Adaptive fuzzy trajectory control for biaxial motion stage system

Abstract

Motion control is an essential part of industrial machinery and manufacturing systems. In this article, the adaptive fuzzy controller is proposed for precision trajectory tracking control in biaxial X-Y motion stage system. The theoretical analyses of direct fuzzy control which is insensitive to parameter uncertainties and external load disturbances are derived to demonstrate the feasibility to track the reference trajectories. The Lyapunov stability theorem has been used to testify the asymptotic stability of the whole system, and all the signals are bounded in the closed-loop system. The intelligent position controller combines the merits of the adaptive fuzzy control with robust characteristics and learning ability for periodic command tracking of a servo drive mechanism. The simulation and experimental results on square, triangle, star, and circle reference contours are presented to show that the proposed controller indeed accomplishes the better tracking performances with regard to model uncertainties. It is observed that the convergence of parameters and tracking errors can be faster and smaller compared with the conventional adaptive fuzzy control in terms of average tracking error and tracking error standard deviation.

Introduction

Design and control of multi-axis motion systems have become a vitally important part in modern industry applications.^1–9 Precise contour control for the X-Y stage mechanism has attracted much attention in many areas, such as work feeder of computer numerical control (CNC) lathe, CNC milling or drill press machines, and the work table of laser cutting and painting. Due to high stiffness and resolution properties, ball-screw architectures are usually employed in linear table which is broadly used in industry automation equipment, semiconductor manufacturing processes, and pharmaceutical machines. In general, X-Y stage is composed of X-axis and Y-axis motion mechanisms, in which each motion axis is driven by individual actuator such as DC and AC servomotor through high precision ball-screw module. There often exist several kinds of disturbance sources in this architecture, such as nonlinear friction, unmodeled dynamics, load disturbance, and coupled interference, and these will affect the precision and tracking performance of the controlled system.

The contour tracking system techniques can be classified as zero-phase error tracking control (ZPETC) methods,^2,3 ${\mathrm{H}}_{\infty }$ control method, ⁵ adaptive neural network control methods,^6–8 adaptive sliding-mode control methods,^9,10 adaptive task coordinate frame-based control,^11,12 and so on. The nonminimum phase model (NMP) ³ is obtained from input and output experiment data, and the ZPETC strategy ² is employed to eliminate the phase error caused by NMP zero effect for low frequencies. However, the mathematical models of X-Y table servo performance are greatly influenced by uncertainties, including parameter variations, external loads, and unmodeled dynamics in the drive system. The contouring control^10–14 is developed using the task coordinate frame to estimate the contour error to detour the direct computation. The contour error is not always the same as the tracking error. The task polar coordinate frame (TPCF) method can calculate and control the estimation error by circular approximation. By transforming the system dynamics from the Cartesian coordinate frame to TPCF, the contour error and angular error can be estimated explicitly, and the corresponding error can be drastically reduced. By integrating the task coordinate frame and the friction model, the observer-based controller ¹⁰ is developed for precision position control. The state observer is designed to solve the frictional characteristic using the position feedback information. A H_∞-based precision control system that deals with $\mathrm{X}-\mathrm{Y}-\theta$ platform is proposed ⁵ for a linear motor direct drive mechanism. According to the reduced order modeling structure, the velocity controller considering three most significant resonance modes is designed. The algorithm is experimented with the selection of the weighting function for robust control design.

The sliding-mode control techniques for X-Y table of the CNC machine and nonlinear systems are presented in Lin et al., ⁹ Wang et al., ¹⁵ Hsueh and Su, ¹⁶ and Ahmed et al. ¹⁷ The adaptive fuzzy sliding approximation system ¹⁷ is proposed to consider the nth-order single-input and single-output (SISO) nonlinear systems. An adaptive law containing an approximate error feedback scheme is utilized, and its stability is also proved via Lyapunov theorems. The recurrent neural network (RNN) integral sliding-mode control system ¹⁸ based on a computed-torque design is developed to track various reference contours. This robust system management controller (SMC) is presented to deal with the uncertainty terms, while the RNN structure can eliminate the disadvantage of high-frequency chattering phenomenon for systems. With regard to the intelligent control methods, neural networks and fuzzy logic systems have been widely employed to the various control systems and mechatronic servo systems.^15,19–24 Wang^4,5 had proposed stable direct and indirect adaptive fuzzy controllers based on Lyapunov synthesis approach for SISO nonlinear systems. Yin and Lee ²⁰ proposed a fuzzy model reference adaptive controller (MRAC) for the system with unknown parameters. Both the tracking error and the prediction error are combined, and the fuzzy MRAC provides more adaptation power than a traditional adaptive control solely based on either tracking or prediction error. Hsueh and Su ¹⁶ proposed a direct adaptive fuzzy control design with the use of the compensative controller. Based on the finite ${\mathrm{L}}_2$ gain property, the compensative controller provided a referable approximate error estimator for the fuzzy system and had the better learning performance. To maintain the performance of fuzzy adaptive control in the presence of external disturbances, some robust schemes based on fuzzy sliding-mode control technique are presented in the literature.^{15–17,22–27} Hsueh and Su ²⁶ and Lu and Chen ²⁷ used an additional switching controller to attenuate the influence of the approximation error and external disturbances. However, these methods may lead to higher frequency switching control signals and excite high-frequency modes of unmodeled dynamics. Also, systems’ variables vary continually by external disturbances and friction force in the practical applications. Kim and Bien ²² proposed the adaptive fuzzy controllers in situations where the system model includes the uncertainties and unknown bound external disturbance. To resolve this problem, we propose an adaptation scheme which can improve the robustness of adaptive fuzzy control in X-Y stage dynamic system.

The aim of this research is to design an intelligent controller that ensures to have a smaller trajectory tracking error. The single-axis dynamics of the servo system will be considered, and the corresponding model included the terms of a lumped uncertainty with parameter variations, external disturbances, and friction effect. This mechanism is actuated by a permanent magnet synchronous motor (PMSM) and controlled using an adaptive direct fuzzy system in this study. To alleviate the effect of unmodeled dynamics and external disturbance in X-Y stage, an output feedback control law and an update law are derived for online tuning the weighting factors of the fuzzy method. Using strictly positive-real Lyapunov theory, the stability of the closed-loop system compensated by the fuzzy control can be verified. The overall adaptive scheme guarantees that all signals involved are bounded, and the output of the closed-loop system asymptotically tracks the desired X-Y plane contour trajectory. It is shown that the tracking trajectory errors of the X-Y stage can converge smoothly and fast than the conventional adaptive fuzzy control methods.

This article is organized as follows: The dynamic model of X-Y stage and fuzzy logic scheme are introduced in section “System model of biaxial motion system.” The direct adaptive fuzzy controller design is described in section “Direct adaptive fuzzy control method.” Section “Proposed direct fuzzy control method” presents the proposed adaptive fuzzy control structure for the various contour tracking performances. Section “Simulation and experimental results” provides the simulation results. Finally, a brief conclusion is given in section “Conclusion.”

System model of biaxial motion system

Stage dynamic model

The typical mathematical modeling of the PMSM^7–9,18,26 in the synchronously rotating rotor reference frames can be derived in the following. The stator voltage equations in the d-q reference frame can be represented as follows

$${\mathrm{V}}_{\mathrm{qi}}={\mathrm{R}}_{\mathrm{si}}{\mathrm{i}}_{\mathrm{qi}}+{\omega }_{\mathrm{si}}{\lambda }_{\mathrm{di}}+\mathrm{\rho }{\lambda }_{\mathrm{qi}}$$

(1)

$${\mathrm{V}}_{\mathrm{di}}={\mathrm{R}}_{\mathrm{si}}{\mathrm{i}}_{\mathrm{di}}-{\omega }_{\mathrm{si}}{\lambda }_{\mathrm{qi}}+\mathrm{\rho }{\lambda }_{\mathrm{di}}$$

