Active Vibration Suppression of Smart Cantilever Beam with Sliding Mode Observer Using Two Piezoelectric Patches

This paper presents a vibration suppression control patch was used as an actuator element, while the other was used as a sensor. The controller design was designed via the balance realization reduction method to elect the reduced order model that is most controllable and observable. sliding mode observer was designed to estimate six states from the reduced order model but three states are only used in the control law. Estimating a number of states larger than that used is in order to increase the estimation accuracy. Moreover, the state estimation error is proved bounde states with the sliding mode observer, to elements. The control spillover problem was avoided, by deriving an avoi stability for the proposed vibration control design. attenuation ability of the proposed optimal control. For 15 mm initial tip able to reduce the tip displacement to about 0.1 experimental results showed a good performance of the proposed LQR control law and the sliding mode obser well a good agreement with theoretical results.


Introduction
Active vibration control is a technique that used for suppression structural vibrations, using this technique to smart structures becomes of great interest and getting much more important. One way of making the structure a smart one is performed by the use of piezoelectric materials which can be used as sensors and actuators are flexible enough to be placed in a variety of places and have the capacity to work ranges of frequencies [1]. Piezoelectric materials have simple mechanical properties, small volume, light weight, and good ability to perform vibration 51 Scarce analytical solutions for specific situations are available. The discretization of these structures is an alternative and basic step for further analysis. The finite element methods (FEM) is an efficient tool for this purpose [5].
In the finite element modeling, the structure is modeled to retain large number of degrees of freedoms. In active vibration control, the use of smaller order model has computational advantages. Therefore, it is essential to apply model reduction techniques for getting a reduced model for designing the control law. One of these techniques is based on balance realization method [6]. The approach taken for reduction the order of a given model based on removal the coordinates, that are the least controllable and observable. To implement this idea, a measure of the degree of controllability and observability is required [7]. For closed-loop system, it is not always possible to get a control law that cause eigenvalues to have the required and desired values. The system is completely controllable if every state variable can be affected in such a way as to cause it to reach a particular value within a finite amount of time by some unbounded control [6]. However, an alternative, more useful measure is provided for asymptotically stable systems of the form given by equations by defining the controllability gramian. Gramian matrices can be used for checking if a system is controllable and observable [6].
The control of the vibration of a smart structure is usually performed via a controller that based on the reduced order model (ROM). Once a ROM based controller is applied to the full order system, actuator forces will reduce the vibration of the lower modes. However, this will also influence the residual modes of the structure, producing undesirable vibration due to the unmodeled dynamics. This phenomenon is known as control spillover [9]. Similarly, the sensor will sense the deflection from the lower modes as well as from the other modes, this is known as observation spillover. Spillover effects are undesirable and may cause performance degradation and even system instability [10]. Indeed, one way to avoid a spillover, which comes from the state estimation or observation process, is to use an observer for estimating the states with minimal error.
Numerous control strategies have been suggested to suppress the vibration of flexible structures for the flexible systems. Some of these studies are state feedback control, linear quadratic regulator (LQR) approach [11], control, ∞ control [12] and sliding mode control [13]. The theory of optimal control is concerned with operating a dynamic system at minimum cost. The case, where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic function, is called the LQR problem. The settings of a regulating controller governing a controlled system are found by using a mathematical algorithm that minimizes a cost function with weighting factors. The cost function is a sum of the deviations of key measurements from their desired values. Often the magnitude of the control action itself is included in this sum so as to bound the energy spent by the control action [11]. The LQR is the most commonly used controller in smart structures [14][15][16][17].
State measurement is required for the controller; an observer is usually designed to estimate the states. Observers are dynamic systems that can be used to estimate the unavailable state variables of a plant. Luenberger observer [18], used a dynamical linear system to generate estimates of the plant states. In some cases, some of the inputs to the system are unknown. This led to the development of the unknown input observer. For the smart beam, which modeled by FEM with a reduced number of states, the remaining states which are acting on the reduced model can be regarded as unknown inputs.
Sliding mode theory is a robust control approach in treating disturbances and modeling uncertainties through the concepts of sliding surface design and equivalent control [13]. Based on the equivalent control, the sliding mode control theory can be used as an observer to estimate the system states. The sliding mode observer (SMO) are able to ensure finite-time convergence of the output estimation error even in the presence of the unknown inputs. Additionally, they can be used to reconstruct unknown inputs using equivalent control methods [19]. The method V-function, [20], has been used to formulate sliding mode observers design which lead the state estimation errors approach to zero asymptotically with matched uncertainties. The main advantage of using sliding-mode observers is that, while in sliding, they are insensitive to the matched unknown inputs. Moreover, they can be used to reconstruct unknown inputs, which could be a combination of system disturbances [21]. Many researchers were interested in designing sliding mode observers for uncertain dynamical systems; such as Utkin et al. [22], Walcot et al. [23], Zak et al. [24], Edwards et al. [19], and Slotine et al. [25].

