Mikael Kurula - Åbo Akademi

4.2 Hamiltonian systems with external ports 33
Let E and F be two Hilbert spaces. By F ×E we denote the standard product of
f
F and E, i.e., the set of pairs , such that f ∈ F and e ∈ E. By F ⊕E we mean the
e
1 f
Hilbert space obtained by equipping F ×E with the inner product
e1
2 f
,
e2
=
F⊕E
(f 1 ,f 2 )F +(e 1 ,e 2 )E.
rx 0
Definition 4.4. Let Ex,E∂,Fx,F∂ be Hilbert spaces and let rE,F = be a uni-
0 −r∂
tary operator from the space E := Ex ⊕E∂ of efforts to the space F := Fx ⊕F∂ of flows.
Let D be a Dirac structure on the bond space B := F ×E with power product (4.5).
The linear port-Hamiltonian system, which is induced by the Dirac structure D and
of functions, such that
the Hamiltonian H(x)= 1
2x2E , is the set of all quadruples
x ∈C 1 (R + ;Ex), f∂ ∈C(R + ;F∂), e∂ ∈C(R + ;E∂), for which the following inclusion holds:
rx ˙x
x
f∂
e∂
⎡ ⎤
rx ˙x(t)
⎢ f∂(t) ⎥
⎣ x(t) ⎦ ∈ D,
e∂(t)
t ≥0. (4.12)
In the case of an electrical circuit, the port effort e∂ has the interpretation of
voltage over the port, whereas the port flow f∂ is the electrical current flowing into
the system. An abstract port-Hamiltonian system is illustrated graphically in Figure
4.1. We will later expand this figure to illustrate the interconnection of two port-
Hamiltonian systems in the next section.
x
D
H
Figure 4.1: The abstract port-Hamiltonian system induced by the
Dirac structure D and the Hamiltonian H.
Remark 4.5. We sometimes need to consider systems which are of port-Hamiltonian
type, i.e., a system described by a subspace D ⊂ B, a Hamiltonian H and the inclusion
(4.12), but where D is not necessarily a Dirac structure. In this case we refer to D as
the interconnection structure of Σ.
Evaluating the power product [·,·] B for a trajectory
⎡⎡
⎤
rx ˙x(t)
⎢⎢
⎢⎢
f∂(t) ⎥
⎣⎣
x(t) ⎦
e∂(t)
,
⎡ ⎤⎤
rx ˙x(t)
⎢ f∂(t) ⎥⎥
⎥⎥
⎣ x(t) ⎦⎦
e∂(t)
=2
f∂
e∂
rx ˙x
f∂ x
e∂
at time t we obtain
rx ˙x(t) rxx(t)
,
f∂(t) −r∂e∂(t)
Fx⊕F∂
B
=2( ˙x(t),x(t)) −2(f∂(t),r∂e∂(t)) Ex F∂ .
(4.13)