Abstract
A finite element model of plate partly treated with ACLD treatments is developed based on the constitutive equations of elastic, piezoelectric, viscoelastic materials and Hamilton’s principle. The Golla-Hughes-Mctavish (GHM) method is employed to describe the frequency-dependent characteristics of viscoelastic material (VEM). A model reduction is completed by using iterative dynamic condensation and balance model reduction method to design an effective control system. The emphasis is concerned on hybrid (combined feedback/feedforward) control system to attenuate the vibration of plates with ACLD treatments. The optimal linear quadratic Gaussian (LQG) controller is considered as a feedback channel and the adaptive filtered-reference LMS (FxLMS) controller is used as a feedforward channel. They can be utilized individually or in a hybrid way to suppress the vibration of plate/ACLD system. The results show that the hybrid controller which combines feedback/feedforward together can reduce the displacement amplitude of plate/ACLD system subjected to a complicated disturbance substantially without requiring more control effort. Furthermore, the hybrid controller has more rapid and stable convergence rate than the adaptive feedforward FxLMS controller. Meanwhile, perfect robustness to phase error of the cancellation path in feedforward controller and the weight matrices in feedback LQG controller is demonstrated in proposed hybrid controller. Therefore, its application in structural engineering can be highly appreciated.
1. Introduction
Vibration and noise control is of great interest in many industrial structures, like airplane passenger cabin, vehicle body structure, and submarine hull. The thin-wall structure is a key part in the body/cabin/hull structure. The vibrating thin-wall structures, like plates and shells, which are mainly disturbed by the harmonic disturbance induced from the engine and other rotating machines, as well as stochastic disturbance, make radiation noise into the passenger cabin, influencing the performance of the structure.
Active vibration control (AVC) [1], which is well-known for its efficiency in low frequency band where traditional passive strategies cannot work well, has been applied to various structures over the past two decades. Active vibration control of simple structure, such as beams [2] and plates [3], was researched broadly in the open literatures. Thakkar and Ganguli [4] investigated the twist vibration control of helicopter rotor blade by using the piezoelectric actuators. Rao et al. [5] designed a controller and demonstrated experimentally the effectiveness of multimode vibration control in the composite fin-tip of aircraft. Kwak and Yang [6] studied the suppression of vibration of ring-stiffened cylindrical shell in contact with external fluid using piezoelectric sensor and actuator.
Active constrained layer damping (ACLD) is a hybrid active-passive vibration control technique [7] which combines the advantages of conventional passive constrained layer damping (PCLD) and active vibration control (AVC). Therefore, in the same damping treatment, the broader band vibration suppression can be achieved through ACLD treatments. In recent decades, extensive efforts had been made to reduce the vibration of the thin-wall structure using ACLD treatments. Baz and Ro, as the pioneering researchers, demonstrated the feasibility of using the ACLD in controlling the vibration of rotating beams with the ultimate goal of extending its application of attenuating the structural vibration of rotor blades [8]. The vibration control of shells [9, 10] was also investigated to extend the application of ACLD in the cabin of aircraft and submarine. In these studies, the PVDF (polyvinylidene fluoride) was served as the sensors and actuators. They use the simple proportional and derivative feedback of the transverse deflection to suppress the vibration of the base structure theoretically and experimentally. To improve the force transmissibility between piezoelectric actuator and the base structure, Liao and Wang [11] proposed an enhanced active constraining layer (EACL), and Gao and Liao [12] extended this configuration. Kumar et al. [13] added the stand-off layer (SOL) between the viscoelastic layer and the base structure to increase the viscoelastic strain and enhance the effect of the active force. In recent years, piezoelectric fiber-reinforced composite (PFRC) materials [14] and piezoelectric composites (PZC) [15, 16], which have a wide range of effective materials properties, were introduced in the ACLD treatments to serve as active constrained layer. Shah and Ray [17] focused on investigating the effect of the piezoelectric fiber orientation angle in the constrained layer on the performance of ACLD patches. In these studies, the emphasis was placed on the vibration characteristics analysis/enhancement of the performance of the ACLD treatments.
