Robust Control Design of Smart Beams

European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004 P. Neittaanmäki, T. Rossi, S. Korotov, E. Oñate, J. Périaux, and D. Knörzer (eds.) Jyväskylä, 24—28 July 2004 ROBUST CONTROL DESIGN OF SMART BEAMS G.E. Stavroulakis*, D. Marinova† , E. Hadjigeorgiou$, G. Foutsitzi$, and C.C. Baniotopoulos+ * † Department of Mathematics, University of Ioannina, GR-45110 Ioannina, Greece and Institute of Applied Mechanics, Technical University of Braunschweig, Germany e-mail: gestavr@cc.uoi.gr , web page: www.math.uoi.gr\~gestavr Department of Applied Mathematics and Informatics, Technical University of Sofia, 8 Kl. Ohridski Str., 1756 Sofia, Bulgaria e-mails: dmarinova@dir.bg $ Department of Material Science and Engineering, University of Ioannina, GR-45110 Ioannina, Greece e-mails: ehadjig@cc.uoi.gr, gfoutsi@cc.uoi.gr + Institute of Metallic Structures, Department of Civil Engineering, Aristotle University, GR-54006 Thessaloniki, Greece e-mails: ccb@civil.auth.gr Key words: Smart composite beams; robust control; structural control;. Abstract. The influence of structural uncertainties on actively controlled smart beams is investigated in this paper. The dynamical problem of the composite beam is based on a simplified modeling of the actuators and sensors, both being realized by means of piezoelectric layers. In particular, a practical robust controller design methodology is developed, which is based on recent theoretical results on H2 and H∞ control theory. Numerical examples demonstrate the vibration-suppression property of the proposed smart beams. 1 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. 1 INTRODUCTION The use of active control techniques in smart structures is an active research area. Vibration control of beams may serve as a model problem, since the beam is a fundamental structural element [1-3]. A number of different control schemes have been proposed, where the main class of controllers is linear feedback laws. There are always differences between the physical plant that is controlled and the model on which the controller design is based (for instance, neglected higher frequency dynamics). Therefore, robustness must be an important goal for any applicable feedback controller design [4-6]. The performance specifications, which the control system must fulfill and the class of uncertainties for which the control system must be robust against, determine the robust controller design methodology for any particular vibration control problem. In this study a vibration control problem in flexible structure (smart beam) is considered and the performance specification is stated in terms of a disturbance attenuation requirement for particular class of external disturbances acting on the structure. The paper illustrates H2 and H∞ robust controller design techniques by considering the problem of active vibration control in a flexible cantilever beam using piezoelectric patches as sensors and actuators. This work demonstrates that the proposed robust control design schemes are suited to vibration broadband disturbances, which can be modelled as Gaussian white noise (e.g., in earthquake modeling), as sinusoidal wind-like pressure, as well as structured uncertainties. The considered robust control design methodologies lead to linear time invariant feedback controllers. The controllers are designed to achieve optimal performance for a nominal model and maintain robust stability and robust performance for a given class of uncertainties. This is achieved by the solution of two algebraic Ricatti equations, while in classical structural control one such equation arises. In this paper, the governing equations of a beam with bonded piezoelectric sensors and actuators are formulated. After the finite element discretization, H2 and H∞ robust control of the beam vibration is investigated. Numerical results obtained by using MATLAB routines demonstrate that these two robust control laws can effectively suppress the vibration of lower modes of the beam as well as avoid spillover from the higher frequency modes. 