Active vibration control of smart piezoelectric beams: Comparison of classical and optimal feedback control strategies

Computers and Structures 84 (2006) 1402–1414 www.elsevier.com/locate/compstruc

Active vibration control of smart piezoelectric beams: Comparison of classical and optimal feedback control strategies C.M.A. Vasques, J. Dias Rodrigues

*

Faculdade de Engenharia da Universidade do Porto, Departamento de Engenharia Mecaˆnica e Gesta˜o Industrial, Rua Dr. Roberto Frias s/n, 4200-465 Porto, Portugal Received 16 June 2005; accepted 14 January 2006 Available online 5 June 2006

Abstract This paper presents a numerical study concerning the active vibration control of smart piezoelectric beams. A comparison between the classical control strategies, constant gain and amplitude velocity feedback, and optimal control strategies, linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) controller, is performed in order to investigate their effectiveness to suppress vibrations in beams with piezoelectric patches acting as sensors or actuators. A one-dimensional finite element of a three-layered smart beam with two piezoelectric surface layers and metallic core is utilized. A partial layerwise theory, with three discrete layers, and a fully coupled electro-mechanical theory is considered. The finite element model equations of motion and electric charge equilibrium are presented and recast into a state variable representation in terms of the physical modes of the beam. The analyzed case studies concern the vibration reduction of a cantilever aluminum beam with a collocated asymmetric piezoelectric sensor/actuator pair bonded on the surface. The transverse displacement time history, for an initial displacement field and white noise force disturbance, and point receptance at the free end are evaluated with the open- and closed-loop classical and optimal control systems. The case studies allow the comparison of their performances demonstrating some of their advantages and disadvantages.  2006 Civil-Comp Ltd. and Elsevier Ltd. All rights reserved. Keywords: Smart piezoelectric beam; Finite element; Active vibration control; Velocity feedback; LQR and LQG controller

1. Introduction The aim of active vibration control is to reduce the vibration of a mechanical system by the automatic modification of the system’s structural response. An active structure consists of a structure provided with a set of sensors (to detect the vibration) and actuators (which influence the structural response of the system) coupled by a controller (to suitably manipulate the signal from the sensor and change the system’s response in the required manner). Structures which have the distinctive feature of having sensors and actuators that are often distributed and have a high degree of integration inside the structure are called *

Corresponding author. Tel.: +351 22 508 1723; fax: +351 22 508 1584. E-mail addresses: [email protected] (C.M.A. Vasques), [email protected] (J. Dias Rodrigues).

smart structures. A review of the state of the art of smart structures was performed by Chopra [1] and related books can be found in Refs. [2,3]. Some of the most widely used distributed sensors and actuators are made of piezoelectric materials. The piezoelectric effect consists of the ability of certain crystalline materials to generate an electrical charge in proportion of an externally applied force (direct piezoelectric effect) and to induce an expansion on the piezoelectric material in proportion to an electric field parallel to the direction of polarization (inverse piezoelectric effect) [4]. There are two broad classes of piezoelectric materials used in vibration control: polymers and ceramics. The piezopolymers are used mostly as sensors because they require extremely high actuating voltages; the piezoceramics are extensively used both as sensors and actuators, since they require lower actuating voltages, and are used for a wide range of frequency.

0045-7949/$ - see front matter  2006 Civil-Comp Ltd. and Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2006.01.026

C.M.A. Vasques, J. Dias Rodrigues / Computers and Structures 84 (2006) 1402–1414

Significant advances in smart material actuators have taken place in the past decade and the most current trends in piezoelectric actuation architectures are presented in a review paper by Niezrecki et al. [5]. Modeling smart structures often requires a coupled modeling between the host structure and the piezoelectric sensors and actuators. They can be modeled as either lumped or distributed parameter systems, and usually these systems have complicated shapes and structural patterns that make the development and solution of descriptive partial differential equations burdensome, if not impossible. Alternatively, various discretization techniques, such as finite element (FE) modeling, modal analysis, and lumped parameters, allow the approximation of the partial differential equations by a finite set of ordinary differential equations. Since the 1970s, many FE models have been proposed for the analysis of smart piezoelectric structural systems. A survey on the advances in piezoelectric FE modeling of adaptive structural elements is presented by Benjeddou [6]. On the development of FE models, different assumptions can be taken into account in the theoretical model when considering the electro-mechanical coupling. These assumptions regard mainly the use (or not) of electric degrees of freedom (DoFs) and the approximations of the through-the-thickness variation of the electric potential. Therefore, they lead to decoupled, partial and fully coupled electro-mechanical theories, which in turn can lead to different modifications of the structure’s stiffness and different approximations of the physics of the system. These different electro-mechanical coupling theories might, however, be considered by the use of effective stiffness parameters, defined according the electric boundary condition considered, as shown by Vasques and Rodrigues [7] for a smart beam. The recent advances in digital signal processing and sensors and actuators technology have prompted interest in active vibration control (see related books [8–13]). In the past two decades, various strategies to actively control the vibrations of structures with piezoelectric layers acting as sensors or actuators have been applied. Some examples of analytical and experimental studies concerning the actuation and vibration control of smart piezoelectric beams can be found in Refs. [14–17]. An analytical and experimental study utilizing classic and optimal feedback control techniques for the vibration control of beams and plates was performed by Han et al. [18] and FE models for plates and shells are presented in Refs. [19,20]. Studies about modal sensors and actuators concerning independent modal space control applied to beams are presented in Refs. [21–23]. The different algorithms utilized in active vibration control can be classified under two general categories: feedforward and feedback control. Variations of the two general methods exist, each with advantages, disadvantages and limitations. A review about the active structural vibration control is presented by Alkhatib and Golnaraghi [24].

