1
EECI-M10: Port-Hamiltonian Systems
Delft University of Technology
Port-Hamiltonian Systems:
From Geometric Network Modeling to Control
Module M10: HYCON-EECI Graduate School on Control Dimitri Jeltsema and Arjan van der Schaft
April 06–09, 2010
April 06–09, 2010 2
EECI-M10: Port-Hamiltonian Systems
Passivity-Based Control (PBC) — RECAP
• Physical systems satisfy
stored energy
=
supplied energy+
dissipated energy • This suggest the natural control objectivedesired stored energy
=
new supplied energy+
desired dissipated energy • Essence of PBC (Ortega/Spong, 1989)PBC
=
energy shaping+
damping assignment • Main objective: rendering the (closed-loop) system passive w.r.t.some desired storage function.
April 06–09, 2010 3
EECI-M10: Port-Hamiltonian Systems
Recall: Passivity of pH Systems
The port-Hamiltonian system
˙
x
= [
J
(
x
)
−
R
(
x
)]
∂
H
∂
x
(
x
) +
g
(
x
)
u
,
y
=
g
T(
x
)
∂
H
∂
x
(
x
)
,
with state
x
∈
R
n, and port-variablesu
,
y
∈
R
m, ispassiveifH
[
x
(
t
)]
−
H
[
x
(
0
)]
!
"#
$
stored energy≤
% t 0u
T(
τ
)
y
(
τ
)
d
τ
!
"#
$
supplied energy,
(
∗
)
for some Hamiltonian
H
:
R
n→
R
+.April 06–09, 2010 4
EECI-M10: Port-Hamiltonian Systems
Stabilization by Damping Injection
• Use storage function (read: Hamiltonian) as Lyapunov function for the uncontrolled system.
• Passive systems can be asymptotically stabilized by adding damping via the control. In fact, for a passive port-Hamiltonian system we have
˙
H
(
x
)
≤
u
Ty
.
Hence letting
u
=
−
K
dy
, withK
d=
K
dT&
0, we obtain
˙
H
(
x
)
≤ −
y
TK
dy
,
⇒
asymptotic stability, provided an observability condition is met (i.e., zero-state detectability of the output).April 06–09, 2010 5 EECI-M10: Port-Hamiltonian Systems
Stabilization by Damping Injection
• If
H
(
x
)
non-negative, total amount of energy that can be extracted from a passive system is bounded, i.e.,−
% t 0u
T(
τ
)
y
(
τ
)
d
τ
≤
H
[
x
(
0
)]
<
∞
.
April 06–09, 2010 6EECI-M10: Port-Hamiltonian Systems
Energy-Balancing PBC
• Usually, the point where the open-loop energy is minimal is not of interest. Instead some nonzero eq. point, say
x
∗, is desired. • Standard formulation of PBC: Findu
=
β
(
x
) +
v
s.t.H
d[
x
(
t
)]
−
H
d[
x
(
0
)]
!
"#
$
stored energy=
% t 0v
T(
τ
)
z
(
τ
)
d
τ
!
"#
$
supplied energy−
d
d(
t
)
! "# $
diss. energy,
where the desired energy
H
d(
x
)
has a minimum atx
∗, andz
isthe new output (which may be equal to
y
).• Hence control problem consist in finding
u
=
β
(
x
) +
v
s.t. energy supplied by the controller is a function of the statex
.April 06–09, 2010 7
EECI-M10: Port-Hamiltonian Systems
Energy-Balancing PBC
• Indeed, from the energy-balance inequality (
∗
) we see that if we can find aβ
(
x
)
satisfying−
% t
0
β
T
[
x
(
τ
)]
y
(
τ
)
d
τ
=
H
a
[
x
(
t
)] +
κ
,
for some
H
a(
x
)
, then the controlu
=
β
(
x
) +
v
will ensurev
(→
y
is passive w.r.t. modified energy
H
d(
x
) =
H
(
x
) +
H
a(
x
)
.April 06–09, 2010 8
EECI-M10: Port-Hamiltonian Systems
Energy-Balancing PBC
Proposition:Consider the port-Hamiltonian system
˙
x
= [
J
(
x
)
−
R
(
x
)]
∂
H
∂
x
(
x
) +
g
(
x
)
u
,
y
=
g
T(
x
)
∂
H
∂
x
(
x
)
.
