Proportional-derivative Control Law Research Articles

Utilizing a control technique for orbital maintenance near [formula omitted] point and Lyapunov exponents

In present article, we have explored the orbit maintenance strategy around an artificial equilibrium point in the Sun–Jupiter system. Two control strategies, proportional control and proportional derivative control laws are discussed for orbit maintenance. The maximum Lyapunov exponent is used to check the chaotic behavior of planetary orbit. We anticipate that the recently established Lyapunov exponent criteria may be used to study motion of planets in generalized stellar binary systems. This research is being guided by the conclusions of recent observations as well as those that are expected from existing and upcoming planet search missions.

Adaptive Sliding Mode Control with Chattering Elimination for Buck Converter Driven DC Motor

The Adaptive Sliding Mode Control (ASMC) that combines a robust proportional derivative control law for use in Buck converter driven DC motor is presented in this paper. Based on the LYAPUNOV theory, the proportional derivative control law is designed to eliminate the chattering action of the control signal. The simplicity of the proposed scheme facilitates its implementation and the overall control scheme guarantees the global asymptotic stability in the LYAPUNOV sense if all the signals involved are uniformly bounded. Simulation studies have shown that the proposed controller shows superior tracking performance.

Anti-disturbance Trajectory Tracking Control for a Quadrotor UAV with Input Constraints⋆

This paper presents the trajectory tracking control for a quadrotor unmanned aerial vehicle(UAV) with disturbances and input constraints. The UAV's system is decomposed into an outer position loop and an inner attitude loop. The position tracking control is achieved by a class of saturated proportional-derivative(PD) control laws. Assuming that the unknown disturbances mainly effect on the translational dynamics, an adaptive control term is investigated to compensate the effects of the disturbances. The position tracking control force implies a coupled-attitude system that connects the inner attitude loop with the outer position loop. An intrinsic PD control law, which is directly designed on the special orthogonal group SO(3), is proposed to track the desired attitude. In order to reduce the calculational difficulty of the attitude tracking control law, an observer-based control law is further developed while the control law and observer can be designed separately. Simulation results complete this work.

Hybrid control technique applied to an aero-servo-viscoelastic simplified wing model

Considering that flutter represents a potential catastrophic event in the context of aerospace structures, numerous studies have evaluated a number of strategies to avoid and/or control this kind of aeroelastic phenomenon. Currently, both active and passive control have been investigated to prevent instabilities induced by the interaction between aerodynamic and structural forces. It is also important to highlight the successful cases in which passive control techniques using viscoelastic materials have been useful to mitigate several types of vibration problems. However, there are still opportunities to explore the potential of control using viscoelastic material in the scope of aeroelasticity, especially when involving its combination with other control techniques. Therefore, this work presents a strategy involving a hybrid approach to aeroelastic control of a simplified unswept and untapered wing, using a combination of passive and active techniques. Passive control is achieved by the use of viscoelastic materials inserted as resilient elements in the aeroelastic model, while active control is performed by means of the deflections of a flap-like aerodynamic control surface, governed by a proportional-derivative control law. The results show that the application of the passive control alone causes an increase of up to 25.4% in critical flutter speed. In addition, the association of passive and active controls lead to higher control performance and the critical speed is increased by a further 6.8%, thus providing a broader safe flight speed range. Hence, the investigation indicates that the hybrid control approach exploring viscoelastic materials can be advantageous in practical applications.

Rub-Impact Force Induces Periodic, Quasiperiodic, and Chaotic Motions of a Controlled Asymmetric Rotor System

This article aims to explore the oscillatory characteristics of a controlled asymmetric rotor system when subjected to rub and impact forces between the rotor and stator. Four electromagnetic poles are used to control the whirling motion of the rotor system through a linear proportional-derivative control law. The equations of motion that govern the whole system dynamics are derived including the rub and impact forces. The derived mathematical model is analyzed in two basic steps. Firstly, the obtained model is treated as a weakly nonlinear system using perturbation analysis to obtain the slow-flow modulating equations when neglecting the rub and impact forces. Depending on the obtained slow-flow equations, different response curves are plotted to explore the system’s periodic vibrations and determine the conditions at which the system can exhibit rub and impact force. Secondly, the whole system model including the rub and impact forces is investigated by using the bifurcation diagrams, Poincare map, frequency spectrums, and temporal oscillations. The obtained results revealed that the applied control law could mitigate the system whirling oscillations and prevent the rub and impact forces if the control gains are tuned properly. However, the system can perform period-n, quasiperiodic, or chaotic motion depending on the shaft spinning speed if the controller fails to eliminate the contact between the rotor and stator.