2 MODELING AND PROBLEM SETUP In the smart beam of Figure 1, the control actuators and the sensors are piezoelectric patches symmetrically bonded to the top and the bottom surfaces of the host beam. The top (sensor) and the bottom (actuator) piezoelectric layers are positioned with identical poling directions for effective sensing and strong actuation. The beam has a rectangular cross section, length L, width b and thickness h. The sensor and the actuator have widths bS and bA and thicknesses hS and hA, respectively. The modeling of such a beam can be found in many papers on vibration control [3,4,7]. The electromechanical parameters of our example are shown in Table 1 (they are comparable to the ones of reference [8]). Thermal effects are neglected. 2 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. Figure 1. The smart beam and schematic control system Beam length, L Beam width, b Beam thickness, h Beam density, ρ Beam Young’s modulus, E Charge constant, d31 Voltage constant, G31 Coupling coefficient, k31 Piezoelectric Young’s modulus, ES=EA Piezoelectric width, bS=bA Piezoelectric thickness, hS=hA 1.023 m 0.04995 m 0.00285 m 2712.6 kg/m 6.94* 1010 N/m2 -210*10-12 m/V -11.5*10-3 Vm/N -0.34 6.9*1010 N/m2 0.0258 m 0.001 m Table 1: Parameters of the composite smart beam. 2.1 Model design The linear theory of piezoelectricity is employed due to small structural vibrations. This theory assumes quasi-static motion indicating that the mechanical and electrical forces are balanced at any given instant. The linear constitutive equations read {σ } = [Q ] ({ε } − [ d ] {E}) {D} = [ d ][Q ]{ε } + [ξ ]{E} T (2.1) (2.2) where {σ}6x1 is the stress vector, {ε}6x1 is the strain vector, {D}3x1 is the electric displacement, {E}3x1 is the strength of applied electric field acting on the surface of the piezoelectric layer, [Q]6x6 is the elastic stiffness matrix, [d]3x6 is the piezoelectric matrix and [ξ]3x3 is the permittivity matrix. Eq. (2.1) describes the inverse piezoelectric effect (actuator). Eq. (2.2) describes the direct piezoelectric effect (sensor). Basic hypothesis of modeling are as follows: (1) Sensor and actuator (S/A) layers are thin compared with the beam thickness. (2) The polarization direction of the S/A is the thickness direction (z axis). (3) The electric field loading of the S/A is uniform uni-axial in the x-direction. (4) Piezoelectric material is homogeneous, transverse isotropic and elastic. Therefore, the set of equations (2.1) and (2.2) is reduced as follows 3 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. σ x  Q11 0    ε x   d 31    =     −   Ez  τ xz   0 Q55   γ xz   0   (2.3) Dz = Q11d 31ε x + ξ 33 E z (2.4) The electric field intensity E z can be expressed as V Ez = (2.5) hA where V is the applied voltage across the thickness direction of the actuator and hA is the thickness of the actuator layer. Since, only strains produced by the host beam act on the sensor layer and no electric field is applied to it the output charge from the sensor can be calculated using eq. (2.4). The charge measured through the electrodes of the sensor is given by      1   (2.6) q ( t ) =  ∫ Dz dS  +  ∫ Dz dS    S  2  S z= h 2   z = h 2 + h   where S ef is the effective surface of the electrode placed on the sensor layer. In the numerical ef ef S examples of this paper the whole length of the sensor layer is covered by the electrode. The current on the surface of the sensor is given by i (t ) = dq ( t ) dt . (2.7) The current is converted into open-circuit sensor voltage output by V = GS i ( t ) S (2.8) where GS is the gain of the current amplifier. Furthermore, we suppose that (5) bending-torsion coupling and the axial vibration of the beam centerline are negligible and (6) the components of the displacement field {u} of the beam are based on the Timoshenko beam theory which, in turn, means that the axial displacement is proportional to z and to the rotation ψ(x,t) of the beam cross section about the positive y-axis and that the transverse displacement is equals to the transverse displacement w(x,t) of the point of the centroidal axis (y=z=0). The strain-displacement relationships read ∂ψ ∂w εx = z , ε xz =ψ + . (2.9) ∂x ∂x The kinetic energy of the beam with the layers can be expressed as L