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Advances in Acoustics and Vibration
Volume 2012 (2012), Article ID 852780, 7 pages
http://dx.doi.org/10.1155/2012/852780
Research Article

Self-Tuning Vibration Control of a Rotational Flexible Timoshenko Arm Using Neural Networks

1Department of Human and Information Systems Engineering, Faculty of Engineering, Gifu University, 1-1 Yanagido, Gifu 501-1193, Japan
2Department of Mechanical Engineering, Faculty of Engineering, Ibaraki University, 4-12-1 Nakanarisawachou, Hitachi 316-8511, Japan
3Department of Mechanical and Systems Engineering, Gifu University, 1-1 Yanagido, Gifu 501-1193, Japan
4School of Mechanical Engineering, Georgia Institute of Technology J. E. Love Jr. Manufacturing Building, 771 Ferst Drive, Atlanta, GA 30332-0405, USA

Received 22 October 2011; Revised 20 February 2012; Accepted 24 February 2012

Academic Editor: Akira Ikuta

Copyright © 2012 Minoru Sasaki et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A self-tuning vibration control of a rotational flexible arm using neural networks is presented. To the self-tuning control system, the control scheme consists of gain tuning neural networks and a variable-gain feedback controller. The neural networks are trained so as to make the root moment zero. In the process, the neural networks learn the optimal gain of the feedback controller. The feedback controller is designed based on Lyapunov's direct method. The feedback control of the vibration of the flexible system is derived by considering the time rate of change of the total energy of the system. This approach has the advantage over the conventional methods in the respect that it allows one to deal directly with the system's partial differential equations without resorting to approximations. Numerical and experimental results for the vibration control of a rotational flexible arm are discussed. It verifies that the proposed control system is effective at controlling flexible dynamical systems.

1. Introduction

In this paper, a self-tuning vibration control of a rotational flexible arm using neural networks is presented. In the past several decades, there has been increasing interest in dynamics and control of flexible structures across a broad spectrum of engineering disciplines [17]. A new and exciting idea has been attracting increasing attention in many engineering-related areas. The idea is that the performance of flexible structures can be greatly improved by the use of active control. Flexible structures arise in several important areas of applications, for instance, antenna control, robotics, and large space structures. Satisfactory control of these systems is hampered by many difficulties related to sensing and identification.

Several approaches are possible when using neural networks for the control of these kinds of plants [810]. Two distinct approaches have been used to control a plant adaptively. There are direct control and indirect control. In direct control, the parameters of the controller are directly adjusted to reduce some norm of the output error. In indirect control, the parameters of the plant are estimated as the elements of a vector at any instant, and the parameter vector of the controller is chosen assuming that it represents the true value of the parameter vector. A self-tuning gain control is one of the indirect approaches for using neural networks. In indirect control, neural networks are learning to get the optimal controller gain for tracking a desired trajectory [11].

To the self-tuning control system, the control scheme consists of gain tuning neural networks and a variable-gain feedback controller. The neural networks are trained so as to make the root moment zero. In the process, the neural networks learn the optimal gain of the feedback controller. The feedback controller is designed based on Lyapunov’s direct method. The feedback control of the vibration of the flexible system is derived by considering the time rate of change of the total energy of the system [1218]. This approach has the advantage over the conventional methods in the respect that it allows one to deal directly with the system’s partial differential equations without resorting to approximations.

Numerical and experimental results for the vibration control of a rotational flexible arm are presented. It verifies that the proposed control system is effective at controlling flexible dynamical systems.

