Structures Journal of Intelligent Material Systems and

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Synthesis of Reference Signal in Adaptive Feedback Controller for Structure Vibration Suppression S.M. Yang, G.J. Sheu and C.C. Li Journal of Intelligent Material Systems and Structures 2008; 19; 727 originally published online Jul 30, 2007; DOI: 10.1177/1045389X07079651 The online version of this article can be found at: http://jim.sagepub.com/cgi/content/abstract/19/6/727

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Synthesis of Reference Signal in Adaptive Feedback Controller for Structure Vibration Suppression S. M. YANG,1,* G. J. SHEU2

AND

C. C. LI3

1

Institute of Aeronautics and Astronautics, National Cheng Kung University, Taiwan, ROC 2

Department of Electrical Engineering, Hsiuping Institute of Technology, Taiwan, ROC 3

Industrial Technology Research Institute, Taiwan, ROC

ABSTRACT: Adaptive control has been known to be desirable to accommodate the system parameter variations and adapt to operational requirements in smart (intelligent) structures. Conventional feedforward controller requires both the reference sensor to measure the disturbance and the error sensor to measure the residual vibration; however, the reference sensor measurement may be impractical because the disturbance is often not known a priori in structural vibration. This study presents an adaptive feedback controller design in which the reference signal is synthesized by the error sensor measurement and the system dynamics identification, which is a prerequisite also in adaptive feedforward controller design. The infinite impulse response (IIR) adaptive filter for system identification and the finite impulse response (FIR) adaptive filter for feedback controller are implemented on digital signal processor for effective on-line vibration suppression. Experimental results show that the controller performance is strongly influenced by the accuracy of system identification. The controller achieves broadband attenuation and remains robust under parameter variations. Key Words: smart structure, adaptive filter, system identification, vibration control.

INTRODUCTION (intelligent) structures with built-in sensor and actuator that can actively change their physical geometry and/or property have been developed for applications in aerospace systems. Integration of composite laminates and piezoelectric materials is an ideal candidate for smart structures in structural health monitoring and control. Many experimental studies of smart structures have been reported. Wang et al. (1996) applied surface-mounted piezoelectric sensor and actuator for vibration control. Yang and Jeng (1997) developed a composite structure with embedded optical fiber sensor and piezoelectric actuator for vibration suppression. Morgan and Wang (2002a,b) recently developed a vibration absorber for systems under timevarying harmonic excitation. Gern et al. (2002) also proposed a morphing airfoil design of smart wing to improve aerodynamic performance.

S

MART

*Author to whom correspondence should be addressed. E-mail: [email protected] Figures 2 and 6 appear in color online: http://jim.sagepub.com

JOURNAL

OF INTELLIGENT

The PID or LQR controllers in the above studies are applicable to systems of constant parameters and under stationary or periodic excitation. They are, however, often ineffective to the parameter variations in smart structure system, such as those caused by the sensor/ actuator embedded inside composite laminates. Mall (2002) and Yang et al. (2005a) have studied the interface mechanics of composite laminates with embedded sensor/actuator, where possible fiber delamination around the sensor/actuator and hence the stiffness change, is inevitable. Adaptive filters have thus been proposed in systems with slowly changing signal conditions and/or unknown disturbance. Gu et al. (1994) applied the filtered-X least-mean-square (LMS) algorithm to control a plate by modal sensor. Vipperman and Burdisso (1995) developed an adaptive filter to control a simply supported plate. Na and Park (1997) and Huang and Shu (1997) applied adaptive feedforward controller to systems under periodic disturbance. Yang et al. (2005b) recently developed a feedforward adaptive controller to vibration suppression of composite smart structures. The above feedforward controllers as shown in Figure 1(a) require at least two sensors: the reference sensor to measure the disturbance and the error sensor to measure the residual vibration response. Aside from

MATERIAL SYSTEMS

AND

STRUCTURES, Vol. 19—June 2008

1045-389X/08/06 0727–8 $10.00/0 DOI: 10.1177/1045389X07079651 ß SAGE Publications 2008 Downloaded from http://jim.sagepub.com at NATIONAL New TAIWANDelhi UNIV LIBand on September 12, 2008 Los Angeles, London, Singapore

727

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S. M. YANG

x(k) reference sensor

d(k)

P

yf (k)

W (adaptive filter)

v(k)

S (actuator)

y(k) S (actuator)

Error sensor e(k)

+

LMS algorithm

(a)

yf (k)

− d(k) +

W (adaptive filter)

v(k)

y(k) S (actuator)

+

Error sensor e(k)

LMS algorithm

S (actuator) (b)

Actuator Error sensor e(t)

Disturbance yf (t) W Adaptive filter

or optimized based on the error signal to compensate the changes in input, output, or system parameters. An ideal adaptive filter for structural control is to represent the transfer function between the actuator input and sensor output. Traditionally the block diagram of an adaptive controller design by adaptive filter is shown in Figure 1(a). The input and output of a finite impulse response (FIR) adaptive filter can be written as: yf ðkÞ ¼ xT ðkÞwðkÞ

d(k)

y(k)

ET AL.

(c)

Figure 1. (a) Block diagram of the adaptive feedforward control with filtered-X LMS algorithm where at least two sensors are required: one for reference signal and another for error signal, (b) block diagram and (c) schematic diagram of adaptive feedback control with only the error sensor.

rotor dynamics system where the disturbance is highly correlated to the rotation speed, the disturbance is often not known a priori in most structural dynamics system so that the reference sensor measurement is impractical. Lee and Elliot (2001) and Herold et al. (2004) applied a controller based on internal model control (IMC) to a cantilever beam of surface-bonded piezoelectric sensor/ actuator. Shaw (2000) also presented an adaptive controller for vibration attenuation. It is known that the performance of an adaptive controller, whether in feedforward or feedback, is strongly dependent on the model discrepancy of system identification (Vaudrey et al., 2003.) This study presents an adaptive feedback controller design based on an improved system identification model. The controller implemented on a digital signal processor is shown stable, effective, and robust to parameter variations.

ADAPTIVE FEEDBACK CONTROL In many active control applications, adaptive control is necessary for there may be uncertainties in system parameters. The adaptive filter for controller design is composed of two dependent parts: the filter and the adaptive algorithm. The filter parameters are adjusted

ð1Þ

where x(k) is the input vector from the reference sensor, xðkÞ ¼ ½xðkÞ xðk 1Þ xðk LÞT , w(k) is the vector representing the filter parameters of adjustable weight, wðkÞ ¼ ½w0 w1 wL T , L is the order of the adaptive filter, and k is the time step index. The weight w(k) is adjusted in the adaptation process such that the system’s output y(k) agree as closely as possible with the desired response d(k). In implementation, the control input yf (k) calculated by the adaptive filter cannot be directly applied to reject the disturbance; it should pass through the dynamics of the low-pass filter, power amplifier, sensor/actuator, and the structural dynamics from the actuator to the sensor location. This transfer function S(z) is termed as the system dynamics. Thus the error signal e(k) is written as: eðkÞ ¼ dðkÞ þ ’T ðkÞwðkÞ

ð2Þ

T

where ’ðkÞ ¼ ½vðkÞ vðk 1Þ vðk LÞ and v(k) is the inverse z-transform of V(z) and V(z) ¼ S(z)X(z), X(z) is the z-transform of the reference input x(k). Since the output of the adaptive filter is coupled with the system dynamics, the algorithm for updating the adjustable weight is in the filtered-X LMS algorithm (Yang et al., 2005b) wðk þ 1Þ ¼ wðkÞ þ 2eðkÞxðkÞ

ð3Þ

where is the rate of convergence of the minimization process. It has been known that the adaptive feedforward control in Figure 1(a) is, in theory, inherently stable and can effectively attenuate disturbances, but such controller design requires at least two sensors: one reference sensor in the upstream to measure the reference input x(k) and another error sensor in the downstream to measure e(k). The contamination of the upstream reference signal by the downstream error signal inevitably leads to an adverse feedback loop towards instability. Except in rotor dynamics system where the vibration signal is strongly influenced by the rotation speed, the requirement of a coherent reference signal unaffected by the control input signal is infeasible in most systems. In addition, the disturbance is often not known a priori so that its measurement by reference sensor is impossible. In engineering applications, the actuator output y(k) depends not only on the adaptive filter W(z) but also on


