DSpace at VNU: Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam

Journal of Sound and Vibration ] (]]]]) ]]]–]]] Contents lists available at ScienceDirect Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam Quoc Chi Nguyen a,n, Slava Krylov b a b Department of Mechatronics, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet, Ho Chi Minh City, Vietnam School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel a r t i c l e i n f o abstract Article history: Received October 2013 Received in revised form September 2014 Accepted 17 October 2014 Handling Editor: D.J Wagg In this paper, a feedback control algorithm to regulate oscillation amplitude of a microelectromechanical (MEMS) cantilever beam operated at parametric resonances is developed The control objective is to drive the oscillation amplitude of the micro-beam, which is amplified using parametric excitation, to the desired values The principle of the control algorithm is to establish an output tracking control based on the nonlinear dynamic model of the microbeam, where the supply voltage is considered as a control input The tracking control algorithm is designed to solve the singularities resulting from the zero-deflection state of the micro-beam The Galerkin method is applied in order to reduce the partial differential equation describing the dynamics of the beam into a set of ordinary differential equations (ODEs) Uniformly ultimate boundedness stability of the control system is proved using Lyapunov method The effectiveness of the proposed control algorithm is illustrated via numerical simulations & 2014 Elsevier Ltd All rights reserved Introduction Microelectromechanical systems (MEMS) have been developed to replace conventional mechanical and electronic components such as actuators, transducers, and gears in microscale devices [1–3] The micro-sensors and micro-actuators have enabled development of smart products that are able to sense and control the environment, applying to mechanical, thermal, biological, chemical, optical, and magnetic systems [4–12] MEMS cantilever is one of the most popular MEMS structures widely used due to its ability to provide high sensitivity, high selectivity, and flexibility of its working environment [7–12] Large part of MEMS cantilever uses mechanical vibrating parts for sensing, where the amplitude of the resonant peak and the bandwidth of resonance are important to achieve high sensitivity To increase the oscillation amplitude, design and fabrication techniques have been developed in order to eliminate the effect of air damping [13,14] In an active manner, the oscillation amplitude of MEMS cantilevers can be increased externally using parametric excitation [4,5,15–23], where the parametric pump voltage that yields electrostatic forces have been employed The periodic voltage signal provides a modulation of the stiffness of the structure that and consequently, results in an increase of the oscillation amplitude progressively at specific frequencies The mechanical parametric excitation of MEMS was introduced for the first time in [15] Turner et al [16] showed the measurement of the five parametric resonances in MEMS and proposed the use of parametric n Corresponding author Tel.: ỵ84 38688611 E-mail addresses: nqchi@hcmut.edu.vn (Q.C Nguyen), vadis@eng.tau.ac.il (S Krylov) http://dx.doi.org/10.1016/j.jsv.2014.10.029 0022-460X/& 2014 Elsevier Ltd All rights reserved Please cite this article as: Q.C Nguyen, & S Krylov, Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.029i Q.C Nguyen, S Krylov / Journal of Sound and Vibration ] (]]]]) ]]]–]]] excitation to improve the quality of the MEMS sensing devices Zhang et al [18] investigated the effect of the nonlinearity in the parametrically excited resonant MEMS mass sensor Baskaran and Turner [19] provided an experimental result that shows parametric resonances of a torsional MEMS resonator, considering two interacting mechanical modes of oscillation Zhang et al [21] presented a study of nonlinear dynamics and chaos of a MEMS resonator under two parametric and external excitations Khirallah [23] proposed a parametric amplification of a comb drive oscillator Recently, Krylov et al [4] and Linzon et al [5] introduced parametric excitation of silicon-on-insulator microcantilever beams by fringing