APPROXIMATION OF THE TELEGRAPHER'S EQUATIONS WITH DISSIPATION

IFAC MCPL 2007 The 4th International Federation of Automatic Control Conference on Management and Control of Production and Logistics September 27-30, Sibiu - Romania

APPROXIMATION OF THE TELEGRAPHER’S EQUATIONS WITH DISSIPATION C. Chera*, A. Nakrachi**, G. Dauphin-Tanguy** *University ”Politehnica” of Bucharest, Romania ** LAGIS, University of Sciences and Technology of Lille Splaiul Independetei, 313, Bucharest, 060041, Romania Phone: +40213169561, Fax: +40213169561 E-mail: [email protected]

Abstract: We have already proposed a representation and spatial discretization of line transmission in terms of Bond Graph (Nakrachi, 2003). In this paper, one shows that we preserve the Dirac structure of a distributed parameter system represented in the form of a port-Hamiltonian, after a space discretization. Indeed, the conservation of the Dirac structure was shown only in the conservative case, without dissipation. The study is applied to the case of a transmission line represented by the telegrapher’s equations. Copyright © 2002 IFAC Keywords: approximation, dissipation, telegrapher’s equations, Dirac structure, distributed systems.

1. INTRODUCTION

simulation. But in the case of distributed parameter systems an important problem concerns the incorporation of the numerical method, like finiteelements and finite-difference, used for resolving the Partial Derivative Equations (PDE), into this framework. We have an infinite-dimensional system that we need to approximate with a finitedimensional one. The numerical methods for solving the PDE assume that the boundary conditions are given. In the case of telegrapher’s equations the boundary conditions are the voltage and the current at both ends of the line. But these values cannot be considered given because the transmission line is connected to the other dynamic systems of the electrical network. So we need to approximate the distributed-parameter systems with a finite dimensional system and to keep the power port structure.

In the literature there are some papers where it is show how port based network modelling of lumpedparameter physical systems naturally leads to a geometrically defined class of systems, called portHamiltonian systems (Van der Schaft, 2002). The Hamiltonian approach starts from the principle of least action, use the Euler-Lagrange equations and the Legendre transformation, and arrive to the Hamiltonian equation of motion. In this approach the dynamics of the systems is defined by a Dirac structure, which represent the power-conserving interconnection structure of the system, and the Hamiltonian which is given by the total energy of the energy storing elements in the system. We can add energy-dissipating elements by terminating some of the system ports with resistive elements.

In the case of the telegrapher’s equation, the previous works have used an ideal model of the transmission line (Golo, et al., 2002) (Golo, et al., 2004). In this paper we take a model that includes also the dissipative elements.

From a simulation point of view this approach is important because give information about the energy function and other conserved quantities in the system, which preferably should be kept in 247

dH = eq ∧ ( −deφ − g (*eq ) ) + dt ∫Ω + eφ ∧ ( −deq − r (*eφ ) ) =

2. DISSIPATIVE TRANSMISSION LINE Let’s consider a transmission line with Ω = [0,1] ⊂ \ and define the energy variables as the charge/length unity q = q(t , x ) ∈ Λ1 (Ω) and the

= ∫ −eq ∧ deφ − eφ ∧ deq − ∫ eq ∧ g (*eq ) + Ω

flux/length unity φ = φ (t , x ) ∈ Λ1 (Ω) where Λ1 (Ω) denotes the space of 1-forms. The energy density (or the Hamiltonian density) at time t in the transmission line is given as: h ( q, φ ) =

= − ∫ d ( eφ ∧ eq ) − ∫ eq ∧ g (*eq ) + eφ ∧ r (*eφ ) = Ω

∂Ω

dH = − ∫ eb ∧ f b − ∫ edq ∧ f dq + edφ ∧ f dφ ∂Ω Ω dt

(8)

where r f dφ = r (*eφ ) = φ (t , x ) ∈ Λ1 (Ω) , l g f dq = g (*eq ) = q(t , x ) ∈ Λ1 (Ω) c

In order to introduce the energy variables, we write the power balance of the transmission line starting with total energy H ( q, φ ) = ∫ h( q, φ )

and edq=eq, edφ=eφ.

