A fuzzy robust control scheme for vibration suppression of a nonlinear electromagnetic-actuated flexible system

Mechanical Systems and Signal Processing 86 (2017) 86–107

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A fuzzy robust control scheme for vibration suppression of a nonlinear electromagnetic-actuated flexible system

cross

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A.R. Tavakolpour-Saleh , M.A. Haddad Department of Mechanical and Aerospace Engineering, Shiraz University of Technology, Shiraz, Iran

A R T I C L E I N F O

ABSTRACT

Keywords: Robust vibration control Fuzzy active force control Acceleration feedback

In this paper, a novel robust vibration control scheme, namely, one degree-of-freedom fuzzy active force control (1DOF-FAFC) is applied to a nonlinear electromagnetic-actuated flexible plate system. First, the flexible plate with clamped-free-clamped-free (CFCF) boundary conditions is modeled and simulated. Then, the validity of the simulation platform is evaluated through experiment. A nonlinear electromagnetic actuator is developed and experimentally modeled through a parametric system identification scheme. Next, the obtained nonlinear model of the actuator is applied to the simulation platform and performance of the proposed control technique in suppressing unwanted vibrations is investigated via simulation. A fuzzy controller is applied to the robust 1DOF control scheme to tune the controller gain using acceleration feedback. Consequently, an intelligent self-tuning vibration control strategy based on an inexpensive acceleration sensor is proposed in the paper. Furthermore, it is demonstrated that the proposed acceleration-based control technique owns the benefits of the conventional velocity feedback controllers. Finally, an experimental rig is developed to investigate the effectiveness of the 1DOF-FAFC scheme. It is found that the first, second, and third resonant modes of the flexible system are attenuated up to 74%, 81%, and 90% respectively through which the effectiveness of the proposed control scheme is affirmed.

1. Introduction The flexible plate structures have received many attentions from the designers owing to their extensive applications in mechanical systems such as aircraft and submarine. However, such flexible plates can be more easily affected by unwanted vibrations. Indeed, the amplitude of unwanted vibrations in the flexible structures should be controlled so that the flexible systems do not experience performance degradation or structural damage. Accordingly, vibration control of the flexible plates is a significant task. The Vibrations of flexible plate structures can be attenuated by either passive or active vibration control techniques. The passive control method consists of mounting passive material on the structure in order to change its dynamic characteristics such as stiffness and damping coefficient. This method is efficient at high frequencies, but expensive and bulky at low frequencies [1,2]. Moreover, passive vibration control usually leads to an increase in the overall weight of the structure. On the other hand, active vibration control (AVC) is a technique to electronically generate an additional vibration field to cancel out the unwanted vibrations at an observation point. Lueg [3] was among the first researchers who introduced the concept of active noise control in pipes, which was later extended to active vibration control problems as well. Generally, there are two well-defined AVC approaches for the flexible

⁎

Corresponding author. E-mail addresses: [email protected], [email protected] (A.R. Tavakolpour-Saleh).

http://dx.doi.org/10.1016/j.ymssp.2016.09.039 Received 22 June 2016; Received in revised form 17 September 2016; Accepted 24 September 2016 0888-3270/ © 2016 Elsevier Ltd. All rights reserved.

