Adaptive Displacement Control With Hysteresis ... - Semantic Scholar
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 3, JUNE 2006
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Adaptive Displacement Control With Hysteresis Modeling for Piezoactuated Positioning Mechanism Hsin-Jang Shieh, Member, IEEE, Faa-Jeng Lin, Senior Member, IEEE, Po-Kai Huang, and Li-Tao Teng
Abstract—An adaptive displacement control with hysteresis modeling for a piezoactuated positioning mechanism is proposed in this paper because the dynamic performance of piezosystems is often severely deteriorated due to the hysteresis effect of piezoelectric elements. First, a new mathematical model based on the differential equation of a motion system with a parameterized hysteretic friction function is proposed to represent the dynamics of motion of the piezopositioning mechanism. As a result, the mathematical model describes a motion system with hysteresis behavior due to the hysteretic friction. Then, by using the developed mathematical model, the adaptive displacement tracking control with the adaptation algorithms of the parameterized hysteretic function and of an uncertain parameter is proposed. By using the proposed control approach on the displacement control of the piezopositioning mechanism, the advantages of the asymptotical stability in displacement tracking, high-performance displacement response, and robustness to the variations of system parameters and disturbance load can be provided. Finally, experimental results are illustrated to validate the proposed control approach for practical applications. Index Terms—Adaptive displacement control, hysteresis effect, hysteretic friction function, piezoactuated positioning mechanism, robustness.
I. I NTRODUCTION
D
UE TO THE requirements of the nanometer resolution in displacement, high stiffness, and fast response, the piezoactuators are often used in high-precision positioning applications, such as scanning probe microscopy [1]. Because the materials of the piezoactuators are ferroelectric, they fundamentally exhibit hysteresis behavior in their response to an applied electric field. This leads to problems of severe inaccuracy, instability, and restricted system performance due to hysteresis nonlinearity if the piezoactuator is operated in an open-loop fashion [2]–[6]. Moreover, the hysteresis characteristics are usually unknown, and the states of representing to hysteresis dynamics are often unmeasured. These usually cause the increasing difficulties in tracking control design with high-performance requirements for piezoactuated mechanisms [2]–[4]. In general, when systems have hysteresis effect in dynamic responses, an established model or function that can accurately represent the hysteresis feature becomes very important for the analysis and control of systems [7]–[17]. Therefore, the Preisach-function-based Preisach models were proposed in Manuscript received October 10, 2003; revised July 29, 2005. Abstract published on the Internet March 18, 2006. The authors are with the Department of Electrical Engineering, National Dong Hwa University, Hualien 97401, Taiwan, R.O.C. (e-mail: hjshieh@ mail.ndhu.edu.tw). Digital Object Identifier 10.1109/TIE.2006.874264
[7]–[12] to represent the hysteresis dynamics. In these studies, the hysteresis dynamics could be approximately described by the Preisach function, but the derived equations and the employed operations were complicated [7]–[10]. This complicated operation often leads to a problem of spending much time on computation during the control process. Moreover, a fast Preisach-based magnetization model that established several characteristics of the hysteresis and its inverse was proposed in [11]. In [11], the equations derived for system control process were simplified by constructing some specified Preisach function in advance. However, some specified conditions needed in the simplification procedure, which was for the derivation of the dynamics of the hysteresis and its inverse, were required. These conditions usually cause a more restricted control design. In [13] and [14], the hysteresis model was approximated by using the motion dynamics constructed by an applied force to one set of the massless bodies paralleling to springs, and moreover, relationship in terms of the applied force, spring constants, and break forces was used to determine the hysteresis dynamics. Unfortunately, from the experimental results provided by [13] and [14], the critical numbers of the employed springs and massless bodies for accurately representing the hysteresis dynamics were very difficult to determine. Furthermore, it seems that the numbers of the constructed springs and massless bodies could be obtained only by using computer simulations or practical experiments. Moreover, an electromechanical model constructed by transduction of the electric charge and discharge behaviors to generate the applied force was proposed in [15]. Although the model composed of a first-order differential equation and a partial differential equation was employed for both shaping the hysteresis loop and describing the mechanical model of piezoactuator, this model was apparently more suitable for vibration control than for displacement control. Similarly, developments in [16] and [17] were mainly focused on the vibration control of piezoactuators. Because compensation for the effect of a linear or nonlinear friction on control systems becomes very important for highperformance control design, studies on compensation techniques with system modeling for various friction phenomena have been published in [18]–[27]. In these studies, the developed friction model that includes the characteristics of the Stribeck effect, hysteresis, and springlike behaviors was assembled in mechanical motion dynamics. Therefore, the hysteretic friction model developed in [18] is extended to represent the dynamics of motion of the piezoactuated positioning mechanism, which has the hysteresis behavior in dynamic response. First, a new mathematical model constructed by a differential equation of a motion system with the addition of a hysteretic
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friction function is proposed to represent the dynamics of motion of the piezopositioning mechanism. As a result, the mathematical model can be viewed as a motion system with hysteresis behavior due to the hysteretic friction. Then, for the purpose of the controller design, we parameterize the hysteretic friction function with respect to the defined state variables, the system output displacement, and velocity. Moreover, an external disturbance load is also considered in the developed mathematical model. With the use of the motion equation, hysteretic friction, and external disturbance load, the dynamics of motion of the piezoactuated positioning mechanism studied in this paper can be completely established. Furthermore, according to the developed model, an adaptive displacement tracking control with the parameter adaptations of the parameterized hysteretic function and of an uncertain parameter is proposed. In this control system, both the asymptotical stability in displacement tracking and the boundedness of the parameter adaptations can be guaranteed. In addition, the robustness to external disturbance load can be obtained because the value of the uncertain disturbance load is adapted by the proposed adaptation laws as well as the parameters of the hysteretic friction function. By using the proposed control approach, the displacement tracking response of the piezoactuated positioning mechanism has the following advantages. 1) The dynamics of the hysteresis present in the piezoactuated positioning mechanism can be completely modeled by the proposed hysteretic friction function. 2) The asymptotic stability in displacement tracking control and the boundedness of the parameters adaptation can be provided. 3) The high-performance displacement control can be achieved. 4) The robustness to the variations of system parameters and external disturbance load can be obtained. Finally, experimental results are illustrated to validate the proposed control approach for practical applications.
shown in [13], [15], and [16]. With the aforementioned reasons, the dynamics of motion of the piezoactuated positioning mechanism with hysteresis behavior can be further developed in this section. The friction model with hysteresis effect named the LuGre model and proposed in [18] can be represented by
II. D YNAMICS OF P IEZOACTUATED P OSITIONING M ECHANISM
According to [18]–[20], the complete dynamics of motion of the one-dimensional piezoactuated positioning mechanism can be equivalently expressed as
In general, there exist two difficulties in modeling the hysteresis nonlinearity of piezoactuators, namely: 1) nonlocal memory phenomenon and 2) asymmetric loop between descending and ascending paths [3], [18]. Therefore, developing a dynamic model to describe the hysteresis behavior becomes an important subject for improving the control performance of piezoactuated positioning mechanisms, as shown in [7]–[9] and [13]–[17]. However, as mentioned in the previous section, there still exist many drawbacks in the past developments for hysteresis compensation. Therefore, we intentionally develop a hysteretic friction function to represent the hysteresis behavior of the piezoactuated positioning mechanism. The reasons that the hysteretic friction function is chosen as a model of the hysteresis in this piezomechanism can be summarized as follows. 1) The qualitative mechanisms [18] of friction have been fairly well understood in modern time. 2) The friction model with hysteresis behavior has been completely developed and validated in [18]–[27]. 3) The hysteresis and piezoelectric effect of piezoactuators can be separately considered in system control design, as
|x| ˙ dz = x˙ − z dt h(x) ˙ dz + σ2 x˙ FH = σ 0 z + σ 1 dt
(1) (2)
where FH denotes the hysteretic friction function; z denotes the internal friction state that represents the average deflection of bristles with contact force applied; x˙ denotes the relative velocity between the two contact surfaces; σ0 , σ1 , and σ2 are positive constants, which are typically unknown and generally difficult to identify, and can be equivalently interpreted as bristle stiffness, bristle damping, and viscous coefficient, respectively [18]; and the function h(x) ˙ denotes the Stribeck effect curve given by 2 − x˙ ˙ = fC + (fS − fC )e x˙ S (3) σ0 h(x) where fc is the Coulomb friction level, fs is the level of the stiction force, and x˙ s is the Stribeck velocity. Moreover, from (1), it can be found that the boundedness of the internal friction ˙ state z and x˙ implies the boundedness of the function h(x). Furthermore, the feasibility of the hysteretic friction model shown in (1)–(3) for a rigid-body system had been confirmed by the results of the theoretical analyses, simulations, and experiments in [18]–[20]. Therefore, by substituting (1) into (2), the hysteretic function shown in (2) can be rewritten as FH = σ 0 z − σ 1
1 z|x| ˙ + (σ1 + σ2 )x. ˙ h(x) ˙
m¨ x + FH + F L = K E u
(4)
(5)
where m denotes the effective mass of the piezoactuated positioning mechanism, x denotes the displacement of the piezopositioning mechanism, x ¨ denotes the velocity, KE denotes the voltage-to-force coefficient of the piezoactuator, u indicates the applied voltage to the piezoactuator, FH denotes the hysteretic friction function, and FL is the external disturbance load. From (5), it can be found that the dynamics of the system hysteresis caused by the hysteretic friction seems to be an individual component of the control system. Therefore, the model-based controller design approach can be used if systems with a nonlinear hysteresis behavior can be represented by (5). Moreover, by substituting (4) into (5), the following dynamics can be obtained: 1 1 KE u− z|x| ˙ + (σ1 + σ2 )x˙ . (σ0 z + FL ) − σ1 x ¨= m m h(x) ˙ (6)
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Fig. 1. Block diagram of the proposed dynamic model for the piezoactuated positioning mechanism.
