Book of Abstracts of
4th IFAC Workshop on
Lagrangian and Hamiltonian Methods for Non Linear Control
Technical Program for Wednesday August 29, 2012
WePP

Sala degli Affreschi

The BraytonMoser Form for PowerShaping Control (Plenary Session)

09:0010:00

WePP.1

The BraytonMoser Form for PowerShaping Control*

Winkin, Joseph J.

Univ. of Namur

Powershaping control is a control design method which can be seen as an extension of energybalancing passivitybased control. This control scheme is characterized by the fact that stabilization is achieved by shaping a function that is related to power instead of energy. One of the most difficult parts in achieving such a control design is the derivation of a specific form of the dynamics, namely the BraytonMoser form, on which Powershaping control is based. More specifically the latter involves the solution of a partial differential equation (PDE) system together with an additional sign constraint.
This talk will include a brief tutorial on the BraytonMoser Form for Power Shaping Control, a short survey of the literature and some examples illustrating the wide applicability of this control design method. In addition, a general methodology will be described for solving the PDE system that is needed for getting the BraytonMoser form. The solution set of this PDE system can be characterized in terms of the solutions of a related system of algebraic linear equations. A necessary condition will also be reported that guarantees the existence of a solution to the algebraic linear system which satisfies the sign condition. The methodology will be illustrated by an example of a chemical reactor where the physical knowledge of the system is used to find a suitable solution.

WeAR

Sala degli Affreschi

Hamiltonian and PortHamiltonian Systems (Regular Session)

10:3010:50

WeAR.1

On the PortHamiltonian Representation of Systems Described by Partial
Differential Equations, pp. 16

Schoberl, Markus

Johannes Kepler Univ. of Linz

Siuka, Andreas

Johannes Kepler Univ. Linz

We consider infinite dimensional portHamiltonian systems. Based on a power balance relation we introduce the portHamiltonian system representation where we pay attention to two different scenarios, namely the nondifferential operator case and the differential operator case regarding the structural mapping, the dissipation mapping and the in/output mapping. In contrast to the wellknown representation on the basis of the underlying StokesDirac structure our approach is not necessarily based on using energyvariables which leads to a different portHamiltonian representation of the analysed partial differential equations.

10:5011:10

WeAR.2

Fast Computation by Simplifications of a Class of HydroMechanical
Systems, pp. 712

Sakai, Satoru

Shinshu Univ.

This paper presents a fast computation method of forward dynamics of a class of hydromechanical systems. First, an exact simplification of the conventional representation of the hydromechanical systems is presented based on Casimir functions. Second, a further simplification is given via an integration by parts which is a new structural property and a new representation of the hydromechanical systems is proposed. Third, the proposed representation and the conventional representation are compared with each other with respect to computational cost and the validity of the proposed representation is confirmed by numerical experiment. 
11:1011:30

WeAR.3

Irreversible Port Hamiltonian Systems, pp. 1318

Ramirez Estay, Hector M.

Univ. Claude Bernard Lyon 1

Maschke, Bernhard

Univ. Claude Bernard of Lyon

Sbarbaro, Daniel G.

Univ. de Concepcion

A class of quasi port Hamiltonian system expressing the first and second principle of thermodynamic as a structural property is defined, namely Irreversible PHS. The IPHS is defined by: a generating function that for physical systems corresponds to the total energy; a constant skewsymmetric structure matrix that represents the network structure of the system; a nonlinear function that depends on the states and costates and on the Poisson bracket of the generating function and some entropy function. For physical systems this Poisson bracket defines the thermodynamic driving force. The IPHS is completed with input and output ports. IPHS encompasses a large set of thermodynamic systems, including heat exchangers and chemical reactors. The nonisothermal CSTR is used to illustrate the formalism.

11:3011:50

WeAR.4

Weak Positive Poisson Stability and Hamiltonian Vector Fields in
Mechanical Systems, pp. 1923

Bayadi, Ramaprakash

Indian Inst. of Tech. Bombay

Banavar, Ravi N.

Indian Inst. of Tech.

We consider mechanical systems whose configuration manifold is $Q = G times S$, where $G$ is a compact Lie group and $S$ is a smooth manifold. Under an additional assumption of symmetry, we show that the dynamics of the system over the phase space $T^*Q$ can be reduced to $G times T^*S$. We then show that the component of the dynamics on $G$ is weakly positively Poisson stable. We apply this result to analyze global attitude controllability of a spacecraft with two rotors.

11:5012:10

WeAR.5

A Hamiltonian Perspective on the Control of Dynamical Distribution
Networks, pp. 2429

van der Schaft, Arjan J.

Univ. of Groningen

Wei, Jieqiang

Univ. of Groningen

We study a basic dynamical distribution network, modeled as a directed graph with storage variables corresponding to the vertices, and unknown but constant inflows and outflows. It is shown how standard PIcontrol, regulating the storage variables irrespective of the inflows and outflows, corresponds to associating with every edge of the graph a controller state variable, yielding a closedloop portHamiltonian system. Furthermore, it will be shown how regulation is proved by modifying the total Hamiltonian of the portHamiltonian system into a Lyapunov function based on the vector of constant inflows and outflows. Subsequently, the results are extended to the case that the input variables are constrained, leading to nonsmooth Lyapunov functions.

12:1012:30

WeAR.6

On PortHamiltonian Modeling of the Synchronous Generator and Ultimate
Boundedness of Its Solutions, pp. 3035

Shaik, Fiaz

Univ. of Groningen

Zonetti, Daniele

Lab. des Signaux et Systemes CNRSSUPELEC

Ortega, Romeo

Supelec

Scherpen, Jacquelien M.A.

Univ. of Groningen

van der Schaft, Arjan J.

Univ. of Groningen

In this paper starting with bond graph techniques a nonlinear mathematical model of the synchronous generator in portHamiltonian framework is derived. This leads to an energybased description of the system which we later use for stability analysis. We use Park’s state transformation to decouple the dynamics of other state variables from the dynamics of rotor angle, resulting in a quotient system admitting equilibria. We show that the solutions of this quotient system are bounded and provide closed form expression for the ultimate bound of these solutions. We will also give some preliminary results on stability analysis of these equilibria using energy shaping techniques.

