Circuit Synthesis of Passive Descriptor Systems A Modified Nodal Approach
Timo Reis†
Abstract
In this work we consider the problem of multiport passive reciprocal network
synthesis by descriptor systems. A method is presented that leads to circuit equations
in modified nodal analysis.
Keywords:
descriptor systems, passivity, reciprocity, positive realness, modified nodal analysis
1
Introduction
The problem of network synthesis has a quite long tradition in systems theory and goes
back to Cauer [7, 8, 9] in the first half on the 20th century (see [6] for historical overview).
He discovered that the positive real and rational functions are exactly those which can be
realized by electrical circuits. Further important contributions in this area are by Brune
[5] and Darlington [12]. The so far discussed works consider the synthesis problem
from a frequency domain point of view, that is, given is a positive real transfer function
G(s) and the aim is to find an electrical circuit with impedance function G(s). In the
1960s, positive realness became well understood by the Positive Real Lemma [1], which is
a criterion on the matrices A ∈ Rn,n , B ∈ Rn,p , C ∈ Rq,n , D ∈ Rq,p for the positive realness
of the transfer function G(s) = D + C(sI − A)−1 B of a standard state-space system
ẋ(t) = Ax(t) + Bu(t),
y(t) = Cx(t) + Du(t),
(1)
where x(t) ∈ Rn is a state vector, u(t) ∈ Rp is a control input and y(t) ∈ Rq is an output.
Based on this result, circuit synthesis problem was then regarded from a time-domain point
Institut für Mathematik, MA 4-5, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin,
Germany (reis@math.tu-berlin.de). Supported by the DFG Research Center Matheon ”Mathematics
for key technologies” in Berlin.
1
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
2
of view [3, 4, 22, 23]. A state space realization is given and it is looked for a circuit whose
input-output transition behaviour coincides with the given system. Circuit synthesis that
is based on state-space models has many advantages in comparison to the pure frequency
domain methods. On the one hand, state-space models provide more insight into the
topology of the circuit and on the other hand, linear state-space models are more useful for
numerical computations. However, the modelling of electrical circuits with standard statespace systems is limited. The reason is that there is a huge class of circuits that containing
physical variables which are formed by the derivative of the input. This class cannot
be described by standard state-space models but one has to deal with with differentialalgebraic systems - also called - descriptor systems
E ẋ(t) = Ax(t) + Bu(t),
y(t) = Cx(t) + Du(t),
(2)
which differ from standard systems by a possibly singular E ∈ Rn,n in front of the derivative of the state vector. In most cases, we consider descriptor systems with D = 0. This is
no restrictive assumption since any this can be included by a suitable choice of the other
matrices [11].
The differential-algebraic approach is much more natural for circuit modelling since besides
differential equations like component relations of reactive elements, pure algebraic equations appear such as Kirchhoff’s laws and the component relations for resistors and ideal
transformers. A very common modelling technique is the so-called modified nodal analysis
(MNA) [17, 24], a method that is based on a graph theoretical consideration of the circuit.
The advantages of the MNA are that the modelling can be done automatically, the circuit
topology can be directly read off from the equations.
In the area of reduced order circuit modelling suitable algorithms for circuit synthesis are
useful. Various works like [14, 10, 19, 20] consider the positive realness preserving model
order reduction of MNA equations. Applying circuit synthesis to the reduced order model,
a reduced order circuit is obtained. Then common circuit simulation packages like for
instance TITANr , SPICEr and PSIMr can be used for the numerical simulation of the
reduced order model.
In this paper, we consider the following problem: Given a descriptor system (2) with D = 0,
˙
the aim is to construct another descriptor system in MNA form Ē x̄(t)
= Āx̄(t) + B̄u(t),
−1
y(t) = C̄ x̄(t) with same transfer function, i.e. C(sE −A) B = C̄(sĒ − Ā)−1 B̄. We restrict
to the class of reciprocal systems, meaning that the transfer function satisfies a certain symmetry property. The class of reciprocal systems contains electrical circuits whose elements
satisfy a coupling symmetry such as it holds for resistances, inductances, capacitances,
ideal transformers and independent voltage and current sources.
This paper is organized as follows: In the remaining part of this section we introduce the
basic notation and definitions. The following section deals with foundations of the modified nodal analysis. Section 3 collects the needed facts about positive real and reciprocal
descriptor systems. The main results and algorithms for the synthesis of electrical circuits
are presented in Section 4. Before this work is concluded in Section 6, the presented results
are illustrated by means of an example in Section 5.
3
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
Throughout the paper Rn,m and Cn,m denote the spaces of n × m real and complex matrices whereas Gln (R) and Gln (C) are the sets of real and complex invertible n × n matrices,
respectively. The open right half-plane is denoted by C+ = { s ∈ C : Re(s) > 0 }
and i is the imaginary unit. The matrices AT and A∗ denote, respectively, the transpose
and the conjugate transpose of A ∈ Cn,m , and A−T = (A−1 )T . An identity matrix of
order n is denoted by In or simply by I. The zero matrix of dimensions n times m is
denoted by 0n,m or simply by 0. We will denote by im A the image and by ker A the
nullspace of a matrix A. Further, for symmetric matrices P, Q ∈ Rn,n we write P > Q
(P ≥ Q) if P − Q is positive (semi-)definite. By writing P > Q or P ≥ Q, we implicitly mean that both P and Q are symmetric. A matrix pair (E, A) is called regular, if there exists s ∈ R such that sE − A ∈ Gln (R). For any regular pair (E, A)
there exist W, T ∈ Gln (R) such that (W ET, W AT ) is in Kronecker normal form [16]
(W ET, W AT ) = diag(Inf , N ), diag(Af , In∞ ) for some Af ∈ Rnf ,nf and a nilpotent
N ∈ Rn∞ ,n∞ . The nilpotency index ν of N is called the index of the pair (E, A).
For a descriptor system (2), we generally assume that (E, A) is regular. We identify the
index of a descriptor system (2) with the index of (E, A). The transfer function of (2) defined by G(s) = C(sE − A)−1 B + D. It is a rational matrix-valued function that describes
the input-output relation of system (2) in the frequency domain. On the other hand, for
any rational function G, one can find a descriptor system whose transfer function is G [11].
We call such a descriptor system (2) a realization of G. We sometimes use the notation
[ E, A, B, C, D ] for a system (2). For a realization with D = 0, we use the notation
[ E, A, B, C ]. Such a realization [ E, A, B, C ] is called minimal if the dimension of the
matrices E and A is as small as possible. The condition that the matrices [ αE +βA , B ],
[ αE T +βAT , C T ] have full row rank for all (α, β) ∈ C2 \{(0, 0)} is equivalent to minimality
of a realization (2) [11]. The transfer function G is called proper if lim kG(s)k < ∞, and
s→∞
improper otherwise. We can additively decompose G(s) =P
Gsp (s) + P (s), where Gsp (s) is
k
p,q
strictly proper and P is a polynomial matrix, i.e., P (s) = ν−1
k=0 Mk s for some Mk ∈ R .
We call Gsp the strictly proper part and
part of G, respectively. DefinPνP the polynomial
k
ing Gp (s) := Gsp (s) + M0 and P s := k=1 Mk s , we have the alternative decomposition
G(s) = Gp (s) + P s (s). We call Gp proper part and P s the strict polynomial part of G.
Now we define the special properties of descriptor systems.
Definition 1 A descriptor system (2) with transfer function G is called reciprocal if p = q
and there exist p1 , p2 ∈ N with p1 + p2 = p such that for Σext = diag(Ip1 , −Ip2 ) and all
s ∈ C for which G has no pole in s holds
G(s)Σext = Σext GT (s).
The matrix Σext ∈ Rp,p is then called external signature of G
Reciprocity is examined in [2, 26] in detail.
4
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
Definition 2 A descriptor system (2) is called passive if p = q and for all sufficiently
smooth and square integrable u : R → Rp , the system satisfies
Z
u(τ )T y(τ ) dτ ≥ 0.
R
System (2) is called lossless passive if the above relation becomes an equality.
Generally speaking, passivity means that system does not produce energy. The claim ”sufficiently smooth” in Definition 2 is stated to guarantee that the output is square integrable.
Indeed, passive systems are may differentiate the input.
Definition 3 A square transfer function G(s) = C(sE − A)−1 B is called positive real if
it has no poles on C+ and, furthermore, for all s ∈ C+ holds G(s) + G(s)∗ ≥ 0.
If the transfer function furthermore satisfies G(iω) + G(iω)∗ = 0 for all ω ∈ R such that
G has no pole at iω, then G is called lossless positive real.
A descriptor system (2) is passive if and only if its transfer function G is positive real.
Furthermore, a system is lossless passive if and only if its transfer function is lossless
positive real. For a detailed proof of these facts, we refer to [4]. We will collect some
further properties of reciprocal, passive and lossless passive systems in Section 3.
2
Modified Nodal Analysis
A general electrical circuit can be modelled as a directional graph whose nodes correspond
to the nodes of the circuit and whose branches correspond to the circuit elements like
capacitors, inductors, resistors, transformers, current sources and voltage sources. The
input vector is given by u(t) = [ iTI (t) , vVT (t) ], where iI and vV are currents of current
sources and voltages of voltage sources, respectively. As output, we chose the voltages of
current sources and currents of voltage sources, i.e. y(t) = [ vIT (t) , iTV (t) ]. Using Kirchhoff’s
current and voltage laws as well as the branch constitutive relations, the modified nodal
analysis leads to a descriptor system (2) with D = 0 and