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The present work is entirely devoted to feedback control where one is particularly concerned with systems in which the excitation of the structure cannot be directly observed and thus cannot be used as a feedforward control signal. Therefore, the control systems discussed here will be those in which the control signal obtained from the piezoelectric sensor, which is affected by both the excitation force (primary excitation) and the piezoelectric actuator voltage (secondary excitation, over which we have control), is fed back to the actuator. The design problem consists of finding the appropriate compensator such that the closed-loop system is stable and behaves in an appropriate manner. As noticed by Preumont [13], one of the objectives of feedback control can be to reduce the resonant peaks of the frequency response function, which is known as active damping. It can be achieved without a model of the structure and with guaranteed stability, provided that the actuator and sensor are collocated and have perfect dynamics. Another objective of the control can be to reduce the effects of external disturbances in order to keep a control variable (e.g., position) to a desired value in some frequency range, which is known as model based feedback. These model based strategies are global methods which manage to attenuate all the disturbances in the frequency range of analysis. In compensation they require a mathematical model of the system (e.g., FE model), have a bandwidth limited by the accuracy of the model and may amplify the disturbances outside the bandwidth (spillover). In the open literature similar studies concerning different active control of vibration strategies (e.g., feedforward wave suppression, proportional and velocity feedback, optimal control) and with different types of actuation (e.g., point forces, pair of moments, piezoelectric patch actuation) in different structural systems (beams with piezoelectric actuators or hybrid treatments also with passive viscoelastic layers) can be found in Refs. [25–27]. Trindade et al. [26] present a study concerning a FE model of sandwich beams with piezoelectric laminated surface layers and viscoelastic cores where the control design and performance are evaluated for a cantilever sandwich beam model using three control algorithms, namely, LQR, LQG and derivative feedback. One of the well-known advantages of having treatments with viscoelastic layers is that they increase the passive damping and consequently the stability margin of the control system; that makes it substantially different from a pure active system with only piezoelectric layers. However, in that work, time domain results for an initial velocity field, for the three algorithms, and frequency response functions, determined by analytical definitions, only for the LQR and derivative feedback algorithms, have been reported and the effects of the statistical content of noise have only been analyzed to assess the state estimator convergence in the time domain. The aims of this paper are the analysis and comparison of the classical and optimal feedback control strategies on the active control of vibrations of smart piezoelectric beams. The mathematical model utilized by Vasques and

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Rodrigues [7] in the development of a one-dimensional smart beam FE is succinctly described. The kinematic and electric potential assumptions of the theoretical and FE spatial model are first presented. Next, the resultant FE equations of motion and electric charge equilibrium are presented and recast into a state variable representation in terms of the physical modes of the beam. Moreover, some theoretical considerations about the classical control strategies, with two distinct velocity feedback control algorithms being utilized, constant amplitude and constant gain velocity feedback (CAVF and CGVF), and optimal control strategies, linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) controller, are presented. The control models assume that one of the piezoelectric layers acts as a distributed sensor and the other as a distributed actuator, and the sensor signal is used as a feedback reference in the closed-loop control systems. Finally, a case study concerning the active vibration control of a cantilever aluminum beam with an asymmetric piezoelectric sensor/actuator pair mounted closed to the clamped edge is analyzed. The displacement time history, for an initial displacement field and white noise force disturbance, and point receptance at the free end are evaluated with the open- and different closed-loop classical and optimal control systems. Furthermore, a white noise force disturbance is considered to simulate a system subjected to a stochastic mechanical disturbance and the driving point frequency response functions are obtained for all the control systems (including the CAVF and LQG) by a frequency response function estimation procedure, as usually done with real measurements in experimental analysis. 2. Mathematical model

ub ðx; zb ; tÞ ¼ u0 ðx; tÞ þ

  hc hb hc ðx; tÞ þ zb þ hb ðx; tÞ; 2 2

uc ðx; zc ; tÞ ¼ u0 ðx; tÞ þ zc hc ðx; tÞ; ua ðx; za ; tÞ ¼ u0 ðx; tÞ 

  hc ha hc ðx; tÞ þ za  ha ðx; tÞ; 2 2

ð1Þ

wk ðx; tÞ ¼ w0 ðx; tÞ; where the subscript k = a, b, c refers to the layer, hk is the thickness of each layer, u0(x, t) and w0(x, t) are the axial and transverse displacements of the beam’s mid-plane, and hk(x, t) is the rotation of each layer. Note that axial displacement continuity at the interfaces of the layers is assured, leading to coupling terms in the axial displacement uk(x, zk, t) of the layers, and that a constant through-thethickness transverse displacement w0(x, t) is considered. According to the displacement definitions in Eq. (1), the extensional and shear mechanical strains of the layers can be determined by the usual linear strain–displacement relations, where the kinematic assumptions lead to null transverse strains and a first-order shear deformation theory (FSDT) for each layer (see [7] for further details). The piezoelectric material considered for the surface layers is orthotropic with the directions of orthotropy coincident with the axes of the beam, x, y, zk, and is polarized in the transverse direction zk. It has the behavior of standard piezoelectric materials with the symmetry properties of an orthorhombic crystal of the class mm2 [28,29]. Furthermore, the electrical model takes the direct piezoelectric effect into account by considering a quadratic throughthe-thickness distribution of the electric potential and an appropriate approximation of the x-component of the electric field leading to a fully coupled electro-mechanical model (see [7,30] for further details).

2.1. Kinematic and electrical assumptions 2.2. Spatial model Consider the three-layered beam illustrated in Fig. 1. The axial and transverse displacements, uk(x, zk, t) and wk(x, t), of the metallic core (c) and two piezoelectric surface layers (a, b) are defined as

The FE spatial model is obtained from the weak form of the equations of motion and electric charge equilibrium presented by Vasques and Rodrigues [7]. It is derived using

Fig. 1. Displacement field of the three-layered smart piezoelectric beam.

