DOI 10.21008/j.1897-0737.2018.93.0002
__________________________________________
* Bialystok University of Technology
Tadeusz KACZOREK*
ANALYSIS OF THE POSITIVITY AND STABILITY OF LINEAR ELECTRICAL CIRCUITS WITH
STATE-FEEDBACKS
The analysis of the positivity and stability of linear electrical circuits by the use of state-feedbacks is addressed. Generalized Frobenius matrices are proposed and their properties are investigated. It is shown that if the state matrix of electrical circuit has generalized Frobenius form then the closed-loop system matrix is not positive and as- ymptotically stable. Different cases of modification of the positivity and stability of linear electrical circuits by state-feedbacks are discussed and necessary conditions for the existence of solutions to the problem are established.
KEYWORDS: positivity, stability, linear, electrical circuit, state-feedback.
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 nonnega- tive inputs. An overview of state of the art in positive theory is given in the monographs [3, 10]. Variety of models having positive behavior can be found in engineering, especially in electrical circuits [16], economics, social sciences, biology and medicine, etc. [3, 10].
The positive electrical circuits have been analyzed in [4-6, 8-13]. The con- structability 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 [8]. A new class of normal positive linear electrical circuits has been introduced in [9]. Positive fractional linear electrical circuits have been investigated in [12] and positive unstable electrical circuits in [13]. Infinite ei- genvalue assignment by output-feedback for singular systems has been analyzed in [7]. Zeroing of state variables in descriptor electrical circuits has been ad- dressed in [14]. Controller synthesis for positive linear systems with bounded controls has been investigated in [1]. Stability of continuous-time and discrete- time linear systems with inverse state matrices has been analyzed in [15].
In this paper the analysis of the positivity and stability of linear electrical cir- cuits by state-feedbacks will be investigated.
The paper is organized as follows. In section 2 the basic definitions and prop- erties of positive electrical circuits are recalled. The generalized Frobenius ma- trices are introduced and their properties are analyzed in section 3. The linear electrical circuits with state-feedbacks are investigated in section 4. Concluding remarks are given and some open problems are formulated in section 5.
The following notation will be used: - the set of real numbers, nm - the set of nm real matrices, nm - the set of nm real matrices with nonnega- tive entries and n n1, M - the set of n n Metzler matrices (real matri-n ces with nonnegative off-diagonal entries), I - the n n identity matrix. n
2.POSITIVEELECTRICALCIRCUITS
Consider the linear continuous-time electrical circuit described by the state equations
) ( ) ( )
(t Ax t Bu t
x , (2.1a) )
( ) ( )
(t Cx t Du t
y , (2.1b) where x(t)n, u(t)m, y(t)p are the state, input and output vectors and Ann, Bnm, Cpn, Dpm.
It is well-known [16] that any linear electrical circuit composed of resistors, coils, capacitors and voltage (current) sources can be described by the state equations (2.1). Usually as the state variables x1(t),…, xn(t) (the components of the state vector x(t)) the currents in the coils and voltages on the capacitors are chosen.
Definition 2.1. [16] The electrical circuit (2.1) is called (internally) positive if t n
x( ) and yy(t)p, t[0,] for any x0 x(0)n and every t m
u( ), t[0,].
Theorem 2.1. [16] The electrical circuit (2.1) is positive if and only if Mn
A , Bnm, Cpn, Dpm. (2.2) Theorem 2.2. [16] The linear electrical circuit composed of resistors, coils and voltage sources is positive for any values of the resistances, inductances and source voltages if the number of coils is less or equal to the number of its linear- ly independent meshes and the direction of the mesh currents are consistent with the directions of the mesh source voltages.
Theorem 2.3. [16] The linear electrical circuit composed of resistors, capacitors and voltage sources is not positive for all values of its resistances, capacitances
and source voltages if each its branch contains resistor, capacitor and voltage source.
Theorem 2.4. [16] The R, L, C, e electrical circuits are not positive for any val- ues of its resistances, inductances, capacitances and source voltages if at least one its branch contains coil and capacitor.
Definition 2.2. [16] The positive electrical circuit is called asymptotically stable if
0 ) (
lim
x t
t for any x0n. (2.3) Theorem 2.5. [16] The positive electrical circuit is asymptotically stable if all coefficients of the characteristic polynomial
0 1 1
1 ...
]
det[InsA sn ansn asa (2.4) are positive, i.e. ak 0 for k 0,1,...,n1.