E =

AC C ATC
0
0
0
0
L
0
0
0
0
0
0
0
0
0
0



,


−AR R−1ATR

AT

A =  T LT T
 ATi −T ATt
ATV
−AL
0
0
0


−ATi +ATt T −AV
−ATI


0
0 
 0
, B = C T = 
 0
0
0 
0
0
0
0
0
0
−I



.

(3)
The state is given by x(t) = [ eT (t) , iTL (t) , iTTi (t) , iTV (t) ]T , where e contains the node potentials, iL and iTi are the vectors of currents of inductances and initial ports of ideal
transformers, respectively. The incidence matrix A′ := [ AC , AL , AR , ATi , ATt , AV , AI ] is
composed of the element-related incidence matrices of capacitances, inductances, resistances, initial ports of ideal transformers, terminal ports of ideal transformers, voltage
sources and current sources. A′ contains the information on the topology of the circuit.
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
5
Furthermore, C, R and L are, respectively, the capacitance, resistance and inductance matrices and T denotes the gain matrix of the transformer. We denote by ne , nL , nTi , nV and
nI the numbers of nodes (except the grounding node), inductances, initial ports of ideal
transformers, voltage sources and current sources, respectively. We now make general
assumptions on the above defined matrices.
Assumptions 4
(i) AV has full column rank.
(ii) [ AC , AL , AR , ATi , ATt , AV ] has full row rank.
(iii) ker[ ATi , ATt , AV ] = {0} × ker[ ATt , AV ].
(iv) ker[ AC , AR , AL , AV ]T = ker[ AC , AR , AL , ATt , AV ]T .
(v) C > 0, R > 0 and L > 0.
The first claim corresponds to the absence of loops of voltage sources while (ii) forbids
cutsets of current sources. Condition (iii) excludes loops of terminal ports of ideal transformers together with initial ports and/or voltage sources, whereas in (iv), cutsets of terminal together with initial ports and/or current sources are excluded. The requirement on
the capacitance, resistance and inductance matrices means that all elements are couplingsymmetric and do not generate energy. For a more detailed mathematical statement of the
above conditions and their connection to topological assertions, we refer to [13, 21].
Proposition 5 If Assumption 4 is valid, then the pair (E, A) with matrices as in (3) is
regular.
Proof:
Let λ ∈ R with λ > 0 and let x ∈ ker(λE − A) with x = [ xT1 , xT2 , xT3 , xT4 ]T partitioned
according to the block structure of E and A in (3). Then we have xT (λ(E + E T ) − (A +
AT ))x = 0 and thus xT1 (λAC CATC + AR R−1 ATR )x1 = 0 and λxT2 Lx2 = 0. Since λ > 0,
(v) implies x1 ∈ ker[ AC , AR ]T and x2 = 0. We further obtain from (λE − A)x = 0
that x1 ∈ ker[ AC , AR , AL , AV ]T . Condition (iv) then implies that ATTt x1 = 0 and thus
ATTi x1 = 0. Hence, we get ATTi x1 = (ATTi − TT ATTt )x1 = 0 and altogether
x1 ∈ ker[ AC , AL , AR , ATi , ATt , AV , ]T .
Then condition (ii) yields x1 = 0 and thus x ∈ ker(λE − A) reduces to x1 = 0, x2 = 0 and
(ATi − ATt T)x3 + AV x4 = 0, i.e.,
[ xT3 , −xT3 TT , xT4 ]T ∈ ker[ ATt , ATi , AV ].
From condition (iii), we obtain x3 = 0 and thus AV x4 = 0. Claim (i) leads to x4 = 0.
Altogether we have ker(λE − A) = {0} for λ > 0 and thus the regularity of (E, A).
We now show that the descriptor system formed by the MNA equations is both passive
and reciprocal under Assumption 4.
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
6
Proposition 6 If Assumption 4 is valid, then the descriptor system (2) with matrices as
in (3) is passive and reciprocal with external signature Σext = diag(InI , −InV ). If the circuit,
moreover, does not contain resistances, the system is lossless passive.
Proof:
To show the reciprocity, consider the matrix Σint := diag(Ine , −InL +nTi +nV ). Then the
reciprocity can be obtained from the relations C = B T , BΣext = Σint B, EΣint = Σint E T
and AΣint = Σint AT .
Passivity follows from E ≥ 0, A + AT ≤ 0 and the identity
G(s) + G∗ (s) = B T (sE − A)−1 (2Re(s)E − (A + AT ))(sE − A)−∗ B.
In the case where resistances are absent, we furthermore have A + AT = 0. The above
relation then implies that G is lossless positive real.
3
Positive Real and Reciprocal Systems
The aim of this section is to show that any passive descriptor system that is reciprocal
with external signature Σext = diag(Ip1 , −Ip2 ) has a realization of the form
E11 0
ẋ1 (t)
A11 A12 x1 (t)
B1 0
u1 (t)
=
+
,
0 E22 ẋ2 (t)
−AT12 0
x2 (t)
0 B2 u2 (t)
T
(4)
y1 (t)
B1
0
x1 (t)
=
y2 (t)
0 B2T x2 (t)
where E11 , A11 ∈ Rn1 ,n1 , E22 ∈ Rn2 ,n2 with E11 ≥ 0, E22 ≤ 0, A11 ≤ 0 and A12 ∈ Rn1 ,n2 ,
B1 ∈ Rn1 ,p1 , B2 ∈ Rn2 ,p2 . One can observe that (4) has a block partition that looks alike
an MNA system. Indeed, descriptor systems of this form will be the basis for further
manipulations leading to a system in MNA form.
First, we collect further properties of reciprocal and passive descriptor systems that we
require for the main result of this section.
Proposition 7 ([4], p. 216) A rational transfer function G(s) is positive real if and only
if its proper part Gp is positive real and its strictly polynomial part is given by P s (s) = sM1
for some M1 ≥ 0.
The previous result implies that for a minimal realization of a positive real transfer function,
the index cannot exceed 2. Next we recall the positive real lemma, a criterion on the
matrices A, B, C and D of a standard system realization (1) such that the transfer function
G(s) = C(sI − A)−1 B + D is positive real.
Theorem 8 (Positive Real Lemma for Standard Systems, [1, 4]) A minimal standard system (1) is passive if and only if there exist K ∈ Rn,p , J ∈ Rp,p and X ∈ Rn,n with
X > 0, that solve the positive real Lur’e equations
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
AT X + XA = −K T K,
XB − C T = −K T J,
D + DT = J T J.
7
(5)
Furthermore, a system is lossless passive if and only if there exists a solution X > 0 of (5)
with K = 0 and J = 0.
In the case were D + DT is regular, the positive real Lur’e equations can be reformulated
as algebraic Riccati equations [1, 4] and they can be solved by common techniques [18].
For singular D + DT , a method for the solution of Lur’e equations is presented in [25].
An extension of the positive real lemma to descriptor systems is given in [15]. For systems (2) satisfying lims→∞ (Gp (s) + GTp (s)) ≤ D + DT , it is shown there that passivity is
equivalent to the existence of X ∈ Rn,n with E T X ≥ 0 and