2. Equation of Motions and Boundary Conditions

The rotational flexible arm depicted in Figure 1 of total length 𝐿, area moment of inertia 𝐼, cross-sectional area 𝐴, density 𝜌, Young’s modulus 𝐸, shear modulus 𝐺, and shear coefficient 𝜅 is attached at one end to a payload of mass 𝑚 and inertia 𝐼𝑐 and on the other end a hub of inertia 𝐼, which in turn is connected to an actuator that supplies a torque 𝑢. We obtain the following equations of motion with the boundary conditions:𝜕𝜌𝐴2𝑦𝜕𝑡2𝜕+𝑥2𝜗𝜕𝑡2𝜕𝜅𝐺𝐴2𝑦𝜕𝑥2𝜕𝜙𝜕𝜕𝑥=0,(1)𝜌𝐼2𝜙𝜕𝑡2+𝜕2𝜗𝜕𝑡2𝜕𝐸𝐼2𝜙𝜕𝑥2𝜅𝐺𝐴𝜕𝑦𝐼𝜕𝑥𝜙=0,(2)𝜕2𝜗𝜕𝑡2=𝐸𝐼𝜕𝜙(𝑡,0)𝑚𝜕𝜕𝑥+𝑢,(3)2𝜕𝑡2(𝑦(𝐿,𝑡)+𝐿𝜗)=𝜅𝐺𝐴𝜕𝑦(𝐿,𝑡)𝐼𝜕𝑥𝜙(𝐿,𝑡),(4)𝑐𝜕2𝜕𝑡2(𝜙(𝐿,𝑡)+𝜗)=𝐸𝐼𝜕𝜙(𝐿,𝑡)𝜕𝑥,(5)𝑦(0,𝑡)=0,(6)𝜙(0,𝑡)=0.(7) If the shear deformation and rotary inertia can be neglected, the model of the flexible arm is presented by the Euler-Bernoulli beam model [5].

852780.fig.001
Figure 1: Rotational flexible arm.

Taking the Laplace transform of (1)–(7), we get the following:𝜌𝐴𝑠2𝜕(𝑌+𝑥Θ)𝜅𝐺𝐴2𝑌𝜕𝑥2𝜕Θ𝜕𝑥=0,(8)𝜌𝐼𝑠2𝜕(Φ+Θ)𝐸𝐼2Φ𝜕𝑥2𝜅𝐺𝐴𝜕𝑌𝐼𝜕𝑥Φ=0,(9)𝑠2Θ=𝐸𝐼𝜕Φ(𝑠,0)𝜕𝑥+𝑈,(10)𝑚𝑠2(𝑌(𝐿,𝑠)+𝐿Θ)=𝜅𝐺𝐴𝜕𝑦(𝐿,𝑠)𝐼𝜕𝑥Φ(𝐿,𝑠),(11)𝑐𝑠2(Φ(𝐿,𝑠)+Θ)=𝐸𝐼𝜕Φ(𝐿,𝑠)𝜕𝑥,(12)𝑌(0,𝑠)=0,(13)Φ(0,𝑡)=0.(14)

3. Stabilizing Feedback Control

The problem is to find feedback control 𝑢 such that the arm’s total energy given by𝑉=𝑎12𝐿0𝜌𝐴𝜕𝑦𝜕𝑡+𝑥𝜕𝜃𝜕𝑡2+𝜌𝐼𝜕𝜙+𝜕𝑡𝜕𝜃𝜕𝑡2+𝑎𝑑𝑥22𝐼𝜕𝜃𝜕𝑡2+𝑎32𝐿0𝐸𝐼𝜕𝜙𝜕𝑥2+𝜅𝐺𝐴𝜙𝜕𝑦𝜕𝑥2𝑑𝑥.(15) The time rate of change of 𝑉 is given by𝜕𝑉𝜕𝑡=𝑎1𝐿0𝜌𝐴𝜕𝑦𝜕𝑡+𝑥𝜕𝜃𝜕𝜕𝑡2𝑦𝜕𝑡2𝜕+𝑥2𝜃𝜕𝑡2+𝜌𝐼𝜕𝜙+𝜕𝑡𝜕𝜃𝜕𝜕𝑡2𝜙𝜕𝑡2+𝜕2𝜃𝜕𝑡2𝑑𝑥+𝑎2𝐼𝜕𝜃𝜕𝜕𝑡2𝜃𝜕𝑡2+𝑎3𝐿0𝐸𝐼𝜕𝜙𝜕𝜕𝑥2𝜙𝜕𝑥𝜕𝑡+𝜅𝐺𝐴𝜙𝜕𝑦𝜕𝑥𝜕𝜙𝜕𝜕𝑡2𝑦𝜕𝑥𝜕𝑡𝑑𝑥.(16) Substituting (1) into (2) reduces to𝜕𝑉𝜕𝑡=𝑎1𝐿0𝜕𝑦𝜕𝑡+𝑥𝜕𝜃𝜕𝜕𝑡𝜅𝐺𝐴2𝑦𝜕𝑥2𝜕𝜙+𝜕𝑥𝜕𝜙+𝜕𝑡𝜕𝜃𝜕𝜕𝑥𝐸𝐼2𝜙𝜕𝑥2𝜅𝐺𝐴𝜙𝜕𝑦𝜕𝑥𝑑𝑥+𝑎2𝜕𝜃𝜕𝑡𝐸𝐼𝜕𝜙(0,𝑡)𝜕𝑥+𝑢+𝑎3𝐿0𝐸𝐼𝜕𝜙𝜕𝜕𝑥2𝜙𝜕𝑥𝜕𝑡+𝜅𝐺𝐴𝜙𝜕𝑦𝜕𝑥𝜕𝜙𝜕𝜕𝑡2𝑦𝜕𝑥𝜕𝑡𝑑𝑥.(17) Equation (17) is then integrated by parts, and boundary conditions (3)–(7) are substituted into the resulting equation. The result is𝜕𝑉=𝜕𝑡𝜕𝜃𝑎𝜕𝑡2𝑎𝑢+3𝑎1𝐸𝐼𝜕𝜙(𝑡,0)𝜕𝑥.(18) Taking into consideration of the fact that there is a feedback control law which gives nonpositive 𝜕𝑉/𝜕𝑡 is𝑢=𝐾1𝜕𝜃𝜕𝑡𝐾2𝐸𝐼𝜕𝜙(𝑡,0)𝐾𝜕𝑥1,𝐾2>0.(19) Substituting (19) into (18) leads to𝜕𝑉𝜕𝑡=𝐾𝜕𝜃𝜕𝑡20(20) which is negative semidefinite. This implies that the closed-loop system is energy dissipative and, hence, stable. This control law is elegant. Notice that the rigorous stability proof does not depend on introducing spatial discretization methods. Of important practical consequence, notice that controllers based on this law are easy to implement since no state estimation is required. The bending moment can be measured by using conventional strain gauges. However, the tuning of a feedback gain needs to be addressed.