If we can find a function
β
(
x
)
and a vector functionK
(
x
)
satisfying[
J
(
x
)
−
R
(
x
)]
K
(
x
) =
g
(
x
)
β
(
x
)
such that i)∂
K
∂
x
(
x
) =
∂
TK
∂
x
(
x
)
(integrability);April 06–09, 2010 9 EECI-M10: Port-Hamiltonian Systems
Energy-Balancing PBC
Proposition (cont’d): ii)K
(
x
∗) =
−
∂
H
∂
x
(
x
∗)
(equilibrium assignment); iii)∂
K
∂
x
(
x
∗)
& −
∂
2H
∂
x
2(
x
∗)
(Lyapunov stability).Then the closed-loop system is a port-Hamiltonian system of the form
˙
x
= [
J
(
x
)
−
R
(
x
)]
∂
H
d∂
x
(
x
)
,
with
H
d(
x
) =
H
(
x
)+
H
a(
x
)
,K
(
x
) =
∂∂Hxa(
x
)
, andx
∗(locally) stable.April 06–09, 2010 10
EECI-M10: Port-Hamiltonian Systems
Energy-Balancing PBC
• Note that (R
(
x
) =
R
T(
x
)
)
0)
˙
H
d(
x
) =
−
∂
TH
d∂
x
(
x
)
R
(
x
)
∂
H
d∂
x
(
x
)
≤
0
.
• Also note that
x
∗is (locally) asymptotically stable if, in addition, the largest invariant set is contained in&
x
∈
D
'
'
'
'
∂
TH
d∂
x
(
x
)
R
(
x
)
∂
H
d∂
x
(
x
) =
0
(
,
whereD
⊂
R
n. April 06–09, 2010 11EECI-M10: Port-Hamiltonian Systems
Mechanical Systems
Consider a (fully actuated) mechanical systems with total energy
H
(
q
,
p
) =
12p
TM
−1(
q
)
p
+
P
(
q
)
,
with generalized mass matrix
M
(
q
) =
M
T(
q
)
&
0. Assume that the
potential energyP
(
q
)
is bounded from below. PH structure:
q
˙
˙
p
=
0
I
k−
I
k0
!
"#
$
J=−JT
∂∂Hq ∂H ∂p
+
0
B
(
q
)
! "# $
g(q)u
,
y
=
-
0
B
T(
q
)
.
∂∂Hq ∂H ∂p
.
April 06–09, 2010 12EECI-M10: Port-Hamiltonian Systems
Mechanical Systems
Clearly, the system has as passive outputs the generalized velocities:
˙
H
(
q
,
p
) =
u
Ty
=
u
TB
T(
q
)
∂
H
∂
p
(
q
,
p
) =
u
T
M
−1(
q
)
p
=
u
Tq
˙
.
Now, the Energy-Balancing PBC design boils down to
JK
(
q
,
p
) =
g
(
q
)
β
(
q
,
p
)
,
K
(
x
) =
∂
H
a∂
x
(
x
)
which in the fully actuated (
B
(
q
) =
I
k) case simplifies toK
2(
q
,
p
) =
0
April 06–09, 2010 13 EECI-M10: Port-Hamiltonian Systems
Mechanical Systems
The simplest way to ensure that the closed-loop energy has a minimum at
(
q
,
p
) = (
q
∗,
0
)
is to selectβ
(
q
) =
∂
P
∂
q
(
q
)
−
K
p(
q
−
q
∗
)
,
K
p
=
K
pT&
0
.
This gives the controller energy
H
a(
q
) =
−
P
(
q
) +
1
2
(
q
−
q
∗
)
TK
p
(
q
−
q
∗) +
κ
,
so that the closed-loop energy takes the form
H
d(
q
,
p
) =
1
2
p
TM
−1(
q
)
p
+
1
2
(
q
−
q
∗)
TK
p(
q
−
q
∗)
.
April 06–09, 2010 14EECI-M10: Port-Hamiltonian Systems
Mechanical Systems
To ensure that the trajectories actually converge to
(
q
∗,
0
)
we need to render the closed-loop asymptotically stable by adding some dampingv
=
−
K
d∂
H
∂
p
(
q
,
p
) =
−
K
dq
˙
,
as shown before. Note that the energy-balance of the system is now
H
d[
q
(
t
)
,
p
(
t
)]
−
H
d[
q
(
0
)
,
p
(
0
)]
!
