Circuit Synthesis - An MNA Approach
Timo Reis
Model Reduction Seminar
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Outline
1
Motivation
2
Descriptor Systems
3
Circuit Equations
4
Circuit Synthesis
5
Example
6
Conclusion and Outlook
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Problem Formulation
Given is a descriptor system
E ẋ(t) = Ax(t) + Bu(t)
y(t) = Cx(t).
Find an electrical circuit such that for the voltages uV (t), uI (t) of voltage and
current sources and currents iV (t), iI (t) of voltage and current sources hold:
–
–
»
»
i (t)
u (t)
and y(t) = V
u(t) = V
uI (t)
iI (t)
fulfill the given descriptor system for some state vector x(t).
Arising questions:
For which class of descriptor systems can we find a circuit that realizes
its input-output-behaviour?
Can we formulate numerical methods for the realization?
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Problem Formulation
Given is a descriptor system
E ẋ(t) = Ax(t) + Bu(t)
y(t) = Cx(t).
Find an electrical circuit such that for the voltages uV (t), uI (t) of voltage and
current sources and currents iV (t), iI (t) of voltage and current sources hold:
–
–
»
»
i (t)
u (t)
and y(t) = V
u(t) = V
uI (t)
iI (t)
fulfill the given descriptor system for some state vector x(t).
Arising questions:
For which class of descriptor systems can we find a circuit that realizes
its input-output-behaviour?
Can we formulate numerical methods for the realization?
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
History of the problem
Wilhelm Cauer (1920s-1940s)
Consideration of the problem from a frequency domain point of
view.
Single-Input-Single-Output-Systems.
Discovered that impedance functions of circuit satisfy some
special conditions (positive realness).
Otto Brune (1930s-1940s) and Sidney Darlington (1950s-1970s) gave
further realization techniques (in frequency domain)
B.D.O Anderson and S. Vongpanidlerd (1960s-1990s) gave methods for
the realization of multiple-input-multiple-output ODE models
ẋ = Ax + Bu, y = Cx + Du. Realization based on matrix operations.
We are interested in time-domain methods for
differential-algebraic models!
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
History of the problem
Wilhelm Cauer (1920s-1940s)
Consideration of the problem from a frequency domain point of
view.
Single-Input-Single-Output-Systems.
Discovered that impedance functions of circuit satisfy some
special conditions (positive realness).
Otto Brune (1930s-1940s) and Sidney Darlington (1950s-1970s) gave
further realization techniques (in frequency domain)
B.D.O Anderson and S. Vongpanidlerd (1960s-1990s) gave methods for
the realization of multiple-input-multiple-output ODE models
ẋ = Ax + Bu, y = Cx + Du. Realization based on matrix operations.
We are interested in time-domain methods for
differential-algebraic models!
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Outline
1
Motivation
2
Descriptor Systems
3
Circuit Equations
4
Circuit Synthesis
5
Example
6
Conclusion and Outlook
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Let a descriptor system
E ẋ(t) = Ax(t) + Bu(t)
y(t) = Cx(t).
be given with E, A ∈ Rn,n , B ∈ Rn,p , C ∈ Rq,n such that the pencil λE − A is
regular.
Definition
The transfer function G ∈ R(s)q,p of this system is given by
G(s) = C(sE − A)−1 B.
The system is called minimal, if the state space dimension n is minimal
among all descriptor systems with transfer function G(s).
Lemma
For i = 1, 2, let two minimal descriptor systems Ei ẋi (t) = Ai xi (t) + Bi ui (t),
yi (t) = Ci xi (t) with Ei , Ai ∈ Rn×n , Bi ∈ Rn×p , Ci ∈ Rq×n be given which have
the same transfer function. Then there exist unique W , T ∈ Rn,n such that
E2 = WE1 T , A2 = WA1 T , B2 = WB1 , C2 = C1 T .
Moreover both W , T are invertible.
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Let a descriptor system
E ẋ(t) = Ax(t) + Bu(t)
y(t) = Cx(t).
be given with E, A ∈ Rn,n , B ∈ Rn,p , C ∈ Rq,n such that the pencil λE − A is
regular.
Definition
The transfer function G ∈ R(s)q,p of this system is given by
G(s) = C(sE − A)−1 B.
The system is called minimal, if the state space dimension n is minimal
among all descriptor systems with transfer function G(s).
Lemma
For i = 1, 2, let two minimal descriptor systems Ei ẋi (t) = Ai xi (t) + Bi ui (t),
yi (t) = Ci xi (t) with Ei , Ai ∈ Rn×n , Bi ∈ Rn×p , Ci ∈ Rq×n be given which have
the same transfer function. Then there exist unique W , T ∈ Rn,n such that
E2 = WE1 T , A2 = WA1 T , B2 = WB1 , C2 = C1 T .
Moreover both W , T are invertible.
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Let a descriptor system
E ẋ(t) = Ax(t) + Bu(t)
y(t) = Cx(t).
be given with E, A ∈ Rn,n , B ∈ Rn,p , C ∈ Rq,n such that the pencil λE − A is
regular.
Definition
The transfer function G ∈ R(s)q,p of this system is given by
G(s) = C(sE − A)−1 B.