Mechanical Systems and Signal Processing 86 (2017) 86–107

A.R. Tavakolpour-Saleh, M.A. Haddad

plates namely feedforward and feedback control schemes [4]. In designing the feedforward vibration controllers, it is assumed that a reference signal representing the primary source of vibration is observable. However, a suitable reference signal is not available in many cases. Besides, the feedforward AVC scheme mostly attenuates the unwanted vibrations in the flexible plates locally. On the other hand, in the feedback AVC schemes, an error signal is fed into the controller and then, the feedback controller generates a control signal to drive the error signal to zero. It is important to note that the stability is assured in the feedback AVC systems containing collocated actuator/sensor pairs [4]. Furthermore, overall vibration attenuation can be found in the flexible plate system using the feedback controllers. Hence, a novel feedback controller was considered in this paper. However, in the digital implementation of the feedback control algorithms, the bandwidth of the control system is usually dependent on the sampling frequency. One should keep in mind that the sampling frequency of a feedback AVC system is restricted by different parameters such as hardware and software limitations. The software limitation is usually the dominant factor. Thus, the increase of sampling frequency results in the increase of bandwidth in the feedback controllers [4]. Obviously, a simple feedback control algorithm helps to reduce the processing time, which leads to a higher sampling frequency. Accordingly, the bandwidth of the control system can be modified. As a result, the simple feedback AVC schemes have a better chance to cope with the high-frequency vibrations. In contrast, more complicated control algorithms are not usually effective for controlling the high-frequency vibrations. Thus, another objective of the paper is to consider the simplicity of the feedback control algorithm because of the mentioned benefits. In general, three observation signals, including displacement, velocity, and acceleration can be employed in the feedback AVC systems [5]. The main drawback of both displacement and acceleration feedbacks is due to the phase shift between the observation signal and the actuating force, which considerably affects the performance of the feedback controller. Whereas, in the velocity feedback AVC systems, the observation signal is in-phase with the actuating force through which the control system performance is improved. Hence, the velocity feedback control is known as the most effective control strategy for suppressing vibrations of the flexible plate structures. However, the velocity sensors, which are the key elements of the velocity feedback controllers, are often massive and expensive compared to the conventional acceleration sensors. Consequently, the next objective of the paper is to present an AVC algorithm based on the acceleration feedback, but with the benefits of the velocity feedback controllers. Many researchers proposed different feedback controllers to suppress unwanted vibrations of the flexible structures. Qiu et al. [6] presented an acceleration sensor-based modal identification and active vibration control methods to suppress the first two bending and the first two torsional modes of vibration of a cantilever plate. Three acceleration sensors and piezoelectric actuator patches were mounted on the structure at optimal locations to decouple the bending and torsional vibrations in sensing and actuating actions. Two acceleration-based control approaches, including the acceleration proportional feedback and a nonlinear control scheme, were proposed and experimentally compared. Finally, effective vibration suppression was found based on the acceleration-based control algorithms. Previdi et al. [7] reported another work on the acceleration feedback AVC. They applied a single acceleration sensor collocated with a piezoelectric actuator to a kitchen hood to reduce the unwanted vibrations in the system. Two different minimum variance control laws were considered. The first controller operated without the information about the hood motor velocity while the second one required the velocity information. Vibration reductions of about 85% and 75% were found for the first and second controllers respectively. An et al. [8] applied a time-delayed acceleration feedback controller to a flexible structure. Due to the introduction of time delay into the controller with acceleration feedback, the presented control system had the feature of not only changing the mass property but also altering the damping property of the controlled system. Thus, the control system behavior was improved considerably with this feature. The proposed controller was applied to a flexible beam and its performance in comparison with a pure acceleration feedback controller was experimentally evaluated. Enríquez-Zárate et al. [9] applied positive acceleration feedback (PAF) and multiple positive acceleration feedback (MPAF) controllers to suppress vibrations of a building-like structure. The experimental results revealed a relatively acceptable performance of the proposed feedback controllers in suppressing the first three dominant modes of vibration of the structure. Jenifene [10] proposed a simple position feedback controller for AVC of a single-link flexible manipulator. A delayed position feedback signal was used to actively control the vibration of the flexible structure. This method was found to be acceptable in lightly damped dynamic systems. Tavakolpour-Saleh et al. [2] proposed a self-learning control technique with displacement feedback for active control of a rectangular flexible plate with clamped edges. Iterative learning algorithm along with the collocated piezoelectric actuator and laser displacement sensor were considered in this investigation. It was demonstrated that the proposed control system effectively suppressed the unwanted vibrations. Gosiewski and Koszewnik [11] investigated the performance of the proportionalderivative (PD) controller to attenuate vibrations of a 3D space truss. They considered Eddy current displacement sensor as the sensing element of the feedback controller. The effectiveness of the proposed active vibration damping system was demonstrated via simulation and experiment. Khorshidi et al. [12] studied active vibration control of a circular plate coupled with piezoelectric layers on both sides using simulation. They proposed linear quadratic regulator (LQR) and fuzzy logic controller (FLC) to control the transverse displacement of a circular plate, which was excited by sound pressure waves. The simulation results revealed the effectiveness of the proposed control techniques. Wu et al. [13] applied independent modal control approach based on negative velocity feedback to a highly flexible beam. They used Hamilton's principle to model the beam dynamics. Piezoelectric patches were mounted on the hosted beam and the first three modes of vibration of the flexible system were investigated. The simulation and experimental results revealed the effectiveness of the proposed control method in suppressing the unwanted vibration. Gupta et al. [14] proposed a negative velocity feedback to control the first mode of vibration of a smart plate at different temperatures. The simulation results showed an effective vibration suppression of the smart plate. However, no experimental results were given to validate the simulation outcomes. Active force control (AFC) is a well-known disturbance rejection technique that was proposed in a compact form by Hewit and Burdess [15] in 1816 for robotic applications. The conventional AFC scheme consisted of two feedback loops that formed a 2 degree87

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of-freedom (DOF) control system. The inner loop with acceleration feedback served as a regulator to modify the effect of disturbance while the outer loop with position feedback was a servo controller for trajectory tracking. Tavakolpour-Saleh [16] presented the first attempt to employ only the inner loop of the conventional AFC scheme as a 1DOF controller for active vibration control of flexible structures. The collocated piezoelectric actuator and sensor with linear characteristics were proposed for implementation of the control algorithm. It was numerically and experimentally proved that the inner loop of the conventional AFC scheme was merely enough to suppress the unwanted vibrations in a flexible plate system with all clamped edges. Other researches on the applications of AFC scheme can be found in [17,18]. In this article, a novel acceleration-based feedback control scheme, namely 1DOF-FAFC incorporating a nonlinear electromagnetic actuator is proposed for active vibration control of a flexible plate structure with CFCF boundary conditions. First, an appropriate computational platform of the flexible plate system is presented and experimentally validated. Then, a hand-made electromagnetic actuator with nonlinear characteristics was developed and experimentally modeled via a system identification scheme. The obtained parametric model of the actuator is then applied to the computational platform of the flexible system and the performance of the proposed fuzzy control scheme incorporating the proposed nonlinear actuator is evaluated through simulation. Finally, an experimental rig is set up and the feasibility of the proposed control system is investigated. 2. Computational model of the flexible system The vibration problem of a flexible thin plate can be formulated as a partial differential equation together with the corresponding boundary conditions. The plate is assumed to undergo a small lateral deflection. Using Kirchhoff's plate theory, the following partial differential equation (PDE) governing the flexible system can be acquired [2,16]:

∂ 4w (x, y, t ) ∂ 4w (x, y, t ) ∂ 4w (x, y, t ) ρh ∂ 2w (x, y, t ) 1 +2 + + = F (x , y , t ) ∂x 4 ∂x 2∂y 2 ∂y 4 D ∂t 2 D

(1) 3

2

where w is the lateral deflection in z direction, ρ is the density of the plate, h is the thickness, D=(Eh )/(12(1−υ )) is the flexural rigidity, F is the transverse external force with dimension of force per unit area, E is the modulus of elasticity, υ is the Poisson ratio, x and y are coordinate variables. To simulate the plate system, the central FD method was used to numerically solve Eq. (1). The obtained central difference equation can be written as:

wi, j, k +1 = −

DΔt 2 (Pwi, j, k ρh

+ Q (wi +1, j, k + wi −1, j, k ) + (Wi, j +1, k + wi, j −1, k ) +

+ S (wi +1, j +1, k + wi −1, j +1, k + wi −1, j −1, k + wi +1, j −1, k ) + T (wi +2, j, k + wi −2, j, k ) + + U (wi, j +2, k + wi, j −2, k )) + 2wi, j, k − wi, j, k −1 +

