Control of Flexible Structures Governed by the Wave Equation Using Infinite Dimensional Transfer Functions

2004 ◽  
Vol 127 (4) ◽  
pp. 579-588 ◽  
Author(s):  
Yoram Halevi
Keyword(s):  
Wave Equation ◽  
Transfer Function ◽  
Transfer Functions ◽  
Controller Design ◽  
Flexible Structures ◽  
Infinite Dimensional ◽  
Second Stage ◽  
Finite Dimensional ◽  
Control Scheme ◽  
Two Stages

A method of noncollocated controller design for flexible structures, governed by the wave equation, is proposed. First an exact, infinite dimension, transfer function is derived and its properties are investigated. A key element in that part is the existence of time delays due to the wave motion. The controller design consists of two stages. The first one is an inner collocated rate loop. It is shown that there exists a controller that leads to a finite dimensional plus delay inner closed loop, which is the equivalent plant for the outer loop. In the second stage an outer noncollocated position loop is closed. It has the structure of an observer-predictor control scheme to compensate for the response delay. The resulting overall transfer function is second order, with arbitrarily assigned dynamics, plus delay.

Control of Two-Link Flexible Structures

Volume 1 ◽  
2004 ◽  
Author(s):  
Clarice Wagner-Nachshoni ◽  
Yoram Halevi
Keyword(s):  
Transfer Function ◽  
Time Delays ◽  
Wave Motion ◽  
Controller Design ◽  
Flexible Structures ◽  
Infinite Dimensional ◽  
Second Stage ◽  
Finite Dimensional ◽  
Control Scheme ◽  
Two Stages

A method of noncollocated controller design for non-uniform flexible structures, governed by the wave equation, is proposed. An exact, infinite dimensional, transfer function, relating the actuation and measurement points, with general boundary conditions, is derived for the multi-link case. Three modeling methods are presented and discussed. A key element of the model is the existence of time delays, due to the wave motion, which play a major role in the controller design. The design consists of two stages. First an inner rate loop is closed in order to improve the system dynamic behavior. It leads to a finite dimensional plus delay inner closed loop, which is the equivalent plant for the outer loop. In the second stage an outer noncollocated position loop is closed. It has the structure of an observer–predictor control scheme to compensate for the response delay. The resulting overall transfer function is second order, with arbitrarily assigned dynamics, plus delay.

An adaptive PID-like controller for vibration suppression of piezo-actuated flexible beams

2017 ◽  
Vol 24 (12) ◽  
pp. 2656-2670 ◽  
Author(s):  
Teerawat Sangpet ◽  
Suwat Kuntanapreeda ◽  
Rüdiger Schmidt
Keyword(s):  
Vibration Suppression ◽  
Flexible Structures ◽  
Flexible Beam ◽  
Flexible Beams ◽  
Present Scheme ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Finite Dimensional ◽  
Control Scheme ◽  
The Stability

Flexible structures have been increasingly utilized in many applications because of their light-weight and low production cost. However, being flexible leads to vibration problems. Vibration suppression of flexible structures is a challenging control problem because the structures are actually infinite-dimensional systems. In this paper, an adaptive control scheme is proposed for the vibration suppression of a piezo-actuated flexible beam. The controller makes use of the configuration of the prominent proportional-integral-derivative controller and is derived using an infinite-dimensional Lyapunov method. In contrast to existing schemes, the present scheme does not require any approximated finite-dimensional model of the beam. Thus, the stability of the closed loop system is guaranteed for all vibration modes. Experimental results have illustrated the feasibility of the proposed control scheme.

scholarly journals Dynamic Characteristics in the Design of Maglev Suspensions

1994 ◽  
Vol 208 (1) ◽  
pp. 33-41 ◽  
Author(s):  
R Goodall
Keyword(s):  
Dynamic Characteristics ◽  
Transfer Functions ◽  
Controller Design ◽  
Design Method ◽  
Vehicle Speed ◽  
Passenger Comfort ◽  
Operational Conditions ◽  
Second Stage ◽  
Maglev Vehicle

The paper reviews the essential functions which apply to any kind of suspension, and distinguishes between the various inputs to which a suspension is subjected. These are used to assess the particular characteristics of an electromagnetically suspended (Maglev) vehicle, and to identify considerations which have important implications for the controller design, irrespective of the design method. Some general equations are developed which interrelate the vehicle speed, the quality of the track and the passenger comfort requirements, and these are used to identify operational conditions for which a second stage of suspension becomes necessary (that is, in addition to that provided by the magnets). The importance of understanding the suspension's response to deterministic track inputs is also highlighted. Although the paper is directed towards Maglev, the analysis is strongly based upon a consideration of the suspension transfer functions, and so many of the principles are applicable to actively controlled supensions in general.