The system is called minimal, if the state space dimension n is minimal
among all descriptor systems with transfer function G(s).
Lemma
For i = 1, 2, let two minimal descriptor systems Ei ẋi (t) = Ai xi (t) + Bi ui (t),
yi (t) = Ci xi (t) with Ei , Ai ∈ Rn×n , Bi ∈ Rn×p , Ci ∈ Rq×n be given which have
the same transfer function. Then there exist unique W , T ∈ Rn,n such that
E2 = WE1 T , A2 = WA1 T , B2 = WB1 , C2 = C1 T .
Moreover both W , T are invertible.
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Given a system E ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) with E, A ∈ Rn×n ,
B ∈ Rn×p , C ∈ Rp×n and transfer function G(s) ∈ R(s)p×p .
Definition
A system is passive if for all (sufficiently
smooth) u(·) ∈ L2 (R+ , Rp )
R∞ T
2
+
p
holds y(·) ∈ L (R , R ) with 0 u (τ )y(τ )dτ ≥ 0.
A system is lossless passive if for all (sufficiently
smooth)
R∞
u(·) ∈ L2 (R+ , Rp ) holds y(·) ∈ L2 (R+ , Rp ) with 0 u T (τ )y(τ )dτ = 0.
A transfer function G(s) ∈ R(s)p×p is positive real if it has no poles in
C+ and for all s ∈ C+ holds G(s) + G ∗ (s) ≥ 0.
A transfer function G(s) ∈ R(s)p×p is lossless positive real if it is
positive real and for all ω ∈ R such that G has no pole is iω holds
G(iω) + G T (−iω) = 0.
Parseval’s identity implies:
A descriptor system (∗) is (lossless) passive if and only if its transfer function
is (lossless) positive real.
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Given a system E ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) with E, A ∈ Rn×n ,
B ∈ Rn×p , C ∈ Rp×n and transfer function G(s) ∈ R(s)p×p .
Definition
A system is passive if for all (sufficiently
smooth) u(·) ∈ L2 (R+ , Rp )
R∞ T
2
+
p
holds y(·) ∈ L (R , R ) with 0 u (τ )y(τ )dτ ≥ 0.
A system is lossless passive if for all (sufficiently
smooth)
R∞
u(·) ∈ L2 (R+ , Rp ) holds y(·) ∈ L2 (R+ , Rp ) with 0 u T (τ )y(τ )dτ = 0.
A transfer function G(s) ∈ R(s)p×p is positive real if it has no poles in
C+ and for all s ∈ C+ holds G(s) + G ∗ (s) ≥ 0.
A transfer function G(s) ∈ R(s)p×p is lossless positive real if it is
positive real and for all ω ∈ R such that G has no pole is iω holds
G(iω) + G T (−iω) = 0.
Parseval’s identity implies:
A descriptor system (∗) is (lossless) passive if and only if its transfer function
is (lossless) positive real.
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Given a system E ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) with E, A ∈ Rn×n ,
B ∈ Rn×p , C ∈ Rp×n and transfer function G(s) ∈ R(s)p×p .
Definition
A system is passive if for all (sufficiently
smooth) u(·) ∈ L2 (R+ , Rp )
R∞ T
2
+
p
holds y(·) ∈ L (R , R ) with 0 u (τ )y(τ )dτ ≥ 0.
A system is lossless passive if for all (sufficiently
smooth)
R∞
u(·) ∈ L2 (R+ , Rp ) holds y(·) ∈ L2 (R+ , Rp ) with 0 u T (τ )y(τ )dτ = 0.
A transfer function G(s) ∈ R(s)p×p is positive real if it has no poles in
C+ and for all s ∈ C+ holds G(s) + G ∗ (s) ≥ 0.
A transfer function G(s) ∈ R(s)p×p is lossless positive real if it is
positive real and for all ω ∈ R such that G has no pole is iω holds
G(iω) + G T (−iω) = 0.
Parseval’s identity implies:
A descriptor system (∗) is (lossless) passive if and only if its transfer function
is (lossless) positive real.
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Auxiliary result:
Positive real lemma for lossless descriptor systems:
A minimal system E ẋ (t) = Ax(t) + Bu(t), y(t) = Cx(t) is lossless passive if
and only if there exists X ∈ Rn,n such that
AT X + X T A = 0,
X T B − C T = 0 and E T X ≥ 0.
Moreover, in the case of solvability X is invertible and unique.
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Definition
A transfer function G ∈ R(s)p,p is reciprocal if there exists p1 , p2 ∈ N with
p1 + p2 = p such that for Σe = diag(Ip1 , −Ip2 ) and all s ∈ C where G has no
pole holds
G(s)Σe = Σe GT (s).
The matrix Σe is called external signature of the system.
A descriptor system is called reciprocal if its transfer function is reciprocal.
The transfer function of a reciprocal system has the form
»
–
G11 (s)
G12 (s)
G(s) =
,
T
(s) G22 (s)
−G12
T
T
where G11 = G11
∈ R(s)p1 ,p1 , G22 = G22
∈ R(s)p2 ,p2 .
