Simple Examples of POG Modeling

Simple Examples of POG Modeling

1. Consider the following electric circuit composed by the inductances L1, L2, the capacities C3, C3, the resistances Ra, Rb, the input voltage Va and the output current Ib:

V_a V3 V4

L1 L2

R_a

Rb

C3 C4 Ib

I1 I2

The POG model of the given electric circuit is the following:

V_a ^{- }

?

1 s

?^φ1

1 L1

?

I1

 -

 -

R_a

6

6 - 

- 

?

1 s

?^φ2

1 L2

?

I2

 -

 -

6

1 s

6^Q3

1 C3

6

V3

- 

- 

1 R_b

?

?

 -

 -

6

1 s

6^Q4

1 C4

6

V4

-  I_b

Let x = 

I1 I2 V3 V4

^T

be the state vector (composed by the output power variables of the dynamic elements ) and let u =

V_a I_b ^T

be the input vector. Write the corresponding dynamic system L ˙x = A ˙x + Bu in the state space:











L1 0 0 0 0 L2 0 0 0 0 C3 0 0 0 0 C4











| {z }

L











˙I1

˙I2

V˙3

V˙4











| {z }

˙x

=











−Ra R_a 0 0

R_a −Ra −1 0 0 1 −_R¹_{b R}¹_b 0 0 _R¹_b −_R¹_b











| {z }

A









 I1

I2

V3

V4











| {z } x

+







 1 0 0 0 0 0 0 −1









| {z }

B

" Va

I_b

#

| {z } u

• Analogy: direct correspondence between Electrical and Mechanical physical domains:

V1 V_a

F_b

V3 V4

E1 E2

b_a

b_b

M3 M4 F_b

F1 F2

The physical domain is different (Mechanical), but the mathematical model is the same if V1 = V a, E¹ = L¹, E² = L², M³ = C³, M⁴ = C⁴, Fb = Ib, F¹ = I¹, F² = I², ba = Ra and b_b = Rb.

2. Consider the following electric circuit composed by the inductances L¹, L³, the capacities C², C3 and the resistances R³ and R⁴. Two inputs act on the system: the voltage Va and the voltage Vb. The outputs of the system are: the current Ia and the current Ib = I⁴.

Ia

I4

Va

L1

I1

V2

C2

I3

L3 R3

V3

V4

C4

R4

Vb

The POG model of the given electric circuit is the following:

V_a

I_a

- 

?

1 s

?^φ1

1 L1

?

I1

 -

 -

6

1 s

6^Q2

1 C2

6

V2

- 

- 

?

1 s

?^φ3

1 L3

?

I3

 -

 

R3

6

6

V3

- -

 -

6

1 s

6^Q3

1 C4

6

V4

- 

- 

1 R4

?

?

I4

 -

V_b

I_b

Let x = 

I1 V2 I3 V4

^T

be the state vector, u = 

V_a V_b ^T

the input vector and y=

Ia Ib

^T

the output vector. Write the corresponding dynamic system L ˙x = Ax + Bu and y = Cx + Du in the state space:









L1 0 0 0 0 C2 0 0 0 0 L3 0 0 0 0 C4









| {z }

L











˙I1

V˙2

˙I3

V˙4











| {z }

˙x

=









0 −1 0 0

1 0 −1 0

0 1 −R³ −1 0 0 1 −_R¹₄









| {z }

A







 I1

V2

I3

V4









| {z } x

+







 1 0 0 0 0 0 0 _R¹₄









| {z } B

"

V_a V_b

#

| {z } u

"

I_a Ib

#

| {z } y

=





1 0 0 0 0 0 0 _R¹₄





| {z }

C

x +" 0 0 0 −_R¹4

#

| {z }

D

" V_a V_b

#

| {z } u

• Analogy: direct correspondence between Electrical and Hydraulic physical domains:

3. Consider the following electric circuit composed by the capacities C¹, C³, C⁴, the inductance L2 and the resistances Ra and Rb. Two inputs act on the system: the current Ia and the voltage Vb. The outputs of the system are: the voltage Va and the current Ib.

Ib

Ib

Va VR

Ia

C1

V1

Ra

I2

L2

V3

C3

Rb

V4

C4

Vb

The POG model of the given electric circuit is the following:

I_a

V_a

- -

?

1 s

?^Q1

1 C1

?^V1

 

- 

R_a

?

?

V_R

 -

 -

6

1 s

6^φ2

1 L2

6

I2

- 

- 

?

