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EECI-M10: Port-Hamiltonian SystemsDelft 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 objective

desired 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


Recall: Passivity of pH Systems

The port-Hamiltonian system

x= [J(x)!R(x)]∂H∂x

(x)+g(x)u,

y= gT (x)∂H∂x

(x),

with state x " Rn, and port-variables u,y " Rm, is passive if

H[x(t)]!H[x(0)]! "# $stored energy

#% t

0uT (τ)y(τ)dτ

! "# $supplied energy

, ($)

for some Hamiltonian H : Rn% R+.

April 06–09, 2010 4


Stabilization by Damping Injection• Use storage function (read: Hamiltonian) as Lyapunov functionfor the uncontrolled system.

• Passive systems can be asymptotically stabilized by addingdamping via the control. In fact, for a passive port-Hamiltoniansystem we have

H(x)# uT y.

Hence letting u=!Kdy, with Kd = KTd & 0, we obtain

H(x)#!yTKdy,

' asymptotic stability, provided an observability condition ismet (i.e., zero-state detectability of the output).

April 06–09, 2010 5


Stabilization by Damping Injection• If H(x) non-negative, total amount of energy that can beextracted from a passive system is bounded, i.e.,

!% t

0uT (τ)y(τ)dτ # H[x(0)] < ∞.

April 06–09, 2010 6


Energy-Balancing PBC• Usually, the point where the open-loop energy is minimal is notof interest. Instead some nonzero eq. point, say x$, is desired.

• Standard formulation of PBC: Find u= β (x)+ v s.t.

Hd[x(t)]!Hd[x(0)]! "# $stored energy

=% t

0vT (τ)z(τ)dτ


! dd(t)!"#$diss. energy

,

where the desired energy Hd(x) has a minimum at x$, and z 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 state x.

April 06–09, 2010 7


Energy-Balancing PBC• Indeed, from the energy-balance inequality ($) we see that if wecan find a β (x) satisfying

!% t

0βT [x(τ)]y(τ)dτ = Ha[x(t)]+κ,

for some Ha(x), then the control u= β (x)+v will ensure v (% yis passive w.r.t. modified energy Hd(x) = H(x)+Ha(x).

April 06–09, 2010 8


Energy-Balancing PBC

Proposition: Consider the port-Hamiltonian system

x= [J(x)!R(x)]∂H∂x

(x)+g(x)u,

y= gT (x)∂H∂x

(x).

If we can find a function β (x) and a vector function K(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



Proposition (cont’d):

ii) K(x$) =!∂H∂x

(x$) (equilibrium assignment);

iii)∂K∂x

(x$)&!∂2H∂x2

(x$) (Lyapunov stability).

Then the closed-loop system is a port-Hamiltonian system of the form

x= [J(x)!R(x)]∂Hd∂x

(x),

with Hd(x) =H(x)+Ha(x), K(x) = ∂Ha∂x (x), and x$ (locally) stable.

April 06–09, 2010 10



• Note that (R(x) = RT (x)) 0)

Hd(x) =!∂THd∂x

(x)R(x)∂Hd∂x

(x)# 0.

• Also note that x$ is (locally) asymptotically stable if, in addition,the largest invariant set is contained in

&x " D

''''∂ THd∂x

(x)R(x)∂Hd∂x

(x) = 0(

,

where D* Rn.

April 06–09, 2010 11


Mechanical Systems

Consider a (fully actuated) mechanical systems with total energy

H(q, p) = 12 p

TM!1(q)p+P(q),

with generalized mass matrixM(q) =MT (q)& 0. Assume that thepotential energy P(q) is bounded from below. PH structure:

)

*qp

+

, =

)

* 0 Ik!Ik 0

+

,

! "# $J=!JT

)

*∂H∂q∂H∂ p

+

,+

)

* 0

B(q)

+

,

! "# $g(q)

u,

y=-0 BT (q)

.)

*∂H∂q∂H∂ p

+

, .