52
This paper focuses on designing an optimal LQR controller by using sliding mode observer to attenuate the vibration of a smart cantilever beam. Piezoelectric elements are used as an actuator and a sensor. The utilized model for control design purpose is the reduced order model that is obtained according to the balance realization method. Based on the equivalent control, the SMO is designed with estimation error related to the ROM. To avoid control spillover, an avoidance condition, using complete system dynamics, was derived.

Modeling of Smart Cantilever Beam
Figure (1) depicting the smart cantilever beam considered in this work. The beam is bonded with a pair of piezoelectric patches, working as a sensor and an actuator, at the indicated position. The piezoelectric material used in this work is the Lead-Zirconate-Titanate (PZT). Using Euler-Bernoulli beam equation, the partial differential equation of the beam can be written as [26]; is the deflection along the -axis, is the Young's modulus, is the moment of inertia, is the cross sectional area, and is the density. Equation (1) can be solved by using the assumed mode approach, which yields finite dimensional ordinary differential equation set. The dynamic equation of the smart structure is obtained by using both regular beam element and piezoelectric beam elements. The mass and stiffness matrices of the smart structure include the mass and stiffness of the sensor/actuator [27]. The smart beam, shown in Figure (1), is modelled in state space form using the FEM, the beam is divided into six equal finite elements. The sensor and actuator were integrated on the top and bottom surfaces at the second element from the fixed end of the beam.
A beam element is considered with two nodes at its end. Each node is considered to have two degrees of freedom (DOF), i.e. translation (indirection) and rotation. The mass and stiffness matrix are derived using shape functions for the beam element. The inertia force appears in the governing differential equation of the beam, Equation (1), involves a fourth order derivative w.r.t. , and a second order derivative w.r.t. time (acceleration). The solution of this equation is assumed as a cubic polynomial function [28].
where is displacement function which satisfies the fourth order partial differential equation, here x is the local coordinate of the element. The constants to be obtained by using the boundary conditions at both the nodal points (fixed end and free end). And it can be represented in a matrix form as, [29] where , , , and are the degree of freedom at node 1 and 2, respectively, and # is the length of the element. Accordingly, the mass matrix is given as, [29] ) * * * + , , -.
where ) * is the mass matrix of regular beam, , is the shape function. Also for the strain energy the stiffeness matrix is given by [29]: where E , E , ) , ) are the forces and the bending moments acting on nodes 1 and 2 respectively Figure (1