Δt 2 F (i , ρh

j, k )

(2)

where i, j, and k are the reference index along x, y, and z directions respectively and Δt is the sampling time. The terms U, P, Q, R, S, and T can be expressed as follows:

P=

6 8 6 + + Δx 4 Δx 2Δy 2 Δy 4

(3)

Q=−

4 4 − Δx 4 Δx 2Δy 2

(4)

R=−

4 4 − Δy 4 Δx 2Δy 2

(5)

S=

2 Δx 2Δy 2

(6)

T=

1 Δx 4

(7)

U=

1 Δy 4

(8)

Details on initial and boundary conditions and their equivalent difference equations have been previously given in [16]. One should keep in mind that Eq. (2) is the final difference equation corresponding to the PDE presented earlier in Eq. (1) through which the rectangular thin plates under different excitation forces can be modeled and simulated. The simulation procedure is discussed in Section 4.2. The simulation platform was designed so that it was possible to apply excitation forces at any desired locations on the flexible system and observe the plate response at the desired observation point. According to the well-known Von Neumann stability conditions [19] for convergence of a finite difference equation, Eq. (9) can be obtained to satisfy the convergence of Eq. (2):

0≤c≤

1 4

(9) 88

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where

⎛ DΔt 2 ⎞ ⎛ Δy 2 Δx 2 ⎞ c=⎜ + ⎟ ⎜2 + ⎟ Δx 2 Δy 2 ⎠ ⎝ ρΔx 2Δy 2 ⎠ ⎝

(9a)

For the implementation of the simulation algorithm, a 1.25 mm thick aluminum plate with CFCF boundary conditions and the parameters given in Table 1 was considered in this study (see Fig. 1). Using the FD method, the plate was divided into 36 and 14 sections along x and y directions respectively such that square elements were obtained. The value of c, in Eq. (9) was chosen as 0.1, which was less than half of its maximum allowable value so that the stability requirement was fulfilled. Fig. 1 represents the geometry of the flexible plate system with CFCF boundary conditions as well as the positions of excitation and observation points considered in the open-loop investigation. In order to investigate the validity of the simulation platform, a finite duration step input was applied to the excitation point and the frequency-domain response of the system at the observation point was investigated using the FD simulation and experimental measurements. Besides, finite element simulation of the flexible system was carried out using ABAQUS software in order to find more evidence about the validity of the proposed technique. The element size was refined to achieve the converged results. Table 2 summarizes the obtained comparative results for the first three natural frequencies of the flexible plate system. As can be seen in the table, the first three natural frequencies of the flexible system obtained from the experiment measurements are close to the finite difference and finite element results. Consequently, the validity of Table 1 Plate specifications. Parameter

Value

Length l (m) Width b (m) Thickness h (m) Densityρ (kg m−3) Modulus of elasticity E (N m−2) Poisson ratio υ

0.9 0.35 0.00125 2690 6.83×1010 0.34

Fig. 1. Geometry of the flexible plate system and positions of the observation and excitation points considered in the open-loop experiment.

Table 2 First three natural frequencies of the flexible system. Mode

ABAQUS (Hz)

Finite difference (Hz)

Experiment (Hz)

1 2 3

8.39 16.2 23.2

8.0 18.5 23.1

8.2 18.0 22.4

89

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Fig. 2. First three resonant modes of the flexible system (a) mode 1 (b) mode 2 (c) mode 3.

the FD simulation platform is verified. Thus, it can be further employed for simulation of the control system in the next sections. Suitable positions of actuator and sensor on the hosted plate is another important issue which must be taken into consideration. It is known that the active vibration control systems with collocated actuator/sensor pairs (the case considered in this work) are inherently stable [4]. However, appropriate location of the sensor on the plate is an important consideration that affects the observability of the target vibratory modes. Consequently, the suitable position of the sensor on the flexible structure must be first determined using simulation. The numerical simulation presented in Fig. 2 revealed that the plate deflection along special lines, namely nodal lines (see Fig. 3) is insignificant. Then, for instance, if one just wants to control the third mode of vibration (see Fig. 2c) it should be avoided to mount the sensor on the vertical line in the middle of the plate. Otherwise, the output signal of the sensor is insignificant when the third mode of vibration is excited. Consequently, to observe the first three resonant modes of vibration, the sensor should not be placed near the nodal lines represented in Fig. 3.

3. System identification of the nonlinear electromagnetic actuator In this section, a hand-made electromagnetic actuator as shown in Fig. 4 is modeled through a system identification approach. As can be seen, the proposed nonlinear actuator contains a permanent magnet attached to the structure and an electromagnet placed near the permanent magnet. Thus, by controlling the current in the electromagnet a controllable force is applied to the flexible structure. A power amplifier was used to generate necessary current corresponding to the applied voltage. In order to design an effective control system, it is essential to obtain an accurate model for the proposed nonlinear actuator. It is obvious that the actuator system considered in this work is a double-input-single-output (DISO) system. The applied voltage (V(k)) to the power amplifier and 90

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Fig. 3. Positions of nodal lines in the first three modes of vibration of the flexible plate.

Permanent magnet

Electromagnet

Fig. 4. The nonlinear electromagnetic actuator.