(2)

with ${\mathrm{\lambda }}_{\mathrm{qi}}={\mathrm{L}}_{\mathrm{qi}}{\mathrm{i}}_{\mathrm{qi}}$ , ${\mathrm{\lambda }}_{\mathrm{di}}={\mathrm{L}}_{\mathrm{di}}{\mathrm{i}}_{\mathrm{di}}+{\mathrm{\lambda }}_{\mathrm{fi}}$ , ${\mathrm{\omega }}_{\mathrm{si}}={\mathrm{n}}_{\mathrm{pi}}{\mathrm{\omega }}_{\mathrm{ri}}$ , where $\mathrm{i}=\mathrm{x},\mathrm{y}$ (x and $\mathrm{y}$ are the stage axes), ${\mathrm{V}}_{\mathrm{qi}}$ and ${\mathrm{V}}_{\mathrm{di}}$ are the d-q-axis voltages, ${\mathrm{i}}_{\mathrm{qi}}$ and ${\mathrm{i}}_{\mathrm{di}}$ are the d-q-axis currents, ${\mathrm{R}}_{\mathrm{si}}$ is the phase winding resistance, ${\mathrm{L}}_{\mathrm{di}}$ and ${\mathrm{L}}_{\mathrm{qi}}$ are the d-q-axis inductances, ${\lambda }_{\mathrm{fi}}$ is the permanent magnet flux linkage, $\rho$ denotes the differential operator, ${\omega }_{\mathrm{ri}}$ is the rotor speed, ${\omega }_{\mathrm{si}}$ is inverter frequency, and ${\mathrm{n}}_{\mathrm{pi}}$ is the number of primary pole pairs. Rearranging equations in matrix form, it can be given as

$$\left[\begin{array}{c}{\mathrm{V}}_{\mathrm{qi}}\hfill \\ \multicolumn{1}{c}{{\mathrm{V}}_{\mathrm{di}}}\end{array}\right]=\left[\begin{array}{cc}{\mathrm{R}}_{\mathrm{si}}+\mathrm{\rho }{\mathrm{L}}_{\mathrm{qi}}\hfill & {\omega }_{\mathrm{ri}}{\mathrm{L}}_{\mathrm{di}}\hfill \\ \multicolumn{1}{c}{-{\omega }_{\mathrm{ri}}{\mathrm{L}}_{\mathrm{qi}}}& {\mathrm{R}}_{\mathrm{si}}+\mathrm{\rho }{\mathrm{L}}_{\mathrm{di}}\hfill \end{array}\right]\left[\begin{array}{c}{\mathrm{i}}_{\mathrm{qi}}\hfill \\ \multicolumn{1}{c}{{\mathrm{i}}_{\mathrm{di}}}\end{array}\right]+\left[\begin{array}{c}{\omega }_{\mathrm{ri}}{\lambda }_{\mathrm{fi}}\hfill \\ \multicolumn{1}{c}{\rho {\lambda }_{\mathrm{fi}}}\end{array}\right]$$

(3)

The developed torque of motor is given as

$${\mathrm{T}}_{\mathrm{ei}}=\frac{3}{2}\left(\frac{\mathrm{P}}{2}\right)\left({\lambda }_{\mathrm{di}}{\mathrm{i}}_{\mathrm{qi}}-{\lambda }_{\mathrm{qi}}{\mathrm{i}}_{\mathrm{di}}\right)=\frac{3\mathrm{P}}{4}\left[{\lambda }_{\mathrm{di}}{\mathrm{i}}_{\mathrm{qi}}+\left({\mathrm{L}}_{\mathrm{di}}-{\mathrm{L}}_{\mathrm{qi}}\right){\mathrm{i}}_{\mathrm{di}}{\mathrm{i}}_{\mathrm{qi}}\right]$$

(4)

Based on realization of field-oriented control, the developed electromagnetic thrust force is given by

$${\mathrm{F}}_{\mathrm{ei}}={\mathrm{K}}_{\mathrm{ti}}{\mathrm{i}}_{\mathrm{q}}^{*}=\left(\frac{3\mathrm{\pi }}{2\mathrm{\tau }}{\lambda }_{\mathrm{fi}}\right){\mathrm{i}}_{\mathrm{q}}^{*}$$

(5)

where ${\mathrm{K}}_{\mathrm{ti}}$ is the force constant, ${\lambda }_{\mathrm{fi}}$ is the permanent magnet flux linkage, and $\tau$ is the pole pitch. The dynamic equation of the single-axis mechanism with the friction model included is represented as

$${\mathrm{M}}_{\mathrm{i}}{\stackrel{··}{\mathrm{x}}}_{\mathrm{i}}+{\mathrm{B}}_{\mathrm{i}}{\stackrel{·}{\mathrm{x}}}_{\mathrm{i}}+{\mathrm{F}}_{\mathrm{Li}}+{\mathrm{F}}_{\mathrm{fi}}\left({\mathrm{v}}_{\mathrm{i}}\right)={\mathrm{F}}_{\mathrm{ei}}$$

(6)

where $\mathrm{i}=\mathrm{x},\mathrm{y}$ (x and y denote the axes), ${\mathrm{x}}_{\mathrm{i}}$ is the platform displacement, ${\mathrm{M}}_{\mathrm{i}}$ is the total mass of the moving stage, ${\mathrm{B}}_{\mathrm{i}}$ is the viscous friction coefficient, ${\mathrm{F}}_{\mathrm{Li}}$ is the external disturbance term including loading and cross-coupled interference, ${\mathrm{F}}_{\mathrm{fi}}({\mathrm{v}}_{\mathrm{i}})$ is the friction torque, and ${\mathrm{v}}_{\mathrm{i}}$ is the linear velocity of X- (Y-) axis. Considering Coulomb friction, viscous friction, and Stribeck effect, the friction force can be formulated as follows

$$\begin{array}{l}{\mathrm{F}}_{\mathrm{fi}}\left({\mathrm{v}}_{\mathrm{i}}\right)={\mathrm{F}}_{\mathrm{Ci}}\mathrm{sgn}\left({\mathrm{v}}_{\mathrm{i}}\right)+\left({\mathrm{F}}_{\mathrm{Si}}-{\mathrm{F}}_{\mathrm{Ci}}\right)\\ \exp \left[-{\left(\frac{{\mathrm{v}}_{\mathrm{i}}}{{\mathrm{v}}_{\mathrm{Si}}}\right)}^2\right]\mathrm{sgn}\left({\mathrm{v}}_{\mathrm{i}}\right)+{\mathrm{K}}_{\mathrm{Vi}}{\mathrm{v}}_{\mathrm{i}}\end{array}$$

(7)

where $F_{\mathrm{C}i}$ is the Coulomb friction, ${\mathrm{F}}_{\mathrm{Si}}$ is the static friction, ${\mathrm{v}}_{\mathrm{si}}$ is the Stribeck velocity parameter, ${\mathrm{K}}_{\mathrm{vi}}$ is the coefficient of viscous friction, and $\mathrm{sgn}(·)$ is a sign function. All the parameters in equation (7) are time varying. The mathematical equation of mechanical X-Y stage servo system can be represented as

$$\stackrel{··}{\mathrm{x}}={\mathrm{A}}_{\mathrm{s}}\stackrel{·}{\mathrm{x}}+{\mathrm{B}}_{\mathrm{s}}[{\mathrm{F}}_{\mathrm{e}}-{\mathrm{F}}_{\mathrm{L}}-{\mathrm{F}}_{\mathrm{fi}}\left({\mathrm{v}}_{\mathrm{i}}\right)]$$

(8)

with ${\mathrm{A}}_{\mathrm{s}}=-\mathrm{B}/\mathrm{M}$ and ${\mathrm{B}}_{\mathrm{s}}=1/\mathrm{M}$

In practical application, the actual system model changes when the drive system operates under different circumstances. The mechanical dynamics in equation (8) with parameter variations, disturbance load, and unpredictable uncertainties will be considered as

$$\begin{array}{l}\stackrel{\mathrm{··}}{\mathrm{x}}=\left({\mathrm{A}}_{\mathrm{sn}}+\Delta {\mathrm{A}}_{\mathrm{s}}\right)\stackrel{.}{\mathrm{x}}+\left({\mathrm{B}}_{\mathrm{sn}}+\Delta {\mathrm{B}}_{\mathrm{s}}\right)\\ \left[{\mathrm{F}}_{\mathrm{e}}-{\mathrm{F}}_{\mathrm{L}}-{\mathrm{F}}_{\mathrm{fi}}\left({\mathrm{v}}_{\mathrm{i}}\right)\right]={\mathrm{A}}_{\mathrm{sn}}\stackrel{.}{\mathrm{x}}+{\mathrm{B}}_{\mathrm{sn}}\mathrm{U+D}\end{array}$$