Ω

dH ( q, φ ) δ H ( q, φ ) ∂q δ H ( q, φ ) ∂φ =∫ ∧ + ∧ Ω ∂t ∂t dt δq δφ

The two terms of the right side of equation (8) represent the power flow at the boundary and the dissipation power in a transmission line. In this way, the equation (6) becomes

(2)

where the variational derivatives are given by

fφ = −deq − f dφ

δ H ( q, φ ) 1 = * q( t , x ) δq c δ H ( q, φ ) 1 = * φ (t , x ) δφ l

f q = −deφ − f dq

(3)

(9)

The resulting port-Hamiltonian dissipative system: ⎛ fφ ⎞ ⎛ 0 − d ⎞ ⎛ eq ⎞ ⎛ 1 0 ⎞ ⎛ f dφ ⎞ ⎜⎜ ⎟⎟ = ⎜ ⎟⎟ ⎟ ⎜⎜ ⎟⎟ − ⎜ ⎟ ⎜⎜ ⎝ f q ⎠ ⎝ −d 0 ⎠ ⎝ eφ ⎠ ⎝ 0 1⎠ ⎝ f dq ⎠ ⎛ f dφ ⎞ ⎛ r * 0 ⎞ ⎛ eφ ⎞ ⎜⎜ ⎟⎟ = ⎜ ⎟ ⎜⎜ ⎟⎟ ⎝ f dq ⎠ ⎝ 0 g * ⎠ ⎝ eq ⎠ ⎛ eb 0 ⎞ ⎛ 1 0 0 0 ⎞ ⎛ eq (t , 0) ⎞ ⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎜ eb1 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ eq (t ,1) ⎟ = ⎜ f ⎟ ⎜ 0 0 1 0 ⎟ ⎜ e (t , 0) ⎟ φ ⎜ b0 ⎟ ⎜ ⎟ ⎟⎜ ⎜ f b1 ⎟ ⎝⎜ 0 0 0 1⎠⎟ ⎜ eφ (t ,1) ⎟ ⎝ ⎠ ⎝ ⎠

We introduce the conjugate energy variables flow (1form) and effort (0-forms) as follows: ∂φ (t , x ) ∂t δ h ( q, φ )

fφ ( t , x ) =

eφ (t , x ) = ∗

Ω

We have the following structure:

with l- inductance and c- capacitance per unit of length and where ^ is a wedge product and * is the Hodge star operator (defined as k n−k *(D ) : Λ (D ) → Λ (D ) where D is an open in Ω , and Ω is a Riemannian manifold.

∂q(t , x ) ∂t δ h ( q, φ ) eq (t , x ) = ∗ δq

Ω

= − ∫ eq ∧ eφ − ∫ ( eq ∧ g (*eq ) + eφ ∧ r (*eφ ))

1⎡ q(t , x ) φ (t , x ) ⎤ + φ (t , x ) ∧ * (1) q(t , x ) ∧ * 2 ⎢⎣ c l ⎥⎦

f q (t , x ) =

(7)

Ω

+ eφ ∧ r (*eφ ) =

δφ

(4a) (4b)

The equation (2) become, dH ( q, φ ) = ∫ f q ∧ eq + fφ ∧ eφ Ω dt

(5)

b0 and b1 denotes, respectively, the left and right boundary.

In addition, we have the telegraph equations written in terms of conjugate variables and differential forms: ⎧⎪ fφ = − deq − r (*eφ ) ⎨ ⎪⎩ f q = − deφ − g (*eq )

(10)

Note : In terms of current and tension, the transmission line with dissipation satisfies the following equation:

(6)

dH = v (t ,0)i (t ,0) − v (t ,1)i (t ,1) dt − ∫ ⎡⎣ ri 2 (t , x ) + gv 2 (t , x ) ⎤⎦

where r- resistance and g- conductance per unit of length and where d is the usual exterior-derivative.

Ω

1 1 where i (t , x ) = * φ (t , x ) and v (t , x ) = * q(t , x ) l c

Substitution in the equation (2) gives:

248

(11)

3. SPATIAL DISCRETIZATION

wqα (α ) = 1 wqα ( β ) = 0 wqβ (α ) = 0 wqβ ( β ) = 1

We will carry out a separation of variables and we will use the Whitney forms which make it possible to preserve the properties of the p-forms at the time of a spatial discretization (Bossavit, 1991).

wφα (α ) = 1 wφα ( β ) = 0 wφβ (α ) = 0 wφβ ( β ) = 1

(16)

This 0-form of Whitney makes it possible to have the following relations:

Next we make the spatial discretization of the telegrapher’s equations. The transmission line is split in m cells. Due to spatial compositionality (i.e. interconnection of two transmission lines via a common boundary once again gives a transmission line), we need to perform discretization to only one cell.