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Synthesis of Reference Signal in Adaptive Feedback Controller

the system dynamics S(z) and adaptive filter output yf (k) which is in-accessible because of the system dynamics. An effective controller design is to use only the error sensor and its measurement to synthesize the reference signal. The concept is similar to the so-called IMC (Morari and Zifiriou, 1989) often seen in H1 controller design to balance system performance and robustness (Yang and Huang, 1996), where a feedback controller can be realized by an adaptive feedforward filter. Figure 1(b) depicts the block diagram of an adaptive feedback control, where the error sensor measures the sum of the primary path vibration signal d(k) and the secondary path canceling output y(k). The objective is to generate y(k) based on the measurement of e(k) so as to minimize the vibration. Figure 1(c) illustrates the vibration suppression of a structure by the adaptive feedback controller with a pair of sensor/actuator. If the transfer function of the system dynamics S(z) is known or can be estimated by system identification, then an estimation of the primary path ^ signal d(k) at the sensor location, denoted by dðkÞ, can be obtained by subtracting the reflected component of the actuator output from the sensor measurement e(k). ^ can then be filtered to yield yf (k). For the adaptive dðkÞ feedback control system as shown in Figure 1(b), the error signal can be written as: ^ EðzÞ ¼ DðzÞ þ SðzÞWðzÞDðzÞ,

If E(z)/D(z) is close to zero, then e(k) converges and W(z) will be a non-causal filter, WðzÞ !

SYSTEM IDENTIFICATION The aforementioned time delay and system dynamics lead to a system of non-minimum phase so that the transfer function S(z) will have zero(s) located outside of the unit circle leading to instability. The controller design by FIR adaptive filter is thus preferable over by IIR filter in terms of system stability. Nevertheless, the adaptive feedback controller may suffer from instability by poor plant estimation causing the adaptation algorithm in Equation (3) to diverge. The selection of model dimension is therefore critical to controller performance. One first needs to identify the system dynamics S(z) when implementing the adaptive controller as described above. A commonly used parametric model is the ARX model

ð4Þ

Bðz1 Þ 1 uðkÞ þ eðkÞ 1 Aðz Þ Aðz1 Þ

EðzÞ : ^ 1 þ SðzÞWðzÞ

ð5Þ

Substituting Equation (5) into Equation (4), EðzÞ ¼ DðzÞ þ

SðzÞWðzÞEðzÞ , ^ 1 þ SðzÞWðzÞ

Aðz1 Þ ¼ 1 þ a1 z1 þ þ an zn 1

^ 1 þ SðzÞWðzÞ : HðzÞ ¼ ^ SðzÞÞWðzÞ 1 þ ðSðzÞ

ð7Þ

þ þ bm z

ð11Þ :

ð12Þ

ð13Þ

where ðkÞ ¼ ½yðk 1Þ yðk nÞ uðk gÞ uðk g m þ 1ÞT ; h ¼ ½a1 an b1 bm :

ð8Þ

mþ1

yðkÞ ¼ T ðkÞ þ "ðkÞ

T

^ 6¼ SðzÞ, H(z) is an infinite impulse response (IIR) If SðzÞ system, i.e., e(k) cannot be completely canceled. System identification is therefore critical to the performance of a feedback system. ^ ¼ SðzÞ and the system dynamics has c Suppose SðzÞ ^ ¼ zc S^0 ðzÞ, where S^0 ðzÞ has time delay samples, i.e., SðzÞ no delay, one may find the overall transfer function EðzÞ ¼ 1 þ zc S^0 ðzÞWðzÞ: DðzÞ

1

n and m are the orders of the respective polynomials, and the coefficients ai and bi are determined by the least square method. Equation (10) can be rewritten in a vector form as:

ð6Þ

one can find the overall transfer function H(z) of the feedback system from d(k) to e(k)

ð10Þ

where g is the number of delays from input to output, A() and B() are polynomials in the delay operator, z1, Bðz Þ ¼ b1 þ b2 z

^ ¼ DðzÞ

ð9Þ

Thus, the adaptive feedback control system is sensitive to both the time delay and the system dynamics.

yðkÞ ¼ zg

^ where E(z), D(z), and DðzÞ are the z-transforms of e(k), ^ d(k), and dðkÞ, respectively. An estimation of the primary path signal d(k) can be obtained by:

zc : S^0 ðzÞ

ð14Þ ð15Þ

y(k) at the current step is a function of /ðkÞ, the input and output of the previous steps, and is the coefficient. An ARX model is equivalent to an IIR filter when m 6¼ 0 and n 6¼ 0, whereas as it is an FIR filter when m ¼ 0. The selection of model dimension for S(z) is often a problem in practice. Higher model dimension often leads to smaller model error, but more computation is ^ required. Figure 2(a) illustrates the model error PNLðhÞ2 at ^ different model dimensions, LðhÞ ¼ ð1=NÞ k¼1 " ðkÞ where N is the sample number. The question is whether


730

Mean square error

S. M. YANG

ET AL.