h +h s [ ] T b 2 1 2 T = ∫ ρ{u} {u}dV = ∫ ∫ ρ ( zψ ) + w 2 dzdx 2V 2 0 − h −h 2 (2.10) A Here is assumed that the densities of the host beam and piezoelectric patches coincide. The strain (potential) energy is given by 4 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. h +h L 2 S 1 b U = ∫ {ε }T {σ }dV = ∫ 2V 20 ∫ h − − hA 2 2   ∂ψ  2 ∂w     dzdx  + Q55 ψ + Q11  z ∂x      ∂x  (2.11) If the only loading consists of moments induced by piezoelectric actuators and since the structure has no bending-twisting couple then the first variation of the work has the form L  ∂ψ δ W = b ∫ M Aδ   ∂x 0  dx  (2.12) where δ is the first variation operator, MA is the moment per unit length induced by the actuator layer and is given by −h M = A ∫ 2 −h zσ dz = A x − h − hA 2 ∫ 2 zQ11d31 EzA dz ( E zA = − h − hA 2 VA hA ) (2.13) Using Hamilton ’s principle the equations of motion of the beam are derived. 2.2 Finite element discretization Beam finite elements are used, with two degrees of freedom at each node: the transversal deflection wi and the rotation ψi.(see Figure 2). They are gathered to form a vector X i = [ wi ψ i ] . Finally, from Hamilton’s principle we obtain the discretized equations of motion for each finite element [M i ]Xi + [K i ]X i = Fe (2.14) [ ][ ] where M i , K i are the element mass and stiffness matrices and F e represents the electrical force vector provided by applied (actuator) voltages and is proportional to them. Figure 2. The i-th beam element After assembling the mass and stiffness matrices for all elements, we obtain the equation of motion in the form (2.15) Μ X + ΛX + ΚX = Fm + Fe where M and K are the generalized mass and stiffness matrices, Fe is the generalized control force vector produced by electromechanical coupling effects, Λ is the viscous damping matrix and Fm is the external loading vector. The computer implementation in MATLAB follows the lines of [10]. 5 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. The main objective is to design robust control laws for the smart beam bonded with piezoelectric S/A subjected to external induced vibrations. For this purpose the following state space representation of the dynamical system will be used x = Ax + B1w + B2u (2.16) as it is common in control problems for general dynamical systems. Here I   0 x = [ X X ]T A= (2.17) −1 −1  − M Κ − M Λ  x is the state vector, A is the state matrix, B1 and B2 are allocation matrices for the disturbances w (corresponding to external forces Fm) and control u (corresponding to Fe). The initial conditions are assumed to be zero. The identity matrix is denoted by I. 3 DEVELOPMENT OF THE ROBUST CONTROLLERS 3.1 Preliminaries We will restrict our attention to a regulator type problem where the system is assumed to be at equilibrium, and tries to maintain it in the presence of internal or external disturbances, by means of a suitable feedback control law. The mathematical model (2.16) provides a linear mapping from the inputs to the responses. The signals sent to the control actuators are functions of the response of the system measured by appropriate sensors. Let us assume the measurement vector y = C2 x + D21w + D22u (3.1) Let us denote by z the second group of outputs that we are interesting in controlling. z = C1 x + D11w + D12u (3.2) The equations (2.16),(3.1),(3.2) are the plant representation in state space form of the smart beam structure. For proper statement of the robust control problem we suppose that the matrix D11 is equal to zero [11]. It is clear that any mathematical representation of a system involves simplifying assumptions. For example, dynamic structures have complicated high frequency dynamics that are often ignored at the design stage or in a finite element model. The particular property that a control system must possess in order to operate properly in realistic situation is the robustness. The underlying concept within control theory in this direction is feedback, which must have the following properties: sensitivity and disturbance rejection. Two popular methods, H2 and H∞ optimal control strategies, can be applied for these purposes. They use as performance measures H2 and H∞ norms, defined