Modeling and control of coupled bending and torsional vibrations offlexible beams

An evolution equation which governs coupled bending and torsional vibrations of flexible beams with a tip body is derived based on Lagrangian dynamics. A feedback control scheme is shown for suppressing the vibrations     

Feedback control of flexible systems

Feedback control is developed for the class of flexible systents described by the generalized wave equation with damping. The control force distribution is provided by a number of point force actuators and the system displacements and/or their velocities are measured at various points. A feedback controller is developed for a finite number of modes of the flexible system and the controllability and observability conditions necessary for successful operation are displayed. The control and observation spillover due to the residual (uncontrolled) modes is examined and the combined effect of control and observation spillover is shown to lead to potential instabilities in the closed-loop system. Some remedies for spillover, including a straightforward phase-locked loop prefilter, are suggested to remove the instability mechanism. The concepts of this paper are illustrated by some numerical studies on the feedback control of a simply-supported Euler-Bernoulli beam with a single actuator and sensor.     

Feedback linearization control of flexible joint robots

In this paper, the control of a two link, flexible joint manipulator is examined. Among external forces, an exogenous constraint force acting on the end-effector is included. The manipulator dynamics are described by:

. On the assumption that f(x), g(x) and J^T f_e(x) are smooth vector fields, it is shown that the inner loop control u is of the form:

u=1/dT_n,g(v−dT_n,(ƒ(x)+J^Tƒ_e(x)))

where u is an outer loop control signal and y = T(x) is a diffleomorphism that transform (a) into linear system. As the position control scheme is adopted, the value of the contact force is not controlled.

The results for the inner loop control are substantiated by simulation of a two-link robot model. The robustness of the control method is examined and a Lyapunov-based control correction, similar to that of the free motion case, is implemented. Results are obtained for parametric errors of up to 50%. In the simulation, the manipulator is required to follow a specified joint trajectory such that the end-effector traces a sinusoidal path along a constraint surface. The results obtained illustrate the tracking of the link reference trajectory and indicate that the inner loop corrections are valid.     

Feedback control of a flexible cylindrical manipulator

Flexible robotic manipulators have been the subject of numerous studies in the past few years. Past studies, however, have concentrated on flexible manipulators with only revolute joints. This paper considers the case of a structurally flexible manipulator with prismatic and revolute joints; specifically, an algorithm for controlling a structurally flexible three-degree-of-freedom cylindrical manipulator is presented. The control algorithm involves two steps. Using nonlinear feedback, the equations of motion are first decoupled into three subsystems representing the three rigid degrees of freedom together with their associated ‘flexible equations’, if any. Then, using linear optimal control theory, controllers are designed for the three subsystems independently. Computer simulated results for an example system are presented.     

Nonlinear output‐feedback control of torsional vibrations in drilling systems

This paper considers the design of a nonlinear observer‐based output‐feedback controller for oil‐field drill‐string systems aiming to eliminate (torsional) stick–slip oscillations. Such vibrations decrease the performance and reliability of drilling systems and can ultimately lead to system failure. Current industrial controllers regularly fail to eliminate stick–slip vibrations under increasingly challenging operating conditions caused by the tendency towards drilling deeper and inclined wells, where multiple vibrational modes play a role in the occurrence of stick–slip vibrations. As a basis for controller synthesis, a multi‐modal model of the torsional drill‐string dynamics for a real rig is employed, and a bit–rock interaction model with severe velocity‐weakening effect is used. The proposed model‐based controller design methodology consists of a state‐feedback controller and a (nonlinear) observer. Conditions, guaranteeing asymptotic stability of the desired equilibrium, corresponding to nominal drilling operation, are presented. The proposed control strategy has a significant advantage over existing vibration control systems as it can effectively cope with multiple modes of torsional vibration. Case study results using the proposed control strategy show that stick–slip oscillations can indeed be eliminated in realistic drilling scenarios in which industrial controllers fail to do so. Moreover, key robustness aspects of the control system involving the robustness against uncertainties in the bit–rock interaction and changing operational conditions are evidenced. Copyright © 2017 John Wiley & Sons, Ltd.     

Feedback control of one-link flexible manipulator with gear train

This article reports experimental results in control of a one-link flexible manipulator. A d.c. drive with gear train rotates the link with a tip-mass in the horizontal plane. A “stiff” hardware servo-system tracks the commanded drive velocity. We describe the experimental setup and develop a dynamic model for the one-mode flexible vibration control. We design and experimentally test controllers for active damping of the vibrations and stabilization of the link-tip position. Results for continuous control implemented with an analog computer and a sampled-data, digital-computer control are reported. A high control performance is achieved despite the gear-train friction influence. In addition, sampled-data digital controllers for tracking control of the link reference motion are designed and tested.     

