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.