C.M.A. Vasques, J. Dias Rodrigues / Computers and Structures 84 (2006) 1402–1414

Hamilton’s principle where the Lagrangian and the applied forces total work are conveniently adapted for the electrical and mechanical contributions [31]. The one-dimensional smart beam FE considers a fully coupled electro-mechanical theory and has two nodes, with five mechanical degrees of freedom (DoFs) per node (the axial and transverse displacements of the beam’s mid-plane and the rotation of each layer), and two electric DoFs per element (the electric potential difference of each piezoelectric layer). The FE assumes linear C0 shape functions for the mechanical DoFs and the electrical DoFs are assumed constant and uniform in the element. Therefore, the elemental mechanical and  e ðtÞ, are defined as electrical DoFs vectors,  ue ðtÞ and /    10 ðtÞ;  u10 ðtÞ; w h1a ðtÞ;  h1c ðtÞ;  h1b ðtÞ; ue ðtÞ ¼  T   20 ðtÞ;  ð2Þ u20 ðtÞ; w h2a ðtÞ;  h2c ðtÞ;  h2b ðtÞ ;   T  a ðtÞ; /  b ðtÞ :  e ðtÞ ¼ / / The resultant global FE spatial model, governing the motion and electric charge equilibrium, is given by  ¼ FðtÞ;  Muu € uðtÞ þ Kuu  uðtÞ þ KT/u /ðtÞ  ¼ QðtÞ; K/u uðtÞ þ K// /ðtÞ

ð3Þ

 are the global mechanical and electrical where  uðtÞ and /ðtÞ DoFs vectors, Muu and Kuu are the global mass and stiffness matrices, Ku/ ¼ KT/u is the global piezoelectric coupling matrix, K// is the global capacitance matrix and F(t) and Q(t) are the global mechanical force and electric charge vectors. The electrical DoFs vector in Eq. (3) can be divided into  ¼ col½/  a ðtÞ; /  s ðtÞ, the actuating and sensing DoFs, /ðtÞ where the subscripts ‘a’ and ‘s’ denote the actuating and sensing capabilities. Furthermore, the stiffness matrix can be written as the sum of the elastic and piezoelectric layers stiffness matrices KEuu and KPuu . Hence, considering open-circuit electrodes, and in that case Q(t) = 0, the non-specified potential differences in (3) can be statically condensed and the equations of motion and charge equilibrium become  a ðtÞ þ FðtÞ;  Muu € uðtÞ þ ðKEuu þ KP uðtÞ ¼ KT/ua / uu Þ  s ðtÞ ¼ K1 K/us  / uðtÞ;

ð4Þ

//s

where P T 1 KP uu ¼ Kuu  K/us K//s K/us :

ð5Þ

It is worthy to mention that a non-null parabolic throughthe-thickness distribution of the electric potential within the piezoelectric layers was already considered in the variational formulation through the use of effective stiffness parameters and therefore the static condensation in Eq. (5) only considers the linear counterpart of the electrical potential distribution, which is the one that in fact contributes to the sensor voltage. Moreover, a second alternative where the equipotential area condition is satisfied by means of a modified static condensation of the non-specified potentials might be utilized, which corresponds to a more realistic approach, that becomes more significant for bare

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piezoelectric beams or as the length of the piezoelectric layers approaches the length of the host beam (see [7,32] for further details). 2.3. Modal model In the process of designing an active control system one can utilize a full model of the system and, consequently, a higher computational effort is needed, or a reduced model of the system, which requires a lower computational effort. However, the structural mathematical model and controller design are not independent aspects of vibration control. Flexible structures are distributed parameter systems that have an infinite number of DoFs, and a feedback controller based on a finite reduced modal model can destabilize the residual modes (unmodeled dynamics) leading to observation and control spillover problems. They both degrade the system’s performance, and the former can even cause the system to become unstable. Methods to reduce the effects of spillover are discussed by Balas [33] and Meirovitch [9]. When excited a structure presents preferable modes of vibration which depend of the spectral content of the excitation. Assuming that the lower order modes, which have lower energy associated and consequently are the most easily excitable ones, are the most significant to the global b can response of the system, a truncated modal matrix W be utilized as a transformation matrix between the generalized coordinates uðtÞ and the modal coordinates g(t). Thus, the displacement vector uðtÞ can be approximated by the modal superposition of the first r modes, uðtÞ ¼

r X

b Wi gi ðtÞ ¼ WgðtÞ;

ð6Þ

i¼1

b ¼ ½W1 ; . . . ; Wr  is the truncated modal matrix and where W g(t) = {g1(t), . . . , gr(t)}T the correspondent modal coordinates vector. Hence, the system’s size is not anymore the total number of DoFs of the FE model but the number of modes chosen to model it. The spatial stiffness and mass matrices obtained with the FE method typically present a band structure, which represents the coupling between the several DoFs, and the spatial damping matrix is usually difficult to be obtained. In this study a proportional viscous damping model is utilized and the spatial damping matrix Duu, according to the orthogonality properties of the modal vectors with respect to the mass and stiffness matrices, is diagonalized by the b truncated modal matrix W, b T Duu W b ¼ K; W

ð7Þ

and a diagonal modal damping matrix K with the generic term 2nixi, where ni is the modal damping ratio and xi the undamped natural frequency of the ith mode, is obtained. Therefore, the inclusion of the inherent damping effects in the model is considerably simplified. Thus, con-

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sidering (6) and a normalization scheme for unitary modal masses, Eq. (4) with the damping effects included become b T KT / bT  _ þ XgðtÞ ¼  W I€ gðtÞ þ KgðtÞ /ua a ðtÞ þ W FðtÞ;  s ðtÞ ¼ K1 K/us WgðtÞ; b / //s

ð8Þ

where I is an identity matrix and X is a diagonal matrix with the generic term x2i . Eq. (8) represent the reduced (truncated) modal FE model of the smart piezoelectric beam. 3. Active control of vibrations 3.1. State space design The state space approach is the basis of the modern control theories and is strongly recommended in the design and analysis of control systems with a great amount of inputs and outputs [34]. In this method, dynamic systems are described by a set of first-order differential equations in variables called the state. According to Eq. (8) the state variables are chosen as _ xðtÞ ¼ col½gðtÞ; gðtÞ;

ð9Þ

and the open-loop system is represented by two first-order matricial differential equations expressed in terms of the state variables vector x(t), _ ¼ AxðtÞ þ B/ u/ ðtÞ þ Bu uu ðtÞ; xðtÞ yðtÞ ¼ CxðtÞ;