Fig. 5. Forward model of the DISO actuator system.

the distance between the two magnets (x(k)) are known as the inputs while the generated force (F(k)) is the system output (see Fig. 5). As mentioned earlier a parametric system identification scheme was proposed in this work to obtain the forward and inverse models of the actuator system. Indeed, the mentioned models were essential in designing the control algorithm presented in the next section. The standard system identification procedure is depicted in Fig. 6. First, the input/output data of the dynamic system is picked up through a measurement system. Then, a suitable model structure should be found. The third step is to estimate the model parameters. Once the system model is obtained based on the experimental data and the considered model structure, it is required to verify whether the model is good enough to represent the system. To do so, a number of validation tests, including correlation tests, one-step-ahead prediction, model-predicted output, estimation, and test data are available in the literature [20,21]. The following subsections are devoted to parametric model identifications of the proposed electromagnetic actuator.

3.1. Experimental data Fig. 7 demonstrates the experimental setup used in the identification scheme. As can be seen in this figure, a 3 kg load cell with sufficient accuracy and an instrument amplifier (Gain: 201) were used to measure the generated force of the actuator. A 14-bit NIUSB6009 data acquisition system (compatible with LabVIEW software) was used to collect and log the input/output data of the actuator system. A graphical program was thus written within the LabVIEW environment such that a random voltage in the range −5 to 5 V (see Fig. 8) was applied to the actuator's power amplifier and the measured force by the load cell (see Fig. 9) was stored in memory. The experimental measurements were carried out considering four distances between the permanent magnet and the electromagnet i.e. 2 mm, 3.5 mm, 5.5 mm, and 6.3 mm as shown in Fig. 9. The Sampling frequency of the data acquisition system 91

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DATA FROM EXPERIMENT

SELECTION OF MODEL STRUCTURE

MODEL STIMATION

MODEL VALIDATION

Not accepted

Accepted

Fig. 6. System identification procedure.

Fig. 7. Experimental setup for system identification of the actuator. 1 0.8

Normalized Voltage

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0

1000

2000

3000

4000

5000

6000

7000

8000

Sample

Fig. 8. Normalized random input voltage.

was adjusted to 100 Sample/s that was the maximum allowable value of the digital to analog converter. Besides, it was sufficient to cover the frequency range of the first five modes of vibration of the flexible plate system based on the Nyquist sampling theorem. The experiment was implemented during a period of 74 s resulting in 7400 data points. Normalized values of the measured data were considered in the following system identification scheme (see Figs. 8 and 9). 3.2. Model structure selection The parametric system identification scheme requires a family of candidate model structures. Since there are well-defined linear model structures such as FIR, ARX, ARMAX, ARMA and OE, the selection of linear model structures is straightforward [21]. 92

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A.R. Tavakolpour-Saleh, M.A. Haddad 1

2 mm

6.3 mm

5.5 mm

3.5 mm

0.8

Normalized Force (N)

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0

1000

2000

3000

4000

5000

6000

7000

Sample

Fig. 9. Normalized force.

Table 3 Estimated parameters using LS method. Parameter Value

a 0.9561

b −0.0258

c −0.01

d −0.0149

e −2.3142

l −0.6387

g −1.239

h −0.4833

i −0.6998

Fig. 10. Normalized force of the actuator.

However, such linear models were not sufficient for the current application because the actuator nonlinearity was significant. Consequently, the black-box system identification was implemented using nonlinear model structures. One should keep in mind that there are numerous nonlinear model structures that can be taken into account. Accordingly, parametric system identification of the nonlinear black-box systems is a difficult task and it is definitely dependent on a tedious trial and error scheme. Therefore, a forward parametric nonlinear model structure for the considered actuator was proposed using a trial and error scheme as follows:

Fˆ (k )=aF (k −1)+bV (k )+cV 2 (k )+dx (k )

(10)

where Fˆ is one-step-ahead (OSA) predicted force, F is the normalized measured force, V is the normalized voltage, x is the normalized distance between the magnets and k is the sample number. The variables a, b, c, and d are the unknown parameters of the forward model which should be identified. Eq. (10) can also be represented in a compact matrix form as:

Fˆ (k / θ )=φ1T (k ) θ1

(11)

where φ1(k) and θ1 are regression and parameter vectors respectively which can be expressed as:

φ1 (k )=[F (k −1) V (k ) V2 (k ) x (k )]T

(12)

θ1=[a b c d ]T

(13)

Subsequently, a nonlinear model structure was proposed as the inverse model of the actuator as follows: 93

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Vˆ (k )=eF (k )+lx (k )+gF (k ) x (k )+hx (k )

F (k ) + i F (k )

1+x 2 (k )

(14)

where Vˆ is the predicted voltage. Eq. (14) can also be written in matrix form as:

Vˆ (k / θ )=φ2T (k ) θ2

(15)

where,

φ2 (k )=[F (k ) x (k ) F (k ) x (k ) x (k )

F (k )

F (k )

T

1+x 2 (k ) ]

θ2=[e l g h i ]T

(16) (17)

It is important to note that the discrete model structures presented by Eqs. (10) and (14) are approximate models. However, they found to be sufficient for the control task in the next steps. Moreover, it is important to note that the forward model structure presented by Eq. (10) is able to provide a one-step-ahead prediction. This prediction can be sufficient for some applications such as adaptive control systems, but, it cannot be considered as the perfect model of the system to predict output over the infinite prediction horizon. In other words, it acts as an ARX model in linear system identification theory. As a result, the perfect model of the actuator system can be acquired by converting the ARX model structure into its equivalent output error (OE) model in which the delayed predicted outputs are used in the regression vector instead of the measured outputs. However, the inverse model presented by Eq. (14) acts as an FIR model structure and hence, it can be employed as a perfect inverse model of the actuator system. More details on the mentioned issues can be found in [20]. 3.3. Estimation of parameters The parameter vectors (see Eqs. (13) and (17)) can be estimated using different techniques. Among them, the famous least squares (LS) method can be selected due to its extensive applications. Thus, the LS approach was used to estimate the parameters of

Fig. 11. Normalized voltage predicted by the inverse model.