Furthermore, by defining the state variables x = x1 and x˙ 1 = x2 , we can rewrite (6) as follows: x˙ 1 = x2
1 1 x˙ 2 = K E u− z|x2 |+(σ1 +σ2 )x2 (σ0 z+FL ) − σ1 m h(x2 ) (7)
where K E = KE /m. From (7), it can be found that the dynamics of motion of the piezoactuated positioning mechanism is nonlinear with respect to the defined state variables, and the corresponding block diagram about (7) can be shown in Fig. 1. To validate the feasibility of the dynamic model shown in (7) of describing the dynamics of motion of the piezopositioning mechanism, the simulation results are illustrated in Fig. 2, in which a sinusoidal input voltage with an amplitude of 5 V and a frequency of 1 Hz is applied to (7), and the values of the system parameters are given as follows: σ0 = 105 N/m, √ σ1 = 105 Ns/m, σ2 = 0.4 Ns/m, fc = 1 N, fs = 1.5 N, xs = 0.001 m/s, m = 1 kg, and KE = 1 N/V. In Fig. 2, we can observe that using the developed dynamic model (7) with hysteresis behavior to represent the dynamics of motion of the piezoactuated positioning mechanism can be confirmed.
Fig. 2. Simulation result of the hysteresis response from the proposed mathematical model.
the piezoactuated positioning mechanism can be reexpressed as follows: F H = σ0 h(x2 )
x2 + σ2 x2 + σ3 x1 |x2 |
(9)
where σ3 x1 can be viewed as an appended term due to piezoelectric effect. Moreover, by replacing (4) with (9), the dynamics of motion of the piezoactuated positioning mechanism can be derived as follows: x˙ 1 = x2
III. A DAPTIVE D ISPLACEMENT C ONTROL To achieve high-performance displacement response from the piezoactuated positioning mechanism, a straightforward technique to compensate the hysteresis effect is to adapt the unknown hysteresis function or component. Therefore, the adaptive displacement tracking control with hysteresis compensation is proposed in this section. Here, we assume that the internal state z and the function h(x2 ) are bounded and piecewise continuous and the change in average deflection of z is small enough to be ignored. With the definition of the steady-state z s , the equation of z in steady state can be derived as follows: x2 h(x2 ). zs = (8) |x2 | Therefore, in steady state, the hysteretic friction function (4) mainly causing the hysteresis effect in motion dynamics of
x˙ 2 = K E u − [θ0 sgn(x2 ) + θ1 x1 + θ2 x2 + θ3 ]
(10)
where sgn(·) denotes a sign function defined as x2 sgn(x2 ) = = |x2 |
+1 0 −1
as x2 > 0 as x2 = 0 as x2 < 0
and θ0 , θ1 , θ2 , and θ3 denote the unknown and bounded parameters defined by θ0 = σ0 h(x2 )/m, θ1 = σ3 /m, θ2 = σ2 /m, and θ3 = FL /m. Therefore, the hysteretic friction function (9) is equivalently parameterized by θ0 , θ1 , θ2 , and θ3 with respect to the system states x1 and x2 . For the tracking controller design, the following states of the tracking errors are defined as z1 = x1 − xm
(11)
z2 = x2 + k1 z1 − x˙ m
(12)
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Fig. 3. System configuration of the controlled piezoactuated positioning mechanism by using (a) the conventional PI controller and (b) the proposed adaptive controller.
where k1 is a properly positive constant and xm and x˙ m denote the reference displacement and its derivative, respectively. Taking the time derivatives of (11) and (12) yields z˙1 = − k1 z1 + z2 z˙2 = K E u − [θ0 sgn(x2 ) + θ1 x1 + θ2 x2 + θ3 ] − k12 z1 + k1 z2 − x ¨m .
(13)
From (13), it can be observed that the values of the system parameters θ0 , θ1 , θ2 , θ3 , and K E must be exactly known for model-based control design with high-performance requirement. However, these values of the parameters are initially unknown and difficult to identify because they are associated with the unfamiliar hysteresis behavior of the piezopositioning mechanism. Therefore, in order to achieve high-performance tracking control, the dynamics (13) is purposely rewritten as follows: z˙1 = − k1 z1 + z2 ˆ u − θˆ sgn(x ) + θˆ x + θˆ x + θˆ + K E u z˙2 = K E 0 2 1 1 2 2 3 − θ0 sgn(x2 )+ θ1 x1 + θ2 x2 + θ3 − k 2 z1 + k1 z2 − x ¨m 1
(14) ˆ , θ = θ − θˆ , θ = θ − θˆ , θ = = KE − K where K E 0 0 0 1 1 1 2 ˆ θ2 − θ2 , and θ3 = θ3 − θˆ3 denote the estimation errors and ˆ , θˆ , θˆ , θˆ , and θˆ denote the estimates of the unknown K E 0 1 2 3 parameters K E , θ0 , θ1 , θ2 , and θ3 , respectively.
According to the dynamics (14), the adaptive displacement tracking control with parameter adaptations is proposed as follows: 1 2 k1 z1 − k1 z2 + x ¨m − z1 − k2 z2 u= ˆ K E + θˆ0 sgn(x2 ) + θˆ1 x1 + θˆ2 x2 + θˆ3 (15) ˆ˙ = β z u K (16) E k 2 ˙ ˙ ˆ ˆ θ0 = − β0 sgn(x2 )z2 , θ1 = −β1 z2 x1 , ˙ ˙ (17) θˆ2 = − β2 z2 x2 , θˆ3 = −β3 z2 where k2 is a positive constant and βk , β0 , β1 , β2 , and β3 denote the adaptation gains. Theorem 1: The control system shown in (13) can be asymptotically stabilized by using the proposed tracking control shown in (15)–(17), where the parameters adaptations given in (16) and (17) are ultimately bounded. Proof: By substituting (15) into (14), (14) can be rewritten as follows: z˙1 = − k1 z1 + z2 E u − θ0 sgn(x2 )+ θ1 x1 + θ2 x2 + θ3 . z˙2 = − z1 − k2 z2 + K (18) According to (18), the following Lyapunov function candidate is chosen: V =
1 2 1 2 1 z + z + 2 1 2 2 2
1 2 1 1 1 1 KE + θ02 + θ12 + θ22 + θ32 . (19) × βk β0 β1 β2 β3
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Fig. 6. Flowcharts of the PC-based system control programs.
Fig. 4. (a) Machine structure and (b) implementation of the PC-based control design for the displacement control of the piezoactuated positioning mechanism.
Taking the time derivative of (19) yields ˙ E + z2 u E 1 K V˙ = − k1 z12 − k2 z22 + K βk 1 ˙ 1 ˙ + θ0 θ0 − sgn(x2 )z2 + θ1 θ1 − z2 x1 β0 β1 1 ˙ 1 ˙ (20) + θ2 θ2 − z2 x2 + θ3 θ 3 − z2 . β2 β3 Moreover, let ˙ E K ˙ θ0 ˙ θ1 ˙ θ2 ˙ θ3
= − βk z2 u = β0 sgn(x2 )z2 = β1 z2 x1 = β2 z2 x2 = β3 z2 .