PD-Based Optimal ADRC with Improved Linear Extended State Observer.

Taking dead-zone nonlinearlity and external disturbances into account, an active disturbance rejection optimal controller based on a proportional-derivative (PD) control law is proposed by connecting the proportional-integral-derivative (PID) control, the active disturbance rejection control (ADRC) and particle swarm optimization (PSO), with the purpose of providing an efficient and practical technology, and improving the dynamic and steady-state control performances. Firstly, in order to eliminate the negative effects of the dead-zone, a class of 2-order typical single-input single-out system model is established after compensating the dead-zone. Following that, PD control law is introduced to replace the state error feedback control law in ADRC to simplify the control design. By analyzing the characteristics of the traditional linear extended state observer, an improved linear extended state observer is designed, with the purpose of improving the estimation performance of disturbances. Moreover, employing PSO with a designed objective function to optimize parameters of controller to improve control performance. Finally, ten comparative experiments are carried out to verify the effectiveness and superiority of the proposed controller.

On Robust Adaptive PD Control of Robot Manipulators

In this study, an adaptive proportional-derivative (PD) control scheme is proposed for trajectory tracking of multi-degree-of-freedom robot manipulators in the presence of model uncertainties and external disturbances whose upper bounds are unknown but bounded. The developed controller takes the advantages of linear control in the sense of simplicity and easy design, but simultaneously possesses high robustness against model uncertainties and disturbances while avoiding the necessity of precise knowledge of the system dynamics. Due to the linear feature of the proposed method, both the transient and steady-state responses are easily controlled to meet desired specifications. Also, an adaptive law for control gains using only position and velocity measurements is introduced so that parameter uncertainties and disturbances are successfully compensated, where the prior knowledge about their upper bounds is not required. Stability analysis is conducted using the Lyapunov’s direct method and brief guidelines on how to select control parameters are also provided. Simulation results corroborate that the adaptive PD control law proposed in this paper can achieve a fast convergence rate, small tracking errors, low control effort, and small computational cost and its performance is compared with that of an existing nonlinear sliding mode control method.

PD Tracking for a Class of Underactuated Robotic Systems With Kinetic Symmetry

In this letter, we study stability properties of Proportional-Derivative (PD) controlled underactuated robotic systems for trajectory tracking applications. Stability of PD control laws for fully actuated systems is an established result, and we extend it for the class of underactuated robotic systems. We will first show some well known examples where PD tracking control laws do not yield tracking; some of which can even lead to instability. We will then show that for a subclass of robotic systems, PD tracking control laws, indeed, yield desirable tracking guarantees. We will show that for a specified time interval, and for sufficiently large enough PD gains (input saturations permitting), local boundedness of the tracking error can be guaranteed. In addition, for a class of systems with the kinetic symmetry property, stronger conditions like convergence to desirable bounds can be guaranteed. This class is not restrictive and includes robots like the acrobot, the cart-pole, and the inertia-wheel pendulums. Towards the end, we will provide necessary simulation results in support of the theoretical guarantees presented.

Control of a quadrotor and a ground vehicle manipulating an object

This paper focuses on the control of a cooperative system composed of an Unmanned Aerial Vehicle (UAV) and an Unmanned Ground Vehicle (UGV) manipulating an object. The two units are subject to input saturations and collaborate to move the object to a desired pose characterized by its position and inclination. The dynamics are derived using the Euler–Lagrange method. A stabilizing control law is proposed where the UGV is tasked to deploy the object to a certain position whereas the UAV adjusts its inclination. In particular, a proportional–derivative control law is proposed for the UGV, and a cascade control is used for the UAV, where the inner loop controls the attitude of the UAV and the outer loop stabilizes the inclination of the object. The stability of the points of equilibrium is proven using small gain arguments. To ensure constraints satisfaction at all times, a reference governor unit is added to this stabilizing control law. Simulations and experimental results are provided to validate the effectiveness of the proposed control scheme.

Spacecraft output feedback attitude control based on extended state observer and adaptive dynamic programming

This paper investigates spacecraft output feedback attitude control problem based on extended state observer (ESO) and adaptive dynamic programming (ADP) approach. For the plant described by the unit quaternion, an ESO is first presented in view of the property of the attitude motion, and the norm constraint on the unit quaternion can be satisfied theoretically. The practical convergence proof of the developed ESO is illustrated by change of coordinates. Then, the controller is designed with an involvement of two parts: the basic part and the supplementary part. For the basic part, a proportional-derivative control law is designed. For the supplementary part, an ADP method called action-dependent heuristic dynamic programming (ADHDP) is adopted, which provides a supplementary control action according to the differences between the actual and the desired system signals. Simulation studies validate the effectiveness of the proposed scheme.