By substitution of (13) and (15) in (9) and by holding account of (10) we obtain:

That is to say the cell delimited by space Ω c = [α , β ] .

f qαβ (t ) 1 wq ( x ) = − eφα (t )dwφα ( x ) − eφβ (t )dwφβ ( x )

wqα ( x ) + wqβ ( x ) = 1 and wφα ( x ) + wφβ ( x ) = 1

fφαβ (t ) 1 wφ ( x ) = − eqα (t )dwqα ( x ) − eqβ (t )dwqβ ( x ) − r ( eφα (*wφα ) + eφβ (*wφα ) )

In order to determine the equations of dynamics inside a cell, we integrate the equations above:

We express the boundary variables in function of the efforts: f bα (t ) = eφ (t , α )

fφαβ (t ) ∫

∫ αβ α f (t ) ∫ w ( x ) = −eφ (t ) ∫ − geα ∫

-

Ωc

α

Ωc

β

(*wq ) − geq

∫

Ωc

(*wqβ )

wq ( x ) = 1 and

∫

Ωc

1

wφ ( x ) = 1

(14)

inside the cell, are approximated by :

wqβ ( x )

Ω

dw =

∫

∂Ω

w and (*w) = wdx , where w is a 0-

1 fφαβ (t ) = eqα (t ) − eqβ (t ) − rαβ ( eφα (t ) + eφβ (t )) 2 (20) 1 αβ α β α β f q (t ) = eφ (t ) − eφ (t ) − gαβ ( eq (t ) + eq (t )) 2

In the same way for the efforts eq (t , x ) and eφ (t , x ) ,

wqα ( x )

∫

form and (*w) is a 1-form, x−β x −α , and wφβ = wqβ = - wφα = wqα = α −β β −α the Whitney 0-form one leads to the following relations:

conditions:

1

(19)

Using the relations (14) and the properties

where wq , wφ ∈ Λ (Ω c ) are the 1-form satisfying the

1

(*wφβ )

dwφα ( x ) − eφβ (t ) ∫ dwφβ

Ωc

q

1

Ωc

q

Ωc

∫

β φ Ω c

(*w ) − re

1

(13)

fφαβ (t , x ) ≈ fφ (t ) ⋅ 1 wφ ( x )

∫

Ωc

α φ

1

f q (t , x ) ≈ f q (t ) ⋅ wq ( x )

1

Ωc

− re

(12)

q

1

wφ ( x ) = −eqα (t ) ∫ dwqα ( x ) − eqβ (t ) ∫ dwqβ α φ Ωc

One considers the size of a sufficiently small cell, to be able to make the following approximations to represent flows inside the cell: αβ

1

Ωc

f bβ (t ) = eφ (t , β )

(18)

− g ( eqα (*wqα ) + eqβ (*wqα ) )

One consider a cell (with the length (β-α)), and we denote the spatial manifold Ω c = [α , β ] .

ebα (t ) = eq (t , α ) ebβ (t ) = eq (t , β )

(17)

1

where rαβ = r ( β − α ) and gαβ = g ( β − α ) . We arrive to the following spatial discretization representation of this typical cell

α

⎛ f bα (t ) ⎞ ⎛ 1 ⎜ ⎟ ⎜ ⎜ f bβ ( t ) ⎟ ⎜ 0 ⎜ ebα (t ) ⎟ ⎜ 0 ⎜ ⎟=⎜ ⎜ ebβ (t ) ⎟ ⎜ 0 ⎜ αβ ⎟ ⎜ ⎜ fφ (t ) ⎟ ⎜ −.5rαβ ⎜⎜ f αβ (t ) ⎟⎟ ⎜⎝ 1 ⎝ q ⎠

β

A Whitney 0-form eq (t , x ) = eqα (t ) ⋅ wqα ( x ) + eqβ (t ) ⋅ wqβ ( x ) eφ (t , x ) = eφα (t ) ⋅ wφα ( x ) + eφβ (t ) ⋅ wφβ ( x )

where

wqα , wqβ , wφα , wφβ ∈ Λ 0 (Ω c )

(15)

are the 0-form

0 1 0 0 −.5rαβ −1

0 0 1 0 1 −.5 gαβ

0 ⎞ α 0 ⎟⎟ ⎛ eφ (t ) ⎞ ⎜ β ⎟ 0 ⎟ ⎜ eφ (t ) ⎟ (21) ⎟⎜ ⎟ 1 ⎟ ⎜ eqα (t ) ⎟ −1 ⎟ ⎜⎜ e β (t ) ⎟⎟ ⎟⎝ q ⎠ −.5 gαβ ⎟⎠

It remains to check that this is a port Hamiltonian system corresponding to a cell which preserves the structure of Dirac. This corresponds to an instantaneous conservation of the power (power net).

satisfying the conditions:

