Energy-based Control Approach Research Articles

Energy-based high frequency oscillation control of linear drives

Linear electromagnetic actuators realizing periodic oscillations have gained significant importance in the compressor and pump technology. This is due to the need of compact drive units to increase the power density of modern pumps and compressors. This work presents an energy-based control approach for harmonic oscillations of linear electromagnetic drives and analyzes the functionality and performance of the oscillation control for periodic actuator movements up to 250 Hz. To obtain a more robust control behavior, the control structure needs to be decoupled, with an amplitude and phase component using a periodical harmonic carrier function. This approach avoids a standstill of the actuator by constantly triggering an oscillation at the fundamental frequency of the mechanical system. The presented control approach is especially beneficial for oscillatory systems which need to adapt their oscillation frequency to mechanical loads.

Stable cargo transportation for underactuated partial space elevator with model uncertainties and disturbances

An adaptive neural network energy-based control approach is proposed in this paper to address the issue of stabilizing a partial space elevator during cargo transportation. The method handles the challenges presented by external disturbances and model uncertainties. The dynamic model of the partial space elevator, with a typical multi-input and multi-output underactuated system, is presented. A control scheme using energy-based techniques is proposed to guarantee the stable configuration of the system, and auxiliary kinematic vector variables are designed to incorporate the uncontrollable variables to constitute controllable state subsets. Under these frameworks, adaptive control schemes are formulated using multi-layer neural networks to mitigate model uncertainty. Robust compensators are designed to effectively counteract the unknown boundaries of external disturbances and approximation errors. The stability of the closed control system is rigorously demonstrated through the application of the Lyapunov function and LaSalle's invariance principle. Numerical simulations demonstrate that the proposed control method is highly effective in suppressing oscillations during the transfer period of a partial space elevator.

Energy based control of a bi-stable and underactuated soft robotic system based on dielectric elastomer actuators*

In this paper, we propose an energy based control approach for a class of underactuated soft robotic systems. The considered case study consists of an elastic structure driven by soft dielectric elastomer actuators, and is able to achieve large bending displacement thanks to a bi-stable design concept. The bi-stability feature, however, causes the system to exhibit an unstable behavior in open-loop. After providing a port-Hamiltonian description of the soft robotic system, sufficient conditions for the existence of an energy based stabilizing controller are provided. A linear matrix inequality approach is then proposed to practically address the design of the controller gain. The effectiveness of the method is verified by means of simulation studies, conducted on an experimentally validated model of the real-life device.

Set-point control for folded configuration of 3-link underactuated gymnastic planar robot: new results beyond the swing-up control

This paper concerns a set-point control for a folded configuration of a 3-link gymnastic planar robot moving in the vertical plane with the first joint being passive and the others being active. To realize the goal configuration of the kip motion of a human gymnast on the high bar, the control objective is to drive the robot from any initial state to any small neighborhood of the up–down–down equilibrium point and then balance the 3-link robot about that point, where link 1 is in the upright position and links 2 and 3 are in the downward position. This paper uses the energy-based control approach and the notion of virtual composite link to design a controller and provides a global motion analysis of the 3-link robot. Different from the swing-up control problem for which three links are fully stretched out in the upright position at the goal configuration, a new result of this paper is that in addition to some conditions on control parameters, a constraint on the mechanical parameters of the 3-link robot is needed for achieving the set-point control of the folded configuration (links 1 and 2 are folded). The proposed constraint guarantees the linear controllability at the up–down equilibrium point of the Acrobot (degenerated from the 3-link robot) whose actuated second link is the merge of links 2 and 3 of 3-link robot with all possible relative angles. The simulation results for two 3-link robots are presented to validate the obtained theoretical results.

Trajectory tracking control of variable length pendulum by partial energy shaping

This paper concerns a trajectory tracking control problem for a pendulum with variable length, which is an underactuated mechanical system of two degrees-of-freedom with a single input of adjusting the length of the pendulum. We aim to study whether it is possible to design a time-invariant control law to pump appropriate energy into the variable length pendulum for achieving a desired swing motion (trajectory) with given desired energy and length of the pendulum. First, we show that it is difficult to avoid singular points in the controller designed by using the conventional energy-based control approach in which the total mechanical energy of the pendulum is controlled. Second, we present a tracking controller free of singular points by using only the kinetic energy of rotation and the potential energy of the pendulum and not using the kinetic energy of the motion along the rod. Third, we analyze globally the motion of the pendulum and clarify the stability issue of two closed-loop equilibrium points; and we also provide some conditions on control parameters for achieving the tracking objective. Finally, we show numerical simulation results to validate the presented theoretical results.