WeBR

Sala degli Affreschi

Control of Lagrangian and Hamiltonian Systems (I) (Regular Session)

14:0014:20

WeBR.1

Equivalence of Immersion and Invariance and IDAPBC for the Acrobot,
pp. 3641

Kotyczka, Paul

Tech. Univ. M\9Fnchen

Sarras, Ioannis

CNRS

In this note the two well known nonlinear control design techniques Interconnection and Damping Assignment Passivity Based Control (IDAPBC) and Immersion and Invariance (I&I) are compared through the example of the socalled Acrobot under actuated mechanical system. Equivalences of both procedures become obvious from the corresponding immersion and matching equations. In particular, the coordinate change which renders the potential energy matching PDE in IDAPBC an ordinary differential equation is used to define the immersion map in I&I. It is shown that the energy shaping part of the IDAPBC controller makes the closedloop system an interconnection of two lowerdimensional portHamiltonian (pH) systems in the on and offmanifold coordinates. The effect of damping injection output feedback can be identified with dissipation in the offmanifold part of the interconnected system. Dissipation is propagated to the onmanifold part which results in asymptotic stability of the system’s equilibrium. The analysis in the present work provides an interesting interpretation of the effect of the IDAPBC control law using the I&I framework.

14:2014:40

WeBR.2

Vibration Suppression of MassSpringDamper System with Dynamic Dampers
Using IDAPBC, pp. 4247

Aoki, Takashi

Hokkaido Univ.

Yamashita, Yuh

Hokkaido Univ.

Tsubakino, Daisuke

Hokkaido Univ.

In this paper, we propose a vibration suppression control method for a massspringdamper system with one or two dynamic damper(s). The feedback is designed by the interconnection and damping assignment passivitybased control, where the system is transformed to a system having a skyhook damper with an artificial modification of the structure matrix. The feedback is expressed by a function of the relative displacements and velocities. The proposed control method can suppress the influences of the floor vibration and the disturbance force acting on the main body, simultaneously.

14:4015:00

WeBR.3

The Matching Equations of Energy Shaping Controllers for Mechanical
Systems Are Not Simplified with Generalized Forces, pp. 4853

Crasta, Naveena

Supelec

Ortega, Romeo

Supelec

Pillai, Harish

Indian Inst. of Tech. Bombay

Romero Velazquez, Jos\8E Guadalupe

Lab. des Signaux et Syst\8Fmes,
CNRSâ€“SUPELEC

Total Energy Shaping is a
controller design methodology that achieves (asymptotic) stabilization of
mechanical systems endowing the closedloop system with a Lagrangian or
Hamiltonian structure with a desired energy function. The success of the
method relies on the possibility of solving two partial dierential equations
(PDE) which identify the kinetic and potential energy functions that can be
assigned to the closedloop. Particularly troublesome is the PDE associated
to the kinetic energy which is quasilinear and inhomogeneous and the
solution, that denes the desired inertia matrix, must be positive denite.
This task is simplied by the inclusion of gyroscopic forces in the target
dynamics, which translates into the presence of a free skewsymmetric matrix
in the matching equations that reduces the number of PDE’s to be solved.
Recently, it has been claimed that considering a more general form for the
target dynamic forces, that relax the skewsymmetry condition, further
reduces the number of PDE’s. The purpose of this paper is to prove that this
claim is wrong.

15:0015:20

WeBR.4

A Remark on Controlled Lagrangian Approach for Completely Integrable
Mechanical Systems, pp. 5459

Shiriaev, Anton

Umea Univ.

Freidovich, Leonid

Ume\8C Univ.

Spong, Mark W.

Univ. of Texas at Dallas

Using an example of cartpendulum
system we give some new insights into the methods of Controlled Lagrangians,
which can be used for planning forced trajectories and their orbital
stabilization. The issue of integrability as a result of preservation and
creation of conserved quantities is emphasized and discussed in detail

15:2015:40

WeBR.5

Simplifying Robust Energy Shaping Controllers for Mechanical Systems
Via Coordinate Changes, pp. 6065

Romero Velazquez, Jos\8E Guadalupe

Lab. des Signaux et Syst\8Fmes,
CNRSâ€“SUPELEC

Donaire, Alejandro

CDSC – Centre for Complex Dynamic Systems and
Control, TheUnive

Ortega, Romeo

Supelec

The problem of robustness
improvement, vis `a vis external disturbances, of energy shaping controllers
for mechanical systems was addressed by the authors in a previous paper. It
was shown thatif the inertia matrix is constant– constant disturbances
(both, matched and unmatched) can be rejected simply adding a suitable integral
action. For systems with nonâ€“constant inertia matrix and nonconstant
disturbances the controller, that adds nonlinear damping and gyroscopic
forces terms, is quite complicated. The purpose of this paper is to show
that, including a partial change of coordinates, the controller can be
significantly simplified, achieving the same robustness property of
input–tostate stability with respect to matched and unmatched disturbances
of the previous controller.

15:4016:00

WeBR.6

Representation and Control of Brayton–Moser Systems Using a Geometric
Decomposition, pp. 6671

Guay, Martin

Queen’s Univ.

Hudon, Nicolas

The Univ. of New South Wales

Hoeffner, Kai

Massachusetts Inst. of Tech.

This paper considers the problem of
representing a sufficiently control affine system as a structured
Brayton–Moser system and to use the obtained structure to stabilize a
desired equilibrium of the system. In recent years, matching conditions,
expressed as partial differential equations, were developed to represent a
general nonlinear systems into Brayton–Moser form. Departing from this
approach, the present note proposes a geometric decomposition technique to
reexpress a given vector field as a Brayton–Moser system. The proposed
method is based on a decomposition of a differential oneform that encodes
the divergence of a given vector field into its exact and antiexact
components, and into its coexact and anticoexact components. The
decomposition method, based on the Hodge decomposition theorem, is rendered
constructive by introducing a dual operator to the standard homotopy
operator. The dual operator inverts locally the codifferential operator, and
is used in the present paper to identify the structure of the dynamics.
Applications of the proposed approach to the control of the threedimensional
rigidbody problem are also presented to illustrate the construction.