electrostatic fields The effectiveness of the parametric excitation to improve the performance of the MEMS cantilevers has been verified As shown in [4,5,17,22], with given amplitude, the specific voltage and frequency of the pumping excitation can be determined through the resonance curves This can be considered as the open-loop control method for MEMS cantilever However, the dynamic nonlinearities affect significantly oscillations of the MEMS devices and consequently, their sensitivity [24] Besides, the imprecise fabrication and mechanical and thermal noises limiting the performance of the MEMS devices are evident Therefore, reasonably prompt feedback control of oscillation amplitude for improvement of the performance of a parametrically excited MEMS cantilever is desirable In recent years, there have been numerous papers published on control of MEMS [25–40] The most focused control topic is to stabilize MEMS resonators [25–35] Alsaleem and Younis [32,33] used delay feedback controllers to stabilize MEMS resonators (especially near pull-in point) The effects of the control gains in [32,33] were later investigated in [34] Siewe and Hegazy [35] indicated that reducing amplitude of parametric excitation can control chaotic motion of a MEMS resonator Seleim et al [36] proposed a closed-loop control of a MEMS resonator, providing optimal operating regions for the resonator There are a lot of researches investigating oscillation amplitude control of MEMS However, only linear resonant MEMS devices have been considered [27,37–40] Leland [27] designed two adaptive controllers for a MEMS gyroscope to regulate the amplitude of the drive axis vibration Batur et al [37] used sliding mode control algorithm to stabilize a MEMS gyroscope Zheng et al [38] proposed an active disturbance rejection control strategy to regulate the output amplitude of the drive axis of a MEMS gyroscope to a fixed level Zhang et al [39] introduced an adaptive vibration control of microcantilever beam to eliminate the useless high amplitude Baghelani et al [40] used analytical formula to design an amplitude control for ring shape micro-disk resonator It is shown that, under parametric excitation, MEMS devices possess much higher amplitude and consequently, higher sensitivity than the ones operated at linear resonances However, the high amplitude is insufficient to obtain the high quality of the MEMS devices In many applications, the amplitude should be controlled For example, in atomic force microscopy (AFM) scanners, the distance between the tip of the AFM beam and the scanned surface should be controlled In optical scanners, the optical scanned angles should be constant under the effects of temperature and voltage fluctuation Actually, even in the MEMS mass sensors, to be able to measure with high precision, the amplitude should be usually known Since the uncertainties such as temperature, voltage fluctuation, uncertain damping, etc usually affect to the MEMS devices, the amplitude need to be controlled In the present work, a control method to regulate vibration amplitude of a parametrically excited microcantilever structure, which is introduced in [4,5], is developed The distributed electrostatic forces are generated by tailored asymmetries in fringing fields provided by single co-planar electrode located symmetrically around the actuated cantilever (see Fig 1) The distributed electrostatic forces depend on the supply voltage and the deflection of the microcantilever Under the effects of the AC supply voltages, the parametric excitation is achieved The Fig Micro-cantilever beam with two symmetrically located electrodes Please cite this article as: Q.C Nguyen, & S Krylov, Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.029i Q.C Nguyen, S Krylov / Journal of Sound and Vibration ] (]]]]) ]]]–]]] control objective is to drive the vibration amplitude of the micro-beam parametrically excited to the desired values The principle of control algorithm is to establish a tracking control based on the nonlinear dynamic model of the micro-beam, where the supply voltage is considered as a control input For the control design purpose, the reduced-order model of the micro-beam obtained using the Galerkin method [41–44] is employed Contributions of this paper are the following First, a nonlinear