#### Invariant Decoupling and Blocking Zeros of Positive Linear Electrical Circuits with Zero Transfer Matrices

Invariant Decoupling and Blocking Zeros of Positive Linear Electrical Circuits with Zero Transfer Matrices
Tadeusz Kaczorek 0
0 Faculty of Electrical Engineering, Bialystok University of Technology , Wiejska 45D, 15-351 Białystok , Poland
The invariant zeros, input-decoupling and output-decoupling zeros and blocking zeros of positive electrical circuits with zero transfer matrices are addressed. It is shown that the positive electrical circuits have no invariant zeros, input-outputdecoupling zeros and blocking zeros and also the list of eigenvalues of the system matrix is the sum of the list of input-decoupling zeros and the list of output-decoupling zeros.
Positive electrical circuit; Decoupling zero; Blocking zero; Invariant zero
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 systems theory is given in the monographs [2, 9].
Variety of models having positive behavior can be found in engineering, economics,
social sciences, biology and medicine, etc.
The notion of controllability and observability and the decomposition of linear
systems have been introduced by Kalman [17, 18]. These notions are the basic concepts
of the modern control theory [1, 8, 16, 19–21]. They have been also extended to positive
linear systems [2, 9]. The positive circuits and their reachability have been investigated
in [10, 12] and controllability and observability of electrical circuits in [4, 15].
B Tadeusz Kaczorek
The reachability of linear systems is closely related to the controllability of the
systems. Specially for positive linear systems, the conditions for the controllability are
much stronger than for the reachability [9,15]. Tests for the reachability and
controllability of standard and positive linear systems are given in [9,15]. The positivity and
reachability of fractional continuous-time linear systems and electrical circuits have
been addressed in [2,7,10,12,15] and the decoupling zeros of positive discrete-time
linear systems and positive electrical circuits in [5,6]. Standard and positive electrical
circuits with zero transfer matrices have been investigated in [14].
The positive linear systems consisting of n subsystems with different fractional
orders have been analyzed in [11]. The constructability and observability of standard
and positive electrical circuits have been addressed in [3].
In this paper the invariant zeros, decoupling zeros and blocking zeros of positive
linear electrical circuits will be investigated. The paper is organized as follows. In
Sect. 2 basic definitions and theorems concerning invariant decoupling and blocking
zeros of linear systems are recalled the positivity; reachability and observability of
linear systems are addressed in Sect. 3. The positive linear electrical circuits with
zero transfer matrices are presented in Sect. 4. The invariant, decoupling and blocking
zeros of positive electrical circuits with zero transfer matrices are analyzed in Sect. 5.
Concluding remarks are given in Sect. 6.
The following notation will be used: is the set of real numbers, n×m represents
tahnedsetn+of=n ×mn+×r1e,aClmiastrthiceesfi,eldn+×omf cdoemnoptleesxthneusmebtoerfsn, ×Mmn smtaantrdicfeosrwthitehsneotnonfegna×tivne
Metzler matrices (real matrices with nonnegative off-diagonal entries), In is the n × n
identity matrix.
2 Invariant, Decoupling and Blocking Zeros of Linear Systems
Consider the linear system
where x = x (t ) ∈ n, u = u(t ) ∈ m , y = y(t ) ∈
vectors and A ∈ n×n, B ∈ n×m , C ∈ p×n .
The system matrix of linear system (2.1) is defined by
p are the state, input and output
S(s) =
Let the matrix
SS(s) =
Definition 2.1 The zero of the polynomial p(s) is called the invariant zero of system
(2.1).
p(s) = det SS(s) = c det SS(s),
where c = det L(s) det R(s) since L(s) and R(s) are unimodular matrices row and
column operations on matrix (2.2).
Theorem 2.2 If m = p then
p(s) = det
= det [Ins − A] det T (s),
T (s) = C [Ins − A]−1 B.
= det
= 1.
Proof It is easy to see that
Consider the submatrix
of system matrix (2.2).
S1(s) = [Ins − A B]
rank [In z − A B] < n
are called the input-decoupling (i.d.) zero of system (2.1).
Let the matrix
S1S(s) =
diag [ p¯1(s) . . . p¯n(s)]
0 ∈
p¯(s) = p¯1(s) . . . p¯n(s).
Therefore, the i.d. zeros of the system are the zeros of polynomial (2.10). The system
has no i.d. zeros if and only if p¯(s) = 1, i.e., the matrix S1(s) has the canonical Smith
form [In 0]. The i.d. zeros represent unreachable modes of system (2.1).
The number of i.d. zeros n1 of system (2.1) is equal to the rank defect of its
controllability matrix, i.e.,
n1 = n − rank Rn,
Rn =
An−1 B .
of system matrix (2.2).
are called the output-decoupling (o.d.) zero of system (2.1).