In the open literatures, much attention was paid on active control methods in the structural active/active-passive vibration control. The optimal feedback control [18], velocity feedback control [19], fuzzy logic control [3], and the control [5] were usually employed in active vibration control (AVC). Viswamurthy and Ganguli [20] compared the global harmonic control with local blade optimal control applied to vibration control of helicopter. For active-passive ACLD/structure, Baz [21] presented a variational mathematical model for beams and a globally stable boundary controller was investigated to get the high damping characteristics over broad frequency range. Meanwhile, Baz [22] developed a robust controller based on the transfer function model of the ACLD beam. The control strategy was stable in the presence of parameter uncertainty, as well as ensured optimal disturbance rejection capabilities. Furthermore, the infinity robust control strategy was utilized by Crassidis et al. [23] to maximize disturbance rejection capabilities over a desired frequency band. Liu et al. [24] developed a infinity controller based on the reduced finite element model. Rodriguez [25] made a comparison of sliding mode control (SMC) with state-feedback control (LQR) applied to a partially treated ACLD beam. In these studies, the control strategies can be classified into feedback control theories. These control strategies have been shown to be more suitable in applications where the structure is disturbed by the stochastic or impulsive disturbances.
Feedforward control, which is more effective in the case where the deterministic or correlated information about the disturbance is known, is another popular control strategy. It was widely used in active vibration and noise control field [26, 27]. However, only a few feedforward theories were utilized in structural vibration control with ACLD treatments. Cao [28] employed the feedforward LMS algorithm to control the vibration of a plate treated with ACLD treatments. The finite element model was developed and ADF method was used to identify the performance of VEM. Furthermore, hybrid control of feedback and feedforward control theories were also successfully used to noise and vibration suppression [29, 30]. However, it was rarely investigated for structural vibration control with ACLD treatments. Vasques validated the effectiveness of the hybrid control strategies for ACLD beam [31].
In this paper, the focus is placed on combining individual feedback control based on linear quadratic Gaussian (LQG) and feedforward control based on adaptive filtered-reference LMS (FxLMS), to develop a hybrid controller for the vibration control of plates treated with ACLD treatments. The performance of vibration suppression subjected to different control strategies is compared and the effect of parameters on vibration suppression of the plate with ACLD treatment is further investigated.
This paper is organized in the following five sections. A brief background is introduced in Section 1. The dynamic models including the finite element model of the plate treated with ACLD patches and GHM model of VEM are developed in Section 2. In Section 3, the feedback controller, the feedforward controller and the hybrid controller are designed. A numerical example is discussed in Section 4. Finally, the main results and future work are summarized in Section 5.
2. The ACLD/Plate System Model Construction
2.1. Finite Element Model
The ACLD/plate system is illustrated in Figure 1. The viscoelastic damping layer is sandwiched between two piezoelectric layers which serve as piezosensor and piezoactuator, respectively. The three layers are bonded to the base plate acting as a smart constraining layer damping. It is assumed that the piezosensor and the base plate are perfectly bonded together such that they can be reduced to a single equivalent layer. Some other assumptions are made in [32].
The ACLD/plate elements considered are four-node two-dimensional elements. Each node has seven degrees of freedom to describe the longitudinal displacements and of the constrained layer, the longitudinal displacements and of the base plate, the transverse deflection , and the slopes and of the deflection line. The longitudinal displacement functions are expressed as 4-term polynomials, and transverse deflection is expressed as a 12-term polynomial. Therefore, the element deflection vector is given by where for .
Applying Hamilton’s principle, the equations of motion for an ACLD element can be written as where , , , , and stand for the element mass matrices for base plate, the constrained layer, and the viscoelastic layer, , and stand for the element stiffness matrices for base plate, the constrained layer, and the viscoelastic layer, is the shear stiffness matrix for viscoelastic layer, is the shear modulus of the viscoelastic materials, is the external force vector, and is the control force produced by the piezoelectric actuators.
Assembling the condensed system for all elements yields the dynamic equation of the plate with ACLD treatments: where is the global stiffness matrix without considering the shear stiffness of the VEM.