T
A X + XT A XT B − CT
≤ 0.
(6)
BT X − C
D + DT
For getting an equivalent criterion for the passivity of descriptor systems with D = 0, we
have to claim that lims→∞ (Gp (s) + GTp (s)) = 0. However, this is true for lossless passive
descriptor systems and we can formulate the following result.
Lemma 9 (Positive Real Lemma for Lossless Descriptor Systems) A minimal descriptor system (2) with D = 0 is lossless passive if and only if there exists a unique
X ∈ Gln (R) such that
AT X + X T A = 0,
X T B − C T = 0 and E T X ≥ 0.
(7)
Proof:
In the case where X ∈ Gln (R) satisfying (7) exists, the losslessness of (2) follows from
Theorem 1 in [15] and the proof therein.
To show that losslessness implies the solvability of (7), we make use of Theorem 2 from
[15] to obtain that there exists an X ∈ Rn,n such that (6) holds for D = 0, that is
AT X + X T A ≤ 0, B T X = C together with E T X = X T E ≥ 0. Furthermore, from [15], p.
72, it follows that for all ω ∈ R such that the transfer function G has no pole in iω, we
have
0 = G(iω)+G(−iω)T = B T (−iωE T −AT )−1 (−AT X −X T A)(iωE −A)−1 B ≥ 0.
The complete controllability of the system then implies that AT X + X T A = 0. It remains
to be shown that X is non-singular.
Since the transposed system is also lossless passive, there exists a Y ∈ Rn,n such that
AY + Y T AT = 0, CY = B T . This implies that Y T X T A = AY X, Y T X T E = EY X and
C = CY X. Therefore, we have that
C(sE − A)−1 = CY X(sE − A)−1 = C(sE − A)−1 X T Y T .
Due to the complete observability of the system, we then have X T Y T = I. Hence, the
solution X is both unique and invertible.
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
8
Due to uniqueness of X, the above result can be reformulated that lossless passivity
of a minimal system is equivalent to E T X ≥ 0 for a matrix X ∈ Rn,n that satisfies
AT X + X T A = 0, X T B − C T = 0, E T X = X T E. For computational issues, this is an
alleviation since X can now be obtained by linear matrix equations instead of linear matrix
inequalities.
In the next result we show that two realizations of the same transfer function are connected by a state space transformation. The existence of the state space transformation is
a well-known result in the theory of descriptor systems [11]. We will furthermore show the
uniqueness of these transformations.
Lemma 10 Let a minimal descriptor system (2) with transfer function G(s) be given
e A,
e B,
e C
e ] be a further minimal realization of G(s). Then there exist unique
and let [ E,
e = W ET , A
e = W AT , B
e = W B, C
e = CT . Moreover, both W
W, T ∈ Rn,n such that E
and T are then invertible.
Proof:
The existence of W, T follows from a result in [11]. It remains to show their uniqueness.
e A = A,
e B = B,
e C = C
e and T, W ∈ Rn,n such
For this, we assume that E = E,
that W ET = E, W AT = A, W B = B, CT = C. If we show that W = T = In ,
the uniqueness result follows immediately. Let λ ∈ R such that λE − A is invertible.
Then we have (λE − A)−1 E(I − T ) = (I − T )(λE − A)−1 E. Assume that T is not the
identity. Then im(I − T ) is a non-trivial (λE − A)−1 E-invariant subspace and therefore
there exists x ∈ im(I − T )\{0} with (λE − A)−1 Ex = µx for some µ ∈ C. This implies
((λE − A)−1 E − µI)x = 0 and thus ((λµ − 1)E − A)x = 0. Since C = CT we, moreover,
have that Cx = 0 and thus x ∈ ker[ (λµ − 1)E T − AT , C ]T . This is a contradiction to the
minimality. The further equation W = In can be obtained by a dual argument.
Lemma 10 implies that for reciprocal systems, there exist unique W, T ∈ Gln (R) with
E T = W ET, AT = W AT, C T = W BΣext , Σext B T = CT.
(8)
This will be the basis for the subsequent results. Before we show the main theorem of
this section, we consider two special cases. The first results considers lossless passive and
reciprocal systems.
Lemma 11 Let a minimal, lossless passive and reciprocal descriptor system (2) with transfer function G(s) and external signature Σext = diag(Ip1 , −Ip2 ) be given. Then there exist
W, T ∈ Gln (R) and Ē11 ∈ Rn1 ,n1 , Ē22 ∈ Rn2 ,n2 with Ē11 ≥ 0, Ē22 ≥ 0 and Ā12 ∈ Rn1 ,n2 ,
B̄1 ∈ Rn1 ,p1 , B̄2 ∈ Rn2 ,p2 , such that
Ē11 0
0
Ā12
B̄1 0
T T
W ET =
, W AT =
, WB =T C =
.
(9)
0 Ē22
−ĀT12 0
0 B̄2
Proof:
By Lemma 10, it is no loss of generality to assume that E ≥ 0, A = −AT and B = C T
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
9
(otherwise, perform a transformation from the left with X T where X solves AT X + X T A =
0, X T B − C T = 0, E T X ≥ 0). Since the system is reciprocal, Lemma 10 leads to the
existence of unique matrices W̄ , T̄ ∈ Gln (R) satisfying (8). The relations (8) lead to
T̄ T E W̄ T = E,
T̄ T AW̄ T = −A,
T̄ T B = BΣext .
The uniqueness of the transformations that is guaranteed by Lemma 10 gives us W̄ = T̄ T .
Moreover, we can multiply (8) from the left with W̄ −1 and from the right with T̄ −1 to
obtain that
W̄ −1 E T̄ −1 = E,
W̄ −1 AT̄ −1 = −A,
W̄ −1 B = BΣext .
The uniqueness of the transformations then leads to the further relations T̄ −1 = T̄ ,
W̄ −1 = W̄ . Hence the spectra of W̄ and T̄ are contained in {1, −1}. Thus, together
with T̄ = W̄ T , there have to exist matrices W ∈ Gln (R) and Σint = diag(In1 , −In2 ) such
that W̄ = W −T Σint W T and T̄ = W Σint W −1 . Now transform with W and T = W T to obtain W ET , W AT , W B and CT satisfying W̄ E T̄ ≥ 0, W̄ AT̄ = −(W̄ AT̄ )T , W̄ B = (C T̄ )T
and furthermore Σint W ET = W ET Σint , Σint W AT = −W AT Σint , Σint W B = (CT )T Σext .
However, this implies that W ET , W AT , W B and CT are structured as in (9).
The matrix Σint has a similar role as Σext for the input and is therefore called an internal signature matrix. Note that for computational issues, it is beneficial to use the
self-invertibility of T̄ and the fact W̄ = T̄ T . Therefore, T̄ is the unique solution of the
linear matrix equations T̄ T E − E T̄ = 0, T̄ T A + AT̄ = 0, T̄ T B = BΣext .
Now, a result for standard systems is followed. In [4], the case of systems with symmetric
transfer functions is considered. We present a slight generalization to reciprocal systems.
Lemma 12 Let a minimal positive real and reciprocal standard system (1) with transfer
function G(s) and external signature Σext = diag(Ip1 , Ip2 ) be given. Then there exists a
system