WeCR

Sala degli Affreschi

Mechanical Systems and Robotics (Regular Session)

16:3016:50

WeCR.1

The EulerPoincare Equations for a Spherical Robot Actuated by a
Pendulum, pp. 7277

Gajbhiye, Sneha

IIT Bombay

Banavar, Ravi N.

Indian Inst. of Tech.

Mechanical systems with rolling
constraints form a class of nonholonomic systems. In this paper we derive the
dynamic model of a spherical robot, which has been designed and realized in
our laboratory, using Lagrangian reduction theory defined on symmetry groups.
The reduction is achieved by applying Hamilton’s variation principle on a
reduced Lagrangian and then imposing the nonholonomic constraints. The
equations of motion are in the EulerPoincare form and are equivalent to
those obtained using Lagranged’Alembert’s principle.

16:5017:10

WeCR.2

Decentralized Global Connectivity Maintenance for Interconnected
Lagrangian Systems with Communication Delays, pp. 7883

Secchi, Cristian

Univ. of Modena and Reggio Emilia

Sabattini, Lorenzo

Univ. of Modena and Reggio Emilia

Fantuzzi, Cesare

Univ. of Modena and Reggio Emilia

In order to accomplish cooperative
tasks, multiâ€“robot systems are required to communicate among each
other. Thus, maintaining the connectivity of the communication graph is a
fundamental issue. In this paper we extend the connectivity maintenance
control strategy introduced in (Sabattini et al., 2012), in order to
explicitly take into account the presence of communication delays. When
dealing with interconnected robotic systems, in fact, assuming instantaneous
exchange of data is often unrealistic. For this reason, we provide a solution
to the connectivity maintenance problem for interconnected Lagrangian
dynamical agents, in the presence of communication delays.

17:1017:30

WeCR.3

Further Results on Virtual Holonomic Constraints, pp.
8489

Jankuloski, Dame

Univ. of Toronto

Maggiore, Manfredi

Univ. of Toronto

Consolini, Luca

Univ. of Parma

This paper continues recent work by
the authors on virtual holonomic constraints (VHCs) for EulerLagrange
control systems with $n$ degreesoffreedom and $m$ control inputs. The focus
of the paper is on implicit constraints of the form $h(q)=0$. Under suitable
regularity conditions, the enforcement of $kleq m$ constraints induces
constrained dynamics that are described by a reducedorder control system of
dimension $2 (nk)$ with $(mk)$ control inputs. When $m=k=n1$, conditions
are given guaranteeing that the constrained dynamics are EulerLagrange. It
is shown that the presence of dissipation may have unexpected consequences on
the constrained dynamics, turning stable equilibria into unstable ones.
Finally, VHCs are applied to the problem of constraining a spherical pendulum
to lie on the upper half plane.

17:3017:50

WeCR.4

A Class of Standard Mechanical System with Force Feedback in the
PortHamiltonian Framework, pp. 9095

MunozArias, Mauricio

Univ. of Groningen

Scherpen, Jacquelien M.A.

Univ. of Groningen

Dirksz, Daniel A.

Eindhoven Univ. of Tech.

In this paper we show force
feedback and position control of a class of standard mechanical system in the
portHamiltonian framework. Furthermore, we show how to derive an extended
portHamiltonian system with structure preservation which can be used for
force feedback purposes besides providing the closedloop system
asymptotically stable. We also show the usefulness of the extended
portHamiltonian system by showing its disturbance attenuation properties.
Finally, we present simulation results obtained for the proposed control
laws.

17:5018:10

WeCR.5

Abstractions for Mechanical Systems, pp. 96101

Sloth, Christoffer

Aalborg Univ.

Wisniewski, Rafal

Aalborg Univ.

This paper proposes a method for
discretizing the state space of mechanical systems. This is a first attempt
in using reduction techniques for mechanical systems in the partitioning of
the state space. The method relies on a combination of transversal and
tangential manifolds for the conservative mechanical system. The tangential
manifolds are generated using constants of motion, which can be derived from
Noether’s theorem. The transversal manifolds are subsequently generated on a
reduced space given by the Routhian, via actionangle coordinates. The method
fully applies for integrable systems.
We focus on a particular aspect of abstraction –
partitioning the state space, as existing methods can be applied on the
discretized state space to obtain an automatabased model. The contribution
of the paper is to show that wellknown reduction methods can be used to
generate abstract models, which can be used for formal verification.

18:1018:30

WeCR.6

On Linearization of Mechanical Control Systems, pp.
102107

Respondek, Witold

Inst. National des Sciences Appliquees

Ricardo, Sandra

UTAD and ISRCoimbra

We discuss linearization, via a
diffeomorphism, of mechanical control systems and study the problem of
whether both structures, linear and mechanical ones, are compatible. The
first problem we consider is: given a mechanical control systems that is
linearizable, can we linearize it preserving, simultaneously, its given
mechanical structure. The second problem is whether a general controlaffine
system that is linearizable and admits a mechanical control structure can be
transformed into a linear mechanical structure. Finally we discuss
equivalence to a subclass of linear mechanical control systems, namely those
subject to positional forces only.

Technical Program for Thursday August 30, 2012
ThPP

Sala degli Affreschi

New Results on EulerLagrange and Port Hamiltonian Systems (Plenary Session)

09:0010:00

ThPP.1

New Results on EulerLagrange and Port Hamiltonian Systems:
State Observers, Robust Control and Synchronization*

Ortega, Romeo

Supelec

In this talk we review some recent
results on control of EL and pH systems. Including (i) globally convergent
speed observers for mechanical systems with nonholonomic constraints; (ii)
design of robust (integral action) controllers to reject (or attenuate the
effect of) external disturbances; (iii) robustification via networking of
energy shaping controllers; (iv) synchronization of distinct agents with
communication delays and uncertain parameters.

