Modeling Flexible Plate Structure System with Free-Free-Clamped-Clamped (FFCC) Edges using Particle

Swarm Optimization

Muhammad Sukri Hadi, Intan Z. Mat Darus and Hanim Yatim Department of Applied Mechanics, Faculty of Mechanical Engineering

Universiti Teknologi Malaysia Johor Bahru, Malaysia

intan@fkm.utm.my

Abstract—This paper presents the performance of system identification for rectangular flexible plate using conventional parametric modeling approach by Recursive Least Squares (RLS) and intelligent parametric modeling approach by Particle swarm Optimization (PSO). The experimental rig of rectangular flexible plate with free-free-clamped-clamped edges boundary condition designed and fabricated in this research. The experimental study was conducted to collect the input-output data using a vibration flexible plate complete with data acquisition and instrumentation system. The input-output data will be used during developed the system identification. The whole developed model using RLS and PSO were validated using mean squares error (MSE), one step ahead prediction (OSA) and correlation test. The estimated of developed models was found are comparable, acceptable and possible to be used as a platform of controller development later on. As a comparison between developed system, it was found that the Particle Swarm Optimization algorithm has perform better in term of the lowest mean squares error which is 0.00032719.

Keywords-Recursive least squares; Particle swarm optimization; Rectangular flexible plate struture; System identification

I. INTRODUCTION Recently, the characteristic posses by the flexible plate

structure have attracted various engineering fields such as marine and aerospace industries to employ the flexible plate. The engineering field currently has focused on how to establish a more reliable, lightweight and efficient flexible structure. The flexible plate used in the industries today is thinner, lighter and larger than the previously used.

However, the vibration is a major problem when lightweight flexible structure is used in the engineering applications because the lighter flexible structure will easily be influenced by the unwanted vibrations. This vibration may lead to structural fatigue, reduction on the performance, instability and human discomfort. Hence, controlling the unwanted vibrations of the flexible structures is needed [1,2].

Finding the appropriate and accurate model of the flexible plate is important in order to control the unwanted vibrations properly. An appropriate model is a key to design the best

controller to suppress the vibration on the flexible plate. System identification is a very robust method on finding the model of the system. To characterize the dynamic of a system using identification technique, input and output responses of the system to be modelled is required. Although, the method has been used by many researchers in many engineering application, a lot of attention are still being received nowadays. The main purpose of system identification developed is to find the appropriate model of a system based on the input and output data was observed [3].

The experimental study was conducted to acquire the input-output data of the system. In this research, the rectangular flexible plate experimental rig with free-free-clamped-clamped (FFCC) boundary condition was used. The flexible plate are clamped at the right and left side while the upper and bottom sides are free to vibrate. The external force was applied on the experimental rig to generate vibration responses which were used as input and output data for identification of the system.

The procedure of system identification started by obtaining input and output data set of the flexible plate structure. Next, the best model structure and its order to represent the system are presented. Then, the model of the system is estimated using several identification techniques. The validation of the system developed will be carried out using one step-ahead (OSA) prediction, mean squares error (MSE) and correlation test [3]. The system identification of rectangular flexible plate in this research developed using conventional parametric modeling approach by Recursive Least Squares (RLS) and intelligent parametric modeling approach by Particle swarm Optimization (PSO).

II. EXPERIMENTAL STUDY The experimental setup as shown in Figure 1 is used to

obtain the input-output data set of flexible plate. Through experimental studies, National Instruments (NI) data acquisition system was used to acquire the input-output of the flexible plate system. The structure used in the experiment are rectangular, thin and flat aluminium plate with the dimension 60 cm × 30 cm × 0.035 cm is considered. The dimensions of the plates are 66 cm x 30 cm x 0.035 cm. In this experimental setup, the flexible plate was clamped at both left and right sides

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while the upper and bottom sides was let free to vibrate. The details of the flexible plate specifications used in this research are listed in the Table I.

Figure 1. Experimental setup to acquire the input-output data.

TABLE I. THE SPECIFICATIONS OF FLEXIBLE PLATE.