L1

L2

n1

n2 Model dimension (a)

PZT sensor

PZT actuator

Low-pass filter

Voltage Amp.

Low-pass filter 586 PC & DSP TMS C32 with 16 bit A/D D/A converter (b)

(c) Figure 2. (a) Illustration of the model error at different model dimension in system identification, (b) schematic diagram, and (c) experimental setup for system identification and vibration suppression of the composite smart structure.


731

Synthesis of Reference Signal in Adaptive Feedback Controller

2

ð16Þ ð17Þ

where n* is the hypothetical dimension, /2 ðkÞ ¼ ½/T1 ðkÞ/~ T2 ðkÞT , n1 ¼ dimðh^ 1 Þ < n2 ¼ dimðh^ 2 Þ, and h^ 2 ¼ ½h^ T1 h~ T2 T . The statement H0 is called the null hypothesis with least square error L1, and H1 is called the alternative hypothesis with error L2. The procedure is to specify the probability of type I error (often called the significance level of the test, denoted by ) and then to verify the hypothesis. If the null hypothesis is rejected when it is true, then type I error has occurred. A common choice of the probability of type I error is ¼ 0.05. Two different sequences of the sample data must be recorded to verify the ARX model: one is the training data for estimating the optimum parameter h^ and another is the testing data for model verification to ^ is acceptable. Assume determine if the model error LðhÞ that the sample data satisfy yðkÞ ¼ /T1 ðkÞh þ "ðkÞ where {"(k)} is the sequence of independent normal distributed random variables, N(0, 2) and is the standard deviation. It has been shown that a variable f defined by f¼

L1 L2 N n2 n2 n1 L 2

ð18Þ

is corollary F-distributed F(n2 n1, N n2) so that f can be used to construct a statistical test for evaluating the model dimension n1 and n2. The number of data N is generally large compared to n2 such that an approximation of F(n2 n1, N n2) is equivalent to chi-square 2 2 distribution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn2 n1 Þ, 0:05 ðn2 n12Þ ðn2 n1 Þþ 8ðn2 n1 Þ for ¼ 0.05. When f < ðn2 n1 Þ, the null hypothesis H0 should be accepted, i.e., the model dimension n1 is appropriate. Conversely when the model dimension n2 is accepted, the hypothesis test of another higher dimension should be conducted. The above analysis on system identification is validated by the experiment of a composite smart structure as shown in Figure 2(b) and (c). The (90/90/ 90/90/0)s composite laminated beam of 265 40.5 1 mm3 is made of S-glass/epoxy uni-directional pre-prag tapes with two embedded 120 30 0.375 mm3 piezoelectric actuators and one 13.5 25 0.375 mm3 piezoelectric sensor. Details of the fabrication process and material properties are given in Yang and Lee (1997). The adaptive control system

(a) 40000 Measured data FIR model Output

H0 : n ¼ n1 , i:e:, y1 ðkÞ ¼ /T1 ðkÞh^ 1 H1 : n > n1 , i:e:, y2 ðkÞ ¼ /T ðkÞh^ 2

consists of a pair of piezoelectric sensor and actuator, two low-pass filters, a power amplifier, and a TMS320C32 digital signal processor (DSP). Random signal is generated in 2 kHz sampling frequency and lowpass filtered at 200 Hz cut-off frequency. The vibration signal measured by the embedded sensor through the anti-aliasing, low-pass filter at 400 Hz cut-off frequency is also recorded by the digital signal processor. The ARX model in Equation (6) can be written by IIR or FIR adaptive filter. The former has both poles and zeros defined by the number of delay element (m and n) in Equations (11) and (12) hence capable of representing the sharp cut-off filter characteristics by a modest number of the delay elements. The limitation, however, is that it is not necessarily stable because the pole may locate outside the unit circle. By comparison, FIR adaptive filter is inherently stable though the model order may be substantially higher. For system identification by FIR filter, the model error of n1 ¼ 101 and n2 ¼ 111 yield f < 20:05 when ¼ 0.05 and N ¼ 20,000. Therefore, n1 ¼ 101 is selected as an appropriate dimension to represent the system dynamics. Figure 3(a) shows the identification result of the system dynamics using an ARX (FIR) model with 101 moving average parameters (m ¼ 0, n ¼ 101), where the model output agrees closely with the measured output. Note that one can select another set of test and the probability to conduct hypothesis test by checking the chi-square distribution. For the common choice of ¼ 0.05, n ¼ 101 is validated. Similar hypothesis test on the ARX (IIR) identification indicates that the model of 31 autoregressive parameters and 31 moving average parameters (n ¼ m ¼ 31) can also accurately identify the system dynamics both on- and off-line as shown in Figure 3(b). One can then efficiently