u
(t)
#
#
"
"
1
e11 0 B
e13 0 u2 (t)
e12 x1 (t)
B
ẋ1 (t)
0
A


+
=
e22 0 B
e24 u3 (t)
eT
x
(t)
ẋ2 (t)
0
B
−A
0
2
12
u4 (t)




  eT

(10)
e 12
e 14
B11 0
0
D
0 D
y1 (t)
u
(t)
1

T 
T
e22
e 12
e 23 0 
y2 (t) 

0 B
0
D
 x1 (t)
−D


=
u2 (t)
+




T
T
y3 (t) B

e
e
 0
0  x2 (t)
−D
0
0  u3 (t)
13
23
y4 (t)
u4 (t)
eT
eT
0 B
−D
0
0
0
24
14
e12 ∈ Rn1 ,n2 , B
eij ∈ Rni ,pj , D
e ij ∈ Rpi ,pj , such that (10) together with the additional
with A
relations u3 (t) = −y3 (t), u4 (t) = −y4 (t) has the same input-output behaviour as (1).
Before the proof is presented, we state an auxiliary result.
Lemma 13 ([4], p. 410) Let a symmetric matrix T ∈ Gln (R) be given. Then there exists
an orthogonal V ∈ Gln (R), Σ = diag(In1 , −In2 ) ∈ Gln (R) and some an R > 0, such that
10
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
1
1
T = RV T ΣV = V T ΣV R. Furthermore, the matrix square root R 2 satisfies R 2 V T ΣV =
1
V T ΣV R 2 and it holds V RV T = diag(R1 , R2 ) for some positive definite R1 ∈ Rn1 ,n1 ,
R2 ∈ Rn2 ,n2 .
The matrix R can be obtained by a polar decomposition of T , i.e. T = RU for some symmetric and positive definite R ∈ Gln (R) and an orthogonal U ∈ Gln (R). It is furthermore
shown in [4] that U 2 = I and therefore, U admits an eigenvalue decomposition U = V T ΣV
for some Σ = diag(In1 , −In2 ) and an orthogonal V ∈ Gln (R). Then R, V, Σint have the
desired properties.
Proof of Lemma 12:
Since the standard system (1) is minimal and positive real, there exists X > 0 satisfying
the positive real Lur’e equations (5). Factorizing P = LTP LP for some square matrix LP ,
−T
−T
−T
T
¯
the matrices Ā = L−T
P ALP , B̄ = LP B, C̄ = CLP , D̄ = D, K̄ = LP K, J = J then
satisfy
Ā + ĀT = −K̄ K̄ T , C̄ T − B̄ = −K̄J T , D̄ + D̄T = JJ T .
(11)
The reciprocity and Lemma 10 implies the existence T ∈ Gln (R) with
ĀT = T −1 ĀT, C̄ T = T −1 B̄Σint , Σint B̄ T = C̄T.
By transposing these relations, multiplying with T −T from the left and with T −1 from the
right, we furthermore have
ĀT = T −T ĀT T , C̄ T = T −T B̄Σint , Σint B̄ T = C̄T T .
Lemma 10 now implies that T = T T . By Lemma 13, we get T = RV T Σint V = V T Σint V R
for some R > 0, an orthogonal matrix V ∈ Gln (R) and Σint = diag(In1 , −In2 ) ∈ Gln (R).
1
Now consider the matrix T1 := R 2 V . Then we have
1
1
T1 Σint T1T = R 2 V Σint V T R 2 = V T Σint V R = T.
e := T1−1 AT
e 1, B
e := T1−1 B,
e C
e := CT
e 1, D
e := D
e satisfy
As a consequence, we have that A
eT = Σint AΣ
e int , B
e T = Σint CΣ
e ext , C
eT = Σext BΣ
e int , D
e T = Σext DΣ
e ext .
A
Hence, according to the block structure of Σext

e11
−A
#
"

e
e
eT12
A
f = −A −B = 
M

e
e
eT
B
C
D
11
T
e12
−B
and Σint , we obtain a partition

e12 −B
e11 −B
e12
−A
e22 −B
e21 −B
e22 
−A

,
eT D
e 11 −D
e 12 

−B
21
T
T
e22
e 12
e 22
B
D
D
e 11 , D
e 22 . We now show that M
f+ M
fT ≥ 0. With
with symmetric Â11 , Â22 , D
"
#
e
e
−V T AV
−V T B
M=
,
e
e
CV
D
11
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
it follows that
#
T 1 "
T 1
e −B
e R− 21 V T 0
2
−
A
V
R
0
V R2V
f
M=
=
e
e
0
I
0
I
0
C
D
1
0
V R− 2 V T 0
M
.
I
0
I
1
1
1
Lemma 13 and the uniqueness of the matrix square root implies V R− 2 V T = diag(R12 , R22 ).
T
Due to (11), we have M + M ≥ 0. Now partition M according to the block structure of
f, i.e. M = [Mij ]i,j=1,...,4 . Then we have
M
"
# −1 T ! e
e
R
0
M
M
M
M
A
−
B
R
0
1
11
14
11
14
11
12
1
f1 :=
M
+
,
e T −D
e 22 = 0 I
M41 M44
M41 M44
0 I
−B
12
"
# −1 T ! e
e
e
e22 0
A
−
B
M
M
M
M
B
0
B
22
21
22
23
22
23
22
f2 :=
M
+
.
T
e21
e 11 = 0 I
M32 M33
M32 M33
0 I
−B
−D
Since the matrices in the middle of the right hand side of the above equations are positive
T
f1 and M
f2 are
semidefinite as submatrices of M + M , we get that the eigenvalues of M
f
f
all non-negative. Together with the symmetry, this implies M1 ≥ 0 and M2 ≥ 0 and thus
f+ M
fT ≥ 0.
M
Now consider full rank factorizations
#
#
"
#
"
#
"
"
h
i
h
i
e
e
e
e
e
e
−A22 −B21
B24 e T e T
−A11 −B12
B13 e T e T
,
B24 D14 =
B13 D23 =
T
eT D
e 11 .
e
e
e
e
−B
D14
−B12 D22
D23
21
e12 , B
e11 , B
e13 , B
e22 , B
e24 , D
e 12 , D
e 14 , D
e 23 , now consider the
With the so far introduced matrices A
system (10). By setting u3 (t) = −y3 (t) and u4 (t) = −y4 (t) and plugging in the relations
e13 B
e T = −Â11 , B
e13 D
e T = −B
e12 , D
e 23 D
eT = D
e 22 , B
e24 B
e T = −A
e22 , B
e24 D
e T = −B
e21 ,
B
13
23
23
24
14
T
e 14 D
e 14 = D
e 11 , we obtain the system
D
e
e
ẋ(t) = Ax(t)
+ Bu(t),
e
e
y(t) = Cx(t)
+ Du(t),
which has the same transfer function as (1).
Based on the previous lemmas, we now present the main result of this section.
(12)
Theorem 14 Let a minimal and positive real descriptor system (2) be given that is reciprocal with transfer function G satisfying Σext G(s) = G(s)T Σext for Σext = diag(Ip1 , −Ip2 ).
Then there exists a realization (4) of G with E11 ≥ 0, E22 ≥ 0, A11 ≤ 0.
Proof:
Without loss of generality, we assume that the system is in Kronecker normal form
I 0
Af 0
Bf
E=
, A=
, B=
, C = Cf C∞
0 N
0 I
B∞
(13)
12
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
where N is nilpotent and B, C are partitioned according to the block structure of E. Due
to the positive realness and the minimality, we have that N 2 = 0 and the system transfer
function has the form
G(s) = −sC∞ N B∞ − C∞ B∞ + Cf (sI − Af )−1 Bf .
A comparison of coefficients yields that the proper part of G is also reciprocal with internal
signature Σext . Therefore, the system
ẋf (t) = Af xf (t) + Bf u(t)
yf (t) = Cf xf (t) − C∞ B∞ u(t)
(14)
is reciprocal and due to Proposition 7 it is moreover positive real. Furthermore, due to
reciprocity and positive realness, we have that
M11 0
−C1 N B1 =
0 M12
for some M11 ∈ Rp1 ,p1 , M12 ∈ Rp2 ,p2 with M11 ≥ 0, M12 ≥ 0. Now Lemma 12 yields that
(14) has the same input-output-behaviour as a system (10) with the additional relation
u3 (t) = −y3 (t) and u4 (t) = −y4 (t). Now consider the transfer function


