An energy-balancing perspective of interconnection and damping assignment control of nonlinear systems

Stabilization of nonlinear feedback passive systems is achieved assigning a storage function with a minimum at the desired equilibrium. For physical systems a natural candidate storage function is the difference between the stored and the supplied energies—leading to the so-called energy-balancing control, whose underlying stabilization mechanism is particularly appealing. Unfortunately, energy-balancing stabilization is stymied by the existence of pervasive dissipation, that appears in many engineering applications. To overcome the dissipation obstacle the method of Interconnection and Damping Assignment, that endows the closed-loop system with a special—port-controlled Hamiltonian—structure, has been proposed. If, as in most practical examples, the open-loop system already has this structure, and the damping is not pervasive, both methods are equivalent. In this brief note we show that the methods are also equivalent, with an alternative definition of the supplied energy, when the damping is pervasive. Instrumental for our developments is the observation that, swapping the damping terms in the classical dissipation inequality, we can establish passivity of port-controlled Hamiltonian systems with respect to some new external variables—but with the same storage function.     

Construction of discrete-time models for port-controlled Hamiltonian systems with applications

The issues of constructing a discrete-time model for Hamiltonian systems are in general different from those for dissipative systems. We propose an algorithm for constructing an approximate discrete-time model, which guarantees Hamiltonian conservation. We show that the algorithm also preserves, in a weaker sense, the losslessness property of a class of port-controlled Hamiltonian systems. An application of the algorithm to port-controlled Hamiltonian systems with quadratic Hamiltonian is presented, and we use this to solve the stabilization problem for this class of systems based on the approximate discrete-time model constructed using the proposed algorithm. We illustrate the usefulness of the algorithm in designing a discrete-time controller to stabilize the angular velocity of the dynamics of a rigid body.     

Simultaneous interconnection and damping assignment passivity-based control of mechanical systems using dissipative forces

To extend the realm of application of the well known controller design technique of interconnection and damping assignment passivity-based control (IDA-PBC) of mechanical systems two modifications to the standard method are presented in this article. First, similarly to Batlle et al. (2009) and Gómez-Estern and van der Schaft (2004), it is proposed to avoid the splitting of the control action into energy-shaping and damping injection terms, but instead to carry them out simultaneously. Second, motivated by Chang (2014), we propose to consider the inclusion of dissipative forces, going beyond the gyroscopic ones used in standard IDA-PBC. The contribution of our work is the proof that the addition of these two elements provides a non-trivial extension to the basic IDA-PBC methodology. It is also shown that several new controllers for mechanical systems designed invoking other (less systematic procedures) that do not satisfy the conditions of standard IDA-PBC, actually belong to this new class of SIDA-PBC.     

Energy-based control for hybrid port-controlled Hamiltonian systems

In this paper we develop an energy-based hybrid control framework for hybrid port-controlled Hamiltonian systems. In particular, we obtain constructive sufficient conditions for hybrid feedback stabilization that provide a shaped energy function for the closed-loop system, while preserving a hybrid Hamiltonian structure at the closed-loop level. Furthermore, an inverse optimal hybrid feedback control framework is developed that characterizes a class of globally stabilizing energy-based controllers that guarantee hybrid sector and gain margins to multiplicative input uncertainty of hybrid Hamiltonian systems.     

An LMI approach to stabilization of linear port-controlled Hamiltonian systems

In this paper, it is shown that controllers for stabilizing linear port-controlled Hamiltonian (PCH) systems via interconnection and damping assignment can be obtained by solving a set of linear matrix inequalities (LMIs). Two sets of (almost) equivalent LMIs are proposed. In the first set, the interconnection and damping matrices do not appear explicitly, which makes it more difficult to directly manipulate those matrices. By requiring the system to have no uncontrollable pole at s=0, the second set of LMIs, explicitly containing the interconnection and damping matrices, can be obtained. Taking into account the physical properties of the system, some prespecified structures can be imposed directly on those matrices.     

Plasma q-profile control in tokamaks using a damping assignment passivity-based approach

The IDA-PBC based on PCH model for tokamak q-profile is investigated. Two scenarios are carried out. The first one is the resistive diffusion model for the magnetic poloidal flux. The second one is extended with the thermal diffusion. A feedforward control is used to ensure the compatibility with the actuator physical ability. An IDA-PBC feedback is proposed to improve the system stabilization and convergence speed. The controllers are validated in the simulation using RAPTOR code and tested in TCV, the result is analyzed and the followed discussion proposed the required improvement for the next experiments.     

Abstractions of Hamiltonian control systems

Given a control system and a desired property, an abstracted system is a reduced system that preserves the property of interest while ignoring modeling detail. In previous work, abstractions of linear and nonlinear control systems were considered while preserving reachability properties. In this paper, we consider the abstraction problem for Hamiltonian control systems, where, in addition to the property of interest we also preserve the Hamiltonian structure of the control system. We show how the Hamiltonian structure of control systems can be exploited to simplify the abstraction process. We then focus on local accessibility preserving abstractions, and provide conditions under which local accessibility properties of the abstracted Hamiltonian system are equivalent to the local accessibility properties of the original Hamiltonian control system.     