1 s

?^Q3

1 C3

?

V3

 -

 -

1 Rb

6

6

I_b

- 

- -

?

1 s

?^Q4

1 C4

?^V4

 

I_b

V_b

Let x = 

V1 I2 V3 V4

^T

be the state vector, u = 

I_a V_b ^T

the input vector and y=

V_a I_b ^T

the output vector. Write the corresponding dynamic system L ˙x = Ax + Bu and y = Cx + Du in the state space:









C1 0 0 0 0 L2 0 0 0 0 C3 0 0 0 0 C4









| {z }

L









 V˙1

˙I2

V˙3

V˙4











| {z }

˙x

=









0 0 0 0

0 −Ra −1 0 0 1 −_R¹_{b R}¹_b 0 0 _R¹_b −_R¹_b









| {z }

A







 V1

I2

V3

V4









| {z } x

+









1 0

R_a 0 0 _R¹_b 0 −_R¹_b









| {z }

B

"

I_a V_b

#

| {z } u

"

V_a Ib

#

| {z } y

=





1 −Ra 0 0 0 0 _R¹_b −_R¹_b





| {z }

C

x +

" R_a 0 0 −_R¹_b

#

| {z }

D

" I_a V_b

#

| {z } u

• Correct the following POG scheme adding minus signs in the summation elements:

Ia

V_a

- -

?

1 s

?^Q1

1 C1

?^V1

 

- 

Ra

?

?

V_R

 -

 -

6

1 s

6^φ2

1 L2

6

I2

- 

- 

?

1 s

?^Q3

1 C3

?

V3

 -

 -

1 R_b

6

6

I_b

- 

- -

?

1 s

?^Q4

1 C4

?^V4

 

Ib

V_b

Solution:

I_a

V_a

- -

?

1 s

?^Q1

1 C1

?^V1

 

- 

R_a

?

?

V_R

 -

 -

6

1 s

6^φ2

1 L2

6

I2

- 

- 

?

1 s

?^Q3

1 C3

?

V3

 -

 -

1 R_b

6

6

I_b

- 

- -

?

1 s

?^Q4

1 C4

?^V4

 

I_b

V_b

• Add the positive directions of the power flows to the following POG scheme:

V_a

I1

- -

1 C1s

6

6

V1

 

- -

1 R3

?

?

IR3

 

- 

1 L2s

?

?

I2

 -

 -

1 C_bs

6

6

V_b

- 

- 

1 R2

?

?

I_R2

 -

 -

1 C3s

6

6

V3

- 

V3

I_b

Solution:

4. Consider the following electric circuit composed by the capacities C¹, Cb, C³ and the resistan- ces R² and R³. Two inputs act on the system: the voltage Vaand the current Ib. The outputs of the system are: the current I¹ and the voltage V³.

Cb

L2

I1 I2

IR3

IR2

Ib

Va Vb

R2

R3

C1

C3

V1

V3

The POG model of the given electric circuit is the following:

V_a

I1

- -

1 C1s

6

6

V1

 

- -

1 R3

?

?

IR3

 

- 

1 L2s

?

?

I2

 -

 -

1 C_bs

6

6

V_b

- 

- 

1 R2

?

?

I_R2

 -

 -

1 C3s

6

6

V3

- 

V3

I_b

Let x = 

V1 I2 V_b V3

^T

be the state vector, u = 

V_a I_b ^T

the input vector and y =

 I1 V3

^T

the output vector. Write the corresponding dynamic system L ˙x = Ax + Bu and y= Cx + Du in the state space:









C1 0 0 0 0 L2 0 0 0 0 C_b 0 0 0 0 C3









| {z }

L









 V˙1

˙I2

V˙_b V˙3











| {z }

˙x

=









−_R¹3 1 0 0

−1 0 −1 0

0 1 −_R¹2

1 R2

0 0 _R¹

2 −_R¹₂









| {z }

A







 V1

I2

V_b V3









| {z } x

+









1 R3 0

1 0

0 0

0 −1









| {z }

B

"

Va

I_b

#

| {z } u

"

I1

V3

#

| {z } y

=

"

−_R¹₃ 1 0 0 0 0 0 −1

#

| {z }

C

x +

" 1 R3 0

0 0

#

| {z } D

" Va

I_b

#

| {z } u

5. Consider the following electric circuit composed by the inductances L¹, L³, the capacities C², C4 and the resistances Ra and Rb. Two inputs act on the system: the voltage V¹ and the current I⁴. The outputs of the system are: the current I¹ and the voltage V⁴.