April 06–09, 2010 12


Mechanical Systems

Clearly, the system has as passive outputs the generalized velocities:

H(q, p) = uT y= uTBT (q)∂H∂ p

(q, p) = uTM!1(q)p= uT q.

Now, the Energy-Balancing PBC design boils down to

JK(q, p) = g(q)β (q, p), K(x) =∂Ha∂x

(x)

which in the fully actuated (B(q) = Ik) case simplifies to

K2(q, p) = 0

!K1(q, p) = β (q, p).

April 06–09, 2010 13


Mechanical Systems

The simplest way to ensure that the closed-loop energy has aminimum at (q, p) = (q$,0) is to select

β (q) =∂P∂q

(q)!Kp(q!q$), Kp = KTp & 0.

This gives the controller energy

Ha(q) =!P(q)+12(q!q$)TKp(q!q$)+κ,

so that the closed-loop energy takes the form

Hd(q, p) =12pTM!1(q)p+

12(q!q$)TKp(q!q$).

April 06–09, 2010 14


Mechanical Systems

To ensure that the trajectories actually converge to (q$,0) we need torender the closed-loop asymptotically stable by adding some damping

v=!Kd∂H∂ p

(q, p) =!Kdq,

as shown before. Note that the energy-balance of the system is now

Hd[q(t), p(t)]!Hd [q(0), p(0)]! "# $stored energy

=

% t

0vT (τ)q(τ)dτ


!% t

0qT (τ)Kpq(τ)dτ

! "# $diss. energy

.

April 06–09, 2010 15


Mechanical Systems

• Observe that the controller obtained via energy-balancing is justthe classical PD + gravity compensation controller.

• However, the design via Energy-Balancing PBC provides a newinterpretation of the controller, namely, that the closed-loopenergy is (up to a constant) equals to

Hd(q, p) = H(q, p)!% t

0uT (τ)y(τ)dτ ,

i.e., the difference between the open-loop and controller energy.

April 06–09, 2010 16


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 admissibleequilibrium.

April 06–09, 2010 17


Dissipation Obstacle• Unfortunately, EnergyBalancing PBC is stymied by the existenceof pervasive dissipation.

• A necessary condition to satisfy the Energy-Balancing PBCproposition is

R(x)K(x) = R(x)∂Ha∂x

(x) = 0 ' dissipation obstacle.

• This implies that no damping is present in the coordinates thatneed to be shaped.

• Appears in many engineering applications.• Limitations and the dissipation obstacle can be characterized viathe control–by–interconnection perspective.

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 ControlApril 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


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) ="JT (x). Suppose that the mapping from the energy variables x to theco-energy variables e is invertible, such that

x = x(e) =!H!

!e(e),

with H!(e) the Legendre transformation of H(x) given by

H!(e) = eT x"H(x).

Then the dynamics (!) can also be expressed in terms of e as

! 2H!

!e2 (e)e = J(x)e.



Assume that we can find coordinates x = (xq,xp)T , with dim xq = k and dimxp = n" k, such that

J(x) =!

0 B(x)"BT (x) 0

",

with B(x) a k# (n" k) matrix. Furthermore, assume that

H(xq,xp) = Hq(xq)+Hp(xp).

In that case, the co-Hamiltonian can be written as

H!(eq,ep) = H!q (eq)+H!

p(ep),

and thus #

$! 2H!

!e2q

0

0 ! 2H!

!e2p

%

&!

eqep

"=

!0 B(x)

"BT (x) 0

"!eqep

".



Defining the (mixed-potential) function

P(eq,ep,x) = eTq B(x)ep,

it follows that !

""#

! 2H!

!e2q

0

0 !! 2H!

!e2p

$

%%&

' () *Q(eq,ep)

+eqep

,=

!

""#

!P!eq!P!ep

$

%%& .

These equations, called the Brayton-Moser equations, can be interpreted as agradient system with respect to P and the indefinite pseudo-Riemannian metric Q.



Note that the mixed-potential can be written as

P(eq,ep,x) = eTq B(x)ep = eT

q xq =!xTp ep,

which represent the instantaneous power flow between !q and !p.