in the frequency domain for a stable transfer matrix. We consider the steady state case. In this case the optimization horizon is considered to take a limit value equal to infinity and the control law is the following linear time invariant function u = Ky (3.3) where K is the controller gain, which must be determined. Let denote by G the transfer matrix of the plant (2.16),(3.1),(3.2) (3.4) 6 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. Hence, S(s) = (I+GK)-1 is the sensitivity transfer matrix and T(s) = (I+GK)-1GK is the complementary sensitivity transfer matrix. For good tracking and disturbance rejection the sensitivity S(s) must be small while for proper suppression of noise the complementary sensitivity T(s) must be small. In order to keep the controller within specified limits and prevent saturation problems, we need to limit the quantity |KS(s)|. 3.2 H2 optimal control law H2 optimal control is considered under the assumption that the exogenous signals are fixed or have fixed power spectrum. Figure 3 shows the corresponding two-port block diagram. Figure 3. Closed-loop system diagram for robust H2 control. Both G and K are supposed to be real, rational and proper. We search the controller in the linear feedback form (3.3) that will keep the regulated outputs z as small as possible despite the exogenous inputs w. The generalized plant G contains the plant plus all weight functions. The signal w contains all external inputs including disturbances, sensor noise and commands; the output z is an error signal; y is the measured variables; and u is the control input. The resulting, closed-loop transfer function from w to z is denoted by Tzw. We get as a performance criterion the minimization of the H2 norm of Tzw , i.e. 12  1 +∞  Tzw 2 =  ∫ trace[Tzw ( jω ) * Tzw ( jω )]dω  (3.5) 2  −∞  over all internally stabilizing controllers K. One way to explain the mechanical meaning of this criterion is the following. Let m denotes the dimension of w and denote by ei, i = 1,…,m, the standard basis in Rm. An impulse at the i-th component of the exogenous signal is achieved by setting wi(t) = δ(t)ei, the resulting output being zi(t) = Tzwδ(t)ei. Then it is easy to derive that 12 12   2 2 Tzw 2 =  ∑ zi  =  ∑ Tzwδei  (3.6) 2 2  i  i the right hand norm being the usual one on £2(-∞,+∞). The H2 norm of Tzw minimizes the worst-case root mean square value of the regulated variables when the disturbances are unit intensity white processes. This circumstance allows for a state-space solution to the frequency domain optimization problem. Under assumptions that can be found in the literature, it can be 7 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. shown that there exists a unique controller K2 which minimizes Tzw with the following transfer matrix representation [10] (3.7) where X and Y are the solutions of the two ARE AT X + XA − XB2 B2T X + C1T C1 = 0 AY + YAT − YC2T C 2T + B1 B1T = 0 (3.8) for a stable matrix A. The controller K2 has a separation structure. It implies reducing the output feedback problem to a combination of the full information and the output estimation problems. From the measurements the whole system is first reconstructed in an optimal way using Kalman – Bucy filter during the estimation phase, and then the optimal control problem is based on this reconstructed state vector. The H2 control design technique provides robustness and allows for the control objectives to be conveniently defined in time domain. Another theory of robust optimal control, based on the H∞ norm is discussed and used in the next section. The optimal state feedback in the full information problem is − B2T X and the result in this case is as follows min Tzw 2 ( = trace( B1T XB1 ) ) 12 Quantity − YC2T is the optimal output injection in the full control case and the result reads min Tzw 2 ( = trace(C1YC1T ) ) 12 Therefore, the controller K2(s) provides 2 min Tzw 2 = trace( B1T XB1 ) + trace(C1YC1T ) (3.9) 3.3 Uncertainty modeling Uncertainty denotes the difference between the model and the reality. By adopting the mechanical model described previously, we consider uncertainties in the parameters of the model. The H∞ approach begins with an uncertain system model for the plant to be controlled. In this section we will consider an uncertain system model whose primary purpose is to account for the uncertainty introduced by varying the nominal plant parameters. Disk-shaped regions on the real axis approximate the variations in the structure system. This uncertainty will be represented by multiplicative uncertainty as shown in Figure 4. 8 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. Figure 4. Uncertain system representation. Let us suppose that the three actual physical parameters M, Λ , and K in the eq. (2.15) are not known exactly, but are believed to lie within known intervals. In particular, the actual mass M is within pM percentages of the nominal mass Μ , the actual damping value Λ is within pΛ percentages of the nominal value Λ , and the spring stiffness K is within pK percentages of its nominal value of Κ . Now by introducing real perturbations ∆Μ = δΜ I ∆Κ = δΚ I ∆Λ = δ Λ I (3.10) which are assumed to be unknown but size-restricted (3.11) (−1 ≤ δ Μ , δ Λ , δ Κ ≤ 1) we can write the actual physical parameters of the system in the following form (3.12) Λ = Λ ( I + pΛ ∆ Λ ) Κ = Κ ( I + pΚ ∆ Κ ) Μ = Μ ( I + pΜ ∆ Μ ) -1 The uncertainty in the matrices M , Λ and K can be represented by the matrix functions linear fractional transformations (LFT) as upper LTF in the perturbations ∆M, ∆Λ and ∆K [10]  − p I Μ −1     0  0  Κ Λ , ∆ Λ  Κ = FU   , ∆ Κ  (3.13) , ∆ Μ  Λ = FU   Μ −1 = FU   Μ    1 −  −p I Μ     pΚ I Κ      pΛ I Λ   Μ  Thus, the considered control design problem will be formulated in a LFT framework The LFT in (3.13) have a nominal mapping (the first members in the first rows in the matrices) that are perturbed by ∆M, ∆Λ, ∆K while the other members of the matrices describe how the perturbations affect the nominal maps. This way the system can be rearranged as a standard one via “pulling out the ∆’s”. For this purpose, we first isolate the uncertainty parameters and denote the inputs of ∆M, ∆Λ, ∆K as yM, yΛ, yK and their outputs as uM, uΛ, uK. The outputs u∆ = [uM, uΛ, uK] from the perturbations are added to the system’s inputs and the inputs y∆ = [yM, yΛ, yK] to the perturbations are added to the system’s outputs (see Figure 4). The model for the uncertain system is obtained in the following matrix form  x  x  y  = G u  u ∆ = ∆. y∆ (3.14)  ∆  ∆  y   u  9 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. A G =  C1 C2 B1 D11 D21  − pΜ I  D11 =  0  0  B2  D12  D22   0 B1 =   − pΜ I − pΛ Μ −1 0 0 0 − pΛ Μ − pΚ Μ −1   0   0   − pΚ Μ  0 −1 −1  Μ −1 H    D12 =  0   0    Μ −1Κ  C1 =  0  Κ  − Μ −1 Λ   Λ  0  D21 = [0 0 0] (3.15) and represents LFT of the natural uncertainty parameters δM, δΛ, δK. The matrix H is a distribution matrix defining the locations of the control forces. The matrix G in eq. (3.14) is known from the nominal parameters of the system. The system model uncertainty matrix in eq. (3.14), denoted by ∆, is a structured matrix. (3.16) ∆ = diag [∆ M ∆ Λ ∆ Κ ] It is H∞ norm bounded, ∆ ∞ ≤ 1 , has a block diagonal structure and influences on the input/output connection between the control u and the output y in a way that can be represented as a feedback by the upper LFT y = FU (G, ∆ )u (3.17) 3.4 Robustness objectives Further we consider the perturbed system (3.14). The performance criterion is to keep the errors as small as possible in some sense for all perturbed models. The performance specifications will be specified in some requirements on the closed loop frequency response of the transfer matrix between the disturbances and the errors, within H∞ design framework. The robust stability and robust performance criteria can be treated in a unified framework using LFT and the structured singular value (SSV) µ∆. We shall consider the real parametric uncertainty with norm-bounded dynamical uncertainty. For the robust stability analysis the controller K can be viewed as a known system component and absorbed into an interconnection structure P together with the plant Gn marked by a dashed line in Figure 4. According to the Nyquist criterion, if the matrices P and ∆ are stable then the interconnection system is stable if and only if det( I − P∆ ) ≠ 0 [11]. For the robust stability we are interested in finding the smallest perturbation ∆, real and norm bounded ||∆||∞ < 1 (that is ensured by means of eq. (3.10)) in the sense of maximal singular value σ (∆) , such that destabilizes the closed loop framework i.e. det( I − P∆) = 0 (3.18) The matrix function SSV is defined as 1 µ ∆ ( P) = (3.19) min{σ ( ∆ ) : ∆ ∈ D, det( I − P∆ ) = 0} SSV µ∆ is bounded by the spectral radius ρ(P) of the matrix P as lower bound and by is the maximal singular value σ (P) of the matrix P as follows 10 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. ρ ( P) ≤ µ ∆ ( P) ≤ σ ( P) (3.20) The interconnection system is well-posed and internally stable for all norm bounded perturbations ∆ if and only if sup µ ∆ ( P( jω )) < 1 (3.21) ω∈R Hence, the peak value on the µ∆ plot of the frequency response determines the size of the perturbations for which the loop is robustly stable against. The quantity 1 (3.22) max µ ∆ [ P ( jω )] ω is a stock of stability with respect to the structured uncertainty influenced P. The robust stability is not the unique feature required for the system with parameter perturbations. Often, exogenous influences acting on the system lead to errors in tracking and regulating. Therefore, we need to test the robust performance of the system. The nominal performance of a system is characterized using H∞ norm of some transfer matrix, here we take the weighted sensitivity transfer matrix of the closed loop. We assume that for good performance the following relationship is satisfied W p ( I + GK ) −1 < 1 (3.23) ∞ The weighting matrix Wp is taken such that to suppress the effect of the disturbance influence on the output. It is utilized to reflect the relative importance of frequency range of interest and takes into account the trade-off between control system robustness and control system performance via an iterative process. Let us assume for Wp the diagonal form (3.24) W p ( s) = wp ( s) I Thus, the assumption (3.23) implies that the maximal singular value of the sensitivity transfer matrix must satisfy the following inequality [ ] σ ( I + GK ) −1 ( jω ) < 1 w p ( jω ) (3.25) The exact values of the parameters in the weighing function wp were chosen iteratively as follows: At any frequency, adjusting wp tends to improve the closed loop performance and decrease the closed loop robustness at that frequency. The order of the weighing function wp will determine how steeply this filter can roll off at low frequencies. However, if the order of Wp is high, this will lead to a high order controller. The final factor wp of the performance weight matrix in the equation (3.24) is chosen as s 2 + 2s + 10 (3.26) w p (s) = 2 s + 70s + 0.01 11 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. 3.5 H∞ optimal control law To obtain a best possible performance in the face of the uncertainties a robust H∞ optimal control is considered. The implementation of H∞ control theory is motivated by the inability of the H2 theory to directly accommodate plant uncertainties. Let us present the considered uncertain system (3.13) by the diagram in Figure 5. Figure 5. General framework for H∞ control problem. where the exogenous input w = [u ∆ d ] includes all signals coming to the system and the T error z = [ y∆ e] includes all signals characterizing the system response. Therefore, the system (4.5) can be represented by the equation  w z (3.27)  y = P u      The aim of this section is to design an admissible controller K∞ satisfying eq. (3.3), which stabilizes internally the system (3.26) and the H∞ norm of the closed loop transfer matrix from w to z is minimized. The closed loop transfer matrix of the system (3.26) from w to z is given as lower LFT in K z = FL ( P, K ) w (3.28) Then the optimal H∞ control design problem can be formulated by the equation FL ( P, K ) ∞ = max σ ( FL ( P, K )( jω )) → min (3.29) T ω The transfer FL ( P, K ) matrix contains measures of nominal performance and stability robustness. Its H∞ norm gives a measure of the worst case response of the system over an entire class of input disturbances. The optimal H∞ controller as just defined is