ð10Þ

where A is the system matrix, Bu and B/ are the mechanical and electrical input matrices, C is the output matrix, uu(t) and u/(t) are the mechanical and electrical input vectors and y(t) is the output vector, given by   0 I A¼ ; X K " #   0 0 Bu ¼ ; B/ ¼ b T KT ; bT W W /ua 2 3 b K WÞ 0 avgðK1 /us //s 5; C¼4 b 0 avgðK1 //s K/us WÞ uu ðtÞ ¼ FðtÞ;

 a ðtÞ; u/ ðtÞ ¼ /

 s ðtÞ; / _ s ðtÞ: yðtÞ ¼ col½/ ð11Þ

It is worthy to mention that in the output matrix C the notation avg(Æ) is utilized to denote that the signal induced by the piezoelectric sensors should be calculated from an average of the electrical DoFs where an electrical FE separation of the electrodes was performed. That is an approximation procedure utilized to estimate the induced voltage if an equipotential area condition had been considered which corresponds to a more realistic case for electroded layers in open-circuit [7]. The open-loop system in Eq. (10) considers two different input vectors, a mechanical disturbance force uu(t) (pri-

mary excitation) and a control voltage u/(t) (secondary excitation), and one output vector y(t) (sensor voltage and its derivative). However, in general one can consider that multiple inputs are applied to the smart beam (multiple forces and piezoelectric actuators) and that multiple outputs are obtained from multiple piezoelectric sensors. 3.2. Classical control With the purpose of reducing the vibrations of smart beams one can establish a feedback control loop where the signals produced by the piezoelectric sensors are amplified and fed back to the actuators in order to produce a secondary excitation that can cancel the primary one. Therefore, the control voltage is given by u/ ðtÞ ¼ GyðtÞ;

ð12Þ

where G is a feedback gain matrix defined according to the control law of interest. Substituting Eq. (12) into (10), the closed-loop system state equation is given by _ ¼ ðA  B/ GCÞxðtÞ þ Bu uu ðtÞ; xðtÞ

ð13Þ

where the control vector u/(t) is condensed into the state equations. From Eq. (13) one can see that the gain matrix G controls the system through the modification of the closed-loop system poles. Therefore, if an adequate gain matrix and control law are established, the system vibration modes are attenuated through an increase in the modal damping ratio (the initial open-loop poles of matrix A are transformed into the highly damped ones of A  B/GC). The gain selection and control system design of such feedback controllers can be done using either a pole-zero representation of the system (e.g., root-locus method, pole placement) or a frequency response representation (e.g., Nyquist method). Control systems where indirect methods are used for the definition of the feedback gains are often referred to as classical control systems [35]. The stability is guaranteed provided the actuator and sensor are collocated [13]. However, perfect collocation with an asymmetric piezoelectric sensor/actuator pair is unfeasible due to the in-plane and out-of plane coupling and can lead to instabilities of the control system. Therefore, some care must be taken in the design of the controller and, as pointed out by Yang and Huang [36], in principle, a symmetric collocated sensor/actuator technique would make the controller absolutely stable. Assuming that a velocity feedback scheme is utilized, where the sensor output is differentiated, amplified, and then fed back into the actuator, the signal used in the velocity feedback is representative of the strain rate of the beam. Thus, a conventional term, velocity feedback, is used. In the velocity feedback two control algorithms are considered [10]. In the first one, which is termed constant amplitude velocity feedback (CAVF), the individual gain of the ith piezoelectric actuator is defined according to the polarity of the ith sensor voltage and, to denote that, the function sign(Æ) is utilized. Hence, the feedback control voltage

C.M.A. Vasques, J. Dias Rodrigues / Computers and Structures 84 (2006) 1402–1414

amplitude is constant, non-linear and discontinuous, and the gain matrix is defined by G ¼ ½0

diagðA1 ; . . . ; Ai ; . . . ; An Þ ;

ð14Þ

where diag(Æ) denotes a diagonal matrix with the individual amplitudes Ai of the ith actuating control voltage, with i = 1, . . . , n, and the control voltage is given by u/(t) = G sign(y(t)). Alternatively, a constant gain velocity feedback (CGVF) can be used, with the gain matrix defined by G ¼ ½0

diagðG1 ; . . . ; Gi ; . . . ; Gn Þ ;

ð15Þ

where Gi is the individual gain of the ith actuator and the control voltage is given by Eq. (12). In this case, the feedback control voltage is linear, continuous and decreases as the vibration velocity decays. It is worthy to mention that in Eq. (13) the output matrix C assumes that all the sensor voltages and its derivatives are known for every sensor, but the zero matrices in Eqs. (14) and (15) indicate that only the voltage velocity is used in the feedback loop.

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the designer chooses to model the system, i.e., the way he suitably chooses the state variables and the system outputs. As noted by Inman [8], modal control can be examined in either state space or physical space form in terms of the physical modes of the mechanical system. The present work considers an approach to control system design where the dynamics of the mechanical system is considered in terms of its modal response. Hence, one can choose as a performance index the cost function minimizing the outputs of the system (sensor voltage and its derivative), Eq. (16), or, alternatively, one can chose to minimize the state variables (modal amplitudes), Eq. (17). By a convenient definition of the state weighting matrix Qx, the modal gain matrix can be ‘tuned’ to give different design objectives. Assuming that all the modes (state variables) are observable and controllable, the cost function in Eq. (17) provides independent control over the natural frequencies and damping ratios of each mode. That strategy is called independent modal-space control, or IMSC [9]. Some convenient choices for the definition of the weighting matrices can be, for example,

3.3. Optimal control

R ¼ diagðI; IÞ;

In the previous section the values of the feedback gains were chosen to achieve some prescribed change in the dynamic properties of the system. However, the ultimate aim of the feedback control is often to reduce the motion of the mechanical system to the greatest possible extent and, in that case, the control system is said to act as a regulator. Systems where direct methods of designing feedback control systems which achieve the greatest possible reduction in the dynamic response are used, are known as optimal control systems [34,37]. In optimal control the feedback control system is designed to minimize a cost function, or performance index, which is proportional to the required measure of the system’s response and to the control inputs required to attenuate the response. The cost function can be chosen to be quadratically dependent on the output response and control input, Z tf h i J¼ yT ðtÞQy yðtÞ þ uT/ ðtÞRu/ ðtÞ dt þ yT ðtf ÞSy yðtf Þ;

where an identity matrix I and diagonal matrix X with the generic term x2i (squared natural frequency of the ith mode) of size (r · r) are utilized. It can be seen in Refs. [34,37] that the feedback control system which minimizes the cost function in Eq. (17) for the linear time-invariant system defined in Eq. (10), uses state feedback with a time-varying feedback gain matrix Kg ðtÞ, so that

