Research Article  Open Access
Improving DelayMargin of Noncollocated Vibration Control of PiezoActuated Flexible Beams via a FractionalOrder Controller
Abstract
Noncollocated control of flexible structures results in nonminimumphase systems because the separation between the actuator and the sensor creates an inputoutput delay. The delay can deteriorate stability of closedloop systems. This paper presents a simple approach to improve the delaymargin of the noncollocated vibration control of piezoactuated flexible beams using a fractionalorder controller. Results of real life experiments illustrate efficiency of the controller and show that the fractionalorder controller has better stability robustness than the integerorder controller.
1. Introduction
Flexible structures have attracted increasing attentions for many applications because of their weights and production costs. However, the flexibility leads to unwanted vibration problems. Thus, vibration control is usually needed. Over the past few decades, active vibration control has drawn more interest from researchers since it can effectively suppress the vibration [1, 2]. A large part of the active vibration control research has used piezoelectric materials for actuation and sensing. Advantages of using piezoactuators/sensors include nanometer scale resolution, high stiffness, and fast response. Piezoactuators have been proven to be useful in suppressing structural vibration [3, 4].
Stabilization of flexible structures can be easily done by collocating the sensors and the actuators. However, the collocated control configuration is not always feasible in practice and its performance is not always satisfactory. Thus, noncollocated control has been investigated [5, 6]. However, noncollocated control results in a nonminimumphase closedloop system because the separation between the actuator and the sensor creates an inputoutput delay, which can deteriorate stability of the closedloop system.
Fractional calculus is a 300yearold mathematical topic [7, 8]. However, its practical applications have just recently been explored. Recently, fractionalorder control has been attracting interest. It has been illustrated that fractionalorder controllers yield superior performance to integerorder controllers [9, 10].
This paper presents a simple approach to improve the delaymargin of the noncollocated vibration control of piezoactuated flexible beams using a fractionalorder proportionalintegralderivative (PID) controller. The rest of the paper is organized as follows. Section 2 provides some preliminaries. Section 3 describes the piezoactuated flexible beam that is used as an experimental test bench. Section 4 presents a finite element model of the experimental beam. The proposed controller design is given in Section 5. Experimental results are presented in Section 6. The last section concludes the paper.
2. Preliminaries
2.1. FractionalOrder Calculus
The frequently used definitions of fractionalorder derivatives are RiemannLiouville definition, GrünwaldLetnikov definition, and Caputo definition. The RiemannLiouville definition is given as where is the fractional order, is the integer that , and is the Gamma function. The GrünwaldLentnikov definition can be written as where is the fractional order and is the usual notation for the binomial coefficients. The Caputo definition is described by where is the fractional order, is the integer that , and is the Gamma function. These three definitions are equivalent to a wide class of functions [1, 2].
The most general expression of the Laplace transform of the fractionalorder derivative is given as where is the integer that and . If all the initial conditions are zero, the Laplace transform of the fractional derivative becomes very simple as
The transfer function of fractional order integrator and the fractional order derivative, respectively, is The transfer function of the fractionalorder proportionalintegralderivative (PID) controller can be expressed as where , , , , and are the controller parameters. It is worth noting that the fractionalorder controller has two extra degrees of freedom compared to the conventional integerorder PID controller. This provides more freedom to better adjust the performance of the closedloop systems.
2.2. PiezoActuator Equation
The equation relating the applied voltage and the axial strain produced by a piezoactuator is given as [11, 12] where is the dielectric constant of the piezoelectric material, is the applied voltage, and is the thickness of the piezoelectric material. When the piezoactuator is bonded on the surface of a beam, the normal stress and the bending moment on the beam produced by the actuator are where is the thickness of the beam, is Young’s modulus of the piezoelectric material, and .
2.3. Finite Element Equation
Consider a simple cantilever beam with a piezoactuator bonded on the surface as shown in Figure 1. It is assumed that the beam is divided into elements. Each element has two nodes and each node has two degrees of freedom: transverse direction and slope .
The transverse displacement at any position on the element can be expressed as [11, 13] where is the nodal displacement vector and is the mode shape vector function. By using the Lagrangian method, the differential equation of motion to be solved can be expressed as [14, 15] where is the mass matrix, is the stiffness matrix, is the damping matrix, and is the force matrix. For Raleigh proportional damping, , where , are the damping constants. The force matrix for the element without the piezoactuator is and for the element bonded with the piezoactuator is , which is directly derived from (11).
