Stabilization and Disturbance Rejection for the Beam Equation

Cataloged from PDF version of article.We consider a system described by the Euler–Bernoulli beam equation. For stabilization, we propose a dynamic boundary controller applied at the free end of the system. The transfer function of the controller is a marginally stable positive real function which may contain poles on the imaginary axis. We then give various asymptotical and exponential stability results. We also consider the disturbance rejection problem

Implementation of Delayed-Feedback Controllers on Continuous Systems and Analysis of their Response under Primary Resonance Excitations

During the last three decades, a considerable amount of research has been directed toward understanding the influence of time delays on the stability and stabilization of dynamical systems. From a control perspective, these delays can either have a compounding and destabilizing effect, or can actually improve controllers\u27 performance. In the latter case, additional time delay is carefully and deliberately introduced into the feedback loop so as to augment inherent system delays and produce larger damping for smaller control efforts. While delayed-feedback algorithms have been successfully implemented on discrete dynamical systems with limited degrees of freedom, a critical issue appears in their implementation on systems consisting of a large number of degrees of freedom or on infinite-dimensional structures. The reason being that the presence of delay in the control loop renders the characteristic polynomial of the transcendental type which produces infinite number of eigenvalues for every discrete controller\u27s gain and time delay. As a result, choosing a gain-delay combination that stabilizes the lower vibration modes can easily destabilize the higher modes. To address this problem, this dissertation introduces the concept of filter-augmented delayed-feedback control algorithms and applies it to mitigate vibrations of various structural systems both theoretically and experimentally. In specific, it explores the prospect of augmenting proper filters in the feedback loop to enhance the robustness of delayed-feedback controllers allowing them to simultaneously mitigate the response of different vibration modes using a single sensor and a single gain-delay actuator combination. The dissertation goes into delineating the influence of filter\u27s dynamics (order and cut-off frequency) on the stability maps and damping contours clearly demonstrating the possibility of effectively reducing multi-modal oscillations of infinite-dimensional structures when proper filters are augmented in the feedback loop. Additionally, this research illustrates that filters may actually enhance the robustness of the controller to parameter\u27s uncertainties at the expense of reducing the controller\u27s effective damping. To assess the performance of the proposed control algorithm, the dissertation presents three experimental case studies; two of which are on structures whose dynamics can be discretized into a system of linearly-uncoupled ordinary differential equations (ODEs); and the third on a structure whose dynamics can only be reduced into a set of linearly-coupled ODEs. The first case study utilizes a filter-augmented delayed-position feedback algorithm for flexural vibration mitigation and external disturbances rejection on a macro-cantilever Euler-Bernoulli beam. The second deals with implementing a filter-augmented delayed-velocity feedback algorithm for vibration mitigation and external disturbances rejection on a micro-cantilever sensor. The third implements a filter-augmented delayed-position feedback algorithm to suppress the coupled flexural-torsional oscillations of a cantilever beam with an asymmetric tip rigid body; a problem commonly seen in the vibrations of large wind turbine blades. This research also fills an important gap in the open literature presented in the lack of studies addressing the response of delay systems to external resonant excitations; a critical issue toward implementing delayed-feedback controllers to reduce oscillations resulting from persistent harmonic excitations. To that end, this dissertation presents a modified multiple scaling approach to investigate primary resonances of a weakly-nonlinear second-order delay system with cubic nonlinearities. In contrast to previous studies where the implementation is confined to the assumption of linear feedback with small control gains; this effort proposes an approach which alleviates that assumption and permits treating a problem with arbitrarily large gains. The modified procedure lumps the delay state into unknown linear damping and stiffness terms that are function of the gain and delay. These unknown functions are determined by enforcing the linear part of the steady-state solution acquired via the Method of Multiple Scales to match that obtained directly by solving the forced linear problem. Through several examples, this research examines the validity of the modified procedure by comparing its results to solutions obtained via a Harmonic Balance approach demonstrating the ability of the proposed methodology to predict the amplitude, softening-hardening characteristics, and stability of the resulting steady-state responses

Power system dynamic security analysis via decoupled time domain simulation and trajectory optimization

Electric power systems are subject to disturbances in the operation, and may encounter system failures such as power outages and blackouts due to disturbances. Power system security analysis plays an important role in improving the survivability to disturbances. This dissertation proposes advanced computational and optimization techniques that can be applied to mitigate instabilities in power systems subject to disturbances. The research work has been integrated into a general framework for power system dynamic security analysis. The proposed methods cover strategies for both power system instability assessment and control, and provide a fast simulation algorithm and coordinated optimization techniques to improve power system security. In the assessment phase of power system security analysis, a fast algorithm is proposed to identify power system dynamic behavior using decoupled time domain simulation method. Traditional time domain simulation algorithms can be categorized as explicit and implicit methods. While explicit methods are fast, the simulation results cannot be guaranteed to be correct for stiff dynamical systems. On the other hand, implicit methods may give correct qualitative behavior with slow performance. As a hybrid method, the proposed decoupled method improves the computational efficiency and achieves numerical stability of time domain simulation by combining the advantages of traditional explicit and implicit methods, and the decomposition is accomplished through invariant subspace partition with rigorous mathematical analysis. In mitigation phase of system security analysis, a coordinated control strategy based on trajectory optimization is proposed. Power system dynamic performance is improved by the proposed method within the constraints imposed on system transition. In addition to the equilibrium conditions, inequality constraints in power system dynamics such as voltage level are considered in the formulation and solved through penalty function method. As one of the applications, power quality such as voltage dip in power system dynamics can be improved. Cascading events may also be prevented by including transitional constraints in the trajectory optimization. Numerical examples of test power systems are presented to demonstrate the applications of the proposed methods