Recall that dissipation can be included in the PH framework via

x = J(x)!H!x

(x)+gR(x) fR,

eR = gTR(x)

!H!x

(x),

with

fR =!!D!

!e(e).

Hence we can write!

#! 2H!

!e2q

0

0 ! 2H!

!e2p

$

&+

eqep

,=

+0 B(x)

!BT (x) 0

,+eqep

,!gR(x)

!D!e

(e),

eR = gTR(x)e.



For simplicity we assume that

gR =!+

Ik 00 In"k

,,

so that!

""#

! 2H!

!e2q

0

0! 2H!

!e2p

$

%%&

+eqep

,=

+0 B(x)

!BT (x) 0

,+eqep

,+

!

""#

!D!

!eq!D!

!ep

$

%%&

eR =!+

eqep

,.



Finally, definingP(eq,ep,x) = eT

q B(x)ep +D!(eq,ep),

we (again) obtain the Brayton-Moser equations

!

""#

! 2H!

!e2q

0

0 !! 2H!

!e2p

$

%%&

'eqep

(=

!

""#

!P!eq!P!ep

$

%%& .

Let us zoom in to electrical circuits . . .



Consider an electrical network ! with nL inductors, nC capacitors, and nR resistors.Let i " RnL and v " RnC , then [Brayton and Moser 1964]

! = !p#!q :

)*+

*,

!L(i)didt

= $iP(i,v)

C(v)dvdt

= $vP(i,v),-$• = !

!•

.,

with mixed-potential P : RnL%RnC & R. For topologically complete networks

P(i,v) = vT Bi+R(i)!G(v)/ 01 2D!(i,v)

,



Example: Tunnel-diode circuit [Moser 1960]

! R(i) = 12 Ri2!Uini

! G(v) =3 v

ig(v")dv"

! vT Bi = vi, (B = 1)

Mixed-potential function: P(i,v) = iv+12

Ri2!Uini!3 v

ig(v")dv"

! :

)*+

*,

!Ldidt

= $iP =!Uin +Ri+ v (!p)

Cdvdt

= $vP = i! ig(v). (!q)


Main Motivation: Stability Theory

Rewrite BM equations asQ(x)x =$xP(x),

with

x =4

iv

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 general P = 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 any M = M! and ! # R, new pairs can be found from

Q(x) :=#$2

xP(x)M +! I$

Q(x)

P(x) := !P(x)+12[$xP(x)]!M$xP(x).

Note that the system behavior is preserved since

x = Q"1$xP(x) % x = Q"1$xP(x).


Example: Tunnel Diode Circuit

Obviously, for the tunnel diode circuitQ+Q! ! 0. However, selecting

M =* 1

R 00 L

RC

+"1

, ! = 1,

yields

Q(v) =

,0 L

R

( LR (C( L

R$vig(v)

-,

and

P(i,v) =) v

0ig(v$)dv$+

L2RC

(i( ig(v))2 +1

2R(v(Uin)2.

Hence, it is easily seen that

minv$vig(v) >(RC

L% Q(v)+ Q!(v)! 0.


State-of-the-Art

! Constructive procedures to obtain stability criteria using the mixed-potentialfunction are given in [Brayton and Moser 1964]& Three theorems, eachdepending on the type of nonlinearities in R or LC part.

! Generalizations can be found in e.g.,! [Chua and Wang 1978] (for cases that |$2

i R|= 0 or |$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...


Energy-Balancing Control [Ortega et al.]

To put our ideas into perspective let us briefly recall the principle ofEnergy-Balancing (EB) Control. Consider general system representation

x = f (x)+g(x)u, y = h(x), x # Rn, u,y # Rm. (')

Assume that (') satisfies

H[x(t)](H[x(0)]% &' (Open-loop stored energy

)) t

0u!(")y(")d"

% &' (Supplied energy

,

with storage function H : Rn * R. If H + 0, then (') is passive wrt (u,y).