"#
$
stored energy=
% t 0v
T(
τ
)
q
˙
(
τ
)
d
τ
!
"#
$
supplied energy−
% t 0q
˙
T(
τ
)
K
pq
˙
(
τ
)
d
τ
!
"#
$
diss. energy.
April 06–09, 2010 15EECI-M10: Port-Hamiltonian Systems
Mechanical Systems
• Observe that the controller obtained via energy-balancing is just the classical PD + gravity compensation controller.
• However, the design via Energy-Balancing PBC provides a new interpretation of the controller, namely, that the closed-loop energy is (up to a constant) equals to
H
d(
q
,
p
) =
H
(
q
,
p
)
−
% t
0
u
T
(
τ
)
y
(
τ
)
d
τ
,
i.e., the difference between the open-loop and controller energy.
April 06–09, 2010 16
EECI-M10: Port-Hamiltonian Systems
Exercise: Linear RLC circuit
+
-y u r L C• Write the dynamics in PH form. • Determine the equilibrium point. • Find the passive input and output.
• Design an Energy-Balancing PBC that stabilizes the admissible equilibrium.
April 06–09, 2010 17 EECI-M10: Port-Hamiltonian Systems
Dissipation Obstacle
• Unfortunately, EnergyBalancing PBC is stymied by the existence of pervasive dissipation.
• A necessary condition to satisfy the Energy-Balancing PBC proposition is
R
(
x
)
K
(
x
) =
R
(
x
)
∂
H
a∂
x
(
x
) =
0
⇒
dissipation obstacle.
• This implies that no damping is present in the coordinates that need to be shaped.
• Appears in many engineering applications.
• Limitations and the dissipation obstacle can be characterized via thecontrol–by–interconnectionperspective.
Port-Hamiltonian Systems:
From Geometric Network Modeling to Control
Dimitri Jeltsema and Arjan van der Schaft Module M10 — Special Topics HYCON-EECI Graduate School on Control
April 06–09, 2010
Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 1
Brayton-Moser Equations
! From Port-Hamiltonian Systems to the Brayton-Moser Equations ! Main Motivation: Stability Theory
! State-of-the-Art
! Passivity and Power-Shaping Control ! Generalization
! Final Remarks
Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 2
From PH Systems to the Brayton-Moser Equations
Consider a port-Hamiltonian system without dissipation and external ports˙
x
=
J(x)
∂
H
∂
x
(x)
,
(
∗
)
with
J(x) =
−
J
T(x). Suppose that the mapping from the energy variables
x
to the co-energy variablese
is invertible, such thatx
=
x(e) =
ˆ
∂
∂
H
∗e
(e)
,
withH
∗(e)
the Legendre transformation ofH(x)
given byH
∗(e) =
e
Tx
−
H(x)
.
Then the dynamics (
∗
) can also be expressed in terms ofe
as∂
2H
∗∂
e
2(e)
e
˙
=
J(x)e.
Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 3
From PH Systems to the Brayton-Moser Equations
Assume that we can find coordinatesx
= (x
q,
x
p)
T, with dimx
q=
k
and dimx
p=
n
−
k, such that
J(x) =
!
0
B(x)
−
B
T(x)
0
"
,
with
B(x)
ak
×
(n
−
k)
matrix. Furthermore, assume thatH(x
q,
x
p) =
H
q(x
q) +
H
p(x
p)
.
In that case, theco-Hamiltoniancan be written asH
∗(e
q,
e
p) =
H
q∗(e
q) +
H
∗p(e
p)
,
and thus
∂2H∗ ∂e2 q0
0
∂2H∗ ∂e2 p
!
˙
e
q˙
e
p"
=
!
0
B(x)
−
B
T(x)
0
"!
e
qe
p"
.
From PH Systems to the Brayton-Moser Equations
Defining the (mixed-potential) functionP
(e
q,
e
p,
x) =
e
TqB(x)e
p,
it follows that
∂
2H
∗∂
e
2 q0
0
−
∂
2H
∗∂
e
2 p
'
()
*
Q(eq,ep)+
˙
e
q˙
e
p,
=
∂
P
∂
e
q∂
P
∂
e
p
.
