International Journal of Control, Automation and Systems 15(6) (2017) 2892-2905 http://dx.doi.org/10.1007/s12555-016-0650-1
ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555
Radial Basis Function Neural Network based Adaptive Fast Nonsingular Terminal Sliding Mode Controller for Piezo Positioning Stage To Xuan Dinh and Kyoung Kwan Ahn* Abstract: This paper presents an adaptive fast nonsingular terminal sliding mode control base on a neural network based approximation technique to control the position of a piezo positioning stage (PSS). The proposed terminal sliding mode control can provide faster convergence and higher precision control while maintain its robustness to uncertainties. In the proposed control scheme, the combination of the fast-nonsingular terminal sliding mode control and neural network, which can precisely estimate the uncertainties in dynamic of the PSS system by employing an online tuning scheme, is a promising control approach for actuator systems. In addition, the robust control term is adopted to compensate the modeling error and ensure the robustness corresponding to a bounded disturbance. Stability of the closed loop system is analyzed and proved by using special Lyapunov functions. Experiment results strongly confirm the effectiveness of the proposed control method. Keywords: Fast nonsingular terminal sliding mode, neural network, piezo positioning stage, robust control.
1.
INTRODUCTION
Piezoelectric ultrasonic motor [1–5] is a recent developed motor, which is a promising alternative to conventional electromagnetic drives for many precision positioning applications owing to their advantages such as high position accuracy in the micrometer and nanometer range, fast dynamic, high force, no electromagnetic interference, noiseless operation and compact design. A piezoelectric ultrasonic motor operating principle is based on excitation of acoustic waves in a solid using the reserve piezoelectric effect. As an electromechanical coupling system, piezo ultrasonic motor is characterized by their energy conversion. The process of energy conversion which takes place in the piezo ultrasonic motor is divided into two stage [6]. In the first stage, piezoelectric components convert applied voltage to ultrasonic frequency mechanical oscillations of the stator (vibrator). An ellipse motion of the surface/driving tip is created due to the excitation of the stator. The conversion of high-frequency vibration into unidirectional rotary or linear motion of a rotor/slider is made when the contact mechanism occurring at the interface between stator and rotor in the second stage. According to the vibration modes used to create the ellipse motion, piezoelectric ultrasonic motors are classified into two forms: one use a single vibration and the other two vibration modes.
The ultrasonic motor utilizing a single vibration mode is furthered divided into two categories: traveling-wave ultrasonic motors (TWUMs) and standing-wave ultrasonic motors (SWUMs). The TWUMs produce the motion by mean of the superposition of standing waves in the stator [7] while the operation of SWUMs are based on standingwave to generate their motion. The SWUMs are simpler to design and build. Ultrasonic motors can be also classified according to the motor function, with two groups of ultrasonic motor, rotary ultrasonic motor and linear ultrasonic motor. The popular design of the linear ultrasonic motor branch is the piezoelectric bimodal ultrasonic motor, which is introduced in the work [8]. The working principle of linear piezoelectric bimodal ultrasonic motor is to generate two orthogonal vibration modes, whose resonant frequencies are nearly the same and the phase relationship results in elliptical trajectories of the attached pusher. This type of ultrasonic motor is used as an actuator in the piezoelectric positioning stage in this research. The advantages of piezoelectric positioning stage are ultra-high resolution and fast response. However, the positioning accuracy of the system is limited owing to the nonlinear properties of ultrasonic piezo actuator and friction behavior of positioning stage. To address the hysteresis characteristics, many control techniques have been presented. Feedforward control is one of the most popular
Manuscript received October 15, 2016; revised January 10, 2017 and February 19, 2017; accepted February 21, 2017. Recommended by Associate Editor Yang Tang under the direction of Editor Hamid Reza Karimi. This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2017R1A2B3004625). To Xuan Dinh and Kyoung Kwan Ahn are with the School of Mechanical Engineering, University of Ulsan, 93 Daehak-ro, Namgu, Ulsan, Korea (e-mails:
[email protected],
[email protected]). * Corresponding author.