(21)
Thus, the following inequality can be obtained by substituting (21) into (20): V˙ = −k1 z12 − k2 z22 ≤ 0.
Fig. 5. Experimental result of the hysteresis response from the piezoactuated positioning mechanism.
(22)
From the Lyapunov stability theory of view, the controlled states z1 and z2 will converge to zero asymptotically, and θ0 , θ1 , θ2 , and θ3 are ultimately the estimation errors K, bounded [28] because V˙ (t) → 0 as t → ∞. Furthermore, the following conditions are assumed. 1) The system parameter K E is unknown but constant. 2) The designated parameters θ0 , θ1 , θ2 , and θ3 are piecewise continuous. 3) The responses of the ˆ , θˆ , θˆ , θˆ , and θˆ are purposely parameter adaptations of K E 0 1 2 3
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Fig. 7. Square displacement response under the conventional PI control with (a) displacement = 0−50 µm and frequency = 0.5 Hz and (b) its control effort, (c) displacement = 0−50 µm and frequency = 1.0 Hz and (d) its control effort, (e) displacement = 0−10 µm and frequency = 0.5 Hz and (f) its control effort, and (g) displacement = 0−10 µm and frequency = 1.0 Hz and (h) its control effort.
designated to be faster than that of the system parameters K E , θ0 , θ1 , θ2 , and θ3 ; that is, these values of the system parameters can be equivalently viewed as piecewise constants during the ˆ , θˆ , θˆ , θˆ , and θˆ . Therefore, the adaptation transient of K E 0 1 2 3 following adaptation laws can be derived: ˙ E K ˙ θ0 ˙ θ1 ˙ θ2 ˙ θ3
= = = = =
ˆ˙ −K E ˙ − θˆ0 ˙ − θˆ1 ˙ − θˆ2 ˙ − θˆ3 .
(23)
Consequently, the asymptotical stability in displacement tracking by using (15)–(17) can be confirmed. Remark 1: The asymptotical stability in displacement tracking of the piezoactuated positioning mechanism represented by (5) can be achieved by means of the proposed control approach given by (15)–(17), which is based on the dynamics (10) with the parameterized hysteresis behavior and the lumped disturbance load. Remark 2: Robustness against the variation of both the system parameter K E and external load FL can be provided because the adaptation of these two parameters are practically considered in the proposed control design.
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Fig. 8. Square displacement response under the proposed adaptive control with (a) displacement = 0−50 µm and frequency = 0.5 Hz and (b) its control effort, (c) displacement = 0−50 µm and frequency = 1.0 Hz and (d) its control effort, (e) displacement = 0−10 µm and frequency = 0.5 Hz and (f) its control effort, and (g) displacement = 0−10 µm and frequency = 1.0 Hz and (h) its control effort.
IV. E XPERIMENTAL R ESULTS A. System Configuration To investigate the effectiveness of the proposed control approach, the personal-computer (PC)-based control system is implemented, and the displacement tracking responses of the piezoactuated positioning mechanism are illustrated. The block diagram of the system implementation using the proportionalintegral (PI) controller and the proposed adaptive control approach, respectively, is shown in Fig. 3(a) and (b). The machine structure of the one-dimensional piezoactuated positioning mechanism, which is manufactured by Piezosystem
Jena Inc., Model PX300, and the corresponding implementation of the PC-based control system are shown in Fig. 4(a) and (b), respectively. In Fig. 4(a), the mover is actuated by piezoactuators inside the mechanism, and a capacitive displacement sensor is preinstalled in this mechanism. The rated values of this experimental piezomechanism are given in the following list: motion distance = 0−240 µm; maximum external load = 10 N; input voltage = 0−150 V; stiffness = 0.08 N/µm; repeatability = 0.03%;
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Fig. 9. Sinusoidal displacement response under the proposed adaptive control with (a) displacement = ±50 µm and frequency = 0.5 Hz and (b) its control effort, (c) displacement = ±50 µm and frequency = 1.0 Hz and (d) its control effort, (e) displacement = ±10 µm and frequency = 0.5 Hz and (f) its control effort, and (g) displacement = ±10 µm and frequency = 1.0 Hz and (h) its control effort.
resonant frequency = 170 Hz; capacitance = 2300 nF. In this control system, the control voltage is outputted by a digital-to-analog (D/A) conversion interface and then amplified through a specified amplifier with around 10 times gains to drive the piezoactuators inside the mechanism. The output displacement signal detected by a capacitive displacement sensor with a resolution of 41.67 mV/µm is fed back to the analogto-digital (A/D) conversion interface. The D/A conversion has an output range of −10 to +10 V with 16-bit resolution, whereas the A/D conversion has an input range of −5 to +5 V with 12-bit resolution. To compare the practical hysteresis
response with the simulated response, the experimental results of the hysteresis response from the piezoactuated positioning mechanism is illustrated in Fig. 5, in which a sinusoidal input voltage with a frequency of 1.0 Hz is applied to the mechanism. From a comparison between Figs. 2 and 5, we can find that the developed dynamics of motion of the piezopositioning mechanism practically can be used. B. Experimentations As shown in Fig. 6, the PC-based system control programs for displacement control of the piezoactuated positioning
SHIEH et al.: ADAPTIVE DISPLACEMENT CONTROL FOR PIEZOACTUATED POSITIONING MECHANISM
mechanism consists of a main program and an interrupt service routine (ISR) procedure, and all are based on the C language. In the main program, initialization of the device parameters and input/output (I/O) are executed first. Then, a time period of 1 ms for operating the ISR procedure is set. After enabling the interrupt request, the main program primarily monitors the control data. The ISR procedure executes the functions of resolving the mover position from the A/D conversion interface, operating the proposed adaptive controller, and sending out the control effort through the D/A conversion interface. The values of the control parameters used in the simulations and experiments for the displacement tracking response of the piezoactuated positioning mechanism are given as follows: kP = 1.5, kI = 25, k1 = 11, k2 = 110, βk = 11, β0 = 105 , and β1 = β2 = β3 = 10−3 . All these values are chosen to meet the requirement of the high-performance tracking response and stable control system. The tracking responses to the square reference commands with amplitudes of 50 and 10 µm and frequencies of 0.5 and 1.0 Hz under the conventional PI control and the proposed adaptive control, respectively, are illustrated in Figs. 7 and 8. From a comparison between Figs. 7 and 8, it can be observed that the tracking performance can be substantially improved by using the proposed adaptive control approach. Moreover, the tracking response to the sinusoidal reference-commands with amplitudes of 50 and 10 µm and frequencies of 0.5 and 1.0 Hz under the proposed adaptive control is illustrated in Fig. 9. In Fig. 9, it can be observed that the transient performance of the sinusoidal displacement tracking response under the proposed adaptive control is also good enough. Furthermore, to show the robustness of the proposed adaptive control approach, the square displacement response with a step disturbance load of 10 N by attaching a rigid body to the mover of the piezopositioning mechanism is illustrated in Fig. 10. In Fig. 10, it can be observed that the robustness to external disturbance load can be effectively improved by using the proposed adaptive control approach. In addition, from an analysis of the system dynamics (7), the deviation of the external disturbance load can be equivalently regarded as the deviation of the system parameters, e.g., the system parameters m and K E . Therefore, experimental result for confirming the robustness of the proposed adaptive control scheme against the system parameters variations can be directly considered the result of the square displacement response illustrated in Fig. 10.
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Fig. 10. Square displacement response with a step disturbance load of 10 N. (a) Tracking response and (b) its control effort using the conventional PI control. (c) Tracking response and (d) its control effort using the proposed adaptive control.
