Keywords: Passivity-based control, Port-Hamiltonian systems

Abstract: 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 non-holonomic 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.

Keywords: Port-Hamiltonian systems, Distributed parameter systems, Modeling

Abstract: A port-Hamiltonian model is derived for the thermo-magneto-hydro dynamics of plasma in tokamaks. Magnetohydrodynamic and energy balance equations are expressed in their covariant form and written in the port-Hamiltonian formalism using Dirac structures. This Dirac structure is established as the interconnection of Stokes-Dirac structures with a specific interconnection structure representing the magneto-hydrodynamical coupling. Finally the problem of current density profile control is defined and potential approaches are discussed.

Keywords: Dirac structures, Port-Hamiltonian systems, Distributed parameter systems

Abstract: Stokes-Dirac structures are infinite-dimensional 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 so-called simplicial Dirac structures, discrete analogues of Stokes-Dirac structures, and thus provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The port-Hamiltonian systems defined with respect to Stokes-Dirac 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 infinite-dimensional Dirac structure to the Poisson structure associated with Stokes-Dirac structures and of a fine-dimensional Dirac structure to simplicial Dirac structures. We demonstrate this Poisson scheme on a physical example of the vibrating string.

Keywords: Distributed parameter systems, Boundary control, Passivity-based control

Abstract: This paper deals with the stabilization via Casimir generation and energy shaping of linear, lossless, distributed port-Hamiltonian systems. Once inputs and outputs of the distributed port-Hamiltonian system have been chosen to obtain a well-defined boundary control systems, conditions for the existence of Casimir functions in closed-loop 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 full-actuation on one side.

Keywords: Boundary control, Distributed parameter systems, Optimal control

Abstract: 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 well-posed. We apply these results to boundary control of parabolic systems to illustrate the ideas and present numerical results.

Keywords: Port-Hamiltonian systems, Mechatronics, Boundary control

Abstract: This paper discusses an energy estimation method in a numerical scheme, the Newmark-beta method for nonlinear partial differential equations in terms of the Stokes-Dirac 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 Newmark-beta method for dynamical nonlinear numerical analyses. Next, we illustrate example applications of the estimation to flexible beams with large deformations.

Keywords: Distributed parameter systems, Passivity-based control, Boundary control

Abstract: We study a time-varying well-posed system resulting from the additive perturbation of the generator of a time- invariant well-posed 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 time-varying system (the perturbed system) is well-posed 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.

Keywords: Passivity-based control, Robotics, Nonlinear control

Abstract: Finite-gain L2 stabilization is achieved locally for the system using PID set position controller with the proposed static anti-windup compensation for Euler-Lagrange systems with actuator saturation and external disturbances. On a closed-loop nonlinear system with feedback and input saturation, L2 stability of the Euler-Lagrange systems is guaranteed based on passivity for anti-windup compensation. The control performance against the external disturbance added to saturate input is verified by numerical simulations and experiments on a two-link robot arm.

Keywords: Passivity-based control, Port-Hamiltonian systems, Modeling

Abstract: A well-known problem in controller design for underactuated mechanical systems using the Interconnection and Damping Assignment (IDA-PBC) technique is friction in unactuated degrees of freedom. For certain equilibria the definiteness requirements on the virtual energy of the port-Hamiltonian (pH) target system and the closed-loop 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 IDA-PBC 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 closed-loop system is not block diagonal, which gives rise to the proposed structure of the energy.

Keywords: Passivity-based control, Nonlinear control, Robotics

Abstract: In this paper the problem of robust coordination of multi-agent systems via energy-shaping 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 equilibrium-which 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.

Keywords: Port-Hamiltonian systems, Passivity-based control, Modeling

Abstract: In this work some recent results on the linearization and passivity-based 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 well-known results on interconnection and damping assignment passivity-based control (IDA-PBC), (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 Euler-Lagrange description.

Keywords: Passivity-based control, Port-Hamiltonian systems, Nonlinear control

Abstract: A prevailing trend in the stabilization of port-Hamiltonian 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 closed-loop 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 semi-definite resistive structure, while guaranteeing stability of both the closed-loop and the controller.

Keywords: Passivity-based control, Port-Hamiltonian systems, Hamiltonian dynamics

Abstract: This paper presents the use of the memristor as a new element for designing passivity-based controllers. From the port-Hamiltonian 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 Assignment-Passivity-based 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 state-modulated gain. Two mechanical examples, in the form of a position control systems are included to show possible applications.

Keywords: Geometric mechanics, Nonlinear control

Abstract: Hamel’s equations are an analogue of the Euler-Lagrange 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.

Keywords: Nonlinear control, Geometric mechanics

Abstract: A simple mechanical system is said be quasi-linearizable if there is a linear transformation of velocity that eliminates all terms quadratic in the velocity from the equations of motion. It is well-known that controller/observer synthesis becomes tractable when the dynamics of a mechanical system are in quasi-linearized form. The quasi-linearization 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 quasi-linearization and partial quasi-linearization, the latter of which is for systems that are not quasi-linearizable. A sufficient condition for full quasi-linearizability is that the Riemannian manifold be locally symmetric. On two dimensional manifolds, the constant scalar curvature condition is necessary and sufficient for full quasi-linearizability. The two conditions extend the zero Riemannian curvature condition by Bedrossian and Spong.

Keywords: Geometric mechanics, Nonlinear control, Hamiltonian dynamics

Abstract: 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 well-pose the equations when not regular -even discontinuous- controls are implemented.

Keywords: Optimal control, Nonlinear control, Hamiltonian dynamics

Abstract: An algorithm to approximate a solution to the Hamilton-Jacobi 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)(N-1)/2 linear first-order 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.

Keywords: Port-Hamiltonian systems, Dirac structures, Passivity-based control

Abstract: This paper derives a standard system representation for passivity-based boundary controls of Euler-Lagrange equations, called a distributed port-Lagrangian (DPL) system from implicit Lagrangian representations and the multi-symplectic 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 multi-symplectic instantaneous systems by defining a Stokes-Dirac differential and a field implicit Lagrangian system. Second, the DPL system is derived from the extended field implicit Lagrangian systems. Finally, we define passivity-based boundary controls based on a power balance equation of DPL systems.

Keywords: Quantum control

Abstract: 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.