c ⃝ICROS, KIEE and Springer 2017
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approach for positioning control of piezo actuator system. The core idea of the feedforward control technique is to develop a mathematical model that describes the complex hysteresis and uses the inverse hysteresis model based feedforward controller to linearize the response of the actuator. The examples of hysteresis models include of the Duhem model [9], Bouc-Wen model [10], PrandtlIshlinskii model [11], Preisach model [12, 13]. Nevertheless, it is difficult or time-consuming to obtain an accurate mathematical model and to identify system parameters based on these techniques due to the complexity and highly nonlinearity of the hysteresis effects. Alternately, many other approaches employed the “black box” models, such as neural network based model [14–17], fuzzy model [18–20], and radial basis function neural network [21–24]. Though, none of these methods applied to approximate the nonlinearities and friction behavior of the linear ultrasonic piezo-driven stage system. Thus, in this brief, a radial basis function neural network with the advantages of simple structure, fast learning algorithm and better approximation ability than conventional neural network, is employed to estimate the nonlinear characteristic of the ultrasonic piezoelectric actuator and the friction behavior of the stage. It is worth to note that a nonlinear sliding mode called terminal sliding mode (TSM) [25–27] is an effectiveness finite-time control technique, especially for the systems with uncertainty and external disturbance. Compare with traditional sliding mode algorithm, TSM can provide a finite-time convergence. To eliminate the singularity problem associated with conventional terminal sliding mode control, Feng [28] introduced a new terminal sliding mode manifold namely nonsingular terminal sliding mode control (NTSM). However, TSM and NTSM have slower convergence to the equilibrium than the sliding mode control based on linear switching hyperplanes when the system state is far away from the equilibrium. To solve this problem, a modified terminal sliding mode control named fast terminal sliding mode control (FTSM) is proposed in the work [29,30], which combines the advantages of the TSM and traditional sliding mode control. In this work, a model-free and accurate controller for piezoelectric positioning stage using radial basis function algorithm and fast nonsingular terminal sliding mode control (FNTSMC) is proposed. First, a fast nonsingular terminal sliding surface is selected to make the system not only avoid the singularity problem, but also exhibit a fast finite time transient convergence both at a distance from and at a close range of the equilibrium. Second, the nonlinearities and friction behavior of the piezoelectric positioning stage is approximated and eliminated by a radial basis function neural network scheme without parameter identification of such a complex system. Experiment results show the superiority of proposed control algorithm over the conventional FNTSMC and PID control in speed
of convergence and accuracy of position tracking. The rest of the paper is presented as follows: The dynamic of a piezoelectric positioning stage is investigated in Section 2. A FNTSMC algorithm combine with radial basis function neural network estimation scheme is designed in the next section. The stability analysis is given in Section 4. In Section 5, experiment results are carried out to verify the highly accuracy tracking performance of the proposed approach. Finally, we concluded our work in Section 6. 2. PIEZO ACTUATED SYSTEM 2.1. Structure and design concept of piezoelectric positioning stage The actuator investigated in this research is a single mode piezoelectric plate actuator for ultrasonic linear motors, which was introduced in the work [8]. The developed piezoelectric ultrasonic actuator and the basic design of a linear stage driven by it are shown in Fig. 1. The actuator is mainly composed of a stator, which is a piezoelectric plate polarized in the Y direction, and a friction element (driving tip) attached at the midpoint of one long edge of the piezoelectric plate. Electrodes are set on the large surfaces (X-Z plane) of the plate. There are two symmetric electrodes, called exciter electrodes, each cover one haft of the front surface. The rear surface has only one electrode, plays a role as a common drain. Only one of electrodes in the front side is excited by placing a signal on it, while the other is left floating. The excited signal has a resonant frequency of an E(3,1) vibration mode of the plate, so that the wave in the X direction exhibits three haft wavelengths and the wave in the Z direction exhibits one haft wave length. To obtain linear motion with the pusher attached in the midpoint of the top edge of the piezo plate, the E(3,1) vibration mode is excited asymmetrically. Only one of two electrodes is excited and the another is left to float. To get the motion in the opposite direction, the excitation is switched to another electrode. As the result, an elliptical motion of the pusher is generated. By pressing a slider against the pusher of actuator, the motion of pusher is transformed into linear motion of slider through friction force.
