Experiments on optimal vibration control of a flexible beam containing piezoelectric sensors and actuators

In this paper, a digital regulator is designed and experimentally implemented for a flexible beam type structure containing piezoelectric sensors and actuators by using optimal control design techniques. The controller consists of a linear quadratic regulator with a state estimator, namely a Kalman observer. The structure is a cantilever beam containing a set of sensor/actuator PVDF/PZT ceramic piezoelectric patches bonded to the beam surface at the optimal location obtained for the first three vibration modes. The equations of motion of the beam are developed by using the assumed modes technique for flexible structures in infinite-dimensional models. This paper uses a method of minimizing the effect of the removed higher order modes on the low frequency dynamics of the truncated model by adding a zero frequency term to the low order model of the system. A measure of the controllability and observability of the system based on the modal cost function for flexible structures containing piezoelectric elements (intelligent structures) is used. The observability and controllability measures are determined especially to guide the placement of sensors and actuators, respectively. The experimental and numerical transfer functions are adjusted by using an optimization procedure. Experimental results illustrate the optimal control design of a cantilever beam structure.


Introduction
A single piezoelectric element called a self-sensing actuator combines actuator and sensing capabilities for collocated control [1].The use of piezoelectric actuator/sensor pairs has been widely used in many vibration control applications of flexible structures [2][3][4].
The French scientists Pierre and Paul-Jacques Curie discovered piezoelectricity in 1880.The piezoelectric effect is observed in many crystalline materials, which strain when exposed to a voltage and produce a voltage when strained.In other words, these materials are capable of transforming mechanical energy into electri-cal energy and vice versa.Piezoelectric actuators and sensors are bonded to flexible structures such as beams and plates, forming the so-called smart structures.
Different methodologies have been used for modeling smart structures, however the most promising technique is the finite element method, as first presented for piezo-mechanical systems by [5].Later this formulation was extended for different structural elements.[6] studied the modeling of one-dimensional piezoelectric patches embedded into the body of beams and formulated the moment generated by a voltage applied to the piezoceramics.[7] presented a piezoelectric finite element approach aiming at applications devoted to distributed dynamic measurement and control of advanced structures.The assumed modes approach has been extensively used throughout the literature to model the dynamics of distributed systems such as flexible beams ISSN 1070-9622/03/$8.00 2003 -IOS Press.All rights reserved and plates [8].This approach is used in this paper for modeling the flexible beam.
In control design problems, one is often interested in designing a controller for a particular frequency range.In such situations, it is common practice to remove the modes that correspond to frequencies that lie out of the bandwidth of interest and keep only the modes that directly contribute to the low frequency dynamics of the system.[4] suggested that the effect of higher frequency modes on the low frequency dynamics of the system can be captured by adding a zero frequency term to the truncated model to account for the compliance of the ignored modes.In the present work, this approach is taken.
The optimal control design applied to smart structures has been studied in several papers by using different methodologies.[9] presented a control system test bed which is used for active vibration control of a plate by using piezoelectric actuators.In that work, the authors applied a localized hybrid fuzzy-PD controller.The gains of the fuzzy-PD controller were obtained by using genetic algorithm techniques.[10] designed an optimal multivariable feedback controller for the active vibration control of a dynamic system consisting of a linear elastic plate with piezoelectric sensors and actuators.That contribution used PD and LQG/LTR control methods.[11] presented a new approach for simultaneous optimization of the intelligent structure together with the control system for vibration suppression.
The choice of the actuator location is an important issue in the design of actively controlled structures.The actuators should be placed at the locations so that the desired modes are excited most effectively [6].Various engineering applications using classical optimization and genetic algorithm schemes for the determination of optimal piezoelectric actuator placement have been reported.[12] worked with genetic algorithms to determine optimal actuator size and location for two piezoelectric actuators bonded to a cantilever beam.[13] used a discrete-continuous optimization technique to determine the position of the actuators along the flexible structure and to obtain the controller gains.In that work the goal was to minimize the control effort applied to a beam type structure.
In this paper, the measures of controllability and observability are based on the modal cost function or modal cost analysis, as proposed by [14].Such measures are used to guide the placement of sensors and actuators in flexible piezo-actuated structures.
To evaluate the usefulness of the proposed method, a digital regulator applied to the active vibration control of a cantilever beam type test structure was designed.The results obtained confirmed the applicability of the above technique and the stability of the control algorithm used.