Sensor and Actuator Equations
The sensor equation is derived from the direct piezoelectric equation, which is used to calculate the total charge created by the strain in the structure. The output charge can be transformed into the sensor current O where, z = 0 J and , T is the second spatial derivative of the shape function, Q is the piezoelectric stress constant.
The output current of the piezoelectric sensor measures the moment rate of the flexible beam. This current is converted into the open circuit sensor voltage U V using a signal-conditioning device with the gain G X . Thus [30]: where ] [ Z [ P Q H and ^ is a constant vector depends on the type of sensor, its characteristics and its location on the beam.
The actuator equation is derived from the converse piezoelectric equation. The strain developed _ by the electric field ` on the actuator layer is given by [33]: where, ` a I I is the electric field, and U J is the input voltage applied to the piezoelectric actuator in the thickness direction J . Then the stress b J that developed by the actuator is given by: .. (15) where F is the Young's modulus of the piezoelectric and . is piezoelectric strain constant.
The bending moment of the piezoelectric element is given by: The resultant moment ) J acting on the beam element due to the applied voltage U J is determined by integrating the stress in Equation (15) throughout the structure thickness as: The control force D [ g/ produced by the actuator that is applied on the beam element is obtained as [30]: Alternatively, D [ g/ can be expressed as:

Dynamic Equation of Smart Structure
The element mass and stiffness matrices, Equation (10) and Equation (11), can be assembled using the standard FEM procedure, to obtain the global mass and stiffness matrices of the entire beam. The beam is divided into six finite elements with two piezo-patches placed at the specified location. The equation of motion of the smart structure is given by [28]: where ), >, D N , D [ g/ and D are the global mass matrix, stiffness matrix, external force vator, the controlling force vector (from the actuator), and the total force coefficient vector, respectively. The generalized structural modal damping matrix j is introduced into Equation (21) by using [35,36]: where k and l are the frictional damping constant and the structural damping constant respectively. When applying the cantilever beam boundary condition, the system equation of motion for the six element cantilever beam is: For free vibration condition D N equal to zero, so the remaining applied force on the system is the controlling force D [ g/ exerted by the controller.

State Space Model of the Smart Beam
Model reduction in modern control theory needs a state space form for the mathematical model of a plant. Consequently, the mathematical model of smart flexible cantilever beam can be written in state space form as follows [37]; let then the smart cantilever beam state space model is; And in a matrix form where ! , m 0 :) q > :) q j n , s m 0 ) q ℎ n and r U J .
with appropriate zero and identity matrices dimensions. The sensor voltage is taken as the output of the system and the output equation is obtained as: Thus, the sensor output equation in state space form is given by: where t [0 ^-]. Eventually, the single input single output state space model of the smart beam is given by Equations (26) and (28):

Model Reduction
Using the FEM the structure is modeled to retain large (but finite) number of DOF. In active vibration control, the use of smaller order model has computational advantages. Therefore, it is necessary to apply a model reduction technique to the state space representation [6]. In this work the 24 th order system model, obtained from the FEM modelling, is reduced to the three order using a model reduction technique based on balance realization.
In this approach the state coordinate basis is selected such that the controllability and observability grammians are diagonal matrix and equal , normally with the diagonal entries of σ in descending order the state space representation is then called balanced realization [6].
To implement this idea, a measure of the degree of controllability and observability is needed. However, an alternative, more useful measure is provided for asymptotically stable systems of the form given by equations by defining the controllability grammian, denoted by { | , as [6]: And the observability grammian, denoted by {~, as [6]: where the matrices , s, and t are defined in Equation (30). The properties of these matrices provide useful information about the controllability and observability of the closedloop system. If the system is controllable (or observable), the matrix { | (or {~) is nonsingular [38]. These grammians characterize the degree of controllability and observability by quantifying just how far away from being singular the matrices { | and {~ are [6].
Applying the idea of singular values as a measure of rank deficiency to the controllability and observability grammians yields a systematic model reduction method. The matrices { | and {~ are real and symmetric and hence are similar to a diagonal matrix. There is equivalent system for which these two grammians are both equal and diagonal. Such a system is called balanced system, and { | and {~ must satisfy the following two Liapunov-type equations: [39] Now to transform the system to a balance realization form, the determination of a transformation matrix € • that will transform the system in Equation (29) to: The controllability and observability grammians matrices are diagonal and equal where { ‚ | and { ‚~ are the controllability and observability grammians for system after applying the transformation € and the numbers bˆ are the singular values of the grammians and are ordered such that bˆ> bˆL , O 1,2, … . . , Š Therefore, the pair ′ , s ′ could be uncontrollable pair since some of b ‹ could be equal to zero. Indeed, there exist a subsystem, i.e. a reduced order model, which is still controllable and observable.
Choosing the matrix € • in the form the grammians { [ and {~ will transformed to become equal and transform the system in Equation (29) to a balanced realization form. Namely, where Σ can be written in terms of two set of the singular values b and b as In this representation b describes the "strong" sub-systems to be retained and b the "weak" sub-systems to be deleted. Conformally partitioning the matrices as (38) and truncating the model, retaining g , s g s and t g t as the reduced system, and deleting the "weak" internal subsystems [6].