Faster Positioning of One Degree-of-Freedom Mechanical Systems With Friction and Actuator Saturation

This paper concerns with faster positioning of one degree-of-freedom (1DOF) mechanical systems with friction and actuator saturation. A very simple but quite effective saturated relay proportional-derivative (PD) control is proposed. The proposed control is conceived within the framework of nonlinear PD methodology. It is accomplished by adding a relay position error for friction compensation to the saturated PD control law. For faster transient with bounded input, a saturated nonlinear function is introduced. Lyapunov's direct method is employed to prove global asymptotic positioning. The appealing advantages of the proposed control are that it is fairly easy to construct without involvement of any modeling parameter and has the ability to avoid the actuator saturation. Numerical example and experiments demonstrate the effectiveness and improved performance of the proposed approach.

Tilt-Prioritized Quadrocopter Attitude Control

This paper presents an attitude tracking control law and control allocation strategy for quadrocopters which prioritizes the vehicle’s ability to achieve a desired translational acceleration. The quadrocopter’s attitude error is split into a reduced attitude error, which describes the misalignment of the thrust direction, and a yaw error, which describes the orientation error about the thrust direction. A model-based proportional–derivative control law is derived, where the proportional action is in terms of the reduced attitude and yaw errors, and the derivative action is in terms of the quadrocopter’s angular velocity error. Almost global asymptotic convergence of the reduced attitude error is established, and the convergence of the yaw error is proven. It is further shown that the attitude control law decouples the reduced attitude error dynamics from the yaw error dynamics. A control allocation strategy is derived which exploits the decoupling in order to prioritize the correction of the reduced attitude over the yaw error. The proposed control strategy is computationally lightweight and therefore well-suited to run on board quadrocopters at high rates. Experimental results demonstrate improved error recovery and position tracking performance.

An Expanded Two-Dimensional Proportional–Derivative Command to Line-of-Sight Guidance Law

This paper deals with an expanded two-dimensional (2D) proportional-derivative (PD) command to line-of-sight (CLOS) guidance law which fights the spiral type trajectory of an anti-tank guided missile (ATGM) in the plane perpendicular to the line-of-sight (LOS) or the so called picture plane, in order to improve the transition process performance while putting the missile onto the LOS. The guidance law acts as a classical PD control law within a small predetermined area around the LOS while at missile deviations pointing a position outside this area the control law includes additional nonlinear components connected with the derivatives of the missile position vector in the plane perpendicular to the LOS. The global asymptotic stability of the guidance loop is established by a specific positive definite Lyapunov function. The effectiveness of the proposed approach is illustrated by simulation results. The guidance law enables to decrease the near-field boundary of the missile operational range.

Four-Bar Mechanism's Proportional-Derivative and Neural Adaptive Control for the Thorax of the Micromechanical Flying Insects

The paper focuses on the dynamics and control of the nondeformable and deformable four-bar mechanism (three of the bars are mobile and one is fixed), this being a subsystem of the micromechanical flying insects' (MFIs) thorax. The control of the mechanism (six-order system described by Lagrange equations) is initially achieved by using a proportional-derivative (PD) control law, a Newton–Raphson type algorithm, and the Lyapunov theory. Because the thorax's dynamics is strongly nonlinear and is characterized by fast time varying coefficients, the PD control law cannot always guarantee small overshoot and angular rates; to overcome this drawback, over the control law PD component we superpose a neural adaptive component which compensate the error of the global nonlinearity's approximation associated to the thorax's dynamics. The two obtained control systems are validated by complex numerical simulations.

Partial Lyapunov Strictification: Smooth Angular Velocity Observers for Attitude Tracking Control

A smooth angular velocity observer is proposed for the attitude tracking control of a rigid body in the absence of angular velocity measurements. The observer design ensures asymptotic convergence of angular velocity state estimation errors irrespective of the control torque or the initial attitude state of the spacecraft. Unlike existing rate observer formulations that attain estimation error convergence by imposing certain switching conditions or hybrid logic, the proposed observer has a smooth structure that ensures continuity of all estimated states. Furthermore, the combined implementation of the proposed observer with a proportional-derivative type of attitude control law leads to an important “separation property.” In particular, an independently designed proportional-derivative control law driven by angular velocity estimates generated from the smooth observer results in (almost) global asymptotic stability of the overall closed-loop tracking error dynamics. The main feature of this key technical result stems from our use of a Lyapunov “strictification” process that enables the closed-loop stability and convergence analysis to proceed along novel lines in a spiral logic fashion. A rigorous analysis of the proposed formulation is provided and numerical simulation studies are presented to help illustrate the effectiveness of the angular velocity observer for rigid-body attitude tracking control.