Electro-mechanical model of a beam containing piezoelectric sensors and actuators
In this section, a model for a piezoelectric laminate beam with an actuator-sensor pair is obtained by using modal analysis techniques.
Figure 1(a) shows a flexible structure with an attached piezoelectric actuator-sensor pair.The piezoelectric patch on the upper side of the beam is used as an actuator, while the film on the bottom side serves as a sensor.The voltage that is applied to the actuator patch is represented by V a (t) and the voltage that is induced to the sensor film is represented by V s (t) (see Fig. 1 A model of the structure is obtained through a modal analysis procedure.This approach exacts the finding of a solution to the partial differential equation that describes the dynamics of the flexible structure: where ρ b , A b , E b and I b represent density, crosssectional area, Young's modulus of elasticity, and moment of inertia about the neutral axis of the beam, respectively.

Actuator dynamics
The total distributed load q(x, t) generated by the piezoelectric actuator when it is deformed is given by: where R is the generalized location function expressed as a function of the Heaviside function (H) [15]: The bending moment acting on the beam (denoted by M ax ) is given by [16]:  where: and where d 31 is the electric charge constant, E a is the Young's modulus and h a is the thickness of the PZT patch as shown in Fig. 1(b).The evaluation of Eq. (3) through Eqs (3.1) and (3.2) gives the following expression: where C a is a constant which depends on the geometry of the composite system: The boundary conditions for the cantilever beam in Fig. 1(a) are: To use the assumed mode technique, the function w(x, t) is expanded as an infinite series as follows [3]: where φ i (x) are the eigenfunctions satisfying the ordinary differential equations resulting from the substitution of Eq. ( 7) into Eqs (1) and ( 6), and η i (t) are the temporal functions.
The mode shapes for clamped-free boundary conditions, Eq. (6), are assumed to be expressed as [17]: where the following equations define α i and β i , respectively: and the constant C i is determined by the condition expressed by the following expression: Substituting Eqs ( 7) and ( 2) into (1) yields: where ( . ) denotes time derivatives and ( ) represents space derivatives.Pre-multiplying Eq. ( 10) by dx, the i-th equation, will be given by: Substituting Eqs (9) and (2.1) in Eq. (11), it is possible to obtain: where: 2) The natural frequencies ω ni are determined by using Eqs (12.1) and ( 8): Modal damping (ζ i ) can be included in Eq. ( 12), as follows: It is a very difficult task to determine modal structural damping by using physical principles.Therefore, ζ i 's are often estimated experimentally, which will be shown later in this paper.Now, by taking the Laplace transform of Eq. ( 13), the transfer function of this system is obtained: This equation describes the elastic deflection of the flexible beam due to a voltage applied to the actuating piezoelectric.

Sensor dynamics
The net forcing of the beam is equivalent to two equal and opposite moments applied to the beam at the endpoints of the actuator, as shown in Fig. 2. Due to the piezoelectric effect, a strain-induced voltage appears in the sensors.
The electric charge distribution q s (t), i.e., the charge per unit area, is given by [18]: where k 31 is the electromechanical coupling factor and g 31 is the piezoelectric voltage constant in the X direction.Using Hooke's law for the beam deflection in the X direction (see Fig. 2), the expression for the strain in the sensor patch is obtained as [3]: The total charge accumulated on the sensing layer can be found by integrating q s (x, t) over the entire length of the piezoelectric element [18]: Substituting Eq. ( 7) into ( 17), yields: where k s is given by: The piezoelectric sensor is similar to an electric capacitor and the voltage across the two layers is given by the following formula: where C s is the capacitance per unit area of the piezoelectric sensor and b s (x s2 − x s1 ) is the surface area of the piezoelectric element.
If the piezo-sensor is connected to a voltage amplifier as depicted in Fig. 3, the sensor output voltage is [19]: where C r stands for the circuit capacitance (see Fig. 3).The circuit configuration shown in Fig. 3 was used in the present contribution.The corresponding parameters will be presented later on.
Substituting Eq. ( 18) in Eq. ( 21), yields: Taking the Laplace transform of Eqs ( 22) and ( 13), the expression for V s (s) in terms of the input voltage V a (s) is obtained: This equation is the transfer function of the circuit, i.e., the relation between the voltage applied to the actuator and the voltage induced in the piezoelectric sensor.