0

ð16Þ where Qy, R and Sy are the output, control input and terminal output condition positive definite weighting matrices, respectively. Alternatively, the cost function J can also be written in another form, Z tf h i J¼ xT ðtÞQx xðtÞ þ uT/ ðtÞRu/ ðtÞ dt þ xT ðtf ÞSx xðtf Þ; 0

ð17Þ where Qx and Sx are the state variable and terminal state condition positive semi-definite weighting matrices. Eq. (17) is the form of cost function generally considered in optimal control. However, that all depends on the way

Qy ¼ diagðI; IÞ;

Qx ¼ diagðX; 0Þ;

u/ ðtÞ ¼ Kg ðtÞxðtÞ:

ð18Þ

ð19Þ

The optimal time-varying feedback gain is given by Kg ¼ R1 BT/ PðtÞ, where P(t) is the solution of the matrix Riccati equation, _ PðtÞ ¼ PðtÞA  AT PðtÞ  Qx þ PðtÞB/ R1 BT/ PðtÞ:

ð20Þ

This control philosophy is called linear quadratic regulator (LQR). That regulator requires the knowledge of all the optimal gain values Kg ðtÞ in the time interval [0, tf]. However, the feedback gains of the LQR usually approach steady-state values far from the final time. The use of the steady-state LQR controller considerably simplifies the controller design and the analog and digital implementation [34]. The steady-state feedback gain matrix is then given by Kg ¼ R1 BT/ P1 ;

ð21Þ

1

where P is the steady-state solution, limt!1 P(t), of the matrix Riccati equation. Therefore, considering a steadystate feedback gain Kg in Eq. (19), the closed-loop state equation is given by _ ¼ ðA  B/ Kg ÞxðtÞ þ Bu uu ðtÞ: xðtÞ

ð22Þ

In the previous equation it was assumed that all the states were completely observable and therefore could be directly

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related to the outputs and used by the control system. However, that is not always the case and a more realistic approach would consider that only the outputs y(t) can be known and measured. In order to be able to use the states information in the control system, it will be necessary to estimate the states from a model of the system and a limited number of observations of the outputs. That estimation is made usually by a state estimator or observer. It is fed by the same known input signals as the mechanical system, u/(t), and has its output ^ yðtÞ constantly compared with the output of the mechanical system, y(t). However, the state variable estimates are very sensitive to any uncorrelated noise in the system, particularly measurement noise from the observed outputs of the mechanical system. Knowing the statistical properties of the various sources of noise in the mechanical system, and assuming white uncorrelated noise uniformly distributed in the bandwidth, an ‘optimal’ state observer which minimizes the effects of plant and measurement noise is known as Kalman–Bucy filter. Considering the open-loop system in Eq. (10) adjoined with plant and measurement noise, w(t) and v(t), yields _ ¼ AxðtÞ þ B/ u/ ðtÞ þ Bu uu ðtÞ þ Bw wðtÞ; xðtÞ yðtÞ ¼ CxðtÞ þ vðtÞ;

ð23Þ

where Bw is a plant noise input matrix. The plant and measurement noise are both assumed to be white, have a Gaussian probability density function and are assumed uncorrelated with the inputs. The correlation properties of the plant and measurement noise vectors are given by the correlation matrices E½Bw wðtÞwT ðtÞBTw  ¼ W; E½vðtÞvT ðtÞ ¼ V;

ð24Þ

where E denotes the expectation operator. Considering a state estimator with the same dynamics as the system under control, which is assumed known, and the same known ^ would thus be governed by inputs, the estimator states x the equation ^_ ðtÞ ¼ A^ xðtÞ; x xðtÞ þ B/ u/ ðtÞ þ Ke ðtÞ½Cx þ vðtÞ  C^

ð25Þ

^ð0Þ ¼ 0 and Ke ðtÞ is known as the Kalman–Bucy where x gain matrix. Defining the error between the true and esti^ðtÞ, the dynamic behavior mated states as eðtÞ ¼ xðtÞ  x of the Kalman–Bucy filter can now be expressed as a coupled set of first order differential equations through the difference of Eqs. (23) and (25), e_ ðtÞ ¼ ½A  Ke ðtÞCeðtÞ þ Bu uu ðtÞ þ Bw wðtÞ  Ke ðtÞvðtÞ:

which is usually solved with the initial condition M(0) = E[x(0)xT(0)]. In a similar way to the steady-state LQR, the ‘optimal’ Kalman–Bucy gain matrix Ke ðtÞ in Eq. (26) typically experiences a transient and then approaches steady-state as time increases from the initial time. In applications where the estimator is designed to operate for time periods that are long compared to the transient times of the Kalman–Bucy gains, it is reasonable to ignore the transient and exclusively use the steady-state gains Ke [34]. The equations for computing the Kalman–Bucy gain have a striking resemblance to the equations for computing the LQR gain. As pointed out by Burl [34] the steady-state Kalman–Bucy filter problem is shown to be mathematically equivalent to the steady-state LQR problem when appropriate substitutions are made. Combining the steady-state Kalman–Bucy filter with the steady state LQR, the inter-related dynamic system will take the form    _ xðtÞ B/ Kg xðtÞ A  B/ Kg ¼ e_ ðtÞ 0 A  Ke C eðtÞ     Bu Bw 0 wðtÞ þ uu ðtÞ þ : ð28Þ Bw Ke vðtÞ Bu Since one we must assume that the random perturbations (force disturbance or measurement noise) are Gaussian, this control philosophy is called linear quadratic Gaussian (LQG) control. The dynamics of the coupled controller and observer system is determined by the eigenvalues of the square system matrix in Eq. (28) and the closed-loop coupled system stability depends of the two sub-systems. The error will asymptotically be stable provided the observer poles have negative real components collocated, in the complex plan, as far as possible from the system poles so that the observer error reduces more rapidly than the system response. 4. Case study In this section the FE model is utilized in the evaluation of the classical and optimal active vibration control of a cantilever aluminum beam with an asymmetric collocated piezoelectric sensor/actuator pair mounted on the surface (Fig. 2). Furthermore, the displacement time history, for an initial displacement field and white noise force disturbance, and point receptance at the free end are evaluated with the open- and closed-loop systems. In this case, the sensing and actuation effects of a collocated sensor/