tracking control of vibration amplitude for a parametrically excited micro-cantilever beam is developed, where the tracking through the singularities are handled Second, uniformly ultimate boundedness stability of the control system is proved mathematically using Lyapunov function Third, simulations results show the advantages of the proposed control method on performance and robustness The rest of this paper is organized as follows Section presents the dynamic model of the microcantilever beam system Section introduces the proposed control algorithm design A Lyapunov function-based stability analysis of the closed-loop system and the proof of stability also are discussed Section includes numerical simulation results that illustrate the effectiveness of the proposed control scheme Finally, Section draws conclusions Dynamic models of the micro-beam 2.1 Continuous model As shown in Fig 1, the rectangular cross-section microcantilever beam is designed such that it can freely move in the out-of-plane direction (z-axis) The planar electrode placed in the area surrounding the micro-beam provides the fringing field, which becomes asymmetric when the micro-beam deflects from the equilibrium position in the x–y plane Under the assumption that the deflections are small in comparison with the length of the beam, the following governing equations of the micro-cantilever beam are derived based on Euler–Bernoulli theory [45]: ∂2 w ∂w ~ w ^ ^ ỵ c^ ỵ EI ỵ V tịf ẳ 0; t x t A (1) where A ¼ bh and I ¼ bh =12 Since the width of the beam b is larger than its thickness h, the effective (plane strain) ^ modulus E~ ẳ E=1 v2 ị is employed, where v is Poisson's ratio The supply voltage is denoted as VðtÞ The beam is subject to the following homogenous boundary conditions (2) and the nonzero initial conditions (3):      ∂w ~ ∂ w ~ ∂ w w0; tị ẳ 0; ẳ 0; EI ẳ 0; EI ¼ 0: (2) ∂x x ¼ ∂x2 x ¼ l ∂x3 x ¼ l wð0; xÞ ¼ w0 xị;  w ẳ 0: t t ẳ (3) The distributed electrostatic force f^ , which is generated due to asymmetry of fringing fields between the beam and the electrode, act as restoring forces that pull the micro-beam back towards the equilibrium position Since the distributed electrostatic force f^ cannot be calculated analytically in a closed form, in this paper, the distributed electrostatic force is approximated as follows [5]: ^ a wị ^ ẳ f^ wị ; ^ 2p ỵ w (4) ^ ẳ w=h, and a, , and p are fitting parameters The unit of the value of a is N/mV2, and the parameters, σ and p, are where w ^ is plotted non-dimensional The relationship between the electrostatic force f^ and the non-dimensional beam deflection w in [5, Fig 2, p 163508-3] It should be noted that the nonzero initial deflection of beam (3) is necessary to enable the actuation by the distributed electrostatic forces at the starting time Remark In this paper, it should be noted that since the contribution of the electrostatic nonlinearity associated with the actuation deflection-dependent force is dominant, the influence of the nonlinear curvature and rotational inertia [46–48], which have only minor effects, is neglected Moreover, since the force function (4) decreases with the large amplitude, it may result in the limitation of the amplitude as well The experiment and simulation results in [4,5] show that the parametric amplitude is limited by the electrostatic nonlinearity Introduce the following non-dimensional variables: x x^ ¼ ; L t t^ ¼ ; T ^ ¼ w w ; h (5) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ is a time scale Substituting Eq (5) into Eqs (1)–(3) and dropping the hats, the non-dimensional where T ¼ ρAL4 =EI equations of motion, the boundary conditions, and the nonzero initial condition that govern the transverse deflection of the micro-beam are obtained as follows: w w w ỵ c ỵ ỵ V tịf wị ẳ 0; t ∂x ∂t (6) Please cite this article as: Q.C Nguyen, & S Krylov, Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.029i Q.C Nguyen, S Krylov / Journal of Sound and Vibration ] (]]]]) ]]]]]] w0; tị ẳ w0; xị ẳ where     ~ ∂ w ~ ∂ w EI ¼ EI ¼ 0;  ∂x2 x ¼ ∂x3 x ¼  ∂w ¼ 0; ∂x x ¼ sffiffiffiffiffiffiffiffiffiffiffi L4 ; c ¼ c^ e EI ρA  ∂w ¼ 0: t t ẳ w0 xị ; h f wị ẳ f^ wị ; a aL4 V ¼ ~ 0; EIh (7) (8) (9) where V0 is the unit voltage Hereafter, for the sake of convenience, V0 ¼1 is adopted As a result, the numerical