No. Parameters Value

1. Length of the plate, (a) 60 cm

2. Width of the plate, (b) 30 cm

3. Density, (ρ) 2690 kgm-3

4. Poison ratio, (v) 0.3

5. Thickness, (t) 0.035 cm

6. Modulus of elasticity, (E) 6.83 x 1010 Nm-2

An actuation force was applied on the experimental rig during conducting the experiment at the excitation point of flexible plate. A piezoelectric patch actuator P876-A12 was used in this study in purpose to give a vibration on the experimental rig. Piezoelectric actuator is made of piezoelectric materials that when subjected to voltage, they show an alternation in length which can be employed to exert forces to the hosted structure. While, Function Generator was employed to create sinusoidal signal in process of creating an actuation force. The sinusoidal signal with amplitude 10 Vpp, frequency at 50 Hz was amplified at the piezoelectric patch connected to the amplifier generated an actuation force to the experimental rig [3].

The choices of usage sensor and actuator used in the experiment need to be considered carefully. The total mass of sensor and actuator attached on the test experimental rig cannot be exceed 1/10 of the mass flexible plate used in this study. The characteristics of experimental rig will be distorted if the total mass used are exceed 1/10 of the mass flexible plate. The mass of accelerometer used at the detection point (S1) and observation point (S2) was 5 gram each while the mass of flexible used is 186.42 gram. Hence, the choices of sensors used in this study not exceed the mass of flexible plate used as follows [5]:

S1 + S2 < 101

M1 (1)

Two pieces of piezo-beam type accelerometer was used in purpose to acquire the acceleration signal in the experiment. Accelerometer directly connected with National Instruments (NI) data acquisition system that consists of NI compact-data acquisition unit (NI cDAQ-9172) equipped with NI-9234 module. NI 9234 is a four channel C Series dynamic signal acquisition module for making high accuracy frequency measurements. The four input channels simultaneously digitize signals at rates up to 51.2 kHz per channel with built-in antialiasing filters that automatically adjust to its sampling rate. Intel CoreTM Duo Processor and LabVIEW software was used to analyse the required signal obtained in the experiment [3]. Figure 2 shows the layout for experimental setup in this research.

Figure 2. Experimental setup layout [2].

III. SYSTEM IDENTIFICATIONS In this study, the parametric modeling was conducted via

conventional and evolutionary algorithms using linear ARX model structure. These include conventional Recursive Least Square (RLS) and evolutionary Particle Swarm Optimization (PSO) techniques.

A. Recursive Least Square (RLS)

Recursive least square (RLS) is one of the conventional parametric modeling based on the mathematical weighted LS criterion which is provide the best fit estimate for a set static data. In this system, a set of unknown parameters need to estimates during real time operation to provide the accurate modeling. RLS algorithm also presents the continuous estimate for a set unknown systems parameters and one of examples an adaptive filtering algorithm [6,7].

)()()1()( iEiKii (2)

)()1()(1

)()1()( 1

1

ixiPixixiPiK T

(3)

)1()()()1()( 11 iPixiKiPiP T (4)

Flexible Plate Accelerometer (Kistler-8636C50)

Accelerometer (Kistler-8636C50

Piezoelectric Patch Actuator

P 876 A12

NI Compact Data Acquisition System type 9172 with NI-9234

Personal Computer with installation of LabVIEW

E-835 OEM Amplifier

Function Generator TG1010A

Programmable

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)1()()()( iixiyiE T (5)

where )(i is current parameter estimation, )1(i is previous estimation parameters, is forgetting factor )(ix is regression vector and )(iy is output system.

B. Particle Swarm Optimization (PSO)

The social behavior of social organisms represents successful animal team behavior such as swarm or flocking intelligence was inspired Kennedy and Elberhart to propose intelligence algorithm of Particle Swarm Optimization (PSO) in 1995. The model is a stochastic and population-based swarm intelligence algorithm. PSO was inspired from the social behavior of organisms such as bird flocks and fish schooling. These last few years, optimization method using PSO become very popular, due to the number of advantages possess by this method as compared to other optimization methods. Some of the advantages are it can easily programme and possess only a few parameters to be optimized [8].