not unique for our beam MIMO system (in contrast with the standard H2 theory, in which the optimal controller is unique). Knowing the optimal H∞ norm is useful theoretically since it sets a limit on what we can achieve. In practice it is often not necessary to design an optimal controller. We consider below the suboptimal H∞ control problem. For given γ > 0 , find an admissible controller Ks(s) such that the H∞ norm of the closed loop transfer matrix of the system (3.26) from w to z is less than γ. FL ( P, K s ) ∞ < γ (4.30) Under some assumptions for the plant G the following three conditions are necessary and sufficient for the existence of an admissible controller satisfying the equation (3.30) [12]. 1. X ∞ ≥ 0 is a stabilizing solution to the algebraic Riccati equation 12 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. AT X ∞ + X ∞ A + C1T C1 + X ∞ (γ −2 B1 B1T − B2 B2T ) X ∞ = 0 2. Y∞ ≥ 0 is a solution of the algebraic Riccati equation AY∞ + Y∞ AT + B1 B1T + Y∞ (γ −2C1T C1 − C2T C2 )Y∞ = 0 3. ρ ( X ∞Y∞ ) < γ 2 . One such controller is (3.31) where A∞ = A + γ −2 B1 B1T X ∞ + B2 F∞ + Z ∞ L∞ C2 4 F∞ = − B2T X ∞ L∞ = −Y∞ C 2T Z ∞ = ( I − γ −2Y∞ X ∞ ) −1 NUMERICAL SIMULATIONS In this section a cantilever beam with four finite element nodes is considered. Four pairs of piezoelectric patches of every quarter are bonded symmetrically at the top and the bottom surfaces of each beam element. The effectiveness of these types of active beam for vibration suppression is investigated below for Euler-Bernoulli model using robust H2 and H∞ control. The material constants are shown in Table 1. Both the top and bottom layers have the same thickness. The disturbances influence the displacements. The allowable voltage of the piezoelectric actuators used for the beam is from -500V to +500V. In the control design process the balance between the vibration control level and control input is considered in order the piezoelectric actuators to endure the limited input voltage. Four kinds of dynamic loading are used as disturbances. The first one is an instantaneous impulsive constant force consisting in 7N distributed in the free end and of the beam and acting in vertical direction. The second and third loadings are periodic sinusoidal vertical loadings acting on every node. The fourth dynamic loading is random white noise with zero mean and unit variance acting along the vertical direction on each node. All cases illustrate asymptotic stability of both control strategies. The figures 6, 8, 10 and 12 show the transient responses of every node of the uncontrolled and H2 controlled beam due the different kinds of loading. Fidures 7, 9, 11 and 13 exhibit the responses of the free end of the beam. Figures 14-17 give the coparision of the responses of the uncontroled and H∞ controlled beam. Figure 6. Response of the four nodes for the free (dot) and H2 (solid) controlled beam tip due white noise. 13 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. Figure 7. Displacement and rotation of the free end for the uncontrolled (dot) and H2 (solid) controlled beam tip due to white noise Figure 8. Displacements and rotations of the four nodes for the free (dot) and H2 (solid) controlled beam tip due to sinusoidal loading . Figure 9. Displacement and rotation of the free end for the uncontrolled (dot) and H2 (solid) controlled beam tip due to sinusoidal loading 14 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. Figure 10. Displacements and rotations of the four nodes for the free (dot) and H2 (solid) controlled beam tip due to sinusoidal periodic impulsive loading. Figure 11. Displacement and rotation of the free end for the uncontrolled (dot) and H2 (solid) controlled beam tip due to sinusoidal periodic impulsive loading Figure 12. Displacements and rotations of the four nodes for the free (dot) and H2 (solid) controlled beam tip due to impulsive loading. 15 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. Figure 13. Displacement and rotation of the free end for the uncontrolled (dot) and H2 (solid) controlled beam tip due to impulsive loading Figure 14. Displacement of the free end for the uncontrolled (dot) and H∞ (solid) controlled beam tip due to white noise. Figure 15. Displacement of the free end for the uncontrolled (dot) and H∞ (solid) controlled beam tip due to periodic sinusoidal loading. 