3.GENERALIZEDFROBENIUSMATRICES Definition 3.1. The following matrices
1 ,..., 1 , 0
, 0 0
0
1 0
0
0 1
0
0 0
1
, 0 0
0
0 0 0
0 0 0
, 0
0
0 0
0 0
0 0
0
, 0
0 0
0 0
0
0 0
0
0 1 2 1
3 4
1 2
1
0 1 2
1
3
1 1
2 2
1 1
0
1 2
1 2
1 0
1 2
1
1
n k
b
a a a a A
A b
b b
a a a
a A
a b
a b
a b
a A
A a a
a a
b b
b A
k
n n
T
n n
n
n n
T
n n
(3.1)
are called the matrices in generalized Frobenius form.
It is easy to verify that
1 1 0 1 2 1 1
1 ... ... ...
]
det[InsAj sn ansn ab bnsa b bn for j1,...,4 (3.2) and the coefficients of the polynomial are positive if and only if ak 0 and
0
bk for k1,...,n1.
Theorem 3.1. 1) The inverse matrix of the generalized Frobenius matrix is also the generalized Frobenius matrix.
2) The inverse matrix of the generalized Frobenius matrix is asymptotically sta- ble if and only if the generalized Frobenius matrix is also asymptotically stable.
Proof. The proof will be given only for the matrix A . The proof for the remain-1 ing matrices (3.1) is similar.
It is easy to verify that
.
0 0
0
0 0
0
0 0
0 0 0 0
0 0
0
0 0
0
1 1 21
1 1
1 0 1 1
1 1 0 2
1 2 1 0 1 1 1 1 0
1
1 2
1 0
1 2
1
11
n n n n
n
b b
b
a a
b a a
b a a b a
a a
a a
b b
b
A
(3.3)
By Theorem 2.5 the matrix A1 is asymptotically stable if and only if ak 0 for 1
,...,
0
n
k . By definition bk 0 and bk10 for k1,...,n1. Therefore, from (3.3) it follows that all coefficients of the first row are negative if and only if all coefficients of the n-th row of the matrix A1 are negative. □
Theorem 3.2. If sji, j1,...,4, i1,...,n are the nonzero eigenvalues of the generalized Frobenius matrix A then j s are the eigenvalues of the inverse ji1 matrix A , j1 j1,...,4.
Proof. Let s , ji j1,...,4, i1,...,n be the zeros of the characteristic equation of the matrix Aj
0 ]
det[Insj Aj for j1,...,4. (3.4) Then multiplying (3.4) by det[sj1Aj1] we obtain
. 0 ] det[
)]
)(
det[(
] det[
]
det[InsjAj sj1Aj1 InsjAj sj1Aj1 Insj1Aj1 (3.5) Therefore, if s , ji j1,...,4, i1,...,n are the eigenvalues of the matrix A then j
1
sji are the eigenvalues of the matrix A , j1 j1,...,4. □
Theorem 3.3. The characteristic polynomial of the inverse matrices in the gen- eralized Frobenius forms (3.1) is given by
11 11 01 1 1
1 1 1 1 1 0
1 1 1 1
1] 0 ... ... ...
det[InsAj sn abasn ab bnansab bn (3.6) for j1,...,4.
Proof. Using (3.3) and developing the determinant with respect to the first row we obtain
.
0 0
0 0
0
0 0
] det[
1 1 1
1
1 0 1 1
1 1 0 2
1 2 1 0 1 1 1 1 0
1 1
s b
s s
b
a a b a a
b a a b a s
A s I
n n n
n
(3.7)
Similar results we obtain for j2,3,4. □
Example 3.1. The characteristic polynomial of the generalized Frobenius matrix
5 4 1
2 0 0
0 2 0
A (3.8)
has the form
4 8 5 5
4 1
2 0
0 2 ]
det[ 3 3 2
s s s
s s s A s
I (3.9)
and its zeros are s1 1, s2 s32. The inverse matrix of (3.8) has the form
0 5 . 0 0
0 0 5 . 0
1 5 . 2 2
A1 (3.10)
and its characteristic polynomial
25 . 0 25 . 1 2 5
. 0 0
0 5
. 0
1 5 . 2 2 ]
det[ 3 1 3 2
s s s
s s s
A s
I (3.11)
with zeros s111, s21s310.5.