e 12
e 14
sM11 D
0 D
 eT
e 23 0 

−D12 sM11 D
Gip (s) = 
(15)
,
T
e
 0
−D23 0
0 
T
e 14
−D
0
0
0
which is reciprocal with external signature Σip = diag(Ip1 , −Ip2 , Ip3 , −Ip4 ) and moreover
lossless positive real. By Lemma 9, we get that there exists a realization Gip (s) =
Cip (sEip − Aip )−1 Bip with
Ē11 0
0
Ā12
B̄11 0 B̄13 0
T
Eip =
, Aip =
, B̂ip = Ĉip =
.
0 Ē22
−ĀT12 0
0 B̄22 0 B̄24
Consider the system
2
Ē11
6 0
6
4 0
0
0
Ē22
0
0
0
0
I
0
3 2
32
3 2
32
0
Ā12
0
0
B̄11
x1 (t)
ẋ1 (t)
0
T
6 0
7
6
6
7
7
6
0
0
0 7 6x2 (t)7
07 6ẋ2 (t)7 6−Ā12
7 6
=
e11
e12 5 4x3 (t)5 + 4B
05 4ẋ3 (t)5 4 0
0
0
A
T
e12
x4 (t)
ẋ4 (t)
I
0
0
0
−A
0
3
32
3 2 T
2
T
e
B̄11
0
B11
0
x1 (t)
y1 (t)
T
T 76
e22
7
6y2 (t)7 6 0
B̄22
0
B
7 6x2 (t)7 .
7=6
6
T
4y3 (t)5 4B̄ T
e13
0
B
0 5 4x3 (t)5
13
T
x4 (t)
y4 (t)
eT
0
B̄24
0
B
24
0
B̄22
0
e22
B
B̄13
0
e13
B
0
3
32
0
u1 (t)
7
6
B̄24 7 6u2 (t)7
7
0 5 4u3 (t)5
e
u4 (t)
B24
Now introducing a new state x̄(t) = y4 (t) and plugging in the relations u2 (t) = −y2 (t),
u4 (t) = −y4 (t), a further reordering of the states leads to a system of type (4) with in
13
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
particular





T
eT
Ā12 0
−B̄13 B̄13
−B̄13 B
0
Ē11 0 0
13
e12 ,
T
T
e13 B̄13
e13 B
e13
E11 =  0 I 0, A11 = −B
−B
0 , A12 =  0 A
T
eT
0 0 0
B̄24
B
0
0
−I
24
 
B̄11
e11 , E22 = Ē22 0 , B2 = B̄22 .
B1 = B
e22
0 I
B
0

4
Synthesis
Up to now, from a realization of a positive real and reciprocal descriptor system, we have
constructed another structured realization (4) that resembles a system in MNA form.
For the construction of an MNA realization some further transformation is necessary.
In principle, we are looking for invertible matrices T1 ∈ Gln1 (R), T2 ∈ Gln1 (R) such
that T1T E11 T1 = AC CATC , T1T A11 T1 = −AR R−1 ATR , T1T A12 T2 = [ AL, ATi − ATt T, −AV ],
T
T2T E22 T2 = diag(L, 0, 0), T1T B11 = −AI , T2T B22 = 0 0 −I for some C > 0, L > 0 a
matrix T and some incidence matrices AC , AR , AL , ATi , ATt , AI . Unfortunately, such a
transformation will be not possible in any case. One reason for this can be that the matrix
B22 does not have full column rank. However, we will see in the following that certain
transformation together with the induction of additional states leads to a system that is
in MNA form. The proof of the following result is constructive and a numerical algorithm
can be easily deduced.
Theorem 15 Let a system (4) with E11 , A11 ∈ Rn1 ,n1 , E22 ∈ Rn2 ,n2 , A12 ∈ Rn1 ,n2 , B1 ∈
Rn1 ,p1 , B2 ∈ Rn2 ,p2 and E11 ≥ 0, E22 ≥ 0, A11 ≤ 0. Then there exists a descriptor system
(2) which has the same transfer function as (4) and is, furthermore, in MNA form with
incidence matrices satisfying Assumption 4.
Proof:
We will successively construct realizations
"
#T 
# "
# "
# "
(i)
(i)
(i)
(i)
(i)
A11
A12
B1
0
0
B1
0

 E11
,
(i)
(i) ,
(i) T
(i) ,
0 B2
0 E22
−(A12 )
0
0 B2
(16)
of (4) such that in the final step, a system in MNA form is obtained. The following steps
either consist of elimination or introduction of states or of transformations of type
(i+1)
(i)
(i)
(i)
(i+1)
(i)
(i+1)
(i)
(i)
(i)
(i+1)
(i)
E11
= T1 E11 (T1 )T, E22
A12 = T1 A12 (T2 )T, B1
(i)
(i)
(i+1)
(i)
(i)
(i+1)
(i)
(i)
(i)
= T2 E22 (T2 )T, A11 = T1 A11 (T1 )T,
= T1 B (i),
B2
= T2 B2
(17)
14
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
(i)
(i)
for some invertible T1 , T2 . Note that each of the following steps preserves the regularity
of the matrix pair formed by the first two matrices in (16). For the block matrices, we will
use the notations
h i
h i
h i
h
i
(i)
(i)
(i)
(i)
(i)
(i)
(i)
(i)
E11 = Cjk , E22 = Ljk , A11 = Gjk , A12 = Tjk .
We will pass on specifying the matrix dimensions, since they can be obtained from the
context.
Step 1:
(1)
Perform a transformation T E22 T T = diag(L11 , 0) with some orthogonal T and a square
(1)
and invertible L11 . Partition T = [ T1 , T2 ] according to the block structure of T E22 T T and

 (1)
define




L11 0 0 0
0
0
0
0
0
0

(1)
(1)
E11 = 0 C(1)
0 , E22 = 
22

0
0
0


0
0
−I
0
 (1)