Fig. 1. Structure of piezo-actuated-driven linear stage.
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2.2. Experimental setup The system configuration and the experiment apparatus are shown in Figs. 2 and 3, respectively. The piezo positioning stage combined linear encoder is fabricated by Physik Instrumente (PI) GmbH & Co. The system operating process is built on a personal computer within Simulink environment integrated real time Windows target toolbox of Matlab (Mathworks Inc.). The sampling time was set to be 0.001 second in all experiments. A laser sensor from Keyence Corp. was used to measure the absolute displacement of moving platform. Two multi-function data acquisition cards, a PCI 6014 from National Instrument (NI) and a Quad04 from Measurement Computing company, were installed on PCI slots of the PC to perform peripheral buses. The displacement feedback signals from the laser sensor were fed to the computer through the PCI 6014 card and the control voltage applied to the piezoelectric actuator was obtained from the PCI 6014 card and a high voltage amplifier. Moreover, the relative displacement feedback signals from the linear encoder were sent to the host by the Quad04 data acquisition card. Firstly, the zero-position of the moving platform was obtained by using feedback signals from the laser sensor. Then, the position tracking controller using the feedback signals from the high accuracy linear encoder to achieve excellent tracking performances. 2.3. The dynamic modeling of piezoelectric positioning stage In this section, a review is given for the modeling approach introduced in [3]. The dynamic model of ultrasonic motor is divided into four subsystems: the electrical subsystem, the stator, the interface mechanism and the moving platform. Each subsystem is described as the following. 2.3.1 The electrical subsystem A simplified driving circuit for the single mode piezoelectric plate actuator is shown in Fig. 4. The electrical excitation of actuator vibrations is applied by a voltage source resonant converter [31]. This resonant tank is formed by piezoelectric elements and inductors in series. The dynamic equation of the resonant converter can be described as follows [3]: diL = VINV − Rc iL −Vp , dt dVp (Ca +Cp ) = iL − imech , dt
Lc
(1) (2)
where Lc and Rc denote the inductance and the resistance of the resonant tank, respectively. Cp and Ca denote the capacitance of the piezoelectric elements and of the added capacitance, respectively. VINV and Vp are the supplied voltage and the voltage across the piezoelectric elements, respectively. iL is the current flows from the source and imech denotes the motion current that is directly coupled to
Fig. 2. System configuration.
Fig. 3. Experiment apparatus.
Fig. 4. Resonant converter. the dynamics of the stator. u denotes the control signal from the personal computer. 2.3.2 The stator The stator is excited by the voltage applied to one pair of electrodes with a single driving frequency. Basing on the inverse piezoelectric effect, the electrical energy is transferred to vibration of piezoelectric plate (stator). According to [8], the mode shape of both oscillations in X and Z direction of the stator is described as the following: ) ( ) 3π ( ( π ) Ux (x, z,t) = −A sin x cos z − 1 sin (ω t), L H ( ( )) ( ) 3π π Uz (x, z,t) = B 1 +C cos x sin z sin (ω t), L H where Ux and Uz are the displacement in the X and Z (3) direction; A, B, C are material and geometrical amplitude functions; L, H are the length and height of piezo plate in the X and Z direction.
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Fig. 5. The static and dynamic deformation of the stator.
2.3.3 The contact mechanism When the piezoelectric plate is excited asymmetrically, the standing wave is shift causing the driving tip with the largest Z displacement and also has an X displacement, that forms the contact mechanism between the driving tip and the moving platform and provide motion for the moving platform. The contact mechanism between driving tip and moving platform is modeled as an elastic spring as shown in Fig. 5. There are two resultant force act on the contact: a normal force and a tangential force. The normal force acting on the interface is calculated as the following: { Fp + ks (σd − σst ) if Fp + ks (σd − σst ) > 0, FN = 0 otherwise, (4)
between driving tip and moving platform is divided into two states: the contact state when FN > 0 and the separating state when FN = 0. In contact state, there are two situations of contact: the driving tip and slider moving with the same velocity (vrel = 0), called the stick phase, or the two objects slide on the surface of each other (vrel = ̸ 0), called the slip phase. As a result, there are two types of friction force occurred in the interface, static friction force Fs with respect to the stick phase and kinetic friction force Fk with respect to the slip phase. From Coulomb friction model, the tangential force acting on the contact is calculated as: FN > 0 and vrel = 0, Fs ∈ [−µs FN , µs FN ] FT = Fk = sign (vrel ) µd FN FN > 0 and vrel ̸= 0, 0, (8)
where Fp denotes the preload force, ks the elastic coefficient of the spring, σst the static elongation of the stator under the preload only, σd the total dynamic longitudinal elongation of the stator when the voltage is applied. The static and dynamic elongation are calculated as follows: ( ) ( ) L L σd = Uz , H,t −Uz , 0,t , (5) 2 2 ηN Fp σst = , (6) Kst
where µs and µd are the static and dynamic friction coefficients of the contact, respectively.