ThAR

Sala degli Affreschi

Structured Modeling and Control of Distributed Parameter Systems (Invited Session)

Organizer: Le Gorrec, Yann

FEMTOST

10:3010:50

ThAR.1

PortHamiltonian Formulation for Systems of Conservation Laws:
Application to Plasma Dynamics in Tokamak Reactors (I), pp.
108113

Vu, Trang

LCIS Lab.

Lefevre, Laurent

Grenoble INP

Maschke, Bernhard

Univ. Claude Bernard of Lyon

A portHamiltonian model is derived
for the thermomagnetohydro dynamics of plasma in tokamaks.
Magnetohydrodynamic and energy balance equations are expressed in their
covariant form and written in the portHamiltonian formalism using Dirac
structures. This Dirac structure is established as the interconnection of
StokesDirac structures with a specific interconnection structure
representing the magnetohydrodynamical coupling. Finally the problem of
current density profile control is defined and potential approaches are
discussed.

10:5011:10

ThAR.2

Reduction of StokesDirac Structures and Gauge Symmetry in
PortHamiltonian Systems (I), pp. 114119

Seslija, Marko

Univ. of Groningen

van der Schaft, Arjan J.

Univ. of Groningen

Scherpen, Jacquelien M.A.

Univ. of Groningen

StokesDirac structures are
infinitedimensional Dirac structures defined in terms of differential forms
on a smooth manifold with boundary. These Dirac structures lay down a
geometric framework for the formulation of Hamiltonian systems with a nonzero
boundary energy flow. Simplicial triangulation of the underlaying manifold
leads to the socalled simplicial Dirac structures, discrete analogues of
StokesDirac structures, and thus provides a natural framework for deriving
finitedimensional portHamiltonian systems that emulate their
infinitedimensional counterparts. The portHamiltonian systems defined with
respect to StokesDirac and simplicial Dirac structures exhibit gauge and a
discrete gauge symmetry, respectively. In this paper, employing Poisson reduction
we offer a unified technique for the symmetry reduction of a generalized
canonical infinitedimensional Dirac structure to the Poisson structure
associated with StokesDirac structures and of a finedimensional Dirac
structure to simplicial Dirac structures. We demonstrate this Poisson scheme
on a physical example of the vibrating string.

11:1011:30

ThAR.3

Boundary Energy Shaping of Linear Distributed PortHamiltonian Systems
(I), pp. 120125

Macchelli, Alessandro

Univ. of Bologna – Italy

This paper deals with the
stabilization via Casimir generation and energy shaping of linear, lossless,
distributed portHamiltonian systems. Once inputs and outputs of the
distributed portHamiltonian system have been chosen to obtain a welldefined
boundary control systems, conditions for the existence of Casimir functions
in closedloop and of the associated semigroup are given, together with a
criterion to be used to check asymptotic stability. Casimir functions suggest
how to select the controller Hamiltonian to introduce a minimum at the
desired equilibrium, while stability is ensured if proper
“pervasive” boundary damping is present. The methodology is
illustrated with the help of a Timoshenko beam with fullactuation on one
side.

11:3011:50

ThAR.4

On the Inclusion of Actuator Dynamics in Boundary Control of
Distributed Parameter Systems (I), pp. 126130

Burns, John A

Virginia Tech.

Zietsman, Lizette

Virginia Tech.

The problem of boundary control in
systems governed by partial differential equations often leads to abstract
control systems of the form% [
dot{z}(t)=mathcal{A}z(t)+mathcal{N}(z(t))+mathcal{B}v(t )+mathcal{G}%
w_{p}(t), ] where $mathcal{A}$ generates a C$_{0}$semigroup, $mathcal{B}$ is
an unbounded operator and the system is defined in a very weak sense. Here
$mathcal{N}(cdot)$ is a nonlinear term and $w_{p}(t)$ represents a
disturbance to the plant. In this setting the unboundedness of the operator
$mathcal{B}$ can lead to theoretical and computational challenges. However,
in most practical settings the input at the boundary $v(t)$ is typically the
output of a dynamic “actuator” and the inclusion of actuator dynamics is a
more realistic representation of the system. Although the inclusion of
actuator dynamics can bring additional complexity to the corresponding
control problem, in some cases the formulation the control system as a
composite system is essential. Moreover, in some cases including the actuator
dynamics can produce theoretical and computational advantages that can be
exploited when introducing approximations. In this paper we discuss various
formulations of boundary control problems with actuator dynamics and suggest
an alternate approach to formulating certain boundary control problems so
that the resulting composite system is wellposed. We apply these results to
boundary control of parabolic systems to illustrate the ideas and present numerical
results.

11:5012:10

ThAR.5

Energy Estimation in Numerical Scheme for Nonlinear Partial
Differential Equations (I), pp. 131136

Yamaguchi, Kyosuke

Nagoya Univ.

Nishida, Gou

Kyoto Univ.

Sakamoto, Noboru

Nagoya Univ.

This paper discusses an energy
estimation method in a numerical scheme, the Newmarkbeta method for
nonlinear partial differential equations in terms of the StokesDirac
structure. First, we show that the total energy of numerical systems can be
detected by a boundary integration of energies distributed on a system domain
in the Newmarkbeta method for dynamical nonlinear numerical analyses. Next,
we illustrate example applications of the estimation to flexible beams with
large deformations.

12:1012:30

ThAR.6

Passive LTI Systems with a TimeVarying Parturbation (I),
pp. 137142

Weiss, George

Tel Aviv Univ.

Chen, JianHua

Faculty of Engineering, Tel Aviv Univ.

We study a timevarying wellposed
system resulting from the additive perturbation of the generator of a time
invariant wellposed system. The associated generator family has the form
$A+G(t)$, where $G(t)$ is a bounded operator on the state space and $G(cdot)$
is strongly continuous. We show that the resulting timevarying system (the
perturbed system) is wellposed and we investigate its properties. In the
particular case when the unperturbed system is scattering passive, we derive
an energy balance inequality for the perturbed system. If the operators
$G(t)$ are dissipative, then the perturbed system is again scattering
passive. We illustrate this theory by using it to formulate the system corresponding
to a conductor moving in an electromagnetic field described by Maxwell’s
equations.