3. Experimental System
A schematic and a photograph of the experimental system used in this paper are shown in Figure 2. The system consists of a flexible cantilever beam, a piezoelectric actuator, a highvoltage power amplifier, a laser displacement sensor, a 12bit A/D and D/A interface board, and a PC. The beam is made of aluminum. The dimensions and properties of the beam are summarized in Table 1. The actuator is leadzirconatetitanate (also known as PZT). It is bonded on the beam at the distance from the clamped point. The dimensions and properties of the actuator are summarized in Table 2. The sensor comprises a sensor head and a sensor amplifier. The sensor head makes use of a red laser semiconductor with a wavelength of 655 nm and an output power of 220 W. The measurement range is ±5 mm. The maximum sampling frequency is 3 kHz, but in our experiments, a default setting of 1 kHz was used. The sensor amplifier outputs an analog signal with a range of ±5 V. The sensor measures the tip displacement of the beam. The displacement signal is fed back to the PC through the interface board. The control signal from the computer is converted to an analogue signal by the interface board and is transmitted to the highvoltage power amplifier. The gain of the amplifier is 150. Figure 3 displays the pulse response and the frequency response function of the beam and shows that the first two natural frequencies of the beam are 41.6952 rad/s and 235.1796 rad/s and the damping ratio is about 0.0097. The magnitude and duration of the pulse input are 1 V and 16 msec, respectively.


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4. Finite Element Model
A finite element (FE) model of the experimental piezoactuated beam was developed for being used in the control design process. For simplicity, was chosen. The pulse response and the frequency response function obtained from the developed FE model are shown in Figure 4. The responses agree with those of the experimental system.
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5. Control Design
The control objective is to drive the tip of the beam to a desired position within 3 seconds. The proposed fractionalorder PID controller can be designed using the following steps.
Step 1. Design a conventional integerorder PID controller based on an approximated lowerorder transfer function of the beam. Here, the transfer function containing only the first natural frequency was employed. The transfer function is expressed as where , , and are the natural frequency, damping ratio, and system gain, respectively. The values found in Section 3 were used; that is, rad/sec, , and . The transfer function of the controller is given by where , , and are the integerorder controller parameters. A root locus technique was used to determine the parameters.
Step 2. Tune the integerorder controller parameters from Step 1 with a FE model of the beam. Here, the FE model developed in Section 4 was employed. The parameters of the controller were found to be , , and . A step response of the control system based on the FE model is shown in Figure 5. The result shows that the control response meets the objective. The Bode characteristic of the openloop transfer function is shown in Figure 6. The delaymargin is 13.9 msec.
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Step 3. Replace the integrator of the tuned integerorder controller by a fractionalorder integrator. The transfer function of the controller becomes where is the fractional order. Here, was chosen.
Step 4. Tune the parameters of the fractionalorder controller from Step 3 with the FE model. Similar to Step 2, the parameters were found to be , , and . A step response of the fractionalorder control system based on the FE model is shown in Figure 7. The result shows that the control response meets the objective with less initial oscillation compared to those of the integerorder controller. The Bode characteristic of the openloop system is shown in Figure 8. The delaymargin was increased to 20.8 msec.
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6. Experimental Results
The integerorder controller (15) and the fractionalorder controller (16) were implemented on the experimental beam for comparison. The parameters of the controllers used in the experiments were obtained in the previous section and are summarized in Table 3. The controllers were implemented digitally on the PC. The control laws were written in . The control sampling period was 1 msec for all experiments.

The control response of the integerorder controller is shown in Figure 9, which shows satisfactory step response. Note that the oscillatory behavior in the experiment is greater than that in the FE model. This is due to the effect of the measurement noise and the uncertainties of the system’s parameters.
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After that, we added a 1.5gram mass to the tip of the beam to examine the stability robustness of the controller. We found that the control system is unstable as shown in Figure 10. This is due to the delay (the effect of the noncollocated configuration). This implies that the closedloop system does not have sufficient delaymargin.
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Now, the fractionalorder controller is evaluated. The continued fraction expansion (CFE) of the AlAlaoui operator [16] is used to discretize the controller. The control response of the fractionalorder controller is shown in Figure 11, which provides the same satisfactory response compared to those of the integerorder controller. Moreover, there is no marked difference compared to the FE result. This implies that the fractionalorder controller is more robust to the noise and the uncertainties than the integerorder counterpart.
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Finally, we evaluated the stability robustness of the fractionalorder controller by adding the 1.5gram mass to the tip of the beam (in the same way that we did with the integerorder case). In this case, the control response is stable, as shown in Figure 12. This is due to the larger delaymargin provided by the fractionalorder controller.
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7. Conclusions
This paper presented the position control of a piezoactuated flexible beam using a fractionalorder PID controller. The control employed a noncollocated configuration, which always produces inputoutput delay. The fractionalorder controller was designed to increase delaymargin of the closedloop system. Experimental results show that the fractionalorder controller provides better stability robustness than the integerorder counterpart.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the Royal Golden Jubilee Ph.D. Program of the Thailand Research Fund.
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Copyright
Copyright © 2014 Teerawat Sangpet et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.