where ηN and Kst are the load coefficient and stiffness of the stator. If the condition Fp + ks (σd − σst ) ≥ 0 happens, the driving tip will separate from the moving platform and the contact normal force is equal to zero. Conversely, the tangential force which resulting from the dynamic friction force at the contact is related to the relative velocity of driving tip and slider in the longitudinal direction. The relative velocity is defined as follows: vrel = vtip − v p f ,
(7)
where vtip and v p f are the tangential velocities of the driving tip and the moving platform (slider), respectively. Basing on the value of the normal force, the contact situation
2.3.4 Dynamic of the moving platform The equation of motion of transitional moving platform can be expressed as follows: Mp f x¨ p f +Cp f x˙ p f = FT + Ff ,
(9)
where Mp f , Cp f , x p f denote the mass, the viscous damping coefficient and the position of the moving platform, respectively. Ff denotes the friction force occurring in the linear guide of the moving platform. The friction force Ff is effected by the gravity force of the moving platform. Moreover, because the normal force pushes the platform against the linear guide, then the friction force also should include the effect of normal force FN . As in [32], the friction force existing in the contact between moving platform and linear guide is calculated as the following: { −FT if |FT | < |Ff max |, Ff = (10) Ff max otherwise, Ff max = sign (−x˙ p f ) µlg (FN + Mp f g) ,
(11)
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sliding mode control is presented to achieve the excellent position tracking performance and compensate the nonlinearities and hysteresis phenomenon of the piezoelectric positioning system. 3. CONTROL ALGORITHM 3.1. Fast nonsingular terminal sliding mode controller A simpler expression of the aforementioned dynamic equation (12) included disturbance can be represented as the following: M¯ p f x¨ p f = ζ (x˙ p f , x p f , u) + u + d, Fig. 6. System response with sinusoidal voltage signal.
(13)
where M¯ p f = Mp f /α ,ζ (x˙ p f , x p f , u) = α −1 (Ff + h(x p f , x˙ p f , u) − Cp f x˙ p f ) denotes all the nonlinearities of the piezo-actuated system including the hysteresis effect and friction behavior of the linear guide, d the unexpected disturbance. As in [33], the conventional terminal sliding surface is described by the following first order terminal sliding variable: s = e˙ + β |e|q/p sign (e) ,
(14)
where β > 0 is a design constant, p and q are positive odd integers and p > q, e = x p f d − x p f which is the error between the desired position (x p f d ) and actual response (x p f ). The sufficient condition for the existence of terminal sliding mode is chosen as: Fig. 7. Hysteresis curve of the experimental piezoelectric positioning stage at frequency 0.25 Hz.