ThBR

Sala degli Affreschi

Control of Lagrangian and Hamiltonian Systems (II) (Regular Session)

14:0014:20

ThBR.1

FiniteGain L2 Stability of PID Set Position Control with AntiWindup
Compensation for EulerLagrange Systems with Actuator Saturation,
pp. 143148

Kanamori, Mitsuru

Maizuru National Coll. of Tech.

Finitegain L2 stabilization is
achieved locally for the system using PID set position controller with the
proposed static antiwindup compensation for EulerLagrange systems with
actuator saturation and external disturbances. On a closedloop nonlinear system
with feedback and input saturation, L2 stability of the EulerLagrange
systems is guaranteed based on passivity for antiwindup compensation. The
control performance against the external disturbance added to saturate input
is verified by numerical simulations and experiments on a twolink robot arm.

14:2014:40

ThBR.2

On a Generalized PortHamiltonian Representation for the Control of
Damped Underactuated Mechanical Systems, pp. 149154

Kotyczka, Paul

Tech. Univ. M\9Fnchen

Delgado Londo\96o, Sergio

Tech. Univ. of Munich, Inst. of
AutomaticControl

A wellknown problem in controller
design for underactuated mechanical systems using the Interconnection and
Damping Assignment (IDAPBC) technique is friction in unactuated degrees of
freedom. For certain equilibria the definiteness requirements on the virtual
energy of the portHamiltonian (pH) target system and the closedloop
dissipation matrix can not be satisfied simultaneously. In this contribution
a modification of the pH target system is proposed, where particularly the
total energy function is augmented by a cross term between coordinates and
momenta. The approach stems from the fact that, although IDAPBC may fail,
unstable equilibria of underactuated mechanical systems are stabilized by
linear state feedback, if the linearization is stabilizable. Then the
solution of a Lyapunov equation for the linearized closedloop system is not
block diagonal, which gives rise to the proposed structure of the energy.

14:4015:00

ThBR.3

Coordination of MultiAgent Systems Via EnergyShaping: Networking
Improves Robustness, pp. 155160

Nu\96o, Emmanuel

Univ. of Guadalajara

Ortega, Romeo

Supelec

Jayawardhana, Bayu

Univ. of Groningen

Basanez, Luis

Univ. Pol. de Catalunya

In this paper the problem of robust
coordination of multiagent systems via energyshaping is studied. The agents
are nonidentical, Euler–Lagrange systems with uncertain parameters. The
control objective is to drive all agents states to the same constant
equilibriumwhich is achieved shaping their potential energy function. It is
assumed that, if the parameters are known, this task can be accomplished with
a decentralized strategy. In the face of parameter uncertainty, the assigned
equilibrium is shifted away from its desired value. It is shown that adding
information exchange between the agents to this decentralized control policy
improves the performance. More precisely, it is proven that if the
communication graph is connected and balanced, the equilibrium of the
networked controller is always closer (in a suitable metric) to the desired
one. If the the potential energy functions are quadratic, the result holds
for all interconnection gains, else, it is true for sufficiently large gains.
The decentralized controller is the well–known energy–shaping proportional
plus derivative controller, extensively used in applications. An additional
advantage of networking is that the control objective is achieved injecting
lower gains into the loop.

15:0015:20

ThBR.4

On the Modeling, Linearization and Energy Shaping Control of Mechanical
Systems, pp. 161166

Sarras, Ioannis

CNRS

Ortega, Romeo

Supelec

van der Schaft, Arjan J.

Univ. of Groningen

In this work some recent results on
the linearization and passivitybased control of mechanical systems are
reviewed from a unified perspective. This is established by adopting a
generalization of the Poisson bracket formalism to more general structures than
smooth functions. In this manner, the corresponding geometric structures as
well as their respective energy terms are all expressed by simple,
identifiable terms. More precisely, the objective consists in illustrating
that the proposed framework captures the essential terms involved in the
conditions of the literature, reveals the connection between the results in
linearization and stabilization, and reduces the cumbersome calculations. In
this direction, the generalized Poisson bracket is shown to be an effective
tool that leads to (i) the refinement of wellknown results on
interconnection and damping assignment passivitybased control (IDAPBC),
(ii) the derivation of a new set of simplified conditions for partial
linearization via a change of coordinates, and (iii) the identification of
certain relationships connecting the Hamiltonian with the EulerLagrange
description.

15:2015:40

ThBR.5

CasimirBased Control Beyond the Dissipation Obstacle,
pp. 167171

Koopman, Johan

Delft Univ. of Tech.

Jeltsema, Dimitri

Delft Univ. of Tech.

A prevailing trend in the
stabilization of portHamiltonian systems is the assumption that the plant
and the controller are both passive. In the standard approach of control by
interconnection based on the generation of Casimir functions, this assumption
leads to the dissipation obstacle, which essentially means that dissipation
is admissible only on the coordinates of the closedloop Hamiltonian that do
not require shaping and thus severely restricts the scope of applications. In
this contribution, we show that we can easily go beyond the dissipation
obstacle by allowing the controller to have a negative semidefinite
resistive structure, while guaranteeing stability of both the closedloop and
the controller.

15:4016:00

ThBR.6

Memristive PortHamiltonian Control: PathDependent Damping Injection
in Control of Mechanical Systems, pp. 172177

DoriaCerezo, Arnau

Tech. Univ. of Catalonia (UPC)

van der Heijden, Laurens

Faculty of Math. & Natural Sciences, Univ.
ofGroningen

Scherpen, Jacquelien M.A.