Fig. 12. Block diagram of the proposed 1DOF-FAFC.

94

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the nonlinear parametric models as [21]:

⎤−1 N ⎡N θˆN = ⎢∑ φ (k ) φT (k ) ⎥ ∑ φ (k ) y (k ) ⎥⎦ k =1 ⎢⎣ k =1

(18)

where N is the number of sampled data. Using the LS method, the parameter vectors θ1 and θ2 were estimated. The values of estimated parameters were given in Table 3. 3.4. Model validation In this subsection, the model capability to reproduce the input/output data of the actuator system is investigated through simulation. As previously mentioned, a perfect model is one that is able to predict the output of the system over the infinite prediction horizon. Thus, the forward model presented by Eq. (10) was first transformed into an OE structure based on the obtained parameters listed in Table 3 and then, it was simulated. Fig. 10 depicts the simulated force of the actuator corresponding to the applied random voltage (see Fig. 8) for the nonlinear-ARX and nonlinear-OE models in a forward manner. It can be seen that the nonlinear-OE model cannot act as well as the nonlinear-ARX model which uses the measured values of the delayed output because the OE model uses the delayed predicted output in the regression vector. However, the nonlinear-OE model structure is the perfect model of the actuator that can be considered as the forward actuator model in control system design. Next, another simulation was carried out using the inverse model of the actuator (Eq. (14)) and the estimated parameters presented in Table 3. The inverse model possessed two inputs, including the normalized force (F) and the normalized distance (x)

Fig. 13. Places of the excitation and control points.

Fig. 14. Membership functions for (a) input variable (b) output variable.

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Disturbing force (q) at x=0.15m and y=0.1m 100 80 60

Force (N/m2)

40 20 0 -20 -40 -60 -80 -100 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time (sec)

acceleration at x=0.67m and y=0.1m 3

uncontrolled vibration controlled vibration

acceleration (m/s2)

2

1

0

-1

-2

-3 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time (sec)

( ) -70

uncontrolled vibration controlled vibration

-80 -90 -100

dB

-110 -120 -130 -140 -150 -160 -170 20

40

60

80

100

120

140

160

180

frequency,(Hz)

( ) Fig. 15. Performance of the control system for first resonant mode of the flexible plate (a) disturbance with frequency of 8 Hz at point E (b) simulated acceleration at point C (c) frequency response at point C.

while the applied voltage was considered as the output. Fig. 11 demonstrates the capability of the inverse model to reproduce the actuator voltage. It can be seen that the inverse model of the actuator acceptably follows the measured voltage and hence, it can be considered as the inverse model of the actuator system in the next section.

96

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Table 4 Fuzzy rules. Input rule If If If If If If If If

input input input input input input input input

is is is is is is is is

mf mf mf mf mf mf mf mf

Output rule (1) (2) (3) (4) (5) (7) (8) (9)

Then Then Then Then Then Then Then Then

output output output output output output output output

is is is is is is is is

mf mf mf mf mf mf mf mf

(5) (4) (3) (3) (3) (2) (2) (1)

4. One degree-of-freedom fuzzy active force control scheme As mentioned earlier, active force control is a well-known 2DOF control method for controlling the robot arms and manipulators in the presence of unknown disturbances and parameter uncertainties that are prevalent in the real world [15]. More details on the conventional active force control can be found in [17]. In this work, the inner loop of the conventional 2DOF active force control scheme was proposed as a 1DOF regulator system for vibration suppression of the CFCF flexible plate system. 4.1. Fundamentals of the control scheme The active force control method relies on the appropriate estimation of the mass parameter of the dynamic system, the measurements of the acceleration, and accurate modeling of the actuator. Consequently, the lack of success to achieve the mentioned requirements will significantly deteriorate the performance of the control algorithm. Thus, it is important to obtain the forward and inverse models of the actuator accurately. Indeed, it can be a challenging problem when a nonlinear actuator is considered in the control scheme. However, such models were found in the previous section using the parametric system identification scheme. Another important parameter that must be tuned in the active force controller is the mass of the dynamic system. In many cases, estimating the mass value is a difficult task (i.e. in distributed systems) and sometimes a tedious trial and error scheme is used to find the mass value. In this investigation, a fuzzy observer is proposed to estimate the mass value based on the acceleration feedback. In order to smooth the input signal to the fuzzy controller a signal processing scheme based on the mean of the absolute values of the acceleration signal was proposed in this study. Fig. 12 represents a schematic block diagram of the proposed 1DOF fuzzy active force controller. In this figure Va, Ga, Ga , Fd, F*d, Fa, Fa , F, F , M, and aact are input voltage to the actuator, real model of the actuator, estimated model of the actuator, disturbance force, estimated disturbance force, actuator force, predicted actuator force, total input force to the dynamic system, total estimated force, estimated mass, and actual acceleration of the dynamic system respectively. Using the control procedure demonstrated in Fig. 12 the unknown disturbance Fd can be estimated using Eq. (19) as F*d and then, fed into the actuator through the inverse actuator model to cancel out the negative effect of disturbance. It is important to note that Fd is an unknown disturbance affecting the control point C on the flexible plate as a result of the applied excitation force at point E (see Fig. 13). However, knowledge about the form of Fd is not significant as the proposed control scheme can estimate every form of disturbance (using Eq. (19)) at the control point C.

