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

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.


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][20][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].
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 the set of n ×m real matrices, n×m + denotes the set of n ×m matrices with nonnegative and n + = n×1 + , C is the field of complex numbers, M n stand for the set of n × n Metzler matrices (real matrices with nonnegative off-diagonal entries), I n is the n × n identity matrix.

Invariant, Decoupling and Blocking Zeros of Linear Systems
Consider the linear systemẋ ∈ p are the state, input and output vectors and A ∈ n×n , B ∈ n×m , C ∈ p×n . The system matrix of linear system (2.1) is defined by Let the matrix Proof It is easy to see that The state vector x of system (2.1) for any input u(t) and zero initial state of system matrix (2.2).

Definition 2.3 [8]
A number z ∈ C for which rank are called the output-decoupling (o.d.) zero of system (2.1). Let the matrix The number of o.d. zeros n 2 of system (2.1) is equal to the rank defect of its observability matrix, i.e., Theorem 2.4 [8] The output y of system (2.1) for any input u (t) = Bu(t) and zero initial condition where [I n 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.

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 equationṡ

input and output vectors and
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.
contains a monomial matrix.
Definition 3.4 [9,15] contains a monomial matrix. The transfer matrix of positive electrical circuit (3.1) is given by

5)
where p×m (s) is the set of p × m rational matrices in s.
This completes the proof. if C = 0. Therefore, the pair (A, B) is unreachable. Similarly, from (3.11) we have ⎡ if B = 0. Therefore, the pair ( A,C) is unobservable.

Linear Electrical Circuits with Zero Transfer Matrices
Following [14] the positive linear electrical circuits with zero transfer matrices will be presented. Fig. 1 with given resistances R 1 , R 2 , R 3 , R 4 , inductance L, capacitance C and voltage source e.

Example 4.1 Consider the electrical circuit shown in
Using Kirchhoff's laws we may write the equations As the output y we choose y = u C . where The transfer function of the electrical circuit is and the electrical circuit is stable for all nonzero values of R 1 , R 2 , R 3 , R 4 , L and C . By Theorems 3.2 and 3.3 the positive electrical circuit with (4.3b) is unreachable and unobservable since the matrices have only one monomial column and one monomial row, respectively. From (4.7) we have 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 R k , inductances L k , capacitances C k and voltage sources e k .
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. In general case the class of positive electrical circuits with zero transfer matrix can be presented in the form shown in Fig. 3 [14].

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 , L [1,3], R [2,3] can be reduced to the form and the positive electrical circuit has one input-decoupling zero z i1 = − 1 R 4 C . The electrical circuit has also one output-decoupling zero z o1 = − R L since rank 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 input-decoupling zeros and output-decoupling zeros and has not input-output-decoupling zeros (n i + n o = 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) n io = 0, i.e., n i + n o = n. Proof Proof follows immediately from (2.20) and the assumption that the transfer matrix of the positive electrical circuit is zero.

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: (1) have no invariant zeros, input-output-decoupling zeros and blocking zeros (Theorems 5.1, 5.2), (2) the list of eigenvalues of the system matrix A is the sum of the list of inputdecoupling zeros and the list of output-decoupling zeros of the electrical circuit (Theorem 5.2).
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.