Univ. of Groningen

This paper presents the use of the
memristor as a new element for designing passivitybased controllers. From
the portHamiltonian description of the electrical circuits with memristors,
a target dynamics is assigned to the matching equation proposed by the
methodology known as Interconnection and Damping AssignmentPassivitybased
Control. The inclusion of the memristor element extends the closed loop
dynamics and it results in an extra term in the control algorithm that can be
seen as a statemodulated gain. Two mechanical examples, in the form of a
position control systems are included to show possible applications.

ThCR

Sala degli Affreschi

Geometric Mechanics and Control (Invited Session)

Organizer: Zenkov, Dmitry

North Carolina State Univ.

Organizer: Fujimoto, Kenji

Nagoya Univ.

16:3016:50

ThCR.1

Variational Structures for Hamel’s Equations and Stabilization (I),
pp. 178183

Ball, Kenneth

North Carolina State Univ.

Zenkov, Dmitry

North Carolina State Univ.

Bloch, Anthony M.

Univ. of Michigan

Hamel’s equations are an analogue
of the EulerLagrange equations of Lagrangian mechanics when the velocity is
measured relative to a frame which is not related to system’s local
configuration coordinates. The use of this formalism often leads to a simpler
representation of dynamics but introduces additional terms in the equations
of motion. The paper elucidates the variational nature of Hamel’s equations
and discusses their utility in control and stabilization. The latter is
illustrated with the problem of stabilization of a falling disk.

16:5017:10

ThCR.2

On the QuasiLinearization of the Equations of Motion of Simple
Mechanical Systems (I), pp. 184187

Chang, Dong Eui

Univ. of Waterloo

McLenaghan, Raymond

Univ. of Waterloo

A simple mechanical system is said
be quasilinearizable if there is a linear transformation of velocity that
eliminates all terms quadratic in the velocity from the equations of motion.
It is wellknown that controller/observer synthesis becomes tractable when
the dynamics of a mechanical system are in quasilinearized form. The
quasilinearization property is equivalent to the property that the Lie
algebra of Killing vector fields is pointwise equal to the tangent space to
the configuration manifold with the metric induced by the mass tensor of the
mechanical system. We show conditions for full quasilinearization and
partial quasilinearization, the latter of which is for systems that are not
quasilinearizable. A sufficient condition for full quasilinearizability is
that the Riemannian manifold be locally symmetric. On two dimensional
manifolds, the constant scalar curvature condition is necessary and
sufficient for full quasilinearizability. The two conditions extend the zero
Riemannian curvature condition by Bedrossian and Spong.

17:1017:30

ThCR.3

Control Foliations for Mechanical Systems (I), pp.
188193

Rampazzo, Franco

Univ. di Padova

The idea of regarding the last M
coordinates of a (N+M)dimensional mechanical system is made intrinsic by
considering a foliation of the configuration space. A control is then a path
in the space of leaves. We review some results concerning the way the control
equations governing the motion on the leaves depend on the derivative of such
a control. In general, these equations are quadratic polynomials of the
derivative of the control. The quadratic term turns out to be interesting for
controllability and (vibrational) stabilizability purposes. On the other
hand, the special case when the quadratic term is vanishing characterizes the
possibility to wellpose the equations when not regular even discontinuous
controls are implemented.

17:3017:50

ThCR.4

Approximate Solutions to the HamiltonJacobi Equations for Generating
Functions with a Quadratic Cost Function with Respect to the Input (I),
pp. 194199

Hao, Zhiwei

Nagoya Univ.

Fujimoto, Kenji

Nagoya Univ.

An algorithm to approximate a
solution to the HamiltonJacobi equation for a generating function for a
nonlinear optimal control problem with a quadratic cost function with respect
to the input is proposed in this paper. An approximate generating function
based on Taylor series up to the order N is obtained by solving (N+2)(N1)/2
linear firstorder ordinary differential equations recursively. A single
generating function is effective to generate optimal trajectories to the same
nonlinear optimal control problem for any different boundary conditions. It
is useful to online trajectory generation problems. Numerical examples
illustrate the effectiveness of the proposed algorithm.

17:5018:10

ThCR.5

Implicit Representation for PassivityBased Boundary Controls (I),
pp. 200207

Nishida, Gou

Kyoto Univ.

Maschke, Bernhard

Univ. Claude Bernard of Lyon

This paper derives a standard
system representation for passivitybased boundary controls of EulerLagrange
equations, called a distributed portLagrangian (DPL) system from implicit
Lagrangian representations and the multisymplectic instantaneous formalism.
The DPL system is a local representation of implicit Lagrangian systems
extended for field equations. First, we extend an induced Dirac structure to
multisymplectic instantaneous systems by defining a StokesDirac
differential and a field implicit Lagrangian system. Second, the DPL system
is derived from the extended field implicit Lagrangian systems. Finally, we
define passivitybased boundary controls based on a power balance equation of
DPL systems.

ThSP

Sala degli Affreschi

Perspectives on Quantum Feedback Control (Plenary Session)

18:3019:20

ThSP.1

Perspectives on Quantum Feedback Control*

James, Matthew R.

Australian National Univ.

Recent theoretical and experimental
advances mean that it is now possible to control physical systems at the
quantum level. Indeed, developments in quantum technology provide strong
motivation for the feedback control of quantum systems. This talk will give
some perspectives on quantum feedback control, including both measurement
feedback as well as fully quantum coherent feedback control. In particular,
we contrast and compare open loop and closed loop quantum control, and
describe some of the significant research challenges.

Technical Program for Friday August 31, 2012
FrPP

Sala degli Affreschi

Contraction Analysis and PortHamiltonian Modeling (Plenary Session)

09:0010:00

FrPP.1

Contraction Analysis and PortHamiltonian Modeling*

Sepulchre, Rodolphe J.

Univ. de Liege

This talk will focus on the concept
of contraction in the context of portHamiltonian control systems. Motivated
and driven by ten years of collaboration with SchneiderToshiba on control of
induction motors and power electronics, we will illustrate the relevance of
this framework to rigorously address two pillars of industrial control: PI
feedback and anti windup. After 40 years of dissipativity theory, physics
keeps providing efficient encounters between control, analysis, and
differential geometry.