values of the non-dimensional voltage V are equal to the actual applied voltages values 2.2 Reduced-order model A reduced-order model is obtained through discretization of the partial differential equation (6) with the boundary conditions (7) into a finite-degree-of-freedom system Based on the Galerkin procedure, the deflection of the beam is approximated by a combination of smooth basis functions as n wðx; tÞ ¼ ∑ qi ðtÞϕi ðxÞ; (10) i¼1 È É where qi ðtÞ is are the generalized displacement, and the set of the basis functions i xị i ẳ 1;;n is orthogonal, i.e., R1 i xịj xịdx ẳ ij , where δij is the Kronecker delta In this paper, ϕi ðxÞ is the ith linear undamped mode shape of the straight micro-beam given as ! sinh i ỵ sin i i xị ẳ Di sin i x sinh i x ỵ cos i x ỵ cosh i x ; (11) cosh i ỵ cos i where Di is a constant, and λi satisfies cos λi cosh λi ¼ 1, correspondingly giving infinite number of frequencies ωi ¼ λ2i ; (12) where ωi is the ith natural frequency of the linearized straight micro-beam The finite dimensional dynamic system will be obtained by carrying out the following procedure: (i) multiplying Eq (6) by a weighting function ϕj ðxÞ, (ii) integrating the resultant equation obtained in step (i) over the domain x ẳ ẵ0; 1, (iii) utilizing the boundary conditions (7) and substituting Eq (10) into the resultant equation obtained in step (ii), and (iv) collecting all terms of the resultant equation obtained in step (iii) with respect to q€ i ðtÞ, q_ i ðtÞ, and qi ðtÞ After step (iv), Eq (6) is rewritten into a set of ordinary differential equations (ODEs) as ỵ C qtị _ ỵKqtị ỵ V Fqị ẳ 0; q0ị ẳ q0 ; M qtị where qtị ẳ ẵ q1 tị are given as … (13) qn ðtÞ Š is the time-dependent vector of generalized coordinates, and the elements of M, C, and K T mij ¼ δij kij ¼ δij Z Z ϕi ϕj dx; (14) ϕ″i ϕ″j dx; (15) cij ¼ cmij : (16) Note that M, C, and K are diagonal matrices The vector βV Fqị represents the distributed electrostatic actuation, where Fqị ẳ ẵ f ðqÞ … f n ðqÞ ŠT with Z f i qị ẳ f q; i ẳ 1;:::;n Þϕi ðxÞdx: (17) It should be noted that the parametric excitation is generated by applying the pumping voltage VðtÞ 2.3 State space model For the control design purpose, it is convenient to rewrite the ODE (13) into a state space model Introduce the state T vector ztị ẳ ðqT ; q_ ÞT A R2n , and the control input utị ẳ V tị, utị Z The reduced-order model of the micro-beam (13) can Please cite this article as: Q.C Nguyen, & S Krylov, Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.029i Q.C Nguyen, S Krylov / Journal of Sound and Vibration ] (]]]]) ]]]–]]] be converted to a state space model: z_ ¼ Az ỵ Bzi ẳ 1;:::;n ịutị; z0ị ẳ qT0 ; q_ ÞT ; (18) ! ; (19) T where InÂn 0nÂn A¼ ÀM À1 " B¼ K ÀM À1 C # 0nÂ1 À β M À Fðzi ¼ 1;:::;n Þ : (20) The control objective is to drive the deflection of the tip of the micro-beam to a desired oscillation In the control design, the deflection of the tip of the micro-beam is considered as a control output, which is given as y ẳ Hz; H ẳ ẵ ð1Þ (21) ϕ2 ð1Þ … ϕn ð1Þ 01Ân Š: (22) Output tracking control problem Consider the dynamic system (18)–(22) with the desired continuous oscillation yd ðtÞ A feedback control input uðtÞ is  à designed to guarantee the tracking of the control output y(t), that is, lim yðtÞ Àyd tị ẳ 0, and ztị A L1 \ L1 for the nonzero t-1 initial condition z0ị ẳ z0 Three assumptions about the desired output are made as follows: (i) the desired oscillation and its derivatives are continuous; (ii) fyðiÞ tịgi ẳ 1;;n are bounded for all t Z 0; and (iii) fyiị tịgi ẳ 1;;n are piecewise continuous d d functions Following the design procedure in [49], the first step of the control design is to seek the nominal control input ud ðtÞ and the desired state zd ðtÞ satisfy the following statements: (i) ud ðtÞ and zd ðtÞ satisfy the differential equation z_ d ẳ Azd ỵBzdi ẳ 1;:::;n Þud ðtÞ: (23) (ii) The output tracking is achieved, i.e., yd ẳ Hzd : (24) (iii) ud tị and zd ðtÞ A L1 \ L1 Taking the first and second derivatives of the control input and using the dynamic equation (18), the following equations is obtained: y_ ¼ H z_ ẳ HAz ỵHBu ẳ HAz; (25a) y ẳ H Az_ ẳ HA2 zỵ HABu: (25b) It should be noted that the expression HB ¼ 0, which obtained from Eqs (20) and (22), is employed to derive Eq (25a) Let   n 1ị tị ẳ HA2 z ẳ i ki zi