−Fd*=Fa−Maact

(19)

As mentioned earlier a single-input-single-output (SISO) fuzzy controller was utilized to estimate the mass parameter. The estimated mass of the system can be expressed as: (20)

M =f (Mean ( aact ))

where f is a nonlinear function defined by the fuzzy logic controller. According to the proposed control scheme and using Eqs. (19) and (20), the applied voltage to the actuator (at time step k) can be found:

Va (k )=Ga−1 [Ga Va (k −1)−f (mean ( aact )) aact (k −1)]==Va (k −1)−Ga−1f (mean ( aact )) aact (k −1)

(21)

Consequently, if one assumes a very accurate actuator model such that Ga≈Ga , the actuator force can be estimated as follows: (22)

Fa (k )=Ga Va (k )=Fa (k −1)−f (mean ( aact )) aact (k −1)

Eqs. (21) and (22) are the final control laws used in the simulation investigation in the next subsection. The rest of the paper is devoted to evaluate the performance of the proposed control scheme using simulation and experiment. 4.2. Simulation study Since, the obtained models for the flexible structure, electromagnetic actuator, and intelligent controller were discrete models they could be coupled and solved simultaneously to simulate the overall behavior of the control system. An accelerometer with a linear characteristic and a very small response time was considered in the simulation study. A flexible thin plate with CFCF boundary 97

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( ) Fig. 16. Performance of the control system for second resonant mode of the flexible plate (a) disturbance with frequency of 18.5 Hz at point E (b) simulated acceleration at point C (c) frequency response at point C.

conditions and specifications given in Table 1 was considered in the simulation study. Fig. 13 demonstrates the geometry of the flexible structure and the positions of the excitation (E) and control (C) points. A control strategy with a collocated actuator/sensor pair was proposed in this work. Hence, the electromagnetic actuator and the accelerometer were assumed to be mounted on the flexible plate at the same place. 98

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( ) Fig. 17. Performance of the control system for third resonant mode of the flexible plate (a) disturbance with frequency of 23.1 Hz at point E (b) simulated acceleration at point C (c) frequency response at point C.

A description of the FD simulation program is given as follows: – – – – –

Start Initial conditions and Inputs (e.g. plate specifications, number of elements, boundary conditions and so on) Loop (1) for time variable (k) { Loop (2) for segment variable (i) { Loop (3) for segment variable (j) { 99

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( ) Fig. 18. Simulated acceleration of the observation point for the first resonant mode of the 2 mm thin plate (a) in time domain (b) in frequency domain.

– – – – – – – –

Apply disturbance force to the excitation point (E) Compute actuator voltage and force using Eqs. (21) and (22) Apply actuator force to the observation point (C) Simulate the flexible plate using Eq. (2) Observe and store the acceleration of observation point (H) for next time iteration End of Loop (3)} End of Loop (2)} End of Loop (1)}

Using the FD simulation framework, different forms of disturbance and actuating forces could be exerted on the flexible plate at the desired nodes and the dynamic response of the system in terms of acceleration or displacement could be observed. As mentioned earlier a SISO fuzzy controller was applied to the control system so that an acceptable mass value was automatically estimated based on the system acceleration. The fuzzy controller was tuned through an extensive simulation study within the FD framework considering different dimensions and specifications of the flexible system. Consequently, the fuzzy logic controller was trained so as to provide an appropriate mass corresponding to system uncertainties and parameter changes. In other words, a self-tuning active force controller was presented in this investigation. Figs. 14 and 15 respectively demonstrate the fuzzy sets for input and output variables of the fuzzy controller. The universe of discourse of the input variable was the system acceleration while the universe of the output variable was the system mass. Nine triangular membership functions with uniform distribution were proposed for the input variable and five membership functions were defined for the output variable. ‘Centroid’ defuzzification technique was used to obtain the resultant crisp set from the output fuzzy set. ‘min’ and ‘max’ operators were respectively proposed as implication and aggregation methods in the fuzzy controller. Finally, the fuzzy rules were defined as shown in Table 4. A simulation was carried out to investigate the performance of the control system to attenuate the first three resonant modes of vibration of the flexible plate system. Accordingly, three sinusoidal disturbing forces were considered to excite the plate system at 100

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( Fig. 19. Simulated acceleration of the observation point for the first resonant mode of the 3 mm thin plate (a) in time domain (b) in frequency domain.

Fig. 20. Experimental rig (front view).

point E and the performance of the control system at point C was investigated (see Fig. 13). The obtained results are demonstrated in Figs. 15–17. As can be seen, the proposed controller effectively attenuates the first three modes of vibration of the flexible plate system as 90%, 93%, and 95% respectively. It is interesting to note that the 1DOF-FAFC scheme shows a better performance at higher frequencies and it is an important feature of the proposed controller. 101

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Fig. 21. Collocated electromagnetic actuator and accelerometer at the observation point (top view).

Fig. 22. Graphical program of the 1DOF-FAFC scheme in Simulink/MATLAB.

Another simulation study was conducted to evaluate the performance of the intelligent controller under parameter changes in the dynamic system. To do so, the performance of the control system to attenuate the first resonant mode of other thin plates with different thickness (e.g. 2 mm and 3 mm) was investigated using the FD simulation platform and the acceleration of the observation point was simulated. Figs. 18 and 19 depict a significant vibration reduction that can be achieved using the proposed fuzzy controller corresponding to the mentioned parameter changes in the flexible system. Since it was demonstrated previously that the proposed 1DOF-FAFC scheme had a better performance at higher frequencies, simulation of the higher resonant frequencies were ignored in this section. Furthermore, other parameter changes such as dimension changes, stiffness changes, and etc. were considerably compensated using the proposed fuzzy controller, but they were neglected in this section to shorten the article. 102

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Fig. 23. Experimental performance of the control system for the first resonant mode (a) in time domain (b) in frequency domain.