In the PSO algorithm, firstly, the number of particles are randomly initialized. The particle flies through the space problem. Every particle will remember the best position it has seen. The position of each particle is represented with i-th particle in d-dimensional vector in problem space as Xi = (xi1, xi2,…,xid) where i = 1, 2,..n (n is number of particles). The velocity vector of ith particle vi=(vi1 ,vi2,…,vid) defined as the change of its position. The algorithm completes the optimization through following the personal best solution of each particle and the global best value of the whole swarm. The particles will use their memories of their best positions to adjust their own velocities and positions. Here, the particles will always flies through the problem space tend towards the better areas than the previous best position. The best fitness value represents as Pi(pi1,pi2,….pid) and the fittest particle found at time t represents as Pg(pg1,pg2,….pgd). The velocities and positions of particles are updated using the following formulas [9, 10]:

vid (t +1) = wvid (t)+c1 rand1 (pid - xid (t))

+c2 rand2 (pid - xid (t)) (6)

xid (t +1) = xid (t) + vid (t +1) (7)

where t is number of iterations; c1 and c2 are acceleration coefficients; rand1 and rand2 are random number in range [0, 1]; w is the inertia weight. In order to reduce the weight over number of iterations, the inertia weight, w is updated following the equation shows below [9]:

w = max

minmaxmax iter

www × t (8)

where wmax , wmin are the maximum and minimum values of inertia weight while itermax is the maximum number of iterations. Figure 3 shows the computational flowchart of

Particle Swarm Optimization algorithm developed in this study [10].

Figure 3. Computational flowchart of PSO Algorithm

IV. EXPERIMENTAL RESULTS

A rectangular aluminum plate was used with free-free-clamped-clamped boundary condition during conducting the experimental procedures to acquire the input-output data of the flexible plate structure. Data acquisition and instrumentation system were designed and integrated with the experimental rig. The input-output data were acquired by disturbing the flexible plate with a sinusiodal force using piezoelectric patch actuator. Then, two pieces of piezo beam types accelerometer were used to acquire acceleration signals.

Figure. 4 shows the locations of sensor and actuator on the rig. Point (X) is the location of the sensor which was used as the primary source of vibration and also known as excitation point. While, point (Y) and point (Z) are locations for the actuators. Point (Y) was used to sense the detection signal and point (Z) was used to sense the observation signal. As stated by Tavakolpour et. al. the locations of sensor and actuator are to be chosen far away from the nodal lines which were defined by the natural frequencies of the flexible plate [11].A sine wave with amplitude 10 Vpp and frequency of 50 Hz was applied as primary source of vibration at point (X) located at x=15 cm and y=7.5 cm as shown in Figure. 4. Figures. 5 and 6 are the lateral deflection at the detection and observation points.

A. Analytical Results for Mode of Vibrations The analytical calculation for the mode of vibrations for

flexible plate with free-free-clamped-clamped edges boundary condition is described in equation (9) [12, 13]. The detail specifications of the flexible plate used in this research are listed in Table I. The values of constant 2 and the natural

Start

Specify the parameters for PSO

Generate initial population

Time-domain simulation

Find the fitness of each particle in the current population

Update the particle position and velocity using equations (6) & (7)

Gen>Max. Gen.? Stop

Gen. = Gen. +1

No

Yes

Gen. = 1

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frequency for different mode of vibration are depicted in Table II.

njivy

Eha

f ijij ....3,2,1,

)1(122 2

3

2

2

(9)

Figure 4. Position of sensor and actuator on the flexible plate.

Figure 5. Lateral deflection at the detection point

Figure 6. Lateral deflection at the observation point.