LQR State Feedback Control Design
The LQR is an optimal control approach to determine control signals that will cause a process to satisfy some physical constraints, and at the same time (maximize or minimize) a chosen performance index or cost function [11]. To design an LQR control, to the reduced order model of the smart beam, the Reduced Model (RM) and the Residual Model (RSM) [9] are presented here as follows; according to the balance realization the linear state model for the cantilever beam, as given in Equation (34)  The pair g , s is the controllable pair and g , t g is the observable pair with highest controllability and observability grammian. From Equation (39), the reduced model of the transformed model, which given in Equation (34), is Note that › represents also the number of the states that will be used in the control design.
In the second step the performance index oe, is defined for the linear regulator problem as [6]; where • and ž are symmetric positive definite weighting matrices. The larger the matrix •, the more emphasis is placed by optimal control on returning the system to zero, since the value of x corresponding to the minimum of the quadratic form g -• g is 0.
On the other hand, increasing ž has the effect of reducing the amount, or magnitude, of the control effort allowed [6].
The optiumal control law that will minimize oe is given by [40].
where € is the solution to the algebraic Riccati Equation [15].
• : €s ž q s -€ g -€ € g 0 ... (43) The linear controller, as in Equation (42), that grantted asymptotic stabilty of the reduce model, may also cause the instability for the system dynamic. This type of instability is named as the control spillover.

Control Spillover Problem and Avoidance Condition.
Spillover phenomenon occurs because that the unmodeled dynamics, which are not included in reduced order model, is excited by the control input as in the case of the reduced model. As a result the complete system model may have positive eigenvalue that leads to system instability. In order to avoid the control spillover, an avoidnace condition is derived as follows; let the matrix ∆ be defined as; which represents the whole model matrix after applying the proposed LQR control. In order to avoid control spillover the matrix Δ must be Hurwitz with its minimum absolute real eigenvalue is larger than theabsolute real eigenvalue of the matrix Equation (29). Namely, if žˆ| represent the real term to eigenvalue of Δ and žˆ~ represent the real term to eigenvalue of A then the avoidance condition is; When difference between ž | and ž~is large,the vibration is attenuated effectively since the eigenvalue placed at ž~ is the dominant one which shaped the cantilever beam response.

