Nonlinear Contour Tracking of a Voice Coil Motors-Driven Dual-Axis Positioning Stage Using Fuzzy Fractional PID Control with Variable Orders

Chen, Syuan-Yi; Yang, Meng-Chen

doi:https://doi.org/10.1155/2021/6697942

Mathematical Problems in Engineering

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Recent Advances in Optimisation Theory, Methods, and Applications in Science and Engineering

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Research Article | Open Access

Volume 2021 | Article ID 6697942 | https://doi.org/10.1155/2021/6697942

Nonlinear Contour Tracking of a Voice Coil Motors-Driven Dual-Axis Positioning Stage Using Fuzzy Fractional PID Control with Variable Orders

Syuan-Yi Chen¹and Meng-Chen Yang²

Academic Editor: Guoqiang Wang

Received04 Nov 2020

Revised01 Dec 2020

Accepted02 Mar 2021

Published25 Mar 2021

Abstract

This study aims to develop a variable-order fuzzy fractional proportional-integral-differential (VOFFPID) control system for controlling the mover position of a newly designed voice coil motors- (VCMs-) driven dual-axis positioning stage. First, the operation principle and dynamics of the stage are analyzed. After that, the design of a fuzzy fractional proportional-integral-differential (FFPID) control system is introduced on the basis of a fractional calculus and fuzzy logic system. With an additional degree of freedom to the control parameters and fuzzy operation, the FFPID control system can upgrade the contour tracking performance of a conventional proportional-integral-differential (PID) control system with respect to the specified dynamics of the stage. Moreover, the VOFFPID control system is designed to further improve the tracking responses of the FFPID control system. In this system, the five control parameters are optimized with the cuckoo search algorithm via an adaptive strategy. Lastly, nominal and payload conditions attributed to two nonlinear contour demands are provided to evaluate the contouring performance of the PID, FFPID, and VOFFPID control systems. The experimental results subjected to different performance measures demonstrate that the proposed VOFFPID controller outperforms PID and FFPID controllers in terms of the designed VCMs-driven dual-axis positioning stage under both conditions.

1. Introduction

Although control engineers prefer a conventional proportional-integral-differential (PID) controller because of its easy implementation, low cost, and uncomplicated structure, they cannot use it to achieve a high-precision control level in a highly nonlinear and disturbed situation. To address this problem, a fractional-order (FO) PID (FPID) control method was developed by adding fractional differential and integral operations. With the consideration of more degrees of freedom for the selection of control parameters, the FPID controller can obtain better control responses and anti-interference characteristic over the integer-order (IO) counterparts because of the additional flexibility to the design of a control system [1]. However, accurately determining numerous control parameters in practical applications is difficult. Therefore, many intelligent strategies were designed for FPID control [2, 3].

The introduction of a fuzzy logic system (FLS) to a PID controller has been widely explored because it provides a ﬂexible and model-free way to determine the PID control parameters through engineering intuitions and experiences [4, 5]. In addition, fuzzy FPID (FFPID) control systems were further developed to enhance the control performances of a FPID controller [1, 6–9]. In the FFPID, the fractional operation of errors introduces an extra degree of flexibility in the input variables of FLS, and it can be tuned similarly to the input-output scaling factors of the FLS to enhance the closed-loop performance. Some experimental results have verified that the FFPID control system outperforms classical PID, fuzzy PID, and FPID control systems because of its FLS and higher degrees of freedom for tuning.

Swarm intelligence algorithms have been widely applied to solve many real-world problems, such as control system design [2], path planning [10], parameter estimation [11], and energy management [12], because these algorithms can obtain a global optimal solution for multidimensional optimization problems by relying on colony behaviors in nature. For example, inspired by the aggressive reproduction behavior of cuckoo bird species, cuckoo search algorithms (CSAs) were developed in [2, 13–16]. In cuckoo reproduction, female cuckoos fly from one nest to another and randomly lay their fertilized eggs inside other host birds’ nests instead of building their own nests. Thus, host birds may unknowingly raise these eggs. In general, female cuckoos choose the best nest so that their eggs have the best chance of hatching and creating a new generation. To enhance the hatching chance, some cuckoo birds strategically lay their eggs in a good position or drop the host bird’s eggs outside its nest. Some cuckoo species even evolve to produce eggs similar to those of other bird species. However, if an alien egg is found, host birds throw it out or even transfer to a new nest with their own brood elsewhere. In this case, the eggs of cuckoo fail to hatch. In the CSA algorithm, cuckoo birds represent the particles assigned to ﬁnd the optimal solution, while cuckoo eggs and host birds’ eggs represent the new and old solutions for the current iteration process, respectively. If a new solution is better than the old one, the worse one is replaced.

A linear voice coil motor (VCM) is a direct drive and hysteresis-free device, which utilizes a magnetic field generated by a permanent magnet with a coil wire to produce an electric driving force [17, 18]. This device has a compact structure, high acceleration, and no hysteresis features, so it has been extensively used in various small range positioning applications, such as servo valves [17], hard disk drives [19, 20], automatic transmitters [21], autofocus actuators [22], and ultrasound scanners [23], which demand high-precision and high-speed control levels. However, effective controls for this device should be designed because external disturbances and operational changes instantly act on a direct drive system. For instance, an intelligent FO sliding-mode control was proposed to control a linear voice coil actuator for the tracking of a reference trajectory [18]. In this control scheme, a fuzzy neural network was designed to compensate for system uncertainties, thereby reducing the chattering phenomena. Moreover, a coupling controller design was proposed by considering the interaction between a VCM and a piezoactuator of a head positioning control system [20]. In another study [24], a direct amplitude control strategy was developed to improve the amplitude accuracy of a reciprocating rig in a high-frequency band compared with that given by a traditional proportional-integral control strategy.

In the direct drive VCM system, there are no mechanical reduction and transmission components so that the mover is directly coupled to the payload. Compared with the conventional rotary motor using mechanical components to translate the rotary motion into linear motion, direct drive device apparently reduces mechanical loss, system nonlinearities, and backlash [25]. Thus, the control accuracy of the VCM system can be enhanced in practical applications. However, it also loses the advantage of using mechanical components attenuating the effects of system parameter variations and external disturbances. With this structure, the system uncertainties are directly transmitted to the payload and then unavoidably affect the control performance of the payload. On the other hand, any change or disturbance in the payload will be directly reflected back to the VCM. Although many control methods have been proposed to control the single-axis VCM systems [17, 18, 21–24], designing effective and robust control methods to meet high-precision requirements for the multi-axis VCM systems is still required. As a result, this study aims to develop a variable-order FFPID (VOFFPID) control strategy for controlling the mover position of a VCM-based dual-axis positioning stage with a high-precision contouring performance. In the VOFFPID controller, control parameters are self-tuned to deal with system uncertainty so that the trivial trials of control parameters are unnecessary. Furthermore, good stability and robustness during the control process can be ensured. Experiments involving the tracking of two nonlinear contour demands were conducted by using PID, FFPID, and VOFFPID under nominal and payload conditions to demonstrate the different control performance and robustness levels.

From the aforementioned studies, the main academic and industrial contributions of this study are summarized as follows: (i) the new VOFFPID controller that optimizes the conventional FFPID controller online is successfully developed; (ii) the new VCMs-driven dual-axis positioning stage is made with operation and dynamic analyses; (iii) the PID, FFPID, and VOFFPID controllers for the VCMs-driven dual-axis positioning stage control system are successfully implemented; and (iv) the experimental results of the three controllers associated with two nonlinear contour tracking commands under two test conditions are compared. The remaining parts of this study are organized as follows. The operation principle of VCMs-driven dual-axis positioning stage is described in Section 2. The CSA with the adaptive strategy used for optimizing the control parameters of the VOFFPID is presented in Section 3. The designs of contour tracking controllers are introduced in Section 4. The experimental setup and results are discussed in Section 5. The conclusions of the proposed work are provided in Section 6.

2. Operation Principle of the VCMs-Driven Dual-Axis Positioning Stage

A circular moving coil-type single-axis VCM that is composed of a moving coil winding and a stationary permanent magnet within a soft iron shell is utilized in this study as shown in Figure 1. In accordance with the interaction between the permanent magnetic field and a drive current perpendicular to the field, the mover of the VCM moves along the direction of the electric driving force, which can be determined with Fleming’s left-hand rule [7]. If the direction of the drive current changes, the moving direction also reverses. Moreover, the generated electric driving force is proportional to the product of the permanent magnetic field and the drive current [18].

In this study, a VCMs-driven dual-axis positioning stage is newly designed and implemented as shown in Figure 2. The dimension of the whole stage is 230 mm × 194 mm × 100 mm. It is composed of three VCMs (Akribis, AVM 40-20), namely, a VCM in the Y-axis and two parallel VCMs in the X-axis. A 100 mm² moving platform is placed on the mover of the Y-axis VCM, and the stator of the Y-axis VCM is mounted on a moving base. With the design of this stage, two VCMs in the X-axis can generate a stronger electromagnetic force to push the moving platform, moving base, Y-axis VCM, and payload along the X-axis. They can even create rotational motion according to the specified mechanism design and different displacements of X-axis VCMs. Two high-resolution linear scales measure the mover displacements for high-precision and repeatability applications. Specifications of the adopted VCMs are listed in Table 1 [26].

The state-space model is given below to describe the dynamics of the VCMs-driven dual-axis positioning stage [17]:where x and y are the mover positions in X-axis and Y-axis, respectively; and indicate the control signals of the VCMs; c_x and c_y represent the linear gains of the current amplifiers; k_ix and k_iy are the force-current coefﬁcients of the VCMs; k_bx and k_by are the equivalent damping coefficients; k_kx and k_ky denote the equivalent elastic coefficients; Δk_kx, Δk_ky, Δk_bx, Δk_by, Δc_xk_ix, and Δc_yk_iy represent the unknown parameter variations of k_kx, k_ky, k_bx, k_by, c_xk_ix, and c_yk_iy, respectively; m_b, m_p, and m_l denote the masses of the moving base, platform, and payload, respectively; F_fx and F_fy are the friction forces; and F_dx and F_dy denote the unmodeled system uncertainties, comprising internal cross-coupled interferences and external disturbances. Thus, the dynamic model of the VCMs-driven dual-axis positioning stage can be reexpressed aswhere M_x = m_b + m_p + m_l and M_y = m_p + m_l; L_x and L_y are the lumped uncertainties regarded as follows:

In equation (1), the practical control characteristics of the VCMs are nonlinear because the system coefficients described above may vary due to the changes in operating temperature and duration, though the VCMs-driven dual-axis positioning stage can be presented with a state-space model. Moreover, the lumped uncertainties L_x and L_y cannot be measured exactly. Therefore, designing a model-free control method is important to control the VCMs-driven dual-axis positioning stage with a stable and precise nonlinear contour tracking performance for the practical applications.

3. CSA with an Adaptive Strategy

CSA is a metaheuristic evolutionary algorithm based on the aggressive reproduction of a cuckoo species with a Lévy ﬂight behavior. Three idealized characteristic rules are assumed as follows to formulate the CSA [13–16]:(a)Each cuckoo bird lays one egg in a randomly selected host nest, representing a solution to the optimization problem.(b)Some of these nests contain high-quality eggs, representing good solutions, which are preserved for the next generation.(c)The number of available host nests is ﬁxed in the ecosystem, and the probability of alien eggs discovered by the host bird is P_a∈[0, 1]. When the host bird ﬁnds the alien eggs, it destroys the egg or abandons the old nest and builds a new one in another place.

3.1. Principle of CSA

From the optimization perspective, cuckoo birds correspond to the particles assigned to ﬁnd solutions, and cuckoo eggs indicate the candidate solutions for an optimization problem. In the CSA, the random step of cuckoo birds is characterized by a Lévy ﬂight, indicating that the step length of the ﬂight behavior follows the Lévy distribution; consequently, the CSA realizes a “random walk” and a “long jump” among their ﬂights [15]. In this regard, the CSA can avoid obtaining an unreliable local optimal solution and shorten the convergence time required to reach a global optimal solution.

An unconstrained optimization problem can be stated as follows:where x is the individual nest position, D is the optimized variable dimension, and J is an objective function. In the CSA, the update of the egg position is given according to a Lévy ﬂight as follows [13–16]:where i = 1, 2, …, N_p is the population size, k is the current index for the generation iteration, ⊕ is entry-wise multiplication, α > 0 is a step size related to the scales of the problem of interest, and 1 ≤ β ≤ 3 is a parameter used to formulate the Lévy distribution and it is considered to be 1.5 in this study. Then, the step length ς is defined aswhere µ and are random numbers derived from normal distribution aswhere σ_μ is derived by using Mantegna’s algorithm for symmetric distributions and Γ(·) is a Gamma function. Then, the step size s is calculated aswhere x_b is the current best solution. Thus, the update of the egg position as shown in equation (5) can be formulated:where r is a random value following the normal distribution N(0, 1). Figure 3 shows the typical trajectory of a three-dimension random Lévy ﬂight path by using equations (5)–(10). Afterward, the ﬁtness values of J(x_i,k+1) and J(x_i,k) are compared. If J(x_i,k+1) > J(x_i,k) holds, the i^th solution is replaced, and the new solution is accepted as x_i,k+1. In addition, the parameter is set as the threshold of discovery probability that the cuckoo’s eggs are found by a host bird. The host bird builds nests at new locations according towhere x_q,k and x_j,k are two randomly selected different solutions in the k^th iteration and is a uniform random number distributed in [0, 1]. Similarly, if the fitness value of the new solution is better than the old one, then the new solution x_i,k+1 is used to replace the old one x_i,k.

3.2. Adaptive Strategy of the CSA

An adaptive strategy based on Rechenberg’s 1/5 criteria is utilized to enhance the evolution and adaptation efficiency of the CSA [14]. With the adaptive strategy, step size and discovery probability are dynamically tuned during optimization. First, the improvement rate ζ is defined as follows:where N_r is the number of all cuckoo birds whose fitness values are improved after evolution. Thus, the step size α and discovery probability can be further updated aswhere α_u and α_l are the upper and lower thresholds of ζ with respect to α; and are the upper and lower thresholds of ζ with respect to the discovery probability; and 1 ≤ f_α ≤ 2 and 1 ≤ f_P ≤ 2 are the learning factors of α and , respectively.

According to equations (13) and (14), α and are increased to strengthen the global exploration ability when ζ is large. This result indicates that the current solution space is relatively monotonous and smooth. On the contrary, α and are decreased to enhance the local exploitation ability when ζ is small. This result suggests that the optimal solution may be in the surrounding search area near the current solution. In this regard, the local exploitation and global exploration abilities of the CSA can be well balanced to deal with the diversiﬁcation and intensiﬁcation of a population, thereby avoiding the premature convergence.

4. Control System Designs of the VCMs-Driven Dual-Axis Positioning Stage

First, typical PID and FFPID control strategies are adopted in this study to control the VCMs-driven dual-axis positioning stage for nonlinear contour tracking. Subsequently, a VOFFPID is proposed to improve the stability and accuracy of the contour tracking performance under system uncertainties, including parameter variations, cross-coupled interferences, and friction forces [27]. With the help of online-tuned control parameters, the system uncertainties can be compensated, and the high-precision nonlinear contour tracking performance can be guaranteed.

4.1. Typical PID Control

The popularity of IO PID (IOPID) controllers as expressed in equation (15) can be attributed partly to their favorable performance in a wide range of applicability and partly to their functional simplicity, which allows engineers to operate them in an easy and straightforward manner. As for PID controller, the proportional (P) action amplifies errors, the integral (I) action accumulates errors, and the differential (D) action calculates the change in errors. In this study, the PID controller compares the actual mover positions x and y with the reference contour positions x_d and y_d to obtain the error signals e_x and e_y. After that, it accumulates the results of the P, I, and D actions as below [7]:where t denotes the current time; j = x, y represent the X-axis and Y-axis of the VCMs-driven dual-axis positioning stage, respectively; u_x and u_y denote the control signals; , K_Ij, and K_Dj denote the P, I, and D control parameters, respectively; and e_x and e_y indicate the tracking errors defined as e_x = x_d − x and e_y = y_d − y, respectively.

4.2. FO Integral and Differential Definitions

FO integral and differential operators are defined in the following [7, 18, 25]:in which D is the fractional calculus operator; λ is the fractional order; and a and t represent the operation range. The three major FO integral and differential deﬁnitions are the Caputo, Grunwald-Letnikov (GL), and Riemann–Liouville (RL) definitions. The operator given in equation (16) applied to the f(t) function leads to an extended Caputo form, which can be derived as follows [28, 29]:where m is an integer such that m > λ. Moreover, the λ^th-order RL FO integral of f(t) is defined as follows [30]:

Similarly, the RL FO differential of f(t) is defined as

By contrast, the λ^th-order GL FO operation based on finite differences is defined as follows [29]:where [.] is the integer part, h is the time increment, and () is the fractional binomial coefﬁcient defined as

Intuitively, integral and differential operations with fractional orders can provide a higher degree of freedom to the control parameters than those with integer orders. As a result, the control performance of PID control system can be enhanced by properly selecting fractional integral and differential orders. For convenience, the FO operator is noted as D^λ in the subsequent sections.