TABLE II. VALUE OF CONSTANT 2 AND NATURAL FREQUENCY FOR DIFFERENT MODE OF VIBRATION

Mode 1 2 3 4 5

2 22.17 36.30 61.15 81.67 97.08

Frequency (Hz) 5.23 8.563 14.424 19.26 22.90

V. RESULTS AND DISCUSSIONS The vibration of flexible plate structure with free-free-clamped-clamped edges boundary condition was modeled

using parametric modeling approaches via conventional LS and evolutionary PSO algorithm. The models have been devised using ARX model structure. There were 4000 input-output data sets acquired from the experimental study. The 4000 data set collected were divided into two parts. The first 3000 data sets were used to train the model, whereas, the last 1000 data sets were used to test the performance of thus developed model. The entire developed models were validated using one step-ahead (OSA) prediction, mean squares error (MSE) of actual and predicted responses and correlation tests. The vibration of the flexible structure in frequency domain was obtained using fast Fourier transform and the first five natural frequencies of the structure were acquired from the frequency response.

A. Modeling using Recursive Least Square (RLS) Heuristic method was used to identify the most suitable

model order since there is no prior knowledge on actual model order for the flexible plate structure. For identification using RLS, the model was tuned properly by varying the orders and the forgetting factor. The best model order was found to be model 8 with forgetting factor of 0.9. The best mean square error of 0.0036 and 0.0014 were achieved for training and testing data respectively. Correlation test are carried out to determine the effectiveness of the developed system. The model developed was found to be within 95% of confidence level. Figures 7, 8, 9, and 10 are the output of the actual and predicted signal, the error of prediction, the spectral density of the prediction, the correlation tests, respectively.

B. Modeling using Particle Swarm Optimization (PSO) Similar to RLS, identification using PSO were also

conducted by heuristic method where the best models were obtained by tuning on different model orders, number of particles and number of iterations. The best model order was found to be 2nd order, which obtained with 60 particles in the algorithm at th100 iteration. The best and lowest mean squares error for PSO identification are 0.0012 and 0.00032719 for training and testing data. Correlation test are carried out to determine the effectiveness of the developed system. The model developed was found to be within 95% of confidence level. Figures 11, 12, 13, 14 and 15 are the output of the actual and predicted signal, the error of prediction, the spectral density of the prediction, the correlation tests and the convergence of identification, respectively.

C. Comparative Assessment of Performance of All Developed Algorithms

By comparing in term of lowest mean squares error (MSE) achieved, it can be seen that model predicted using PSO is better as compared to the model predicted using conventional modeling of RLS. The best and lowest mean squares error performed by PSO identification is 0.00032719 as compared to MSE achieved by RLS identification is 0.0036. Besides that, PSO identification also has performed better as compared to RLS in estimating the first mode of vibration which is the dominant mode of the structure with 5.45% error. The first five mode of vibration between analytical and developed system were listed in Table III.

60 cm

30cm

15 cm 30 cm

7.5

15 c

m

X

Y

Z

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Figure 7. The outputs of actual and RLS predicted model

Figure 8. Error between actual and RLS predicted model

Figure 9. The spectral density of RLS predicted model

(a) Auto correlation (b) Cross correllation

Figure 10. Correlation test for RLS algorithm

Figure 11. The outputs of actual and PSO predicted model

Figure 12. Error between actual and PSO predicted model

Figure 13. The spectral density of PSO predicted model

(a) Auto correlation (b) Cross correllation

Figure 14. Correlation test for PSO algorithm

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Figure 15. The convergence of the system identification using PSO

TABLE III. COMPARISON MODE OF VIBRATION BETWEEN ANALYTICAL AND DEVELOPED SYSTEM.

Mode

Measured Value (Hz)

RLS PSO

(Hz) % Error (Hz) % Error

1 5.230 4.94 5.54 4.945 5.45 2 8.563 9.876 15.33 9.893 15.53 3 14.42 14.81 2.70 15.05 4.37 4 19.26 9.14 0.93 19.355 0.49 5 22.90 0.87 9.70 24.94 8.91

The transfer function of the plate system obtained during developed the system identification are described as in Table IV. Using ARX model, transfer function of system described as equation below:

)(

)()()()()( 11

1

zAttu

zAzBty (10)

where: n

nzazazA ...1)( 11

1 ; )1(110

1 ...)( nn zbzbbzB ;

0)(t (white noise)

TABLE IV. ESTIMATED PARAMETERS USING ARX MODEL STRUCTURE.