These equations, called theBrayton-Moser equations, can be interpreted as a gradient systemwith respect to
P
and the indefinitepseudo-RiemannianmetricQ.
Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 5
From PH Systems to the Brayton-Moser Equations
Note that the mixed-potential can be written asP
(e
q,
e
p,
x) =
e
TqB(x)e
p=
e
Tqx
˙
q=
−
x
˙
Tpe
p,
which represent the instantaneous power flow betweenΣ
qandΣ
p.Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 6
From PH Systems to the Brayton-Moser Equations
Recall that dissipation can be included in the PH framework via˙
x
=
J(x)
∂
∂
H
x
(x) +
g
R(x)
f
R,
e
R=
g
TR(x)
∂
∂
H
x
(x)
,
withf
R=
−
∂
D
∗∂
e
(e)
.
Hence we can write
∂2H∗ ∂e2 q0
0
∂∂2He2∗ p
+
˙
e
q˙
e
p,
=
+
0
B(x)
−
B
T(x)
0
,+
e
qe
p,
−
g
R(x)
∂
∂
D
e
(e)
,
e
R=
g
TR(x)e
.
Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 7
From PH Systems to the Brayton-Moser Equations
For simplicity we assume thatg
R=
−
+
I
k0
0
I
n−k,
,
so that
∂
2H
∗∂
e
2 q0
0
∂
∂
2H
∗e
2 p
+
˙
e
q˙
e
p,
=
+
0
B(x)
−
B
T(x)
0
,+
e
qe
p,
+
∂
D
∗∂
e
q∂
D
∗∂
e
p
e
R=
−
+
e
qe
p,
.
From PH Systems to the Brayton-Moser Equations
Finally, defining
P
(
e
q,
e
p,
x
) =
e
TqB
(
x
)
e
p+
D
∗(
e
q,
e
p)
,
we (again) obtain the Brayton-Moser equations
∂
2H
∗∂
e
2 q0
0
−
∂
∂
2e
H
2∗ p
'
˙
e
q˙
e
p(
=
∂
P
∂
e
q∂
P
∂
e
p
.
Let us zoom in to electrical circuits . . .Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 9
Brayton-Moser Equations
Consider an electrical network
Σ
withn
Linductors,n
Ccapacitors, andn
Rresistors. Leti
∈
R
nL andv
∈
R
nC, then [Brayton and Moser 1964]Σ
=
Σ
p∪
Σ
q:
−
L
(
i
)
d
d
t
i
=
∇
iP
(
i
,
v
)
C
(
v
)
d
v
d
t
=
∇
vP
(
i
,
v
)
,
-∇
•=
∂∂•.
,
with mixed-potential
P
:
R
nL×
R
nC→
R
. For topologically complete networksP
(
i
,
v
) =
v
TBi
+
R
(
i
)
−
G
(
v
)
/
01
2
D∗(i,v)
,
Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 10
Brayton-Moser Equations
Example:Tunnel-diode circuit [Moser 1960]
!
R
(
i
) =
1 2Ri
2−
U
ini
!G
(
v
) =
3 vi
g(
v
")d
v
" !v
TBi
=
vi
, (B
=
1
) Mixed-potential function:P
(
i
,
v
) =
iv
+
1
2
Ri
2−
U
ini
−
3 vi
g(
v
")
d
v
"Σ
:
−
L
d
i
d
t
=
∇
iP
=
−
U
in+
Ri
+
v
(
Σ
p)
C
d
v
d
t
=
∇
vP
=
i
−
i
g(
v
)
.
(
Σ
q)
Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 11
Main Motivation: Stability Theory
Rewrite BM equations asQ
(
x
)
x
˙
=
∇
xP
(
x
)
,
withx
=
4
i
v
5
,
Q
(
x
) =
4
−
L
(
i
)
0
0
C
(
v
)
5
.
Observation:˙
P
=
x
˙
#Q
(
x
)
x
˙
⇒ P
?(
x
)
candidate Lyapunov function. Special cases:! RL networks (
x
=
i
,Q
(
x
) =
−
L
(
i
)
):P
˙
=
R
˙
≤
0;
! RC networks (x
=
v
,Q
(
x
) =
C
(
v
)
):−
P
˙
=
−
G
˙
≤
0
.