ỵczi ỵ n ; (26) m i iẳ1 n tị ẳ HAB ẳ i 1ịmi f i zi ẳ 1;;n ị: (27) y ẳ tị ỵ tịutị: (28) i¼1 Eq (25) can be rewritten as From Eq (28), choose the nominal control input ud ðtÞ as ud ðtÞ ẳ tịẵy d tị tị: (29) It should be noted that when zi ¼ 1;:::;n ¼ (i.e., q ẳ 0), tị becomes zero, and consequently, ud ðtÞ-1 Therefore, to avoid È É the unboundedness of the control input, for the neighborhood of the singular point S qð:ÞjqðtÞ A Rn ; jjqjj1 r r , the Please cite this article as: Q.C Nguyen, & S Krylov, Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.029i Q.C Nguyen, S Krylov / Journal of Sound and Vibration ] (]]]]) ]]]–]]] following nominal control input is used: uSd tị ẳ t r ịẵy d ðtÞ À θðtފ; (30) where t r is the time when jjqjj1 ¼ r Finally, the following control law is proposed: utị ẳ ptịfy d tị tịị kp ztị zd tịịg; (31) where ptị ẳ sgnjjqjj1 r ị ỵ1 sgnjjqjj1 r ị ỵ ; 2ψ ðtÞ 2ψ ðt r Þ (32) and the control gain vector is given as  kp ¼ k1 à k2n : … (33) Remark The singularities of the system occur when the deflection of micro-beam is zero, that is, the beam reaches the x-axis At singular points, the electrostatic forces become zero, and the system is temporarily uncontrollable Remark The control law (31) includes the feedfoward term pðtÞðy€ d ðtÞ À θðtÞÞ and feedback term pðtÞkp ðzðtÞ À zd ðtÞÞ With the feedfoward term, the system is partially linearized, as shown in Eqs (36)–(38) below Meanwhile, the feedback term È É provides a stabilizing effect by adjusting the control gains ki i ¼ 1;…;2n Remark It should be noted that the control law (31) is a combination of the exact tracking control law utị ẳ ψ À ðtÞ fðy€ d ðtÞ À θðtÞÞ À kp ztị zdẩtịịg and the approximate tracking utị ẳ ψÉÀ ðt r Þfðy€ d ðtÞ À θðtÞÞ Àkp ðzðtÞ Àzd ðtÞÞg, which É È control law n n work in the regions S qð:ÞjqðtÞ A R ; jjqjj1 4r and S qð:ÞjqðtÞ A R ; jjqjj1 r r , respectively Remark From the control law (31), it is shown that the large amplitude and the high frequency of the desired oscillation result in the large y€ d ðtÞ and consequently, the large control input Since utị ẳ V tị, the control input has a limit determined by the maximum voltage, which can be applied to the micro-beam The amplitude and the frequency of the desired oscillation are limited by the maximum voltage Introduce the change of the state variables y 6 y_ 6 7 6 7 60 ¼ z3 ¼ ξ¼6 7 6 ⋮ ⋮ 6⋮ 5 4⋮ η2n À z2n χ1 χ2 η1 32 z 76 z2 76 76 ⋮ ⋮ 76 76 9Nz: ⋮ ⋮7 76 76 ⋮ 56 z2n À z2n H HA ⋮ ⋱ ⋮ ⋱ … … … (34) Since HA ẳ ẵ 01n 1ị 1ị … ϕn ð1Þ Š, it is concluded that the matrix N has full rank and consequently, is invertible Therefore, Eq (34) yields the following inverse transformation: z ¼ N À ξ: (35) Using the new state variables, the system dynamics becomes 8_ χ ¼ χ2 > < χ_ ẳ tị ỵ tịutị > : _ ẳ s; ị ẳ A N ỵ B utị (36) where 0n 2n ỵ A ẳ ÀM " Bη ¼ À1 K In À 2Ân À ÀM À1 C 0n À 2Â1 À β M Fi ẳ 1;:::;n ị ! ; (37) # ; (38) _ ẳ s; ị is the internal dynamics of system (36) and η Define the tracking error e as follows: e ẳ ẵ y yd y_ y_ d ŠT (39) Please cite this article as: Q.C Nguyen, & S Krylov, Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.029i Q.C Nguyen, S Krylov / Journal of Sound and Vibration ] (]]]]) ]]]–]]] Using control law (31), the feedback control of the nonlinear system (36) is now considered as follows: < e_ ẳ Ae e ỵ1 pị ỵ p 1ịy d ỵ ke À ψ pkp ÞN À ðξ À ξ Þ d _ ẳ s; ị ẳ A N ỵB pfy d tị tịị kp N À ðξ À ξd Þg :η where " Ae ¼ (40) # À ke Àke ke ¼ ½ ke ϕ1 ðLÞ … ; ke 40; (41) ke ϕ2n ðLÞ Š: (42) The Jacobian matrix of the internal dynamics Jð0; ηÞ is obtained as  ∂s  J0; ị ẳ  ẳ J ỵJc1 ỵ Jc2 ; (43) _ ẳ s0; ị The matrices Jη , Jc1 , and Jc2 are given in Appendix A where η0 is the equilibrium point of the internal dynamics η It is shown that the following Lipschitz condition holds for all t Z0: jjfsðχa ; ηa Þ ÀJð0; η0 Þηa g À fsðχb ; ηb Þ À Jð0; ịb gjj r jja b jj ỵ μ2 jjηa À ηb jj; where n o μ1 Z max jjAη N À jj1 ; jjBHA2 jj1 ; jjkp N À jj1 ; n μ2 Z max jjAη N À jj1 ; jjBHA2 jj1 ; jjkp N À jj1 ; Jð0; η0 Þ (44) (45) o (46) Theorem Consider systems (36)–(38) The control gain vector kp is chosen to satisfy the following conditions: (i) Jð0; η0 Þ has no eigenvalues on the imaginary axis (the method to determine kp can be refer to [50]), i.e., the origin of the _ ẳ s0; ị is hyperbolic system _ ẳ s0; ị is asymptotically