Taking into account out of bandwidth modes
As an infinite number of modes are taken into account in the above formulation, Eqs ( 14) and ( 23) represent infinite-dimensional transfer functions.
In a typical control design scenario, the designer is often interested only in a particular bandwidth.Therefore, an approximate model of the system that best represents its dynamics in the prescribed frequency range is needed.A natural choice in this case is to simply ignore the modes which correspond to the frequencies that lie outside of the bandwidth of interest.Equation ( 23) is rewritten by limiting the number of modes considered to N .
where F i is given by: The drawback of this approach is that the truncated higher order modes may contribute to the low frequency dynamics in the form of distorting zero locations [20] suggests a way of dealing with this problem.The idea is to allow for a constant feed through term in Eq. (24) to account for the compliance of omitted higher order modes of Eq. (23).That is, to approximate Eq. ( 24) or G(s) by: Ĝ where K c is a constant that considers the effect of dynamical responses of higher order modes.[20] found the optimal value of K c so that the effect of higher order modes on the low frequency dynamics is minimized in some measure.The optimal value of K c was found to be: where ω c represents the cut-off frequency chosen to lie within the interval: In the experimental part of this paper, the above approach for a testbed cantilever beam type structure will be shown.

State space representation and controller design
Using Eq. ( 13) and the corrected model for Eq. ( 23), the state space model of the system is found to be: where: where I is the identity matrix, and k a and k s are the actuator and sensor constants defined by Eqs (12.2) and ( 19), respectively.Since a digital computer is used to implement the proposed controller, it is necessary to discretize the continuous time system.
The state-space system given by Eqs ( 27) and ( 28) is used to design an optimal controller.The control algorithm used here is an iterative version of the Linear Quadratic Regulator (LQG).The goal is to find an optimal linear feedback controller while respecting a prescribed maximum voltage applied to the piezoelectric actuators to prevent their depoling.Hence, a full state feedback control is considered to minimize the cost function defined by Eq. (30) subject to the linear constraints Eq. ( 27) and to the voltage limitation Evidently, the performance of this controller depends on the state Q and input R weight matrices, where Q defines the relative weight of each state variable and R stands for the relative weight of each actuator voltage.The latter is supposed to be in the form R = γI and the factor γ is then adjusted automatically to limit the maximum voltage, through an iterative algorithm.Consequently, the control gain matrix [G c ] is evaluated, for each value of γ, by solving the algebraic Riccati equation [21].For the linear feedback of the controlled modes, the control law is: However, the control law cannot be implemented since all the states are not usually available.So a dynamic observer to estimate the states was designed [21]: where [L] is the observer gain.
The dynamic observer gain [L] is obtained by solving an algebraic Riccati equation corresponding to the assumed state noise covariance matrix Q e and measurement noise matrix R e (see Eq. ( 30)).Since the system is observable, the gain matrix [L] is calculated by using pole placement techniques.The location of the estimator poles should be carefully chosen so that fast convergence of the estimated states to the actual states is achieved.In doing this, the observer should not be very sensitive to noise (that may accompany the sensor signals).

Modal cost function
In this section the problem of choosing the optimum locations for the piezoelectric elements used in the active vibration control of the structure is discussed.For this purpose, modal cost techniques will be used, i.e., how the controllability/obsevability of the system is changed as the piezoelectric actuators/sensors move along the structure [14].
Assuming the pair (A, B) controllable and (A, C) observable (see Eqs (29.2) to (29.4)), the controllability and observability are measured according to the grammians, as defined in the following equations [22]: where t f is some fixed final time and W c and W o satisfy the algebraic Lyapunov equations: [23] suggested a measure of the effect of the actuators/sensors positions in the dynamical system by using a quadratic cost function: where the vector y i (t) is composed of output variables x i due to impulsive inputs u i (t) = δ(t) (with u j (t) = 0, i = j), applied at t = 0, with zero initial conditions, and Q c is a weight matrix.Substituting the relation: The unit impulse response for zero initial condition in the ith-direction is given by [24]: Considering zero initial conditions, Eq. (35.8) is given by: Substituting Eq. ( 36) into (35.7),yields: e At BB T e A T t dt C T (37) Fig. 8. Scheme of the experimental setup for system model identification.Substituting Eq. (35.1) into (37), the cost function V is derived as follows: where trace{ } denotes the trace (diagonal sum) of the matrix { }.
If the cost function (V ) of a particular sensor/actuator location is zero, the sensor/actuator has no authority over those modes.On the other hand, if a given sensor/actuator location has a maximum cost value, the actuator/sensor has maximum authority over those modes.
Based on these principles it is possible [23]: -to compare different sensor and actuator configurations and to choose the one that contributes the most to the cost function, and, -to analyze the importance of the contribution of a given sensor/actuator in the cost function V with respect to the others.