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Definition
A transfer function G ∈ R(s)p,p is reciprocal if there exists p1 , p2 ∈ N with
p1 + p2 = p such that for Σe = diag(Ip1 , −Ip2 ) and all s ∈ C where G has no
pole holds
G(s)Σe = Σe GT (s).
The matrix Σe is called external signature of the system.
A descriptor system is called reciprocal if its transfer function is reciprocal.
The transfer function of a reciprocal system has the form
»
–
G11 (s)
G12 (s)
G(s) =
,
T
−G12
(s) G22 (s)
T
T
where G11 = G11
∈ R(s)p1 ,p1 , G22 = G22
∈ R(s)p2 ,p2 .
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Outline
1
Motivation
2
Descriptor Systems
3
Circuit Equations
4
Circuit Synthesis
5
Example
6
Conclusion and Outlook
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Equations for an electrical circuit:
uR (t) = R · iR (t)
iC (t) = C ·
d
u (t)
dt C
uL (t) = L ·
d
i (t)
dt L
(resistances)
(capacitances)
(inductances)
uTi (t) = TT · iTi (t), iTt (t) =
TTT
· uTi (t)
(ideal transformers)
AR iR (t) + AL iL (t) + ATi iTi (t) + ATt iTt (t) + AV iV (t) + AI iI (t) = 0
(Kirchhoff’s current law)
ATR e(t) = uR (t),
ATV e(t) = uV (t),
ATL e(t) = uL (t),
ATI uI (t) = e(t)
ATTi e(t) = uTi (t),
ATTt e(t)uTt (t),
(Kirchhoff’s voltage law)
where
e(t):
vector of node potentials
R, C, L:
resistance, capacitance and inductance matrix (symm. pos. def)
TT :
transformer gain matrix,
AR , AC , AL , AI , Element incidence matrix of branches of resistances, capacitances,
AV , ATi , ATt :
inductances, voltage/current source and initial/terminal port
of ideal transformers,
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Conclusion
Example
Then modified nodal analysis leads to:
2
6
6
6
6
4
AC C AT
C
0
0
0
L
0
0
0
0
0
0
0
3
0 2 ė(t)
7
6
0 7
76 i̇L (t)
74
0 5 i̇Ti (t)
i̇V (t)
0
»
2
−AR R −1AT
R
6
6
AT
7 6
L
7=6
5 6 T
T T
4 ATi −TT ATt
3
– "
−AT
uI (t)
I
=
iV (t)
0
AT
V
0
0
0
0
−AL
−ATi +ATtTT
−AV
0
0
0
0
0
0
0
0
0
3
2
7
76
76
74
7
5
2
−AI
3
6 0
e(t)
6
6
iL (t) 7
7+6 0
iTi (t) 5 6
6
4 0
iV (t)
0
3
0
0 7
7»
–
7 i (t)
7 I
0 7
7 vV (t)
0 5
−I
3
2
# e(t)
7
0 6
i
(t)
6 L 7,
5
4
−I iTi (t)
iV (t)
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Conclusion
Example
Then modified nodal analysis leads to:
2
6
6
6
6
4
|
AC C AT
C
0
0
0
L
0
0
0
0
0
0
{z
0
=:E
3
0 2 ė(t)
7
0 76 i̇L (t)
76
74
0 5 i̇Ti (t)
i̇V (t)
0
}
3
2
−AR R −1AT
R
6
AT
7 6
L
7=6
5 6
6 AT −T T AT
4 Ti
T Tt
AT
V
=:y (t )
−ATi +ATtTT
−AV
0
0
0
0
0
0
0
0
0
{z
|
– "
−AT
uI (t)
I
=
iV (t)
0
| {z } |
»
−AL
=:A
0
0
0
0
{z
=:C
2
3
# e(t)
6 i (t) 7
6 L 7,
4
5
−I iTi (t)
} iV (t)
0
3
2
7
76
76
74
7
5
|
}
e(t)
iL (t)
iTi (t)
iV (t)
{z
3
2
6
6
76
7+6
56
6
4
}
=:x (t )
|
0
−AI
0
0
0
0
3
0 7
7»
–
7 i (t)
I
0 7
7 vV (t)
7
0 5| {z }
=:u(t )
−I
{z
=:B
}
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Motivation
Descriptor Systems
Circuit Equations
Example
Conclusion
2
−AI
7
6
7
6 0
7
7, B = C T = 6
6 0
7
4
5
0
3
0
7
0 7
7
0 7
5
−I
Circuit Synthesis
Observations for the circuit descriptor system:
2
6
6
E =6
6
4
AC C AT
C
0
0
0
L
0
0
0
0
0
0
0
2
−AR R −1AT
R
6
7
6
AT
0 7
L
7, A = 6
6
7
6 AT −T T AT
0 5
4 Ti
T Tt
0
3
0
−AL
−ATi +ATt TT
−AV
0
0
0
0
0
0
0
0
0
AT
V
3
E and −(A + AT ) are symmetric and positive semidefinite,
if the circuit does not contain resistances, then A + AT = 0,
B = CT .
Conclusion:
The transfer function G(s) = C(sE − A)−1 B of the circuit satisfies for s ∈ C+ :
G(s) + G ∗ (s)
= B T (sE − A)−1 B + B T (sE − AT )−1 B
= 2B T (sE − A)−1 (2 Re(s)E − (A + AT ))(sE − A)−∗ B ≥ 0,
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i.e., the circuit system is positive real.