where Ff max denotes the maximum value of friction force between moving platform and linear guide, µlg the friction coefficient of the linear guide, g the gravitational acceleration. It is noted that, if the tangential force is not able to overcome the friction force, that means |FT | < |Ff max |, the platform remains no motion. In this case, the friction force Ff is set to be the −FT . Therefore, this characteristic makes the ultrasonic motor generate a deadzone area. The experimental data with the displacement response of the moving platform with respect to a sinusoidal input voltage signal at 0.25 Hz is shown in Fig. 6. It can be seen that the system has large time delay, large deadzone and uncertain parameters. Also, from the input-output relationship, the hysteresis curve of the system is shown in Fig. 7. The dynamic equation of the moving platform included hysteresis effects can be expressed as the following: Mp f x¨ p f +Cp f x˙ p f + d = Ff + h (x p f , x˙ p f , u) + α u, (12) where h(x p f , x˙ p f , u) denotes the hysteresis nonlinear term, α the voltage to force coefficient of the piezo-actuated system. In the next section, a novel fast nonsingular terminal
1 d ( 2) s < −η |s| . (15) 2 dt Also, the work [33] showed that, the system states will converge to zero in the finite time tr , which meets the following condition: tr
0 denotes the learning rate. Then the stability of the system can be guaranteed. Proof: Let W˜ = W ∗ − Wˆ be the estimation error of weight vector W . Consider a Lyapunov function candidate as follows: 1 1 V2 (t) = M¯ p f sT s + W˜ T W˜ . 2 2γ
)] q 1− qp v ˆ ¯ −Mp f β |e| ˙ sign (e) ˙ (λ1 s + λ2 |s| ) p ( ( ) 1 p qp −1 q 2− qp ¯ |e| − ˙ ∆Mp f |x¨ p f d | + β |e| ˙ βq p ⌢ ˙ 1 +D + εN ) |s| − W˜ T W γ = − Mˆ¯ p f sT (λ1 s + λ2 |s|v ) ) ( )( 1 p p −1 T ˆ q ¯ ¯ |e| + s Mp f −Mp f e˙ + ˙ sign (e) ˙ x¨ p f d βq ) ( 1 p qp −1 |e| |x¨ p f d | ˙ − |s|T ∆M¯ p f |e| ˙+ βq 1 p qp −1 |e| ˙ − sT sign (e) ˙ (ε − ε0 + d) βq 1 p qp −1 |e| − |s|T ˙ (D + εN ) βq ( ) ⌢ 1 p qp −1 T T 1 ˙ ˜ |e| −W ˙ sign (e) ˙ s Φ . (43) W+ γ βq Substituting the update law (40) into (43), we have
and the update law for weight vector is chosen by: 1 p qp −1 |e| ˙ sign (e) ˙ , W˙ˆ = −γ sT Φ βq
+ uRBF − d
(41)
Calculating the time derivative of V (t), we have ⌢ ˙ 1 V˙2 (t) =M¯ p f sT s˙ − W˜ T W γ [ 1 p qp −1 |e| =sT M¯ p f e˙ + ˙ sign (e) ˙ (M¯ p f x¨ p f d βq ] ⌢ ˙ 1 −ζ (x˙ p f , x p f , u) − u − d) − W˜ T W (42) γ
By substituting the formula of control signal u in (37)(39) and the definition from (35), (36) into (42), one can obtain [ 1 p qp −1 |e| V˙2 (t) =sT M¯ p f e˙ + ˙ sign (e) ˙ (M¯ p f x¨ p f d βq
V˙2 (t) ≤ −Mˆ¯ p f sT (λ1 s + λ2 |s|v ) .
(44)
Inequation (44) implies that both s and W˜ are bounded. Meanwhile, considering (17), it can be seen that both e and e˙ are bounded. Moreover, due to the boundedness of x p f d and x˙ p f d , then x p f and x˙ p f are bounded; hence, s˙ is bounded. Therefore, all signals of the closed loop system are bounded or the stability of the closed loop system has been proved. □ Remark 8: It is noted that the control signal given in (37)-(39) can guarantee the robust stability of the closed loop system even the value of the term M p f can not be accurately defined and only the bounded values are needed. Remark 9: An adaptive approximator is presented in the proposed control approach and the prior knowledge of upper bound of the uncertainties in the system is not required in the proposed control design. Remark 10: The adaptive estimator using RBFNN helps reducing the magnitude of discontinuous control signal, hence the chattering phenomenon is reduced in the control signal. 5. EXPERIMENTAL RESULTS To evaluate the effectiveness of the proposed RBF based fast nonsingular terminal sliding mode control, a set of three tests was investigated and the control performances of the designed controller were compared to the results of FNTSMC and conventional PID controller. There are three different reference signals applied in the
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Fig. 9. The proposed RBF based fast nonsingular terminal sliding mode control.