(9)

with $\mathrm{D}=\Delta {\mathrm{A}}_{\mathrm{s}}\stackrel{·}{\mathrm{x}}-{\mathrm{B}}_{\mathrm{sn}}[{\mathrm{F}}_{\mathrm{L}}+{\mathrm{F}}_{\mathrm{fi}}({\mathrm{v}}_{\mathrm{i}})]+\Delta {\mathrm{B}}_{\mathrm{s}}[{\mathrm{F}}_{\mathrm{e}}-{\mathrm{F}}_{\mathrm{L}}-{\mathrm{F}}_{\mathrm{fi}}({\mathrm{v}}_{\mathrm{i}})]$ , where ${\mathrm{A}}_{\mathrm{sn}}$ and ${\mathrm{B}}_{\mathrm{sn}}$ are nominal values of ${\mathrm{A}}_{\mathrm{s}}$ and ${\mathrm{B}}_{\mathrm{s}}$ , respectively; $\Delta {\mathrm{A}}_{\mathrm{s}}$ and $\Delta {\mathrm{B}}_{\mathrm{s}}$ are the uncertainties of the mechanical parameters, respectively; and D is the lumped disturbance parameter which is bounded. It is assumed that $\left|\mathrm{D}\right|<{\mathrm{D}}^{\mathrm{U}}$ .

Fuzzy controller structure

Adaptive fuzzy systems combine adaptive control theory and fuzzy systems cooperated with training algorithms used to adjust the parameters according to numerical input or output data. Many attractive features of fuzzy systems are presented such as ability of real-time learning, universal function approximation, and the identification of input and output for unknown systems. The proposed fuzzy system with a static system mapping from ${\mathrm{R}}^2$ to $\mathrm{R}$ is presented. The fuzzy rules are composed of the following structure. It is given as

$$\begin{array}{l}{\mathrm{R}}^{\mathrm{l}}:\mathrm{IF}{\mathrm{x}}_1\mathrm{is}{\mathrm{A}}_1^{\mathrm{l}}\mathrm{and}{\mathrm{x}}_2\mathrm{is}{\mathrm{A}}_2^{\mathrm{l}},\\ \mathrm{THEN}{\mathrm{y}}_{\mathrm{F}}\mathrm{is}{\theta }^{\mathrm{l}},\mathrm{for}\mathrm{l}=1,2,\dots ,\mathrm{M}\end{array}$$

where $\mathbf{x}=[{\mathrm{x}}_1,{\mathrm{x}}_2]$ is the input vector, ${\mathrm{y}}_{\mathrm{F}}$ is the output the fuzzy approximator, ${(\mathrm{A}}_1^{\mathrm{l}}{,\mathrm{A}}_2^{\mathrm{l}})$ and ${\theta }^{\mathrm{l}}$ are the corresponding fuzzy sets, $\mathrm{l}$ is the rule index, and M is the total number of rules. The fuzzy system ⁵ with a singleton fuzzier, product inference, and a center-average defuzzier is designed as follows

$${\mathrm{y}}_{\mathrm{F}}\left(\mathbf{x}\right)=\frac{\sum _{\mathrm{i}=1}^{\mathrm{M}}{\theta }^{\mathrm{l}}\left(\underset{\mathrm{j}=1}{\overset{\mathrm{N}}{\Pi }}{\mu }_{{\mathrm{A}}_{\mathrm{i}}}^{{\mathrm{l}}_{\mathrm{i}}}\left({\mathrm{x}}_{\mathrm{i}}\right)\right)}{\sum _{\mathrm{i}=1}^{\mathrm{M}}\left(\underset{\mathrm{j}=1}{\overset{\mathrm{N}}{\Pi }}{\mu }_{{\mathrm{A}}_{\mathrm{i}}}^{{\mathrm{l}}_{\mathrm{i}}}\left({\mathrm{x}}_{\mathrm{i}}\right)\right)}$$

(10)

where ${\mu }_{{\mathrm{A}}_{\mathrm{i}}}^{{\mathrm{l}}_{\mathrm{i}}}({\mathrm{x}}_{\mathrm{i}})$ is the membership function of the fuzzy variable. Each input variable is transformed strength based on the corresponding Gaussian membership functions. The membership function is formulated as

$${\mu }_{{\mathrm{A}}_{\mathrm{i}}}^{{\mathrm{l}}_{\mathrm{i}}}\left({\mathrm{x}}_{\mathrm{i}}\right)=\exp [-\frac{{({\mathrm{x}}_{\mathrm{i}}-{\mathrm{v}}_i)}^2}{{\sigma }_{\mathrm{i}}^2}]\mathrm{for}\mathrm{i}=1,2,\dots ,\mathrm{N}$$

(11)

where ${\mathrm{v}}_{\mathrm{i}}$ and ${\sigma }_{\mathrm{i}}$ are the mean and variance of the ith Gaussian function, respectively. N is the number of Gaussian membership function. The sum of the weight vector yields the fuzzy control output. The fuzzy control is given as

$${\mathrm{y}}_{\mathrm{F}}\left(\mathbf{x}\right)={\mathbf{\theta }}^{\mathrm{T}}\mathbf{\xi }\left(\mathbf{x}\right)$$

where $\mathbf{\theta }=[{\theta }^1,{\theta }^2,\dots ,{\theta }^{\mathrm{M}}{]}^{\mathrm{T}}$ is a vector of adjustable parameters and $\mathbf{\xi }(\mathbf{x})=[{\mathbf{\xi }}^1,{\mathbf{\xi }}^2,\dots ,{\mathbf{\xi }}^{\mathrm{M}}{]}^{\mathrm{T}}$ is the fuzzy basis function vector which is defined as

$${\mathbf{\xi }}^{\mathrm{l}}\left(\mathbf{x}\right)=\frac{\underset{\mathrm{j}=1}{\overset{\mathrm{N}}{\Pi }}{\mu }_{{\mathrm{A}}_{\mathrm{i}}}^{{\mathrm{l}}_{\mathrm{i}}}\left({\mathrm{x}}_{\mathrm{i}}\right)}{\sum _{\mathrm{i}=1}^{\mathrm{M}}\left(\underset{\mathrm{j}=1}{\overset{\mathrm{N}}{\Pi }}{\mu }_{{\mathrm{A}}_{\mathrm{i}}}^{{\mathrm{l}}_{\mathrm{i}}}\left({\mathrm{x}}_{\mathrm{i}}\right)\right)}$$

(13)

The fuzzy system above can be proven in Lemma 1 to be a universal approximation in our controller design.

Lemma 1 ⁴

Supposed that the input universe of discourse U is a compact set in ${\mathrm{R}}^{\mathrm{n}}$ . For any given real continuous function $\mathrm{g}(\mathbf{x})$ on U and arbitrary $\epsilon >0$ . The fuzzy system output variable in the form of equations (10) and (12) is defined as

$$\underset{\mathbf{x}\in \mathrm{U}}{sup}|{\mathrm{y}}_{\mathrm{F}}\left(\mathbf{x}\right)-\mathrm{g}\left(\mathbf{x}\right)|<\epsilon$$

(14)

It is said that the fuzzy system with product inference engine, singleton fuzzier, Gaussian membership function, and center-average defuzzier is universal approximator. It can be used as an adaptive fuzzy controller for nonlinear motion stage system.

Direct adaptive fuzzy control method

In this section, a conventional direct adaptive fuzzy controller is described. A second-order dynamic system is represented by the following form

$$\stackrel{··}{\mathrm{x}}={\mathrm{A}}_{\mathrm{sn}}\stackrel{·}{\mathrm{x}}+{\mathrm{B}}_{\mathrm{sn}}\mathrm{U}$$

(15)

$$\mathrm{y}=\mathrm{x}$$

(16)

where x is the position variable of stage; ${\mathrm{A}}_{\mathrm{sn}}$ is the unknown system function; ${\mathrm{B}}_{\mathrm{sn}}$ is unknown positive constant; and $\mathrm{u}\in \mathrm{R}$ and $\mathrm{y}\in \mathrm{R}$ are input and output of the system plant, respectively. Let $\mathbf{x}=[\mathrm{x},\stackrel{·}{\mathrm{x}}{]}^{\mathrm{T}}=[{\mathrm{x}}_1,{\mathrm{x}}_2{]}^{\mathrm{T}}\in {\mathrm{R}}^2$ is the vector of states, and the reference output vector is ${\mathbf{y}}_{\mathrm{m}}=[{\mathrm{y}}_{\mathrm{m}},{\stackrel{·}{\mathrm{y}}}_{\mathrm{m}}{]}^{\mathrm{T}}\in {\mathrm{R}}^2$ . Define the output tracking error $\mathrm{e}={\mathrm{y}}_{\mathrm{m}}-\mathrm{y}={\mathrm{y}}_{\mathrm{m}}-\mathrm{x}$ , and the tracking error vector is $\mathbf{e}={[\mathrm{e},\stackrel{·}{\mathrm{e}}]}^{\mathrm{T}}={[{\mathrm{e}}_1,{\mathrm{e}}_2]}^{\mathrm{T}}$ . The tracking system can be rewritten as

$$\stackrel{·}{\mathbf{x}}={\mathbf{A}}_{\mathrm{s}}\mathbf{x}+{\mathbf{B}}_{\mathrm{s}}[{\mathrm{A}}_{\mathrm{sn}}\stackrel{·}{\mathrm{x}}+{\mathrm{B}}_{\mathrm{sn}}\mathrm{U}]$$

(17)

$$\mathrm{y}={\mathbf{C}}_{\mathrm{s}}^{\mathrm{T}}\mathbf{x}$$

(18)

with

$${\mathbf{A}}_{\mathrm{s}}=\left[\begin{array}{cc}0\hfill & 1\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]$$

$${\mathbf{B}}_{\mathrm{s}}=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]$$

$${\mathbf{C}}_{\mathrm{s}}=\left[\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]$$

The optimal control law can be defined as follows

$${\mathrm{u}}^{*}=\frac{1}{\mathrm{b}}[-{\mathrm{A}}_{\mathrm{sn}}{\stackrel{·}{\mathrm{x}}+\mathrm{y}}_{\mathrm{m}}^{(2)}+{\mathbf{K}}^{\mathrm{T}}\mathbf{e}]$$

(19)

where $\mathbf{K}=[{\mathrm{k}}_2,{\mathrm{k}}_1{]}^{\mathrm{T}}$ is the feedback control gain vector. It is chosen such that the characteristic polynomial ${\mathbf{A}}_{\mathrm{s}}-{\mathbf{B}}_{\mathrm{S}}\mathbf{K}$ is Hurwitz because $({\mathbf{A}}_{\mathrm{s}},{\mathbf{B}}_{\mathrm{S}})$ is controllable. The closed-loop dynamic equation becomes $\stackrel{··}{\mathrm{e}}+{\mathrm{k}}_1\stackrel{·}{\mathrm{e}}+{\mathrm{k}}_2\mathrm{e}=0$ . Based on choosing vector $\mathbf{K}$ , we can obtain the error which will be approaching zero when the time approaches infinity. Since ${\mathrm{A}}_{\mathrm{sn}}$ and ${\mathrm{B}}_{\mathrm{sn}}$ are unknown function and variable in the control system, the ideal control law in equation (19) cannot be implemented. A feedback controller can be designed based on the fuzzy method and an adaptation law such that the plant output can track the ideal output variable ${\mathrm{y}}_{\mathrm{m}}$ . The control input of a direct fuzzy method is

$$\mathrm{u}={\mathrm{u}}_{\mathrm{D}}(\mathbf{x}|\theta )={\mathrm{y}}_{\mathrm{F}}\left(\mathbf{x}\right)$$

(20)

where ${\mathrm{u}}_{\mathrm{D}}$ is a fuzzy system and $\theta$ is the adjustable parameter sets. Substituting equations (19) and (20) into equation (15), the following state error dynamic equations are obtained

$${\mathrm{x}}^{(2)}=-\mathrm{b}{\mathrm{u}}^{*}{+\mathrm{y}}_{\mathrm{m}}^{(\mathrm{n})}+{\mathbf{K}}^{\mathrm{T}}\mathbf{e}+\mathrm{b}{\mathrm{u}}_{\mathrm{D}}$$

(21)

$${\mathrm{e}}^{(2)}=-{\mathbf{K}}^{\mathrm{T}}\mathbf{e}+\mathrm{b}[{\mathrm{u}}^{*}-{\mathrm{u}}_{\mathrm{D}}(\mathbf{x}|\mathbf{\theta })]$$

(22)

So, the closed-loop dynamic equation can be rewritten in vector form as follows

$$\stackrel{·}{\mathbf{e}}=\mathbf{Ae}+\mathbf{b}[{\mathrm{u}}^{*}-{\mathrm{u}}_{\mathrm{D}}(\mathbf{x}|\mathbf{\theta })]$$

(23)

with

$$\mathbf{A}=\left[\begin{array}{cc}0\hfill & 1\hfill \\ \multicolumn{1}{c}{-{\mathrm{k}}_2}& -{\mathrm{k}}_1\hfill \end{array}\right]$$

$$\mathbf{b}=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{\mathrm{b}}\end{array}\right]$$

Since matrix $\mathbf{A}$ is a Hurwitz stable matrix, there exists a unique positive definite symmetric $2\times 2$ matrix. $\mathbf{P}$ is a positive definite matrix. Also, matrix $\mathbf{P}$ needs to satisfy the Lyapunov function ${\mathbf{A}}^{\mathrm{T}}\mathbf{P}+\mathbf{P}\mathrm{A}=-\mathbf{Q}$ , where $\mathbf{Q}=\mathrm{q}\mathbf{I}$ , in which $\mathbf{I}$ is an identity matrix with order 2 and $\mathrm{q}>0$ is a constant. Defining optimal parameter is as follows

$${\theta }^{*}=\arg \underset{\theta \in {\Omega }_{\theta }}{min}[\underset{x\in {\Omega }_{\mathrm{s}}}{sup}|{\mathrm{u}}_{\mathrm{D}}(\mathbf{x}|\mathbf{\theta })-{\mathrm{u}}^{*}|]$$

(24)

where ${\Omega }_{\mathrm{s}}$ and ${\Omega }_{\theta }$ are the constant sets of suitable bound of $\mathbf{x}$ and $\mathbf{\theta }$ , respectively. As a matter of fact, the fuzzy method with a finite number of rules can only provide a limited accuracy to the approximation of a nonlinear system. The closed-loop dynamic equation can be written as

$$\stackrel{.}{e}=Ae+\mathbf{b}\left({\mathrm{u}}_{\mathrm{D}}\left(x|\theta \right)-{\mathrm{u}}_{\mathrm{D}}\left(x|\theta \right)\right)-\mathbf{b}\left({\mathrm{u}}_{\mathrm{D}}\left(x|\theta \right)-{\mathrm{u}}^{\mathrm{*}}\right)=\mathbf{Ae}+\mathbf{b}{\left({\mathrm{\theta }}^{\mathrm{*}}-\mathrm{\theta }\right)}^{\mathrm{T}}\xi \left(\mathrm{x}\right)-\mathbf{b}\mathrm{w}$$

(25)

with $\mathrm{w}={\mathrm{u}}_{\mathrm{D}}(\mathbf{x}|{\mathbf{\theta }}^{*})-{\mathrm{u}}^{*}$

Consider the continuous differentiable Lyapunov function

$$\mathrm{V}=\frac{1}{2}{\mathbf{e}}^{\mathrm{T}}\mathbf{Pe}+\frac{\mathrm{b}}{2{\mathrm{r}}_1}{({\mathbf{\theta }}^{*}-\mathbf{\theta })}^{\mathrm{T}}({\mathbf{\theta }}^{*}-\mathbf{\theta })$$

(26)

where ${\mathrm{r}}_1$ is a positive constant. The differential operator is used in Lyapunov equation. That is

$$\stackrel{.}{\mathrm{V}}=\frac{1}{2}{\stackrel{.}{e}}^{\mathrm{T}}Pe+\frac{1}{2}{e}^{\mathrm{T}}P\stackrel{.}{e}-\frac{\mathrm{b}}{\mathrm{r}}{\left({\theta }^{*}-\theta \right)}^{\mathrm{T}}\stackrel{.}{\theta }=-\frac{1}{2}{e}^{\mathrm{T}}Qe+\frac{b}{{\mathrm{r}}_1}{\left({\theta }^{*}-\theta \right)}^{\mathrm{T}}\left[{\mathrm{r}}_1{e}^{\mathrm{T}}{p}_{\mathrm{n}}\xi \left(\mathrm{x}\right)-\stackrel{.}{\theta }\right]-e^{\mathrm{T}}{p}_{\mathrm{n}}b\mathrm{w}$$

(27)

The last one row of $\mathbf{P}$ matrix is ${\mathbf{p}}_{\mathrm{n}}$ , so we can obtain the relation ${\mathbf{e}}^{\mathrm{T}}\mathbf{Pb}={\mathbf{e}}^{\mathrm{T}}{\mathbf{p}}_{\mathrm{n}}\mathrm{b}$ . If $\stackrel{·}{\mathrm{V}}<0$ , the adaptive law $\stackrel{·}{\mathbf{\theta }}={\mathrm{r}}_1{\mathbf{e}}^{\mathrm{T}}{\mathbf{p}}_{\mathrm{n}}\mathbf{\xi }(\mathrm{x})$ is obtained. The differential Lyapunov equation can be rewritten as follows

$$\stackrel{·}{\mathrm{V}}=-\frac{1}{2}{\mathbf{e}}^{\mathrm{T}}\mathbf{Qe}-{\mathbf{e}}^{\mathrm{T}}{\mathbf{p}}_{\mathrm{n}}\mathrm{bw}<0$$

(28)

where matrix $\mathbf{Q}>0$ is positive definite. By developing the fuzzy control input ${\mathrm{u}}_{\mathrm{D}}(\mathbf{x}|\theta )$ with a larger number of fuzzy rules, the approximated error w can be smaller based on $\left|{\mathbf{e}}^{\mathrm{T}}{\mathbf{p}}_{\mathrm{n}}\mathbf{b}\mathrm{w}\right|<(1/2){\mathbf{e}}^{\mathrm{T}}\mathbf{Q}$ . Then, the differential Lyapunov equation $\stackrel{·}{\mathrm{V}}$ is <0 to satisfy the stability criterion.

Proposed direct fuzzy control method

Fuzzy control has been widely applied in many practical engineering applications since it uses as a mean of capturing the human knowledge and experiences. An adaptive fuzzy system with the online learning characteristic is developed to compensate for the finite approximation errors and system dynamic uncertainties. Figure 1 shows the block diagram of our proposed direct fuzzy controller for a single-axis dynamic system. This adaptive learning method using the tracking errors and sliding-mode variables are adopted to implement the fuzzy-based controller. Consider a second-order dynamic system of single-axis feed drive mechanism which is given as

$$\stackrel{··}{\mathrm{x}}={\mathrm{A}}_{\mathrm{sn}}\stackrel{·}{\mathrm{x}}+{\mathrm{B}}_{\mathrm{sn}}\mathrm{U}+\mathrm{D}$$

(29)

$$\mathrm{y}=\mathrm{x}$$

(30)

where D is the external disturbance and $\mathrm{U}\in \mathrm{R}$ and $\mathrm{y}\in \mathrm{R}$ are the input and system output, respectively. Let us define a time-varying surface $\mathrm{s}(\mathrm{t})$ used in this study as follows ²²

$$\mathrm{s}(\mathrm{t})={\mathbf{c}}^{\mathrm{T}}\mathbf{e}$$

(31)

with $\mathbf{c}=[{\mathrm{c}}_1,1{]}^{\mathrm{T}}$ , $\mathbf{e}=[\mathrm{e},\stackrel{·}{\mathrm{e}}{]}^{\mathrm{T}}=[{\mathrm{e}}_1,{\mathrm{e}}_2{]}^{\mathrm{T}}$ , and $\mathrm{e}={\mathrm{y}}_{\mathrm{m}}-\mathrm{y}$ , where ${\mathrm{c}}_1$ is selected as a positive constant such that all root of $\mathrm{s}+{\mathrm{c}}_1=0$ is in the left half of s-plane. If the system dynamics parameters ${\mathrm{A}}_{\mathrm{sn}}$ and ${\mathrm{B}}_{\mathrm{sn}}$ are known, and disturbance $\mathrm{D}$ is zero, the ideal control input can be represented as

$${\mathrm{U}}^{*}=\frac{1}{{\mathrm{B}}_{\mathrm{sn}}}[-{\mathrm{A}}_{\mathrm{sn}}\stackrel{·}{\mathrm{x}}+{\stackrel{··}{\mathrm{y}}}_{\mathrm{m}}+{\mathrm{k}}_{\mathrm{d}}\mathrm{s}+{\mathrm{c}}_1{\mathrm{e}}_2]$$

(32)

where ${\mathrm{k}}_{\mathrm{d}}$ is a positive constant such that all roots of ${\mathrm{s}}^2+({\mathrm{k}}_{\mathrm{d}}+{\mathrm{c}}_1)\mathrm{s}+{\mathrm{k}}_{\mathrm{d}}{\mathrm{c}}_1=0$ are in the left half of s-plane. Let vector $\mathbf{k}$ be $[\begin{array}{cc}{\mathrm{k}}_{\mathrm{d}}{\mathrm{c}}_1\hfill & {\mathrm{k}}_{\mathrm{d}}+{\mathrm{c}}_1\hfill \end{array}{]}^{\mathrm{T}}$ . Since ${\mathrm{A}}_{\mathrm{sn}}$ and ${\mathrm{B}}_{\mathrm{sn}}$ are unknown and $\mathrm{D}\ne 0$ , the ideal control input ${\mathrm{U}}^{*}$ cannot be implemented. The adaptive controller comprised a direct fuzzy method is used to approximate the ideal control law and can be formulated as

$$\mathrm{U}=\mathrm{u}(\mathbf{x}|\mathbf{\theta })={\mathbf{\theta }}^{\mathrm{T}}\mathbf{\xi }\left(\mathbf{x}\right)$$

(33)

Figure 1.

Block diagram of the proposed direct fuzzy controller for single-axis motion stage

Substituting equation (35) into equation (32) and by some mathematical rearrangement, the following can be obtained

$$\stackrel{··}{\mathrm{e}}=-{\mathrm{k}}_{\mathrm{d}}{\mathrm{c}}_1\mathrm{e}-({\mathrm{k}}_{\mathrm{d}}+{\mathrm{c}}_1)\stackrel{·}{\mathrm{e}}+{\mathrm{B}}_{\mathrm{sn}}[{\mathrm{U}}^{*}-\mathrm{u}(\mathbf{x}|\mathbf{\theta })]-\mathrm{D}$$

(34)

Define the optimal parameters ${\mathbf{\theta }}^{*}$

$${\mathbf{\theta }}^{*}=\arg \underset{\mathbf{\theta }\in {\Omega }_{\theta }}{min}\left[\underset{\mathrm{x}\in {\Omega }_{\mathrm{x}}}{sup}|{\mathrm{U}}^{*}-\mathrm{u}(\mathbf{x}|\mathbf{\theta })|\right]$$

(35)

where ${\Omega }_{\theta }$ and ${\Omega }_{\mathrm{x}}$ are the sets of suitable bound of $\mathbf{\theta }$ and $\mathbf{x}$ , respectively. The minimum approximation error of control input becomes

$$\omega =\mathrm{u}(\mathbf{x}|{\mathbf{\theta }}^{*})-{\mathrm{u}}^{*}$$

(36)

The approximation error is bounded, that is $\left|\omega \right|<{\omega }^{\mathrm{U}}$ . The error equation can be rewritten as

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{··}{\mathrm{e}}}& =-{\mathrm{k}}_{\mathrm{d}}{\mathrm{c}}_1\mathrm{e}-({\mathrm{k}}_{\mathrm{d}}+{\mathrm{c}}_1)\stackrel{·}{\mathrm{e}}+{\mathrm{B}}_{\mathrm{sn}}[\mathrm{u}(\mathbf{x}|{\mathbf{\theta }}^{*})-\mathrm{u}(\mathbf{x}|\mathbf{\theta })]\hfill \\ \multicolumn{1}{c}{}& +[{\mathrm{U}}^{*}-\mathrm{u}(\mathbf{x}|{\mathbf{\theta }}^{*})]-\mathrm{D}\hfill \\ \multicolumn{1}{c}{}& =-{\mathrm{k}}_{\mathrm{d}}{\mathrm{c}}_1\mathrm{e}-({\mathrm{k}}_{\mathrm{d}}+\mathrm{c}{}_1)\stackrel{·}{\mathrm{e}}+{\mathbf{\xi }}^{\mathrm{T}}\left(\mathbf{x}\right)\mathbf{\phi }-(\omega +\mathrm{D})\hfill \end{array}$$

(37)

with $\mathbf{\phi }={\mathbf{\theta }}^{*}-\mathbf{\theta }$ and $\stackrel{·}{\mathbf{\phi }}=-\stackrel{·}{\mathbf{\theta }}$ .

The adaptation law can be chosen as follows

$$\stackrel{·}{\mathbf{\theta }}={\mathrm{r}}_2(\stackrel{·}{\mathrm{s}}+{\lambda }_{\mathrm{d}}\mathrm{s})\mathbf{\xi }\left(\mathbf{x}\right)$$

(38)

where ${\mathrm{r}}_2$ is the learning rate and ${\lambda }_d$ is a positive constant and is chosen as ${\mathrm{k}}_{\mathrm{d}}+{\mathrm{k}}_{\mathrm{d}}^{-1}$ . The adaptive law above can be also expressed as

$$\stackrel{.}{\phi }=-{\mathrm{r}}_2\left({\stackrel{.}{\mathrm{s}}+\lambda }_{\mathrm{d}}\mathrm{s}\right)\xi \left(x\right)=-{\mathrm{r}}_2\left(-{\mathrm{k}}_{\mathrm{d}}\mathrm{s}+{\xi }^{\mathrm{T}}\left(x\right)\phi -\left(\mathrm{\omega +D}\right)+{\mathrm{k}}_{\mathrm{d}}\mathrm{s}+{\mathrm{sk}}_{\mathrm{d}}^{-1}\right)\xi \left(x\right)=-\mathrm{\alpha }\phi +\mathrm{\beta }$$

(39)

with $\alpha ={\mathrm{r}}_2\mathbf{\xi }(\mathbf{x}){\mathbf{\xi }}^{\mathrm{T}}(\mathbf{x})$ and $\beta =-{\mathrm{r}}_2({\mathrm{sk}}_{\mathrm{d}}^{-1}-\omega -\mathrm{D})\mathbf{\xi }(\mathbf{x})$ , where $\stackrel{·}{\mathbf{\phi }}=-\alpha \mathbf{\phi }$ is the nominal system and $\beta$ can considered as persistently exciting input vector. The nominal system has a exponentially stable equilibrium point at $\mathbf{\phi }=0$ . Since the perturbation term $\beta$ is bounded, the system response $\mathbf{\phi }$ with the stable nominal dynamics is also bounded. ²² According to the relation between $\mathbf{\phi }$ and $\mathbf{\theta }$ , the fuzzy vector $\mathbf{\theta }$ is bounded. The time derivative of the $s(t)$ is given as

$$\stackrel{·}{\mathrm{s}}\left(\mathrm{t}\right)=-{\mathrm{k}}_{\mathrm{d}}\mathrm{s}+{\mathbf{\xi }}^{\mathrm{T}}\left(\mathbf{x}\right)\mathbf{\phi }-\omega -\mathrm{D}$$

(40)

with $\omega +\mathrm{D}<{\omega }^{\mathrm{U}}+{\mathrm{D}}^{\mathrm{U}}<{\epsilon }_{\mathrm{d}}$

where the parameter ${\epsilon }_{\mathrm{d}}$ is bounded. To prove the stability of the closed-loop system, the Lyapunov function candidate is considered as

$$\mathrm{V}(\mathrm{s},\mathrm{\phi })=\frac{1}{2}(\frac{1}{{\mathrm{k}}_{\mathrm{d}}}{\mathrm{s}}^2+\frac{1}{{\mathrm{r}}_2}{\mathbf{\phi }}^{\mathrm{T}}\mathbf{\phi })$$

(41)

The time derivative of Lyapunov function of V is given as

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{·}{\mathrm{V}}}& =\frac{1}{{\mathrm{k}}_{\mathrm{d}}}\mathrm{s}\stackrel{·}{\mathrm{s}}+\frac{1}{{\mathrm{r}}_2}{\stackrel{·}{\mathbf{\phi }}}^{\mathrm{T}}\mathbf{\phi }=\frac{1}{{\mathrm{k}}_{\mathrm{d}}}\mathrm{s}\stackrel{·}{\mathrm{s}}-(\stackrel{·}{\mathrm{s}}+{\lambda }_{\mathrm{d}}\mathrm{s})\mathbf{\xi }\left(\mathbf{x}\right)({\mathbf{\theta }}^{*}-\mathbf{\theta })\hfill \\ \multicolumn{1}{c}{}& =-{\mathrm{s}}^2-\frac{1}{{\mathrm{k}}_{\mathrm{d}}}(\omega +\mathrm{D})\mathrm{s}-{({\mathrm{u}}_{\mathrm{D}}^{*}-{\mathrm{u}}_{\mathrm{D}})}^2+(\omega +\mathrm{D})({\mathrm{u}}_{\mathrm{D}}^{*}-{\mathrm{u}}_{\mathrm{D}})\hfill \\ \multicolumn{1}{c}{}& <-[{\mathrm{s}}^2-\frac{1}{{\mathrm{k}}_{\mathrm{d}}}{\epsilon }_{\mathrm{d}}\left|\mathrm{s}\right|+{({\mathrm{u}}_{\mathrm{D}}^{*}-{\mathrm{u}}_{\mathrm{D}})}^2-{\epsilon }_{\mathrm{d}}|{\mathrm{u}}_{\mathrm{D}}^{*}-{\mathrm{u}}_{\mathrm{D}}|]<0\hfill \end{array}$$

(42)

Appendix 1 gives a detailed derivation of the function $\stackrel{·}{\mathrm{V}}$ . It is shown that the tracking performance $s(t)$ can converge to zero, and the closed-loop tracking error $\mathrm{e}(\mathrm{t})$ also converges to zero. That is, $\mathrm{e}(\mathrm{t})\to 0$ , as $\mathrm{t}\to \infty$ . Hence, the proposed control system is asymptotically stable. The stability of the proposed fuzzy control method is guaranteed. It needs to be emphasized that the precise values of the system parameters are not necessary. The unmodeled errors can be included in the uncertainty. Also, our design method can be applied to the system that their dynamic models are not precisely known.

Simulation and experimental results

This section presents simulation and experimental results of our proposed algorithm to demonstrate the tracking capabilities under different contours. The direct adaptive fuzzy controller is presented to verify the performances of smaller tracking error and the robustness to disturbance uncertainties. The system parameters and friction models of X-Y table are listed in Table 1. Two types of control methods are compared, and those are (a) the conventional direct fuzzy method and (b) the proposed direct fuzzy method. The controller parameters and fuzzy membership functions are listed in Table 2. Let the initial state $\mathbf{x}(0)=[\begin{array}{cc}1\hfill & 0\hfill \end{array}{]}^{\mathrm{T}}$ , and all initial value of $\mathbf{\theta }(0)$ is set to zero vector. It is assumed that the number of Gaussian membership function is 6, and the total number of adjusted variable $\mathbf{\theta }$ is 36 in our design. The spanning range of fuzzy membership function is in the interval $[-3,3]$ . The control sampling frequency of the position loop is designed as 1 kHz.

Table 1.

Simulation parameters of dynamic model and friction model.

System		Model parameters
System dynamics	X-axis	${\mathrm{K}}_{\mathrm{tx}}=0.96\mathrm{N}/\mathrm{A}$ , ${\mathrm{M}}_{\mathrm{x}}=3\times {10}^{-3}\mathrm{N}{\mathrm{s}}^2/\mathrm{m}$ , ${\mathrm{B}}_{\mathrm{x}}=0.1\mathrm{N}\mathrm{s}/\mathrm{m}$ , ${\mathrm{F}}_{\mathrm{Lx}}=10\mathrm{N}$
	Y-axis	${\mathrm{K}}_{\mathrm{ty}}=0.96\mathrm{N}/\mathrm{A}$ , ${\mathrm{M}}_{\mathrm{y}}=2.8\times {10}^{-3}\mathrm{N}{\mathrm{s}}^2/\mathrm{m}$ , ${\mathrm{B}}_{\mathrm{y}}=0.12\mathrm{N}\mathrm{s}/\mathrm{m}$ , ${\mathrm{F}}_{\mathrm{Lx}}=10\mathrm{N}$
Friction model	X-axis	${\mathrm{F}}_{\mathrm{cx}}=0.15\mathrm{N}$ , ${\mathrm{F}}_{\mathrm{sx}}=0.24\mathrm{N}$ , ${\mathrm{V}}_{\mathrm{sx}}=10\mathrm{m}/\mathrm{s}$
	Y-axis	${\mathrm{F}}_{\mathrm{cy}}=0.15\mathrm{N}$ , ${\mathrm{F}}_{\mathrm{sy}}=0.2\mathrm{N}$ , ${\mathrm{V}}_{\mathrm{sy}}=5\mathrm{m}/\mathrm{s}$

Table 2.