Motivation
Descriptor Systems
Circuit Equations
Example
Conclusion
2
−AI
7
6
7
6 0
7
7, B = C T = 6
6 0
7
4
5
0
3
0
7
0 7
7
0 7
5
−I
Circuit Synthesis
Observations for the circuit descriptor system:
2
6
6
E =6
6
4
AC C AT
C
0
0
0
L
0
0
0
0
0
0
0
2
−AR R −1AT
R
6
7
6
AT
0 7
L
7, A = 6
6
7
6 AT −T T AT
0 5
4 Ti
T Tt
0
3
0
−AL
−ATi +ATt TT
−AV
0
0
0
0
0
0
0
0
0
AT
V
3
E and −(A + AT ) are symmetric and positive semidefinite,
if the circuit does not contain resistances, then A + AT = 0,
B = CT .
Conclusion:
The transfer function G(s) = C(sE − A)−1 B of the circuit satisfies for s ∈ C+ :
G(s) + G ∗ (s)
= B T (sE − A)−1 B + B T (sE − AT )−1 B
= 2B T (sE − A)−1 (2 Re(s)E − (A + AT ))(sE − A)−∗ B ≥ 0,
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i.e., the circuit system is positive real.
Motivation
Descriptor Systems
Circuit Equations
Example
Conclusion
2
−AI
7
6
7
6 0
7
7, B = C T = 6
6 0
7
4
5
0
3
0
7
0 7
7
0 7
5
−I
Circuit Synthesis
Observations for the circuit descriptor system:
2
6
6
E =6
6
4
AC C AT
C
0
0
0
L
0
0
0
0
0
0
0
2
−AR R −1AT
R
6
7
6
AT
0 7
L
7, A = 6
6
7
6 AT −T T AT
0 5
4 Ti
T Tt
0
3
0
−AL
−ATi +ATt TT
−AV
0
0
0
0
0
0
0
0
0
AT
V
3
E and −(A + AT ) are symmetric and positive semidefinite,
if the circuit does not contain resistances, then A + AT = 0,
B = CT .
Conclusion:
The transfer function G(s) = C(sE − A)−1 B of the circuit satisfies for s ∈ C+ :
G(s) + G ∗ (s)
= B T (sE − A)−1 B + B T (sE − AT )−1 B
= 2B T (sE − A)−1 (2 Re(s)E − (A + AT ))(sE − A)−∗ B ≥ 0,
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i.e., the circuit system is positive real.
Motivation
Descriptor Systems
Circuit Equations
Example
Conclusion
2
−AI
7
6
7
6 0
7
T
6
7, B = C = 6
7
4 0
5
0
3
0
7
0 7
7
0 7
5
−I
Circuit Synthesis
Observations for the circuit descriptor system:
2
6
6
E =6
6
4
AC C AT
C
0
0
0
L
0
0
0
0
0
0
0
2
−AR R −1AT
R
6
7
6
AT
7
0 7
6
L
7, A = 6
6 AT −T T AT
0 5
4 Ti
T Tt
0
AT
0
3
V
−AL
−ATi +ATt TT
−AV
0
0
0
0
0
0
0
0
0
3
For
Σi = diag(Ine , −InL , −InV , −InT ),
Σe = diag(InI , −InV ),
the MNA matrices satisfy
Σi E = Σi E T , Σi A = −Σi AT , Σi B = C T Σe , B T Σi = Σe C.
Hence we have that the transfer function G(s) = C(sE − A)−1 B satisfies
G(s)Σi = Σi GT (s),
i.e. circuit system is reciprocal.
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Motivation
Descriptor Systems
Circuit Equations
Example
Conclusion
2
−AI
7
6
7
6 0
7
T
6
7, B = C = 6
7
4 0
5
0
3
0
7
0 7
7
0 7
5
−I
Circuit Synthesis
Observations for the circuit descriptor system:
2
6
6
E =6
6
4
AC C AT
C
0
0
0
L
0
0
0
0
0
0
0
2
−AR R −1AT
R
6
7
6
AT
7
0 7
6
L
7, A = 6
6 AT −T T AT
0 5
4 Ti
T Tt
0
AT
0
3
V
−AL
−ATi +ATt TT
−AV
0
0
0
0
0
0
0
0
0
3
For
Σi = diag(Ine , −InL , −InV , −InT ),
Σe = diag(InI , −InV ),
the MNA matrices satisfy
Σi E = Σi E T , Σi A = −Σi AT , Σi B = C T Σe , B T Σi = Σe C.
Hence we have that the transfer function G(s) = C(sE − A)−1 B satisfies
G(s)Σi = Σi GT (s),
i.e. circuit system is reciprocal.
