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.     

Gait generation and control for biped robots with underactuation degree one

The paper develops a unified feedback control law for n degree-of-freedom biped robots with one degree of underactuation so as to generate periodic orbits on different slopes. The periodic orbits on different slopes are produced from an original periodic orbit, which is either a natural passive limit cycle on a specific slope or a stable periodic walking gait on level ground generated with active control. First, inspired by the controlled symmetries approach, a general result on gait generation on different slopes based on a periodic orbit on a specific slope is obtained. Second, the time-scaling control approach is integrated to reproduce geometrically same periodic orbits for biped robots with one degree of underactuation. The degree of underactuation is compensated by one degree-of-freedom in the temporal evolution that scales the original periodic orbit. Necessary and sufficient conditions are investigated for the existence and stability properties of periodic orbits on different slopes with the proposed control law. Finally, the proposed approach is illustrated by two kinds of underactuated biped robots: one has a passive gait on a specific ground slope and the other does not have a natural passive gait.     

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.     

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.     

An introduction to interconnection and damping assignment passivity-based control in process engineering

In recent years so-called Port-Hamiltonian systems became popular in the literature. This is mainly due to the passivity properties and the evident system structure of Port-Hamiltonian systems that allow an energy interpretation of this system class. Based on these properties Port-Hamiltonian systems constitute an active research area and several control design methods for Port-Hamiltonian systems have been developed. Especially the Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) method has been proven to be very successful. In the application of IDA-PBC and other Port-Hamiltonian system techniques to real-world examples so far mainly electro-mechanical systems have been considered, because for those examples Port-Hamiltonian systems are the standard modeling framework. This paper gives an introduction to the framework of Port-Hamiltonian systems and the IDA-PBC design and shows with different examples that it is also applicable in process control.     

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.     

Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment

We consider the application of a formulation of passivity-based control (PBC), known as interconnection and damping assignment (IDA) to the problem of stabilization of underactuated mechanical systems, which requires the modification of both the potential and the kinetic energies. Our main contribution is the characterization of a class of systems for which IDA-PBC yields a smooth asymptotically stabilizing controller with a guaranteed domain of attraction. The class is given in terms of solvability of certain partial differential equations. One important feature of IDA-PBC, stemming from its Hamiltonian formulation, is that it provides new degrees of freedom for the solution of these equations. Using this additional freedom, we are able to show that the method of "controlled Lagrangians"-in its original formulation-may be viewed as a special case of our approach. As illustrations we design asymptotically stabilizing IDA-PBCs for the classical ball and beam system and a novel inertia wheel pendulum.     

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.     

Stabilization of electromechanical systems via interconnection and damping assignment

In this article, we propose a general controller structure for asymptotic position regulation of electromechanical systems derived using the Interconnection and Damping Assignment Passivity‐Based Control methodology recently proposed in the literature. The controller is applicable to arbitrary fully actuated electromechanical systems with linear magnetic materials consisting of inductances, permanent magnets, and one mechanical co‐ordinate. We assume linear magnetic materials and fully actuated electrical dynamics; however, no restrictions are imposed on the particular form of the parameters that define the system dynamics, i.e. the inductance matrix, the magnetic coupling or the potential energy function. This allows us to treat—in a unified framework and without any additional simplifying assumptions—very diverse applications, including magnetic suspensions, and stepper and permanent magnet synchronous motors. Instrumental for our developments is the inclusion of ‘virtual’ couplings between the electrical and the mechanical subsystem, which is naturally suggested in this control methodology. Copyright © 2003 John Wiley & Sons, Ltd.     

Continuous sliding-mode control for underactuated systems: Relative degree one and two

This paper deals with the design of sliding-mode controllers for the stabilization of some types of underactuated systems. The proposed method takes advantage of the mechanical properties of three different types of underactuated systems instead of using nonlinear transformations. In order to design the controller, some sliding variables with relative degree one and two are introduced. The theoretical differences between these approaches are discussed and some simulation results show the practical differences. Experimental results on a cart–pole system are presented to validate the proposed control strategy.     