Adaptive fuzzy controller parameters.

Controller		Parameters
Conventional method	X-axis	${\mathrm{r}}_{1\mathrm{x}}=20$ , ${\mathrm{k}}_{1\mathrm{x}}=20$ , ${\mathrm{k}}_{2\mathrm{x}}=0.1$ , ${\mathrm{Q}}_{\mathrm{x}}=20\mathbf{I}$
	Y-axis	${\mathrm{r}}_{1\mathrm{y}}=17$ , ${\mathrm{k}}_{1\mathrm{y}}=20$ , ${\mathrm{k}}_{2\mathrm{y}}=0.1$ , ${\mathrm{Q}}_{\mathrm{y}}=18\mathbf{I}$
Proposed method	X-axis	${\mathrm{r}}_{2\mathrm{x}}=3$ , ${\mathrm{c}}_{1\mathrm{x}}=20$ , ${\mathrm{c}}_{2\mathrm{x}}=1$ , ${\mathrm{k}}_{\mathrm{dx}}=100$ , ${\mathrm{\lambda }}_{\mathrm{x}}=100.01$
	Y-axis	${\mathrm{r}}_{2\mathrm{y}}=3$ , ${\mathrm{c}}_{1\mathrm{y}}=20$ , ${\mathrm{c}}_{2\mathrm{y}}=1$ , ${\mathrm{k}}_{\mathrm{dy}}=100$ , ${\mathrm{\lambda }}_{\mathrm{y}}=50.02$
Fuzzy membership functions: ${\mathrm{\mu }}_{\mathrm{N}3}({\mathrm{x}}_{\mathrm{i}})=1/(1+\exp (5({\mathrm{x}}_{\mathrm{i}}+2)))$ , ${\mu }_{\mathrm{N}2}({\mathrm{x}}_{\mathrm{i}})=\exp (-({\mathrm{x}}_{\mathrm{i}}+1.5{)}^2)$ , ${\mu }_{\mathrm{N}1}({\mathrm{x}}_{\mathrm{i}})=\exp (-({\mathrm{x}}_{\mathrm{i}}+0.5{)}^2)$ , ${\mu }_{P1}({\mathrm{x}}_{\mathrm{i}})=\exp (-({\mathrm{x}}_{\mathrm{i}}-0.5{)}^2)$ , ${\mu }_{P2}({\mathrm{x}}_{\mathrm{i}})=\exp (-({\mathrm{x}}_{\mathrm{i}}-1.5{)}^2)$ , ${\mu }_{\mathrm{P}3}({\mathrm{x}}_{\mathrm{i}})=1/(1+\exp (-5({\mathrm{x}}_{\mathrm{i}}-2)))$

Performance index and contour planning

In this section, the effectiveness of the proposed adaptive fuzzy control method is examined to confirm the tracking characteristics. Two indexes ⁷ are employed to demonstrate the motion performances. They are the average tracking error and tracking error standard deviation. They are defined as follows:

• 1. Average tracking error

$$\mathrm{m}=\sum _{\mathrm{k}=1}^{\mathrm{n}}\frac{\mathrm{E}\left(\mathrm{k}\right)}{\mathrm{n}}$$

(43)

with $\mathrm{E}(\mathrm{k})=\sqrt{{\mathrm{e}}_{\mathrm{x}}^2{(\mathrm{k})+\mathrm{e}}_{\mathrm{y}}^2(\mathrm{k})}$ , where ${\mathrm{e}}_{\mathrm{x}}(\mathrm{k})$ is the tracking error in the X-axis, ${\mathrm{e}}_{\mathrm{y}}(\mathrm{k})$ is the tracking error in the Y-axis, and n is the total simulation points.

• 2. Tracking error standard deviation

$${\mathrm{E}}_{\mathrm{s}}=\sqrt{\frac{\sum _{\mathrm{k}=1}^{\mathrm{n}}{(\mathrm{E}\left(\mathrm{k}\right)-\mathrm{m})}^2}{\mathrm{n}}}$$

(44)

The average tracking error, maximum tracking error, and tracking error standard deviation are usually used in contour performance comparison. In this research, four kinds of contours are involved to evaluate the performance of the proposed fuzzy biaxial motion control system. The motion commands of the X-axis and Y-axis are always designed individually, and the various contour planning are described in the following equations. A total of four reference contours are designed and shown in Figure 2. The mathematic equations are given in Table 3. In this contour planning, ${\mathrm{X}}_{\mathrm{i}}$ is the position command of the X-axis, and ${\mathrm{Y}}_{\mathrm{i}}$ is the position command of the Y-axis; S is the position increment of the trajectories, and it is selected as $0.1\mu \mathrm{m}$ . R is the radius of the circle, and $\Delta \phi$ is the angle increment. The parameter R is selected as 10 mm, and the parameter $\Delta \phi$ is set to be $6.28\mathrm{e}-4$ radius here.

Figure 2.

Two-dimensional reference trajectories: (a) square contour, (b) triangle contour, (c) star contour, and (d) circle contour.

Table 3.

Contour planning.

Contour	Mathematic equations
Square contour	Trajectory a: ${\mathrm{X}}_{\mathrm{i}}={\mathrm{X}}_{\mathrm{i}-1}+\mathrm{S}$ , ${\mathrm{Y}}_{\mathrm{i}}={\mathrm{Y}}_{\mathrm{i}-1}$ Trajectory b: ${\mathrm{X}}_{\mathrm{i}}={\mathrm{X}}_{\mathrm{i}-1}$ , ${\mathrm{Y}}_{\mathrm{i}}={\mathrm{Y}}_{\mathrm{i}-1}+\mathrm{S}$ Trajectory c: ${\mathrm{X}}_{\mathrm{i}}={\mathrm{X}}_{\mathrm{i}-1}-\mathrm{S}$ , ${\mathrm{Y}}_{\mathrm{i}}={\mathrm{Y}}_{\mathrm{i}-1}$ Trajectory d: ${\mathrm{X}}_{\mathrm{i}}={\mathrm{X}}_{\mathrm{i}-1}$ , ${\mathrm{Y}}_{\mathrm{i}}={\mathrm{Y}}_{\mathrm{i}-1}-\mathrm{S}$
Triangle contour	Trajectory a: ${\mathrm{X}}_{\mathrm{i}}={\mathrm{X}}_{\mathrm{i}-1}+\mathrm{S}$ , ${\mathrm{Y}}_{\mathrm{i}}={\mathrm{Y}}_{\mathrm{i}-1}$ Trajectory b: ${\mathrm{X}}_{\mathrm{i}}={\mathrm{X}}_{\mathrm{i}-1}-0.5*\mathrm{S}$ , ${\mathrm{Y}}_{\mathrm{i}}={\mathrm{Y}}_{\mathrm{i}-1}+0.865*\mathrm{S}$ Trajectory c: ${\mathrm{X}}_{\mathrm{i}}={\mathrm{X}}_{\mathrm{i}-1}-0.5*\mathrm{S}$ , ${\mathrm{Y}}_{\mathrm{i}}={\mathrm{Y}}_{\mathrm{i}-1}-0.865*\mathrm{S}$
Star contour	Trajectory a: ${\mathrm{X}}_{\mathrm{i}}={\mathrm{X}}_{\mathrm{i}-1}+\mathrm{S}$ , ${\mathrm{Y}}_{\mathrm{i}}={\mathrm{Y}}_{\mathrm{i}-1}$ Trajectory b: ${\mathrm{X}}_{\mathrm{i}}={\mathrm{X}}_{\mathrm{i}-1}-0.809*\mathrm{S}\pm$ , ${\mathrm{Y}}_{\mathrm{i}}={\mathrm{Y}}_{\mathrm{i}-1}-0.588*\mathrm{S}$ Trajectory c: ${\mathrm{X}}_{\mathrm{i}}={\mathrm{X}}_{\mathrm{i}-1}+0.309*\mathrm{S}$ , ${\mathrm{Y}}_{\mathrm{i}}={\mathrm{Y}}_{\mathrm{i}-1}+0.951*\mathrm{S}$ Trajectory d: ${\mathrm{X}}_{\mathrm{i}}={\mathrm{X}}_{\mathrm{i}-1}+0.309*\mathrm{S}$ , ${\mathrm{Y}}_{\mathrm{i}}={\mathrm{Y}}_{\mathrm{i}-1}-0.951*\mathrm{S}$ Trajectory e: ${\mathrm{X}}_{\mathrm{i}}={\mathrm{X}}_{\mathrm{i}-1}-0.809*\mathrm{S}$ , ${\mathrm{Y}}_{\mathrm{i}}={\mathrm{Y}}_{\mathrm{i}-1}+0.588*\mathrm{S}$
Circle contour	Trajectory: ${\phi }_i={\phi }_{i-1}+\Delta \phi$ , $X_i=R\cos {\phi }_i$ , $Y_i=R\sin {\phi }_i$