Sliding Mode Observer
For a large scale system, like the vibration control problem, the control spillover is not the only source for instability. There is another source of instability comes from imperfect or unprecise estimation for the states that are required for feeding back in the control law [13]. In fact the only way to avoid spillover that comes from the state estimation or observation process is to use an observer that estimates the states with minimal error. In the present work the Sliding Mode Observer (SMO) is used to estimate the states which it required in the control law Equation (42). The advantage of using SMO is to ensure that the difference between the system and observer outputs convergence to zero in a finite-time even in the presence of the unknown inputs. Additionally, the SMO can be used also to reconstruct unknown inputs using equivalent control method [21]. In the present work, the unknown input represents the residual term • • for the g dynamics Equation (39) and the corresponding reduced order model in Equation (40).
For the linear time invariant system; Equation (39) R g g g s r j¦ t g g u …(45) where the unknown input ¦ • ∈ ℛ ‡qg and j • ∈ ℛ g× ‡qg . Now decompose g to g [ ]and t g to t g [t t ], then the output is: where ∈ ℛ gq , ∈ ℛ , t ∈ ℛ × gq t ∈ ℛ , where one of the attached piezoelectric patches is used as a sensor, and det t ≠ 0. Note that › represents the number of the estimated states which it selected later equal to six. Accordingly Equation (45) where s ∈ ℛ gq × , s ∈ ℛ , j ∈ ℛ gq × ‡qg and j ∈ ℛ × ‡qg . Equation (47) can be written in terms of and as follows; where is replaced by, t q : t …(49) and, Since |t | ≠ 0, the transformation matrix between , and , , m n ® m n o gq ¯ gq × t t p m n …(50) is nonsingular.
To this end the proposed SMO to the system dynamic model, as given in Equation (48) where Q ° : and Q ³ °: .
To examine the stability of the error dynamics, the equivalent control is utilized as follows; first the output error Q ³ is guaranteed to reach zero value in a finite time if ² is selected as a discontinuous function of Q ³ ² ª Q ³ ´ * µ"Š Q ³ …(53) 58 where µ"Š Q ³ is the signum function defined as follows; µ"Š ¶Q ³ · ¸ 1 OD Q ³ > 0 :1 OD Q ³ < 0 …(54) and ´ is the discontinuous gain that must satisfy the following condition ´> ¥ ª Q : j ¬ ¦ ¥ …(55) If ´ satisfies the inequality (55), then the sliding motion will ocuure on the sliding surface Q ³ 0 after a finite time. By taking the initial condition for observer design as; and with ² as in Equation (53) where ² N½ is computed at Q ³ QR ³ 0.
From Equation (57) the unknown input term ¶±j ¬ : j ¬ ·¦ in the estimation error dynamics will prevent the error Q from decaying exponantionaly to zero where it represents the error source in the observation process. In fact, this term is the source of observer spillover that may cause instability in system response. Minimizing the error in the observation process is not an easy task since it influenced by the size of the reduced order model and the selected ± matrix as stated by the following proposition.
Proposition (1) the estimation error for the sliding mode observer as proposed in Equation (57) will converge to a region around the origin bounded by Remark (1): in order to minimize the state estimation error we must select the ± matrix such that the norm ¿ ¶±j ¬ : j ¬ · -€¿ is minimal and, in the same time, the matrix ¶ ª : ± ª · is Hurwitz as mentioned above. This task cannot easily be handled by try and error but we try in this work to select a suitable ± such that we get an appropriate estimation accuracy.

Simulation Results and Discussion
The simulation results for a cantilever beam, which is subjected to an initial tip deflection, are presented in this section where the MATLAB software is used to simulate the cantilever beam system.
The physical and ge specifications for the beam are given in Table (1). Three steps are performed in the present work toward designing an active vibration control to the cantilever beam. In the first step, the natural frequencies for derived mathematical model of th beam Equation (30) are calculated and compared with the natural frequencies obtained from the ANSYS program. The results are presented in Table (2) which show a good agreement. This proves that the derived model represents system dynamics at least with respect to the dominant natural frequencies.
Also the balance realization and order reduction process for the system model had been performed to reduce its states from twenty-four states to three states, without affecting its dominant mode. This is demonstrated in Figure (2) in the Bode plot.

