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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

Representation of a general composition of Dirac structures

Carles Batlle, Imma Massana, Ester Sim´

• Universitat Polit`

ecnica de Catalunya

IEEE CDC-ECC 2011, Orlando, FL, December 11-15 2011

1 / 19 IEEE CDC-ECC 2011

Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions 1

Introduction.

2

Dirac structures and port-Hamiltonian systems.

3

General composition of Dirac structures. Kernel and image representations of the composed structure.

4

Examples.

5

Conclusions.

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

Port-Hamiltonian systems

Port-Hamiltonian systems in explicit form (explicit PHS) ˙ x = J(x)∂xHT + g(x)u, y = g(x)T ∂xHT , where x ∈ Rn is the state , JT = −J is the interconnection matrix, g is the port matrix and H is the energy function, describe a class of control systems in many areas of interest. Interconnecting several explicit PHS by identifying some of their input/output variables u and y does not, in general, yield again an explicit PHS, but a system of DAE (differential-algebraic equations).

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

Dirac structures

This resulting DAE is, however a generalized PHS, and it can be described in terms of an underlying mathematical structure, known as a Dirac structure. Circuit theory provides the paradigmatic example of Dirac structure by means of Kirchhoff laws i ∈ N(D), v ∈ R(DT ) where D is the incidence matrix of the digraph associated with the circuit. From these, Tellegen’s theorem vT i = 0, stating the power-preserving nature of the interconnection, can be deduced.

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

Main contribution of the paper

The suitable interconnection of Dirac structures yields again a Dirac structure (ditto for generalized PHS) (Dalsmo & van der Schaft, 1998), but the practical matter of explicitly constructing the resulting Dirac structure has only been addressed recently (Cervera, van der Schaft & Ba˜ nos, 2007) for a simple feedback interconnection of two systems. The main contribution of the present work is the explicit construction of the image/kernel representation of the Dirac structure resulting from an arbitrary interconnection (by means of an interconnecting Dirac structure) of any number

• f Dirac structures.

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

Dirac structures

F, a finite-dimensional vector space, the space of flows, and F∗, its dual space, the space of efforts. e|f, which has dimensions of power, denotes the action of the form e ∈ F∗ on the vector f ∈ F. An indefinite, non-degenerate symmetric bilinear form can be defined on F × F∗ by means of ⋖(fa, ea)|(fb, eb)⋗ = ea|fb + eb|fa. A constant Dirac structure on D ⊂ F × F∗ is a subspace such that D = D⊥, with ⊥ the orthogonal complement with respect to ⋖ | ⋗.

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

If the components of vectors and forms are given with respect to dual basis, then e|f = eT f, and then D⊥ =

• ( ˜

f, ˜ e) ∈ F × F∗ | ˜ eT f + eT ˜ f = 0 ∀ (f, e) ∈ D

• .

If follows from the definition of D that, for any (f, e) ∈ D e|f = 0, which encodes the energy conservation property of Dirac structures, also known as power continuity, and which generalizes Tellegen’s theorem from circuit theory.

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

Every Dirac structure D admits a kernel representation D = {(f, e) ∈ F × F∗ | Ff + Ee = 0}, for linear maps F : F → V and E : F∗ → V satisfying EF ∗ + FE∗ = 0 with rank (F + E) = dim F, where V is a vector space with dim V ≥ dim F, and F ∗ : V∗ → F∗ and E∗ : V∗ → F denote the adjoint maps, which are represented, for given basis, by the transposed matrix of the corresponding map. Dirac structures can also be given by means of an image representation D = {(f, e) ∈ F ×F∗ | ∃u ∈ V∗ s.t. f = E∗u, e = F ∗u} (1) with the same maps and spaces of the kernel representation.

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

Generalized PHS

Consider a lumped-parameter physical system defined on a manifold M, with local coordinates x ∈ Rn. The total energy

• f the system is given by the Hamiltonian function H(x), and

the system is assumed to have m open ports. For each x we consider Fx = TxM × Rm and F∗

x = T ∗ xM × Rm, and define a Dirac structure at each

x ∈ M, D(x) ⊂ Fx × F∗

x.

We will write the elements of Fx × F∗

x as (fx, fb, ex, eb),

where b stands for boundary, and the fx and ex are power variables associated to state ports, i.e. ports connected to energy storing elements.

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

A port-Hamiltonian system on M can be defined by

• − ˙

x, fb, ∂T

x H, eb

• ∈ D(x) ∀ x ∈ M.

In a kernel representation one gets F(x) − ˙ x fb

• + E(x)

∂T

x H

eb

• = 0.

This is, in general, a set of differential and algebraic equations (DAE), and may include the definition of some boundary variables as inputs or outputs.