(1)
(1)
(1)
0 ,
A12 = T21 T22 T23
(1)
(1)
T31 T32
0
−I
(1)
(1)
(1)
0 0 0
 , A(1) = 0 G(1)
11
22
0 0 0
0
0
0 0 0
 
 
0
I
0 
(1)
(1)

.
B1 = 0 , B2 =  
0 
0
−I
0
0
0
(1)
0 ,
0
(1)
(1)
where C22 = E11 , G22 = A11 , T21 = A12 T1 , T22 = A12 T2 , T23 = B1 , T31 = −B2T T1 ,
(1)
T32 = −B2T T2 . The resulting systems has the same transfer function as the original one.
Step 2:
Consider an orthogonal T̄ such that
(1)
T̄ C22 T̄ T

(2)
(2)
C22
(C(2) )T
23
=
 0
0
C23
(2)
C33
0
0
0
0
0
0

 (2)
0
G22


0 , T̄ G(1) T̄ T =  0
22
(G(2) )T
0
24
0
0
(1)
Perform a transformation (17) with i = 1, T1

0
0
(2)
0
C
22

(2) T

(2) 0 (C
)
23
E11 = 
0
0
0
0
0
0

0
0
(2)
0
G22


0
0
(2)
A11 = 
0 (G(2) )T

24
0
0
0
0
0
(2)
C23
(2)
C33
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
(2)
0 G24
0
0
(2)
0 G44
0
0
0
0
0
0
0
0
0
0

0
0

0
,
0

0
0

0
0

0
,
0

0
0
(2)
0 G24
0
0
(2)
0 G44
0
0
(1)
= diag(I, T̄ , I), T2

(1)
L11
(2) 
0
E22 = 
 0
0

0
T(2)
 21
 (2)
(2) T31
A12 =  (2)
T41
 (2)
T51
(2)
T61
0
0
0
0
0
0
0
0
0
(2)
T22
(2)
T32
(2)
T42
(2)
T52
(2)
T62
0
= I and obtain

0
0
,
0
0
−I
(2)
T23
(2)
T33
(2)
T43
(2)
T53
0

0
0
.
0

0
0

0

,
0

0
−I
 
I
0 
 
0
(2) 

B1 = 
0  ,
 
0 
0

0
(2)  0 

B2 = 
 0 .
−I

15
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
Step 3:
(2)
(2)
Factorize T̄1 T51 T̄2T = diag(I, 0) and perform a transformation (17) with matrices T1 =
(2)
diag(I, I, I, I, T̄1 , I), T2 = diag(T̄2 , I, I, I, I). Afterwards, perform a further row operation
(2) (2)
to T1 A12 that eliminates the first block row. This yields a form (16) with

0
0
(2)
0
C22

0 (C(2) )T
23
(3) 
E11 =
0
0
0
0

0
0
0
0

0
0
(2)
0
G22

0
0
(3) 
(2) T
A11 =
0
(G

24 )
0
0

0
0
0
0
0
0
(2)
C23 0
(2)
C33 0
0
0
0
0
0
0
0
0
0
0
(2)
0 G24
0
0
(2)
0 G44
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
0
0
0
0
0
0

0
0

0

0
,
0

0
0

0
0

0

0
,
0

0
0

(3)
L11
(L(3) )T
 12
(3)
E22 =


(3)
L12
(3)
L22
0
0
0

0
0


0
(3) 
A12 =
0
I


0
0
0
0
0
0
0
0
0
0
0
(3)
T22
(3)
T32
(3)
T42
0
0
(3)
T72
0
(3)
T23
(3)
T33
(3)
T43
(3)
T53
(3)
T63
(3)
T73
0
0
0
0
0
−I
(3)
T24
(3)
T34
(3)
T44
(3)
T54
(3)
T64
(3)
T74

0
0

0
,
0
0

0
0


0

0
,
0


0
−I
 
I
0 
 
0 

(3) 

B1 =
0,
0 
 
0 
0
 
0
0
 
0
(3) 

B2 =
 0 .
 
0
−I
Step 4:
(3)
The regularity of the matrix pair in the system (16) for i = 3 leads to the fact that T63
(3)
has full row rank. Hence we can factorize T̄1 T63 T̄2T = [ I, 0 ] and perform a transformation
(3)
(3)
(17) with T1 = diag(I, I, I, I, T̄1 , I) and T2 = diag(T̄2 , I, I, I, I). Afterwards, perform a
(3) (3)
row operation that eliminates the entries in the third block row of T1 A12 and a column
(3) (3)
operation that eliminates the entries in the sixth block row of T1 A12 . Then we get a form
(16) with

0
0
(2)
0
C
22

(2)
0 (C )T

23
(4)
E11 = 
0
0
0
0

0
0
0
0

0
0
(2)
0
G22

0
0
(4) 
(2) T
A11 = 
0
(G

24 )
0
0

0
0
0
0
0
(2)
C23
(2)
C33
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
(2)
0 G24
0
0
(2)
0 G44
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
0

0

0
,
0

0
0

(3)
L11
(L(3) )T
 12
(4) 
E22 = 




0
0

0

(4) 
A12 = 0

I

0
0
0
0
0
0
0
(4)
T22
(4)
T32
(4)
T42
0
0
(4)
T72
(3)
L12
(3)
L22
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
(4)
T24
(4)
T34
(4)
T44
(4)
T54
0
(4)
T74
0
0
0
0
0
0
0
0
0
0
0
0
−I
(4)
T25
(4)
T35
(4)
T45
(4)
T55
0
(4)
T75

0
0

0
,
0

0
0

0
0

0


0 ,

0

0
−I
 
I
 0
 
 0
(4)  

B1 = 
 0 ,
 0
 
 0
0

0
 0
 
0
(4) 

B2 = 
 0 .
 
 0
−I

Step 5:
(4)
Since the state corresponding to the sixth block row and the third block column of A12 is
both uncontrollable and unobservable, we can simply cancel these block rows and columns
16
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
and obtain

0
0

(5) 0
E11 = 
0

0
0

0
0

0
(5) 
A11 = 
0

0
0
0
(2)
C22
(2) T
(C23 )
0
0
0
0
(2)
G22
0
(2) T
(G24 )
0
0
0
0
(2)
C23 0
(2)
C33 0
0
0
0
0
0
0
0
0
(2)
0 G24
0
0
(2)
0 G44
0
0
0
0
Step 6:
Introduce a new state and obtain

0
0
(2)
0
C
22

(2)
0 (C )T

23
(6)
E11 = 
0
0
0
0

0
0
0
0

0
0
(2)
0
G
22

0
0

(6)
(2) T
A11 = 
0 (G24 )
0
0

0
0
0
0
0
0
(2)
C23 0
(2)
C33 0
0
0
0
0
0
0
0
0
0
0
(2)
0 G24
0
0
(2)
0 G44
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
 (3)
(3)
L11
L12

0
(L(3) )T L(3)

22
0
, E (5) =  120
0
22


0
 0
0

0
0
0
0


0
0
0
0
(4)
(4)

0
T
T
22
24
0


0 T(4) T(4)
0

(5)
32
34
, A = 
(4)
12
0 T(4)
T44
0

42

(4)
I
0
0
T54
(4)
(4)
0
0 T72 T74
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

0

0
,
0

0
0

0
0

0

0
,
0

0
0

(3)
L11
(L(3) )T
 12
(6) 
E22 = 




0
0

0

(6) 
A12 = 0

I

0
0
(3)
L12
(3)
L22
0
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
(4)
T22
(4)
T32
(4)
T42
0
(4)
T72
−I
0
0
0
0
0
 

I
0
 0
 
0


(5) 
 0 ,

,
B
=
0
1
 0
 
0
 0
0
0

0
 
0
0

0 


0
(5) 

, B2 = 
 0 .
0

0 

0
−I
−I
0
0
0
0
0
−I
(4)
T25
(4)
T35
(4)
T45
(4)
T55
(4)
T75
0
0
0
0
0
0
0
0
0
0
0
0
0
(4)
T24
(4)
T34
(4)
T44
(4)
T54
(4)
T74
0
0
0
0
0
0
0
−I
(4)
T25
(4)
T35
(4)
T45
(4)
T55
(4)
T75
0

0
0

0
,
0

0
0

0
0

0


0 ,

0

−I 
0
 
I
 0
 
 0
(6)  

B1 = 
 0 ,
 0
 
 0
0

0
0
 
(6)  0 

B2 = 
 0 .
 