FrAR

Sala degli Affreschi

Quantum Control (Invited Session)

Organizer: Sarlette, Alain

Ghent Univ.

Organizer: Ticozzi, Francesco

Univ. di Padova

10:3010:50

FrAR.1

Robust OpenLoop Stabilization of Fock States by TimeVarying Quantum
Interactions (I), pp. 208213

Sarlette, Alain

Ghent Univ.

Rouchon, Pierre

MinesParisTech

A quantum harmonic oscillator
(spring subsystem) is stabilized towards a target Fock state by reservoir
engineering. This passive and openloop stabilization works by consecutive
and identical Hamiltonian interactions with auxiliary systems, here
threelevel atoms (the auxiliary ladder subsystem), followed by a partial
trace over these auxiliary atoms. A scalar control input governs the
interaction, defining which atomic transition in the ladder subsystem is in
resonance with the spring subsystem. We use it to build a timevarying
interaction with individual atoms, that combines three noncommuting steps.
We show that the resulting reservoir robustly stabilizes any initial spring
state distributed between 0 and 4n+3 quanta of vibrations towards a pure
target Fock state of vibration number n. The convergence proof relies on the
construction of a strict Lyapunov function for the Kraus map induced by this
reservoir setting on the spring subsystem. Simulations with realistic
parameters corresponding to the quantum electrodynamics setup at Ecole
Normale Superieure further illustrate the robustness of the method.

10:5011:10

FrAR.2

On the Stability of Pointer States Using Lyapunov Theory (I),
pp. 214219

Somaraju, Ram Abhinav

Vrije Univ. Brussel

Petersen, Ian R

Univ. of New South
WalesattheAustralianDefenceForceAcademy

Thienpont, Hugo

Vrije Univ. Brussel

Pointer states are states of an
open quantum system that are able to survive the constant monitoring of the
system by an environment. It has been shown that open systems that are
prepared in superpositions of such pointer states quickly decohere and evolve
into classical statistical mixtures of (pure) pointer states. In this paper
we demonstrate, using appropriate modeling assumptions for the system
environment interaction, the following result: An individual trajectory of
the system state involves towards a specific pointer state (and not just a
statistical mixture of the same) if one monitors the environment state by
measuring environmental observables even if only a fraction of these
measurement outcomes are known to the observer. The central tool used to
demonstrate this is the identification of conserved quantities that
correspond to the eigenprojections of the systemenvironment Hamiltonian. We
construct Lyapunov functions using this Hamiltonian to demonstrate the
stability of the pointer states.

11:1011:30

FrAR.3

On the Role of Hamiltonians for Dissipative Entanglement Engineering
(I), pp. 220225

Ticozzi, Francesco

Univ. di Padova

Viola, Lorenza

Dartmouth Coll.

Determining whether an entangled
state of interest can be asymptotically prepared by realistic opensystem dynamics
has important applications across quantum engineering. This problem has been
recently solved for purely dissipative quasilocal dynamics described by a
continuoustime Markovian semigroup. Here, we extend our previous analysis by
addressing the role of internal Hamiltonian dynamics as well as of
Hamiltonian control resources for achieving the same task. We show how
Hamiltonians that are not frustrationfree can genuinely extend the class of
stabilizable states. In particular, we present stabilizing Hamiltonians,
along with necessary and sufficient conditions for their existence, for
maximally entangled GHZstates and translationally invariant Wstates, none
of which are generally stabilizable by dissipation alone.

11:3011:50

FrAR.4

Which Notion of Energy for Bilinear Quantum Systems? (I),
pp. 226230

Boussa\95d, Nabile

Lab. de Math\8Ematiques, Univ. de FrancheComt\8E

Caponigro, Marco

INRIA Nancy – Grand Est

Chambrion, Thomas

Univ. de Lorraine

In this note we investigate what is
the best L^pnorm in order to describe the relation between the evolution of
the state of a bilinear quantum system with the L^pnorm of the external
field. Although L^2 has a structure more easy to handle, the L^1 norm is more
suitable for this purpose. Indeed for every p>1 it is possible to steer,
with arbitrary precision, a generic bilinear quantum system from any
eigenstate of the free Hamiltonian to any other with a control of arbitrary
small L^p norm. Explicit optimal costs for the L^1 norm are computed on an
example.

11:5012:10

FrAR.5

Exploring the Cooling Limit of Quantum Mechanical Oscillators Via
Optimization (I), pp. 231236

Wang, Xiaoting

Univ. of Massachusetts Boston

Vinjanampathy, Sai

Univ. of Massachusetts Boston

Strauch, Frederick

Williams Coll.

Jacobs, Kurt

Univ. of Massachusetts Boston

In this work, we have studied the
optimal cooling limit when cooling a quantum mechanical oscillator by
coupling it to an auxiliary. Since this limit is necessarily confined by the
energy transfer rate of the control, i.e., the magnitude of the spectrum of
the control Hamiltonian, we search for the optimal control Hamiltonian that
can achieve the maximal cooling fidelity, given a bounded energy transfer
rate. First we analyze and present a class of Hamiltonian which generates the
perfect cooling result in the absence of decoherence. Then with perturbation
analysis, we further show that such Hamiltonian is also optimal at least to
the first order of the dynamical dissipative terms. With the help of
numerical optimization, we have studied cases with different damping rates of
the system and the auxiliary, and derived the plot of the optimal cooling
fidelity curve versus time as well as the optimal time point to achieve the
maximal fidelity.

12:1012:30

FrAR.6

Geometric Optimal Control of the Contrast Problem in Magnetic Resonance
Imaging, pp. 237241

Sugny, Dominique

Univ. of Bourgogne

Lapert, Marc

Univ. of Bourgogne

Glaser, Steffen J.