V. C ONCLUSION An adaptive displacement tracking control for a piezoactuated positioning mechanism was proposed in this paper. A new mathematical model that uses a rigid-body motion system with the addition of a parameterized hysteretic friction was proposed to represent the dynamics of motion of the piezoactuated positioning mechanism. Moreover, we parameterize the hysteretic friction function with respect to the defined state variables, the system output displacement, and its velocity. According to the developed dynamics of motion of the piezoactuated positioning mechanism, an adaptive displacement tracking control with the parameter adaptation is proposed for the displacement
tracking control. From the theoretical analyses and experimental results, the following advantages of using the proposed control approach to the displacement tracking of the piezoactuated positioning mechanism can be provided. 1) The dynamics of hysteresis present in the piezoactuated positioning mechanism can be effectively represented by the proposed hysteretic friction model with reparameterization. 2) The asymptotic stability in displacement tracking can be obtained. 3) The transient performance of the tracking response of the piezoactuated positioning mechanism can be improved. 4) The robustness to the system parameter variations and external disturbance load
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can be provided. Finally, experimental results of the displacement tracking responses are illustrated to validate the proposed control approach for practical applications. R EFERENCES [1] M. A. Paesler and P. J. Moyer, Near-Field Optics: Theory, Instrumentation, and Applications. New York: Wiley, 1996. [2] S. B. Jung and S. W. Kim, “Improvement of scanning accuracy of PZT piezoelectric actuator by feed-forward model-reference control,” Precis. Eng., vol. 16, no. 1, pp. 49–55, 1994. [3] P. Ge and M. Jouaneh, “Tracking control of a piezoceramic actuator,” IEEE Trans. Control Syst. Technol., vol. 4, no. 3, pp. 209–216, May 1996. [4] S. N. Huang, K. K. Tan, and T. H. Lee, “Adaptive motion control using neural network approximations,” Automatica, vol. 38, no. 2, pp. 227–233, 2002. [5] G. Tao and P. V. Kokotovic, “Adaptive control of plants with unknown hystereses,” IEEE Trans. Autom. Control, vol. 40, no. 2, pp. 200–212, Feb. 1995. [6] C. L. Hwang, C. Jan, and Y. H. Chen, “Piezomechanics using intelligent variable-structure control,” IEEE Trans. Ind. Electron., vol. 48, no. 1, pp. 47–59, Feb. 2001. [7] I. D. Mayergoyz, “Dynamic Preisach models of hysteresis,” IEEE Trans. Magn., vol. 24, no. 6, pp. 2925–2927, Nov. 1988. [8] Y. Bernard, E. Mendes, and F. Bouillault, “Dynamic hysteresis modeling based on Preisach model,” IEEE Trans. Magn., vol. 38, no. 2, pp. 885– 888, Mar. 2002. [9] D. Song and C. J. Li, “Modeling of piezo actuator’s nonlinear and frequency dependent dynamics,” Mechatronics, vol. 9, no. 4, pp. 391–410, Jun. 1999. [10] R. B. Gorbet, K. A. Morris, and D. W. L. Wang, “Stability of control for the Preisach hysteresis model,” in Proc. IEEE Int. Conf. Robot. Autom., 1997, pp. 241–247. [11] A. Reimers and E. D. Torre, “Fast Preisach-based magnetization model and fast inverse hysteresis model,” IEEE Trans. Magn., vol. 34, no. 6, pp. 3857–3866, Nov. 1998. [12] S. Mittal and C. H. Menq, “Hysteresis compensation in electromagnetic actuators through Preisach model inversion,” IEEE/ASME Trans. Mechatronics, vol. 5, no. 4, pp. 394–409, Dec. 2000. [13] M. Goldfarb and N. Celanovic, “Modeling piezoelectric stack actuators for control of micromanipulation,” IEEE Control Syst. Mag., vol. 17, no. 3, pp. 69–79, Jun. 1997. [14] G. S. Choi, H. S. Kim, and G. H. Choi, “A study on position control of piezoelectric actuators,” in Proc. IEEE Int. Symp. Ind. Electron., 1997, vol. 3, pp. 851–855. [15] H. J. M. Adriaens, W. L. de Koning, and R. Banning, “Modeling piezoelectric actuators,” IEEE/ASME Trans. Mechatronics, vol. 5, no. 4, pp. 331–341, Dec. 2000. [16] T. S. Low and W. Guo, “Modeling of a three-layer piezoelectric bimorph beam with hysteresis,” J. Microelectromech. Syst., vol. 4, no. 4, pp. 230– 237, Dec. 1995. [17] J. J. Tzen, S. L. Jeng, and W. H. Chieng, “Modeling of piezoelectric actuator for compensation and controller design,” Precis. Eng., vol. 27, no. 1, pp. 70–86, Jan. 2003. [18] C. C. de Wit, H. Olsson, K. J. Åström, and P. Lischinsky, “A new model for control of systems with friction,” IEEE Trans. Autom. Control, vol. 40, no. 3, pp. 419–425, Mar. 1995. [19] E. Panteley, R. Ortega, and M. Gäfvert, “An adaptive friction compensator for global tracking in robot manipulators,” Syst. Control Lett., vol. 33, no. 5, pp. 307–313, Apr. 1998. [20] M. Feemster, P. Vedagarbha, D. M. Dawson, and D. Haste, “Adaptive control techniques for friction compensation,” Mechatronics, vol. 9, no. 2, pp. 125–145, Mar. 1999. [21] B. Friedland and Y. J. Park, “On adaptive friction compensation,” IEEE Trans. Autom. Control, vol. 37, no. 10, pp. 1609–1612, Oct. 1992. [22] M. Krsti´c and P. V. Kokotovi´c, “Adaptive nonlinear design with controlleridentifier separation and swapping,” IEEE Trans. Autom. Control, vol. 40, no. 3, pp. 426–440, Mar. 1995. [23] J. Amin, “Implementation of a friction estimation and compensation technique,” in Proc. IEEE Int. Conf. Control Appl., Sep. 1996, pp. 804–808. [24] C. C. de Wit and S. S. Ge, “Adaptive friction compensation for systems with generalized velocity/position friction dependency,” in Proc. IEEE 36th Conf. Decision Control, Dec. 1997, pp. 2465–2470. [25] M. Gäfvert, “Dynamic model based friction compensation on the Furuta pendulum,” in Proc. IEEE Int. Conf. Control Appl., Aug. 1999, pp. 1260–1265.
[26] H. Olsson and K. J. Åström, “Observer-based friction compensation,” in Proc. IEEE 35th Conf. Decision Control, Dec. 1996, pp. 4345–4350. [27] Z. Wang, H. Meikote, and F. Khorrami, “Robust adaptive friction compensation in servo-drives using position measurement only,” in Proc. IEEE Int. Conf. Control Appl., Sep. 2000, pp. 178–183. [28] M. Krsti´c, I. Kanellakopoulos, and P. Kokotovi´c, Nonlinear and Adaptive Control Design. New York: Wiley, 1995.
Hsin-Jang Shieh (M’03) received the B.S. and Ph.D. degrees in electrical engineering from National Central University, Chung-Li, Taiwan, R.O.C., in 1992 and 1997, respectively. From 1997 to 2002, he was with the Mechanical Industry Research Laboratories, Industrial Technology Research Institute, Hsinchu, Taiwan, R.O.C., as a Researcher in the Micro-Electro-Mechanical Systems (MEMS) Division. Since August 2002, he has been with National Dong Hwa University, Hualien, Taiwan, R.O.C., where he is currently an Assistant Professor in the Department of Electrical Engineering. His research interests include piezoelectric mechanisms, power converters, photovoltaic systems, and control theory applications.
Faa-Jeng Lin (M’93–SM’99) received the B.S. and M.S. degrees from National Cheng Kung University, Tainan, Taiwan, R.O.C., in 1983 and 1985, respectively, and the Ph.D. degree from National Tsing Hua University, Hsinchu, Taiwan, R.O.C., in 1993, all in electrical engineering. From 1993 to 2001, he was an Associate Professor and was promoted to Professor in the Department of Electrical Engineering, Chung Yuan Christian University, Chung Li, Taiwan, R.O.C. He is currently a Professor in the Department of Electrical Engineering and the Dean of the Office of Research and Development at National Dong Hwa University, Hualien, Taiwan, R.O.C. His research interests include alternating current and ultrasonic motor drives, digital-signal-processor-based computer control systems, fuzzy and neural network control theories, nonlinear control theories, power electronics, and micromechatronics.
Po-Kai Huang was born in Taipei, Taiwan, R.O.C., in 1979. He received the B.S. and M.S. degrees in electrical engineering from Chung Yuan Christian University, Chung Li, Taiwan, R.O.C., in 2001 and 2003, respectively. He is currently working toward the Ph.D. degree in the Department of Electrical Engineering, National Dong Hwa University, Hualien, Taiwan, R.O.C. His research interests include alternating current motor drives, artificial intelligence control theories, and micromechatronics.