Nonlinear Proportional-Derivative Control Incorporating Closed-Loop Control Allocation for Spacecraft

A novel saturated proportional-derivative control law incorporated with closed-loop control allocation is proposed for spacecraft attitude stabilization in this paper. More specifically, a saturated proportional-derivative-based baseline nonlinear controller is designed to guarantee the globally asymptotic stability under control input/signal constraints. Then, a closed-loop constrained optimal control allocation scheme is employed to distribute the moments synthesized by the baseline controller over the redundant actuators in the presence of constraints due to actuator amplitude and rate constraints. The optimal control solution is to be found by penalizing the control allocation errors and rates using optimal quadratic programming algorithms. A significant feature of this work is that the asymptotic stability with the closed-loop control allocation is guaranteed theoretically. Simulation results are presented to illustrate the performance of the proposed scheme.

Dynamics, stability, and actuation methods for powered compass gait walkers

In this paper, methods to achieve actively powered walking on level ground using a simple 2-dimensional walking model (compass-gait walker) are explored. The walker consists of 2 massless legs connected at the hip joint, a point mass at the hip, and an infinitesimal point mass at the feet. The walker is actuated either by applying equal joint torques at the hip and ankle, by an impulse applied at the toe off, immediately before the heel strike, or by the combination of both. It is shown that actuating the walker by equal joint torques at the hip and ankle on level ground is equivalent to the dynamics of the passive walker on a downhill slope. The gait cycle for the simplified walker model is determined analytically for a given initial stance angle. Stability of the gait cycle by an analytical approximation to the Jacobian of the walking map is calculated. The results indicate that the short-period cycle always has an unstable eigenvalue, whereas stability of the long-period cycle depends on selection of the initial stance angle. The effect of the torso mass by adding a third link attached at the hip joint is investigated. The torso link is kept in the vertical position by controlling the torque applied to it. The proportional-derivative control law is utilized to regulate the angular position error of the torso link. Using linearized dynamics for this walker, active control is applied to the ankle, which reduces the dynamics of the walker to the passive walker without the torso. The proposed walker is capable of producing stable walking while keeping the torso in an upright position.

CONTROLADOR ADAPTATIVO PD POR MODELO DE REFERENCIA PARA UNA MESA VIBRATORIA BIAXIAL BASADA EN EL MECANISMO BIELA-MANIVELA

This work shows the validation of a model reference adaptive controller with proportional-derivative control law MRAC-PD. This model is designed as a control system of a shaking table with two axis, based on a slider-crank mechanism, and it is used to generate earthquakes on scale models of civilian structures. The MRAC-PD design was based on the MIT optimization rule as adaptation mechanism for the proportional and derivative parameters. The results are better than those obtained with MRAC controllers with one and two proportional adaptation parameters, adaptive controllers with Smith predictor to compensate the delay errors introduced by the systems and a Fuzzy controller. The validation of the control systems are implemented on a 32 bits hardware platform from Microchip and it is remotely accessed through the national RENATA network.

Tuning and noise attenuation of a second order system using Proportional Retarded control

AbstractThis paper presents a tuning strategy for Proportional Retarded (PR) control laws in closed loop with second order systems and experimental results on the noise attenuation performance of a DC servomotor. The PR controller is compared with other commonly employed strategies for avoiding the time-derivative measurement in Proportional Derivative control laws. The experiments show that the PR controller combines good noise attenuation and tracking performance with a simple implementation.

A Dynamic Positioning Method for Single Point Mooring System of a Catamaran

The main purpose of this paper is to solve the yawing problem of a single point mooring catamaran which designed for some special offshore operations. Dynamic characteristics of mooring system had been analyzed with the effect of wind, wave and current in early research. The results show that the yawing motion would be occurred for that system and can not be avoided without external control methods. These results are supported by sea tests of the real catamaran. Then, an azimuth thruster, set aft the ship, is used for a simple dynamic positioning (DP) system of the catamaran. The system is not a complete DP system because only one freedom can be controlled for the horizontal plane motions of the catamaran. The yawing angle is selected as control variable to meet the design requirement. The thrust is perpendicular to the longitudinal axis of the ship and can provide a maximizing yaw motion for DP system. Proportional-derivative control law is applied and the feedback variables can be obtained by global positioning system in further practices. The control effectiveness is validated by numerical results and comparisons with the non-controlled situations.