0

−40000

0

100

200

300

400

Sample (b) 40000

Output

the model dimensions n1 or n2 is preferred. A statistical analysis method called the hypothesis testing can be applied to determine a suitable dimension of the ARX model. The hypothesis is a statement about the parameters of a probability distribution. In identification of the system dynamics, one first assumes that a smaller model dimension n1 is appropriate, and this may be stated in hypothesis testing as:

Measured data ARX model

0

−40000

0

100

200 Sample

300

400

Figure 3. Identification results of the secondary path dynamics using (a) ARX model (FIR) with 101 parameters (m ¼ 0, n ¼ 101) and (b) ARX model (IIR) with 62 parameters (m ¼ n ¼ 31).


S. M. YANG

(a)

Adaptive control can automatically adapt to the optimum condition in changing environments or system requirements. However, the controller is required to process a large number of data in an even greater number of operations. A digital signal processor with the ability of parallel multipliers, pipelines, and fast input/output is preferable. Vibration suppression experiment is conducted on the composite smart structure and the filtered-X LMS algorithm in Equation (3) is implemented on DSP. The embedded piezoelectric sensor is employed to measure the structural vibration at 2 kHz sampling rate, and the measured signal is transmitted into the DSP via the low-pass filter (KrohnHite 3905B) for anti-aliasing and the A/D converter. The required control input is evaluated by the DSP and then sent to the low-pass filter and D/A converter so as to prevent from undesired high frequencies in the actuator. Since the input and output voltage of the DSP is limited to 3V, an amplifier (Krohn-Hite 7500) is needed to drive the embedded actuator for effective control performance. The function generator (HP 3324A) is employed to generate the disturbance signal to another piezoelectric actuator. For the ARX identification model of 62 parameters (m ¼ n ¼ 31), the adaptive feedback controller in FIR filter has 51 adjustable parameters, L ¼ 50 in Equation (1). Figure 4(a) shows the vibration response of the smart structure under an initial tip displacement of about 2 cm. The transient vibration of the close-loop system is effectively subdued in about 0.8 s as compared to more than 5 s in the open-loop system. The vibration response under two-mode resonance excitation at 10.5 and 52.5 Hz is also shown in Figure 4(b). Although the error signal remains, the adaptive feedback controller achieves significant attenuation of the two harmonics in steady-state vibration. The power spectrum of the error signal shows that the controller achieves the reduction of about 20 dB in transient excitation and about 30–40 dB in resonance. The controller is also applicable to the smart structure under random excitation from 0 to 100 Hz, and the peak-to-peak vibration amplitude is reduced by 60% as shown in Figure 4(c). The adaptive feedback controller proposed in Figure 1(b) is shown effective to transient, steady state, and broadband attenuation. It is known that both adaptive feedforward and feedback controller design require system identification. An adaptive feedback controller can become unstable from feedback loop instability and/or adaptation

Close-loop

1.0 0.0 −1.0 −2.0 0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

Time (s) (b)

Output (Volt)

CONTROLLER DESIGN AND IMPLEMENTATION

2.0 Open-loop

Output (Volt)

use the remaining computation resource(s) for controller design.

ET AL.