Simulation Results and Discussion
The simulation results for a cantilever beam, subjected to an initial tip deflection, are presented in this section where the MATLAB 14 the cantilever beam system.
The physical and geometrical specifications for the beam are given in Table (1). Three steps are performed in the present work toward designing an active vibration control to the cantilever beam. In the first step, the natural mathematical model of the ) are calculated and compared with the natural frequencies obtained from the ANSYS program. The results are presented in Table (2) which show a good agreement. This represents the th respect to the Also the balance realization and order reduction process for the system model had been performed four states to three states, without affecting its dominant mode.
in the Bode and geometrical specification for the flexible cantilever beam and the piezoelectric. In the second step, the sliding mode observer is designed to the reduced order model. The dimension of the RM for the SMO is selected in this work equal to six while the LQR control will use only three of them. The idea behind this step is that the estimation error is decreased by using larger dimension to the reduced order model than that used in the control law.
The estimated output using the SMO states ( t g g , g is the estimated six states) is compared with actual output in the discontinuous term injected Equation (53) with ´ 130000 output is chatters around the actual piezoelectric output. This is more clarified in Figure the error between the actual output and the estimated output Equations ( plotted. The chattering effect can be removed by replacing the signum function Eq a continuous approximate function (the arc tan function). The signum function µ"Š ¶Q ³ · È É * tan q Š * Q ³ , Replacing µ"Š Q ³ by the approximation given above will prevent chattering and give smooth values for the estimated states shown in Fig  to 6). By using only three of the estimated states, the designed control law based on the is applied to the cantilever beam dynamic and the system is simulated for 15 mm initial tip displacement. The system eigenvalues after using the proposed LQR control are found in Table (3). Table (4) shows that by using reduced model of dimension three, four or six that can be selected according to the singular values, the control LQR change the value of the minimum absolute eigenvalue to the same value. So the best choice of the reduce model dimension is three. Journal, Vol. 13, No. 1, P.P. 50-65(2017)

Bode plot.
In the second step, the sliding mode observer is designed to the reduced order model. The dimension of the RM for the SMO is selected in this work equal to six while the LQR control will use only three of them. The idea behind this step ion error is decreased by using larger dimension to the reduced order model than The estimated output using the SMO states is the estimated six states) is compared with actual output in Figure (3). Due to in the SMO (² in 130000)), the estimated output is chatters around the actual piezoelectric . This is more clarified in Figure (4) where actual output and the (46) & (51)) is plotted. The chattering effect can be removed by function Equation (54) with a continuous approximate function (the arc tan function is replaced by; Š 0.1 by the approximation given above will prevent chattering and give smooth values for the estimated states shown in Figures (5 By using only three of the estimated states, the designed control law based on the LQR approach is applied to the cantilever beam dynamic and the system is simulated for 15 mm initial tip displacement. The system eigenvalues after using the proposed LQR control are found in Table (3).
(4) shows that by using reduced model of sion three, four or six that can be selected according to the singular values, the control LQR change the value of the minimum absolute eigenvalue to the same value. So the best choice of the reduce model dimension is three.    The first set of numerical simulation to the control system and the SMO uses a 0.00001 second as a period of integration and with the approximate signum function in the observer design as defined above. In Figure (7) the controlled tip displacement with the actual system output is compared, the ability of the controller in stabilizing the tip displacement is clarified in this figure where it required about 3 second only. In addition, the control input voltage to the piezoelectric element is plotted in Figure (8), where, as can be seen, the control input is saturated to the maximum value 200V which is the maximum design value to the piezoelectric voltage. In order to approach the real situation, the time period for the observer is taken equal to 0.0001 second (i.e., the change in the estimated six states happened after each 0.0001 second), while 0.0025 second is the chosen time period where the control input changes. Consequently, the second set of numerical simulation is based on these time numerical values for both; the observer simulation period of time and for the time period required for the control input voltage to change while the time period for the control system simulation is still equal to 0.0001 second. Figure (9) plots the actual output and the estimated output with time. The effectiveness of the SMO can be detected from this figure where the idea is to force the estimated output to follow the actual one in a short time. After that, the estimated states will be used in the control law Equation (42) which will attenuate the beam vibration. The tip displacement of the open and closed loop are plotted with time in Figure  (10), additionally, the control input voltage is shown in Figure (11), where, as can be seen, the vibration suppression ability is nearly the same as in the first set of simulation. This enhance the applicability of the proposed controller.