Optimal placement of piezo sensors and actuators by using the modal cost function
In order to test the proposed optimization method to find the optimal placement of sensors and actuators, a cantilever aluminium beam type structure containing one set of sensor/actuator PVDF/PZT ceramic piezoelectric elements bonded to the beam surface is considered.The characteristics of the resulting mechatronic structure is shown in Table 1.
The main purpose of this section is to find the optimal placement of sensors (x s1 , x s2 ) and actuators (x a1 , x a2 ) bonded to the upper and bottom surfaces of the beam.
The output observability/controllability, that is the participation of each sensor/actuator in the output cost function is computed by using Eq.(38).It means that the configuration presenting the largest cost function index is the one whose output is the largest.
The computed output (Eq.( 38)) of the system for different placements of the sensor (x s1 ) and actuator (x a1 ) are summarized in Fig. 4, where the weight matrix Q c = I (identity matrix).Matrices A, B and C are given by Eqs (29.2), (29.3), and (29.4), respectively, C r = 10 −9 F and the number of modes (Eq.24) is N = 20.The x s2 and x a2 values are obtained by using the following relations: where L s and L a are the sensor and actuator lengths as given in Table 1.
As can be seen in Fig. 4, the magnitude of the modal cost function V for the sensor and actuator locations given by x s1 = 0.0m; x a1 = 0.0m, is larger than those of the remaining locations.This result confirms that the selected configuration should require less control energy, and therefore, the associated optimal controller design will entail smaller gain values in matrices [G p ] and [G d ].

Correlation of numerical and experimental results for the system transfer function
To validate the methodology presented in the previous sections, an experimental apparatus was constructed (see Fig. 5).It is constituted by a flexible cantilever aluminium beam structure with piezoelectric sensor and actuator elements symmetrically bonded to both sides of the beam.The characteristics of the experimental model (dimensions and material properties) are listed in Table 1.
The PZT actuator patch [25] (model QP10N) and PVDF sensor film were attached to the beam by using the optimal placement results as obtained in Section 6 (see Figs 6 and 7).
In the scheme of the experimental setup presented in Fig. 8, a signal generator was used to produce a bandwidth noise signal (0-100 Hz) applied to the voltage amplifier [25] (model 1224/5).The actuator [25] (model QP10N) is excited by the voltage amplifier and the structural vibrations are measured by using the   PVDF sensor film.In this experiment, a Spectral Analyzer (Scientific Atlanta -model SD380) is used to determine the frequency response of the beam.The numerical (Eqs ( 24) and ( 25)), and experimental diagrams for the transfer functions are shown in Fig. 9.
The first three simulated (Eq.(12.3)) and experimentally obtained frequencies and modal dampings [26] are shown in Table 2.
Figure 9 presents the transfer function of the piezobeam, by taking into account the first three modes of the beam (Eq.( 24)).The remaining twenty modes are calculated through Eq. ( 26) and their influence is added to the reduced transfer function according to Eq. (25).The above described situation is compared with two other cases, namely: 1 -the transfer function is obtained by using Eq. ( 24) (N = 3) in which the influence of the remaining modes of the system is disregarded; 2 -experimental case.
The influence of the voltage amplifier was considered in the results presented in Fig. 9.This means that Ĝ(s) and G(s) from Eqs (25) and (24) were previously multiplied by the voltage amplifier gain (g a = 30V/V ).It is also important to mention that for the presented application: K c = 0.031472 and ω c = (ω 3 + ω 4 )/2 = 145.33Hz.
It can be concluded from Fig. 9 that the numerical model must include the effect of the dynamical responses of higher order modes.However, some part of the results presented in Fig. 9, such as those in the range from 0 to 10 Hz, show that the results from Eq. ( 24) seem to be better.In the author's opinion this behavior come from other unknown effects, such as the characteristics of the bounding layer of the piezoelectric patches.This particular influence will be discussed later on this paper.
Finally, it can be added that the numerical model used is accurate enough to furnish a good representation of the dynamical system in the frequency range analyzed.Figure 10 presents a comparison between the case for which 150 modes are considered as in Eq. ( 24) and the case for which the higher order modes are taken into account according to Eq. ( 25).It can be observed in Fig. 10 that the two models are very close.
Another important aspect that can be observed in Fig. 9 is that the experimental transfer function differs from the numerical model.Besides, an accurate dynamic model of a flexible structure containing piezoelectric actuators and sensors is difficult to obtain, due to the unknown properties of the bonding layer material [6] discussed the influence and presented the formulation of a finite bonding layer in a flexible beam type structure.In the present work, a simpler model -that of a perfectly bonded piezoelectric -was used (Sec- tion 2.1).However, the adhesive (bonding layer) modify the interface properties (damping and stiffness) between the piezoelectric element and the flexible structure.Therefore, a model adjusting technique (curve fitting) to determine the elastic modulus of the piezoelectric actuator bonding layer (E a ) was used.Then, the experimental and numerical transfer functions were used together in the following error expression to be minimized: were Ĝ(ω) and G exp (ω) are the numerical and experimental transfer functions.A classical optimization method (from the Optimization Toolbox for Matlab) was used to minimize E(ω).The optimal value for the elastic modulus of the piezoelectric material bonding layer was found to be It should be pointed out that the adjusted value for the piezoelectric bonding layer modulus of elasticity is about 50% of the value specified in Table 1.The obvious consequence is that the actuator efficiency is reduced when it is bonded to the structure.
Figure 11 shows the transfer functions corresponding to the experimental and adjusted models.

