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Proc. of Int. Conf. on Control, Communication and Power Engineering 2010

Implementation of information filter for vibration control of a smart cantilever beam D. Ezhilarasi1, J. Arunshankar2, M. Umapathy3 Department of Instrumentation and Control Engineering.

National Institute of Technology Trichy, India e-mail: ezhil@nitt.edu1, j_arunshankar@yahoo.com2, umapathy@nitt.edu3 Abstract—This paper addresses the design of information filter and state feedback controller for vibration suppression of a flexible beam structure at first two modes. The model of the system which includes the dynamics of the structure together with the sensor/actuator dynamics is obtained through on line system identification technique. The performance of the estimator and the controller is evaluated experimentally using dSPACE controller board.

the states so that they can be used for control. The kalman filter has been one of the most widely used tools for solving estimation problems during the last 50 years [5]. However, early after its introduction, it was noticed that the original algorithms presented some drawbacks related to practical implementation issues. Information filtering has been considered as an alternative approach to the covariance recursions of the original Kalman filter. The filter algorithm, in information form computes the inverse of the covariance matrix (the so-called information matrix), and computes the state information estimate. The application of this approach is justified when; there exists poor information on the initial condition of the state to be estimated. In this case, the information filter can be easily initiated with information matrix zero, whereas the covariance filter would invert very large covariance matrices [8][9]. It can reduce dramatically the storage and computation involved with the estimation of certain classes of large interconnected systems [6]. Although

Keyword: piezoelectric, information filter, state feedback control

I.

INTRODUCTION

More than three decades of research in the field of smart structures has shown the viability and potential of this technology. Numerous applications are proposed and several have been conceived experimentally such as vibration control of plates, beams, shape control and buckling control, while other innovations such as smart skis have been commercially realized. A smart structure consists of actuators and sensors integrated into the main structure with a control unit [1]. There are varieties of adaptive materials which can be integrated with smart structures, among them piezoelectric materials found to be used in wide applications. Over the last two decades, the usage of piezoceramics as actuators and sensors has considerably increased and they provide effective means of high quality actuation and sensing mechanism. The advantages of Piezoceramics include low cost, absence of moving parts, rapid response, compactness and easy implementation.

Signal conditioning, placement, and bonding issues are easy to resolve with piezoceramics compared to other smart materials.

previous works have clearly shown the tremendous potential of information filter, its applications to vibration control of piezoelectric bonded structures have been limited. Therefore the objective of the present experimental work is to demonstrate the information filter with state feedback control for smart structures. The authors believe that the implementation of information filter for multi mode vibration control of smart structures in real time is the first of its kind.

This paper is organized as follows. In Section II, the review of information filter is given. In Section III experimental design and modeling of smart structure is presented.

Controller design including state estimation and experimental evaluation is given in section IV. Conclusions are drawn in section V.

Vibration control of flexible structures by distributed sensors and actuators has been widely studied in the past decade and more dimensions are introduced to improve the control of structural behavior [2] [3]. The main design approach for systems described in state-space form is the use of state feedback. One selects pole locations to achieve a satisfactory dynamic response and develops the control law for the closed-loop system that corresponds to satisfactory dynamic response. One has to design an estimator for the states, because these are generally not measurable. This estimator is an observer that delivers the information about

II.

Brief review of information filter algorithm [5,7] is presented here. Consider a system described in linear form (1) x ( k + 1) = Ax ( k ) + w( k ) where x(k) are states of interest at time k, A the state transition matrix from time (k) to (k+1), and w(k) the associated process noise modeled as an uncorrelated white sequence with

137 Š 2009 ACEEE

REVIEW OF INFORMATION FILTER

Proc. of Int. Conf. on Control, Communication and Power Engineering 2010

T E [ w(i ) w ( j )] = δ ij Q (i )

III.

(2)

STRUCTURE

Where Q(i) is process noise covariance matrix.

The system is observed according to the linear equation z ( k ) = Hx ( k ) + v ( k ) (3) where z(k) is the vector of observations made at time k, H the observation matrix and v(k) the associated observation noise modeled as an uncorrelated white sequence with T E [v (i )v ( j )] = δ ij R (i ) (4) Where R(i) is measurement noise covariance matrix.

T (5) It is also assumed that E[v (i ) w ( j )] = 0 Information filter is essentially a Kalman filter expressed in terms of measures of information about the states of interest, rather than direct state estimates and their associated covariances. The two key information-analytic variables are ) the information matrix and information state vector y (i | j ) . The information matrix (Y) is the inverse of the covariance matrix (P). −1 Y (i | j ) = P ( i | j ) (6) The information state vector is the product of the inverse of the covariance matrix and the state estimate xˆ (i | j ) . ) −1 y (i | j ) = P (i | j ) xˆ (i | j ) (7) The update equation for the information state vector ) ) T −1 y ( k | k ) = y ( k | k − 1) + H R ( k ) z ( k ) (8) The expression for information matrix associated with the above estimate is T −1 Y ( k | k ) = Y ( k | k − 1) + H R ( k ) H (9) The information state contribution i(k) from an observation z(k), and its associated information matrix I(k) are defined respectively as, T −1 i (k ) = H R (k ) z (k ) (10) T −1 I (k ) = H (k ) R (k ) H (k ) (11)

A. Experimental set up

Figure 1 Schematic diagram of the experimental set up

A flexible aluminum beam with fixed clamped end as shown in figure 1 is considered in this paper. Two piezoceramic patches are surface bonded at a distance of 5mm from the fixed end of the beam. The patch bonded on the bottom surface acts as a sensor and the one on the top surface acts as an actuator. To apply an excitation to the structure another piezoceramic patch is bonded on the tip of the beam. The dimensions and properties of the beam and piezoceramic patches are given in table I. The piezoceramics are electrode with fixed-on adherent silver of solderable quality. Electrical contacts to the electrodes are made by soldering. This makes fragile piezoceramics much easier to work with and easier to integrate into the structure.

TABLE1 PROPERTIES AND DIMENSIONS OF ALUMINUM BEAM AND PIEZOCERAMIC SENSOR/ACTUATOR Aluminum Beam Piezoceramic Length 0.0765 0.4 Length (m) lp l (m) Width (m) 0.0135 b 0.0135 Width (m) b Thickness (m) Young’s modulus (Gpa) Density (kg/m3) First natural frequency (Hz) Second natural frequency (Hz)

The information propagation coefficient L ( k | k − 1) , which is independent of the observation made, is given by the expression −1 L( k | k − 1) = Y ( k | k − 1) AY ( k − 1 | k − 1) (12) Prediction ) ) y ( k | k − 1) = L ( k | k − 1) y ( k − 1 | k − 1) Y ( k | k − 1) = ⎡ AY

⎣

−1

(13)

−1 T ( k − 1 | k − 1) A + Q ( k ) ⎤ (14)

Estimation ) ) y ( k | k ) = y ( k | k − 1) + i ( k ) Y ( k | k ) = Y ( k | k − 1) + I ( k )

⎦

tb

0.001

Thickness (m)

ta

0.0005

Eb

71

Young’s modulus (Gpa)

Ep

47.62

ρb

2700

ρp

7500

f1

5.04

Density (kg/m3) Piezoelectric strain constant (m V-1)

d 31

-247x10-12

f2

32.84

g 31

-9x10-3

Piezoelectric stress constant (V m N-1)

The sensor output is given to the piezo sensing system which consists of high quality charge to voltage converting signal conditioning amplifier with variable gain. The conditioned piezosensor signal is given as analog input to dSPACE1104 controller board. The control algorithm is developed using simulink software and implemented in real time on dSPACE 1104 system using RTW and dSPACE real

(15) (16)

138 © 2009 ACEEE

EXPERIMENTAL DESIGN AND MODELING OF SMART (3)

Proc. of Int. Conf. on Control, Communication and Power Engineering 2010

time interface tools. The simulink software is used to build control block diagrams and real time workshop is used to generate C code from the simulink model. The C code is then converted to target specific code by real time interface and target language compiler supported by dSPACE1104. This code is then deployed on to the rapid prototype hardware system to run hardware in-the-loop simulation. The control signal generated from simulink is interfaced to piezo actuation system through configurable analog input/output unit of dSPACE 1104 system. The piezo actuation system drives the actuator and the excitation signal is applied from simulink environment through a DAC port of dSPACE system.

Figure 2 :Experimental setup

A. Design of information filter The information filter is designed to estimate the unknown states of the structure by following the procedure given in section II. For the design of estimator the continuous system model given in equation is discretized at a sampling interval of 0.01 sec. The initial values are chosen to be R=1,Q=1xI4 and zero initial states.

B. Modeling of the structure Control design of flexible structures relies on accurate modeling of the system dynamics. The analytical-model-based approaches have been highly doubtful under high precision requirements because of the difficulty in simulating the properties of these complicated systems. The finite-element model- based methods are usually time consuming and their applications for accurate control are sometimes hindered by factors such as the assumption of perfect bonding at the interface between the structure and transducers. In most cases, these traditional modeling approaches are intractable and even impossible for highly complex structures. Hence, the unknown parameters of the smart structure dynamics are estimated using an online system identification method, which is proven to be more universal and feasible than analytical and numerical models for the present system [3]. The recursive least squares (RLS) method based on the Auto Regression with extra inputs (ARX) model is used here for linear system identification, which is easy to implement and has fast parameter convergence. In this paper, the state space model of the system based on the work in [4] is considered for the design of controller.

B. State feedback gain calculation The state feedback controller is designed using pole placement to reduce the amplitude of vibration of a cantilever beam at resonance. Let (Φ, Γ, c ) be the discrete T

time system obtained by sampling the system in equation (17) at a sampling interval of 0.01 seconds. A stabilizing state feedback gain is obtained such that the eigen values of (Φ + ΓF ) are not at origin [10]. The state feedback gain obtained is F = [16.2346 -7.7462 11.0017 61.2536] The corresponding closed loop poles of the system

(Φ + ΓF ) are-0.38±j0.88, 0.887±j0.29.

.

x = A x + b u+ er ; y = c T x ⎡ 76.9893 71.5731 ⎢−136.1042 6.1271 a=⎢ ⎢ 115.7932 −116.2021 ⎢ ⎣ − 70.8876 45.1268

c = [1 0 0 0],

d = [0]

− 45.5632 71.9048 ⎤ 116.6837 −116.7537⎥⎥ , − 6.5425 136.6781⎥ ⎥ − 71.2161 − 77.5364⎦

C. Experimental results The state feedback controller designed using pole placement in section IV to suppress the vibration at first two vibration modes is implemented using Kalman information filter. The sensor output is sampled at 0.01 sec through ADC port of dSPACE and MATLAB/simulink to estimate the unknown states. The sampling time is chosen to provide approximately six measurements per cycle (sampling frequency 200 Hz) of largest frequency signal used. The estimated output of the filter is compared with the plant output in open loop by exciting the structure with mixed mode frequency signal. The estimated states, comparison between the plant output and estimated output are shown in figure 4. One can see that the estimated output exactly matches with the plant output and showing the better performance and ease of implementation with information filter. The control signal is generated by multiplying the states with state feedback gain and is applied to the control actuator through DAC port of dSPACE controller board.

The controller is implemented by developing a real time simulink model using MATLAB RTW in simulink as shown in Figure 3. To show the performance of estimator

⎡ 0.0029⎤ ⎡ 0.2046⎤ ⎢ 0.0265⎥ ⎢ 0.1955⎥ ⎥ ⎢ ⎥ b= , e=⎢ ⎢− 0.4427⎥ ⎢− 0.0664⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 0.0588⎦ ⎣− 0.0299⎦

(17) where x is the state vector, u is the control signal, y is the sensor output, A, B, C are respectively the state, the input, the output matrices.

IV.

CONTROLLER DESIGN AND EXPERIMENTAL EVALUATION

The experimental set up to measure and control the vibration of a cantilever beam is shown in figure 2. The controller design to suppress the vibration at resonance involves two steps. First is the design of state estimator and the state feedback gain calculation.

139 © 2009 ACEEE

Proc. of Int. Conf. on Control, Communication and Power Engineering 2010

and the controller at both the modes, the excitation frequency was changed from first mode to second mode after 9 sec as shown in Figure 5 the results are also shown for mixed mode frequency excitation in figure 6. The responses show that the estimator and controller continue to perform well even after large changes in the excitation frequency.

Figure 6 Experimental results with state feedback controller when mixed mode frequency excitation is applied.

V.

CONCLUSION

This paper presents an experimental evaluation of information filter and state feedback controller for vibration suppression of smart cantilever beam using the model obtained from identification. From the experimental results it is observed that vibration reduction is 93 % at the first resonance and 89.4% at the second resonance. The experimental results demonstrate very good closed loop performance and simplicity of the information filter.

Figure 3 Real-time simulink model for state feedback controller

REFERENCES [1]

Inderjit Chopra, “Review of state of art of smart structures and integrated systems”, AIAA Journal, 40(11), 2002, 2145-2187.

[2] D.Ezhilarasi, M.Umapathy, B.Bandyopadhyay,“Design and experimental evaluation of simulaneous fast output sampling feedback control for smart structures”, International journal of Automation and Control, vol 4(1),pp 42-64,2010.

[3] Xiongzhu Bu, Lin Ye, Zhongqing Su and Chunhui Wang, Active control of a flexible smart beam using a system identification technique based on ARMAX, Smart materials and Structures, (12), 2003, 845-850.

[4] D.Ezhilarasi, M.Umapathy, B.Bandyopadhyay,“Design and experimental evaluation of piecewise output feedback control for structural vibration suppression”, Smart Materials and Structures, 2006, 15: 1927–1938.

[5] Mutambara Arthur, G. O., Decentralized Estimation and Control for Multisensor Systems, CRC Press Inc., Boca Raton, FL, USA ,1998.

[6] G.J. Bierman, “ An application of the square root filter to large-scale linear interconnected systems”, IEEE Trans. Automat. Contr., vol 22, pp 989-991,1977.

[7] Girija, G. and Raol, J., “Application of information filter for Sensor Data Fusion”, 38th Aerospace Sciences Meeting & Exhibit 10 – 13, Reno, NV,2000.

[8] Y.S. Kim and K.S. Hong, “ Federated information mode-matched filters in ACC environment”, Int. J. Contr. Automat., Syst., Vol3, No 2, pp. 173-182,2005.

[9] Ying Zhanga,Yeng Chai Sohb,Weihai Chen , “Robust information filter for decentralized estimation”, Automatica 41, pp. 2141 – 2146, 2005.

[10] Kumar Rajiv, Khan Moinuddin,” Pole placement techniques for active vibration control of smart structures: A feasibility study”, Journal of vibration and acoustics, vol 129, No 5, pp. 601-615, 2007.

Figure 4 Estimator output under openloop condition

Figure 5 Experimental results with state feedback controller when excitation switched from first mode to second mode

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