2.2. Viscoelastic Damping Model
The frequency- or time-dependent behavior of the viscoelastic material is characterized by using Golla-Hughes-Mctavish (GHM) method [24]. In this approach, the shear modulus of VEM is expressed as a series of “minioscillator” terms in the Laplace domain where corresponds to the equilibrium value of the modulus, constants , and govern the shape of the modulus function over the complex -domain. A column matrix of dissipation coordinates is introduced:
Considering a three-term GHM expression, (3) can be rewritten as follows: where each vector and matrix can be expressed as where , , , , is a diagonal matrix of the positive eigenvalues of matrix , and is the eigenvectors matrix corresponding to the eigenvalues. For simplicity the global equation of motion can be rewritten as
2.3. Model Reduction and State Space Design
As well known, the scale of the finite element model is increased largely due to the introduction of the dissipation coordinates from GHM model. This requires much computational effort to calculate the dynamic response of the plates with ACLD treatments. Otherwise, not all the state variables in the model are controllable and observable. Therefore, model reduction is necessary to design an effective controller. Here, an iterative dynamic condensation [33] is used to reduce the scale of the finite element model.
Equation (8) can be rewritten as
Here, a matrix is defined to relate the master d.o.f.s with the slave d.o.f.s and, after iterations, the reduced dynamic equation is where
Based on above reduced model, the state space form of the ACLD/plate is derived as follows: where the state space vector is chosen as , is system matrix, and and are disturbance input matrices and the control force distribution matrices. is the output matrix and is the output vector. represents the measurement noise.
A further model reduction is performed with the balance method in the state space to guarantee the control system controllable and observable. The details are shown in [34].
3. Control System Architecture Design
3.1. Feedback Control
The ultimate aim of the feedback control is to reduce the motion of the system to the greatest possible extent; in that case, the control system is said to act as a regulator. Here, the linear quadratic regulator (LQR) is designed to control the vibration of plates with ACLD treatments. Cost function or performance index is given by or where , , and are the state variable, the output, and the control input weighting matrices, respectively.
The control input can be calculated using the state feedback so that
The optimal feedback gain matrix is given by where is the solution of the famous Riccati equation.
Therefore, considering the control law in (15) and substituting it into the state space equation of the reduced model, the closed-loop state equation is given by
In above equation, it is assumed that all the state variables can be measured. However that is not always available in a real system; an optimal design methodology that only utilizes noise, partial state information is required. The linear quadratic Gaussian (LQG) methodology provides a means of designing such controllers. The LQG controller is a combination of a LQR and a Kalman filter, a fact known as the separation principle.
The Kalman filter is an optimal state estimator which can minimize the effects of process and measurement noise. In (18), the process noise and measurement noise are both assumed to be white and have a Gaussian probability density function. Meanwhile, they are assumed to be uncorrelated with the inputs. The correlation properties of the plant and measurement noise vectors are given by the correlation matrices where denotes the expectation operator. These matrices are used as weighting matrices of the steady-state Kalman filter problem [20].
A state estimate that minimizes the steady-state error covariance is constructed by the Kalman filter
In this way, a new state space model is generated where is the filter gain; it is determined by solving an algebraic Riccati equation; is the sum of the output and the measurement noise.
3.2. Feedforward Control
The filtered-reference LMS (FxLMS) algorithm is widely studied and has been used in many active vibration control areas due to its easy implementation and remarkable performance. In a practical vibration control system, a cancellation path exists, which has an important effect on the control results. In FxLMS algorithm, the model of the cancellation path is established. The primary disturbance passes through the estimated model of the cancellation path, and a filtered-reference signal , which is correlated with the disturbance, is produced. This signal is then adaptively filtered to generate the necessary control action to cancel the effect of the primary disturbance.
An adaptive finite impulse response (FIR) filter, whose th coefficient at the th sample time is updated using the LMS algorithm, is used in filtering process. The coefficients vector and the filter output are obtained from where is the step size, which has a significant influence on the convergence rate of the LMS algorithm, and is the number of filter coefficients (or the filter length).
The control signal has to pass through a part of the physical system before the error sensor measures the output. This physical path, , is called the cancelation path; this transfer characteristics also affect the performance of the FxLMS algorithm. Therefore, the output of the plant due to the feedforward control input only, , is given by where is the discrete impulse response of the cancelation path which is assumed to be of order . Considering the measurement noise , the output of the system, , can be written as where is the response due to the effects of the primary disturbance alone.
3.3. Hybrid Control
In general, the feedback control is more appropriate for suppression of stochastic disturbance. Meanwhile, it is susceptible to modeling errors. By contrast, the adaptive feedforward control is not susceptible to modeling errors and inherently stable, and it is more suitable to reduce the periodic or limited-band disturbance. Therefore, it is necessary to develop a hybrid (combined feedback/feedforward) control to attenuate vibration of plates with ACLD treatments against stochastic and harmonic disturbance simultaneously. The schematic diagram of the discrete-time generalized plant for a hybrid SISO control system is shown in Figure 2. It is seen that LQG control is considered as feedback channel and an adaptive filtered-reference LMS control is used as feedforward channel.