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Conclusion
Example
Reformulation of the task:
Given is a passive and reciprocal descriptor system with transfer function
G(s). Find
incidence matrices AR , AC , AL , ATi , ATt , AI , AV ,
positive definite matrices R, C, L,
a transformer gain matrix TT
such that
2
6
6
G(s) = 6
4
−AI
0
0
0
3T 2
sAC CATC + AR R −1ATR
6
7
−ATL
0 7 6
7 6
6
T
0 5 4
−ATi +TTT ATTt
−I
−ATV
0
AL
ATi −ATt TT
sL
0
0
0
0
0
AV
3−1 2
7
0 7
7
0 7
5
0
6
6
6
4
−AI
0
0
0
0
3
0 7
7
7
0 5
−I
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Outline
1
Motivation
2
Descriptor Systems
3
Circuit Equations
4
Circuit Synthesis
5
Example
6
Conclusion and Outlook
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
First auxiliary result:
Theorem
Let a minimal and passive descriptor system E ẋ (t) = Ax(t) + Bu(t),
y(t) = Cx(t) that is reciprocal with external signature Σe = diag(Ip1 , −Ip2 ) be
given. Then, there exist W , T ∈ Gln (R) such that
–
–
»
–
»
»
A11 A12
B1 0
E11
0
T
. WAT =
,
WB
=
(CT
)
=
WET =
0 B2
0
E22
−AT11
0
where Eij , Aij ∈ Rni ,nj , Bi ∈ Rni ,pi with further E11 ≥ 0, E22 ≥ 0, A11 ≤ 0.
If the system is moreover lossless, then we can find a transformation with
A11 = 0.
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Conclusion
Example
Next task:
Given is a system with
–
»
»
A11
0
E11
, A=
E=
0
E22
−AT12
A12
0
–
, B = CT =
»
B11
0
0
B22
–
Introduce
new states
and perform blockdiagonal congruence transformations
–
»
T11
0
, such that T T ET , TAT , T T B = C T T is in desired form.
T =
0
T22
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Conclusion
Example
We demonstrate the technique for the lossless case:
E=
»
E11
0
0
E22
–
, A=
T
Transforming T11
E11 T11 =
»
I
0
»
–
»
0
A12
B11
T
,
B
=
C
=
0
−AT12
0
–
–
»
0
I 0
T
, T22
...
E22 T22 =
0
0 0
0
B22
–
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Conclusion
Example
We demonstrate the technique for the lossless case:
E=
»
E11
0
0
E22
–
, A=
T
Transforming T11
E11 T11 =
»
I
0
»
–
»
0
A12
B11
T
,
B
=
C
=
0
−AT12
0
–
–
»
0
I 0
T
, T22
...
E22 T22 =
0
0 0
0
B22
–
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
... leads to
2
I
6 0
6
E =4
0
0
0
0
0
0
0
0
I
0
2
2
3
0
B11
6
0 7
7 , B = C T = 6 B21
4 0
0 5
0
0
0
6 0
A=6
4 −AT13
−AT14
Introducing new states
0
0
−AT23
−AT24
A13
A23
0
0
3
0
0 7
7
B32 5
B42
.
3
A14
A24 7
7
0 5
0
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
... leads to
2
I
6 0
6
E =4
0
0
0
0
0
0
0
0
I
0
2
2
3
0
B11
6
0 7
7 , B = C T = 6 B21
4 0
0 5
0
0
0
6 0
A=6
4 −AT13
−AT14
Introducing new states
0
0
−AT23
−AT24
A13
A23
0
0
3
0
0 7
7
B32 5
B42
.
3
A14
A24 7
7
0 5
0
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
... leads to
2
6
6
6
6
6
E =6
6
6
6
4
0
0
0
0
0
0
0
0
2 0
6 0
6 0
6
6
6 0
A=6
6 0
6
6 0
6
4
I
0
0
I
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
AT
13
AT
14
T
−B11
0
Transforming T1T A23 T2 =
»
I
0
0
0
0
0
I
0
0
0
0
0
0
0
−AT
23
−AT
24
T
−B21
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−B32
−B42
0
I
0
0
0
0
0
0
0
0
2
I
6 0
7
6
7
6 0
7
6
7
6 0
7
7 , B = CT = 6
6 0
7
6
7
6 0
7
4 0
5
0
3
0
A13
A23
T
B32
0
0
0
0
0
A14
A24
T
B42
0
0
0
0
−I
B11
B21
0
0
0
0
0
0
0
0
−I
0
0
0
0
0
0
0
0
0
0
0
−I
3
7
7
7
7
7
7
7
7
7
5
3
7
7
7
7
7
7.
7
7
7
7
5
–
0
and eliminating some other blocks
0
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
... leads to
2
6
6
6
6
6
E =6
6
6
6
4
0
0
0
0
0
0
0
0
2 0
6 0
6 0
6
6
6 0
A=6
6 0
6
6 0
6
4
I
0
0
I
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
AT
13
AT
14
T
−B11
0
Transforming T1T A23 T2 =
»
I
0
0
0
0
0
I
0
0
0
0
0
0
0
−AT
23
−AT
24
T
−B21
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−B32
−B42
0
I
0
0
0
0
0
0
0
0
2
I
6 0
7
6
7
6 0
7
6
7
6 0
7
7 , B = CT = 6
6 0
7
6
7
6 0
7
4 0
5
0
3
0
A13
A23
T
B32
0
0
0
0
0
A14
A24
T
B42
0
0
0
0
−I
B11
B21
0
0
0
0
0
0
0
0
−I
0
0
0
0
0
0
0
0
0
0
0
−I
3
7
7
7
7
7
7
7
7
7
5
3
7
7
7
7
7
7.