ð26Þ The Kalman–Bucy gain matrix which leads to the optimal feedback controller is shown in Ref. [34] to be of the form Ke ðtÞ ¼ MðtÞCT V1 , where M(t) is the correlation matrix of the estimation error which is the solution of another Riccati equation, _ MðtÞ ¼ MðtÞAT þ AMðtÞ þ W  MðtÞCT V1 CMðtÞ;

ð27Þ

Fig. 2. Cantilever smart piezoelectric beam.

C.M.A. Vasques, J. Dias Rodrigues / Computers and Structures 84 (2006) 1402–1414

160 elements of equal length and a shear correction factor equal to 5/6 was utilized. All the numerical computing and FE implementation were performed with MATLAB.

Table 1 Material properties of the aluminum and PXE-5 Aluminum E m q

PXE-5 9

70 · 10 Pa 0.3 2710 kg m3

cE11 cE12 cE13 cE33 cE44 cE66

131.1 · 109 Pa 7.984 · 109 Pa 8.439 · 109 Pa 12.31 · 109 Pa 2.564 · 109 Pa 2.564 · 109 Pa

215 · 1012 m V1 500 · 1012 m V1 515 · 1012 m V1 1800 2100 7800 kg m3

d31 d33 d15 eT11 =e0 eT33 =e0 q

4.1. Initial displacement field

Tip displacement [mm]

actuator pair are asymmetric with respect to the mid-plane of the beam. Since they are not perfectly collocated due to the in-plane and out-of-plane coupling of the piezoelectric sensor [36], that might bring some instability problems if the upper limit of the bandwidth of interest is not well below the natural frequency of the first extensional mode. Moreover, for a non-collocated sensor and actuator pair stability is not guaranteed and the control system might even destabilize the whole system. The cantilever beam is 400 mm long, 2 mm thick and 15 mm wide, and the two piezoelectric patches (Philips Components – PLT 30/15/1-PX5-N [38]) are 30 mm long, 1 mm thick and 15 mm wide, and were mounted at 5 mm from the clamped edge. The mechanical and electrical material properties of the aluminum beam and piezoelectric patches, PXE-5, are presented in Table 1 (see the IEEE standard [29] for details about notation). In the analysis, a truncated modal model with the first four flexural modes was considered. The correspondent modal damping ratios were determined experimentally and their values are as follows: 1.71%, 0.72%, 0.42% and 0.41%. Furthermore, the beam with the sensor/actuator pair was discretized into

Control voltage [V]

Consider an initial displacement field applied to the beam which is obtained by a mechanical force applied at the free tip that induces a tip displacement equal to 1.5 mm. The tip displacement time history and control voltage for the CGVF and CAVF control systems are presented in Fig. 3. The control gain G = 0.4 for the CGVF was chosen from a root locus analysis in order to increase the damping of the first mode as much as possible and, simultaneously, by a trial-and-error method in order to use control voltages within the desired range. According to the limit electric field strength of the piezoelectric patches (300 V mm1), the control voltage should be lower than 300 V. Exceeding that voltage may result in depolarization of the material so that the piezoelectric properties become less pronounced or disappear completely. For the CAVF, a constant amplitude A = 250 V was chosen and the control voltage was turned on at t = 0 s and turned off a t = 0.2 s. Furthermore, in the CAVF only the velocity polarity of the first-mode was considered in the feedback loop (perfect band-pass filter tuned to the first mode). As can be seen in Fig. 3, the CGVF control manages to significantly attenuate the free tip displacement with an admissible control voltage. The closed-loop 5% settling time is equal to 0.5 s, which reveals a great improvement on the response attenuation when compared with the openloop one (2.3 s). Moreover, it presents a better performance

1.5

1.5

1

1

0.5

0.5

0

0

–0.5

–0.5

–1

–1

–1.5 10 500

(a) –3

1409

(b)

–1.5 –2

10

–1

10

10

0

–3

10 500

250

250

0

0

–250

–2

10

10

–1

0

10

–250 (c)

–500 –3 10

–2

–1

10 10 Time (Log) [s]

10

0

(d) –500 –3 10

–2

–1

10 10 Time (Log) [s]

Fig. 3. Tip displacement and control voltage for an initial displacement field with CGVF (a, c) and CAVF (b, d):

0

10

, open-loop;

, closed-loop.

C.M.A. Vasques, J. Dias Rodrigues / Computers and Structures 84 (2006) 1402–1414

Tip displacement [mm]

1410

1.5

1.5

1

1

0.5

0.5

0

0

–0.5

–0.5

–1

–1

Control voltage [V]

–1.5 10 500

(a) –3

(b)

–1.5 –2

10

10

–1

0

10

–3

10 500

250

250

0

0

–250

–2

10

–1

10

10

0

–250 (c)

–500 –3 10

–2

–1

10 10 Time (Log) [s]

0

10

(d) –500 –3 10

–2

–1

10 10 Time (Log) [s]

10

0

Fig. 4. Tip displacement and control voltage for an initial displacement field with output y (a, c) and state x (b, d) weighting: closed-loop (LQR); , closed-loop (LQG).