4.ELECTRICALCIRCUITSWITHSTATE-FEEDBACKS The problem under consideration can be formulated as follows: given the un- stable electrical circuits described by (2.1a) with Annand Bnm find the state-feedback
) ( ) (t Kx t
u , Kmn (4.1) such that the closed-loop system
) ( )
(t A x t
x C , (4.2a) where
n
C A BK M
A (4.2b) is asymptotically stable.
Theorem 4.1. If Ann has the generalized Frobenius form then does not exist a state-feedback matrix K1n for any Bn1 such that the closed- loop system (4.2a) is positive and asymptotically stable.
Proof. Let A has the generalized Frobenius form A , i.e. 1 AA1. Then
n
C A BK M
A 1 1 has also the same form if and only if B[0 0 b]T and
n
n n
n
n
n n C
M
a bk a
a bk a bk
b b
b
k k
k
b a
a a a
b b
b
A
1 2
1 2 0 1
1 2
1
2 1
1 2
1 0
1 2
1
1
0 0
0
0 0
0
0 0
0
] [
0 0 0
0 0 0
0 0
0
0 0
0
(4.3)
if and only if
10
i
i a
bk for i1,...,n. (4.4) If the condition (4.4) is satisfied then by Theorem 2.5 the matrix (4.3) is unsta- ble. □
Example 4.1. Consider the electrical circuit shown in Figure 4.1.
Fig. 4.1. Electrical circuit of Example 4.1
As the state variable we choose the voltage u on the capacitor with given capaci- tance C and the current i in the coil with given inductance L. Using the Kirch- hoff’s laws we obtain the equations
dt u Ldi Ri
e , (4.5a)
dt Cdu
i , (4.5b) which can be written in the form
i Be A u i u dt
d
, (4.6a) where
L R L A C
1 0 1
,
L
B 1
0
. (4.6b)
Note that the matrix A has the generalized Frobenius form.
For R0 the matrix has the form
1 0 0 1
L
A C (4.7)
and it is unstable.
For state-feedback matrix K [k1 k2] we obtain
2 2 1
2
1 1( 1)
0 1 ]
1 [ 0 1 0
0 1
M L k k
L k C k L L
BK C A
AC
(4.8)
for k11. In this case the matrix (4.8) is unstable since the characteristic poly- nomial
LC s k L s k L s k L k
s C A
s
I C 2 2 1
1 2 2
1 )
1 1(
1 det
]
det[
(4.9)
for k11 has nonpositive coefficient LC
k1 1
.
The problem under the consideration can be divided into the following two subproblems:
Subproblem 1. Given the matrix Ann with some negative off-diagonal entries and Bnm find a gain matrix K1mn of the state-feedback such that
n
C A BK M
A 1 1 . (4.10) Subproblem 2. Given AC1Mn and Bnm find a gain matrix K2mn of the state-feedback such that the matrix
n C
C A BK M
A 2 1 2 (4.11) is asymptotically stable (is Hurwitz matrix).
The following theorem gives a necessary condition for the existence of the solu- tion of Subproblem 1.
Theorem 4.2. There exists a gain matrix K1mn such that (4.10) holds only if to each row with at least one off-diagonal negative entry of A the correspond- ing row of B is nonzero.
Proof. Let the i-th row of A has at least one off-diagonal negative entry and the corresponding i-th row of B is zero. Then from (4.10) it follows that by suitable choice of K we are not able to eliminate this negative entry of A. □
Example 4.2. Consider the electrical circuit shown in Figure 4.2 with given resistance R, inductances L1, L2, capacitance C and source voltage e.
Fig. 4.2. Electrical circuit of Example 4.2
As the state variables we choose the currents i , 1 i and voltage u. Using the 2 Kirchhoff’s laws we can write the equations
, ,
,
2 1
2 2 1 1 1
dt Cdu i i
dt L di u
dt u L di Ri e
(4.12)
which can be written in the form
Be i i u A i i u dt
d
2 1 2
1 , (4.13a)
where
0 1 0
1 0
1 0 1
2
1 1
L
L R L
C C
A ,
0 1 0
L1
B . (4.13b)
Note that in the first row of A one entry is negative
C
1 and the first row of B is zero. Therefore, by Theorem 4.2 does not exist a gain matrix K113 such that (4.10) holds.