Experimental evaluation of the optimal control performance
To analyze the performance of the LQG controller, an experimental verification was conducted, as shown schematically by Fig. 12.
The controller was implemented by using a dSpace DS1104 rapid prototyping controller board together with the Matlab and Simulink Software.The flexible beam was excited by using a shaker localized next to the clamp point.The electromechanical exciter was driven by a noise signal (0-100 Hz).A high voltage amplifier (capable of driving highly capacitive loads) was used to supply the necessary voltage to the actuating piezoelectric patch.Since the output voltage of the interface board is limited to ±5 [V ], a voltage amplifier have rank 2N , where N is equal to 3.Then, the control design is carried out by using the iterative LQG algorithm with the following state weight matrices: and where Ω = [1, ω1 ω2 , . . ., ω1 ωN ], and γ = 7.0 × 10 −6 .The control gain matrix [G c ] (Eq.( 31)) is then calculated for the maximum voltage applied to the piezoactuator (150 V ): Using the following weight matrices: and Next step is to design a dynamic observer for the stability of the closed-loop system: The open-loop response of this flexible structure is obtained for an impulsive disturbance force generated by a hammer at the tip of the beam and the voltage output of the sensor V s (t) is shown in Fig. 13.
The active vibration control for the experimental case provides the sensor voltage output as shown by Fig. 14. Figure 15 presents the corresponding control voltage.Consequently, the flexible structure used in the experiments was stabilized by using the feedback control law (LQG).
Figure 16 compares the experimental transfer functions (magnitude and phase) of the open-loop and closed-loop systems (shaker signal to sensor voltage).
It can be observed that the control system was able to reduce the vibrations in the frequency range containing the first three modes.This corresponds to the maximum possible number of controlled modes for one sensor and one actuator, according to Eqs (42) and (43).The vibration reduction level was smaller for the third mode due to its corresponding weight element (see the weight matrix Ω -Eq.( 44)). Figure 16 also shows the appearance of a torsion mode of the beam (≈ 71 Hz).This problem can be overcome by substituting the electromechanical shaker by a PZT actuator.

Conclusions
An LQG controller was designed and implemented on a cantilever flexible beam containing a pair of piezoelectric sensor and actuator.A feed through term was added for minimizing the effect of the removed higher order modes in the low frequency range of the truncated model.The procedure used for placing sensors and actuators along the smart structure (modal cost technique) was proven to be effective.Besides, the modal cost technique has a strong intuitive appeal.It was observed that such a controller resulted in the reduction of the transverse vibration of the flexible structure.The controller was obtained by solving a standard optimal control problem for a finite-dimensional system.A number of experiments was performed, which demonstrated the effectiveness of the developed controller in reducing the vibrations of a flexible beam.It is impor- tant to say that the methodology presented in this paper can be extended to more sophisticated structures, such as thin plates.
(b)).Consider a homogeneous Euler-Bernoulli beam (dimensions: L b , b b , h b ) as shown in Figs 1(a)-(b).The piezoelectric actuator and sensor have dimensions of (x a2 −x a1 )×b a ×h a and (x s2 −x s1 )×b s ×h s , respectively (see Fig.1(b)).The beam transverse deflection at point x and time t is denoted by w(x, t), assuming the beam as a one-dimensional system.

Fig. 1 .
Fig. 1.(a) Flexible cantilever beam containing piezo-sensor and actuator and (b) Geometrical properties of the piezoelectric elements.

Fig. 14 .
Fig. 14.Closed-loop sensor voltage response of the beam.isneeded to drive the actuator in the range of ±150 [V ] for effective control performance.In this case, the observability O and controllability C o matrices:

Fig. 16 .
Fig. 16.Open and closed-loop transfer functions of the beam.

Table 2
First three modes of the cantilever beam