than the CAVF (0.58 s). However, if we look at the 10% settling time, the CAVF control shows a faster attenuation capacity (0.24 s) than the CGVF (0.38 s). In Fig. 4 the LQR and LQG controllers with output weighting (sensor voltage and its derivative) and state weighting (modal amplitudes and modal velocities) are utilized. It is assumed in the state estimation that only the sensor voltage, and not its derivative, is measured. That assumption corresponds to a more realistic approach and avoids the necessity of differentiating the sensor signal. For the output weighting case, the weighting matrices Qy = diag(1, 1) and R = 15 were utilized, and for the state weighting case, the matrix Qx ¼ 1  1010 diagðx21 ; 0; . . . ; 0Þ was defined in order to mainly damp the first mode, and R = 30 was chosen. For the LQG controller design, a white noise mechanical disturbance applied in the free end of the beam is modeled as plant noise. Therefore, in order to define the correlation matrix in Eq. (24), the equality Bw = Bu should be considered. A plant noise vector (force disturbance) with E[w(t)wT(t)] = 4 · 104 N2 and a sensor white noise disturbance with E[v(t)vT(t)] = 1 V2 are considered for the definition of the noise correlation matrices and Kalman–Bucy gain design. However, the effects of the stochastic mechanical disturbance in the displacement time history will be considered only in the following sections. The results in Fig. 4(a) and (b) show that the state estimator of the LQG controller with state weighting (only the first mode) has a better performance than the one with output weighting. Furthermore, in Fig. 4(a) the displacements with the LQR and LQG have a significative difference in attenuation, with 5% settling times equal to 0.49 s and

, open-loop;

,

0.89 s, and the control voltage of the LQG is too high with the risk of depolarizing the actuator. Moreover, the LQR and LQG with state weighting in Fig. 4(b) have the same 5% settling time (0.34 s), with the LQG requiring a lower control voltage and the LQR requiring an inadmissible control voltage. The LQG with state weighting, which corresponds to the realistic approach, when compared with the velocity feedback results presents a better performance. 4.2. White noise disturbance As referred in the previous section, a white noise point force disturbance with variance equal to 4 · 104 N2 is applied in the free end of the beam. The gains, amplitude and control parameters for all the control strategies are the same that were utilized in the previous section and only the output and state weighting matrices are changed to R = 1 and Qx ¼ 1  1010 diagðx21 ; x22 ; x23 ; x24 ; 0; . . . ; 0Þ. This choice for the state-weighting matrix is made in order to distribute the vibration control effort in bandwidth. However, the analysis is limited to the modal model bandwidth (only the first four flexural modes are considered). The open- and closed-loop tip displacements and control voltages for the white noise force disturbance evaluated with the CGVF and CAVF are presented in Fig. 5. In Fig. 6 the displacement and control voltages are evaluated with the LQR and LQG with output and state weighting. It can be seen that all the control systems manage to reduce the tip displacement significantly. When compared to the open-loop response standard deviation, which is equal to 0.57 mm, the closed-loop response standard

Tip displacement [mm]

C.M.A. Vasques, J. Dias Rodrigues / Computers and Structures 84 (2006) 1402–1414

1.5

1.5

1

1

0.5

0.5

0

0

–0.5

–0.5

–1

–1

–1.5

(a)

Control voltage [V]

0

–1.5 0.5

1

1.5

2

500

250

250

0

0

–250

(b)

0

500

1411

0.5

1

1.5

2

0.5

1 Time [s]

1.5

2

–250 (c)

–500 0

(d) 0.5

1 Time [s]

1.5

–500 2 0

Tip displacement [mm]

Fig. 5. Tip displacement and control voltage for white noise point force disturbance with CGVF (a, c) and CAVF (b, d): closed-loop.

1.5

1.5

1

1

0.5

0.5

0

0

–0.5

–0.5

–1

–1

–1.5

(a)

Control voltage [V]

0

–1.5 0.5

1

1.5

2

500

250

250

0

0

–250 0

0.5

1

1.5

2

0.5

1 Time [s]

1.5

2

–250 (c)

–500

,

(b)

0

500

, open-loop;

(d) 0.5

1 Time [s]

1.5

2

–500

0

Fig. 6. Tip displacement and control voltage for white noise point force disturbance with output y (a, c) and state x (b, d) weighting: , closed-loop (LQR); , closed-loop (LQG).

deviations (with the respective control voltage standard deviations) for the CGVF and CAVF, 0.26 mm (79.6 V) and 0.20 mm (250 V), demonstrate the vibration control efficiency. Moreover, the LQR and LQG with output weighting present responses with standard deviations equal to 0.26 mm (97.4 V) and 0.28 mm (118 V), and the LQR

, open-loop;

and LQG with state weighting have their values equal to 0.16 mm (80 V) and 0.20 mm (72 V). The most effective control system is the LQR with state weighting. However, a state estimator is necessary, and the results achieved with the LQG (0.20 mm) are the realistic ones. Furthermore, they are equal to the ones obtained with the CAVF.

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However, if we look at the control voltages utilized, the most interesting one is the LQG with state weighting, with only 72 V, and the most inefficient, with the higher control voltage (250 V), is the CAVF. 4.3. Frequency response function

Receptance [dB] (ref. 1m/N)

In order to analyze the capacity of the various control strategies in the frequency domain, the point receptances at the free end of the beam evaluated with the open- and closed-loop control systems are presented. With that purpose, the displacement time histories and white noise mechanical disturbance of the previous section are utilized to estimate the frequency response functions by a signal processing analysis procedure through a fast Fourier trans-

form (FFT) algorithm. The open- and closed-loop point receptances for the CGVF and CAVF control are presented in Fig. 7. The CAVF control significantly reduces the magnitude of the first resonance in approximately 27 dB, and increases the magnitudes of the other modes. In comparison, the CGVF does not manage to have such a good performance in the attenuation of the first mode, with a 8.6 dB reduction, but in compensation attenuation in all modes is achieved, the fourth mode is completely eliminated and the natural frequencies are shifted. Furthermore, a severe spillover effect is visible in Fig. 7(b) above the first natural frequency. That is related with the fact that for the CAVF algorithm a band pass filter was considered and only the polarity of the sensing voltage related with the first mode contribution was used in the closed-loop

0

0

–20

–20

–40

–40

–60

–60

–80

–80

–100

–100

(a)

–120 0

100 200 300 Frequency [Hz]

400

(b)

–120 0

100 200 300 Frequency [Hz]

Receptance [dB] (ref. 1m/N)

Fig. 7. Point receptance with CGVF (a) and CAVF (b):

0

0

–20

–20

–40

–40

–60

–60

–80

–80

–100

–100

(a)

–120

0

, open-loop;

100 200 300 Frequency [Hz]

400

0

Receptance [dB] (ref. 1m/N)

0

0

–20

–20

–40

–40

–60

–60

–80

–80

–120

0

–100

(a) 100 200 300 Frequency [Hz]

400

, closed-loop.