4. Numerical Simulation and Discussion
To the first bending mode of the plate, a cantilever ACLD/plate as shown in Figure 3 is used as numerical example to demonstrate the effectiveness of the above controllers. The emphasis is on the performance discussion of the controllers.
4.1. The Numerical Model of the ACLD/Plate System
The base plate, with the size of , is partly treated with ACLD treatments. The main physical parameters of the base plate (aluminum), viscoelastic layer (ZN-1), and piezoelectric layer (P-5H) are listed in Table 1.
The cantilever plate/ACLD is divided into 8 × 4 elements. Four ACLD composite elements are applied in the root. The locations of the ACLD patches are decided according to the modal energy of the first mode [21], which is shown in Figure 4. A model reduction in which the deflection of the structure and displacements of the piezoelectric layer are selected to be master d.o.f.s and all others are slave d.o.f.s is performed according to the model reduction procedures in Section 3. The bold point in Figure 3 is used as the excited point and also the response point. The comparison of responses in frequency domain and time domain between the original model and the reduced model is presented in Figure 5. It is clear that the dynamic response of the reduced model is nearly coordinate with the dynamic response of the original model. A SISO control system is considered to design individual and hybrid controller. The input is the control voltage applied to the ACLD treatments, and the output is selected to be the displacement of the response point of the plate, which is assumed to be measurable.
4.2. The Control Results of the Feedback Controller
The LQG feedback controller is tested to mitigate the effects of stochastic disturbances. A white noise mechanical disturbance is modeled as process noise (input force disturbance). The process noise is such a signal with and the measurement noise is a signal with . The output weighting matrix is set to be and the control weighting matrix . In order to avoid the depolarization, the maximum control voltage applied to piezoelectric actuators is set to be 150 V. The control results and the control voltage are presented in Figure 6.
(a)
(b)
As can be seen, when compared to the PCLD/plate system, the ACLD treatments can reduce the displacement of the response point of the plate significantly. The control voltage is limited within 30 V. This means that the proposed feedback control is effective against the stochastic disturbance.
4.3. The Control Results of the Feedforward Controller
The feedforward controller based on adaptive FxLMS algorithm is designed to attenuate the effects of the harmonic point-force disturbance on the ACLD/plate system in Figure 2. The disturbance frequency is close to the first natural frequency of the ACLD/plate system. For convenience, the estimation model of the plant cancellation path is substituted by the control path (control voltage to measured displacement) in the state space model. After that, the FxLMS algorithm is implemented. The filter length is set to be 32 and the step size in the FxLMS controller. The results are presented in Figure 7.
(a)
(b)
It can be seen that the displacement of the plate with ACLD treatment is attenuated from 9.4 mm to 0.08 mm at the end time of 20 s. The control voltage is close to 50 V. The vibration attenuation is improved substantially by ACLD treatments when the adaptive FxLMS algorithm is applied. It is clear that the FxLMS algorithm is very useful to attenuate the vibration of ACLD/plate system excited by harmonic disturbance.
4.4. The Control Results of the Hybrid Controller
Furthermore, it is particularly true that the system is often subject to both the harmonic and stochastic disturbances in engineering practice. Therefore, a hybrid controller, which combines the LQG feedback control and the FxLMS feedforward control together, is developed. Such a hybrid controller is further designed to attenuate the vibration of the plate/ACLD system subjected to the complicated disturbance. In the hybrid controller, the FxLMS control is employed as the feedforward channel, which aims to damp out the harmonic disturbance. Meanwhile, the LQG feedback control is used as a compensator. The mixed disturbance is the sum of the two previous disturbances. The results of individual LQG feedback control, feedforward control, and their combination are presented in Figures 8, 9, and 10, respectively.