Li-Tao Teng was born in Taipei, Taiwan, R.O.C., in 1982. He received the B.S. degree in electrical engineering in 2003 from National Dong Hwa University, Hualien, Taiwan, R.O.C., where he is currently working toward the Ph.D. degree in the Department of Electrical Engineering. His research interests include nonlinear control theories, artificial intelligence control theories, and magnetic levitation systems.
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Adaptive Displacement Control With Hysteresis Modeling for Piezoactuated Positioning Mechanism Hsin-Jang Shieh, Member, IEEE, Faa-Jeng Lin, Senior Member, IEEE, Po-Kai Huang, and Li-Tao Teng
Abstract—An adaptive displacement control with hysteresis modeling for a piezoactuated positioning mechanism is proposed in this paper because the dynamic performance of piezosystems is often severely deteriorated due to the hysteresis effect of piezoelectric elements. First, a new mathematical model based on the differential equation of a motion system with a parameterized hysteretic friction function is proposed to represent the dynamics of motion of the piezopositioning mechanism. As a result, the mathematical model describes a motion system with hysteresis behavior due to the hysteretic friction. Then, by using the developed mathematical model, the adaptive displacement tracking control with the adaptation algorithms of the parameterized hysteretic function and of an uncertain parameter is proposed. By using the proposed control approach on the displacement control of the piezopositioning mechanism, the advantages of the asymptotical stability in displacement tracking, high-performance displacement response, and robustness to the variations of system parameters and disturbance load can be provided. Finally, experimental results are illustrated to validate the proposed control approach for practical applications. Index Terms—Adaptive displacement control, hysteresis effect, hysteretic friction function, piezoactuated positioning mechanism, robustness.
I. I NTRODUCTION
D
UE TO THE requirements of the nanometer resolution in displacement, high stiffness, and fast response, the piezoactuators are often used in high-precision positioning applications, such as scanning probe microscopy [1]. Because the materials of the piezoactuators are ferroelectric, they fundamentally exhibit hysteresis behavior in their response to an applied electric field. This leads to problems of severe inaccuracy, instability, and restricted system performance due to hysteresis nonlinearity if the piezoactuator is operated in an open-loop fashion [2]–[6]. Moreover, the hysteresis characteristics are usually unknown, and the states of representing to hysteresis dynamics are often unmeasured. These usually cause the increasing difficulties in tracking control design with high-performance requirements for piezoactuated mechanisms [2]–[4]. In general, when systems have hysteresis effect in dynamic responses, an established model or function that can accurately represent the hysteresis feature becomes very important for the analysis and control of systems [7]–[17]. Therefore, the Preisach-function-based Preisach models were proposed in Manuscript received October 10, 2003; revised July 29, 2005. Abstract published on the Internet March 18, 2006. The authors are with the Department of Electrical Engineering, National Dong Hwa University, Hualien 97401, Taiwan, R.O.C. (e-mail: hjshieh@ mail.ndhu.edu.tw). Digital Object Identifier 10.1109/TIE.2006.874264
[7]–[12] to represent the hysteresis dynamics. In these studies, the hysteresis dynamics could be approximately described by the Preisach function, but the derived equations and the employed operations were complicated [7]–[10]. This complicated operation often leads to a problem of spending much time on computation during the control process. Moreover, a fast Preisach-based magnetization model that established several characteristics of the hysteresis and its inverse was proposed in [11]. In [11], the equations derived for system control process were simplified by constructing some specified Preisach function in advance. However, some specified conditions needed in the simplification procedure, which was for the derivation of the dynamics of the hysteresis and its inverse, were required. These conditions usually cause a more restricted control design. In [13] and [14], the hysteresis model was approximated by using the motion dynamics constructed by an applied force to one set of the massless bodies paralleling to springs, and moreover, relationship in terms of the applied force, spring constants, and break forces was used to determine the hysteresis dynamics. Unfortunately, from the experimental results provided by [13] and [14], the critical numbers of the employed springs and massless bodies for accurately representing the hysteresis dynamics were very difficult to determine. Furthermore, it seems that the numbers of the constructed springs and massless bodies could be obtained only by using computer simulations or practical experiments. Moreover, an electromechanical model constructed by transduction of the electric charge and discharge behaviors to generate the applied force was proposed in [15]. Although the model composed of a first-order differential equation and a partial differential equation was employed for both shaping the hysteresis loop and describing the mechanical model of piezoactuator, this model was apparently more suitable for vibration control than for displacement control. Similarly, developments in [16] and [17] were mainly focused on the vibration control of piezoactuators. Because compensation for the effect of a linear or nonlinear friction on control systems becomes very important for highperformance control design, studies on compensation techniques with system modeling for various friction phenomena have been published in [18]–[27]. In these studies, the developed friction model that includes the characteristics of the Stribeck effect, hysteresis, and springlike behaviors was assembled in mechanical motion dynamics. Therefore, the hysteretic friction model developed in [18] is extended to represent the dynamics of motion of the piezoactuated positioning mechanism, which has the hysteresis behavior in dynamic response. First, a new mathematical model constructed by a differential equation of a motion system with the addition of a hysteretic
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friction function is proposed to represent the dynamics of motion of the piezopositioning mechanism. As a result, the mathematical model can be viewed as a motion system with hysteresis behavior due to the hysteretic friction. Then, for the purpose of the controller design, we parameterize the hysteretic friction function with respect to the defined state variables, the system output displacement, and velocity. Moreover, an external disturbance load is also considered in the developed mathematical model. With the use of the motion equation, hysteretic friction, and external disturbance load, the dynamics of motion of the piezoactuated positioning mechanism studied in this paper can be completely established. Furthermore, according to the developed model, an adaptive displacement tracking control with the parameter adaptations of the parameterized hysteretic function and of an uncertain parameter is proposed. In this control system, both the asymptotical stability in displacement tracking and the boundedness of the parameter adaptations can be guaranteed. In addition, the robustness to external disturbance load can be obtained because the value of the uncertain disturbance load is adapted by the proposed adaptation laws as well as the parameters of the hysteretic friction function. By using the proposed control approach, the displacement tracking response of the piezoactuated positioning mechanism has the following advantages. 1) The dynamics of the hysteresis present in the piezoactuated positioning mechanism can be completely modeled by the proposed hysteretic friction function. 2) The asymptotic stability in displacement tracking control and the boundedness of the parameters adaptation can be provided. 3) The high-performance displacement control can be achieved. 4) The robustness to the variations of system parameters and external disturbance load can be obtained. Finally, experimental results are illustrated to validate the proposed control approach for practical applications.
shown in [13], [15], and [16]. With the aforementioned reasons, the dynamics of motion of the piezoactuated positioning mechanism with hysteresis behavior can be further developed in this section. The friction model with hysteresis effect named the LuGre model and proposed in [18] can be represented by
II. D YNAMICS OF P IEZOACTUATED P OSITIONING M ECHANISM
According to [18]–[20], the complete dynamics of motion of the one-dimensional piezoactuated positioning mechanism can be equivalently expressed as
In general, there exist two difficulties in modeling the hysteresis nonlinearity of piezoactuators, namely: 1) nonlocal memory phenomenon and 2) asymmetric loop between descending and ascending paths [3], [18]. Therefore, developing a dynamic model to describe the hysteresis behavior becomes an important subject for improving the control performance of piezoactuated positioning mechanisms, as shown in [7]–[9] and [13]–[17]. However, as mentioned in the previous section, there still exist many drawbacks in the past developments for hysteresis compensation. Therefore, we intentionally develop a hysteretic friction function to represent the hysteresis behavior of the piezoactuated positioning mechanism. The reasons that the hysteretic friction function is chosen as a model of the hysteresis in this piezomechanism can be summarized as follows. 1) The qualitative mechanisms [18] of friction have been fairly well understood in modern time. 2) The friction model with hysteresis behavior has been completely developed and validated in [18]–[27]. 3) The hysteresis and piezoelectric effect of piezoactuators can be separately considered in system control design, as
|x| ˙ dz = x˙ − z dt h(x) ˙ dz + σ2 x˙ FH = σ 0 z + σ 1 dt
(1) (2)
where FH denotes the hysteretic friction function; z denotes the internal friction state that represents the average deflection of bristles with contact force applied; x˙ denotes the relative velocity between the two contact surfaces; σ0 , σ1 , and σ2 are positive constants, which are typically unknown and generally difficult to identify, and can be equivalently interpreted as bristle stiffness, bristle damping, and viscous coefficient, respectively [18]; and the function h(x) ˙ denotes the Stribeck effect curve given by 2 − x˙ ˙ = fC + (fS − fC )e x˙ S (3) σ0 h(x) where fc is the Coulomb friction level, fs is the level of the stiction force, and x˙ s is the Stribeck velocity. Moreover, from (1), it can be found that the boundedness of the internal friction ˙ state z and x˙ implies the boundedness of the function h(x). Furthermore, the feasibility of the hysteretic friction model shown in (1)–(3) for a rigid-body system had been confirmed by the results of the theoretical analyses, simulations, and experiments in [18]–[20]. Therefore, by substituting (1) into (2), the hysteretic function shown in (2) can be rewritten as FH = σ 0 z − σ 1
1 z|x| ˙ + (σ1 + σ2 )x. ˙ h(x) ˙
m¨ x + FH + F L = K E u
(4)
(5)
where m denotes the effective mass of the piezoactuated positioning mechanism, x denotes the displacement of the piezopositioning mechanism, x ¨ denotes the velocity, KE denotes the voltage-to-force coefficient of the piezoactuator, u indicates the applied voltage to the piezoactuator, FH denotes the hysteretic friction function, and FL is the external disturbance load. From (5), it can be found that the dynamics of the system hysteresis caused by the hysteretic friction seems to be an individual component of the control system. Therefore, the model-based controller design approach can be used if systems with a nonlinear hysteresis behavior can be represented by (5). Moreover, by substituting (4) into (5), the following dynamics can be obtained: 1 1 KE u− z|x| ˙ + (σ1 + σ2 )x˙ . (σ0 z + FL ) − σ1 x ¨= m m h(x) ˙ (6)
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Fig. 1. Block diagram of the proposed dynamic model for the piezoactuated positioning mechanism.