Example 4.3. Consider the electrical circuit shown in Figure 4.3 with known resistances R , 1 R , 2 R , inductances 3 L , 1 L , capacitances 2 C , 1 C and source 2 voltages e , 1 e . 2
Fig. 4.3. Electrical circuit of Example 4.3
Using the Kirchhoff’s laws we may write the equations
, )
(
, )
(
,
,
2 2 2 1 3 2 3 2 2
1 1 1 2 3 1 3 1 1
2 2 1 1
1 1
dt u L di i R i R R e
dt u L di i R i R R e
dt C du dt i
C du i
(4.14)
which can be written in the form
2 1
2 1 2 1
2 1 2 1
e B e
i i u u A i i u u
dt
d , (4.15a)
where
2 3 2 2
3 2
1 3 1
3 1 1
2 1
0 1 1 0
0 1 0
0
1 0 0
0
L R R L
R L
L R L
R R L
C C
A ,
2
1 1
0 1 00 0
0 0
L
B L . (4.15b)
Note that the third and fourth rows of the matrix A contain negative off-diagonal entries but the corresponding rows of B contain nonzero entries. Therefore, the necessary condition of Theorem 4.2 is satisfied.
Let
4
44 43
42 41
34 33 32 31
2 1
1
0 1 0 0
1 0 0 0
M
a a
a a
a a a a
C C
AC
for aij 0, i3,4; j1,2,3,4 (4.16)
then from (4.10) omitting the first zero rows we obtain
1
2 1
44 2
3 2 2
43 3 2
42 41
1 34 3 33
1 3 32 1
1 31
0 1 1 0 1
1
K L L L a
R R L
a R a L
a
L a R L a
R a R
a L
(4.17)
and
44 2 3 2 3 43 2 42 2 41
2
3 34 1 33 1 3 1 32 1 31 1
1 1
1
a L R R R a L a
L a
L
R a L a
L R R a
L a
K L . (4.18)
Note that the matrix (4.16) is unstable for all aij0, i3,4; j1,2,3,4. Theorem 4.3. By suitable choice of the matrix K1mn it is possible to modi- fy not more than q rank B rows of the matrix A.
Proof. By Kronecker-Capelli theorem the equation
1
1 A BK
AC (4.19) has a solution K if and only if 1
B B
A
AC ] rank
rank[ 1 . (4.20)
Therefore, the maximal number of rows of the matrix A which can be modified by suitable choice of K is 1 q rank B. □
For example in Example 4.3 the rank of the matrix B given by (4.15b) is two and by suitable choice of the matrix K only two rows of A have been modified. 1 Example 4.4. Consider the electrical circuit shown in Figure 4.4 with given resistances R , k k1,2,3, inductances L , 1 L and source voltage e. 2
Fig. 4.4. Electrical circuit of Example 4.4
Using the Kirchhoff’s laws we may write the equations
, )
(
, )
(
2 2 2 2 2 1 3
1 1 1 1 2 1 3
i dt R L di i i R e
i dt R L di i i R e
(4.21)
which can be written in the form
i Be A i i i dt
d
2 1 2
1 , (4.22a)
where
2 3 2 2
3
1 3 1
3 1
L R R L
R
L R L
R R
A ,
2
11
1
L
B L . (4.22b)
The matrix A is asymptotically stable but it is not the Metzler matrix. It will be shown that the gain matrix
] [k1 k2
K (4.23) of the state-feedback can be chosen so that the closed-loop electrical circuit will be positive and asymptotically stable.
Using (4.22b) and (4.23) we obtain
) . (
) (
] 1 [
1
2 3 2 2 2
3 1
1 3 2 1
3 1 1
2 1
2 1
2 3 2 2
3
1 3 1
3 1
L R R k L
R k
L R k L
R R k
k k L L L
R R L
R
L R L
R R BK A AC
(4.24)
Note that the matrix (4.24) is a Metzler matrix if and only if
3
1 R
k and k2 R3. (4.25) If the condition (4.25) is satisfied then the closed-loop electrical circuit is as- ymptotically stable since the coefficients of the characteristic polynomial
2 1
1 3 2 2 3 1 2 1
2 1
1 2 3 2 2 1 3 1 2
2 2 3 2 2
1 3
1 2 3 1
1 3 1
2
) (
) (
) (
) (
] det[
L L
k R R k R R R R
L s L
L k R R L k R s R
L k R s R
L k R
L k R L
k R s R
A s
I C
(4.26)
are positive.
In the analysis of Subproblem 2 the crucial role plays the following theorem [10].