Trajectory tracking control of port-controlled Hamiltonian systems via generalized canonical transformations

This paper addresses trajectory tracking control of port-controlled Hamiltonian systems via generalized canonical transformations and passivity-based control. The main strategy adopted in this paper is to construct an error system, which describes the dynamics of the tracking error, by a passive port-controlled Hamiltonian system. After obtaining the error system, tracking control of the original system can be achieved by stabilizing the error system via passivity-based approach. First, a fundamental framework is provided for constructing the error system via generalized canonical transformations. Then a concrete design procedure is derived for a class of electro-mechanical systems. Furthermore, the proposed method is applied to a magnetic levitation system and laboratory experiments demonstrate its effectiveness.     

Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one

Interconnection and damping assignment passivity-based control is a new controller design methodology developed for (asymptotic) stabilization of nonlinear systems that does not rely on, sometimes unnatural and technique-driven, linearization or decoupling procedures but instead endows the closed-loop system with a Hamiltonian structure with a desired energy function-that qualifies as Lyapunov function for the desired equilibrium. The assignable energy functions are characterized by a set of partial differential equations that must be solved to determine the control law. We prove in this paper that for a class of mechanical systems with underactuation degree one the partial differential equations can be explicitly solved. Furthermore, we introduce a suitable parametrization of assignable energy functions that provides the designer with a handle to address transient performance and robustness issues. Finally, we develop a speed estimator that allows the implementation of position-feedback controllers. The new result is applied to obtain an (almost) globally stabilizing scheme for the vertical takeoff and landing aircraft with strong input coupling, and a controller for the pendulum in a cart that can swing-up the pendulum from any position in the upper half plane and stop the cart at any desired location. In both cases we obtain very simple and intuitive position-feedback solutions.     

Further constructive results on interconnection and damping assignment control of mechanical systems: the Acrobot example

Interconnection and damping assignment passivity‐based control is a controller design methodology that achieves (asymptotic) stabilization of mechanical systems endowing the closed‐loop system with a Hamiltonian structure with a desired energy function—that qualifies as Lyapunov function for the desired equilibrium. The assignable energy functions are characterized by a set of partial differential equations that must be solved to determine the control law. A class of underactuation degree one systems for which the partial differential equations can be explicitly solved—making the procedure truly constructive—was recently reported by the authors. In this brief note, largely motivated by the interesting Acrobot example, we pursue this investigation for two degrees‐of‐freedom systems where a constant inertia matrix can be assigned. We concentrate then our attention on potential energy shaping and give conditions under which an explicit solution of the associated partial differential equation can be obtained. Using these results we show that it is possible to swing‐up the Acrobot from some configuration positions in the lower half plane, provided some conditions on the robot parameters are satisfied. Copyright © 2006 John Wiley & Sons, Ltd.     

Repetitive control of Hamiltonian systems based on variational symmetry

This paper is concerned with repetitive control of Hamiltonian systems, which is based on iterative learning control utilizing the variational symmetry of those systems. Variational symmetry allows us to obtain an algorithm to solve a certain class of optimal control problems in a repetitive control framework. Therefore, the proposed method can deal with not only trajectory tracking control problems but also optimal trajectory generation problems, never before considered in a repetitive control framework. A convergence analysis of this algorithm is also discussed. Furthermore, some numerical simulations demonstrate the effectiveness of the proposed method.     

On the addition of integral action to port-controlled Hamiltonian systems

A technique that provides closed loop integral action depending on the passive outputs of port-controlled Hamiltonian systems is already available. This paper addresses a new method that allows us to add integral action also on system variables having relative degree higher than one, while still preserving the Hamiltonian form and, thus, closed loop stability. The new approach is applied to design speed regulation controllers for the permanent magnet synchronous motor. Closed loop stability and asymptotic rejection of unknown piecewise constant load torques are formally proved. This theoretically predicted control system performance is illustrated via simulation experiments, which also show that the properties hold under parameter uncertainties. This is in line with the usual practice of including integral action in a controller with the aim of improving its closed loop robustness. The fact that the method enhances the range of possible integral actions in the controller, enriched with this robustness property, allows us to assess it as a practically important complement to the well-known interconnection and damping assignment techniques developed in the framework of port-controlled Hamiltonian systems.     

Interconnection and damping assignment passivity‐based control of a class of underactuated mechanical systems with dynamic friction

This paper considers the extension of the interconnection and damping assignment passivity‐based control methodology for a class of underactuated mechanical systems with dynamic friction. We present a new damping assignment approach to compensate friction by means of a nonlinear observer. Friction at the actuated joints is assumed to be captured by a bristle deflection model: the Dahl model. Based on the Lyapunov direct method we show that, under some conditions, the overall closed‐loop system is stable and, by invoking the theorem of Barbashin–Krasovskii, we arrive to asymptotic stability conditions. Experiments with an underactuated mechanical system, the Furuta pendulum, show the effectiveness of the proposed scheme when friction is compensated. Copyright © 2010 John Wiley & Sons, Ltd.     