Furthermore, by defining the state variables x = x1 and x˙ 1 = x2 , we can rewrite (6) as follows: x˙ 1 = x2
1 1 x˙ 2 = K E u− z|x2 |+(σ1 +σ2 )x2 (σ0 z+FL ) − σ1 m h(x2 ) (7)
where K E = KE /m. From (7), it can be found that the dynamics of motion of the piezoactuated positioning mechanism is nonlinear with respect to the defined state variables, and the corresponding block diagram about (7) can be shown in Fig. 1. To validate the feasibility of the dynamic model shown in (7) of describing the dynamics of motion of the piezopositioning mechanism, the simulation results are illustrated in Fig. 2, in which a sinusoidal input voltage with an amplitude of 5 V and a frequency of 1 Hz is applied to (7), and the values of the system parameters are given as follows: σ0 = 105 N/m, √ σ1 = 105 Ns/m, σ2 = 0.4 Ns/m, fc = 1 N, fs = 1.5 N, xs = 0.001 m/s, m = 1 kg, and KE = 1 N/V. In Fig. 2, we can observe that using the developed dynamic model (7) with hysteresis behavior to represent the dynamics of motion of the piezoactuated positioning mechanism can be confirmed.
Fig. 2. Simulation result of the hysteresis response from the proposed mathematical model.
the piezoactuated positioning mechanism can be reexpressed as follows: F H = σ0 h(x2 )
x2 + σ2 x2 + σ3 x1 |x2 |
(9)
where σ3 x1 can be viewed as an appended term due to piezoelectric effect. Moreover, by replacing (4) with (9), the dynamics of motion of the piezoactuated positioning mechanism can be derived as follows: x˙ 1 = x2
III. A DAPTIVE D ISPLACEMENT C ONTROL To achieve high-performance displacement response from the piezoactuated positioning mechanism, a straightforward technique to compensate the hysteresis effect is to adapt the unknown hysteresis function or component. Therefore, the adaptive displacement tracking control with hysteresis compensation is proposed in this section. Here, we assume that the internal state z and the function h(x2 ) are bounded and piecewise continuous and the change in average deflection of z is small enough to be ignored. With the definition of the steady-state z s , the equation of z in steady state can be derived as follows: x2 h(x2 ). zs = (8) |x2 | Therefore, in steady state, the hysteretic friction function (4) mainly causing the hysteresis effect in motion dynamics of
x˙ 2 = K E u − [θ0 sgn(x2 ) + θ1 x1 + θ2 x2 + θ3 ]
(10)
where sgn(·) denotes a sign function defined as x2 sgn(x2 ) = = |x2 |
+1 0 −1
as x2 > 0 as x2 = 0 as x2 < 0
and θ0 , θ1 , θ2 , and θ3 denote the unknown and bounded parameters defined by θ0 = σ0 h(x2 )/m, θ1 = σ3 /m, θ2 = σ2 /m, and θ3 = FL /m. Therefore, the hysteretic friction function (9) is equivalently parameterized by θ0 , θ1 , θ2 , and θ3 with respect to the system states x1 and x2 . For the tracking controller design, the following states of the tracking errors are defined as z1 = x1 − xm
(11)
z2 = x2 + k1 z1 − x˙ m
(12)
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Fig. 3. System configuration of the controlled piezoactuated positioning mechanism by using (a) the conventional PI controller and (b) the proposed adaptive controller.
where k1 is a properly positive constant and xm and x˙ m denote the reference displacement and its derivative, respectively. Taking the time derivatives of (11) and (12) yields z˙1 = − k1 z1 + z2 z˙2 = K E u − [θ0 sgn(x2 ) + θ1 x1 + θ2 x2 + θ3 ] − k12 z1 + k1 z2 − x ¨m .
(13)
From (13), it can be observed that the values of the system parameters θ0 , θ1 , θ2 , θ3 , and K E must be exactly known for model-based control design with high-performance requirement. However, these values of the parameters are initially unknown and difficult to identify because they are associated with the unfamiliar hysteresis behavior of the piezopositioning mechanism. Therefore, in order to achieve high-performance tracking control, the dynamics (13) is purposely rewritten as follows: z˙1 = − k1 z1 + z2 ˆ u − θˆ sgn(x ) + θˆ x + θˆ x + θˆ + K E u z˙2 = K E 0 2 1 1 2 2 3 − θ0 sgn(x2 )+ θ1 x1 + θ2 x2 + θ3 − k 2 z1 + k1 z2 − x ¨m 1
(14) ˆ , θ = θ − θˆ , θ = θ − θˆ , θ = = KE − K where K E 0 0 0 1 1 1 2 ˆ θ2 − θ2 , and θ3 = θ3 − θˆ3 denote the estimation errors and ˆ , θˆ , θˆ , θˆ , and θˆ denote the estimates of the unknown K E 0 1 2 3 parameters K E , θ0 , θ1 , θ2 , and θ3 , respectively.
According to the dynamics (14), the adaptive displacement tracking control with parameter adaptations is proposed as follows: 1 2 k1 z1 − k1 z2 + x ¨m − z1 − k2 z2 u= ˆ K E + θˆ0 sgn(x2 ) + θˆ1 x1 + θˆ2 x2 + θˆ3 (15) ˆ˙ = β z u K (16) E k 2 ˙ ˙ ˆ ˆ θ0 = − β0 sgn(x2 )z2 , θ1 = −β1 z2 x1 , ˙ ˙ (17) θˆ2 = − β2 z2 x2 , θˆ3 = −β3 z2 where k2 is a positive constant and βk , β0 , β1 , β2 , and β3 denote the adaptation gains. Theorem 1: The control system shown in (13) can be asymptotically stabilized by using the proposed tracking control shown in (15)–(17), where the parameters adaptations given in (16) and (17) are ultimately bounded. Proof: By substituting (15) into (14), (14) can be rewritten as follows: z˙1 = − k1 z1 + z2 E u − θ0 sgn(x2 )+ θ1 x1 + θ2 x2 + θ3 . z˙2 = − z1 − k2 z2 + K (18) According to (18), the following Lyapunov function candidate is chosen: V =
1 2 1 2 1 z + z + 2 1 2 2 2
1 2 1 1 1 1 KE + θ02 + θ12 + θ22 + θ32 . (19) × βk β0 β1 β2 β3
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Fig. 6. Flowcharts of the PC-based system control programs.
Fig. 4. (a) Machine structure and (b) implementation of the PC-based control design for the displacement control of the piezoactuated positioning mechanism.