Table 1. Control parameters of FNTSMC and the proposed controller. FNTSMC
RBF based FNTSMC Sliding manifold parameters: p = 3, q = 5, β = 3, λ1 = 10, ν = 0.6, δ = 0.05 λ2 = 6 λ2 = 2 Number of hidden nodes: n = 11 Gaussian variance: bi = 0.5 (i = 1, 2, . . . , 11) Center matrix of hidden layer: −1 −1 −1 −0.8 −0.8 −0.8 −0.6 −0.6 −0.6 C= ... ... ... 0.8 0.8 0.8 1 1 1 11×3 Initial output weight vector of RBF: W = [ω0 , ω1 , ω2 , ..., ω11 ] =0.1 ∗ ones(12, 1) Scaling factors: sc1 = 0.03, sc2 = 0.15, sc3 = 6 Updating factor: γ = 0.01
experimental process: sinusoidal, multi-sin, and multistep signals. The structure of the proposed controller applied for the piezoelectric positioning stage system is described in Fig. 9. The designed parameters for FNTSMC algorithm and RBF based FNTSMC algorithm are shown in Table 1, while the setting parameters of PID controller were selected basing on the trial and error methodology in the real system. To reduce chattering in the control signal, the sign function in (32) was replaced by a pseudo-sliding function: sign(s) ≈ ps(s) =
s , |s| + δ
(45)
where δ is a small positive scalar. The bounds on the uncertain parameter and unexpected disturbances are defined
Fig. 10. Position control with respect to a 0.25 Hz sinusoidal reference signal and its enlargement. as follows: ( ) ( ) 3.75 V s2 /m ≤ M¯ p f ≤ 7.5 V s2 /m , D = 0.01 (N) . First, a sinusoidal reference signal with frequency of 0.25 Hz was chosen to investigate the tracking performance of the designed controller, PID controller and FNTSMC controller. Incidentally, the three parameters of PID controller were obtained with their values of Kp = 19, Ki = 0.1 and Kd = 0.02. Fig. 10 shows a comparison of control performances of PID controller, FNTSMC and the proposed controller with respect to the 0.25 Hz sinusoidal signal, the magnified position performances for the time period from 0.5 to 1.5 second and the error comparison. As can be seen, the proposed controller gave a high tracking performance and the position error values mostly lied within ±0.01 (mm), while the FNTSMC gave larger errors with the bound values of ±0.04 (mm). PID controller provided the worst results, with the maximum error of ±0.1 (mm). The control signals of FNTSMC and the proposed controller are shown in Fig. 11. It can be shown that, the control chattering was almost eliminated when using the RBF estimation. Comparing the control performance and control signal with reference to frequencies of 0.5 Hz and 1 Hz are seen in Figs. 13-14 and Figs. 16-17, respectively. As can be seen, the bound of errors of PID controller increased from 0.1 mm to 0.15 mm with frequency of 0.5 Hz and 0.2 mm with frequency of 1 Hz. The FNTSMC gave a better robust performance with the bound of errors changed from 0.04 mm to 0.05 mm and 0.06 mm with the frequency of
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Fig. 11. Control signal of FNTSMC and the proposed controller with respect to 0.25 Hz sinusoidal reference.
Fig. 12. Hysteresis curve with 0.25 Hz sinusoidal reference signal after implementing the proposed controller.
0.5 Hz and 1 Hz, respectively. The proposed controller provided the best performance with high tracking accuracy and robust performance as frequency become faster, with the errors kept within ±0.01 mm. In order to investigate the robustness of the RBF based FNTSMC against hysteresis effect of the piezoelectric actuator, hysteresis curves of the closed loop system with respect to 0.25 Hz, 0.5 Hz and 1 Hz sinusoidal reference signal after employing the proposed controller are illustrated in Figs. 12, 15 and 18, respectively. It is shown that the hysteresis effect was almost compensated at both low and high frequencies. Second, a multi-step reference signal was utilized to
Fig. 13. Position control with respect to a 0.5 Hz sinusoidal reference signal and its enlargement.