stable (ii) Jð0; η0 Þ has no positive eigenvalue, i.e., the origin of the system η Then, the control law (31) guarantees the uniformly ultimate boundedness of the tracking error e Proof Since the hyperbolic system _ ẳ J0; ị and the Lipschitz condition (44) guarantee that the approximation of sðχ; ηÞ by Jð0; η0 Þη satisfies Condition in [49, p 932], for given bounded χd (i.e., the desired output yd and its first derivative y_ d ), it follows from Theorem in [49, p 932] that the unique bounded solution, Z ỵ1 ẩ ẫ d ẳ t À τÞ sðχ; ηÞ À Jð0; η0 Þη dτ; (47) exists and satisfies lim d ẳ 0, where tị is the solution of the following equation: t- _ ẳ J0; ịX; X X 1ị ¼ 0: (48) If condition (ii) is satisfied, referring to [51,52], there exists a locally Lyapunov function W ðtÞ such that κ jjηjj2 r W ðtÞ r κ jjηjj2 ; (49) ∂W sð0; ηÞ r À κ jjηjj2 ; ∂η (50) ∂W sð0; ηÞ r À κ jjηjj; ∂η (51) where κ i i ẳ 1; ; 4ị are the positive constants (see Theorem 4.16, p 167 in [51]) From Eqs (49)–(51), the following inequality is obtained [52,53]: ∂W sð0; ηÞ r jjjj2 ỵ jjjjjjejjỵ yd ị; (52) Wtị ẳ eT Peỵ W ðtÞ; (53) À Á where jjzjj r κ jjejj ỵjjjj and jyd j r yd are utilized Consider the following Lyapunov function candidate: Please cite this article as: Q.C Nguyen, & S Krylov, Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.029i Q.C Nguyen, S Krylov / Journal of Sound and Vibration ] (]]]]) ]]]–]]] where γ 0, and the positive definite matrix P is the solution of the following equation: Ae P ỵATe P ¼ ÀI: (54) where I is identity matrix È É In the region S ẳ q:ịjqtị A Rn ; jjqjj1 r , the error dynamics of system (40) becomes e_ ẳ Ae e ỵ ke kp ịN ðξ À ξd Þ: (55) The time derivative of the Lyapunov function (53) is obtained as n  o _ Wtị ẳ jjejj2 ỵeT Pke kp ịN d ị ỵ jjjj2 ỵ jjjj jjejj ỵ yd n  o r jjejj2 ỵ jjejj2 ỵ jjjj2 ỵ jjjj jjejj ỵ yd ! & '2 κ Þ2 γκ κ4 κ5 jjejj2 À jjηjjÀ jjejj r À À μ3 À 2κ κ3  2 & κ κ μyd '2 γ κ κ μyd γκ jjηjjÀ À þ κ3 2κ (56) where μ3 ¼ jjðke À kp ÞN À jj jjPjj: (57) There exist kp , ke , and γ satisfying the following inequality: À μ3 À γ ðκ κ Þ2 40: 2κ (58) Then, one is obtained _ WðtÞ o S Wtị ỵ where yd ị2 ; & s ẳ À μ3 À γ ðκ κ Þ2 =2κ κ ; (59) ' γκ : 2κ (60) Inequality (59) implies that WðtÞ r Wð0Þe S t ỵ yd Þ2 : 2κ μS (61) From Eq.È(61), it is concluded that É jjejj is bounded in the region S Let q:ịjqtị A Rn ; jjqjj1 ẳ r be the set of singularity points, and T ¼ ft i gi ¼ 1;…;m be the set of the time when the ỵ system is at the singularity points It is assumed that È there exist two sets,É fεi gi ¼ 1;…;m and fεi gi ¼ 1;…;m , such that jjqt i i ịjj1 ẳ r and jjqt i ỵ i ịjj1 ẳ r In the region S ¼ qð:ÞjqðtÞ A Rn ; jjqjj1 rr , the time derivative of WðtÞ is obtained as n o _ WðtÞ ẳ jjejj2 ỵ pị ỵ p 1ịy d ỵ ke pkp ịN À ðξ À ξd Þ eT P n  o ỵ jjjj2 ỵ jjjj jjejj ỵ yd n  o r jjejj2 ỵ jjjjjjejj ỵ jjejj ỵ jjejj2 þ γ À κ jjηjj2 þ κ κ jjjj jjejj ỵ yd !2 23 μθ À γκ μyd 4μ4 jjejj À rÀ À2μ3 À μθ À γκ μyd !2 ðμθ þ γκ μyd ÞÀ γκ μyd Á2 κ γ À μθ À γκ μyd jjηjj À jjejj À jjηjj À À κ γ À μθ À γκ μyd γ κ 24 2yd 24 ỵ yd ỵ ỵ ψ p jjkp jjjjejj2 À2μ3 À μθ À γκ μyd κ γ À μθ À γκ μyd γ κ 24 μ2yd μ24 r À μS Wtị ỵ ỵ ỵ yd 23 À γκ μyd κ γ À μθ À yd ỵ p jjkp jj jjejj2 ; (62) where À Á jjzjj r jjejj ỵjjjj ; ẳ jjHA2 jj jjPjj; jyd j r μy€ d ; μ4 ¼ μy€ d jjPjj (63) Please cite this article as: Q.C Nguyen, & S Krylov, Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.029i Q.C Nguyen, S Krylov / Journal of Sound and Vibration ] (]]]]) ]]]–]]] ud control input u + + Micro beam k p(zd -z) Desired oscillation of deflection yd Nonlinear inversion Stabilizing compensation Reference state z d Actual state z + - Fig Control scheme of the micro-beam system are employed kp , ke , and γ are selected from sets of the values of kp , ke , and γ satisfying inequality (58) such that the following inequalities hold: À 2μ3 À μθ À γκ μyd 40; (64) κ γ À μθ À γκ μyd 0: (65) Then, it is concluded that _ Wtị r S Wtị ỵ where Z ti ỵ m 24 2yd i 24 ỵ ỵ yd ỵ jjkp jj jjejj2 d; ỵ 23 μyd κ γ À μθ À γκ μyd i ¼ t i À εi È À Á É μs ¼ ð2 À 2μ8 À μθ ị=4 ; ỵ =2κ ; From Eq (66), the boundedness of WðtÞ is obtained as follows:   γ μ26 μ23 27 Wtị rW0ịe S t ỵ ỵ ỵ γμyd μS À 2μ8 À μθ À γμ6 μ3 Z ti ỵ i jjkp jj m jjejj2 d; ỵ S iẳ1 t i iỵ (66) (67) (68) From inequalities (61) and (68), the uniformly ultimate boundedness of the tracking