Taking the time derivative of (19) yields ˙ E + z2 u E 1 K V˙ = − k1 z12 − k2 z22 + K βk 1 ˙ 1 ˙ + θ0 θ0 − sgn(x2 )z2 + θ1 θ1 − z2 x1 β0 β1 1 ˙ 1 ˙ (20) + θ2 θ2 − z2 x2 + θ3 θ 3 − z2 . β2 β3 Moreover, let ˙ E K ˙ θ0 ˙ θ1 ˙ θ2 ˙ θ3
= − βk z2 u = β0 sgn(x2 )z2 = β1 z2 x1 = β2 z2 x2 = β3 z2 .
(21)
Thus, the following inequality can be obtained by substituting (21) into (20): V˙ = −k1 z12 − k2 z22 ≤ 0.
Fig. 5. Experimental result of the hysteresis response from the piezoactuated positioning mechanism.
(22)
From the Lyapunov stability theory of view, the controlled states z1 and z2 will converge to zero asymptotically, and θ0 , θ1 , θ2 , and θ3 are ultimately the estimation errors K, bounded [28] because V˙ (t) → 0 as t → ∞. Furthermore, the following conditions are assumed. 1) The system parameter K E is unknown but constant. 2) The designated parameters θ0 , θ1 , θ2 , and θ3 are piecewise continuous. 3) The responses of the ˆ , θˆ , θˆ , θˆ , and θˆ are purposely parameter adaptations of K E 0 1 2 3
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Fig. 7. Square displacement response under the conventional PI control with (a) displacement = 0−50 µm and frequency = 0.5 Hz and (b) its control effort, (c) displacement = 0−50 µm and frequency = 1.0 Hz and (d) its control effort, (e) displacement = 0−10 µm and frequency = 0.5 Hz and (f) its control effort, and (g) displacement = 0−10 µm and frequency = 1.0 Hz and (h) its control effort.
designated to be faster than that of the system parameters K E , θ0 , θ1 , θ2 , and θ3 ; that is, these values of the system parameters can be equivalently viewed as piecewise constants during the ˆ , θˆ , θˆ , θˆ , and θˆ . Therefore, the adaptation transient of K E 0 1 2 3 following adaptation laws can be derived: ˙ E K ˙ θ0 ˙ θ1 ˙ θ2 ˙ θ3
= = = = =
ˆ˙ −K E ˙ − θˆ0 ˙ − θˆ1 ˙ − θˆ2 ˙ − θˆ3 .
(23)
Consequently, the asymptotical stability in displacement tracking by using (15)–(17) can be confirmed. Remark 1: The asymptotical stability in displacement tracking of the piezoactuated positioning mechanism represented by (5) can be achieved by means of the proposed control approach given by (15)–(17), which is based on the dynamics (10) with the parameterized hysteresis behavior and the lumped disturbance load. Remark 2: Robustness against the variation of both the system parameter K E and external load FL can be provided because the adaptation of these two parameters are practically considered in the proposed control design.
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Fig. 8. Square displacement response under the proposed adaptive control with (a) displacement = 0−50 µm and frequency = 0.5 Hz and (b) its control effort, (c) displacement = 0−50 µm and frequency = 1.0 Hz and (d) its control effort, (e) displacement = 0−10 µm and frequency = 0.5 Hz and (f) its control effort, and (g) displacement = 0−10 µm and frequency = 1.0 Hz and (h) its control effort.
IV. E XPERIMENTAL R ESULTS A. System Configuration To investigate the effectiveness of the proposed control approach, the personal-computer (PC)-based control system is implemented, and the displacement tracking responses of the piezoactuated positioning mechanism are illustrated. The block diagram of the system implementation using the proportionalintegral (PI) controller and the proposed adaptive control approach, respectively, is shown in Fig. 3(a) and (b). The machine structure of the one-dimensional piezoactuated positioning mechanism, which is manufactured by Piezosystem
Jena Inc., Model PX300, and the corresponding implementation of the PC-based control system are shown in Fig. 4(a) and (b), respectively. In Fig. 4(a), the mover is actuated by piezoactuators inside the mechanism, and a capacitive displacement sensor is preinstalled in this mechanism. The rated values of this experimental piezomechanism are given in the following list: motion distance = 0−240 µm; maximum external load = 10 N; input voltage = 0−150 V; stiffness = 0.08 N/µm; repeatability = 0.03%;
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Fig. 9. Sinusoidal displacement response under the proposed adaptive control with (a) displacement = ±50 µm and frequency = 0.5 Hz and (b) its control effort, (c) displacement = ±50 µm and frequency = 1.0 Hz and (d) its control effort, (e) displacement = ±10 µm and frequency = 0.5 Hz and (f) its control effort, and (g) displacement = ±10 µm and frequency = 1.0 Hz and (h) its control effort.
resonant frequency = 170 Hz; capacitance = 2300 nF. In this control system, the control voltage is outputted by a digital-to-analog (D/A) conversion interface and then amplified through a specified amplifier with around 10 times gains to drive the piezoactuators inside the mechanism. The output displacement signal detected by a capacitive displacement sensor with a resolution of 41.67 mV/µm is fed back to the analogto-digital (A/D) conversion interface. The D/A conversion has an output range of −10 to +10 V with 16-bit resolution, whereas the A/D conversion has an input range of −5 to +5 V with 12-bit resolution. To compare the practical hysteresis
response with the simulated response, the experimental results of the hysteresis response from the piezoactuated positioning mechanism is illustrated in Fig. 5, in which a sinusoidal input voltage with a frequency of 1.0 Hz is applied to the mechanism. From a comparison between Figs. 2 and 5, we can find that the developed dynamics of motion of the piezopositioning mechanism practically can be used. B. Experimentations As shown in Fig. 6, the PC-based system control programs for displacement control of the piezoactuated positioning
SHIEH et al.: ADAPTIVE DISPLACEMENT CONTROL FOR PIEZOACTUATED POSITIONING MECHANISM
mechanism consists of a main program and an interrupt service routine (ISR) procedure, and all are based on the C language. In the main program, initialization of the device parameters and input/output (I/O) are executed first. Then, a time period of 1 ms for operating the ISR procedure is set. After enabling the interrupt request, the main program primarily monitors the control data. The ISR procedure executes the functions of resolving the mover position from the A/D conversion interface, operating the proposed adaptive controller, and sending out the control effort through the D/A conversion interface. The values of the control parameters used in the simulations and experiments for the displacement tracking response of the piezoactuated positioning mechanism are given as follows: kP = 1.5, kI = 25, k1 = 11, k2 = 110, βk = 11, β0 = 105 , and β1 = β2 = β3 = 10−3 . All these values are chosen to meet the requirement of the high-performance tracking response and stable control system. The tracking responses to the square reference commands with amplitudes of 50 and 10 µm and frequencies of 0.5 and 1.0 Hz under the conventional PI control and the proposed adaptive control, respectively, are illustrated in Figs. 7 and 8. From a comparison between Figs. 7 and 8, it can be observed that the tracking performance can be substantially improved by using the proposed adaptive control approach. Moreover, the tracking response to the sinusoidal reference-commands with amplitudes of 50 and 10 µm and frequencies of 0.5 and 1.0 Hz under the proposed adaptive control is illustrated in Fig. 9. In Fig. 9, it can be observed that the transient performance of the sinusoidal displacement tracking response under the proposed adaptive control is also good enough. Furthermore, to show the robustness of the proposed adaptive control approach, the square displacement response with a step disturbance load of 10 N by attaching a rigid body to the mover of the piezopositioning mechanism is illustrated in Fig. 10. In Fig. 10, it can be observed that the robustness to external disturbance load can be effectively improved by using the proposed adaptive control approach. In addition, from an analysis of the system dynamics (7), the deviation of the external disturbance load can be equivalently regarded as the deviation of the system parameters, e.g., the system parameters m and K E . Therefore, experimental result for confirming the robustness of the proposed adaptive control scheme against the system parameters variations can be directly considered the result of the square displacement response illustrated in Fig. 10.
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Fig. 10. Square displacement response with a step disturbance load of 10 N. (a) Tracking response and (b) its control effort using the conventional PI control. (c) Tracking response and (d) its control effort using the proposed adaptive control.