scholarly journals Boundary Control for a Certain Class of Reaction-Advection-Diffusion System

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1854
Author(s):  
Eduardo Cruz-Quintero ◽  
Francisco Jurado
Keyword(s):  
Boundary Conditions ◽  
Boundary Control ◽  
Distribution Systems ◽  
Controller Design ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Finite Dimensional ◽  
Advection Diffusion

There are physical phenomena, involving diffusion and structural vibrations, modeled by partial differential equations (PDEs) whose solution reflects their spatial distribution. Systems whose dynamics evolve on an infinite-dimensional Hilbert space, i.e., infinite-dimensional systems, are modeled by PDEs. The aim when designing a controller for infinite-dimensional systems is similar to that for finite-dimensional systems, i.e., the control system must be stable. Another common goal is to design the controller in such a way that the response of the system does not be affected by external disturbances. The controller design for finite-dimensional systems is not an easy task, so, the controller design for infinite-dimensional systems is even more challenging. The backstepping control approach is a dominant methodology for boundary feedback design. In this work, we try with the backstepping design for the boundary control of a reaction-advection-diffusion (R-A-D) equation, namely, a type parabolic PDE, but with constant coefficients and Neumann boundary conditions, with actuation in one of these latter. The heat equation with Neumann boundary conditions is considered as the target system. Dynamics of the open- and closed-loop solution of the PDE system are validated via numerical simulation. The MATLAB®-based numerical algorithm related with the implementation of the control scheme is here included.

Closed-Loop Behavior of a Feedback-Controlled Flexible Arm: A Comparative Study

1991 ◽  
Vol 10 (3) ◽  
pp. 263-275 ◽  
Author(s):  
Sabri Cetinkunt ◽  
Wen-Lung Yu
Keyword(s):  
Mode Shape ◽  
Closed Loop ◽  
Transfer Functions ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Dimensional Model ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Shape Models ◽  
Free Mode

The dynamics of mechanical systems with distributed flexi bility are described by infinite-dimensional mathematical models. In order to design afinite-dimensional controller, a finite-dimensional model of the system is needed. The con trol problem of a flexible beam is a typical example. The general practice in obtaining a finite-dimensional model is to use modal approximation for distributed flexibility, retain a finite number of modes, and truncate the rest. In this approx imation, the appropriate selection of the mode shape func tions and the number of modes is not clearly known. Mostly standard pinned-free and clamped-free mode shapes are used for the flexible beam model, retaining only two or three modes and truncating the rest. The actual system, on the other hand, is infinite-dimensional, and the modes describing its flexible behavior under feedback control would be neither pinned-free nor clamped-free boundary condition modes. Rather, the mode shapes themselves are a function of the feedback control. The infinite-dimensional transcendental transfer functions for a flexible beam are formulated without any modal ap proximation. Finite-dimensional transfer functions with different shapes and numbers of modes are formulated. The closed-loop performance predictions of different models under the same colocated and noncolocated controllers, which attempt to achieve high closed-loop bandwidth, are compared. Results are surprisingly consistent in all cases; the predictions of clamped-free mode shape models are much more accurate than the predictions of the pinned-free mode shape models.

Active Control of Flexible Structures Subject to Distributed and Seismic Disturbances

1993 ◽  
Vol 115 (4) ◽  
pp. 649-657 ◽  
Author(s):  
Akira Ohsumi ◽  
Yuichi Sawada
Keyword(s):  
Active Control ◽  
Flexible Structures ◽  
Distributed Parameter ◽  
Dimensional System ◽  
Random Disturbance ◽  
Modal Expansion ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Real Earthquake ◽  
The Mathematical Model

The purpose of this paper is to present a method of active control for suppressing the vibration of a mechanically flexible cantilever beam which is subject to a distributed random disturbance and also a seismic input at the clamped end. First, the mathematical model of the flexible structure is established by a stochastic partial differential equation which describes the Euler-Bernoulli type distributed parameter system with internal viscous damping and subject to the seismic and distributed random inputs. Second, the distributed parameter model, which is considered as an infinite-dimensional system, is reduced to a finite-dimensional one by using the modal expansion, and split into the controlled part and the uncontrolled (residual) one. The principal approach is to regard the observation spillover due to uncontrolled part as a colored observation noise and construct an estimator, and then we construct the optimal control system. Finally, simulation studies are presented by using a real earthquake accelerogram data.