Usually desired operating point, say x#, not minimum of H. Idea is to look foru : Rn * Rm st

() t

0u![x(")]h[x(")]d" = Ha[x(t)](Ha[x(0)],

for some Ha : Rn * R.Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 16


Hence, the control u = u(x)+ v will ensure that the closed-loop system satisfies

Hd[x(t)]!Hd[x(0)]! "# $Closed-loop stored energy

"% t

0v!(!)y(!)d!

! "# $Supplied energy

,

where Hd = H +Ha is the closed-loop energy storage. If, furthermore,

x" = arg min Hd(x),

then x" will be stable equilibrium of the closed-loop system (with Lyapunov functionthe difference between the stored and the control energies.)



However, applicability of EB severely stymied by the existence of pervasivedissipation. Indeed, since solving for u is equivalent to solving the PDE

[ f (x)+g(x)u(x)]!#xHa(x) =!u!(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

u!(x")h(x") = 0.

$ This is known as the dissipation obstacle.


Passivity: Power-Balance Inequality [Jeltsema et al. 2003]

Extract sources (controls) and rewrite BM equations as

Q(x)x =#xP(x)+G(x)u.

We then observe that if Q(x)+Q!(x)% 0 the network satisfies thepower-balance inequality

P[x(t)]!P[x(0)]"% t

0u!(!)y(!)d!,

with outputs y = h(x,u) =!G!(x)Q#1(x)[#xP(x)+G(x)u].

! Power as storage function instead of energy.

! Trivially satisfied by all RL and RC networks with passive elements.

! Notice that y =!G!(x)x$ natural derivatives in the output!


Control: Power-Shaping Stabilization [Ortega et al. 2003]

The open-loop mixed-potential is shaped with the control u = u(x), where

G(x)u(x) =#xPa(x),

for some Pa : Rn & R. This yields the closed-loop system Q(x)x =#xPd(x),with total power function

Pd(x) = P(x)+Pa(x).

The equilibrium x" will be stable if x" = arg min Pd(x).

$ Power-Balancing: closed-loop power function equals difference betweenopen-loop power function and power supplied by controller, i.e.,

Pa =!u!(x)h(x, u(x)) = u!(x)G!(x)x.

$ No dissipation obstacle since u!(x")G!(x")x" = 0!


Power-Shaping Stabilization

However, as mentioned before, in general Q(x)+Q!(x) ! 0, which requires firstthe generation of a new pair

!Q, P

", such that

Q(x)+ Q!(x)! 0, |Q| "= 0.

Proposition. For any M(x) = M!(x) and ! # R, new pairs can be found from

Q(x) :=#

12$2

xP(x)M(x)+12$x

$M(x)$xP(x)

%+! I

&Q(x)

P(x) := !P(x)+12[$xP(x)]!M(x)$xP(x).

Hence,

x = Q"1$xP(x)+G(x)u % x = Q"1$xP(x)+ G(x)u,

with G(x) = Q(x)G(x).Module M10, Special topics EECI-M10: Port-Hamiltonian Systems 21


Again, selecting M = diag$ 1

R , LRC

%"1and ! = 1 yields

Q(v) =

'0 L

R

& LR &C& L

R$vig(v)

(, G =

)0& L

R

*,

and

P(i,v) =+ v

0ig(v#)dv#+

L2RC

(i& ig(v))2 +1

2Rv2.

Assumption.$

minv$vig(v) >&RC

L.

!This assumption can be relaxed applying a preliminary feedback!Rai, with

Ra >!#

LC

minv"vig(v)+R&.



Using the Power-Shaping procedure we obtain:

Proposition. The control

u =&K(v& v$)+u$, (= Uin)

with control parameter K > 0 satisfying

K >& [1+R$vig(v$)] ,

globally asymptotically stabilizes x$ = col(i$,v$) with Lyapunov function

P(i,v) =+ v

0ig(v#)dv#+

L2RC

(i& ig(v))2 +K2R

(v& v$)2 +1

2R(v&u$)2.


Poincare’s Lemma

Existence of P follows from Poincare’s Lemma. Indeed, suppose the network isdescribed by

x = f (x)+g(x)u,

with f : Rn ' Rn and f # C1, then there exists a P : Rn ' 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 exists Q : Rn ! Rn!n, |Q| "= 0, satisfying

#x(Q(x) f (x)) = [#x(Q(x) f (x))]",

and Q(x)+Q"(x)$ 0. (A.2) There exists Pa : Rn ! R verifying! g#(x)Q$1(x)#xPa = 0, with g#(x)g(x) = 0, rank{g#(x)} = n%m;! x% = arg min Pd(x), where

Pd(x) :=! x

[#x!(Q(x&) f (x&))]"dx&+Pa(x).

Proposition. Under A.1 and A.2, the control law

u ="g"(x)Q"(x)Q(x)g(x)

#$1g"(x)Q"(x)#xPa(x)

ensures x% is (locally) stable with Lyapunov function Pd(x).


Remark:

! Observe that Assumption A.1 includes the class of port-Hamiltonian systemswith invertible interconnection and damping matrices.

Indeed, recall

x = [J(x)%R(x)]#xH(x)+g(x)uy = g"(x)#xH(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) =#xH(x).


Final Remarks

! Power-Shaping applicable to general nonlinear systems.! Similar to Energy-Balancing. However, no dissipation obstacle involved.! Tunnel diode example shows that Power-Shaping yields a simple linear

(partial) state-feedback controller that ensures robust global asymptoticstability of the desired equilibrium point.

! Current research includes:• Solvability of the PDE

#x(Q(x) f (x)) = [#x(Q(x) f (x))]" subject to Q(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)


...

“The real voyage of discovery consists not in seekingnew 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. FranklinInst.,. vol. 296, no. 2, pp. 91-114, Aug. 1973.

• Marten, W., Chua, L.O., and Mathis, W., “On the geometrical meaning of pseudo hybridcontent 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 potentialfunction”, IEEE Trans. Circuits and Systems - I, April 1998.

• Jeltsema, D. and Scherpen, J.M.A., “Tuning of Passivity-Preserving Controllers forSwitched-Mode Power Converters”, IEEE Trans. Automatic Control, Vol. 49, No. 8, August2004, pp. 1333- 1344.


Selected References

• Jeltsema, D., Ortega, R. and Scherpen, J.M.A., “On Passivity and Power-Balance Inequalitiesof 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 forstabilization 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 ofnonlinear systems: a benchmark example”, In Proc. IFAC Workshop on Lagrangian andHamiltonian Methods for Nonlinear Control, Nagoya, Japan (July 2006).

• Brayton, R.K. and Miranker, W.L., “A Stability Theory for Nonlinear Mixed Initial BoundaryValue 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 ofActuators,” Journal of Nonlinear Science, Vol. 11, No. 4, January, 2001.

• Jeltsema, D., and Van der Schaft, A.J., “Pseudo-Gradient and Lagrangian Boundary ControlFormulation 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 ofphysical systems: two case studies”, in proc. 17th IFAC World Congress, Seoul, Korea, July2008.


Some Recent Developments

• Favache Audrey and Dochain Denis, “Analysis and control of the exothermic control stirredtank reactor: the power-shaping approach”, in proc. 48th IEEE Conference on Decision andControl (CDC ’09), 2009, pp.1866-1871.

• Favache Audrey and Dochain Denis, “Power-shaping control of reaction systems: the CSTRcase study”, to appear in Automatica, 2010.

• R. Ortega, A.J. van der Schaft, F. Castanos and A. Astolfi, “Control by interconnection andstandard passivity-based control of port-Hamiltonian system”, IEEE Transactions on AutomaticControl, vol.53, pp. 2527–2542, 2008.

• E. Garcia-Canseco, D. Jeltsema, R. Ortega and J.M.A. Scherpen, “Power-based control ofphysical systems”, Automatica, Volume 46, Issue 1, January 2010, Pages 127-132.


s - dimitri jeltsema/eeci_slides_lecture4+5.pdf · port-hamiltonian systems: from geometric network...

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