On simultaneous control of the energy and actuated variables of underactuated mechanical systems – example of the acrobot with counterweight

The energy-based control approach aiming to control both the total mechanical energy and actuated variables of underactuated mechanical systems has generated renewed interest in recent years. Different from the reports of successful applications of this approach, we investigate whether there exists an underactuated mechanical system for which we fail to control both the total mechanical energy and actuated variable(s) to some given desired values by studying a CWA (Counter-Weighted Acrobot), which is a modified Acrobot with its first link having a counterweight and only its second link being actuated. By analyzing globally the solution of the closed-loop system consisting of the CWA and the controller designed via the energy-based control approach, we show that unless the CWA is linearly controllable at the up–up equilibrium, where links 1 and 2 are in the upright position, the controller fails to achieve the goal of controlling the energy to the potential energy at the equilibrium and controlling the actuated joint variable to zero. We also provide corresponding results for the up–down, down–up, and down–down equilibriums of the CWA, where up and down denote that the link is in the upright and downward positions, respectively. We present numerical simulation results to validate the theoretical results.

A Set-Point Control for a Two-link Underactuated Robot With a Flexible Elbow Joint

This paper concerns a set-point control problem for a two-link underactuated robot moving in the vertical plane with a single actuator at the first joint and a spring between the two links (flexible elbow joint). First, we present two new properties of such a flexible robot about the linear controllability at the upright equilibrium point (UEP, where two links are in the upright position) and about its equilibrium configuration. Second, we show that for the robot with a certain range of spring constant, the proportional derivative (PD) control on the angle of the first joint can globally stabilize the robot at the UEP. Third, for the robot not satisfying the above range of spring constant, we study how to extend the energy-based control approach, which aims to control the total mechanical energy and the angle and angular velocity of the first joint of the robot, to design a swing-up controller. We provide a necessary and sufficient condition for avoiding the singularity in the controller, and we analyze the motion of the robot under the presented controller by studying the convergence of the total mechanical energy and clarifying the structure and stability of the closed-loop equilibrium points. We present the results of numerical investigation, which support our theoretical conclusions.

Energy-based modeling and control for grid-side converter of doubly fed wind generator

To avoid dealing with the zero dynamics limitation brought by the bidirectional power flow through the back-to-back converter in a doubly fed wind generator, a new energy-based modeling and control approach for the grid-side converter is presented. During the modeling process, the grid-side converter is divided into two subsystems with the feedback interconnection structure, and the interactive matrix of the model takes into account the concrete port structure. Then, an energy-based controller is proposed to realize the grid-side control objectives based on the new model. Simulation studies are carried out in MATLAB/Simulink. Comparative results between the proportional-integral controller and the energy-based controller show that the latter one can obtain faster convergence rate and global stability as the load current varies. Moreover, the energy-based controller is also competent for grid-side control when simulated in a 2 MW wind energy conversion system with random wind.

Set-Point Control for Folded Posture of 3-Link Underactuated Gymnastic Planar Robot: Beyond the Swing-up Control

AbstractThis paper concerns a set-point control for a folded posture of a 3-link gymnastic planar robot moving in the vertical plane with the first joint being passive and the others being active. To realize the goal posture of the kip motion of a gymnast on the high bar, the control objective is to drive the robot from any arbitrary initial state to any arbitrarily small neighborhood of the up-down-down equilibrium point and then balance the robot about that point, where the first link is in the upright position and the second and third links are in the downward position. This paper uses the energy-based control approach and the notion of virtual composite link to design a controller and provides a global motion analysis of the robot. This paper shows that the control toward such a folded posture is different from the swing-up control problem, for which all three links are fully stretched out in the upright position at the goal posture. A new result of this paper is that in addition to some conditions on the control parameters, a constraint on the mechanical parameters of the robot is needed for achieving the set-point control of the folded posture. The simulation results for two 3-link robots are provided to validate the effectiveness of the proposed control law and the necessity of the constraint.

Trajectory Tracking Control of Pendulum with Variable Length by Partial Energy Shaping

AbstractThis paper concerns a trajectory tracking control problem for a pendulum with variable length, which is an underactuated mechanical system of two degrees-of-freedom with a single input of adjusting the length of the pendulum. We aim to study whether it is possible to design a time-invariant control law to pump appropriate energy into the variable-length pendulum for achieving a desired swing motion (trajectory) with given desired energy and length of the pendulum. First, we show that it is difficult to avoid singular points in the controller designed by using the conventional energy-based control approach in which the total mechanical energy of the system is controlled. Second, we propose a tracking controller free of singular points by shaping only the kinetic energy of rotation and the potential energy of the pendulum and not utilizing the kinetic energy related to the motion along the rod. Third, we analyze globally the motion of the pendulum and clarify the stability issue of some closed-loop equilibrium points; and we also provide some conditions on control parameters for achieving the tracking objective. Finally, we present numerical simulation results to validate the presented theoretical results.