5. Experimental verification In this section, practical performance of the proposed 1DOF-FAFC scheme is first investigated through experiment and then, a comparative study between the proposed controller and a pure acceleration feedback control scheme is presented to highlight the effectiveness of the proposed controller. 5.1. Development of the experimental rig In this subsection, an experimental rig was developed and evaluated. Fig. 20 shows the experimental rig considered in this work. An analog accelerometer (model: Freescale 7361) along with the electromagnetic actuator was used so that a collocated actuator/ sensor configuration was acquired at the observation point (see Fig. 21). Another electromagnetic actuator with the same specification was developed to generate the disturbance at the excitation point. The permanent magnets of the electromagnetic actuators were mounted on one side of the hosted plate and the accelerometer was placed on the other side. The electromagnets were installed on a rigid stand in the vicinity of the permanent magnets. The initial distance between the electromagnet and the permanent magnet was adjusted to 3.5 mm. A two-channel power amplifier was utilized to provide adequate current for the electromagnets. A PCI-1716 data acquisition (DAQ) card which was fully compatible with SIMULINK was used in the experiment. Resolution of the data acquisition system was 16-bit that was sufficient to measure the output voltage of the accelerometer with acceptable accuracy. An analog low-pass filter was used along with the accelerometer to avoid aliasing phenomenon. A sampling frequency of 100 Hz was selected in the data acquisition system (as it was considered in the system identification section). One analog input of the DAQ system was used to read the output voltage of the accelerometer and two analog outputs were utilized to control the electromagnets via the power amplifier unit. The proposed 1DOF-FAFC scheme was graphically coded in Simulink/MATLAB as shown in Fig. 22. It is important to note that the proposed control scheme acts as a discrete integrator to some extent (see Eq. (21)) and therefore, the unavoidable offset voltage of the accelerometer must be eliminated before feeding back into the controller. Otherwise, the integration of such a biased signal can lead to the saturation of the controller output voltage and consequently, the termination of the control action. Thus, a signal 103

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Fig. 24. Experimental performance of the control system for the second resonant mode (a) in time domain (b) in frequency domain.

processing strategy based on the ‘mean’ operator (see Fig. 22) was proposed in the control algorithm such that the offset voltage of the accelerometer was automatically eliminated. In addition, a second order digital low-pass filter with a cutoff frequency of 100 Hz was used to further smooth the acceleration signal. Indeed, the mentioned filter could enhance the controller performance and remove acoustical disturbance affecting the flexible system. A special care must be taken when one wants to select the cutoff frequency of a filter. It must be selected such that the low-pass filter doesn’t eliminate the useful frequencies in the measured signal, i.e. the first three natural frequencies of the flexible structure considered in this work. 5.2. Experimental evaluation of the proposed 1DOF-FAFC To investigate the performance of the control system three sinusoidal disturbances with the same frequencies as the first three natural frequencies of the flexible plate system were applied to the excitation point. Consequently, the first three resonant modes of the flexible plate structure were excited. Finally, the performance of the 1DOF-FAFC scheme in suppressing the first three resonant modes of vibration of the flexible plate was investigated. Figs. 23–25 show the effectiveness of the proposed controller to attenuate the first three resonant modes of the flexible plate system. As can be seen, the first, second, and third resonant modes of the plate system were attenuated up to 74%, 81%, and 90% respectively. Consequently, the controller had a better performance at the higher frequencies as it was reported in the simulation study. The discrepancies between the simulation and experimental attenuations can be attributed to the considered approximate model for the nonlinear actuator. However, the experimental results revealed a considerable vibration reduction in the flexible plate using the proposed 1DOF-FAFC scheme. 5.3. A comparative study In this subsection, the performance of the presented control scheme was compared to a conventional AVC algorithm. Since the proposed 1DOF-FAFC scheme was based on the acceleration feedback, it was reasonable to compare it with the proportional acceleration feedback controller. The pure acceleration feedback control scheme is a standard technique considered in many references [6–9] and consequently, the theoretical issues on this control algorithm are not given in this work. More theoretical details on the pure acceleration feedback AVC can be found in [6,8]. Fig. 26 demonstrates the practical implementation of the proportional acceleration feedback AVC in Simulink/MATLAB. According to Fig. 26, the proportional gain of the acceleration feedback controller was chosen through a trial and error scheme so that the best performance of the control system was obtained 104

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(b) Fig. 25. Experimental performance of the control system for the third resonant mode (a) in time domain (b) in frequency domain.

experimentally. Besides, in the practical implementation of the pure acceleration feedback controller, the inverse model of the nonlinear actuator (obtained in Section 3) was applied to the control algorithm to compensate the nonlinearity presented in the considered actuator. The second control algorithm was practically implemented on the developed experimental rig without any changes in its mechanical configuration such that a suitable comparison platform was obtained. Finally, the comparative results for both control schemes were given in Table 5. It is obvious that the proposed 1DOF-FAFC technique is superior to the pure acceleration feedback controller through which the effectiveness of the proposed 1DOF-FAFC is affirmed. Furthermore, it is interesting to note that both controllers show a better performance at higher frequencies. It can be attributed to the fact that a better resolution of the acceleration signal can be acquired at higher frequencies. 6. Conclusions In this work, a fuzzy robust control scheme incorporating collocated electromagnetic actuator and accelerometer was proposed for active vibration control of a flexible thin plate with CFCF boundary conditions. First, the flexible plate was modeled using a finite difference approach and the validity of the obtained model was demonstrated through an experimental work. Then, the forward and inverse models of the nonlinear electromagnetic actuator were presented through a parametric system identification scheme and the validity of the obtained models was investigated. A fuzzy observer was applied to the control system so that an intelligent controller was obtained. Consequently, the controller gain (i.e. the system mass) could be tuned intelligently using the acceleration signal and considering the parameter changes in the dynamic system. Although the acceleration signal was used in the proposed fuzzy robust control scheme to estimate the unknown disturbance, it was shown that indeed the integration of the acceleration signal was used in the control scheme. Thus, it was important to eliminate the offset voltage of the accelerometer (if any). Besides, it was essential to define the initial conditions for the integration action in the control algorithm. These important issues were fully considered in the control program (as shown in Fig. 22). As a result, a velocitylike signal was obtained based on the acceleration sensor and via the proposed control action. Thus, all benefits of the velocity feedback controllers were obtained using the proposed 1DOF-FAFC incorporating the inexpensive acceleration sensor. It is well known that the velocity signal is in-phase with the excitation force in a flexible system, which results in the modifications of the control system performance. In contrast, there is an unavoidable phase difference of 90° between the acceleration signal and the excitation force, which results in deteriorating the performance of the acceleration-based controllers. In spite of the mentioned 105

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Fig. 26. Graphical program of the pure acceleration feedback AVC in Simulink/MATLAB.