However, in generalP
˙
=
x
˙
#Q
(
x
)
x
˙
!
0
, or equivalently,Q
(
x
) +
Q
#(
x
)
"
0
.
A Family of BM Descriptions
The key observation is to generate a new pair, say
!
Q
˜
,
P
˜
"
, such that˜
Q
(
x
) +
Q
˜
!(
x
)
!
0,
|
Q
˜
| "
=
0.
For anyM
=
M
!andλ
∈
R
, new pairs can be found from˜
Q
(
x
)
:=
#
∇
2xP
(
x
)
M
+
λ
I
$
Q
(
x
)
˜
P
(
x
)
:=
λ
P
(
x
) +
1
2
[
∇
xP
(
x
)]
!M
∇
xP
(
x
).
Note that the system behavior is preserved since˙
x
=
Q
−1∇
x
P
(
x
)
⇔
x
˙
=
Q
˜
−1∇
xP
˜
(
x
).
Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 13
Example: Tunnel Diode Circuit
Obviously, for the tunnel diode circuitQ
+
Q
!!
0. However, selecting
M
=
*
1 R0
0
RCL+
−1,
λ
=
1,
yields˜
Q
(
v
) =
,
0
LR−
RL−
C
−
LR∇
vi
g(
v
)
-,
and˜
P
(
i
,
v
) =
) v 0i
g(
v
$)d
v
$+
L
2
RC
(
i
−
i
g(
v
))
2+
1
2
R
(
v
−
U
in)
2.
Hence, it is easily seen that
min
v
∇
vi
g(
v
)
>
−
RC
L
⇔
Q
˜
(
v
) +
Q
˜
!(
v
)
!
0.
Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 14
State-of-the-Art
! Constructive procedures to obtain stability criteria using the mixed-potential function are given in [Brayton and Moser 1964]
⇒
Three theorems, each depending on the type of nonlinearities in R or LC part.! Generalizations can be found in e.g.,
! [Chua and Wang 1978] (for cases that|∇2iR|=0or|∇2vG|=0); ! [Jeltsema and Scherpen 2005] (RLC simultaneously nonlinear).
! Over the past four decades several notable generalizations of the BM equations itself have been developed, e.g.,
! [Chua 1973] (non-complete networks: Pseudo-Hybrid Content); ! [Marten et al. 1992] (on the geometrical meaning);
! [Weiss et al. 1998] (on the largest class of RLC networks).
! PBC of power converters [Jeltsema and Scherpen 2004];
! Extension to other domains, e.g., [Jeltsema and Scherpen 2007];
! Passivity and control: Power-Shaping stabilization...
Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 15
Energy-Balancing Control [Ortega et al.]
To put our ideas into perspective let us briefly recall the principle of
Energy-Balancing (EB) Control. Consider general system representation
˙
x
=
f
(
x
) +
g
(
x
)
u
,
y
=
h
(
x
),
x
∈
R
n,
u
,
y
∈
R
m.
(
∗
)
Assume that(
∗
)
satisfiesH
[
x
(
t
)]
−
H
[
x
(0)]
%
&'
(
Open-loop stored energy
≤
) t 0u
!(
τ
)
y
(
τ
)d
τ
%
&'
(
Supplied energy,
with storage function
H
:
R
n→
R
. IfH
≥
0, then
(
∗
)
is passive wrt(
u
,
y
).
Usually desired operating point, sayx
∗, not minimum ofH
. Idea is to look forˆ
u
:
R
n→
R
m st−
) t 0u
ˆ
![
x
(
τ
)]
h
[
x
(
τ
)]d
τ
=
H
a[
x
(
t
)]
−
H
a[
x
(0)],
for someH
a:
R
n→
R
.Energy-Balancing Control [Ortega et al.]
Hence, the control
u
=
u
ˆ
(
x
) +
v
will ensure that the closed-loop system satisfiesH
d[
x
(
t
)]
−
H
d[
x
(0)]
!
"#
$
Closed-loop stored energy≤
% t 0v
!(
τ
)
y
(
τ
)d
τ
!
"#
$
Supplied energy,
where
H
d=
H
+
H
ais the closed-loop energy storage. If, furthermore,x
∗=
arg minH
d
(
x
),
then
x
∗will be stable equilibrium of the closed-loop system (with Lyapunov functionthe difference between the stored and the control energies.)
Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 17
Energy-Balancing Control [Ortega et al.]
However, applicability of EB severely stymied by the existence ofpervasive
dissipation. Indeed, since solving for
u
ˆ
is equivalent to solving the PDE[
f
(
x
) +
g
(
x
)
u
ˆ
(
x
)]
!∇
x
H
a(
x
) =
−
u
ˆ
!(
x
)
h
(
x
),
where the left-hand side is equal to zero at
x
∗, it is clear that the method is onlyapplicable to systems verifying
ˆ
u
!(
x
∗)
h
(
x
∗) =
0.
⇒
This is known as thedissipation obstacle.Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 18
Passivity: Power-Balance Inequality [Jeltsema et al. 2003]
Extract sources (controls) and rewrite BM equations asQ
(
x
)
x
˙
=
∇
xP
(
x
) +
G
(
x
)
u
.
We then observe that if
Q
(
x
) +
Q
!(
x
)
%
0
the network satisfies the power-balance inequalityP
[
x
(
t
)]
− P
[
x
(0)]
≤
% t 0u
!(
τ
)
y
(
τ
)d
τ
,
with outputsy
=
h
(
x
,
u
) =
−
G
!(
x
)
Q
−1(
x
)[
∇
xP
(
x
) +
G
(
x
)
u
].
! Power as storage functioninstead of energy.
! Trivially satisfied by all RL and RC networks with passive elements.
! Notice that
y
=
−
G
!(
x
)
x
˙
⇒
natural derivativesin the output!Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 19
Control: Power-Shaping Stabilization [Ortega et al. 2003]
Theopen-loopmixed-potential is shaped with the controlu
=
u
ˆ
(
x
), where
G
(
x
)
u
ˆ
(
x
) =
∇
xP
a(
x
),
for some
P
a:
R
n→
R
. This yields theclosed-loopsystemQ
(
x
)
x
˙
=
∇
xP
d(
x
),
with total power function
P
d(
x
) =
P
(
x
) +
P
a(
x
).
The equilibrium
x
∗will be stable ifx
∗=
arg minP
d(
x
).
⇒
Power-Balancing: closed-loop power function equals difference between open-loop power function and power supplied by controller, i.e.,˙
P
a=
−
u
ˆ
!(
x
)
h
(
x
,
u
ˆ
(
x
)) =
u
ˆ
!(
x
)
G
!(
x
)
x
˙
.
⇒
No dissipation obstaclesinceu
ˆ
!(
x
∗)
G
!(
x
∗)
x
˙
∗=
0!
Power-Shaping Stabilization
However, as mentioned before, in general
Q
(
x
) +
Q
!(
x
)
!
0, which requires first
the generation of a new pair
!
Q
˜
,
P
˜
"
, such that˜
Q
(
x
) +
Q
˜
!(
x
)
!
0
,
|
Q
˜
| "
=
0
.
Proposition.For any
M
(
x
)
=
M
!(
x
)
andλ
∈
R
, new pairs can be found from˜
Q
(
x
)
:
=
#
1
2
∇
2xP
(
x
)
M
(
x
)
+
1
2
∇
x$
M
(
x
)
∇
xP
(
x
)
%
+
λ
I
&
Q
(
x
)
˜
P
(
x
)
:
=
λ
P
(
x
) +
1
2
[
∇
xP
(
x
)]
!M
(
x
)
∇
xP
(
x
).
Hence,˙
x
=
Q
−1∇
xP
(
x
) +
G
(
x
)
u
⇔
x
˙
=
Q
˜
−1∇
xP
˜
(
x
) +
G
˜
(
x
)
u
,
withG
˜
(
x
) =
Q
˜
(
x
)
G
(
x
)
.Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 21
Example: Tunnel Diode Circuit
Again, selecting
M
=
diag$
1R,
RCL%
−1andλ
=
1
yields˜
Q
(
v
) =
'
0
L R−
LR−
C
−
RL∇
vi
g(
v
)
(
,
G
˜
=
)
0
−
LR*
,
and˜
P
(
i
,
v
) =
+ v 0i
g(
v
#)
d
v
#+
L
2
RC
(
i
−
i
g(
v
))
2+
1
2
R
v
2.