Let the matrix
S2(s) =
S2S(s) =
pˆ(s) = pˆ1(s) . . . pˆn(s).
Therefore, the o.d. zeros of the system are the zeros of polynomial (2.16). The system
has no o.d. zeros if and only if pˆ(s) = 1, i.e., the matrix S2(s) has the canonical Smith
form I0n . The o.d. zeros represent unobservable modes of system (2.1).
n2 = n − rank On,
On = ⎢⎢
⎢
⎣
C An−1
Theorem 2.4 [8] The output y of system (2.1) for any input u (t ) = Bu(t ) and zero
initial condition x (0) = 0 is independent of the o.d. zeros of the system.
Definition 2.4 [8] A number z ∈ C for which both conditions (2.8) and (2.14) are
satisfied are called the input–output-decoupling (i.o.d.) zero of system (2.1).
Therefore, z ∈ C is an i.o.d. zero if and only if it is both an i.d. zero and an o.d.
zero of the system.
The number of i.o.d. zeros nio of system (2.1) is equal to
C [In z − A]ad B = 0,
where [In z − A]ad is the adjoint matrix.
Theorem 2.5 [8] A number z ∈ C is an uncontrollable and/or unobservable mode of
the system if and only if z is a blocking zero of the system.
3 Positivity, Reachability and Observability of Electrical Circuits
Consider linear electrical circuits composed of resistors, capacitors, coils and voltage
(current) sources. As the state variables [the components of the state vector x (t )] we
choose the voltages on the capacitors and the currents in the coils. Using Kirchhoff’s
laws we may describe the linear circuits in transient states by the state equations
x˙(t ) = Ax (t ) + Bu(t ),
y(t ) = C x (t ),
where x (t ) ∈ n, u(t ) ∈ m , y(t ) ∈ p are the state, input and output vectors and
A ∈ n×n, B ∈ n×m , C ∈ p×n.
It is assumed that the initial conditions are zero since the system matrix S(s) and
the transfer matrix T (s) are defined for zero initial conditions.
Definition 3.1 [9,13,15] Linear electrical circuit (3.1) is called (internally) positive
if the state vector x (t ) ∈ n+ and output vector y(t ) ∈ +p, t ≥ 0 for any initial
conditions x (0) ∈ n+ and all inputs u(t ) ∈ m+, t ≥ 0.
Theorem 3.1 [2, 9, 15] The linear electrical circuit is positive if and only if
A ∈ Mn ,
B ∈
C ∈
Definition 3.2 [2, 9, 15] Positive electrical circuit (3.1) is called reachable in time t ∈
[0, t f ] if for every given final state x f ∈ n+ there exists an input u(t ) ∈ m+, t ∈ [0, t f ]
which steers the state of the electrical circuit from zero initial conditions x (0) = 0 to
the final state x f .
Definition 3.3 [9] A matrix A ∈ n+×n is called monomial if each its row and each
its column contain only one positive entry and the remaining entries are zero.
Theorem 3.2 [9, 13, 15] Positive electrical circuit (3.1) is reachable if and only if the
reachability matrix
Rn =
An−1 B
contains a monomial matrix.
Definition 3.4 [9, 15] Positive electrical circuit (3.1) is called observable in time t ∈
[p0o,stsifb]leiftkonfionwdi nitgs uitnsiqinupeuitnuit(ita)l c∈ondm+itioanndx0its=inxp(u0t) y∈(t )n+∈. +p for t ∈ [0, t f ] it is
Theorem 3.3 [9, 15] The positive electrical circuit (3.1) is observable in time t ∈
[0, t f ] if and only if the matrix A ∈ Mn is diagonal and the matrix
On = ⎢⎢
⎢
⎣
contains a monomial matrix.
The transfer matrix of positive electrical circuit (3.1) is given by
T (s) = C [In s − A]−1 B ∈
p×m (s) is the set of p × m rational matrices in s.
Theorem 3.4 If for electrical circuit (3.1)
where On and Rn are defined by (3.4) and (3.3), respectively.
T (s) = C [In s − A]−1 B = 0
On Rn = 0
Proof Note that (3.6) holds if and only if
where L−1 denotes the inverse Laplace transform.
Substitution of
L−1[T (s)] = C e At B = 0
e At =
∞ ( At )k
k=0
into (3.8) yields
∞ C ( At )k B
k=0
= 0 and C Ak B = 0 for k = 0, 1, . . . .
Using (3.3), (3.4) and (3.10) we obtain
On Rn = ⎢⎢
⎢
⎣
⎢
= ⎢⎢
⎣
C An−1 B
C A2 B
C An B
An−1 B
C An−1 B ⎤
C A2(n−1) B ⎦
This completes the proof.
Theorem 3.5 Let for standard electrical circuit (3.1) condition (3.6) be satisfied. Then
(1) the pair ( A, B) is unreachable if C = 0,
(2) the pair ( A, C ) is unobservable if B = 0.