7
7
7
7
5
–
0
and eliminating some other blocks
0
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Conclusion
Example
... leads to
2
6
6
6
6
6
6
6
E =6
6
6
6
6
6
4
0
0
0
0
0
0
0
0
0
0
2
6
6
6
6
6
6
6
A=6
6
6
6
6
6
6
4
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−AT
27
−AT
28
−AT
29
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
E55
E56
0
T
E56
E66
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
−I
0
0
0
0
−AT
57
−AT
−AT
−AT
38
48
58
T
T
−A39
−A49
−AT
59
0
0
I
ˆ
Transforming T1T A48 T2 = I
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
0
0
0
0
0
3
2
I
7
6 0
7
6
7
6 0
7
6
7
6 0
7
6
7
6 0
T
7, B = C = 6
7
6 0
7
6
7
6 0
7
6 0
7
6
5
4 0
0
0
0
−I
0
A27
A28
A29
0
0
A38
A39
0
0
A48
A49
0
A57
A58
A59
−I
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
−I
3
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
7
7
7
7
7
7
7
7.
7
7
7
7
7
7
5
˜
0 and eliminating some other blocks
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Conclusion
Example
... leads to
2
6
6
6
6
6
6
6
E =6
6
6
6
6
6
4
0
0
0
0
0
0
0
0
0
0
2
6
6
6
6
6
6
6
A=6
6
6
6
6
6
6
4
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−AT
27
−AT
28
−AT
29
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
E55
E56
0
T
E56
E66
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
−I
0
0
0
0
−AT
57
−AT
−AT
−AT
38
48
58
T
T
−A39
−A49
−AT
59
0
0
I
ˆ
Transforming T1T A48 T2 = I
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
0
0
0
0
0
3
2
I
7
6 0
7
6
7
6 0
7
6
7
6 0
7
6
7
6 0
T
7, B = C = 6
7
6 0
7
6
7
6 0
7
6 0
7
6
5
4 0
0
0
0
−I
0
A27
A28
A29
0
0
A38
A39
0
0
A48
A49
0
A57
A58
A59
−I
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
−I
3
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
7
7
7
7
7
7
7
7.
7
7
7
7
7
7
5
˜
0 and eliminating some other blocks
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Conclusion
Example
... leads to
2
6
6
6
6
6
6
6
6
E =6
6
6
6
6
6
6
4
0
0
0
0
0
0
0
0
0
0
0
2
6
6
6
6
6
6
6
6
A=6
6
6
6
6
6
6
6
4
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−AT
27
0
−AT
29
−AT
210
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
E55
E56
0
T
E56
E66
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
−I
0
0
0
0
−AT
57
0
−I
0
T
−A39
0
−AT
59
−AT
0
−AT
310
410
0
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
A27
0
0
A57
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
2
I
7
6 0
7
6
7
6 0
7
6
7
6 0
7
6
7
6 0
7
6
T
7, B = C = 6 0
7
6
7
6 0
7
6
7
6 0
7
6 0
7
6
5
4 0
0
0
0
−I
0
0
A29
A210
0
0
A39
A310
0
I
0
0
0
0
A59
A510
−I
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
−I
3
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
7
7
7
7
7
7
7
7
7.
7
7
7
7
7
7
7
5
Eliminating some uncontrollable/onobservable states
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Conclusion
Example
... leads to
2
6
6
6
6
6
6
6
6
E =6
6
6
6
6
6
6
4
0
0
0
0
0
0
0
0
0
0
0
2
6
6
6
6
6
6
6
6
A=6
6
6
6
6
6
6
6
4
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−AT
27
0
−AT
29
−AT
210
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
E55
E56
0
T
E56
E66
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
−I
0
0
0
0
−AT
57
0
−I
0
T
−A39
0
−AT
59
−AT
0
−AT
310
410
0
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
A27
0
0
A57
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
2
I
7
6 0
7
6
7
6 0
7
6
7
6 0
7
6
7
6 0
7
6
T
7, B = C = 6 0
7
6
7
6 0
7
6
7
6 0
7
6 0
7
6
5
4 0
0
0
0
−I
0
0
A29
A210
0
0
A39
A310
0
I
0
0
0
0
A59
A510
−I
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
−I
3
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
7
7
7
7
7
7
7
7
7.
7
7
7
7
7
7
7
5
Eliminating some uncontrollable/onobservable states
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
... leads to
2 0
6 0
6 0
6
6 0
6
6
E =6 0
6
6 0
6
6 0
4
0
0
2
6
6
6
6
6
6
A=6
6
6
6
6
6
4
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−AT
26
−AT
27
−AT
28
0
0
0
0
0
0
0
0
0
0
0
0
0
E55
E56
0
T
E56
E66
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−I
0
0
−AT
46
−AT
−AT
37
47
T
−A38
−AT
48
0
I
0
0
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0 3
0 7
0 7
7
0 7
7
0 7
7, B =
7
0 7
7
0 7
5
0
0
0
0
A26
A27
0
A37
A46
A47
0
0
0
0
0
0
0
0
0
0
2
I
0
0
0
0
0
0
0
0
0
0
0
−I
0
0
0
0
0
6
6
6
6
6
6
T
C =6
6
6
6
6
4
−I
A28
A38
A48
0
0
0
0
0
0
0
0
0
0
0
0
0
−I
3
3
7
7
7
7
7
7
7
7
7
7
7
5
7
7
7
7
7
7
7.