Feedback scheduling of model-based networked control systems with flexible workload

In this paper, a novel control structure called feedback scheduling of model-based networked control systems is proposed to cope with a flexible network load and resource constraints. The state update time is adjusted according to the real-time network congestion situation. State observer is used under the situation where the state of the controlled plant could not be acquired. The stability criterion of the proposed structure is proved with time-varying state update time. On the basis of the stability of the novel system structure, the compromise between the control performance and the network utilization is realized by using feedback scheduler.Examples are provided to show the advantage of the proposed control structure.     

Design and robust optimal control of smart beams with application on vibrations suppression

This paper presents the design of a vibration control mechanism for a beam with bonded piezoelectric sensors and actuators and an application of the arising smart structure for vibrations suppression. The mechanical modeling of the structure and the subsequent finite element approximation are based on Hamilton's principle and classical engineering theory for bending of beams in connection with simplified modeling of piezoelectric sensors and actuators. Two control schemes LQR and H₂ are considered. The latter robust controller takes into account uncertainties of the dynamical system and moreover incompleteness of the measured information, it therefore leads to applicable design of smart structures. The numerical simulation shows that sufficient vibration suppression can be achieved by means of the proposed general methods.     

Nonlinear free bending vibrations of plates

A general solution for the Helmholtz differential equations is obtained in the complex domain and applied to the nonlinear, free, bending vibrations of plates. The analysis is based on the decoupled nonlinear von Karman field equations by Berger assumption for the large deformations of plates. The decoupled differential equation in terms of the deflection function is a fourth order Helmholtz differential equation. Its solution, called the dynamic deflection function, is obtained in the complex domain by means of newly defined first and second kind and modified Bessel functions. The dynamic deflection function can be applied to any plates having any shape and any boundary condition under any arbitrary dynamic loads. For plates with smooth boundary, the parameters of the dynamic deflection function are determined from the boundary conditions of the plates and the initial conditions of the vibrations. The analyses of plates with piece-wise smooth boundaries are obtained on the mapped planes. The nonlinear, free vibration of circular plates are investigated by the dynamic deflection function. The effect of stretching on the natural circular frequencies are illustrated.     

Feedback control design with vibration suppression for flexible air-breathing hypersonic vehicles

This paper investigates the problem of feedback control design with vibration suppression for a flexible air-breathing hypersonic vehicle （FAHV）. FAHV includes intricate coupling between the engine and flight dynamics, as well as complex interplay between flexible and rigid modes, which results in an intractable system for the control design. In this paper, a longitudinal model, which is described by a coupled system of ordinary differential equations （ODEs） and partial differential equations （PDEs）, is adopted. Firstly, a linearized ODE model for the rigid part is established around the trim condition, while vibration of the fuselage is described by PDEs. Secondly, based on the Lyapunov direct method, a control law via ODE state feedback and PDE boundary output feedback is designed for the system such that the closed-loop exponential stability is ensured. Finally, simulation results are given to illustrate the effectiveness of the proposed design method.     

Optimal distribution of multimaterial composites for torsional beams

In this paper, we consider the optimal design of torsional beams using an arbitrary number of materials. The problem is initially ill-posed, and must be relaxed by the introduction of multimaterial composites. The optimization algorithm for multimaterial composites is described and computational results for both perturbations and asymptotical cases are presented. It is noticed that the case for three or more composites undergoes a phenomenon very much like a phase transition when certain conditions are satisfied.     

A torque estimator-based control strategy for oil-well drill-string torsional vibrations active damping including an auto-tuning algorithm

Considerable compliance of the lengthy oil-well drill string combined with low tool inertia and stick-slip friction between the tool and the rock bed causes notable undesired torsional vibrations of the drill-string electrical drive. In order to attenuate the torsional vibrations and, thus, improve both the quality and the productivity of the drilling process, an automatically tuned active damping control strategy based on drill-string torque estimation is proposed in the paper. The core of the strategy includes a proportional-integral (PI) controller extended with estimator-based drill-string torque feedback loop. Furthermore, an appropriate algorithm that prevents drill-string back-spinning caused by the limited braking power of the power converter is presented. Finally, an auto-tuning algorithm is proposed, which is built around an adaptive Kalman filter-based estimator of drill-string drive natural frequencies. The drill-string control strategy is verified experimentally using a drill string hardware-in-the-loop setup under the laboratory conditions, as well as on an actual oil-drilling rig.     