Tech. Univ. Muenchen (TUM)

The control of the dynamics of spin
systems by magnetic fields has opened intriguing possibilities in quantum
computing and in Nuclear Magnetic Resonance spectroscopy. In this framework,
optimal control theory has been used to design control fields able to realize
a given task while minimizing a prescribed cost such as the energy of the
field or the duration of the process. However, some of the powerful tools of
optimal control had not been used yet for NMR applications in medical
imagery. Here, we show that the geometric control theory approach can be
advantageously combined with NMR methods to crucially optimize the imaging
contrast. This approach is applied to a benchmark problem but it gives a
strong evidence for the possibility of using optimal control theory for
enhancing the contrast and the resolution of medical images.

FrBR

Sala degli Affreschi

Hamiltonian Methods for the Modelling and Control of Multidomain
Distributed Parameter Systems (Invited Session)

Organizer: Le Gorrec, Yann

FEMTOST

14:0014:20

FrBR.1

On Damping Models Preserving the Eigenfunctions of Conservative
Systems: A PortHamiltonian Perspective (I), pp. 242247

Matignon, Denis

ISAE

Helie, Thomas

CNRS UMR 9912 IRCAM

In this paper, a special class of
damping model is introduced for second order dynamical systems. This class is
built so as to leave the eigenfunctions invariant, while modifying the
dynamics: for mechanical systems, wellknown examples are the standard fluid
and structural dampings.
In the finitedimensional case, the socalled
Caughey series are a general extension of these standard damping models; the
damping matrix can be expressed as a polynomial of a matrix, which depends on
the mass and stiffness matrices. Damping is ensured whatever the eigenvalues
of the conservative problem if and only if the polynomial is positive for
positive scalar values.
This can be recast in the portHamiltonian framework
by introducing a port variable corresponding to internal energy dissipation
(resistive element). Moreover, this formalism naturally allows to cope with
systems including gyroscopic effects (gyrators).
In the infinitedimensional case, the previous
polynomial class can be extended to rational functions and more general
functions of operators (instead of matrices), once the appropriate functional
framework has been defined. In this case, the resistive element is modelled
by a given static operator, such as an elliptic PDE. These results are
illustrated on several PDE examples: the Webster horn equation, the Bernoulli
beam equation; the damping models under consideration are fluid, structural,
rational and generalized fractional Laplacian or biLaplacian.

14:2014:40

FrBR.2

Infinite Dimensional Port Hamiltonian Representation of Chemical
Reactors (I), pp. 248253

Couenne, Francoise

Univ. of Lyon 1

Hamroun, Boussad

Lab. d’Automatique et G\8Enie des Proc\8Ed\8Es

Le Gorrec, Yann

FEMTOST

Zhou, Weijun

Univ. Claude Bernard Lyon 1

Infinite dimensional Port
Hamiltonian representation of non isothermal chemical reactors is proposed in
the case of mass transport diffusion and chemical reaction without
convection. The proposed approach uses thermodynamic variables. The
presentation is given for one dimensional spatial domain by using the
internal energy and the opposite of the entropy as hamiltonian functions.

14:4015:00

FrBR.3

Diffusive Systems Coupled to an Oscillator: A Hamiltonian Formulation
(I), pp. 254259

Le Gorrec, Yann

FEMTOST

Matignon, Denis

ISAE

The aim of this paper is to study a
conservative wave equation coupled to a diusion equation : this coupled
system naturally arises in musical acoustics when viscous and thermal eects
at the wall of the duct of a wind instrument are taken into account. The
resulting equation, known as WebsterLokshin model, has variable coecients in
space, and a fractional derivative in time. The portHamiltonian formalism
proves adequate to reformulate this coupled system, and could enable another
wellposedness analysis, using classical results from portHamiltonian
systems theory. First, an equivalent formulation of fractional derivatives is
obtained thanks to socalled diusive representations: this is the reason why
we rst concentrate on rewriting these diusive representations into the
portHamiltonian formalism; two cases must be studied separately, the
fractional integral operator as a lowpass lter, and the fractional
derivative operator as a highpass lter. Second, a standard nitedimensional
mechanical oscillator coupled to both types of dampings, either lowpass or
highpass, is studied as a coupled pHs. The more general PDE system of a wave
equation coupled with the diusion equation is then found to have the same
structure as before, but in an appropriate innitedimensional setting, which
is fully detailed.

15:0015:20

FrBR.4

Port Hamiltonian Modeling of MSMA Based Actuator: Toward a
Thermodynamically Consistent Formulation (I), pp. 260264

Calchand, Nandish

FEMTOST AS2M

Hubert, Arnaud

FEMTOST AS2M

Le Gorrec, Yann

FEMTOST

This paper presents a
thermodynamically consistent model of MSMA (Magnetic Shape Memory Alloys)
under port Hamiltonian framework. It is based on previous works on MSMA
proposed in (Gauthier et al., 2008; Calchand et al., 2011). The main
difference lies in the choice of the state variables and manipulated
thermodynamic forces. Furthermore in (Gauthier et al., 2008), subsequent
experiments revealed a highly hysteretic behavior of these materials. Here,
the simplified hysteretic behavior is incorporated into the porthamiltonian
model to obtain a finer and more precise model. Such modeling will allow the
use of a wide range of energy based methods to design the associated control
system. The paper ends with some extensions to more complex hysteresc
phenomena by using Preisach like model. First ideas are proposed to extend
the previous physical model to systems with internal hysteretic loops.

15:2015:40

FrBR.5

Boundary Port Hamiltonian Systems of Conservation Laws Coupled by a
Moving Interface (I), pp. 265270

Diagne, Mamadou

Lab. d’Automatique et de G\8Enie dEs Proc\8Ed\8Es

Maschke, Bernhard

Univ. Claude Bernard of Lyon

In this paper we consider the
portHamiltonian formulation of systems of two conservation laws defined on
two complementary intervals of some interval of the real line and coupled by
some moving interface. We recall first how two port Hamiltonian systems
coupled by an interface may be expressed as an port Hamiltonian systems
augmented with two variables being the characteristic functions of of two
spatial domains. Then we consider the case of a moving interface and show
that it may be expressedas the preceeding port Hamiltonian system augmented
with an input, being the velocity of the interface and define a conjugated
output variable. We conclude by giving some definition of the passivity of
interface relations coupling the external variables associated with the
interface.