error e is concluded Since e and η are bounded, the boundedness of the control law (31) is guaranteed □ Remark When the micro-beam operates in the region S, it is proved that W(t) converges to the ball of radius γ ðκ κ μyd Þ2 =2κ μS , that is, W(t) can be pushed in an arbitrarily small boundedness region by setting a sufficiently small γ Meanwhile, in the region S, the convergence ball (see Eq (68)) cannot easily be adjusted by changing γ Therefore, to improve the control performance, the region S should be decreased by choosing small r However, the small r makes the small ψ ðt r Þ and consequently, the large control input uðtÞ, which yields the difficulty in implementing the control scheme due to the supply voltage saturation in practice Therefore, the value of r yielding the best control performance is determined by the maximum voltage, which could be applied (Fig 2) Simulation results To verify the effectiveness of the proposed control algorithm, numerical simulations were carried out with the system parameters listed in Table The single mode model was used to describe the micro-beam Let the initial condition of the micro-beam system (6)(9) be w0 xị ẳ ϕ1 ðx=LÞ μm The control gain vector kp is selected as follows: kp ¼ 120 10 The dynamical responses of the beam were simulated in two cases: open-loop and closed-loop controls in which the closed-loop control is proposed in this research In the case of open loop control, the resonance curves (see Fig in [5]) for the micro-beam that can predict amplitude of an oscillation of the beam with respect to the excitation voltage is used to 2 determine the control input (i.e., u ¼ V^ AC =2 ỵ V^ AC =2ị cos tị), corresponding to the given desired deflection As shown in Figs and 4, the output tracking can be achieved with either the open-loop or closed-loop controllers However, the settling time and the robustness are different Please cite this article as: Q.C Nguyen, & S Krylov, Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.029i 10 Q.C Nguyen, S Krylov / Journal of Sound and Vibration ] (]]]]) ]]]–]]] Table System parameter values used in numerical simulations Parameter Value ρ E L b h a σ p 2330 kg/m3 169  106 N/m2 500 μm 16 μm μm 1.3  10 À N/m V2 1.45 Fig Dynamic response of the micro-beam in the case of the closed-loop control (the control law (31) used) Fig Dynamic response of the micro-beam in the case of the open-loop control It is believed that the closed-loop controller can drive the deflection of the beam tip to the desired oscillation faster than the use of the open loop controller (0.36 ms in the case of the closed-loop vs 1.08 ms in the case of the open-loop, as shown in Figs and 4, respectively) To illustrate the robustness of the two methods to the variation of the system parameters, the micro-beam dynamics was simulated with the assumption: both the open-loop and closed-loop control laws are obtained by using the nominal Young modulus while the actual Young modulus in the beam system is different It should be noted that the difference of the Young modulus is chosen to represent typically the variation of the beam stiffness due to the change in environmental thermal conditions, materials, geometric shapes of the beam With the deviation of Young modulus ½ À 10% 10% Š, the steady tracking error in the case of closed-loop control is less than percent (see Fig 5), whereas the one in the case of open-loop control reaches À50 and 80 percent (see Fig 6) It is shown that the closed-loop control is more robust than the open-loop control The feedback term in the control law (31) compensates the difference of the system parameters and consequently, makes the control system more robust Figs and shows the results on the settling time of the dynamical responses of the micro-beam in the cases of the closed-loop and open-loop controls, in which the results were collected from the numerical simulations illustrating the robustness of the closed-loop and open-loop controls With the deviation of Young modulus [ À 10% 10%], the settling time in the case of closed-loop control (see Fig 7) varies from 0.27 ms (a decrease of À 23 percent, in comparison to the case of no Please cite this article as: Q.C Nguyen, & S Krylov, Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.029i Q.C Nguyen, S Krylov / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 11 Amplitude error (%) -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 10 11 Young modulus difference (%) Fig The amplitude error vs Young modulus difference in the case of the closed-loop control 100 Amplitude error (%) 80 60 40 20 -20 -40 -60 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 10 11 Young modulus difference (%) Fig The amplitude error vs Young modulus difference in the case of the open-loop control Fig The settling time vs Young modulus difference in the case of the closed-loop control Fig The settling time vs Young modulus difference in the case of the open-loop control Please cite this article as: Q.C Nguyen, & S Krylov, Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.029i Q.C Nguyen, S Krylov / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 12 difference between the nominal and actual values) to 0.53 ms (an increase of 47 percent) As shown in Fig 8, in the case of the open-loop control, the settling time takes 0.83 ms (a decrease of À22 percent) to 2.68 ms (an increase of 48 percent) Comparison between Figs and indicates that the micro-beam system in the case of closed-loop control reaches the steady state significantly faster than the one in the case of open-loop control even there exits difference between the nominal (used in control) and actual values of Young modulus Moreover, it can be observed that the nominal Young modulus that is bigger than the actual one reduces the settling time in both cases of the closed-loop and open-loop controls, whereas the settling time increases when the nominal Young modulus is lower than the actual one Conclusions A control algorithm of vibration amplitude for a parametrically excited microcantilever beam has been developed The proposed control algorithm has been designed based on the nonlinear model of the microcantilever beam The tracking through singularities has been solved Uniformly ultimate boundedness stability of the control system has been proved using Lyapunov method Through the simulation results, the advantages of the closed-loop control have been shown The proposed control algorithm can drive the oscillation of the beam tip to the desired one faster than the open-loop control The use of the feedback control has consolidated the robustness of the proposed control method One of the main advantages of the proposed closed-loop control is its ability to provide the desired vibrational amplitude even in the case of wide variation of the system parameters Acknowledgments The research was supported by the consortium PARM-2, FP7-PEOPLE-2011-IAPP, and Marie Curie Actions, Project no 284544 and by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 107.04-2012.37 Appendix A 0n À 2Ân 60 … 60 … 6 Jη ¼ À k … m3 ⋱ ⋮ 6⋮ … Àmknn 6 6 ϕ3 k3 n 6 m1 m3 ∑ ϕ m i i Jc1 ¼ 6 i¼1 6 ⋮ ϕ3 k3 6 n m m ∑ϕm n i i Àc m1 ⋮ In À 2Ân À … … … ⋱ ⋮ ⋱ ⋮ ⋱ … … … 7 ⋮ 7 ⋮ 7 ⋮ Àc (A.1) mn 0n À 2Â2n À … … 0 … m1 mn ⋱ … ⋮ n m1 m1 n ⋮ i¼1 ⋮ ∑ ϕi mi ⋮ ϕn kn ∑ ϕi mi i¼1 … … … ⋱ ⋱ ⋱ ϕ1 c n mn m1 ∑ ϕ i mi … … 0n À 2Ân kpn m1 ∑ni ¼ ϕi mi ⋮ 7 n ∑ ϕi mi 7 i¼1 7 ⋮ 7 ⋮ 7 ⋮ 7 ϕn c n mn mn ∑ ϕi mi ϕn c m1 mn … i¼1 0n À 2Ân À 6 Jc2 ¼ 6 0nÂn ∑ ϕ i mi i¼1 ϕ3 k3 n mn mn i¼1 ϕ1 c … ⋱ ⋮ … kp2n mn ∑ni ¼ ϕi mi (A.2) i¼1 7 7 7 (A.3) References [1] J Bryzek, Impact of MEMS technology on society, Sensors and Actuators A: Physical 56 (1–2) (1996) 1–9 [2] B Vigna, Future of MEMS: an industry point of view, Proceedings of the Seventh International Conference on Thermal, Mechanical and Multiphysics Simulation and Experiments in Micro-Electronics and Micro-Systems, EuroSimE 2006, Como, Italy, 2006 Please cite this article as: Q.C Nguyen, & S Krylov, Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.029i Q.C Nguyen, S Krylov / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 13 [3] N.S Barker, The future of N/MEMS, IEEE Microwave Magazine (6) (2007) 52–55 [4] S Krylov, N Molinazzi, T Shmilovich, U Pomerantz, S Lulinsky, Parametric excitation of 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beam, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.029i ... electrodes Please cite this article as: Q.C Nguyen, & S Krylov, Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam, Journal of Sound and Vibration (2014),... 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