Fig. 14. Control signal of FNTSMC and the proposed controller with respect to 0.5 Hz sinusoidal reference.
verify the control performance of the tracking irregular commands. The comparison between conventional PID controller, FNTSMC and the proposed controller are depicted in Fig. 19. The enlargement of control performances for the time period between 4 and 4.2 second, and the time period between 6 and 6.2 second are shown in Fig. 20 while the control signals of FNTSMC and the proposed controller are represented in Fig. 21. From the experimental results, it can be seen that the designed controller provided higher tracking performances than those of the conventional PID controller and FNTSMC. At
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Fig. 15. Hysteresis curve with 0.5 Hz sinusoidal reference signal after implementing the proposed controller.
Fig. 17. Control signal of FNTSMC and the proposed controller with respect to 1 Hz sinusoidal reference.
Fig. 18. Hysteresis curve with 1 Hz sinusoidal reference signal after implementing the proposed controller.
Fig. 16. Position control with respect to a 1 Hz sinusoidal reference signal and its enlargement.
steady states, the position tracking errors of the conventional PID controller and FNTSMC were about ±0.02 mm and ±2E−3 mm, respectively, while those of the design controller was less than ±5E−4 mm. In addition, the proposed controllergave the fast response with the average rise time is about 0.03 second, in comparison to 0.04 second of FNTSMC and 0.06 second of the conventional PID controller. Finally, a multi-frequency sinusoidal reference signal is employed as the desired trajectory to investigate the efficacy of the designed controller in difference working frequencies and amplitudes. As shown in Fig. 22, the proposed controller has ability to smoothly track the de-
sired trajectory with high and robust tracking accuracy. The position tracking error of the proposed controller was bounded within ±0.02 mm, while those of FNTSMC was within ±0.06 mm and those of the conventional PID controller was mostly within ±0.3 mm. By the RBF approximation, the control signals in Fig. 23 show the effectiveness of the proposed controller in attenuating the control chatters of the sliding mode controller. 6.
CONCLUSION
In this brief, a robust position control using the combination of FNTSMC and RBF estimation was developed and successfully applied to a nonlinear piezoelectric positioning stage with uncertainties. A dynamic modeling of piezoelectric positioning stage included hysteresis effect was carried out. To deal with nonlinearities and external disturbances in the system, a fast-nonsingular terminal
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Fig. 19. Position control with respect to a multi-step reference signal.
Fig. 21. Control signal of FNTSMC and the proposed controller with respect to multi-step reference.
Fig. 22. Position control with respect to a multi-frequency sinusoidal reference signal. Fig. 20. (a) Magnified position response for t = 4-4.2 (s) and (b) t = 6-6.2 (s). REFERENCES
sliding mode surface and controller were designed, while a RBF based estimator was utilized to compensate the uncertainties of piezoelectric stage. The control structure and the adaptive law of RBF weight vector were developed basing on a special Lyapunov function. Experiment results show that the proposed controller could obtain the high tracking accuracy and fast response with respect to many types of reference signal and the hysteresis effect of piezoelectric actuator and deadzone caused by friction of linear guide were almost compensated. The proposed control method is effective not only for piezoelectric positioning stage but also for other systems that exhibit deadzone and hysteresis phenomenon.
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Kyoung Kwan Ahn received his B.S. degree from the Department of Mechanical Engineering at Seoul National University, Seoul, Korea, in 1990, an M.S. degree in Mechanical Engineering from the Korea Advanced Institute of Science and Technology (KAIST) in 1992, and a Ph.D. degree with the thesis entitled “A study on the automation of out-door tasks using a two-link electro-hydraulic manipulator” from the Tokyo Institute of Technology in 1999. He is currently a professor in the School of Mechanical and Automotive Engineering, University of Ulsan, Ulsan, Korea. His research interests are the design and control of smart actuators using smart materials, fluid power control, rehabilitation robots, and active damping controls. He is a member of IEEE, ASME, SICE, RSJ, JSME, KSME, KSPE, KSAE, KFPS, and JFPS. To Xuan Dinh received his B.S. degree from the Department of Mechanical Engineering at Le Quy Don Technical University, Hanoi, Vietnam, in 2012. He is currently a Ph.D. candidate in the School of Mechanical and Automotive Engineering, University of Ulsan, Ulsan, Korea. His research interests include smart actuators/materials, robotics, and nonlinear
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