Adaptive Vibration Control for an Active Mass Damper of a High-rise Building

As a kind of large flexible structure, high-rise buildings need to consider wind-resistant and anti-seismic problems for the safety of occupants and properties, especially in coastal areas. This paper proposes an infinite dimensional model and an adaptive boundary control law for an active mass damper(AMD) on this question. The dynamic model of the high-rise building is a combination of some storeys which have flexible walls and rigid floors under a series of physical conditions. Then the adaptive boundary controller is acted on an AMD which is equipped on the top floor, in order to suppress the vibration of every floor and guarantee the comfort of residents. Moreover, simulations and experiments are carried out on a two-floor flexible building to illustrate the effectiveness of the proposed control strategy

Modelling, system identification and control of a fibre optic accelerometer

A research report submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in partial ful lment of the requirements for the degree of Master of Science in Engineering. Johannesburg, 2015Control of systems are important in most industrial sectors, they nd applications in electronics, machine design and navigation. These control systems often use sensors to work e ectively. One such sensor is an accelerometer, which is used to measure acceleration with one or more degrees of freedom. This research report investigates the modelling, system identi cation and controller design for an accelerometer, a Fibre Optic Accelerometer (FOA). Such a device may be applied in many applications such as anti-skid control, structural failure in buildings and bridges, as well as strategic missile guidance. This report presents a model of a FOA demonstrator which crudely models an industrially developed accelerometer, the demonstrator is made of a jig consisting of a guitar string and electromagnets. Such a model needs to account for a distributed parameter beam combined with a permanent magnet and four electromagnets. The guitar string is modelled using three beam models, namely a spring/damper model, an Assumed Modes Model (ASM) and a Transfer Function Model (TFM). The parameters for these beam models are identi ed using the Nelder-Mead simplex algorithm and the least squares method. The electromagnets within the jig, are modelled using a mathematical model obtained through curve tting of experimental data. The overall FOA sensor is optimised using a lead-lag controller. Five cost functions where investigated, these cost functions are H1, Integral Square Error (ISE), Integral Absolute Error (IAE), Integral Time Square Error (ITSE) and Integral Absolute Time Error (IATE). It was found that the guitar string may be modelled using a single degree of freedom beam model. This is based on a number of reasons, such as the aperture size - through which the tip Light Emitting Diode (LED) projects, the tip mass (permanent magnet) - acting as a natural damper and the fact that Position Sensing Device (PSD) only measures the tip position. It was found that a single degree of freedom model in two orthogonal axes, with a single link beam spring/damper model was the most suitable representation of the guitar string. And the IAE lead-lag controller was found to be the most e ective in controlling a guitar string, this e ectiveness was due to least settling time.MT201

Dynamic modeling, property investigation, and adaptive controller design of serial robotic manipulators modeled with structural compliance

Research results on general serial robotic manipulators modeled with structural compliances are presented. Two compliant manipulator modeling approaches, distributed and lumped parameter models, are used in this study. System dynamic equations for both compliant models are derived by using the first and second order influence coefficients. Also, the properties of compliant manipulator system dynamics are investigated. One of the properties, which is defined as inaccessibility of vibratory modes, is shown to display a distinct character associated with compliant manipulators. This property indicates the impact of robot geometry on the control of structural oscillations. Example studies are provided to illustrate the physical interpretation of inaccessibility of vibratory modes. Two types of controllers are designed for compliant manipulators modeled by either lumped or distributed parameter techniques. In order to maintain the generality of the results, neither linearization is introduced. Example simulations are given to demonstrate the controller performance. The second type controller is also built for general serial robot arms and is adaptive in nature which can estimate uncertain payload parameters on-line and simultaneously maintain trajectory tracking properties. The relation between manipulator motion tracking capability and convergence of parameter estimation properties is discussed through example case studies. The effect of control input update delays on adaptive controller performance is also studied

Stabilization and disturbance rejection for the beam equation

We consider a system described by the Euler-Bernoulli beam equation. For stabilization, we propose a dynamic boundary controller applied at the free end of the system. The transfer function of the controller is a marginally stable positive real function which may contain poles on the imaginary axis. We then give various asymptotical and exponential stability results. We also consider the disturbance rejection problem

Modeling and Design Implications of Noncollocated Control in Flexible Systems,"

Introduction One of the most important problems in control system design for large flexible structures is simultaneously achieving high performance and robustness. Since there is a trade-off between robustness and model accuracy, accurate modeling of the structure is essential to successful control system design [1], Most existing control design methods for flexible system, such as the Independent Modal Space Control (IMSC) Noncollocated control systems, in contrast, lack these inherent stability characteristics. Here, inaccuracies resulting from model uncertainties and modal truncation present fundamental difficulties in both performance and stability. These limitations directly result from the fact that a noncollocated system is always nonminimum phase above some finite frequency. Nonminimum phase behavior is an inescapable result of the finite propagation speed of elastic deformation waves in the structure. Performance of nonminimum phase systems has inherent limitations; physical readability requirements on the compensator impose definite restrictions on achievable closed loop characteristic

Backstepping-Based Exponential Stabilization of Timoshenko Beam with Prescribed Decay Rate

This is an open access article under the CC BY-NC-ND license.In this paper, we present a rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using PDE backstepping. We introduce a transformation to map the Timoshenko beam states into a (2+2) × (2+2) hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the H1 sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters. Finally, a numerical simulation shows that the proposed controller can rapidly stabilize the Timoshenko beam. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate