MULTI-MODAL VIBRATION CONTROL USING ADAPTIVE POSITIVE POSITION FEEDBACK Keun-Ho Rew, Jae-Hung Han, and In Lee Division of Aerospace Engineering, Department of Mechanical Engineering Korea Advanced Institute of Science and Technology 373-1 Kusong-dong, Yusong-gu, Taejon, 305-701, Korea Tel. : (82-42) 869-3717, Fax. : (82-42) 869-3710 E-mail :
[email protected]
< Article Info. > Publication Title
Journal of intelligent material systems and structures
Journal Homepage
http://jim.sagepub.com/
Publication Year
2008
Volume/Issue
v.13 no.31
Paginations
pp.13-22
DOI
http://dx.doi.org/10.1177/1045389X02013001866
Further Info.
http://sss.kaist.ac.kr/
Remark This PDF file is based on the final submission to the publisher, and there might be slight change from the final form provided by the publisher.
Revised for Journal of Intelligent Material Systems and Structures
MULTI-MODAL VIBRATION CONTROL USING ADAPTIVE POSITIVE POSITION FEEDBACK
Keun-Ho Rew, Jae-Hung Han, and In Lee
Division of Aerospace Engineering, Department of Mechanical Engineering Korea Advanced Institute of Science and Technology 373-1 Kusong-dong, Yusong-gu, Taejon, 305-701, Korea Tel. : (82-42) 869-3717, Fax. : (82-42) 869-3710 E-mail :
[email protected]
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MULTI-MODAL VIBRATION CONTROL USING ADAPTIVE POSITIVE POSITION FEEDBACK
Keun-Ho Rew*, Jae-Hung Han†, and In Lee‡ Division of Aerospace Engineering, Department of Mechanical Engineering Korea Advanced Institute of Science and Technology 373-1 Kusong-dong, Yusong-gu, Taejon, 305-701, Korea
ABSTRACT An adaptive controller, Adaptive Positive Position Feedback (APPF) is proposed for the multi-modal vibration control of frequency varying structures. Spillover phenomena and real-time system identification have been obviously difficult obstacles for the multi-modal adaptive vibration control. To overcome these problems, a fast and powerful algorithm is proposed to identify the frequencies of time-varying structures. Variable PPF controllers are adjusted with estimated natural frequencies at every time step. A composite plate with a bonded piezoelectric sensor and an actuator was prepared as an experimental model, and the natural frequencies of the model are changed by attaching masses. The experimental results show that natural frequencies are estimated quite accurately and that the vibration of controlled modes is significantly reduced. No significant performance reduction has been observed with respect to approximately 10 % frequency changes of the corresponding modes.
On the contrary, the performance of the conventional LQG
controller is significantly degraded due to frequency variations.
Keywords:
smart composite structures, multi-modal vibration control, Adaptive Positive Position Feedback (APPF), real-time frequency estimation
*
Currently Researcher at Mirae Inc.
†
Currently Senior Researcher at Electronics and Telecommunications Research Institute.
‡
Professor, To whom correspondence should be addressed.
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INTRODUCTION
The adaptive vibration control has been an interesting topic for the realization of smart structures, since the adaptation to changing environments is one of the core functions of smart structures.
There are many examples of lightweight structures with varying dynamic
characteristics during operation such as deployable space structures and robot manipulators dealing with time-varying payloads.
These changes in dynamic characteristics can cause
degraded control performances or sometimes yield unstable closed-loop behaviors for timeinvariant controllers.
One of the possible approaches to overcome these problems is the
application of the adaptive controls. Many researchers have studied the adaptive vibration and noise control. Filtered-x-LMS algorithm (Fuller et al., 1996; Kim and Park, 1999) and recursive singular value decomposition technique (Dehandschutter et al., 1998) have been successfully applied to adaptive noise control. Adaptive pole placement control (Sidman, 1986; Park, 1991) and artificial neural network (Yang and Lee, 1997; Yoon et al., 2000) have been used to various active vibration control problems. However, the implementation of real-time adaptive control for multi-modal vibration of structures has not been well explored yet. One reason is the spillover phenomenon, where the control energy for the target modes flows into residual modes. Heavy computational burden of the algorithm is another barrier for the real-time realization of the adaptive control. The adjustment of the controller should be carried out at every time step, comprising mode decoupling (inverting matrices or solving eigenvalue problem that appears in system identification) and controller updates. Therefore, a fast adaptive multi-modal algorithm with less spillover is essential for the practical real-time control.
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In this paper, an efficient adaptive multi-modal control algorithm, named Adaptive Positive Position Feedback (APPF), is suggested for the vibration control of frequency-varying structures. The APPF control is motivated by Positive Position Feedback (PPF) control (Fanson and Caughey, 1987; Friswell and Inman, 1999) and Independent Modal Space Control (IMSC) (Baz et al., 1992), but the heart of the proposed algorithm stems from adaptive signal processing for real-time frequency estimation. The proposed algorithm is applied to experimental vibration suppression of a composite plate with a piezoelectric sensor and an actuator. The natural frequencies of the specimens are changed by adding masses. The performances of the proposed APPF controller are experimentally compared to those of Linear Quadratic Gaussian (LQG) controller.