Contour tracking performances

In this section, the simulation results with the conventional fuzzy method and the proposed method are conducted and compared to verify the tracking performances. Figure 3 shows the tracking results corresponding to the square trajectory case in the X-axis and Y-axis, respectively. A reference square contour of 10 mm width is applied for two-dimensional motion validations. The simulation results of the position response, tracking error, force command, and motion trajectory are depicted in Figure 3(a)–(g). It demonstrates that the proposed direct fuzzy control method is effective for trajectory tracking performances. On average, the average tracking error is 41.452 µm, and the tracking error standard deviation is 2.491 µm. In the second case, a triangle contour of 10 mm side length is considered and shown in Figure 4. The start point of position is set at (0, 0). Our proposed scheme also performs well in triangle displacement response, tracking error, force command, and trajectory, as shown in Figure 4(a)–(g). The transient response provides the robustness fact of the direct fuzzy controller under parameter variations and external disturbances. On average, the average tracking error is 39.646 µm, and the tracking error standard deviation is 2.468 µm in the triangle tracking case.

Figure 3.

Two-dimensional displace response for the square contour tracking: (a) X-axis position tracking response, (b) Y-axis position tracking response, (c) X-axis position tracking error, (d) Y-axis position tracking error, (e) X-axis force command, (f) Y-axis force command, and (g) square contour tracking trajectory.

Figure 4.

Two-dimensional displace response for the triangle contour tracking: (a) X-axis position tracking response, (b) Y-axis position tracking response, (c) X-axis position tracking error, (d) Y-axis position tracking error, (e) X-axis force command, (f) Y-axis force command, and (g) triangle contour tracking trajectory.

The third practical application is the star trajectory. A reference star contour of 10 mm side length is illustrated in Figure 5. It shows that the trajectory response is very smooth, and the steady-state error reduces significantly during the tracking process for direct drive biaxial X-Y system. On average, the average tracking error is 38.980 µm, and the tracking error standard deviation is 3.593 µm in the star contour case. Furthermore, the reference circle contour of 20 mm diameter is shown in Figure 6. It can be observed that the displacement error can be reduced significantly within 1 s. Figure 6(b) shows the Y-axis tracking response, and its plant model is different from X-axis one. It can be seen that the tracking error can be substantially improved, and the error is diminished within 2 s. In the beginning, trajectory route has deviated from the path of the reference. The adaptive fuzzy control architecture can accomplish the better tracking performance, and the trajectory error converges at steady state. On average, the average tracking error of is 20.164 µm, and the standard deviation of error is 4.395 µm.

Figure 5.

Two-dimensional displace response for the star contour tracking: (a) X-axis position tracking response, (b)Y-axis position tracking response, (c) X-axis position tracking error, (d) Y-axis position tracking error, (e) X-axis force command, (f) Y-axis force command, and (g) star contour tracking trajectory.

Figure 6.

Two-dimensional displace response for the circle contour tracking: (a) X-axis position tracking response, (b) Y-axis position tracking response, (c) X-axis position tracking error, (d) Y-axis position tracking error, (e) X-axis force command, (f) Y-axis force command, and (g) circle contour tracking trajectory.

Table 4 shows the tracking error values of our proposed method compared with the traditional direct fuzzy method. It can easily find that the proposed adaptive fuzzy control system has the least average and standard deviation of tracking error and is more suitable to control the X-Y table under the occurrence of uncertainties at various reference trajectories. One can find that our method achieves the better accuracy performances in terms of 23.1% improvement for average tracking error and 54.2% improvement for tracking error standard deviation compared to the conventional one. In summary, we can conclude that the proposed direct fuzzy approach allows the desired tracking performances to be attained and also ensure the robustness of the closed-loop system.

Table 4.

Simulation results of trajectory tracking error.

	Average tracking error (µm)		Tracking error standard deviation (µm)
Trajectory contour	Conventional method	Proposed method	Conventional method	Proposed method
Square contour	43.956	41.452	4.030	2.491
Triangle contour	44.110	39.646	6.570	2.468
Star contour	43.649	38.980	7.977	3.593
Circle contour	50.660	20.164	9.696	4.395
Average	45.594	35.061	7.068	3.237

Experimental results

To verify the proposed control structure, the X-Y stage with ball-screw system driven by AC servo motors is shown in Figure 7. The optical shaft encoder is used to measure the position information, and the resolution of the encoders is set to 1 µm. Our system is designed using a PC-based controller with sampling rate of 1 kHz. The PC includes an analog-to-digital (AD) or digital-to-analog (DA) card with multichannels of analog-to-digital converter (ADC) and a motion control card. The stage has 260-mm travel for the X-axis and 350-mm travel for the Y-axis.

Figure 7.

Experimental control stage system.

The circle contour of 100 mm diameter is tested in our experiments. The resulting contouring trajectory using the proposed method is shown in Figure 8, and the corresponding position tracking response for each axis is presented in Figure 8(a) and (b). It can be seen that the tracking error can be substantially reduced, and the error is diminished quickly. The adaptive fuzzy control architecture can accomplish the better tracking performance, and the trajectory error converges at steady state. On average, the average tracking error is 51.087 µm, and the standard deviation of error is 27.163 µm. Table 5 presents the tracking error values of our proposed method compared with the traditional direct fuzzy method in our experimental test. The proposed system can provide the smaller average and standard deviation of tracking error compared to the conventional one. It is found that our method achieves the better accuracy performances in terms of 22.137 µm for average tracking error and 12.99 µm for tracking error standard deviation for the four kinds of contour trajectories. The experimental results show that our approach can achieve a feasible solution to the contour control in stage system.

Figure 8.

Experimental results of two-dimensional displace response for the circle contour tracking: (a) X-axis position tracking response, (b) Y-axis position tracking response, and (c) circle contour trajectory.

Table 5.

Experimental results of trajectory tracking error.

	Average tracking error (µm)		Tracking error standard deviation (µm)
Trajectory contour	Conventional method	Proposed method	Conventional method	Proposed method
Square contour	33.614	11.340	45.081	6.841
Triangle contour	45.249	12.572	37.052	8.162
Star contour	46.006	13.547	49.580	9.816
Circle contour	91.524	51.087	47.872	27.163
Average	50.098	22.137	44.896	12.99

Conclusion

This article presents the adaptive fuzzy strategy with online training algorithm used for the precision motion control of X-Y stage. The nonlinear friction effect and external disturbance behavior are included in the dynamic system model. Based on the Lyapunov theorem, the adaptive laws of fuzzy coefficients can be obtained and utilized for contour application. The effectiveness of the proposed control system is illustrated by tracking of four motion trajectories, namely, (a) square contour, (b) triangle contour, (c) star contour, and (d) circle contour. The stability of the closed-loop direct drive X-Y system is guaranteed, and all control signals utilized are bounded. To validate the tracking performances, the control results obtained using our method are compared with the traditional direct fuzzy control structure under different contour conditions. The adaptive control scheme significantly reduces the contour error due to the friction and external disturbance, and it can achieve 23.1% and 54.2% improvement for average tracking error and tracking error standard deviation, respectively. The simulation and experimental investigations indicate that the proposed adaptive fuzzy controller can indeed perform well in precision control of X-Y stage system.