(a)
(b)
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(b)
(a)
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When the hybrid control system is engaged, the feedback controller with the same weights and the feedforward controller with the same filter length are used, and the step size is set to be 0.5 to accelerate the convergence rate. It can be seen from Figures 8(a) and 10(a) that at the end of 20 s, the displacement amplitude produced by the hybrid controller is much smaller than that of the LQG feedback controller. It is also apparent that the fluctuation disappears during the convergence process and the convergence rate is improved from 10 s to 5 s when the hybrid controller is applied instead of the feedforward controller. Figure 11 shows that the hybrid control produces a smaller MSE than the feedforward control. The maximum control voltage in the feedforward controller achieves 210 V. However, the control effort in the hybrid controller is only about 110 V, a little larger than the stable control voltage of the two single controller.
4.5. The Performance Discussions to the Controllers
Table 2 lists the convergence situations for hybrid controller and FxLMS controller under different step sizes. It can be seen that the hybrid controller can converge over a larger range of step size. This implies that the hybrid controller has a better robustness than the FxLMS controller.
The effect of different step sizes on the performance of ACLD/plate with the FxLMS and hybrid controller is discussed. It is seen in Figure 12 that when the step size is increased, MSE decreases rapidly; meanwhile, the fluctuation increases. This means that convergence process is not smooth anymore, which results in the fluctuation of the control voltage shown in Figure 9(b). On the other hand, at the end of the iteration process, the MSE approaches to the same value. Figure 13 shows that MSE varies with iteration number in hybrid control under different step sizes. It can be seen clearly that the fluctuation is reduced substantially in the convergence process and the convergence rate is improved greatly when the step size increases. This results in a stable control voltage in Figure 10(b).
In general, the cancellation path model of the feedforward controller is obtained by the system identification algorithm. Therefore, the errors including amplitude error and phase error in the cancellation path has a significant effect on the controller performance. Figures 14 and 15 show that the effect of phase errors of the cancellation path on the performance of ACLD/plate is considerable. It is seen that when the phase error achieves 30°, the control process in feedforward control will not converge anymore. However, when the phase error achieves 60°, the control process is still convergent in the hybrid control. It can be concluded that the hybrid control is not sensitive to the phase error, compared with the feedforward control.
Figure 16 shows the MSE using LQG feedback control under different matrix. Figure 17 demonstrates the MSE by using hybrid control under different matrix. It can be found that the performance of the feedback control has a distinct fluctuation when different matrix is applied. However, the performance of the hybrid control changes indistinctively. That implies that the hybrid control is more stable than individual LQG control.
Based on the above discussions, it can be inferred that the hybrid control will be a potential for vibration control of aerospace and other structures. The hybrid controller demonstrates a better performance than that of individual feedback/feedforward controller with different parameters. Meanwhile, even though one of individual controllers is out of work, the other will still be effective on suppressing the vibration of structure. However, it is reasonable that the modeling accuracy of the cancellation path will make an influence on performance for vibration control in practical applications.
5. Conclusions
In this paper, a finite element model of ACLD/plate system is developed. The behavior of the viscoelastic material is modeled by the Golla-Hughes-Mctavish (GHM) method. The model reduction is carried out by means of dynamic condensation and balance reduction methods, respectively, to design an effective control system. The emphasis has been placed on vibration control of ACLD/plate using the hybrid controller which is composed of the LQG feedback controller and the adaptive FxLMS feedforward controller. The performance of these controllers with different parameters is further discussed in detail.
The results show that the reduction process is effective in low frequency range. The LQG feedback controller is very useful to attenuate the vibration of the ACLD/plate system against stochastic disturbance, while the adaptive FxLMS feedforward controller is more appropriate to attenuate the vibration of the ACLD/plate system induced by harmonic disturbance. Furthermore, the LQG feedback controller and the adaptive FxLMS feedforward controller are combined as the hybrid controller, which can reduce substantially the displacement amplitude of plate/ACLD system subjected to a complicated disturbance without requiring more control effort and is more stable and smooth. On the other hand, the hybrid controller demonstrates a rapider and more stable convergence rate than the adaptive feedforward controller. Meanwhile, the hybrid controller demonstrates much better robustness than individual LQG feedback control or adaptive feedforward control.
It is worth mentioning that although the hybrid controller presented is shown theoretically with better performance for a single mode vibration control, the research for SISO/MIMO vibration control of several modes using hybrid control strategies is more applicable in practical flexible structures of aerospace and other engineering fields. An attempt to address these issues is recommended to future studies.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by Nature Science Foundation of China (no. 50775225), the State Key Laboratory of Mechanical Transmission, Chongqing University (no. 0301002109165). These financial supports are gratefully acknowledged.