Experimental Setup and Results
The beam used in the current analysis was a stainless steel of length 276 mm, width 33 mm, and thickness 1 mm. Two piezoelectric (PZT) patches (type QP20W), of dimensions 46×33 mm and thickness 0.762 mm, were used as sensor and actuator, as depicted in Figure (1). The parameters of the flexible beam and PZT patches are given in Table (1). The control objective is to show the effectiveness of the designed LQR controller, via the sliding mode observer, to suppress the vibration of the smart cantilever beam by means of piezoelectric elements. To achieve this control objective an experiment is set up. The experimental set up is depicted in Figure (12), which consists of a stainless beam, two PZT patches (bonded on two sides of the beam at the location indicated), accelerometer (ADXL335), two data acquisition systems (NI-PCI 6036 and NI-USB 6259), high voltage power amplifier (Trek-601B-3), and a PC to control the system via LabVIEW program.
The beam is initially excited by 15mm tip displacement, as the beam vibrate, the stress induced in the PZT patches will generate voltages proportional to these stresses, in this case one PZT patches is working as a sensor. As the voltage signal sent to the data acquisition, type NI-6036E, the LabVIEW program, designed for this purpose using sliding mode observer, will use these signals to observe the states of the system and then fed-back to the controller. The control signals are sent to the data acquisition and then to high voltage power amplifier, type Trek-601B-3 to apply this action to the other PZT element, in this case the other PZT is used as an actuator. As a result, the vibration in the beam will be suppressed. The controller system sent signals between ±5 Volt, then these signals are amplified 40 times, by power amplifier, therefore the control signals are bounded within ±200 Volt.
The tip point displacement of the beam was measured by using the accelerometer, as shown in Figure (12). The measured signals are sent to the data acquisition type NI-6259, and then to the PC computer to the control algorithm programmed in the LabVIEW software. The acceleration signals were converted to displacement by double integral, and then plotted with time. The controlled and uncontrolled tip displacement are plotted in Figure (13). It can be seen that the current control action is having the ability to suppress the tip displacement within 3 second. The PZT patch sensor output (the actual output) and the sliding mode observer output are given in Figure (14), while the voltage inputs to PZT patch actuator is shown in Figure (15). Finally, a good agreement are clear between the experimental results see Figures (13 to 15) and the results obtained from the numerical simulations Figures (9 to 11).

Conclusions
In this paper, the state space model is obtained using the finite element approach and the modal analysis resulting after appropriate modal reduction. During the theoretical calculations, the 24 th order system model obtained from the finite element model is reduced to the three order using a model reduction technique based on balance realization without affecting its dominant modes.
As a basic requirement to the control design, the sliding mode observer is designed to estimates six states of the reduced model. In spite of the presence of the unknown inputs, which due to the residual model in the observer dynamics, the SMO forces the output, which is determined from the estimated states, to follow the actual output. This is taken place after approximately two seconds. After that, the reduced model states are estimated with bounded error.
To overcome the chattering problem in observer dynamics the signum function is replaced with the approximation given by the arctan function with appropriate parameters. As a result, the chattering is prevented and the estimated states values become more smooth.
Using the estimated states, an LQR approach was designed based on the reduced order model. The control spillover was avoided by satisfying the avoidance condition where the minimum absolute real eigenvalue is three times to that for the matrix Equation (30). With the proposed LQR control and the SMO, the results show that for 15 mm initial tip displacement, the piezoelectric actuator reduces the tip displacement to about 0.1 mm during 3 s.
The performance of the designed controller is examined experimentally where the results are shown very satisfactory and very close to the theoretical analysis.