Improved potential energy-shaping for port-controlled Hamiltonian systems

This paper investigates the asymptotical stabilization of port-controlled Hamiltonian(PCH)systems via the improved potential energy-shaping(IPES)method.First,a desired potential energy introduced by a transitive Hamiltonian function is added to the original kinetic energy to yield a desired Hamiltonian function.Second,an asymptotically stabilized controller is designed based on a new matching equation with the obtained Hamiltonian function.Finally,a numerical example is given to show the effectiveness of the proposed method.     

A brand new nonlinear robust control design of SSSC for transient stability and damping improvement of multi-machine power systems via pseudo-generalized Hamiltonian theory

Nonlinear robust control of static synchronous series compensator (SSSC) is investigated in multi-machine multi-load power systems by using the pseudo-generalized Hamiltonian method. First, the uncertain nonlinear differential algebraic equation model is constructed for the power system. Then, the dissipative pseudo-generalized Hamiltonian realization of the system is established by means of variable transformation and prefeedback control. Finally, based on the obtained dissipative pseudo-generalized Hamiltonian realization, a brand new nonlinear robust controller is put forward. The proposed controller can effectively use the internal structure and the energy balance property of the power system. Simulation results demonstrate the effectiveness and robustness of the control scheme.     

Construction of control Lyapunov functions for damping stabilization of control affine systems

This paper considers the stabilization of nonlinear control affine systems that satisfy Jurdjevic–Quinn conditions. We first obtain a differential one-form associated to the system by taking the interior product of a non vanishing two-form with respect to the drift vector field. We then construct a homotopy operator on a star-shaped region centered at a desired equilibrium point that decomposes the system into an exact part and an anti-exact one. Integrating the exact one-form, we obtain a locally-defined dissipative potential that is used to generate the damping feedback controller. Applying the same decomposition approach on the entire control affine system under damping feedback, we compute a control Lyapunov function for the closed-loop system. Under Jurdjevic–Quinn conditions, it is shown that the obtained damping feedback is locally stabilizing the system to the desired equilibrium point provided that it is the maximal invariant set for the controlled dynamics. The technique is also applied to construct damping feedback controllers for the stabilization of periodic orbits. Examples are presented to illustrate the proposed method.     

On output feedback stabilization of Euler-Lagrange systems with nondissipative forces

This paper provides a solution to the problem of output feedback stabilization of systems described by Euler-Lagrange equations perturbed by nondissipative forces. This class of forces appears in some applications where we must take into account the interaction of the system with its environment. The nonlinear dependence on the unmeasurable part of the state and the loss of the fundamental passivity property render most of the existing results on stabilization of nonlinear systems unapplicable to this problem. The technique we use consists of finding a dynamic output feedback controller and a nonlinear change of coordinates such that the closed loop can be decomposed as a cascade of an asymptotically stable system and an input-to-state stable system. This should be contrasted with the well-known passivity-based technique that aims at a feedback interconnection of passive systems. We believe this design methodology to be of potential applicability to other stabilization problems where passivity arguments are unapplicable.     

On passivity-based output feedback global stabilization of euler-lagrange systems

It is well known that in systems described by Euler-Lagrange equations the stability of the equilibria is determined by the potential energy function. Further, these equilibria are asymptotically stable if suitable damping is present in the system. These properties motivated the development of a passivity-based controller design methodology which aims at modifying the potential energy of the closed loop and the addition of the required dissipation. To achieve the latter objective measurement of the generalized velocities is typically required. Our main contribution in this paper is the proof that damping injection without velocity measurement is possible via the inclusion of a dynamic extension provided the system satisfies a dissipation propggation condition. This allows us to determine a class of Euler-Lagrange systems that can be globally asymptotically stabilized with dynamic output feedback. We illustrate this result with the problem of set-point control of elastic joints robots. Our research contributes, if modestly, to the development of a theory for stabilization of nonlinear systems with physical structures which effectively exploits its energy dissipation properties.     

Canonical transformations used to derive robot control laws from a port-controlled Hamiltonian system perspective

This paper addresses the problem of deriving control laws for robot manipulators in the framework of port-controlled Hamiltonian systems via canonical transformations and passivity-based control. The control design is focused on the presentation of a new energy-shaping methodology for tracking control based on the introduction of virtual non-homogeneous fields where a desired energy is defined to compensate for the actual energy of the robot manipulator while a virtual field forces the system to track a general reference trajectory. This requires use of the Legendre-Fenchel transformation and allows for the derivation standard control laws in the robotics field such as PD control with gravity compensation or PD with precompensation. Finally, the passivity of the input-output mapping of the non-autonomous Hamiltonian system is analyzed in detail, resulting in new Lyapunov candidate functions having their roots in physics.     

Energy-shaping for Hamiltonian control systems with time delay

This paper investigates the asymptotical stabilization of Hamiltonian control systems with time delay. First, Hamiltonian control systems with time delay are proposed. Second, a two-to-one (TTO) principle is introduced that two different Hamiltonian functions are simultaneously energy-shaping by one desired energy function. Third, a novel matching equation is built via the TTO principle for the Hamiltonian control systems with time delay, which generates an effective control law for the Hamiltonian control systems with time delay. Finally, a numerical example shows the effectiveness of proposed method.