Assumption.∗min
v∇
vi
g(
v
)
>
−
RC
L
.
∗This assumption can be relaxed applying a preliminary feedback−Rai, with Ra>− # L Cminv∇vig(v) +R & .
Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 22
Example: Tunnel Diode Circuit
Using the Power-Shaping procedure we obtain:
Proposition.The control
u
=
−
K
(
v
−
v
∗) +
u
∗,
(=
U
in)
with control parameter
K
>
0
satisfyingK
>
−
[
1
+
R
∇
vi
g(
v
∗)]
,
globally asymptotically stabilizes
x
∗=
col(
i
∗,
v
∗)
with Lyapunov function˜
P
(
i
,
v
) =
+ v 0i
g(
v
#)
d
v
#+
L
2
RC
(
i
−
i
g(
v
))
2+
K
2
R
(
v
−
v
∗)
2+
1
2
R
(
v
−
u
∗)
2.
Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 23
Poincare’s Lemma
Existence of
P
follows from Poincare’s Lemma. Indeed, suppose the network is described by˙
x
=
f
(
x
) +
g
(
x
)
u
,
with
f
:
R
n→
R
nandf
∈ C
1, then there exists aP
:
R
n→
R
s.t.∇
xP
=
Q
(
x
)
f
(
x
)
iff∇
x(
Q
(
x
)
f
(
x
)) = [
∇
x(
Q
(
x
)
f
(
x
))]
!.
⇒
Power-Shaping can be applied to general nonlinear systems!Power-Shaping of Nonlinear Syst. [Garcia-Canseco et al. ’06]
Assumption.(A.1)There existsQ
:
R
n→
R
n×n,|
Q
| "
=
0
, satisfying∇x
(
Q
(
x
)
f
(
x
)) = [
∇x
(
Q
(
x
)
f
(
x
))]
",
and
Q
(
x
) +
Q
"(
x
)
$
0
.(A.2)There existsP
a:
R
n→
R
verifying !g
⊥(
x
)
Q
−1(
x
)
∇xP
a
=
0
, withg
⊥(
x
)
g
(
x
) =
0
, rank{
g
⊥(
x
)
}
=
n
−
m;
!
x
∗=
arg minP
d(
x
)
, whereP
d(
x
)
:=
! x
[
∇x
!(
Q
(
x
&)
f
(
x
&))]
"d
x
&+
P
a(
x
).
Proposition.Under A.1 and A.2, the control law
u
=
"
g
"(
x
)
Q
"(
x
)
Q
(
x
)
g
(
x
)
#
−1g
"(
x
)
Q
"(
x
)
∇xP
a
(
x
)
ensures
x
∗is (locally) stable with Lyapunov functionP
d(
x
)
.Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 25
Remark:
! Observe that Assumption A.1 includes the class of port-Hamiltonian systems with invertible interconnection and damping matrices.
Indeed, recall
˙
x
= [
J
(
x
)
−
R
(
x
)]
∇x
H
(
x
) +
g
(
x
)
u
y
=
g
"(
x
)
∇x
H
(
x
).
Now, if
|
J
(
x
)
−
R
(
x
)
| "
=
0
a trivial solution for the PDE∇x
(
Q
(
x
)
f
(
x
)) = [
∇x
(
Q
(
x
)
f
(
x
))]
",
is obtained by settingQ
(
x
) = [
J
(
x
)
−
R
(
x
)]
−1,
and
f
(
x
) =
∇x
H
(
x
)
.Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 26
Final Remarks
! Power-Shaping applicable togeneral nonlinear systems.
! Similar to Energy-Balancing. However,no dissipation obstacleinvolved.
! Tunnel diode example shows that Power-Shaping yields a simplelinear
(partial) state-feedback controller that ensuresrobustglobal asymptotic stability of the desired equilibrium point.
! Current research includes:
•
Solvability of the PDE∇x
(
Q
(
x
)
f
(
x
)) = [
∇x
(
Q
(
x
)
f
(
x
))]
"subject toQ
(
x
) +
Q
"(
x
)
$
0
for different kind of systems (e.g., mechanical, electromechanical, hydraulic, etc.).•
Connections with IDA-PBC.•
Distributed-parameter systems....! Control of chemical reactors (see pub list)
Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 27
...
“The real voyage of discovery consists not in seeking
new landscapes but in having new eyes.”