System RLS PSO

Transfer Functions

a1=-0.9329 a2=-0.3994 a3=0.0262 a4=0.3598

a1=-1.8491 a2=0.8751

b1=0.0109 b2=-0.0119 b3=-0.0947 b4=0.0247

b1=-0.0371 b2=-0.0233

VI. CONCLUSIONS This paper presented and discussed the dynamic

characterization of flexible plate structure using conventional identification (RLS) and intelligent identification (PSO). The models were evaluated and the performances of each model have been assessed. The estimated models using RLS and PSO have been obtained, verified, acceptable, and possible to be used further for active vibration controller. From this research, it was found the performances of PSO algorithm is better than

RLS algorithm in term of the lowest mean squares error and also in term of the best in estimating the lowest percentage error of first mode which is the dominant mode of vibration.

ACKNOWLEDGEMENT The authors wish to thank the Ministry of Higher Education

(MOHE) and Universiti Teknologi Malaysia (UTM) for providing the research grant and facilities. This research is supported using Fundamental Research Grant Vote No. 4F098 and UTM Research University Grant Vote No. 04H17.

REFERENCES

[1] R. Ismail, “Neuro modelling and vibration control of flexible rectangular plate structure”, Master Thesis Department of Applied Mechanics, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 2006.

[2] M. R. Safizadeh, “Optimal Placement of Sensor and Actuator for Active Vibration Control of Flexible Structures”, Master Thesis Department of Applied Mechanics, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 2009.

[3] A. A. M. Al-khafaji, “System identification of flexible plate structure”, Master Thesis Department of Applied Mechanics, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 2010.

[4] I. Z. Mat Darus, and M.O.Tokhi, “Soft computing-based active vibration control of flexible structure”, Journal of Engineering Application of Application of Artificial Intelligent 18, pp. 93-114, 2005.

[5] M. S. Hadi, M. H. Hashim and I. Z. Mat Darus, “Genetic modeling of a rectangular flexible plate system with free-free-clamped-clamped (FFCC) edges”, Proceeding of 2012 IEEE Conference on Control, Systems and Industrial Informatics, Bandung, Indonesia, 2012.

[6] A. Z. Mohamad Sofi, “Parametric system identification and active vibration control of vibrational structures using genetic algorithm”, Master Thesis Department of Applied Mechanics, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 2007.

[7] I. Z. Mat Darus, F. M. Aldebrez and M. O. Tokhi, “Parametric Modeling of a Twin Rotor System using Genetic Algorithms”, Proceedings of 1st IEEE International Symposium on Control, Communications and Signal Proceeding, Hammamet, Tunisia, 21-24 March 2004.

[8] M. Marinaki, Y. Marinakis and G. E. Stavroulakis, “Vibration control of beams with piezoelectric sensors and actuators using particle swarm optimization”, Journal of Systems with Applications 38, pp. 6872-6883, 2011.

[9] S. Julai, M. O. Tokhi, M. Mohamad and I.Abd. Latiff, “Active vibration control of a flexible plate structure using particle swarm optimization”, Proceeding of IEEE 9th International Conference on Cybernetic Intelligent Systems (CIS), Reading, United Kingdom, 2010.

[10] S. Panda, and N. P. Padhy (2008), “Comparison of Particle Swarm Optimization and Genetic Algorithm for FACTS-based Controller Design”, International Journal of Applied Soft Computing, 8, pp. 1418-1427. Science Direct, Elsevier Ltd.

[11] A. R. Tavakolpour, “Mechatronic design of an intelligent active vibration control for flexible structures”, Ph. D. Thesis Department of Apllied mechanics, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 2010.

[12] E. Md. Yusup, “Evaluation of an active force control for flexible structure”, Master Thesis Department of Applied Mechanics, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 2010.

[13] M. R. Safizadeh, I. Z. Mat Darus, and M. Mailah, “Calculating the frequency modes of flexible square plate using finite element and finite difference methods.” Proceedings of the IEEE Asia Modelling Symposium 2010 : 4th Asia International Conference on Mathematical Modelling and Computer Simulation, KLCC, Kuala Lumpur, 2010.

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