(b)

–120

100 200 300 Frequency [Hz]

Fig. 8. Point receptance with the LQR (a) and LQG (b) controller with output y weighting:

–100

400

–120

0

400

, open-loop;

, closed-loop.

(b) 100 200 300 Frequency [Hz]

Fig. 9. Point receptance with the LQR (a) and LQG (b) controller with state x weighting:

, open-loop;

400 , closed-loop.

C.M.A. Vasques, J. Dias Rodrigues / Computers and Structures 84 (2006) 1402–1414

response to the white noise force disturbance. Therefore, since the control voltage is non-linear and discontinuous and a filter is being used for the sensor signal (the controller does not consider the dynamics of the higher modes), the effects of the control voltage originate a control spillover problem, which leads to instabilities outside the bandwidth under control (first mode). Since a truncated modal model has been used, which can for instance be implemented by an adequate filter design, the problem of observation spillover might be prevented and is not at issue here. In Figs. 8 and 9 the point receptance for output y and state x weighting with the LQR and LQG controller are presented. The results show that the effects of the state estimation do not compromise the stability of the system, and that the output weighting LQG controller can have even a better performance than the LQR. A 8.8 dB and 9.4 dB attenuation of the first mode is achieved with the LQR and LQG controller with output weighting, respectively, and the resonant frequencies are shifted. If one now looks at the results with state weighting (Fig. 9), one can see that only the peak amplitudes are affected by the state estimator, and the natural frequencies remain the same. Furthermore, all the modes are damped, and a 12.2 dB and 10.7 dB reductions of the first mode are achieved with the LQR and LQG controller with state weighting, respectively.

5. Conclusion In this paper an analysis and comparison of the classical control strategies, constant amplitude and constant gain velocity feedback (CAVF and CGVF), and optimal control strategies, linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) controller, concerning the active vibration control of smart piezoelectric beams, was presented. The control models assumed that one of the piezoelectric layers acts as a distributed sensor and the other one as a distributed actuator, and the sensor signal was used as a feedback reference in the closed-loop control systems. A case study concerning the active vibration control of a cantilever aluminum beam with an asymmetric pair of collocated piezoelectric patches mounted on the surface was analyzed. The transverse displacement time history, for an initial displacement field and white noise force disturbance, and driving point receptance, both measured at the free end of the beam, allowed to compare the performances of the classical and optimal strategies in the time and frequency domains. It was shown that for an initial displacement field the CAVF and LQG with the first mode weighted are the most interesting solutions. However, for a white noise disturbance, the CAVF only manages to reduce the mode under control and the others are destabilized. Furthermore, the LQG with all the modes weighted presents a better control in bandwidth with a lower control voltage. Moreover, the output weighting LQG can also be an interesting strategy in situations where attenuation of the outputs is of interest and a resemblance in

1413

performance with the CGVF was shown in the present analysis. One of the major advantages of the classical techniques is that they can avoid the necessity of digital control, reducing the time delays in the control system, and that stability is guaranteed provided the actuator/sensor pair is perfectly collocated. However, the CGVF and CAVF strategies require the differentiation of the sensor voltage, which for noisy measurements can become troublesome. If one looks for adaptability in design, the optimal techniques, with the disadvantage of requiring a model of the system, can be more interesting. The quantification of the control system’s performance by a quadratic cost function provides the designer with lots of flexibility to perform trade-offs among various performance criteria. The relationship between cost function weights and performance criteria hold even for high-order and multiple input–output systems, where classical control becomes cumbersome. A major limitation of the LQR is that all the states must be measured when generating the control. The LQG controller overcomes that by estimating the states using a Kalman–Bucy filter, which utilizes noisy partial output information, and some modifications in the LQR performance, which sometimes can improve the efficiency, often occur. However, observation and control spillover problems related with the reduced (truncated) or unobserved model dynamics may compromise stability. References [1] Chopra I. Review of state of art of smart structures and integrated systems. AIAA J 2002;40(11):2145–87. [2] Tzou HS, Anderson GL, editorsIntelligent structural systems. Solid mechanics and its applications, vol. 13. Dordrecht: Kluwer Academic Publishers; 1992. [3] Srinivasan AV, McFarland DM. Smart structures: analysis and design. Cambridge: University Press; 2001. [4] Cady WG. Piezoelectricity: an introduction to the theory and applications of electromechanical phenomena in crystals. New York: Dover Publications; 1964. [5] Niezrecki C, Brei D, Balakrishnan S, Moskalik A. Piezoelectric actuation: state of the art. Shock Vib Dig 2001;33(4):269–80. [6] Benjeddou A. Advances in piezoelectric finite element modelling of adaptive structural elements: a survey. Comput Struct 2000;76(1–3): 347–63. [7] Vasques CMA, Rodrigues JD. Coupled three-layered analysis of smart piezoelectric beams with different electric boundary conditions. Int J Numer Meth Eng 2005;62(11):1488–518. [8] Inman DJ. Vibration: with control, measurement and stability. Englewood Cliffs, London: Prentice Hall; 1989. [9] Meirovitch L. Dynamics and control of structures. New York: John Wiley & Sons; 1990. [10] Tzou HS. Piezoelectric shells: distributed sensing and control of continua. Dordrecht: Kluwer Academic Publishers; 1993. [11] Fuller CR, Elliott SJ, Nelson PA. Active control of vibration. London: Academic Press; 1996. [12] Clark RL, Saunders WR, Gibbs GP. Adaptive structures: dynamics and control. New York: John Wiley & Sons; 1998. [13] Preumont A. Vibration control of active structures: an introduction. 2nd ed. Dordrecht: Kluwer Academic Publishers; 2002. [14] Burke SE, Hubbard JE. Active vibration control of a simplysupported beam using a spatially distributed actuator. IEEE Control Syst Mag 1987;7(4):25–30.

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