V1

L1

I1

Ia

Ra

C2

V2 I3

L3 V4

C4

Ib

Rb

I4

The POG model of the given electric circuit has the following structure:

V1

I1

- 

?

1 s

?^φ1

1 L1

?

I1

 -

 

1 R_a

?

?I_a

- -

 

6

1 s

6^Q2

1 C2

6^V2

- -

 

?

1 s

?^φ3

1 L3

?^I3 - -

 -

6

1 s

6^Q4

1 C4

6

V4

- 

- -

1 R_b

?

?I_b

 

V4

I4

Let x = 

I1 V2 I3 V4

^T

be the state vector, u = 

V1 I4

^T

the input vector and y =

 I1 V4

^T

the output vector. Write the corresponding dynamic system L ˙x = Ax + Bu and y= Cx + Du in the state space:









L1 0 0 0 0 C2 0 0 0 0 L3 0 0 0 0 C4









| {z }

L











˙I1

V˙2

˙I3

V˙4











| {z }

˙x

=









0 −1 0 −1

1 −_R¹a 0 −_R¹a

0 0 0 1

1 −_R¹a −1 −_R¹_a − _R¹_b









| {z }

A







 I1

V2

I3

V4









| {z } x

+







 1 0 0 0 0 0 0 −1









| {z } B

"

V1

I4

#

| {z } u

"

I1

V4

#

| {z } y

=

" 1 0 0 0 0 0 0 1

#

| {z }

C

x +" 0 0 0 0

#

| {z } D

" V1

I4

#

| {z } u

6. Consider the following electric circuit composed by the capacities C¹, C⁴, the inductances L², L3 and the resistances Ra, Rb and Rc. Two inputs act on the system: the voltage Va and the current Ib. The outputs of the system are: the current Ia and the voltage Vb.

I2

I2

Va

Ra

V1

C1

Ic

Rc

L2

L3

I3

V4

C4

Ib

The POG model of the given electric circuit has the following structure:

V_a

I_a

- 

1 R_a

?

?

Ia

 -

 -

6

1 s

6^Q1

1 C1

6

V1

- 

- 

?

1 s

?^φ3

1 L2

?

I2

 -

 

R_c

6

6

I_c

- -

 

?

1 s

?^φ3

1 L3

?^I3 - -

 -

6

1 s

6^Q4

1 C4

6

V4

- 

V_b

I_b

Let x = 

V1 I2 I3 V4

^T

be the state vector, u = 

V_a I_b ^T

the input vector and y =

 I_a V_b ^T

the output vector. Write the corresponding dynamic system L ˙x = Ax + Bu and y= Cx + Du in the state space:









C1 0 0 0 0 L² 0 0 0 0 L³ 0 0 0 0 C⁴









| {z }

L









 V˙1

˙I2

˙I3

V˙4











| {z }

˙x

=









−_R¹_a −1 0 0 1 −Rc 0 −1

0 0 0 1

0 1 −1 0









| {z }

A







 V1

I2

I3

V4









| {z } x

+









1 Ra 0

0 0

0 0

0 −1









| {z }

B

"

V_a I_b

#

| {z } u

"

I_a V_b

#

| {z } y

=

"

−_R¹a 0 0 0 0 0 0 −1

#

| {z }

C

x +

" 1 R^a 0

0 0

#

| {z } D

" V_a I_b

#

| {z } u

7. Consider the following hydraulic circuit composed by the hydraulic inductances L¹, L³, the hydraulic capacities C², C⁴ and the hydraulic resistances R¹, R², R⁴, R⁵ and R⁶. Two inputs act on the system: the pressione Pa and the volume flow rate Qb. The outputs of the system are: the volume flow rate Qa= Q¹+ Q⁵ and the pressione Pb = P⁴.

Pa

P1

L1

R1

P2

C2 Q2 C4 Q4

Qb

R2

L3

R4

P4

Q1

Q3

The POG model of the given hydraulic circuit has the following structure:

P_a

Q_a

- -

R1

6

6

P1

 

- 

?

1 s

?^φ1

1 L1

?

Q1

 -

 -

6

1 s

6^V2

1 C2

6

P2

- 

- -

1 R2

?

?^Q

2

 

- 

?

1 s

?^φ3

1 L3

?