Adaptive passivity-based control for maximum power extraction of stand-alone windmill systems

This paper addresses the high performance regulation of stand-alone windmill systems consisting of a wind turbine coupled to a generator and a battery charging system, which is a challenging problem for at least two reasons. First, the dynamics of the overall system are described by a highly coupled set of nonlinear differential equations. Since the range of operating points of the system is very wide, classical linear controllers yield below par performances. Second, in many applications it is desirable to extract from windmill systems their maximum power. This operating point is a nonlinear function of the wind speed, which is hard to measure. In this paper, a nonlinear passivity-based controller that ensures asymptotic convergence to the maximum power extraction point, which is rendered adaptive combining it with a wind speed estimator previously proposed by the authors, is proposed. Detailed computer simulations are presented to validate the approach.     

Impulsive control of continuous‐time homogeneous positive delay systems of degree one

This paper investigates the impulsive control of continuous‐time homogeneous positive delay systems of degree one. By using max‐separable Lyapunov functions and Razumikhin technique, a number of stability criteria for continuous‐time homogeneous impulsive positive delay systems of degree one are obtained. It should be noted that it is the first time that impulsive stabilization results for the addressed systems are given. Three numerical examples are presented to demonstrate the effectiveness of the control strategy.     

Simple mechanical control systems with constraints

We apply some recently developed control theoretic techniques to the analysis of a class of mechanical systems with constraints. Certain simple aspects of the theory of affine connections play an important part in our presentation. The necessary background is presented in order to illustrate how the methods may be applied. The bulk of this paper is devoted to a detailed analysis of some examples of nonholonomic mechanical control systems. We look at the Heisenberg system, the upright rolling disk, the roller racer, and the snakeboard     

Control via interconnection and damping assignment of linear time-invariant systems: a tutorial

Interconnection and damping assignment is a controller design methodology that regulates the behaviour of dynamical systems assigning a desired port-Hamiltonian structure to the closed-loop. A key step for the application of the method is the solution of the so-called matching equation that, in the case of nonlinear systems, is a partial differential equation. It has recently been shown that for linear systems the problem boils down to the solution of a linear matrix inequality that, moreover, is feasible if and only if the system is stabilisable – making the method universally applicable. It has also been shown that if we narrow the class of assignable structures – e.g. to mechanical instead of the larger port-Hamiltonian – the problem is still translated to a linear matrix inequality, but now stabilisability is not sufficient to ensure its feasibility. It is additionally required that the uncontrolled modes are simple and lie on the jω axis, which is consistent with the considered scenario of mechanical systems without friction. The purpose of this article is to present these important results in a tutorial, self-contained form – invoking only basic linear algebra methods.     

Distributed formation control of fractional-order multi-agent systems with relative damping and communication delay

The distributed formation control of fractional-order multi-agent systems is mainly studied under directed communication graphs in this paper. Firstly, a control law with relative damping and communication delay are proposed. Then, some sufficient conditions for achieving formation control are derived using matrix theory, graph theory and the frequency domain analysis method. Finally, based on the numerical method of predictor-corrector, several simulations are presented to illustrate the effectiveness of the obtained results.     

Distributed formation control of fractional-order multi-agent systems with absolute damping and communication delay

The distributed formation control of fractional-order multi-agent systems is mainly studied under directed interaction topology in this paper. First, the control algorithm with absolute damping and communication delay is proposed to achieve the formation control. Then, some sufficient conditions are derived by using the matrix theory, graph theory and the frequency-domain analysis method. Finally, based on the numerical method of predictor–corrector, several simulations are presented to illustrate the effectiveness of the obtained results.     

On mechanical control systems with nonholonomic constraints and symmetries

This paper presents a computationally efficient method for deriving coordinate representations for the equations of motion and the affine connection describing a class of Lagrangian systems. We consider mechanical systems endowed with symmetries and subject to nonholonomic constraints and external forces. The method is demonstrated on two robotic locomotion mechanisms known as the snakeboard and the roller racer. The resulting coordinate representations are compact and lead to straightforward proofs of various controllability results.     

受正弦扰动时滞非线性系统的近似最优减振控制

研究时滞非线性系统在正弦扰动作用下的最优减振控制问题,给出一种无时滞近似最优减振控制律的迭代方法．通过假设Lagrange算子,将由原系统最优控制问题得到的既含时滞项又含有超前项的非线性两点边值问题转换为新的有利于求解的形式,再通过构造序列将其转化为不舍时滞项和超前项的线性非齐次两点边值问题序列．证明了该序列的收敛性．通过交替迭代序列得到了系统最优减振控制律．仿真结果表明,该方法在不同时滞下对扰动都具有很好的鲁棒性．