Marcel Proust
Selected References
• Brayton, R.K. and Moser, J.K., A theory of nonlinear networks - I,Quart. of App. Math., vol. 22, pp 1–33, April, 1964.
• Moser, J.K., “Bistable systems of differential equations with aplications to tunnel diode circuits”,
IBM Journal of Res. Develop., Vol. 5, pp. 226–240, 1960.
• Chua, L.O., Wang, N.N.; “Complete stability of autonomous reciprocal nonlinear networks,”Int. Journal of Circ. Th. and Appl., July 1978, vol. 6, (no. 3): 211-241.
• Jeltsema, D. and Scherpen, J.M.A., “On Brayton and Moser’s Missing Stability Theorem”,
IEEE Transactions on Circuits and Systems-II, Vol. 52, No. 9, 2005, pp. 550-552.
• Chua, L.O., “Stationary principles and potential functions for nonlinear. networks,”.J. Franklin Inst.,. vol. 296, no. 2, pp. 91-114, Aug. 1973.
• Marten, W., Chua, L.O., and Mathis, W., “On the geometrical meaning of pseudo hybrid content and mixed-potential”,Arch. f. Electron. u. Übertr., Vol. 46, No. 4, 1992.
• Weiss, L., Mathis, W. and Trajkovic, L., “A generalization of Brayton-Moser’s mixed potential function”,IEEE Trans. Circuits and Systems - I, April 1998.
• Jeltsema, D. and Scherpen, J.M.A., “Tuning of Passivity-Preserving Controllers for Switched-Mode Power Converters”,IEEE Trans. Automatic Control, Vol. 49, No. 8, August 2004, pp. 1333- 1344.
Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 29
Selected References
• Jeltsema, D., Ortega, R. and Scherpen, J.M.A., “On Passivity and Power-Balance Inequalities of Nonlinear RLC Circuits”,IEEE Trans. on Circ. and Sys.-I: Fund. Th. and Appl.,Vol. 50, No. 9, 2003, pp. 1174-1179.
• Ortega, R., Jeltsema, D. and Scherpen, J.M.A., “Power shaping: A new paradigm for stabilization of nonlinear RLC circuits”,IEEE Trans. Aut. Contr., Vol. 48, No. 10, 2003.
• Garcia-Canseco, E., Ortega, R., Scherpen, J.M.A. and Jeltsema, D., “Power shaping control of nonlinear systems: a benchmark example”,In Proc. IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Nagoya, Japan (July 2006).
• Brayton, R.K. and Miranker, W.L., “A Stability Theory for Nonlinear Mixed Initial Boundary Value Problems,”Arch. Ratl. Mech. and Anal. 17, 5, 358-376, December 1964.
• Justh, E.W., and Krishnaprasad, P.S., “Pattern-Forming Systems for Control of Large Arrays of Actuators,”Journal of Nonlinear Science, Vol. 11, No. 4, January, 2001.
• Jeltsema, D., and Van der Schaft, A.J., “Pseudo-Gradient and Lagrangian Boundary Control Formulation of Electromagnetic Fields”, J. Phys. A: Math. Theor. , vol. 40, pp. 11627-11643, 2007.
• Garcia-Canseco, E., Jeltsema, D., Scherpen, J.M.A., and Ortega, R., “Power-based control of physical systems: two case studies”, in proc. 17th IFAC World Congress, Seoul, Korea, July 2008.
Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 30
Some Recent Developments
• Favache Audrey and Dochain Denis, “Analysis and control of the exothermic control stirred tank reactor: the power-shaping approach”, in proc. 48th IEEE Conference on Decision and Control (CDC ’09), 2009, pp.1866-1871.
• Favache Audrey and Dochain Denis, “Power-shaping control of reaction systems: the CSTR case study”, to appear in Automatica, 2010.
• R. Ortega, A.J. van der Schaft, F. Castanos and A. Astolfi, “Control by interconnection and standard passivity-based control of port-Hamiltonian system”, IEEE Transactions on Automatic Control, vol.53, pp. 2527–2542, 2008.
• E. Garcia-Canseco, D. Jeltsema, R. Ortega and J.M.A. Scherpen, “Power-based control of physical systems”, Automatica, Volume 46, Issue 1, January 2010, Pages 127-132.