Theorem 4.4. The positive electrical circuit (2.1a) with AMn, Bnm is asymptotically stable if and only if there exists a vector vn with positive components vT [v1 vn], vk 0, k 1,...,n such that
0
Av . (4.27) From Theorem 4.4 it follows that the positive electrical circuit is unstable if at least one diagonal entry of AMn is not negative.
The following theorem gives a necessary condition for the stabilization by the state-feedbacks of the positive electrical circuits.
Theorem 4.5. The positive electrical circuit with AMn and Bnm can be stabilized by a state-feedback matrix Kmn only if to all rows of A with nonnegative diagonal entries the corresponding rows of B are nonzero.
Proof. Note that if the k-th row of the matrix B is zero then the k-th diagonal entry of the closed-loop matrix AC ABK is equal to the diagonal entry of A.
In this case the positive electrical circuit cannot be stabilized by the state- feedback matrix K. □
Example 4.5. (Continuation of Example 4.3) The matrix (4.16) is unstable Metzler matrix since its two diagonal entries are zero. The matrix cannot be stabilized by state-feedback since first two rows of the matrix B (defined by (4.15b)) are zero.
To find the state-feedback matrix Kmn for given A[aij]Mn and
m n T
bn
b
B[ 1 ] we may use the following procedure . Procedure 4.1.
Step 1. Knowing A and B find 0
1
dn
d
d and Nkm, k 1,...,n,
n
k Nk
N
1
(4.28)
such that
0
BN
Ad (4.29) and
0
i j
j ijd bN
a for i , j i,j1,...,n. (4.30) Step 2. Knowing d and N , k k 1,...,n find
] [
]
[ 1 11 1
1 1 1
1
n n
n
n N d N d
d d N N
K . (4.31)
Proof. From (4.29) we have BKdBN and by Theorem 4.4 the electrical cir- cuit is asymptotically stable if and only if
0 )
(ABK d for d0. (4.32) The matrix ABKMn if and only if
0 )
1 (
i j j ij i j ij
ij bN d a bN A BK
a for i , (4.33) j
where (ABK)ij denotes the ( ji, ) entry of the matrix ABK. □
The procedure can be solved as a standard linear programming problem [2].
Example 4.6. Consider the electrical circuit shown in Figure 4.5 with given resistance R, inductance L, capacitance C and source voltage e.
Fig. 4.5. Electrical circuit of Example 4.6
Using the Kirchhoff’s laws we may write the equations
, ,
dt Cdu R i u
dt u Ldi e
(4.34)
which can be written in the form
i Be A u i u dt
d
, (4.35a) where
1 0
1 1
L C
A RC , 1 2
0
L
B . (4.35b)
Note that the matrix A has the generalized Frobenius form. Applying the state- feedback
i K u
e , K [k1 k2] (4.36) to the electrical circuit we obtain
L k L k RC C k
k L L
C BK RC
A AC
2 2 1
1 1
1 1 ]
1 [ 0 1 0
1 1
. (4.37)
From (4.37) it follows that for k11 and k2 0 the closed-loop electrical cir- cuit is positive and asymptotically stable since the diagonal entries are negative and off-diagonal entries are nonnegative.
5. CONCLUDING REMARKS AND OPEN PROBLEMS The analysis of the positivity and stability of linear electrical circuits by the use of state-feedbacks has been investigated. The notion of Frobenius matrices has been extended and the properties of the generalized Frobenius matrices have been analyzed. It has been shown that if the state matrix of linear electrical cir- cuit has the generalized Frobenius form then the closed-loop electrical circuit is not positive and asymptotically stable (Theorem 4.1). Different necessary condi- tions for positivity and stabilization by state-feedbacks of linear electrical cir- cuits have been established (Theorems 4.2, 4.5). Procedure for computation of the state-feedback matrix stabilizing the closed-loop electrical circuit has been proposed (Procedure 4.1). The considerations have been illustrated by examples of the linear electrical circuits.
Open problems:
1) Find a class or classes of linear electrical circuits of order n2 with state matrices of the forms of the generalized Frobenius matrices. In this paper are presented only such electrical circuits of order n2 (Examples 4.1 and 4.6).
2) Find a class or classes of positive linear electrical circuits of order n2 with Metzler state matrices.
3) It is easy to show that the state matrix (4.35b) of the electrical circuit shown in Figure 4.5 is the same if the coil and the capacitor are interchanged. Find a class or classes of linear electrical circuits with this feature for n2.
Acknowledgment. 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
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(Received: 03.02.2018, revised: 02.03.2018)