Proof From (3.11) we have
An−1 B
= 0
· · ·
if C = 0. Therefore, the pair ( A, B) is unreachable.
Similarly, from (3.11) we have
rank B
An−1 B
C A...n−1 ⎥⎦⎥⎥ B = 0
rank ⎢⎢
⎢
⎣
4 Linear Electrical Circuits with Zero Transfer Matrices
Following [14] the positive linear electrical circuits with zero transfer matrices will
be presented.
Example 4.1 Consider the electrical circuit shown in Fig. 1 with given resistances
R1, R2, R3, R4, inductance L, capacitance C and voltage source e.
Using Kirchhoff’s laws we may write the equations
As the output y we choose
Equations (4.1) and (4.2) can be rewritten in the form
+ uC = 0.
y = uC .
Fig. 1 Electrical circuit of
Example 4.1
= A1
+ B1e, y = C1
A1 =
B1 =
Fig. 2 Positive electrical circuit with zero transfer matrix
By Theorem 3.1 the electrical circuit is positive for all values of R1, R2, R3, R4, L
and C since from (4.3b) we have
A1 ∈ M2,
B1 ∈
2+, C1 ∈
The transfer function of the electrical circuit is
T (s) = C1 [I2s − A1]−1 B1 = [1 0]
for all values of R1, R2, R3, R4, L and C .
Note that
−1
= 0
det[Ins − A1] =
1 R
, s1 = − R4C , s2 = − L
and the electrical circuit is stable for all nonzero values of R1, R2, R3, R4, L and C .
By Theorems 3.2 and 3.3 the positive electrical circuit with (4.3b) is unreachable
and unobservable since the matrices
R2 = [B1 A1 B1] =
O2 =
have only one monomial column and one monomial row, respectively. From (4.7) we
have
O2 R2 =
The outputs of the positive electrical circuits shown in Fig. 1 are zero for all values
of the resistances, inductances, capacitances and all inputs.
Note that the positive electrical circuits shown in Fig. 1 are particular case of the
general positive electrical circuit shown in Fig. 2 with any positive part with resistances
Rk , inductances Lk , capacitances Ck and voltage sources ek .
Fig. 3 Positive electrical circuit with zero transfer matrix
If the common part (CP) of the electrical circuit is not a positive electrical circuit,
then the whole class of electrical circuits is not positive one with zero transfer function.
From the considerations we have the following theorem.
Theorem 4.1 The class of electrical circuits shown in Fig. 2 is positive electrical
circuits with zero transfer functions if and only if their common parts are positive
electrical circuits.
In general case the class of positive electrical circuits with zero transfer matrix can
be presented in the form shown in Fig. 3 [14].
5 Invariant, Decoupling and Blocking Zeros of Positive Electrical Circuits with Zero Transfer Matrices
The following operations on polynomial matrices are called elementary row (column)
operations [8]:
(1) Multiplication of the i -th row (column) by scalar (number) c. This operation will
be denoted by L (i × c)( R(i × c)).
(2) Addition to the i -th row (column) of the j -th row (column) multiplied by any
polynomial b(s). This operation will be denoted by L (i + j ×b(s))( R(i + j ×b(s))).
(3) Intercharge of the i -th and j -th rows (columns). This operations will be denoted
by L (i, j )( R(i, j )).
Applying the elementary row and column operations to identity matrices we obtain
unimodular matrices. The elementary row (column) operations are equivalent to
premultiplication (postmultiplication) of the matrix by suitable unimodular matrices. The
elementary row and column operations do not change the rank of the matrices.
First we shall consider the invariant decoupling and blocking zeros of the example
of positive electrical circuits with zero transfer matrices presented in Sect. 4.
Example 5.1 (Continuation of Example 4.1) The positive electrical circuit described
by (4.3) has no invariant zeros since its system matrix
= ⎣
using the elementary operations L 1 + 3 ×
, L[1, 3], R[2, 3] can be reduced to the form
, R[3 × L], R [2 + 3×
and the invariant zero polynomial p(s) = 1.
Note that
rank I2s − A1
B1 = rank
= rank ⎣
By Definition 3.4 the electrical circuit has no input–output-decoupling zeros since
the input-decoupling zero zi1 = − R41C and the output-decoupling zero zo1 = − LR
list of input-decoupling zeros
of the matrix A1 is the sum of the
and of the list of output-decoupling zeros
C1 [I2s − A1]ad B1 = [1 0]
= 0
for all values of R and L.
The list of eigenvalues of the matrix A is the sum of the list of the input-decoupling
zeros and of the list of output-decoupling zeros of the positive electrical circuit.