4. Self-Tuning Neural Controller

The controllers are often poorly tuned. The reason is that it is difficult to tune a parameter by trial and error. We can embed algorithms inside computers that “learn from experience” and self-tune the controllers so as to improve closed-loop performance [16]. Often this learning process builds up a mathematical model based on experimental input/output data; this operation is known as system identification or parameter estimation. The model could be a complete transfer function or simply the gain and phase of the plant at a gain input frequency. A full process model is estimated using system identification methods, and an analytic design procedure uses the model to self-tune the coefficients of a fixed control law. A self-tuning gain control is one of the indirectly approaches for using a neural networks [5]. In indirectly control as shown in Figure 2, neural network is learning to get the optimal controller gain for suppressing vibration of the flexible arm. The control scheme consists of a gain tuning neural network and a variable-gain feedback controller. In the process, the neural network learns the optimal gain of the feedback controller. The backpropagation method is a gradient descent method that establishes the weight in a multilayer, feedforward adaptive neural network. Learning is accomplished by successively adjusting the weight based on a set of input patterns and a corresponding set of desired output patterns. During this iterative process, an input pattern is presented to the network and propagated forward to determine the resulting signal at the output units. The differences between the actual resulting output signal and the predetermined desired output signal in each output unit represents an error that is backpropagated through the network in order to adjust the weights. The learning process continues until the network responds with an output signal the sum of whose root-mean square errors from the desired output signals are less than a preset value. The training process using backpropagation is a difficult process. It is necessary to find an appropriate architecture, adequate size and quality of training data, satisfactory initialization, learning parameter values, and to avoid overtraining effects. To speed up the convergence behavior, the selection of parameters such as the learning rates is done by using the utilization of a momentum factor. The learning rule utilized consists of a weight update using momentum 𝛼𝑐 with the exception that each weight has its own learning rate parameter 𝜂𝑐. To minimize the cost function, the updating equation of the weights is defined byΔ𝑤𝑟𝑝𝑞(𝑡)=𝜂𝑐𝜕𝐸𝜕𝑤𝑟𝑝𝑞(𝑡)+𝛼𝑐Δ𝑤𝑟𝑝𝑞(𝑡1),(21) where 𝑤𝑟𝑝𝑞 is the weight value at 𝑟th layer located between nodes 𝑝 and 𝑞, 𝑡 is the present iteration, and 𝑤𝑝𝑞(𝑡) is the weight increment which is equal to the product of 𝜂𝑐 and the partial derivative of the objective function with respect to the weight, that is,𝑤𝑟𝑝𝑞(𝑡+1)=𝑤𝑟𝑝𝑞(𝑡)Δ𝑤𝑟𝑝𝑞(𝑡),(22) where1𝐸=2𝑎𝑥(𝑡)2+12𝑏𝑒(𝑡)2,𝑒(𝑡)=𝑢(𝑡)𝐾𝑥(𝑡),(23) where 𝑥(𝑡) is the output moment, 𝑢(𝑡) is the control input, and 𝑎 and 𝑏 are the weighting coefficients.