249

PΩc = ∫ eq ∧ f q + ∫ eφ ∧ fφ + ∫ Ωc

Ωc

∂Ωc

eb ∧ f b

∫

(22)

Ωc

wqα 1 wq + ∫ wφα 1 wφ = − ∫ wqα dwφα − ∫ wφα dwqα Ωc

One replaces eq (t , x ) , eφ (t , x ) , f q (t , x ) and fφ (t , x )

Ωc

(e w + e w ) f + ∫ (e w + e w ) f Ωc

α

α

β

β

αβ 1

q

q

q

q

q

α φ

Ωc

α φ

β φ

β φ

(23)

wφ

Ωc

)

Ωc

)

+ ebβ f bβ − ebα f bα

wφ = − dwqα = dwqβ

1

wq = − dwφα = dwφβ

(25)

eαβ = ( eφαβ

(

f αβ = f φαβ

(26)

(27)

∫

wφα 1 wφ = 1 − γ

Ωc

∫

Ωc

f qαβ

ebβ )

f bα

t

f bβ

)

F αβ

⎛ 0 0 γ − 1 −γ ⎞ ⎛ ⎜ ⎜0 0 0 0 ⎟⎟ ⎜ +⎜ ⎜ 1 0 .5rαβ .5rαβ ⎟ ⎜ ⎜⎜ ⎟⎜ 0 1 −1 1 ⎟⎜ ⎝  ⎠ ⎝

Ωc

wqβ 1 wq = 1 − γ

ebα

(35)

0 0 ⎞ ⎛ eφαβ ⎞ ⎛1 0 ⎜ ⎟ ⎜ 0 1 −γ γ − 1 ⎟⎟ ⎜ eqαβ ⎟ ⎜ + ⎜ 0 0 −1 1 ⎟ ⎜ ebα ⎟ ⎟ ⎜⎜ ⎟⎜ 0 0 .5gαβ .5gαβ ⎟⎠ ⎜⎝ ebβ ⎟⎠ ⎝  

We take γ = ∫ wqα 1 wq , and result: Ωc

eqαβ

From (32) and (33) obtain

= ( wqα + wqβ ) 1 wφ = 1 wφ

∫

(33)

where

wq = wqα dwqβ − wqβ dwqα = = wqα 1 wφ + wqβ 1 wφ =

(32)

PΩc = eαβ f αβ = eqαβ f qαβ + eφαβ f φαβ + ebβ f bβ − ebα f bα (34)

In addition, the use of the Whitney forms enables us to have the following relation: 1

(31)

the instantaneous power is written then:

From them results: 1

f bβ = eφβ

eqαβ = γ ebα + (1 − γ )ebβ et eφαβ = (1 − γ ) f bα + γ f bβ

β

1 ( eφα − eφβ − gαβ ( eqα + eqβ )) 1 wq = −eφα dwφα − eφβ dwφβ 2 − g ( eqα (*wqα ) + eqβ (*wqα ) )

f bα = eφα

One poses also the efforts of the cell :

Combination of the equations (18) and (20) give : 1 ( eq − eq − rαβ ( eφα + eφβ )) 1 wφ = −eqα dwqα − eqβ dwqβ 2 − r ( eφα (*wφα ) + eφβ (*wφα ) )

ebβ = eqβ

1 fφαβ (t ) = ebα − ebβ − rαβ ( f bα + f bβ ) 2 1 αβ f q (t ) = f bα − f bβ − gαβ ( ebα + ebβ ) 2

Before developing calculations, we establish initially some relations between the various 1-forms brought into play.

α

ebα = eqα

What makes it possible to write the equations (20) in the form :

+ eφα ∫ wφα 1 wφ + eφβ ∫ wφβ 1 wφ fφαβ (24) Ωc

(30)

We say that

PΩc = eqα ∫ wqα 1 wq + eqβ ∫ wqβ 1 wq f qαβ

(

α

+ ebβ f bβ − ebα f bα

or Ωc

Ωc

α φ

PΩc = (γ eqα + (1 − γ )eqβ ) f qαβ + ( (1 − γ )eφα + γ eφβ ) fφαβ

+ ebβ f bβ − ebα f bα

(

α

The equation (24) become :

wq

αβ 1 φ

α φ

=1

by their approximations to obtain the following expression: PΩc = ∫

Ωc

= − ∫ d ( wq w ) = − ( wq ( β ) w ( β ) − wq (α ) wφα (α ) ) (29) α

fφαβ ⎞ ⎟ f qαβ ⎟ =0 f bα ⎟ ⎟ f bβ ⎟⎠

(36)

Eαβ

(28)

With γ=1/2 in the case of the approximations of Whitney for the 0-forms and the 1-forms.