2.0 1.0 0.0 −1.0 −2.0 0.0

0.2

0.4

0.6

0.8

1.0

Time (s) (c) Output (Volt)

732

2.0 1.5 1.0 0.5 0.0 −0.5 −1.0 −1.5 −2.0 6.0

6.2

6.4

6.6

6.8 Time (s)

7.0

7.2

7.4

Figure 4. Experimental results of the adaptive feedback control under: (a) an initial tip displacement, (b) the two-mode resonance excitation, and (c) the random excitation at 0–100 Hz.

algorithm divergence. The former can be prevented by increasing the number of the filter parameter, provided within the computation limitation of the DSP. The latter can be solved by better system identification for the plant model. The requirement of sufficiently accurate system identification can be determined systematically by the statistical hypothesis test. By comparison, if one were to use an identification model of insufficient accuracy, say the ARX model of 54 parameters (m ¼ n ¼ 27), the model error may be similar to that in Figure 3(b) but the closeloop system response is otherwise. Figure 5(a) illustrates the application of the same feedback controller in suppressing the resonance excitation. Unstable ‘beating’ comes mainly from inadequate system identification. Figure 5(b) shows that the controller, if it is developed based on inadequate system identification, is futile to broadband disturbances. In order to examine the robustness of the adaptive feedback controller, a mass is attached to the tip of the structure to simulate parameter variations. The first mode is reduced from 10.5 to 8.5 Hz. The adaptive feedback controller based on the ARX identification model of 62 parameters, without further training, can effectively reduce the tip vibration from more than 10 s


733

Synthesis of Reference Signal in Adaptive Feedback Controller (a) 2.5

Open-loop Close-loop

0.8

Output (Vol.)

Output (V)

(a)

0.4 0.0 −0.4


1.5 0.5 −0.5 −1.5

−0.8 0

1

2

3

4

5

−2.5

0

2

4

Time (s) 2.0 Open-loop Close-loop

Output (Vd)

1.0 0.0 −1.0 −2.0 6.0

6.4

6.6

6.8

10

7.0

7.2

7.4

Time (s)

Figure 5. Controller performance based on the ARX system identification model of m ¼ n ¼ 27 in suppressing (a) vibration at resonance excitation and (b) vibration at random excitation.

in the open-loop to within 2 s in the close-loop system. Figure 6 indicates that the adaptive controller remains effective under system parameter variations, and it can attenuate not only transient but also non-stationary, broadband disturbances.

CONCLUSIONS 1. Adaptive filter is indispensable to systems of slowly changing signal conditions and/or system parameters variations. Conventional feedforward adaptive control for structure vibration suppression, even if controlling a single mode, requires at least two sensors: one reference sensor and another error sensor. Except in rotor dynamics system where the vibration signal is strongly influenced by the rotation speed, the requirement of a coherent reference signal unaffected by the control input signal is infeasible in most systems. An adaptive feedback controller design in one-sensor configuration is developed in this study. The controller in filtered-X LMS algorithm employs only the error sensor and its measurement to synthesize the reference signal. It is applied successfully to vibration suppression of a composite smart structure with embedded piezoelectric sensor/ actuator. 2. Implementation of the adaptive control by adaptive filter, whether feedforward or feedback, requires that the transfer function of system dynamics S(z) can be estimated by identification. Analysis and experiment show that the ARX model in IIR filter with 31 autoregressive parameters and 31 moving average


20 0 −20 −40

6.2

8

(b) 40 Power spectrum (dB)

(b)

6 Time (s)

0

4

8

12 Frequency (Hz)

16

20

Figure 6. Vibration control experiment of the smart structure with an attached mass: (a) time response and (b) power spectrum.

parameters can faithfully represent the system dynamics. The experimental results show that the adaptive controller can effectively minimize the vibration of the composite smart structure under transient and harmonic excitations. The controller is also robust to system parameter variations and effective to broadband excitation. The adaptive feedback controller with a reduced number of sensor/actuator shall facilitate broader smart structure applications. It should be noted that the time delay has been known critical to system stability in engineering implementation. The tradeoff between the model dimension of system identification S(z) and the controller (adaptive filter) W(z) has to be carefully evaluated. A desirable practice is to have the controller size large enough to better ‘predict’ the system response when computing the control action (input) at this instant and yet applying to the actuator some time delays later.

ACKNOWLEDGMENT The work was supported in part by the National Science Council, Taiwan, ROC under NSC90-2212E006-140 and by the Industrial Technology Research Institute.

REFERENCES Gern, F.H., Inman, D.J. and Kapania, R.K. 2002. ‘‘Structural and Aeroelastic Modeling of General Platform Wings with Morphing Airfoils,’’ AIAA J., 40:628–637.


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ET AL.

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Structures Journal of Intelligent Material Systems and

Structures Journal of Intelligent Material Systems and

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