852780.fig.002
Figure 2: Block diagram of the proposed control system.

According to the generalized 𝛿-rule, neural network learning is performed for each sampling using (21) to minimize the cost function 𝐸. The momentum 𝛼𝑐 is changed dynamically, because each problem has a range of optimal 𝛼𝑐 values to avoid oscillations.

5. Numerical Simulation and Experimental Results and Discussion

The physical parameters of the system are presented in Table 1. A multilayer feedforward network, consisting of three layers with no inner feedback loop, was used for the self-tuning feedback controller. The size of the neural-network is defined as the minimum size in which the weights between the neurons are such that the neural-network’s output matches with the optimal feedback gain; however, it is hard to define that analytically. In this study, the input layer has 6 neurons, the hidden layer has 6 neurons, and the output layer has 1 neurons. Neurons in the input layer represent the reference signals 𝑢(𝑡) and 𝑢(𝑡1), the control signals 𝑒(𝑡) and 𝑒(𝑡1), the feedback gain 𝐾, and the plant outputs 𝑥(𝑡) and 𝑥(𝑡1). The neuron in the output layer represents the feedback gain. The input neuron activation function was assumed to be the linear function 𝑓(𝑥)=𝑥, and all the thresholds were assumed to be zero. To allow for the nonlinear effect, the sigmoid function 𝑓(𝑥)=1/{1+exp(𝑥)} was used to the hidden layer neurons. The output neuron activation function was assumed to be a function 𝑓(𝑥)=ln{1+exp(𝑥)}. The neural network controller design has been carried out on the MATLAB platform. The desired tip acceleration is widely accepted that bang-bang acceleration profiles lead to time optimal trajectories for the case of rigid body arm. We set the desired rise time and displacement from initial point to the desired end-point to 1.0 [sec] and 1.15 [m]. Figure 3 shows the gain variation of the self-tuning controller. Figure 4 shows the time response of the system. Figure 5 shows the learning curve of the self-tuning control system. Figure 6 shows the time response of the self-tuning control system for the flexible arm with best parameter values of 𝛼𝑐 and 𝜂𝑐. The best parameter value of 𝛼𝑐 is 0.6 and 𝜂𝑐 is 0.8. The weighting constants 𝑎 and 𝑏 are fixed to 1 and 2.6, respectively. It can be clearly seen that the vibration suppressing performance of the self-tuning control system using neural networks is better than that without control.

tab1
Table 1: System parameters.
852780.fig.003
Figure 3: Gain variations of the self-tuning controller.
852780.fig.004
Figure 4: Time response of the tip displacement.
852780.fig.005
Figure 5: Learning curve of the self-tuning system.
852780.fig.006
Figure 6: Time response of the root moment.

In the experiment, same neural network is used as the numerical simulation. For the implementation of fast controllers, we routinely use the TMS320C31-based digital signal processing system (DSP-CIT) along with a set of design and implementation software tools, including an automatic code generator. The design in the analogue domain was carried out, and the controller was discretized, after checking for the effects of discretization, computational delays, AD- and DA quantization, the signal processor code was generated and downloaded. The sampling period was 10 [ms]. Figure 7 is the experimental setup for implementation and assessment of the neural network control system. Figure 8 shows the experimental results of the root moment. Figure 9 shows the input voltage of the motor. This control system can suppress the vibrations of the flexible arm within a short time in comparison with no moment feedback. From these results it is concluded that using the neural network may be possible in some cases to tune an optimal gain of the feedback controller.

852780.fig.007
Figure 7: Experimental setup.
852780.fig.008
Figure 8: Time response of the root moment.
852780.fig.009
Figure 9: Input voltage of the motor.

6. Conclusions

A self-tuning control system of a rotational flexible arm using neural networks was presented. The neural networks using new cost function learned optimal feedback gain to avoid excessive control input. Numerical and experimental results show that the proposed controller is useful for vibration control of the rotational flexible arm.