V. C ONCLUSION An adaptive displacement tracking control for a piezoactuated positioning mechanism was proposed in this paper. A new mathematical model that uses a rigid-body motion system with the addition of a parameterized hysteretic friction was proposed to represent the dynamics of motion of the piezoactuated positioning mechanism. Moreover, we parameterize the hysteretic friction function with respect to the defined state variables, the system output displacement, and its velocity. According to the developed dynamics of motion of the piezoactuated positioning mechanism, an adaptive displacement tracking control with the parameter adaptation is proposed for the displacement
tracking control. From the theoretical analyses and experimental results, the following advantages of using the proposed control approach to the displacement tracking of the piezoactuated positioning mechanism can be provided. 1) The dynamics of hysteresis present in the piezoactuated positioning mechanism can be effectively represented by the proposed hysteretic friction model with reparameterization. 2) The asymptotic stability in displacement tracking can be obtained. 3) The transient performance of the tracking response of the piezoactuated positioning mechanism can be improved. 4) The robustness to the system parameter variations and external disturbance load
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 3, JUNE 2006
can be provided. Finally, experimental results of the displacement tracking responses are illustrated to validate the proposed control approach for practical applications. R EFERENCES [1] M. A. Paesler and P. J. Moyer, Near-Field Optics: Theory, Instrumentation, and Applications. New York: Wiley, 1996. [2] S. B. Jung and S. W. Kim, “Improvement of scanning accuracy of PZT piezoelectric actuator by feed-forward model-reference control,” Precis. Eng., vol. 16, no. 1, pp. 49–55, 1994. [3] P. Ge and M. Jouaneh, “Tracking control of a piezoceramic actuator,” IEEE Trans. Control Syst. Technol., vol. 4, no. 3, pp. 209–216, May 1996. [4] S. N. Huang, K. K. Tan, and T. H. Lee, “Adaptive motion control using neural network approximations,” Automatica, vol. 38, no. 2, pp. 227–233, 2002. [5] G. Tao and P. V. Kokotovic, “Adaptive control of plants with unknown hystereses,” IEEE Trans. Autom. Control, vol. 40, no. 2, pp. 200–212, Feb. 1995. [6] C. L. Hwang, C. Jan, and Y. H. Chen, “Piezomechanics using intelligent variable-structure control,” IEEE Trans. Ind. Electron., vol. 48, no. 1, pp. 47–59, Feb. 2001. [7] I. D. Mayergoyz, “Dynamic Preisach models of hysteresis,” IEEE Trans. Magn., vol. 24, no. 6, pp. 2925–2927, Nov. 1988. [8] Y. Bernard, E. Mendes, and F. Bouillault, “Dynamic hysteresis modeling based on Preisach model,” IEEE Trans. Magn., vol. 38, no. 2, pp. 885– 888, Mar. 2002. [9] D. Song and C. J. Li, “Modeling of piezo actuator’s nonlinear and frequency dependent dynamics,” Mechatronics, vol. 9, no. 4, pp. 391–410, Jun. 1999. [10] R. B. Gorbet, K. A. Morris, and D. W. L. Wang, “Stability of control for the Preisach hysteresis model,” in Proc. IEEE Int. Conf. Robot. Autom., 1997, pp. 241–247. [11] A. Reimers and E. D. Torre, “Fast Preisach-based magnetization model and fast inverse hysteresis model,” IEEE Trans. Magn., vol. 34, no. 6, pp. 3857–3866, Nov. 1998. [12] S. Mittal and C. H. Menq, “Hysteresis compensation in electromagnetic actuators through Preisach model inversion,” IEEE/ASME Trans. Mechatronics, vol. 5, no. 4, pp. 394–409, Dec. 2000. [13] M. Goldfarb and N. Celanovic, “Modeling piezoelectric stack actuators for control of micromanipulation,” IEEE Control Syst. Mag., vol. 17, no. 3, pp. 69–79, Jun. 1997. [14] G. S. Choi, H. S. Kim, and G. H. Choi, “A study on position control of piezoelectric actuators,” in Proc. IEEE Int. Symp. Ind. Electron., 1997, vol. 3, pp. 851–855. [15] H. J. M. Adriaens, W. L. de Koning, and R. Banning, “Modeling piezoelectric actuators,” IEEE/ASME Trans. Mechatronics, vol. 5, no. 4, pp. 331–341, Dec. 2000. [16] T. S. Low and W. Guo, “Modeling of a three-layer piezoelectric bimorph beam with hysteresis,” J. Microelectromech. Syst., vol. 4, no. 4, pp. 230– 237, Dec. 1995. [17] J. J. Tzen, S. L. Jeng, and W. H. Chieng, “Modeling of piezoelectric actuator for compensation and controller design,” Precis. Eng., vol. 27, no. 1, pp. 70–86, Jan. 2003. [18] C. C. de Wit, H. Olsson, K. J. Åström, and P. Lischinsky, “A new model for control of systems with friction,” IEEE Trans. Autom. Control, vol. 40, no. 3, pp. 419–425, Mar. 1995. [19] E. Panteley, R. Ortega, and M. Gäfvert, “An adaptive friction compensator for global tracking in robot manipulators,” Syst. Control Lett., vol. 33, no. 5, pp. 307–313, Apr. 1998. [20] M. Feemster, P. Vedagarbha, D. M. Dawson, and D. Haste, “Adaptive control techniques for friction compensation,” Mechatronics, vol. 9, no. 2, pp. 125–145, Mar. 1999. [21] B. Friedland and Y. J. Park, “On adaptive friction compensation,” IEEE Trans. Autom. Control, vol. 37, no. 10, pp. 1609–1612, Oct. 1992. [22] M. Krsti´c and P. V. Kokotovi´c, “Adaptive nonlinear design with controlleridentifier separation and swapping,” IEEE Trans. Autom. Control, vol. 40, no. 3, pp. 426–440, Mar. 1995. [23] J. Amin, “Implementation of a friction estimation and compensation technique,” in Proc. IEEE Int. Conf. Control Appl., Sep. 1996, pp. 804–808. [24] C. C. de Wit and S. S. Ge, “Adaptive friction compensation for systems with generalized velocity/position friction dependency,” in Proc. IEEE 36th Conf. Decision Control, Dec. 1997, pp. 2465–2470. [25] M. Gäfvert, “Dynamic model based friction compensation on the Furuta pendulum,” in Proc. IEEE Int. Conf. Control Appl., Aug. 1999, pp. 1260–1265.
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Hsin-Jang Shieh (M’03) received the B.S. and Ph.D. degrees in electrical engineering from National Central University, Chung-Li, Taiwan, R.O.C., in 1992 and 1997, respectively. From 1997 to 2002, he was with the Mechanical Industry Research Laboratories, Industrial Technology Research Institute, Hsinchu, Taiwan, R.O.C., as a Researcher in the Micro-Electro-Mechanical Systems (MEMS) Division. Since August 2002, he has been with National Dong Hwa University, Hualien, Taiwan, R.O.C., where he is currently an Assistant Professor in the Department of Electrical Engineering. His research interests include piezoelectric mechanisms, power converters, photovoltaic systems, and control theory applications.
Faa-Jeng Lin (M’93–SM’99) received the B.S. and M.S. degrees from National Cheng Kung University, Tainan, Taiwan, R.O.C., in 1983 and 1985, respectively, and the Ph.D. degree from National Tsing Hua University, Hsinchu, Taiwan, R.O.C., in 1993, all in electrical engineering. From 1993 to 2001, he was an Associate Professor and was promoted to Professor in the Department of Electrical Engineering, Chung Yuan Christian University, Chung Li, Taiwan, R.O.C. He is currently a Professor in the Department of Electrical Engineering and the Dean of the Office of Research and Development at National Dong Hwa University, Hualien, Taiwan, R.O.C. His research interests include alternating current and ultrasonic motor drives, digital-signal-processor-based computer control systems, fuzzy and neural network control theories, nonlinear control theories, power electronics, and micromechatronics.
Po-Kai Huang was born in Taipei, Taiwan, R.O.C., in 1979. He received the B.S. and M.S. degrees in electrical engineering from Chung Yuan Christian University, Chung Li, Taiwan, R.O.C., in 2001 and 2003, respectively. He is currently working toward the Ph.D. degree in the Department of Electrical Engineering, National Dong Hwa University, Hualien, Taiwan, R.O.C. His research interests include alternating current motor drives, artificial intelligence control theories, and micromechatronics.
Li-Tao Teng was born in Taipei, Taiwan, R.O.C., in 1982. He received the B.S. degree in electrical engineering in 2003 from National Dong Hwa University, Hualien, Taiwan, R.O.C., where he is currently working toward the Ph.D. degree in the Department of Electrical Engineering. His research interests include nonlinear control theories, artificial intelligence control theories, and magnetic levitation systems.