β 1 φ

w wφ = γ

We denote the space of admissible efforts with e, and the domain of admissible flows with f, such that the following relation it’s satisfied:

Indeed,

D = {( f , e ) ∈ \ 4 : E αβ eαβ + F αβ f αβ = 0}

250

(37)

D is a Dirac structure with respect to the bilinear form if and only if the following two conditions are rank ⎡⎣ E αβ F αβ ⎤⎦ = 4 and satisfied

αβ φ

e

E αβ ( F αβ )t + F αβ ( E αβ )t = 0 .

d φαβ dt

4. CONSTITUTIVES RELATIONS

dqαβ

To complete calculations, we will determine the expressions of the load qαβ(t) and the flow φαβ(t) and their variations on the cell level.

q(t , x ) = qαβ (t ) 1w( x )

dt

(38)

1 φαβ 1 w ∧ φαβ (*1 w) 2l (39) ⎛ φ 2 (t ) q 2 (t ) ⎞ = ⎜ αβ + αβ ⎟ ∫ 1 w ∧ (*1 w) ⎜ 2l 2c ⎟⎠ Ωc ⎝ φ 2 (t ) q 2 (t ) = αβ + αβ 2lαβ 2cαβ Ωc

∫

∫

1

Ωc

w ∧ (*1 w)

c Ωc

w ∧ (*1 w)

(40)

G. Golo, V. Talasila, A.J. van der Schaft,(2002). “Approximation of the telegrapher’s equations” 41st IEEE conf. on decision and control, Las Vegas

= c( β − α )

(41)

G. Golo, V. Talasila, A.J. van der Schaft, B.M. Maschke, (2004). “Hamiltonian discretization of boundary control systems”, Automatica, vol. 40/5, pp. 757--711.

∂q(t , x ) dqαβ 1 w( x ) = f qαβ (t ) 1 w( x ) = ∂t dt

B.M. Maschke, A.J. van der Schaft, (2005). "Compositional Modelling of DistributedParameter Systems pp. 115--154 in /Advanced Topics in Control Systems Theory,/ Lecture Notes from FAP 2004, Springer Lect. Notes in Control and Information Sciences 311, Springer, London,.

(42)

then

dt d φαβ dt eqαβ (t ) = αβ φ

e (t ) =

= f qαβ (t ) = f

αβ φ

∂hαβ ∂q ∂hαβ ∂φ

cαβ

= l(β − α )

In addition, we have the following bonds:

dqαβ

(46)

A.Bossavit, C. Emson and I. Mayergoyz, (1991). “Méthodes numériques en électromagnétisme“ , Eyrolles (Paris).

and cαβ =

= f bα − f bβ − gαβ

qαβ

REFERENCES

with l

lαβ

In this paper we have shown that we can preserve the Dirac structure of a distributed parameter system represented in the form of a port-Hamiltonian, after a space discretization. For this, we have used the case of a transmission line represented by the telegrapher’s equations.

1 1 1 ∫Ωc 2c qαβ w ∧ qαβ (* w)

1

φαβ

5. CONCLUSIONS

+∫

lαβ =

= ebα − ebβ − rαβ

It is known that connection of two Dirac structures gives a Dirac structure. Thus the whole transmission line can be reconstructed by the connection of a fixed number of cells in advance.

The total energy of the cell is given by hαβ (φαβ , qαβ ) =

(45)

The dynamics of the cell is given then by:

After computation we show that this is true and the two conditions are satisfied.

φ (t , x ) = φαβ (t ) 1w( x )

1 ( ebα + ebβ ) 2 1 = ( f bα + f bβ ) 2

eqαβ =

(43)

A. Nakrachi, G. Dauphin-Tanguy (2003). “Bond Graph for distributed parameter systems : The telegrapher equation case” IMACS-IEEE "CESA'03" Lille

(t )

= =

qαβ cαβ

φαβ

P. Ramkrishna, (2006). “On Analysis and Control of Interconnected Finite and Infinite-dimensional Physical Systems” Phd Thesis, University of Twente, Nederland.

(44)

lαβ

and from equation (33) we have : 251

A.J. van der Schaft, (2002). "Port-Hamiltonian systems: network modeling and control of nonlinear physical systems'', pp. 127--168 in Advanced Dynamics and Control of Structures and Machines, CISM Courses and lectures No. 444, CISM International Centre for Mechanical Sciences, Udine, Italy, April 15--19, 2002 (Eds. H. Irshik, K. Schlacher), Springer, Wien, New York, 2004.

252