Energy-balancing Passivity-based Control Research Articles

Port-Hamiltonian Passivity-Based Control on SE(3) of a Fully Actuated UAV for Aerial Physical Interaction Near-Hovering

In this work, we approach the control problem of fully-actuated UAVs in a geometric port-Hamiltonian framework. The UAV is modeled as a floating rigid body on the special Euclidean group SE(3). A unified near-hovering motion and impedance controller is derived by the energy-balancing passivity-based control technique. A detailed analysis of the closed-loop system's behavior is presented for both the free-flight stability and contact stability of the UAV. The robustness of the control system to uncertainties is validated by several experiments, in which the UAV is controlled near its actuator limits. The experiments show the ability of the UAV to hover at its maximum allowed roll angle and apply its maximum allowed normal force to a surface, without the input saturation destabilizing the system.

Reinforcement learning for port-hamiltonian systems.

Passivity-based control (PBC) for port-Hamiltonian systems provides an intuitive way of achieving stabilization by rendering a system passive with respect to a desired storage function. However, in most instances the control law is obtained without any performance considerations and it has to be calculated by solving a complex partial differential equation (PDE). In order to address these issues we introduce a reinforcement learning (RL) approach into the energy-balancing passivity-based control (EB-PBC) method, which is a form of PBC in which the closed-loop energy is equal to the difference between the stored and supplied energies. We propose a technique to parameterize EB-PBC that preserves the systems's PDE matching conditions, does not require the specification of a global desired Hamiltonian, includes performance criteria, and is robust. The parameters of the control law are found by using actor-critic (AC) RL, enabling the search for near-optimal control policies satisfying a desired closed-loop energy landscape. The advantage is that the solutions learned can be interpreted in terms of energy shaping and damping injection, which makes it possible to numerically assess stability using passivity theory. From the RL perspective, our proposal allows for the class of port-Hamiltonian systems to be incorporated in the AC framework, speeding up the learning thanks to the resulting parameterization of the policy. The method has been successfully applied to the pendulum swing-up problem in simulations and real-life experiments.

Passivity-Based Control of Implicit Port-Hamiltonian Systems

The main contribution of this paper is the generalization of well-known energy-based control techniques (i.e., energy-balancing passivity-based control, passivity-based control with state-modulated source, and interconnection and damping assignment passivity-based control) to the case in which the plant is a port-Hamiltonian system in implicit form. A typical situation is when (part of) the system is obtained from the spatial discretization of an infinite dimensional port-Hamiltonian system: in this case, the dynamics is not given in standard input-state-output form but as a set of DAEs. Consequently, the most popular control by energy-shaping techniques have to be extended to deal with dynamical systems with constraints, and geometrical conditions for the existence of the regulator based on the properties of Dirac and resistive structures of the plant are also necessary. The methodological results are discussed with the help of simple but illustrative examples in which the port-Hamiltonian systems in implicit form are the finite-element models of a lossless transmission line and of a Timoshenko beam.

Boundary energy shaping of linear distributed port-Hamiltonian systems

This paper deals with the energy-balancing passivity-based control of linear, lossless, distributed port-Hamiltonian systems. Once inputs and outputs have been chosen to obtain a well-defined boundary control system, the problem is tackled by determining, at first, the class of energy functions that can be employed in the energy-shaping procedure, together with the corresponding boundary state-feedback control actions. To verify the existence of solutions for the closed-loop system, the equivalence between energy-balancing and energy-Casimir methods is shown. For the latter approach, the conditions for having a particular set of Casimir functions in closed-loop are given, and then the existence of the associated semigroup is studied. Since both the methods provide the same control action, the existence result determined for the energy-Casimir method is valid also for the energy-balancing controller. Simple stability is obtained by shaping the open-loop Hamiltonian, while asymptotic stability is ensured if proper “pervasive” (boundary) damping is present. In this respect, a stability criterion is discussed. The methodology is illustrated with the help of a simple example, i.e. a Timoshenko beam with full-actuation on one side, and an inertia on the other side.

Power-shaping control: Writing the system dynamics into the Brayton–Moser form

Power-shaping control is an extension of energy-balancing passivity-based control that is based on a particular form of the dynamics, the Brayton–Moser form. One of the main difficulties in this control approach is to write the dynamics in the suitable form since this requires the solution of a partial differential equation (PDE) system with an additional sign constraint. Here a general methodology is described for solving this partial differential equation system. The set of all solutions to the PDE system is given as the solution of a linear equation system. Furthermore a necessary condition is given so that a solution of the linear system which meets the sign condition exists. This methodology is illustrated on the case of a chemical reactor, where the physical knowledge of the system is used to find a suitable solution.

Stabilisation of nonlinear chemical processes via dynamic power-shaping passivity-based control

It is well known that energy-balancing passivity-based control is stymied by the presence of pervasive dissipation. To overcome this problem in electrical circuits, some authors have used power-shaping techniques, where stabilisation is achieved by shaping a function akin to power instead of energy. Some extensions of the techniques to general nonlinear systems, yielding static state-feedback control laws, have also been reported. In this article, we extend these techniques to dynamic feedback control and apply them to nonlinear chemical processes. The stability analysis is carried out using the shaped power function as Lyapunov function. The proposed technique is illustrated with two nonlinear chemical process examples.

Energy-balancing passivity-based control is equivalent to dissipation and output invariance

Passivity-based controllers (PBCs) achieve stabilization of nonlinear systems, rendering the closed-loop passive with a desired energy (storage) function. A natural question is, under which conditions is it possible to make this function equal to the difference between the plant and controller energies—when the controller is said to be energy-balancing. In this paper we prove that a necessary and sufficient condition for energy-balancing is that the open and the closed-loop systems have the same dissipation functions and passive outputs. A second contribution of our work is the identification of a new passive output for Port–Hamiltonian systems, which is invariant to the action of PBCs that modify only the energy function–so-called basic interconnection and damping assignment PBCs–proving that they are energy-balancing. To establish these results a new algebraic framework for analysis and design of PBCs, centered around the principles of output and dissipation invariance, is developed. Using this framework several PBC schemes reported in the literature are compared. Also, we present a systematic procedure to generate new passive outputs, this result is of interest on its own, since it allows to extend the applicability of PBC to systems that are non-minimum phase and/or have relative degree larger than one.