Singularity-robust decoupled control of dual-elbow manipulators

The ability of a robot manipulator to move inside its workspace is inhibited by the presence of joint limits and obstacles and by the existence of singular positions in the configuration space of the manipulator. Several kinematic control strategies have been proposed to ameliorate these problems and to control the motion of the manipulator inside its workspace. The common base of these strategies is the manipulability measure which has been used to: (i) avoid singularities at the task-planning level; and (ii) to develop a singularity-robust inverse Jacobian matrix for continuous kinematic control. In this paper, a singularity-robust resolved-rate control strategy is presented for decoupled robot geometries and implemented for the dual-elbow manipulator. The proposed approach exploits the decoupled geometry of the dual-elbow manipulator to control independently the shoulder and the arm subsystems, for any desired end-effector motion, thus incurring a significantly lower computational cost compared to existing schemes.     

Youla-Kučera design of decoupled control systems

A multivariable linear controller based upon pole placement design is developed from the family of all stabilizing controllers. In addition, decoupling behaviour is achieved. The decoupling conditions are established and guarantee the existence of a stabilizing controller that will give rise to a closed-loop transfer function matrix which is non-singular and diagonal. Further requirements may be imposed on the controller - here we discuss H 2 norm minimization.     

On decoupled or coupled control of bank-to-turn missiles

In this paper, the differences between decoupled and coupled control of bank-to-turn(BTT) missiles have been studied in detail from three aspects: stabilizability, linear quadratic regulator(LQR) control and tracking control. It is found that for some parameters the BTT missile can be controlled by a decoupled way,especially for stabilizability and LQR control. However, for some others, the BTT missile cannot be decoupled and a coupled control method is needed. Therefore, for a practical flight control system, the decouplability should be studied before giving a decoupled control method. In addition, in order to avoid the disadvantages of LQR control method, an improved H∞controller design method is given. Combined with tracking control problem, a new H∞control method is established for trajectory tracking.     

Optimal coupled and decoupled perimeter control in one-region cities

Perimeter controllers, located at a regional border, can manipulate the transfer flows across the border to optimize the regional operational performance. The macroscopic fundamental diagram (MFD), that relates average flow with accumulation, is used to model the traffic flow dynamics. In this paper, two cases of perimeter control inputs are considered: coupled and decoupled control. For both cases, the explicit formulations of the optimal feedback control policies and proofs of optimality are provided for three criteria. The proofs are based on the modified Krotov-Bellman sufficient conditions of optimality, where the upper and lower bounds of state variables are calculated.     

Design of decoupled non-linear multivariable stochastic control systems

A technique is presented for decoupling a non-linear system with input disturbance, The class of matrices which decouple the nominal system is first obtained. Then, the free parameters in the feedback matrix are selected to minimize the effect of disturbances on the output by minimizing a performance index which involves the input-output error magnitude. An example is presented to illustrate the approach.     

Nonlinear bending of beams using the finite element method

In this paper a finite element formulation for determining the finite deflection of thin bars is presented. The nonlinear stiffness equations are generated after simple approximate expressions involving the nodal parameters are used to replace the nonlinear terms in the energy functional. The procedure used results in a simplified set of nonlinear algebraic equations which are more amenable to solution than the equations usually presented. The applicability and accuracy of the method together with an evaluation of three incremental solution techniques, a step by step method, a one step Newton-Raphson procedure, and a variable interpolation technique is demonstrated by solving a cantilever beam with a point load acting on the end. Curves showing the sensitivity to increment size and to the number of elements are also presented. The results indicate that the formulation is accurate and inexpensive in terms of computational effort.     

Investigation of the torsion and bending effects on static stability of electrostatic torsional micromirrors

In this paper the electromechanical behavior of a torsional micromirror was investigated using of a static model with considering torsion and bending characteristics of micro-beams. A set of nonlinear equations based on the parallel plate capacitor model was derived to represent the relationships between the applied voltage, torsion angle, and vertical displacement of the torsional micromirror. Step by step linearization method (Newton’s method) was used to calculate the rotation angle and vertical displacement of the micromirror due to the applied voltage. This method is fast and gave acceptable and accurate results which were in good agreement with the experimental data.