** Intelligent Control and Automation ** Vol. 3 No. 4 (2012) , Article ID: 24885 , 14 pages DOI:10.4236/ica.2012.34043

Stable Adaptive Fuzzy Control with Hysteresis Observer for Three-Axis Micro/Nano Motion Stages

Department of Mechanical Engineering, National Chung Hsing University, Taichung, Chinese Taipei

Email: lclin@mail.nchu.edu.tw

Received July 31, 2012; revised August 31, 2012; accepted September 7, 2012

**Keywords:** Micro/Nano Stage; Adaptive Fuzzy Control; Hysteresis Observer; Fuzzy Function Approximator

ABSTRACT

This paper considers the analytical dynamics with simplified Dahl hysteresis model for a three-axis piezoactuated micro/ nano flexure stage. An adaptive controller with nonlinear dynamic hysteresis observer is proposed using Lyapunov stability theory. In the controller, a fuzzy function approximator with parameters update law is included to compensate for the identification inaccuracy, model uncertainty, and flexure coupling effects. Simulation results are used to demonstrate the control performance.

1. Introduction

Recently, control of micro/nano stages considering the piezoactuator hysteresis effects has found great interests in the literature. Effective ultrafine-resolution trajectory tracking performance of stages is limited by the intrinsic hysteretic behavior of the piezoceramic material and the structural vibration of the devices [1].

Many efforts were trying to decrease the hysteresis effect of piezoactuators. Newcomb and Flinn [2] found that the relationship between the extension of a piezoceramic actuator and its applied electric charge has significantly less hysteresis nonlinearity than that between deformation and applied voltage. Furutani et al. [3] proposed an induced charge feedback control for the piezoactuators. The approach needs measurement of the induced charge and a specially designed charge drive amplifier, and will cause an increase in the response time of the actuator.

In order to linearize the control system, many researches focused on the inverse feedforward compensation based on some inverse hysteresis model. Several models have been suggested for describing the complex hysteretic behavior, for example, the Preisach model in Ge and Jouaneh [4,5], Yu et al. [6], and Liu et al. [7], the generalized Preisach model in Ge and Jouaneh [8], the dynamic Preisach model in Yu et al. [9]; the generalized Maxwell elasto-slip model in Goldfarb and Celanovic [10]; the variable time-relay hysteresis model in Tsai and Chen [11]; the Prandtl-Ishlinskii (PI) model (a subclass of the Preisach model) in Ang. et al. [1] and Hassani and Tjahjowidodo [12]; the Duhem model in Stepanenko and Su [13]; the polynomial approximation method in Croft and Devasia [14]; and the Jiles-Atherton model in Dupre et al. [15]. Ge and Jouaneh [5] proposed a PID feedback control using the classical Preisach model for the hysteresis. Song et al. [16] proposed a cascaded PD/lead-lag feedback controller based on a linear model for the piezoactuator with hysteresis being compensated via the feedforward cancellation using the inverse classical Preisach model. Recently, Maslan et al. [17] presented a discrete-time transfer function and its inverse for a highly nonlinear and hysteretic piezoelectric actuator, and traditional PID controller and PID with active force control were considered.

To mitigate the effects of the unknown hysteresis, Wang et al. [18] suggested a model reference control for linear systems with unknown input hysteresis using an inverse KP (Krasnosel’skii-Pokrovskii) hysteresis model [19]. Hwang et al. [20] proposed a neural-network nonlinear model for learning the hysteretic behavior of a piezoelectric actuator, and suggested a discrete-time variable-structure control for enhancing the nonlinear model-based feedforward control performance. Based on the learned nonlinear model of piezoelectric actuator systems in [20], Hwang and Jan [21] proposeed a controller including a nonlinear inverse control and a discrete neuroadaptive sliding mode control using a recurrent neural network to compensate for the residue dynamic uncertainty. Wai and Su [22] presented a supervisory genetic algorithm (SGA) control system for a piezoelectric ceramic motor. The controller consists of a GA control to search an optimum control effort online via gradient descent training process and a supervisory control to stabilize the system states around a predefined bound region.

Recently, Ronkanen et al. [23] presented a two-input (velocity and voltage) one-output (current) feedforward backpropagation network to model the inverse nonlinear velocity-current relation of a piezoelectric actuator, and then introduced a feedforward charge control scheme.