In general case it is assumed that matrices of positive electrical circuit (3.1) satisfy
the following assumption
A ∈ Mn, B ∈
Theorem 5.1 The invariant zero polynomial of positive electrical circuit (3.1) with
zero transfer, matrix satisfying (5.6) is equal to one, i.e., p(s) = 1 and the electrical
circuit has no invariant zeros.
Proof Proof is based on the elementary row and column operations.
Theorem 5.2 Positive electrical circuit (3.1) with zero transfer matrix, satisfying
(5.6), has
input-decoupling zeros and
ni = n − rank Ins − A
B forall s ∈ C
no = n − rank
forall s ∈ C
output-decoupling zeros and has not input–output-decoupling zeros (ni + no = n).
The list of eigenvalues of the matrix A is the sum of the list of input-decoupling
zeros and of the list of output-decoupling zeros of the electrical circuit.
Proof From Definition 2.2 and (2.8) it follows that the number of input-decoupling
zeros of the positive electrical circuit is given by (5.7). Similarly, from Definition 2.3
and (2.14) it follows that the number of output-decoupling zeros is given by (5.8).
From assumption that the transfer matrix is zero by Theorem 3.4 we have (3.11) and
from (2.19) nio = 0, i.e., ni + no = n.
Theorem 5.3 Positive electrical circuit (3.1) with zero transfer matrix, satisfying
(5.6), has no blocking zeros.
Proof Proof follows immediately from (2.20) and the assumption that the transfer
matrix of the positive electrical circuit is zero.
6 Concluding Remarks
Using polynomial matrices and the elementary row and column operations it has been
shown that the positive electrical circuits with zero transfer matrices:
The considerations have been illustrated by examples of positive electrical circuits
with zero transfer matrices. The considerations can be extended to fractional positive
linear electrical circuits and to descriptor linear electrical circuits.
Acknowledgements This work was supported under Work No. S/WE/1/16.
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1. E. Antsaklis , A. Michel , Linear Systems (Birkhauser, Boston, 2006 )
2. L. Farina , S. Rinaldi , Positive Linear Systems: Theory and Applications (Wiley, New York, 2000 )
3. T. Kaczorek , Constructability and observability of standard and positive electrical circuits . Electr. Rev . 89 ( 7 ), 132 - 136 ( 2013 )
4. T. Kaczorek , Controllability and observability of linear electrical circuits . Electr. Rev . 87 (9a), 248 - 254 ( 2011 )
5. T. Kaczorek , Decoupling zeros of positive discrete-time linear systems . Circuits Syst . 1 , 41 - 48 ( 2010 )
6. T. Kaczorek , Decoupling zeros of positive electrical circuits . Arch. Electr. Eng . 62 ( 4 ), 553 - 568 ( 2013 )
7. T. Kaczorek , Fractional positive continuous-time linear systems and their reachability . Int. J. Appl. Math. Comput. Sci . 18 ( 2 ), 223 - 228 ( 2008 )
8. T. Kaczorek , Linear Control Systems , vol. 1 (Wiley, New York, 1993 )
9. T. Kaczorek , Positive 1D and 2D Systems , London (Springer, New York, 2002 )
10. T. Kaczorek , Positive electrical circuits and their reachability . Arch. Electr. Eng . 60 ( 3 ), 283 - 301 ( 2011 )
11. T. Kaczorek , Positive linear systems consisting of n subsystems with different fractional orders . IEEE Trans. Circuits Syst . 58 ( 6 ), 1203 - 1210 ( 2011 )
12. T. Kaczorek , Reachability and observability of fractional positive electrical circuits . Comput. Probl. Electr. Eng . 3 ( 2 ), 28 - 36 ( 2013 )
13. T. Kaczorek , Selected Problems of Fractional Systems Theory (Springer, Berlin, 2011 )
14. T. Kaczorek , Standard and positive electrical circuits with zero transfer matrices . Pozn. Univ. Technol. Acad. J. Electr. Eng . 85 , 11 - 28 ( 2016 )
15. T. Kaczorek , K. Rogowski , Fractional Linear Systems and Electrical Circuits, Studies in Systems, Decision and Control , vol. 13 (Springer, New York, 2015 )
16. T. Kailath , Linear Systems (Prentice Hall, Englewood Cliffs , 1980 )
17. R. Kalman , Mathematical description of linear systems . SIAM J. Control 1 ( 2 ), 152 - 192 ( 1963 )
18. R. Kalman , On the general theory of control systems , in Proceedings of First International Congress on Automatic Control (International Federation of Automatic Control (IFAC) , Butterworth, London, 1960 ), pp. 481 - 493
19. H. Rosenbrock , State-Space and Multivariable Theory (Wiley, New York, 1970 )
20. W.A. Wolovich , Linear Multivariable Systems (Springer, New York, 1974 )
21. L.A. Zadeh , C.A. Desoer , Linear System Theory: The State Space Approach (Dover Publications , New York, 2008 )