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

A sufficient (although not necessary) condition to get an explicit PHS is that F has full rank. In this case the fb are the outputs of the system while the eb are the inputs. The minus sign in fx = − ˙ x is introduced in order to get ˙ H = −∂xH ˙ x = eT

b fb.

Source or dissipative terms can be added to the system through some of the boundary ports.

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

Composition of Dirac structures

Consider N Dirac structures Di given in kernel/image representation by matrices ˆ Ei = (Ei EiI), ˆ Fi = (Fi FiI) where the i columns correspond to internal (or maybe open port) flow-effort pairs (fi, ei), fi, ei ∈ Rni, while the iI ones are those of the flow-effort pairs (fiI, eiI), fiI, eiI ∈ Rmi that are to be interconnected. We now compose these N Dirac structures by means of an interconnecting Dirac structure, with matrices EC = (EC1 · · · ECN), FC = (FC1 · · · FCN), where the columns have been split according to the subsystem to which they are to be attached.

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

The set (DC)N

i=1 Di is made up of those (f1, e1, . . . , fN, eN)

for which there exist interconnecting variables (f1I, e1I, . . . , fNI, eNI) ∈ DC such that (fi, fiI, ei, eiI) ∈ Di, ∀ i = 1, . . . , N. The following result can then be proved (DC)N

i=1 Di is a Dirac structure

This result was established in (Dalsmo & van der Schaft, 1998), and proved again, for a simple feedback interconnection, in (Cervera, van der Schaft & Ba˜ nos, 2007) using different techniques, which have been generalized in the present work.

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

Kernel and image representations

From the matrices corresponding to the interconnected ports

• f each subsystem and the matrices of the interconnecting

Dirac structure one can construct MT

i = FCiET iI + ECiF T iI, i = 1, . . . , N,

and MT = (MT

1

· · · MT

i

· · · MT

N).

Now choose a matrix L such that R(M) = N(L). This is equivalent to LM = 0, or MT LT = 0.

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

From the columns of L corresponding to each subsystem and the columns of the matrices of the individual subsystems corresponding to the ports that are not connected one can construct E = (L1E1 · · · LNEN), F = (L1F1 · · · LNFN). Then the main result of this paper can be proved: E, F provide a kernel/image representation for the Dirac structure of the interconnected system. The proof follows that in (Cervera, van der Schaft & Ba˜ nos, 2007), with the appropriate generalizations due to the fact that more than two subsystems are considered, and with an arbitrary interconnection instead of a simple feedback one.

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

Parallel interconnection

Consider 3 Dirac structures D1, D2, D3, with a common number m1 of ports connected in parallel: f1I + f2I + f3I = 0, e1I = e2I = e3I, with eiI, fiI ∈ Rm1, i = 1, 2, 3. A kernel form is given by   1 1 1     f1I f2I f3I   +   1 −1 1 −1     e1I e2I e3I   =     . Matrices FCi and ECi can be read off from this and one gets M =   E1I F1I E2I −F2I F2I E3I −F3I   .

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

Interconnection of PHS

Consider the following circuits, where the third one is just a stub with two open ports, introduced so that an open port can be

• btained at the point of connexion of the other two.

vC1 iC1 v1 i1 vL1 iL1 vC2 iC2 v2 i2 v3 i3 v4 i4 v5 i5

Combining the individual Dirac structures and the matrix M corresponding to the parallel interconnection of the three systems at the ports 1, 2 and 4 one gets M T =   −1 −1 −1 1 −1 −1 1 −1   . The flows and efforts of the interconnected system are f = (− ˙ q1, − ˙ λ, − ˙ q2, i3, i5) and e = (∂q1H, ∂λH, ∂q2H, v3, v5).

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

A matrix L such that M T LT = 0 can be obtained, and the interconnected system turns out to be given in kernel form by Ff + Ee = 0, with F =       −1 1 1 1 1       , E =       −1 1 −1 −1 1 −1 1 −1       . The five component equations yield the DAE of the interconnected system, with inputs i3 and i5 and outputs v3 and v5: ˙ q1 + ˙ q2 = −∂λH + i3 + i5, ˙ λ = ∂q1H, ∂q1H = ∂q2H, v3 = ∂q2H, v5 = ∂q1H.

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Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions

Conclusions

An algorithm to compute kernel/image representations for the Dirac structure resulting from the interconnection of several Dirac structures has been provided, generalizing a simple special case previously published in the literature. A new proof of the basic result concerning the interconnection

• f Dirac structures has been obtained.

Several particular cases have been worked out in detail. The generalizations introduced in this work might be relevant, among other applications, for

the modeling of complex and large scale systems, where nontrivial interconnections of subsystems are often considered. the control by interconnection approach, due to the new topologies now available.

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