Q3

 -

 -

6

1 s

6^V4

1 C4

6

P4

- 

- -

1 R4

?

?^Q

4

 

P_b

Q_b

Let x = 

Q1 P2 Q3 P4

^T

be the state vector, u = 

Pa Qb

^T

the input vector and y=

Q_a P_b ^T

the output vector. Write the corresponding dynamic system L ˙x = Ax + Bu and y = Cx + Du in the state space:









L1 0 0 0 0 C² 0 0 0 0 L³ 0 0 0 0 C⁴









| {z }

L









 Q˙1

P˙2

Q˙3

P˙4











| {z }

x˙

=









−R¹ −1 0 0 1 −_R¹2 −1 0

0 1 0 −1

0 0 1 −_R¹₄









| {z }

A







 Q1

P2

Q3

P4









| {z }

x

+







 1 0 0 0 0 0 0 −1









| {z }

B

"

P_a Q_b

#

| {z }

u

"

Q_a #

=

" 1 0 0 0 #

x +" 0 0 # " Pa #

8. Consider the following mechanical system composed by masses m¹, m³, elasticity E⁴ and dampers b¹, b² and b⁵. Two inputs act on the system: the force Fa and the velocity Vb. The outputs of the system are: velocity V¹ and the force F⁵.

F2 F5

F1 F3

Fa

V1 V3 Vb

m1 m3

b1 b3

b5

E2

The POG model of the given mechanical system has the following structure:

F_a

V1

- 

?

1 s

?^p1

1 m1

?

V1

 -

 

b1

6

6

F1

- -

 -

6

1 s

6^x2

1 E2

6

F2

- 

- 

?

1 s

?^p3

1 m3

?

V3

 -

 

b3

6

6

F3

- -

 -

b5

6

6

F5

- 

F5

V_b

Let x = 

V1 F2 V3

^T

be the state vector, u = 

F_a V_b ^T

the input vector and y =

 V1 F5

^T

the output vector. Write the corresponding dynamic system L ˙x = Ax + Bu and y= Cx + Du in the state space:







m1 0 0 0 E² 0 0 0 m³







| {z }

L





 V˙1

F˙2

V˙3







| {z }

x˙

=







−b¹ −1 0

1 0 −1

0 1 −b³−b⁵







| {z }

A





 V1

F2

V3







| {z } x

+





 1 0 0 0 0 b⁵







| {z }

B

"

F_a V_b

#

| {z }

u

"

V1

F5

#

| {z }

y

=

" 1 0 0 0 0 b5

#

| {z }

C

x +" 1 0 0 −b⁵

#

| {z }

D

" F_a V_b

#

| {z }

u

9. Consider the following electric circuit composed by the inductances L¹, L², the capacities C³, C4 and the resistances R¹, R⁴ and R⁵. Two inputs act on the system: the voltage Va and the current Ib. The outputs of the system are: the current I¹ and the voltage Vb.

Va

I1

L1

I2

L2

V3

C3

R4

Vr4

V4

C4

R5

Vb

Vr5

Ib

The POG model of the given electric circuit has the following structure:

V_a

I1

- 

?

1 s

?^φ1

1 L1

?

I1

 -

 

?

1 s

?^φ2

1 L2

?^I2

- -

 

R4

6

6

Vr4

- -

 

6

1 s

6^Q3

1 C3

6^V3

- -

 -

6

1 s

6^Q4

1 C4

6

V4

- 

- -

R5

6

6

Vr5

 

V_b

I_b

Let x = 

I1 I2 V3 V4

^T

be the state vector, u = 

Va Ib

^T

the input vector and y =

 I1 V_b ^T

the output vector. Write the corresponding dynamic system L ˙x = Ax + Bu and y= Cx + Du in the state space:









L1 0 0 0 0 L² 0 0 0 0 C³ 0 0 0 0 C⁴









| {z }

L











˙I1

˙I2

V˙3

V˙4











| {z }

˙x

=









−R⁴ R4 −1 −1 R4 −R⁴ 1 1

1 −1 0 0

1 −1 0 0









| {z }

A







 I1

I2

V3

V4









| {z } x

+







 1 0 0 0 0 0 0 −1









| {z } B

"

Va

I_b

#

| {z } u

"

I1

V_b

#

| {z } y

=

" 1 0 0 0 0 0 0 1

#

| {z }

x +" 0 0 0 −R⁵

#

| {z }

" Va

I_b

#

| {z }