n
A
D
A
,D
diag[
d
1
d
n]
,d
k
0
,n
k
1
,...,
(4.6) has also only real eigenvaluesk k k
d
fork
1
,...,
n
.
(4.7)Proof. By Theorem 4.5 the symmetric matrix
A
has only real eigenvalues
k,k
1
,...,
n
. Note that] det[ det ]} [ det{ ] det[ 1 1 A D D A D D A D In (4.8) and
0
]
det[
I
n
D
A
if and only if0
]
det[
]
det[
D
1
A
I
n
A
. (4.9)The equality (4.9) implies (4.7). □
Example 4.1. Consider the electrical circuit shown on Figure 4.1
with given resistances
R
1,R
2,R
3, inductancesL
1,L
2 and source voltagese
1,e
2.Fig. 4.1. Electrical circuit with inductances
)
(
1 2 3 1 1 1 1 1R
i
i
dt
di
L
i
R
e
, (4.10a))
(
2 1 3 2 2 2 2 2R
i
i
dt
di
L
i
R
e
(4.10b) which can be written in the form
2 1 1 2 1 1 2 1e
e
B
i
i
A
i
i
dt
d
, (4.11) where
2 3 2 2 3 1 3 1 3 1 1L
R
R
L
R
L
R
L
R
R
A
,.
1
0
0
1
2 1 1
L
L
B
(4.12)The electrical circuit is positive since the matrix
A
1 is a Metzler matrix and the matrixB
1 has nonnegative entries.Using (4.12) for
R
1 R
2
2
,
R
3
1
,
L
1 L
2
2
we obtain
2
3
2
1
2
1
2
3
2 3 2 2 3 1 3 1 3 1 1L
R
R
L
R
L
R
L
R
R
A
(4.13) and ). 2 )( 1 ( 2 3 2 3 2 1 2 1 2 3 ] det[ 2 1 2 A I (4.14) The eigenvalues of the matrix (4.13) are:
1
1
,
2
2
.
Using the results of Example 3.1 we obtain
2 2 2 2 2 / 3 2 / 1 2 / 1 2 / 3 1 0 0 1 ) 2 ( ) ( ) ( 2 2 2 2 2 2 1 1 2 0 1 t t t t t t t t t t t t t A e e e e e e e e e e e e A t c I t c e (4.15)
Example 4.2. Consider the electrical circuit shown on Figure 4.2
with given resistances
R
k,k
1
,...,
4
, inductancesL
1,L
2, capacitanceC
and source voltagese
j,j
1
,
2
,
3
.
Fig. 4.2. Electrical circuit of Example 4.2
Using the Kirchhoff’s laws we can write the equations
dt
du
C
R
u
e
e
dt
di
L
i
R
i
R
R
e
dt
di
L
i
R
i
R
R
e
e
4 3 2 1 2 1 3 2 3 2 2 1 1 2 3 1 3 1 3 1,
)
(
,
)
(
(4.16)which can be written in the form
3 2 1 2 2 1 2 2 1e
e
e
B
u
i
i
A
u
i
i
dt
d
, (4.17) where C R L R R L R L R L R R A 4 2 3 2 2 3 1 3 1 3 1 2 1 0 0 0 0 , C R C R L L L B 4 4 2 1 1 2 1 1 0 0 1 0 1 0 1 . (4.18)The electrical circuit is positive since
A
2M
3 and3 3 2
B
for all values of the resistances, inductances and capacitance.Using (4.18) for
R
1 R
2
2
,
R
3 R
4
1
,
L
1 L
2
1
,
1
1
1
0
0
0
3
1
0
1
3
1
0
0
0
0
4 2 3 2 2 3 1 3 1 3 1 2C
R
L
R
R
L
R
L
R
L
R
R
A
(4.19) and ). 4 )( 2 )( 1 ( ) 8 6 )( 1 ( 1 0 0 0 2 1 0 1 3 ] det[ 2 2 3 A I (4.20) The eigenvalues of the matrix (4.19) are:
1
1
,
2
2
,
.
4
3
Using the results of Example 3.2 we obtain
2 3 2 3 2 3 2 2 2 2 2 1 3 0 1 0 0 0 3 1 0 1 3 2 1 2 1 1 0 0 0 3 1 0 1 3 2 3 4 2 5 1 0 0 0 1 0 0 0 1 ) 3 3 ( ) ( ) ( ) ( 2 t t t t t t t t t t A e e e e e e e e e A t c A t c I t c e t t t t t t t t t t t t t t t e e e e e e e e e e e e e e e 3 2 3 2 3 2 3 2 3 2 3 3 0 0 0 2 23 31 2 41 2 15 20 2 25 0 2 15 20 2 25 2 23 31 2 41 (4.21) Concluding remarks
It has been shown that every matrix function
f
(
A
,
t
)
well-defined on its spectrum can be presented by (3.2) and the coeffi-cients are given by (3.3) (Theorem 3.1). If the matrixA
M
n has distinct eigenvalues then the coefficients in the formula (3.6) satisfy the conditions (3.7) - (3.9) (Theorem 3.2). The formula (3.6) allows to compute the transition matrix Ate
for a given matrixA
M
nwith distinct eigenvalues. The efficiency of the method has been demonstrated on examles of positive electrical circuits. The consid-erations can be extended to matrices with multiple real and complex eigenvalues. An extension of these considerations is also possible for positive fractional linear electrical circuits.
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„The studies have been carried out in the framework of work No. S/WE/1/2016 and financed from the funds for science by the Polish Ministry of Science and Higher Education”
Obliczanie macierzy tranzycji dodatnich liniowych obwodów elektrycznych
Podano metodę obliczania macierzy tranzycji dodatnich obwodów elektrycznych. Wykazano, że jeżeli macierz tranzycji ma postać skończonego szeregu macierzy Metzlera o różnych rzeczywistych wartościach własnych to współczynniki tego szeregu są nieujemny-mi funkcjanieujemny-mi czasu. Metoda ta została zastosowana do analizy dodatnich obwodów elektrycznych.
Słowa kluczowe: dodatni, liniowy, obwód elektryczny, obliczanie ,macierz tranzycji.
Author:
Tadeusz KACZOREK - Bialystok University of Technology, Faculty
of Electrical Engineering Wiejska 45D, 15-351 Bialystok, e-mail: kaczorek@isep.pw.edu.pl