0
−I

Step 7:
(6)
(4)
(4)
(4)
(4)
Consider the matrix T4∗ := [ (T24 )T , (T34 ))T , (T44 )T , (T54 )T ]T . Regularity implies that
(6)
T4∗ has full column rank. Denoting l ∈ N to be the number of its columns and Eij to be
the matrix entry 1 at the position (i, j) and zero elsewhere, this property leads to the fact
(6)
that a column-echelon form of T4∗ has the form



(4)
T24
A24
 (4) 
 A34
T34 )
 (4)  · V = 
 A44
 T44 
(4)
A54
T54



(7)
T24
A24
(7) 
 
 A34
T
)


 +  34  , where 
  T(7) 
 A44
44
(7)
A54
T54

(7)

l
 X
=
Elj ,j for some lj ∈ N.

j=1
Particularly, each row of Tj4 equals zero if Aj4 has a one in that row.

0
0
(2)
0
C22

0 (C(2) )T
23
(7) 
E11 = 
0
0
0
0

0
0
0
0
0
(2)
C23
(2)
C33
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

0
,
0

0
0

(3)
L11
(L(3) )T
 12
(7) 
0
E22 = 
 0

 0
0
(3)
L12
(3)
L22
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
 
I
0
 
0

(7) 

B1 = 
0 ,
0
 
0
0
17
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH

0
0
(2)
0
G
22

0
0

(7)
(2) T
A11 = 
0 (G24 )
0
0

0
0
0
0
0
0
(2)
0 G24
0
0
(2)
0 G44
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
0

0

(7) 
A12 = 0

I

0
0
0
0
0
0
0
0
I
0
(7)
T23
(7)
T33
(7)
T43
(7)
T53
(7)
T63
−I
0
(7)
A24 + T24
(7)
A34 + T34
(7)
A44 + T44
(7)
A54 + T54
(7)
T64
0
−I
(7)
T25
(7)
T35
(7)
T45
(7)
T55
(7)
T65
0

0
 
0
0

 0



0
0

(7) 

0  , B2 =  0 
.

 

0
 0
−I 
−I
0
(7)
Furthermore, by a transformation (16) that adds multiples of the fourth block row of A12
to the third one, we can make sure that the matrices
(7)
(7)
(7)
(7)
(7)
(7)
(7)
(7)
(7)
(7)
T3∗ := [ (T23 )T , (T33 )T , (T43 )T , (T53 )T ]T ,
T5∗ := [ (T25 )T , (T35 )T , (T45 )T , (T55 )T ]T
have zero entries in each row in which [ AT24 , AT34 , AT44 , AT54 ]T contains a one.
Step 8:
Now construct a full column rank matrix Ā2 = [ ĀT23 , ĀT33 , ĀT43 , ĀT53 ]T , where Āi3 has as
many rows as Ai3 for i = 2, 3, 4, 5, each column of Ā2 consists of a canonical unit vector
and each row of Ā2 contains a one if and only if A2 has only zeros in the corresponding
row. Further construct T̄ with the property that

 (7)
(7)
(7)
T23 T24 T25
 (7)
(7)
(7) 
T34 T35 
 T
Ā2 T̄ =  33
(7)
(7)  .
 T(7)
T44 T45 
43
(7)
(7)
(7)
T53 T54 T55
We now obtain an MNA system fulfilling Assumptions 4 with matrices
T
T
AV = 0 0 0 0 0 I 0 ,
AI = −I 0 0 0 0 0 0 ,
T
T
0 I 0 0 0 0 0
0 I 0 0 0 0 0
AC =
,
AR =
,
0 0 I 0 0 0 0
0 0 0 I 0 0 0
T
T
0 0 0 0 −I 0 0
0 0
0
0
0 0 I
AL =
,
ATt =
,
0 0 0 0 0 0 −I
0 ĀT23 ĀT33 ĀT43 ĀT53 0 0

T
#
"
0
0
0
0
0 0 −I
(2)
(2)
C
,
C
22
23
ATi = −  0 AT23 AT33 AT43 AT53 0 0  ,
C=
(2) ,
(2) T
)
C
(C
33
23
−I 0
0
0
0 0 0
#−1
#
"
"
(7)
(2)
(2)
(3)
(3)
(7)
(7)
G22
G24
L11
L12
T63 T64 T65 0
T=
.
R=
L=
(2)
(2) ,
(3)
(3) ,
0
0
0 T̄
(G24 )T , G44
(L12 )T L22
18
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
5
Examples
In this section we present two examples of descriptor systems that will be realized by circuits. The techniques presented in the proofs of Lemma 11, Lemma 12, Theorem 14 and
Theorem 15 were implemented in MATLAB 7.
Example 1:
Consider a descriptor

0
E = 0
0
system (2) with


1 0
1
0 0  , A = 0
0 1
0
matrices
 

0 0
0
1 0  , B = −1 , C = 1 1 1 .
1
0 −1
2
and it can be shown that G is positive
The transfer function is given by G(s) = s +2s+2
s+1
real. Reciprocity is a trivial consequence of the scalarity of G. However, it can be chosen
whether one takes p1 = 1 and p2 = 0 or p1 = 0 and p2 = 1. The first choice leads to
a realization with one current source and no voltage source, whereas the second choice
correspond to a circuit with no current source and one voltage source. Aiming to construct
a circuit with one voltage source, we apply Theorem 14 to obtain a system (4) with in
particular
0.5 0
0 0
−0.5 0
−0.7101 0
−1
E11 =
, E22 =
, A11 =
, A12 =
, B2 =
,
0 0
0 1
0
−1
0
−1
1
and B1 is an empty matrix. Now applying Theorem 15, we obtain a system in MNA form
with empty AI and
 
 
 






0
1
0
1 0
0 0
0 −1
0
0
0
0 1
1 0
0 0 

 
 






AV = 
1 , AC = 0 , AL =  0  , AR = 0 0 , ATt = 0 1 , ATi = 0 0  ,
0
0
−1
0 0
0 0
1 0
2
0
1
0
R=
, T=
, L = 1, C = 0.5.
0 1.000
−1 −1.4142
The corresponding circuit can be read off from the above matrices and is presented below.
1
0
−1−1.4142
uV (t)
1
1
0.5
0.5
19
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
Example 2:
Consider a positive real descriptor system (2) with matrices

1
0

0
E =
0

0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0


0
0
−1
0



0
 , A= 0

0
0


0
1
0
0
1 0 0 0
0 0 0 0
0 0 1 0
0 −1 0 0
0 0 0 1
0 0 0 0
The transfer function is given by
s + s22s+1
G(s) =
− s21+1