7
7
7
7
7
5
Introducing a new state
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
... leads to
2 0
6 0
6 0
6
6 0
6
6
E =6 0
6
6 0
6
6 0
4
0
0
2
6
6
6
6
6
6
A=6
6
6
6
6
6
4
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−AT
26
−AT
27
−AT
28
0
0
0
0
0
0
0
0
0
0
0
0
0
E55
E56
0
T
E56
E66
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−I
0
0
−AT
46
−AT
−AT
37
47
T
−A38
−AT
48
0
I
0
0
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0 3
0 7
0 7
7
0 7
7
0 7
7, B =
7
0 7
7
0 7
5
0
0
0
0
A26
A27
0
A37
A46
A47
0
0
0
0
0
0
0
0
0
0
2
I
0
0
0
0
0
0
0
0
0
0
0
−I
0
0
0
0
0
6
6
6
6
6
6
T
C =6
6
6
6
6
4
−I
A28
A38
A48
0
0
0
0
0
0
0
0
0
0
0
0
0
−I
3
3
7
7
7
7
7
7
7
7
7
7
7
5
7
7
7
7
7
7
7.
7
7
7
7
7
5
Introducing a new state
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Conclusion
Example
... leads to
2
6
6
6
6
6
6
6
6
6
E =6
6
6
6
6
6
6
6
4
0
0
0
0
0
0
0
0
0
0
0
0
2
6
6
6
6
6
6
6
6
A=6
6
6
6
6
6
6
6
4
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
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
−AT
26
−AT
27
T
−A28
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−I
0
0
−AT
37
−AT
38
0
0
0
0
0
0
0
E55
T
E56
0
0
0
0
0
0
0
0
0
0
E56
E66
0
0
0
0
0
0
0
0
0
0
0
−AT
46
−AT
47
−AT
48
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−I
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
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
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
7
2
7
I
7
6 0
7
6
7
6 0
7
6
7
6 0
7
6
7
7 , B = CT = 6 0
6
7
6 0
7
6
7
6 0
7
7
4 0
7
7
0
5
0
A26
0
A46
−I
0
0
0
0
0
0
0
A27
A37
A47
0
0
0
0
0
0
0
−I
A28
A38
A48
0
0
0
0
0
0
0
0
0
0
−I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−I
3
7
7
7
7
7
7
7
7
7
7
7
5
3
7
7
7
7
7
7
7
7
7.
7
7
7
7
7
7
7
5
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Conclusion
Example
Column-echelon-form:
»
A27
A37
–
·V =
»
–
– »
R29
P29
,
+
R39
P39
»
P29
P39
–
ˆ
= ei1
···
eik
˜
... leads to
2
2
6
6
6
6
6
6
6
6
A=6
6
6
6
6
6
6
6
4
6
6
6
6
6
6
6
6
6
E =6
6
6
6
6
6
6
6
4
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
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
−AT
28
T
T
−P29
− R29
T
−A210
0
0
0
0
0
0
0
0
0
0
0
0
0
0
E66
T
E67
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−I
0
0
T
T
−P39
− R39
−AT
310
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
E67
E77
0
0
0
0
0
0
0
0
0
0
0
−AT
48
−AT
49
T
−A410
I
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
−I
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
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
0
0
0
0
3
7
2
7
I
7
6 0
7
6
7
6 0
7
6
7
6 0
7
6
7
7 , B = CT = 6 0
6
7
6 0
7
6
7
6 0
7
7
4 0
7
7
0
5
0
A28
0
A48
−I
0
0
0
0
0
0
0
P29 + R29
P39 + R39
A49
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−I
−I
A210
A310
A410
0
0
0
0
0
0
0
3
7
7
7
7
7
7
7
7
7
7
7
5
0
0
0
−I
0
0
0
0
0
0
0
3
7
7
7
7
7
7
7
7
7.
7
7
7
7
7
7
7
5
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Conclusion
Example
Column-echelon-form:
»
A27
A37
–
·V =
»
–
– »
R29
P29
,
+
R39
P39
»
P29
P39
–
ˆ
= ei1
···
eik
˜
... leads to
2
2
6
6
6
6
6
6
6
6
A=6
6
6
6
6
6
6
6
4
6
6
6
6
6
6
6
6
6
E =6
6
6
6
6
6
6
6
4
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
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
−AT
28
T
T
−P29
− R29
T
−A210
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
E66
T
E67
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−I
0
0
T
T
−P39
− R39
−AT
310
0
0
0
0
0
0
0
E67
E77
0
0
0
0
0
0
0
0
0
0
0
−AT
48
−AT
49
T
−A410
I
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
−I
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
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
0
0
0
0
3
7
2
7
I
7
6 0
7
6
7
6 0
7
6
7
6 0
7
6
7
7 , B = CT = 6 0
6
7
6 0
7
6
7
6 0
7
7
4 0
7
7
0
5
0
A28
0
A48
−I
0
0
0
0
0
0
0
P29 + R29
P39 + R39
A49
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−I
−I
A210
A310
A410
0
0
0
0
0
0
0
3
7
7
7
7
7
7
7
7
7
7
7
5
0
0
0
−I
0
0
0
0
0
0
0
3
7
7
7
7
7
7
7
7
7.