Table 5 Measured vibration reduction. Mode No.

1DOF-FAFC

Acceleration feedback proportional-AVC

1 2 3

74% 81% 90%

54% 61% 77%

drawbacks, an accelerometer was used in the proposed control scheme as the sensing element, because the accelerometers were light, small, and inexpensive and thus, they were excellent choices for active vibration control applications. Consequently, an effective intelligent control scheme was proposed in the paper that used an accelerometer to effectively attenuate the unwanted vibrations in the flexible plate structures while it possessed the benefits of velocity feedback controllers. It was found that the proposed controller was more effective at higher frequencies. Finally, an experimental rig was developed and tested to investigate the performance of the proposed control scheme. The experimental results clearly revealed the effectiveness of the 1DOF-FAFC technique. Furthermore, the superiority of the proposed AVC scheme over the pure acceleration feedback control was demonstrated experimentally. Acknowledgment The authors wish to express their deep gratitude to Shiraz University of Technology and National Elites Foundation for providing research facilities and funding. References [1] M.O. Tokhi, M.A. Hossain, A unified adaptive active control mechanism for noise cancellation and vibration suppression, Mech. Syst. Signal Process. 10 (1996) 667–682. [2] A.R. Tavakolpour-Saleh, M. Mailah, I.Z. Mat Darus, Self-learning active vibration control of a flexible plate structure with piezoelectric actuator, Simul. Model. Pract. Theory 18 (2010) 516–532. [3] P. Lueg, Process of Silencing Sound Oscillations. US Patent No. 2,043,416, 1936. [4] A. Preumont, Vibration Control of Active Structures: An Introduction, 3rd ed., Springer, 2011. [5] S. Kim, M.J. Brennan, G.L.C.M. Abreu, Narrowband feedback for narrowband control of resonant and non-resonant vibration, Mech. Syst. Signal Process. 76– 77 (2016) 47–57. [6] Z. Qiu, H. Wu, C. Ye, Acceleration sensors based modal identification and active vibration control of flexible smart cantilever plate, Aerosp. Sci. Technol. 13 (2009) 277–290. [7] F. Previdi, C. Spelta, M. Madaschi, D. Belloli, S.M. Savaresi, F. Faginoli, F. Silani, Active vibration control over the flexible structure of a kitchen hood, Mechatronics 24 (2014) 198–208. [8] F. An, W. Chen, M. Shao, Dynamic behavior of time-delayed acceleration feedback controller for active vibration control of flexible structures, J. Sound Vib. 333 (2014) 4789–4809. [9] J. Enríquez-Zárate, G. Silva-Navarro, H.F. Abundis-Fong, Active vibration suppression through positive acceleration feedback on a building-like structure: an experimental study, Mech. Syst. Signal Process. 72–73 (2016) 451–461. [10] A. Jenifene, Active vibration control of flexible structures using delayed position feedback, Syst. Control Lett. 56 (2007) 215–222. [11] Z. Gosiewski, A.P. Koszewnik, Fast prototyping method for the active vibration damping system of mechanical structures, Mech. Syst. Signal Process. 36 (2013) 136–151. [12] K. Khorshidi, E. Rezaei, A.A. Ghadimi, M. Pagoli, Active vibration control of circular plates coupled with piezoelectric layers excited by plane sound wave, Appl. Math. Model. 39 (2015) 1217–1228. [13] D. Wu, L. Huang, B. Pan, Y. Wang, S. Wu, Experimental study and numerical simulation of active vibration control of a highly flexible beam using piezoelectric intelligent material, Aerosp. Sci. Technol. 37 (2014) 10–19.

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[14] V. Gupta, M. Sharma, N. Thakur, Active structural vibration control: robust to temperature variations, Mech. Syst. Signal Process. 33 (2012) 167–180. [15] J.R. Hewit, J.S. Burdess, Fast dynamic decoupled control for robotics using active force control, Trans. Mech. Mach. Theory 16 (1981) 535–542. [16] A.R. Tavakolpour-Saleh, Mechatronic Design of Intelligent Active Vibration Control Systems for Flexible Structures (Ph.D. thesis), Universiti Teknologi Malaysia, Malaysia, 2009. [17] M. Mailah, Intelligent Active Force Control of a Rigid Robot Arm Using Neural Network and Iterative Learning Algorithms (Ph.D. thesis), University of Dundee, UK, 1998. [18] M. Mailah, E. Pitowarno, H. Jamaluddin, Robust motion control for mobile manipulator using resolved acceleration and proportional-integral active force control, Int. J. Adv. Robot. Syst. 2 (2005) 125–134. [19] A.R. Tavakolpour-Saleh, I.Z. Mat Darus, M.O. Tokhi, M. Mailah, Genetic algorithm-based identification of transfer function parameters for a rectangular flexible plate system, Eng. Appl. Artif. Intell. 23 (2010) 1388–1397. [20] A.R. Tavakolpour-Saleh, S.A.R. Nasib, A. Sepasyan, S.M. Hshemi, Parametric and nonparametric system identification of an experimental turbojet engine, Aerosp. Sci. Technol. 43 (2015) 21–29. [21] L. Ljung, System Identification Theory for the User, Prentice-Hall, Inc., New Jersey, 1999.

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