0
0
1
0



0
 , B = 0

0
0


0
0
1
1
1
s2 +1
2s
s2 +1


0
0
2
0



0
 , C = 1

0
1


1
0
0
0
T
0
1

0
 .
0

1
0
and therefore, the system is lossless passive and additionally reciprocal with internal signature Σext = diag(1, −1). Applying the method of Lemma 11, we obtain a system
2
0.5570
6−0.6436
6
6−0.1587
E =6
0
6
4
0
0
2
0
0
6
6
0
6
A=6
6−0.6258
4−1.0549
1.1931
0
0
0
−1.0516
1.2702
−0.1807
−0.6436
1.9137
−0.9081
0
0
0
0
0
0
0.7625
0.2525
0.1607
−0.1587
−0.9081
1.0632
0
0
0
0.6258
1.0516
−0.7625
0
0
0
3
0
0
0
0
7
7
0
0
7
7,
0.1564
−0.54007
2.0000
−0.77095
−0.7709
1.1901
2
3
−0.2120
−1.1931
0.1807 7
6 1.7390
6
7
−0.16077
6 0.3937
7 , B = CT = 6
0
0
6
7
4
5
0
0
0
0
0
0
0
1.0653
0.1564
−0.5400
1.0549
−1.2702
−0.2525
0
0
0
3
0
0
7
7
0
7
7,
0.9232 7
−1.00005
0.5371
Now constructing the circuit matrices according to Theorem 15, we obtain
2
0
6 0
6
6 0
6
AL = 6−1
6
6 0
4 0
0
3
2
0
0
0 7
61
7
6
0 7
60
7
6
0 7 , AC = 60
7
6
0 7
60
40
0 5
0
−1
2
»
–
1
3 0
C=
, L = 40
0 1
0
0
0
0
0
0
−1
0
2 3
2
3
2 3
−1
0
0
0
07
6 0 7
61
607
6 7
6
7
6 7
17
6 0 7
60
607
6 7
6
7
6 7
07 , AV = 607 , AI = 6 0 7 , ATt = 60
6 7
6
7
6 7
07
6 0 7
60
617
4 0 5
40
405
05
0
0
1.0198
−0.1981
0 0
0 0
1 0
0 1
0 0
0 0
0 0 0
0
0 2
3
0.2986
−2.9851
0
6−0.9950 −0.0995
6
5
−0.1981 , T = 4
0
0
2.9802
−0.4925 −2.1812
3
2
0
0
07
60
7
6
07
60
7
6
07 , ATi = 60
7
6
17
60
41
05
0
3 0
−2.1213
0.7071 7
7
5
1
0
0
0
0
0
0
0
1
3
1
07
7
07
7
07 ,
7
07
05
0
1
0
−2.1812
0.7071
−0.0995
0
0
−0.4925
uV (t)
−0.9950
1
−2.1213
1
0.2986
3
−2.9851
A realizing circuit is therefore given by
iI (t)
−0.1981
1.0198
2.9802
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
6
20
Conclusion
In this paper we have considered the problem of circuit synthesis. For the class of reciprocal
and passive time-invariant linear descriptor systems, it is shown that a circuit can be
found that realizes the descriptor system, i.e. their input-output behaviours coincide. The
proofs are constructive and numerical techniques can be derived. The methods have been
implemented in Matlab and examples have been presented.
References
[1] B.D.O. Anderson. A system theory criterion for positive real matrices. SIAM J.
Control, 1967; 5(2), pp. 171-182.
[2] B.D.O. Anderson and R.W. Newcomb. On reciprocity in linear time-invariant networks. Technical Report 6558-3, Systems Theory Laboratory, Stanford Electronics
Laboratories, 1965.
[3] B.D.O. Anderson and R.W. Newcomb. Impedance synthesis via state-space techniques. In Proc. IEE, 1968; 115, pp. 928-936.
[4] B.D.O. Anderson and S. Vongpanitlerd. Network Analysis and Synthesis. Prentice
Hall, Englewood Cliffs, New Jersey, 1973.
[5] O. Brune. Synthesis of a finite two-terminal network whose driving-point impedance
is a prescribed function of frequency. J. Math. Phys., 1931; 10, pp. 191-236.
[6] E. Cauer, W. Mathis, and R. Pauli. Life and work of Wilhelm Cauer (1900-1945).
In Proceedings of the 14-th International Symposium on Mathematical Theory of Networks and Systems, Perpignan, France, June 19-23 2000.
[7] W. Cauer. Die Verwirklichung der Wechselstromwiderstände vorgeschriebener Frequenzabhängigkeit. Archiv für Elektrotechnik, 1926; 17, pp. 355-388.
[8] W. Cauer. Über Funktionen mit positivem Realteil. Mathematische Annalen, 1932;
106(1), pp. 369-394.
[9] W. Cauer. Theorie der linearen Wechselstromschaltungen, Vol. I. Akad. Verlagsgesellschaft Becker und Erler, Leipzig, 1941.
[10] X. Chen and J.T. Wen. Positive realness preserving model reduction with H∞ norm
error bounds. Systems Control Lett., 2005; 54(4), pp. 361-374.
[11] L. Dai. Singular Control Systems. Lecture Notes in Control and Information Sciences,
118. Springer-Verlag, Berlin/Heidelberg, 1989.
CIRCUIT SYNTHESIS - A MODIFIED NODAL APPROACH
21
[12] S. Darlington. A history of network synthesis and filter theory for circuits composed
of resistors, inductors, and capacitors. IEEE Trans. Circuits Syst. Regul. Pap., 1999;
46(1), pp. 4-13.
[13] D. Estévez Schwarz and C. Tischendorf. Structural analysis for electric circuits and
consequences for MNA. Int. J. Circ. Theor. Appl., 2000; 28(2), pp. 131-162.
[14] R.W. Freund. SPRIM: structure-preserving reduced-order interconnect macromodeling. In Technical Digest of the 2004 IEEE/ACM International Conference on
Computer-Aided Design, Los Alamos, 2004, pp. 80-87.
[15] R.W. Freund and F. Jarre. An extension of the positive real lemma to descriptor
systems. Optim. Methods Softw., 2004, 19(1), pp. 69-87.
[16] F.R. Gantmacher. Theory of Matrices. Chelsea Publishing Company, New York,
1959.
[17] C.W. Ho, A. Ruehli, and P.A. Brennan. The modified nodal approach to network
analysis. IEEE Trans. Circuits Syst., 1975; CAS-22(6), pp. 504-509.
[18] P. Lancaster and L. Rodman. Algebraic Riccati equations. Clarendon Press, Oxford,
1995.
[19] T. Reis and T. Stykel. Passive and bounded real balancing for model reduction of
descriptor systems. in preparation, 2007.
[20] T. Reis and T. Stykel. Passivity-preserving model reduction of electrical circuits. in
preparation, 2007.
[21] R. Riaza and C. Tischendorf. Qualitative features of matrix pencils and DAEs arising
in circuit dynamics. Dyn. Syst., 2007; 22(2), pp. 107 - 131.
[22] R.Yarlagadda. Network synthesis - a state-space approach. IEEE Trans. Circuit
Theory, 1972; CT-19(3), pp. 227-232.
[23] S. Vongpanitlerd. Reciprocal lossless synthesis via state-variable techniques. IEEE
Trans. Circuit Theory, 1970; CT-17(4), pp. 630-632.
[24] L.M. Wedepohl and L. Jackson. Modified nodal analysis: An essential addition to
electrical circuit theory and analysis. Engineering Science and Education Journal,
2002, 11(3), pp. 84-92.
[25] H. Weiss, Q. Wang, and J.L. Speyer. System characterization of positive real conditions. IEEE Trans. Automat. Control, 1994; 39(3), pp. 540-544.
[26] J.C. Willems. Dissipative dynamical systems. II: Linear systems with quadratic supply
rates. Arch. Ration. Mech. Anal., 1972, 45, pp. 352-393.