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.     

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.     

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.     

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.     

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 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.     

Controllability for single-input mechanical control systems with dissipation

Within the affane connection framework of Lagrangian control systems, based on the results of Sussmann on small-time locally controllability of single-input affine nonlinear control systems, the controllability results for mechanical control systems with single-input are extended to the case of the systems with isotropic damping, where the Lagrangian is the kinetic energy associated with a Riemannian metric, A sufficient condition of negative small-time locally controllability for the system is obtained.Then,it is demonstrated that such systems are small-time locally configuration controllable if and only if the dimemion of the configuration manifold is one. Finally, two examples are given to illustrate the results. Lie bracketting of vector fields and the symmetric product show the advantages in the discussion.     

无源性控制在有源电力滤波器中的应用

线路或负载参数未知可能会导致有源滤波器系统失稳，为解决系统的稳定性问题，将无源性控制应用于有源电力滤波器来保证系统的稳定性。将有源滤波器和非线性负载视为一个整体，对负载进行诺顿等效，把负载电压和电源电流作为状态变量，把补偿电流作为控制变量，建立系统在dp轴上的模型。获得系统的控制律也就获得了补偿电流的给定值，验证系统满足无源性条件之后，再利用无源性控制方法获得系统的控制律。虽然算法在理论上比较复杂，但是实现上相当简单。在仿真中，对比了容性负载，感性负载以及变动负载情况下的控制效果，仿真结果表明算法具有很好的鲁棒性，并能很好的保证系统的稳定性。     

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.     

State covariance assignment for sampled-data feedback control systems

This paper proposes a new performance criterion of ‘covariance’ for sampled-data systems. A covariance of sampled-data systems is defined by taking account of inter-sample behaviour. An SCA (state covariance assignment) problem for sampled-data feedback control systems is also discussed, which is the counterpart of that for purely continuous or discrete-time feedback control systems. The SCA problem for sampled-data systems will be solved as a discrete-time SCA problem, where the discretization preserves the state covariance and is in two steps. In the first step, a certain sample time performance is required to ensure the inter-sample performance, and the output signal results to be discretized. The second step is to discretize the input signal.     

Robust fuzzy control of mechanical systems

This paper considers the problem of controlling a mechanical system described by Euler-Lagrange equations to follow a desired trajectory in the presence of uncertainties. A fuzzy logic system (FLS) is used to approximate the unknown dynamics of the system. Based on the a priori information, the premise part of the FLS as well as a nominal weight matrix are designed first and are fixed. A compensation signal to the weight matrix error is designed based on Lyapunov analysis. To further reduce the tracking error due to the function reconstruction error, a second compensation signal is also synthesized. By running two estimators online for weight matrix error bound and function reconstruction error bound, the implementation of the proposed controller needs no a priori information on these bounds. Exponential tracking to a desired trajectory up to a uniformly ultimately bounded error is achieved with the proposed control. The effectiveness of this control is demonstrated through simulation and experiment results. These results also show that by incorporating a priori informations about the system, the fuzzy logic control can result good tracking behavior using a few fuzzy IF- THEN rules.     

Decomposition-based continuous control of mechanical systems

For Lagrangian dynamical systems with inexactly known parameters, a continuous feedback control is constructed using ideas of the decomposition method on the major segment of the trajectory and linear feedbacks with the coefficients depending on the phase variables in the vicinity of the terminal state. It is supposed that the matrix of kinetic energy of the system is close to a constant matrix in the motion domain and that the system is affected by uncontrollable bounded disturbances. The proposed controls make it possible to steer the system from an arbitrary initial state into a given terminal state in a finite amount of time using generalized forces that are bounded by magnitude. An upper estimate on the total time of system motion is given and techniques for its reduction are described. The efficiency of the proposed algorithms is demonstrated by the numerical simulation of the dynamics of some simple mechanical systems.     

Decentralized robust control of mechanical systems

For mechanical systems described by Euler-Lagrange equations and involving high-order interconnections, this paper proposes a decentralized control, which guarantees the uniform ultimate stability when the control objective is tracking a smooth desired trajectory. The controller design is based on some physical properties of such systems and Lyapunov design methodology. The application of the proposed controller to robot manipulators is presented, and experiment results are included to illustrate the performance of the proposed control algorithm     

Pulse-Width-Modulation control of some mechanical systems

Pulse-Width-Modulation (PWM) control strategies have been of limited interest, and use, in the regulation of a large class of mechanical systems, such as robotic manipulators and rigid spacecrafts. The chattering associated to discontinuous bang-bang feedback control input signals has traditionally posed serious concerns when considering the selection of this robust and efficient control technique over some other smooth control alternatives. This article proposes a means of circumventing the inconveniences caused by the PWM control signal discontinuities in such a class of systems, while retaining the robust features of the PWM control approach.