16 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. Figure 16. Displacement of the free end for the uncontrolled (dot) and H∞ (solid) controlled beam tip due to impulsive periodic loading. Figure 17. Displacement of the free end for the uncontrolled (dot) and H∞ (solid) controlled beam tip due to single impulsive loading. With MATLAB tools in accordance with the equation (3.20) an upper and a lower bound of SSV µ∆ are calculated. The conclusions concerning the robust stability are made in terms of these bounds. Satisfying the equation (3.21) to achieve robust stability the µ∆ upper bound must be less than 1. For the considered smart beam the results displaced in Figure 18 show that the beam achieves robust stability. In the Figure 18 the frequency response of the maximal singular value of the transfer matrix characterizing the robust stability with respect to unstructured uncertainties is displaced. This bound also achieves robust stability but gives pessimistic results with respect to the structured uncertainties. 17 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. Figure 18. µ∆ upper bound (dashed), µ∆ lower bound (dotted), H∞ bound (solid). In accordance with the relation (3.25) the magnitude of the maximal singular value of the sensitivity transfer matrix must lie under the inverse of the performance weight function wp for all frequencies. The result displayed in Figure 19 shows that the system satisfies this requirement what indicates good robust performance with good disturbance rejection and transient response. Figure19. Sensitivity transfer matrix for Ks (solid) and inverse weight matrix (dashed) The maxima of the frequency responses on the nominal and robust performance are 0.13 and 0.859, respectively. Therefore, the system with Ks controller achieves nominal and robust performance. Frequency response of the sensitivity and complementary sensitivity transfer matrices of the closed loop system is shown in Figure 20. Their shapes corroborate good feedback properties. 18 G.E. Stavroulakis, D. Marinova, E. Hadjigeorgiou, G. Foutsitsi and C.C. Baniotopoulos. Figure 20. Sensitivity(solid) and complementary sensitivity (dashed) transfer matrix frequency responses. 5 CONCLUSIONS This paper presents the mathematical formulation and the computational model for the active vibration control of a slender beam bonded with piezoelectric sensors and actuators. Furthermore, the problem of active control is studied by using the robust H2 approach in order to achieve robustness with respect to external disturbances and uncertainties of the system or of the loading. Proper selection of the involved parameters is very important for a successful design of this controller. Structured uncertainties addressed to the main physical parameters (mass, damper and stiffness matrices) have been introduced to reflect the errors between the model and the reality. The model of the uncertain system has been presented in a linear fractional transformation framework. Then robust a control design problem within a similar linear fractional transformation framework using the H∞ technique has been formulated. A suboptimal controller has been used for numerical modeling. A certain amount of trial-anderror iterations are necessary during the design process. The comparison between the response of the uncontrolled beam and the response of the controlled beam using the two proposed control laws shows that both strategies are effective. The numerical simulations show that the proposed robust methods are usable for vibration suppression of a laminated beam subjected to different loadings. ACKNOWLEDGEMENTS The work reported here has been partially supported by the European Union Research and Training Network (RTN) “Smart Systems. New Materials, Adaptive Systems and their Nonlinearities. Modeling, Control and Numerical Simulation”, with contract number HPRNCT-2002-00284. REFERENCES [1] M.A. Trindade, A. Benjeddou and R. 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