Other analytical types of nonlinear differential hysteresis models include the simplified Dahl model used in Lyshevski [24], Sun and Chang [25], Sain et al. [26], and the Bouc-Wen model in Low and Guo [27], Chen et al. [28], and Gomis-Bellmunt et al. [29]. Chen et al. [28] proposed an H_{∞} almost disturbance decoupling robust control based on the Bouc-Wen hysteretic model. Shieh et al. [30] proposed an adaptive displacement control for a piezopositioning mechanism with the LuGre (hysteretic) friction model suggested by De Wit et al. [31]. Gu and Zhu [32] suggested a new mathematic model to describe the frequency-dependent and amplitude-dependent hysteresis in a piezoelectric actuator using a family of ellipses. These analytical hysteresis models will be much easier for precision positioning control design.

In this work, we consider the precision control of a three-axis piezoactuated micro/nano stage. An adaptive controller with simplified Dahl model-based hysteresis variables observer is designed using the Lyapunov stability theory. In the adaptive controller, a fuzzy function approximator with parameters update law is included to compensate for the identification inaccuracy, model uncertainty, and flexure coupling effects. Simulation results are used for illustrating the possible control performance.

2. Dynamic Model for a Three-Axis Micro/Nano Motion Stage

The dynamic model for a single-axis piezoactuated flexure stage with analytic simplified Dahl hysteresis model is as below [24]:

(1)

(2)

where is the output displacement of the flexure stage; is the mass of the flexure mover; is the damping coefficient;, and are the stiffness constants; u is the input voltage of the piezoelectric actuator; is the input gain; is the hysteresis variable; and govern the scale and the shape of the hysteresis loop.

Consider a three-axis flexure micro/nano stage (P-517.3CL, Physik Instrumente, PI) [33] driven by piezoelectric actuators shown in Figure 1. The hysteresis phenomena and the coupling effects among the three axes induced by the flexure structure, can be taken into account via the following complete matrix-vector model:

Figure 1. Three-axis flexure stage.

(3)

(4)

where is the output displacements vector;

is used to consider the coupling effects among the axes and the model uncertainty;

For ease of numerical simulation and implementation, the system parameters in SI units could be scaled in terms of more suitable units: displacement in nm, mass in g, time in ms, and input voltage in mV. After scaling, the scaled models keep the same forms as Equations (3) and (4). The parameters of the stage are identified, based on input/output data pairs via genetic algorithms by Chang [34], and are given as follows:

After defining the state vector as, , the stage’s dynamic model can be written in the following vector state equations:

(5)

where

3. Stable Fuzzy Approximator-Based Adaptive Control for Micro/Nano Stages

3.1. Control Design Using Backstepping Method

Based on the nonlinear dynamics model (5), this subsection considers the backstepping-based stable control law design for the three-axis flexure stage.

First consider the subsystem,. Let

(6)

where is a virtual input. Define the tracking error signal as

(7)

where is the desired trajectory for the three-axis motion. Differentiating Equation (7), we have

(8)

Considering the Lyapunov function candidate

(9)

where is symmetric and positive definite, and differentiating Equation (9), we have

(10)

Thus, we can choose the virtual input as

(11)

with positive definite feedback gain matrix

such that

(12)

and that is, the subsystem is asymptotically stable.

Further, the actual whole nonlinear system is considered:

(13)

After introducing new error signal

(14)

we can obtain

(15)

Then by considering the Lyapunov function candidate as

(16)

where, ,

is symmetric and positive definite, and taking the time derivative of Equation (16), we have

(17)

Thus we can choose the nonlinear control law as follows:

(18)

and obtain

(19)

where .

If further choose with, then we can have

(20)

and. Thus, the equilibrium point of the closed-loop system is exponentially stable.

The internal state variables can also be shown to be bounded. Consider the Lyapunov function

(21)

By choosing the class- functions

then since

, (22)

we know that is positive definite, decrescent, and radially unbounded [35]. Differentiating Equation (21) and substituting in the internal dynamics

(23)

we have

(24)

Since

if the desired trajectory satisfies, then we can have Thus, f is bounded and the overall closed-loop system is stable.

Let the output vector, , we can obtain the system’s zero dynamics as follows:

(25)

That is, the hysteresis variables will become constants when the flexure mover returns to the origin and remains there.

In order to further enhance the system’s active damping capability, we can introduce a nonlinear damping term

where into the control law (18). That is, the control law can be modified as

(26)

where and are the nominal matrices for and, respectively, obtained by substituting in the estimated parameters, and represents the discrepancy due to the estimate error. Let be the integral uncertainty, we can further design a fuzzy function approximator to compensate for its effect. The modified control law can be written as follows:

(27)

where is the observed hysteresis vector for, z is the input vector of the controller, and is the parameters vector to be updated for the fuzzy compensator. Here

and

are used in Equation (27). The hysteresis observer and the fuzzy compensator design will be considered in the sequel.