7
7
7
7
7
7
7
5
tu-logo
Motivation
2
6
6
E =6
6
4
Descriptor Systems
AC C AT
C
0
0
0
L
0
0
0
0
0
0
0
Circuit Equations
2
0
6
7
6
AT
0 7
L
6
7, A = 6
7
6 AT −T T AT
0 5
T Tt
4 Ti
0
0
3
Circuit Synthesis
−AL
−ATi +ATt TT
−AV
0
0
0
0
0
0
0
0
0
AT
V
Conclusion
Example
3
2
−AI
7
6
7
6 0
7
T
7, B = C = 6
6 0
7
4
5
0
0
3
7
0 7
7
0 7
5
−I
with
2
2 3
0
0
60
6I 7
6
6 7
AC = 607 , AL = 6−I
40
405
0
0
20
3
I
07
60
6
7
07 , ATt = 60
4
05
I
0
– 0
»
E66
E67
.
C = I, L =
T
E67
E77
2
3
0
0
07
60
6
7
0 7 , ATi = 60
40
05
I
−I
0
P29
P39
0
0
2 3
2 3
0 3
−I
0
P̄29 7
607
607
6 7
6 7
7
P̄39 7 , AV = 607 , AI = 6 0 7 ,
405
4I 5
5
0
0
0
0
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Outline
1
Motivation
2
Descriptor Systems
3
Circuit Equations
4
Circuit Synthesis
5
Example
6
Conclusion and Outlook
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Consider
Circuit Synthesis
"
s + s22s+1
G(s) =
− s21+1
As descriptor system:
2
1 0 0 0
60 1 0 0
6
60 0 1 0
E =6
60 0 0 1
6
40 0 0 0
0 0 0 0
0
0
0
0
0
0
2
3
0
0
6−1
7
07
6
6
07
7, A = 6 0
60
07
6
7
40
15
0
0
»
0 2 1
C=
0 1 0
1
s2 +1
2s
s2 +1
1
0
0
0
0
0
0
0
#
Example
Conclusion
.
0
0
0
0
0
1
−1 0
0
0
0 –0
1 0
.
1 0
0
0
0
0
1
0
2
3
0
0
61
7
07
6
6
07
7 , B = 60
60
07
6
7
40
05
1
1
3
0
07
7
07
7,
17
7
05
0
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
0
0
0
1.0653
0.1564
−0.5400
3
0
7
0
7
0
7
7,
−0.54007
−0.77095
1.1901
Conclusion
Bringing into block form
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
1.0549
−1.2702
−0.2525
0
0
0
0
0
0
0.1564
2.0000
−0.7709
3
−1.1931
0.1807 7
7
−0.16077
7,
0
7
5
0
0
2
−0.2120
6 1.7390
6
6 0.3937
T
B =C =6
0
6
4
0
0
3
0
0
7
7
0
7
7,
0.9232 7
−1.00005
0.5371
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Realization procedure leads to
2
0
6 0
6
6 0
6
AL = 6−1
6
6 0
4 0
0
2
3
0
0
0 7
61
6
7
0 7
60
6
7
0 7 , AC = 60
6
7
0 7
60
40
5
0
0
−1
0
0
0
0
0
−1
0
»
3
C=
0
2
–
1
0
, L = 40
1
0
2
3
2
2 3
3
0
−1
0
0
61
6 0 7
607
07
6
7
6
6 7
7
17
60
6 0 7
607
6
7
6
6 7
7
07 , AV = 607 , AI = 6 0 7 , ATt = 60
6
7
6
6 7
7
07
60
6 0 7
617
40
5
4
5
4
5
0
0
0
0
0
0
0
0
1.0198
−0.1981
2
3
0.2986
0
6−0.9950
−0.19815 , T = 6
4
0
2.9802
−0.4925
0
0
1
0
0
0
0
0
0
0
1
0
0
0
−2.9851
−0.0995
0
−2.1812
2
3
0
0
60
07
6
7
07
60
6
7
07 , ATi = 60
6
7
17
60
41
5
0
0
0
3
−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
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Conclusion
Example
0.7071
1
0
0
−2.1812
0
−0.4925
uV (t)
−2.9851 −2.1213
1
−0.0995
1
0.2986
3
−0.9950
A realizing circuit is therefore given by
iI (t)
−0.1981
1.0198
2.9802
tu-logo
Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Outline
1
Motivation
2
Descriptor Systems
3
Circuit Equations
4
Circuit Synthesis
5
Example
6
Conclusion and Outlook
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Motivation
Descriptor Systems
Circuit Equations
Circuit Synthesis
Example
Conclusion
Conclusion
Passive and Reciprocal Descriptor Systems
Transformation into MNA form
Interpretation as an electrical circuit.
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Thanks a lot for your attention!