Keywords: Nonlinear
control, Passivity-based
control, Port-Hamiltonian
systems
Abstract:Power-shaping control is a control
design method which can be seen as an extension of energy-balancing
passivity-based 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 Brayton-Moser form, on which Power-shaping
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 Brayton-Moser
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 Brayton-Moser
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.

Keywords: Distributed
parameter systems, Port-Hamiltonian
systems, Modeling
Abstract: We consider infinite dimensional
port-Hamiltonian systems. Based on a power balance relation we
introduce the port-Hamiltonian system representation where we pay
attention to two different scenarios, namely the non-differential
operator case and the differential operator case regarding the
structural mapping, the dissipation mapping and the in/output mapping.
In contrast to the well-known representation on the basis of the
underlying Stokes-Dirac structure our approach is not necessarily based
on using energy-variables which leads to a different port-Hamiltonian
representation of the analyzed partial differential equations.

Keywords: Port-Hamiltonian
systems, Modeling,
Conservation
laws
Abstract:This paper presents a fast
computation method of forward dynamics of a class of hydro-mechanical
systems.
First, an exact simplification of the conventional representation of
the hydro-mechanical 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
hydro-mechanical 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.

Keywords: Port-Hamiltonian
systems, Irreversible
thermodynamics, Modeling
Abstract: 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 skew-symmetric structure matrix that
represents the network structure of the system; a non-linear function
that depends on the states and co-states 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 non-isothermal CSTR is used to illustrate the formalism.

Keywords: Hamiltonian
dynamics, Geometric
mechanics, Nonlinear
control
Abstract: 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.

Keywords: Nonlinear
control, Port-Hamiltonian
systems, Distributed
parameter systems
Abstract: 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 PI-control, 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 closed-loop port-Hamiltonian system.
Furthermore, it will be shown how regulation is proved by modifying the
total Hamiltonian of the port-Hamiltonian 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 non-smooth Lyapunov functions.

Keywords: Power
systems, Port-Hamiltonian
systems, Nonlinear
control
Abstract: In this paper starting with bond
graph techniques a nonlinear mathematical model of the synchronous
generator in port-Hamiltonian framework is derived. This leads to an
energy\96based description of the system which we later use for
stability analysis. We use Park\92s 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.

Keywords: Nonlinear
control, Port-Hamiltonian
systems, Passivity-based
control
Abstract: In this note the two well known
nonlinear control design techniques Interconnection and Damping
Assignment Passivity Based Control (IDA-PBC) and Immersion and
Invariance (I&I) are compared through the example of the so-called
Acrobot underactuated 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 IDA-PBC an ordinary differential
equation is used to define the immersion map in I&I. It is shown
that the energy shaping part of the IDA-PBC controller makes the
closed-loop system an interconnection of two lower-dimensional
port-Hamiltonian (pH) systems in the on- and off-manifold coordinates.
The effect of damping injection output feedback can be identified with
dissipation in the off-manifold part of the interconnected system.
Dissipation is propagated to the on-manifold part which results in
asymptotic stability of the system\92s equilibrium. The analysis in the
present work provides an interesting interpretation of the effect of
the IDA-PBC control law using the I&I framework.

Keywords: Passivity-based
control, Mechatronics,
Port-Hamiltonian
systems
Abstract: In this paper, we propose a
vibration suppression control method for a mass-spring-damper system
with one or two dynamic damper(s). The feedback is designed by the
interconnection and damping assignment passivity-based 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.

Keywords: Nonlinear
control, Passivity-based
control, Port-Hamiltonian
systems
Abstract: Total Energy Shaping is a controller
design methodology that achieves (asymptotic) stabilization of
mechanical systems endowing the closed-loop 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 closed-loop. Particularly
troublesome is the PDE associated to the kinetic energy which is
quasi-linear and inhomogeneous and the solution, that denes the
desired inertia matrix, must be positive denite. This task is
simplied by the inclusion of gyroscopic forces in the target dynamics,
which translates into the presence of a free skew-symmetric 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 skew-symmetry condition,
further reduces the number of PDE’s. The purpose of this paper is to
prove that this claim is wrong.

Keywords: Hamiltonian
dynamics, Nonlinear
control, Conservation
laws
Abstract: Using an example of cart-pendulum
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

Keywords: Nonlinear
control, Passivity-based
control, Port-Hamiltonian
systems
Abstract: 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 that-if the inertia matrix is constant–
constant disturbances (both, matched and unmatched) can be rejected
simply adding a suitable integral action. For systems with
non\96constant inertia matrix and non-constant 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–to-state
stability with respect to matched and unmatched disturbances of the
previous controller.

Keywords: Geometric
mechanics, Nonlinear
control, Port-Hamiltonian
systems
Abstract: 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 re-express a given vector field as a
Brayton–Moser system. The proposed method is based on a decomposition
of a differential one-form that encodes the divergence of a given
vector field into its exact and anti-exact components, and into its
co-exact and anti-coexact 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 co-differential 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
three-dimensional rigid-body problem are also presented to illustrate
the construction.

Keywords: Geometric
mechanics, Robotics,
Modeling
Abstract: 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 Euler-Poincare form and are equivalent to those obtained using
Lagrange-d’Alembert’s principle.

Keywords: Robotics,
Passivity-based
control
Abstract: In order to accomplish cooperative
tasks, multi\96robot 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.

Keywords: Nonlinear
control, Hamiltonian
dynamics, Robotics
Abstract: This paper continues recent work by
the authors on virtual holonomic constraints (VHCs) for Euler-Lagrange
control systems with n degrees-of-freedom 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 reduced-order
control system of dimension 2 (n-k) with (m-k) control inputs. When
m=k=n-1, conditions are given guaranteeing that the constrained
dynamics are Euler-Lagrange. 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.

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

Keywords: Modeling,
Conservation
laws
Abstract: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 action-angle 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 automata-based model. The contribution of the paper
is to show that well-known reduction methods can be used to generate
abstract models, which can be used for formal verification.

Keywords: Geometric
mechanics
Abstract: 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
control-affine 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.