3.2. Hysteresis Observer Design

Since the hysteresis variables are difficult to measure for feedback, a nonlinear observer can be suggested as:

(28)

where are the estimated hysteresis variables, are the observer’s input variables to be defined later in the derivation of the stable control law and parameters update law, and are the input gains. Define estimate errors as

we have

(29)

And Equation (29) can be written in the following vector form:

(30)

where

is the estimate error vector, and

,

.

3.3. Fuzzy Function Approximators Design

This subsection will construct the fuzzy function approximators using T-S fuzzy systems to compensate for the modeling errors and coupling effects among the three axes. The tracking errors are chosen respectively as the input variable of the fuzzy approximator for each axis, and the compensating voltage of each axis is the output variable. In the universe of discourse of each input variable, five fuzzy sets are defined as in Figure 2. The rule base of the fuzzy approximator for the i-th () axis is considered as follows: Rule j: If is

Then

(31)

where are the fuzzy sets defined over the universe of discourse of each input variable, , stands for the, y, andaxis, respectively.

Using singleton fuzzifier, product inference engine, and center average defuzzifier [36], the mapping of the fuzzy approximator for the i-th axis is

(32)

where is the degree of firing of the j-th rule’s antecedent. Let

Figure 2. Membership functions for each axis.

then

(33)

Defining the regressor vector

and the unknown parameter vector

Equation (33) can be written as

(34)

And the fuzzy approximators for the three axes can be written in the vector form as

(35)

where

3.4. Derivation of Parameters Update Law and Stability of Overall System

In this subsection, the input signals

of the hysteresis observer, and the parameters update laws of the fuzzy function approximators will be selected in the stability consideration of the overall adaptive feedback control system for a three-axis piezoelectric flexure stage.

Consider the following Lyapunov function candidate,

(36)

where is symmetric and positive definite, Taking the time derivative, we have

(37)

After substituting in Equations (30) and (17), Equation (37) becomes (38).

Let

where

and

are the error matrices, is the identity matrix. Since, where

we have

and Equation (38) can be written as

(38)

(39)

where is defined as (40).

Choosing the input vector of the hysteresis observer as:

(40)

(41)

That is,

(42)

we can obtain

(43)

By further representing the uncertainty as:

and substituting

in Equation (43), we have

(44)

Thus, we can choose the parameters adaptation law of the fuzzy approximators as:

(45)

If further choose, , and assume the approximation error be bounded, i.e., , then we can obtain

(46)

Letting

(47)

and defining functions:

and Equation (46) can be rewritten as

(48)

Hence, when

or

or , and thus the overall adaptive control system is boundedly stable.

4. Results and Discussion

In this section computer simulation will be used to illustrate the performance of the proposed adaptive fuzzy control with hysteresis observer for a three-axis flexure stage. Triangular uncertainties for the x, y, and z axes () shown in Figure 3 are selected in the simulation. The desired trajectories for the x, y, and z axes are selected as follows (t in ms):

(a)(b)(c)

Figure 3. Triangular uncertainties for the x, y, and z axes. (a) D_{x}, (b) D_{y}, (c) D_{z}.

Controller parameters are selected as follows:

The simulation results are shown in Figure 4. From Figures 4(a)-(c), we know that the tracking performances are very good. The tracking errors of xand y-axes are within –2.5 nm - 2.2 nm, and the tracking error of z-axis is within ±2 nm. From Figures 4(d)-(f), the hysteresisvariable estimate errors of xand y-axes are within ±0.5 nm, and the estimate error of z-axis is within ±1 nm. The control voltages are shown in Figure 4(g), and the fuzzy compensation voltages, , and are shown in Figure 4(h). And the parameters update processes of the function approximators for x-, y-, and z-axes are shown in Figures 4(i)-(k), respectively. The parameters of the first and fifth rules are not updated since the tracking errors are small and they are nearly not fired. Although the persistent exciting of the system signals of this considered simulation case are not sufficient enough to let the other parameters converge to constants, the adaptive control system can guarantee the tracking control performance to be still very good.

5. Conclusion

In this work, a stable adaptive control law with nonlinear dynamic hysteresis observer for a three-axis flexure stage

(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)

Figure 4. Control results. (a) x-axis, (b) y-axis, (c) z-axis tracking errors; Hysteresis estimate errors, (d); (e); (f); (g) Control voltages, and; (h) Compensating voltages, and; Parameters adaptation process (i) x-axis; (j) y-axis; (k) z-axis.

is proposed. Fuzzy function approximators are included in the control law to compensate for the identification inaccuracy, model uncertainty, and flexure coupling effect. The stability of the overall closed-loop system is guaranteed using the Lyapunov theory. Simulation results are shown to illustrate the effectiveness of the suggested control approach. In the future study, actual implementation can be considered for the development of a precision stage for testing the control performance.

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