ADAPTIVE POSITIVE POSITION FEEDBACK
Variable Positive Position Feedback The PPF controller has been successfully applied to multi-modal vibration control (Fanson and Caughey, 1987; Baz et al., 1992; Han and Lee, 1999; Friswell and Inman, 1999), since it shows robustness to system parameter changes and spillover prevention characteristics. It also has a simple decentralized structure so that controller design is relatively simple. Design conflict between the robustness and performance can be easily tuned by the adjustment of the controller damping ratios. Here is the transfer function of a PPF controller for a single mode vibration.
ω 2f H PPF ( s )i = 2 ⋅ K ci s + 2ζ f ω f s + ω 2f
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(1)
where H PPF (s) i is the PPF controller for a specific mode; ( )i is an index for the mode to be
controlled; ωf is the filter frequency of the PPF controller, which is generally set to be the same as the natural frequency of the structure. ζf is the damping ratio of the PPF controller. When ζf increases, the robustness with respect to natural frequency variations increases, but the performance is degraded. The value of ζf , which is typically selected between 0.2 and 0.3, was chosen to be 0.3 in this study. Kci’s are the adjusting gains for each mode. Han and Lee (1999) showed that control performances are degraded when the natural frequencies of structures changes. The key idea of the proposed controller, APPF, is that the filter frequency, ωf, of the controller is continuously updated according to the identified natural frequency of the strcutre in order to prevent performance degradation due to structural variations. Since the PPF has inherently decentralized structure, we can extend the above idea for multi-modal vibration control if we can estimate multi-modal natural frequencies of the strcutre in real-time. The proposed APPF controller is composed of an on-line multi-modal frequency estimation block and a variable PPF block as shown in Figure 1. Since the frequency estimation reply on digital signal processing, we need the discrete version of variable PPF controller, which was obtained using the following pole-zero-mapping method (Ogata, 1995).
z = e sTcon
(2)
s = poles , zeros
The obtained discrete PPF controller is
H PPF ( z )i = K di where
z −1 1 + Pi z −1 + Qi z − 2
(
K di = K ci ⋅ ω f Qi sin ω f 1 − ζ f2 Tcon
Pi = −2e
−ζ f ω f Tcon
)
(3)
1 − ζ f2 ,
)
(
cos ω f 1 − ζ 2f Tcon ,
Qi = e
−2ζ f ω f Tcon
(4) -5-
where Tcon is the time step of the controller, which may be different from Ts, the time step of the frequency estimation. Since the vibration of three lowest modes is targetted in this study, three variable PPF controllers corresponding to each mode are connected in parallel as shown in Figure 1. However, the present APPF is easily extendable to control more modes. At every time step, Tcon, the filter parameters, Kdi, Pi, and Qi, in Equation (4) are updated using the estimated frequencies of the structure.
Frequency estimation algorithm will be
described in the next section.
Estimation of Multi-modal Frequencies
Since 1970’s, many algorithms have been developed for the on-line estimation of multimodal frequencies in the fields of radar, sonar, communications, and damage detection (Haykin, 1994; Chan, Lavoie and Plant, 1981; Varkonyi-Koczy et al., 1998; Sastry and Bodson, 1989). There are two main approaches for the frequency estimation. Non-parametric methods are generally believed to be more accurate, but suffer from heavy computational burden and leakage, because the algorithms are based on the fast Fourier transform. Parametric methods use pre-determined structures of the system model so that they are relatively fast and have high resolution despite of short data records. The frequency estimation algorithm should have light computational burden, good convergence characteristics, and the adaptation ability in the noisy environment. In general, it is quite difficult to detect natural frequencies from the vibration signal, since each mode has different damping and amplitude level (Regalia, 1995; Chan, Lavoie, and Plant, 1981; Varkonyi-Koczy et al., 1998). The present frequency estimation method is composed of Recursive Least Squares (RLS) method and the Bairstow method.
RLS recursively estimates the coefficients of the -6-
characteristic equation of a linear system without matrix inversion and gives good convergence from arbitrary initial parameters. The Bairstow method decouples the characteristic equation into each mode (Chapra and Canale; 1998). A transfer function of an arbitrary linear structure can be described in even order discrete form as follows:
b1 z −1 + b2 z −2 + ⋅⋅⋅ + b2 p z −2 p B(z −1 ) H (z ) = = A(z −1 ) 1 + a1 z −1 + a2 z −2 + ⋅⋅⋅ + a2 p z −2 p −1
(5)
where A(z-1) and B(z-1) are called Auto Regressive (AR) and Moving Average (MA) parts, respectively. The roots of A(z-1) and B(z-1) are poles and zeros of the system. p is the number of vibration modes. We are only interested in the denominator, A(z-1), since the natural frequencies are correlated only with AR part. Let us define the measured signal vector as φ(k) and the unknown parameter vector as θ(k) to simplify the characteristic equation.
φ (k ) = [− y (k − 1) − y (k − 2) " − y (k − 2 p)]T , θ (k ) = [a1 (k ) a2 (k ) " a2 p (k )]T
(6)
where k is an index denoting the current state. Then, the estimated signal at the current state can be described as 2p
ye (k ) = −∑ a j y (k − j ) + ε (k ) = θ (k )T ⋅ φ (k ) + ε (k )
(7)
j =1
where ε is an error between the measured and estimated signals. Estimation of the characteristic equation means that the square of error in Equation (7) is to be minimized with respect to the following objective function using RLS method. 2p
V = ∑ λ 2 p − jε 2 ( j ) ,
( 0 < λ