Rational transfer function models for nitrifying trickling filters

1999 ◽  
Vol 39 (4) ◽  
pp. 121-128 ◽  
Author(s):  
T. Wik
Keyword(s):  
Transfer Function ◽  
Dynamic Models ◽  
Transfer Functions ◽  
Controller Design ◽  
Ammonium Concentration ◽  
Trickling Filters ◽  
Rational Transfer ◽  
Simulated Effluent ◽  
Function Models ◽  
Transfer Function Models

An important step towards optimization and control of wastewater treatment plants is the development of dynamic models and efficient methods of simulation. Using standard simplifying assumptions, non-rational transfer function models describing the fast dynamics of nitrifying trickling filters, are derived. With a method based on the location of their singularities, it is shown how low order rational transfer functions can approximate the non-rational ones. These transfer functions can be used in efficient simulation routines and in standard methods of controller design. Effluent concentrations from trace substance pulse response experiments and an experiment with varying flow and varying influent ammonium concentration carried out on a large pilot plant NTF show close agreement with simulated effluent concentrations using the rational transfer functions.

Modeling and control of flexible link manipulators for unmodeled dynamics effect

2018 ◽  
Vol 233 (3) ◽  
pp. 245-263 ◽  
Author(s):  
Berk Altıner ◽  
Akın Delibaşı ◽  
Bilal Erol
Keyword(s):  
Controller Design ◽  
Flexible Structures ◽  
Lumped Parameter Model ◽  
Unmodeled Dynamics ◽  
Experimental Setup ◽  
Flexible Link ◽  
Linear Quadratic ◽  
Partial Differential ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems

Flexible link manipulators are mostly prefered in applications where energy consumption and faster operation are critically important. Since distributed nature of flexibility makes the system depend on not only time variable but also a spatial variable, the dynamics of flexible structures are expressed by partial differential equations. In the virtue of this kind of modeling, the designers encounter with infinite dimensional systems which means that the system has an infinite number of degrees of freedom. To cope with infinite dimensional systems, one of the most relevant techniques is to truncate the model into a definite order. However, this may yield the unmodeled dynamics that cause performance degradation and even instability. In this paper, the main motivation is to propose control techniques to compensate unwanted effects of unmodeled dynamics which may occur in truncation process. In order to achieve this goal, the linear quadratic Gaussian and the weighted [Formula: see text] controller design are adopted. The performances of the designed controllers are demonstrated on the experimental setup. Besides this motivation, traditional lumped parameter model of the flexible link manipulator which is widely seen in the literature is considered and the superiority of the partial differential equation model is shown on the experimental setup.

Modeling and Control of Acoustic Ducts

2000 ◽  
Vol 123 (1) ◽  
pp. 2-10 ◽  
Author(s):  
H. R. Pota ◽  
A. G. Kelkar
Keyword(s):  
Controller Design ◽  
Parametric Uncertainty ◽  
Theoretical Models ◽  
Unmodeled Dynamics ◽  
Modeling And Control ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Dimensional Models ◽  
And Control ◽  
Acoustic Ducts

This paper presents closed-form mathematical models for an acoustic duct with general boundary conditions. These infinite-dimensional models are derived using symbolic computations. A new method to obtain finite dimensional approximations of infinite-dimensional models using quartic functions is presented. The theoretical models are compared with the experimental data obtained for the KSU duct. The experimental results of a new robust broadband feedback controller, designed using passivity-based techniques, are presented. The controller design is shown to be robust to the unmodeled dynamics and parametric uncertainty.

Synchronization controller design for a network of boundary-controlled anti-stable wave equation with time-varying perturbations

2021 ◽  
pp. 014233122110232
Author(s):  
Jian Yang ◽  
Xiju Zong ◽  
Zhenzhen Chen ◽  
Shuying Yang
Keyword(s):  
Wave Equation ◽  
Controller Design ◽  
Feedback Controller ◽  
Time Varying ◽  
Backstepping Method ◽  
Infinite Dimensional ◽  
Active Disturbance Rejection ◽  
Disturbance Rejection Control ◽  
The Status ◽  
Virtual Leader

In this paper, a output feedback controller based on an infinite dimensional disturbance observer and a state feedback controller based on backstepping method are proposed to solve the synchronous control problem of network anti-stable wave equation with time-varying disturbance at the boundary. One agent in the network wave equation as the virtual leader, and all remaining agents need to track the status of the virtual leader incrementally. Here, the design of synchronous controller is divided into three parts. Firstly, backstepping method is used to design a set of controllers that makes all the systems stable. Secondly, infinite dimensional disturbance observers based on the idea of the active disturbance rejection control (ADRC) technology are used to estimate the disturbance. Finally, the synchronization controllers are designed, and the error between the following system and the virtual leader system converges to 0 in the appropriate sense. The applicability of the closed-loop system is analyzed and proved. The simulation results show the effectiveness of the controller design.

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