Appendix

In the case of the rotational Timoshenko arm, the resulting solution to (8), (9), and (10) is given by𝑌(𝑥,𝑠)=Δ𝑘1Δ𝑒𝜆1𝑥+Δ𝑘2Δ𝑒𝜆2𝑥Δ𝑘3Δ𝑒𝜆1𝑥Δ𝑘4Δ𝑒𝜆2𝑥𝜆𝑥Θ,Φ(𝑥,𝑠)=1𝜌𝑠2𝜆1𝑘𝜅𝐺1𝑒𝜆1𝑥𝑘3𝑒𝜆1𝑥+𝜆2𝜌𝑠2𝜆2𝑘𝜅𝐺2𝑒𝜆2𝑥𝑘4𝑒𝜆2𝑥Θ,(A.1) where𝑎11=𝑚𝑠2+𝜌𝐴𝑠2𝜆1𝑒𝜆1𝐿,𝑎12=𝑚𝑠2+𝜌𝐴𝑠2𝜆2𝑒𝜆2𝐿,𝑎13=𝑚𝑠2𝜌𝐴𝑠2𝜆1𝑒𝜆1𝐿,𝑎14=𝑚𝑠2𝜌𝐴𝑠2𝜆2𝑒𝜆2𝐿,𝑎21=𝜆1𝜌𝑠2𝜅𝐺𝜆1𝐼𝑐𝑠2+𝐸𝐼(1+𝐶𝑠)𝜆1𝑒𝜆1𝐿,𝑎22=𝜆2𝜌𝑠2𝜅𝐺𝜆2𝐼𝑐𝑠2+𝐸𝐼(1+𝐶𝑠)𝜆2𝑒𝜆2𝐿,𝑎23𝜆=1𝜌𝑠2𝜅𝐺𝜆1𝐼𝑐𝑠2𝐸𝐼(1+𝐶𝑠)𝜆1𝑒𝜆1𝐿,𝑎24𝜆=2𝜌𝑠2𝜅𝐺𝜆2𝐼𝑐𝑠2𝐸𝐼(1+𝐶𝑠)𝜆2𝑒𝜆2𝐿,𝑎31=𝑎32=𝑎33=𝑎34𝑎=1,41=𝐼𝑠2𝜆1𝜌𝑠2𝜅𝐺𝜆1𝜆𝐸𝐼(1+𝐶𝑠)21𝜌𝑠2,𝑎𝜅𝐺42=𝐼𝑠2𝜆2𝜌𝑠2𝜅𝐺𝜆2𝜆𝐸𝐼(1+𝐶𝑠)22𝜌𝑠2,𝑎𝜅𝐺43𝐼=𝑠2𝜆1𝜌𝑠2𝜅𝐺𝜆1𝜆+𝐸𝐼(1+𝐶𝑠)21𝜌𝑠2,𝑎𝜅𝐺44𝐼=𝑠2𝜆2𝜌𝑠2𝜅𝐺𝜆2𝜆+𝐸𝐼(1+𝐶𝑠)22𝜌𝑠2,𝑎𝜅𝐺Δ=11𝑎12𝑎13𝑎14𝑎21𝑎22𝑎23𝑎24𝑎31𝑎32𝑎33𝑎34𝑎41𝑎42𝑎43𝑎44,Δ𝑘1=0𝑎12𝑎13𝑎140𝑎22𝑎23𝑎240𝑎32𝑎33𝑎34𝑈𝑎42𝑎43𝑎44,Δ𝑘2=𝑎110𝑎13𝑎14𝑎210𝑎23𝑎24𝑎310𝑎33𝑎34𝑎41𝑈𝑎43𝑎44,Δ𝑘3=𝑎11𝑎120𝑎14𝑎21𝑎220𝑎24𝑎31𝑎320𝑎34𝑎41𝑎42𝑈𝑎44,Δ𝑘4=𝑎11𝑎12𝑎130𝑎21𝑎22𝑎230𝑎31𝑎32𝑎330𝑎41𝑎42𝑎43𝑈,1𝜆=±21+1𝐸(1+𝐶𝑠)𝜅𝐺𝜌𝑠2±𝜌2𝑠4𝐸2(1+𝐶𝑠)22𝜌2𝑠4𝐸+𝜌(1+𝐶𝑠)𝜅𝐺2𝑠4𝜅2𝐺24𝜌𝐴𝑠2𝐸𝐼(1+𝐶𝑠)1/21/2,𝜆=±𝜆1,±𝜆2.(A.2)

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