Energy-based balance control approach to the ball and beam system

This article investigates the set-point balance control problem of the ball and beam system acting on the ball which is an underactuated system. The control law is designed, based on the total mechanical energy and the passivity properties of the system. According to the desired set-point, the zero equilibrium point and non-zero equilibrium points are studied respectively. For the zero case, it is proved that a single PD feedback controller is sufficient to bring the state to zero from any initial condition, on condition that the control parameters satisfy an inequality. For the non-zero case, the control problem is much more complicated. Unlike the previous energy-based control laws, a new form of Lyapunov function candidate is constructed. A complete analysis of the convergence of the energy and the dynamics is given, and the characteristics of the closed-loop system with the proposed feedback control law are illustrated. Moreover, it is proved that with the parameter choice rules proposed in this article, the ball and beam system will eventually converge exactly to the desired non-zero equilibrium point. Furthermore, since the length of the beam is not unlimited, the trajectory of the ball should be restricted within a limited range. The balance control laws are modified to avoid the ball running beyond the joint range limitation for the zero case and the non-zero case respectively. Simulation results show that the control laws proposed in this article are effective for the set-point balance control problem of the ball and beam system.

Analysis of the Energy Based Swing-up Control for a Double Pendulum on a Cart

Designing and analyzing controllers for underactuated systems with underactuation degree greater than one is still an open and challenging problem. In this paper, we study an unsolved problem of analyzing energy based swing-up control for a double pendulum on a cart, which has three degrees of freedom and only one control input. We present an original analysis of the convergence of the energy of the cart-double pendulum system. We show that for all initial states of the cart-double pendulum system, if the convergent value of the energy is not equal to the energy at the upright (up-up) equilibrium point, then the cart-double pendulum remains at its up-down, down-up, and down-down equilibrium points. Moreover, we show that these three equilibrium points are unstable. This shows that for almost all initial states of the cart-double pendulum system, as time approaches infinity, the energy of the cart-double pendulum system can be controlled to its energy at the upright equilibrium point. This paper provides insight into the energy based control approach to underactuated systems with underactuation degree greater than one.

Swing-Up Control for a 3-DOF Gymnastic Robot With Passive First Joint: Design and Analysis

This paper concerns a swing-up control problem for a three-link gymnastic planar robot in a vertical plane with its first joint being passive (unactuated) and the rest being active (actuated). The objectives of this paper are to: (1) design a controller under which the robot can be brought into any arbitrarily small neighborhood of the upright equilibrium point, where all three links of the robot remain in their upright positions; and (2) attain a global analysis of the motion of the robot under the controller. To tailor the energy-based control approach to achieve the aforementioned objectives, first, this paper considers the links 2 and 3 as a virtually composite link, and proposes a coordinate transformation of the angles of active joints. Second, this paper constructs a novel Lyapunov function based on the transformation, and devises an energy-based swing-up controller. Third, this paper carries out a global analysis of the motion of the robot under the controller, and establishes some conditions on control parameters for achieving the swing-up control objective. To validate the theoretical results obtained, this paper provides simulation results for a three-link robot with its mechanical parameters being obtained from a human gymnast.

Approximate dissipative Hamiltonian realization and construction of local Lyapunov functions

The key in applying energy-based control approach is to be able to express the system under consideration as a dissipative Hamiltonian system, i.e., to obtain Dissipative Hamiltonian Realization (DHR) for the system. In general, the precise DHR form is hard to obtain for nonlinear dynamic systems. When a precise DHR does not exist for a dynamic system or such a precise realization is difficulty to obtain, it is necessary to consider its approximate realization. This paper investigates approximate DHR and construction of local Lyapunov functions for time-invariant nonlinear systems. It is shown that every nonlinear affine system has an approximate DHR if linearization of the system is controllable. Based on the diagonal normal form of nonlinear dynamic systems, a new algorithm is established for the approximate DHR. Finally, we present the concept of kth degree approximate Lyapunov function, and provide a method to construct such a Lyapunov function. Example studies show that the methodology presented in this paper is very effective.

Stabilization of the Furuta pendulum around its homoclinic orbit

We present an energy based control approach to control the Furuta pendulum. A controller is proposed for swinging the pendulum and raise it to its uppermost unstable equilibrium position. The passivity properties of the system are used as a guideline in the control strategy. The stability analysis is carried out by using a Lyapunov technique.

Stabilization of the Furuta Pendulum Around Its Homoclinic Orbit

We present an energy based control approach to control the Furuta pendulum. A controller is proposed for swinging the pendulum and raise it to its uppermost unstable equilibrium position. The passivity properties of the system are used as a guideline in the control strategy. The stability analysis is carried out by using a Lyapunov technique.

An adaptive filtered-x algorithm for energy-based active control

A control algorithm based on the filtered-x LMS algorithm is developed that has the property of being able to control the energy in the acoustic field. The algorithm is useful for providing greater global control of the field than is obtained by controlling the pressure. This is due to the fact that the energy-based approach effectively overcomes the observability problems that often limit the performance possible when controlling the pressure field. The control approach is developed for the cases of a predominantly standing wave field, and a predominantly propagating wave field. The approach is applicable for both broadband and narrow-band excitations. A simple example demonstrating the increased performance possible with the energy-based control approach is presented for the case of single frequency excitation.