4.3. Developed FFPID Control System

In the case of a nonlinear and disturbed system, the conventional IOPID control strategy is difficult to concurrently obtain a high control performance level and maintain good robustness because of its linear structure [7]. To improve the control performances, smoothness and robustness of the PID control system, the FFPID control, which combines the merits of PID control, FO operations, and FLS, is adopted and illustrated in Figure 4 in this study. In Figure 4, a_j and b_j are the fractional differential and integral orders, respectively; and K_Dj can be considered the input scaling factors; and K_Ij can be regarded as the output scaling factor. The inputs of the FLS are the tracking error e_j multiplied by and the fractional differential of the tracking error D^aje_j multiplied by K_Dj, which can be regarded as a FO proportional-differential (FOPD) controller. The relationship between the inputs and output of the FLS is specified with the table of the fuzzy rules as given in Table 2 in which the fuzzy linguistic values NL, NM, NS, ZO, PS, PM, and PL indicate negatively large, negatively medium, negatively small, zero, positively small, positively medium, and positively large, respectively [7]. Figure 5 illustrates the membership functions for the inputs and output of FLS in which the horizontal range was designed on the basis of the prior experimental tests to effectively cover the input and output signals [7]. In this study, the triangular membership functions, which can be easily configured with regard to the linear shape and fewer parameters, were selected to ease the computational burden and speed up the control process. Thus, the output of the FLS u_FPDj can be derived according to the designed fuzzy rules with the center of gravity defuzzification method as follows:where c indicates a logical union set of the conclusion fuzzy sets of the fired fuzzy rules; σ_k is a value between the minimum and maximum values of the abscissa of c defined on the universe of discourse; μ_c(σ_k) is the firing strength of c for the point σ_k; and n is the number of the samples.

The final control signal of the FFPID control system u_j is the sum of the output of FLS u_FPDj multiplied by λ_j and the fractional integral of the output of FLS u_FPDj multiplied by K_Ij:

In equations (22) and (23), the whole FFPID controller can be considered a combination of the fuzzy FOPD controller u_FPDj in the first half and the FO proportional-integral (FOPI) controller u_j in the second half. The benefits of the FFPID controller are adjustability and flexibility when these two controllers are combined. On the other hand, as seen from Figure 4, the integral operator D^−bje_j can be regarded a low-pass filter of the error signal e_j. When b_j is appropriately selected, the steady-state error can be suppressed effectively [18]. Besides, the differential operator D^aje_j can be regarded a high-pass filter of e_j. A proper a_j can accelerate the dynamic response of the VCMs-driven dual-axis positioning system. Therefore, the contour tracking responses with a conventional IOPID controller can be enhanced by adding the well-defined fractional orders a_j and b_j regarding the specified dynamics of the VCMs-driven dual-axis positioning stage.

4.4. Proposed VOFFPID Control System

The control gains (i.e., , K_Ij, and K_Dj), along with fractional orders of differentiation (i.e., a_j) and integration (i.e., b_j), are tuned to obtain the optimum contour tracking performance of the VCMs-driven dual-axis positioning system. Hence, a VOFFPID controller is further proposed, in which the control parameters {, K_Ij, K_Dj, a_j, b_j} are dynamically tuned with the CSA with an adaptive strategy. In the CSA application, the most crucial step is to choose the objective function for evaluating the ﬁtness value of each host nest. In this study, an absolute tracking error is employed to design the objective function. Thus, the optimization problem arising in this study can be expressed by rewriting equation (4) as follows:where ε is a small positive constant and e_c is a contour tracking error defined as follows:

According to the design of the object function shown in equation (24), , K_Ij, K_Dj, a_j, and b_j can be updated dynamically to minimize the contour tracking error e_c via the CSA.

In the beginning of the VOFFPID control system, several nest positions x are selected randomly within the specific searching ranges. Then, each vector x is sequentially applied to the VOFFPID controller, and the corresponding tracking performance is evaluated via the object function J. Lastly, the vector with the highest fitness value is selected for the VCMs-driven dual-axis positioning system. As a result, the VOFFPID controller can achieve favorable robustness against uncertainties and external disturbances.

5. Experimental Results

5.1. Experimental Setup

Figure 6 shows the experimental setup of the VCMs-driven dual-axis positioning system, which consists of a newly developed dual-axis positioning stage, power supplies, servo drivers (Elmo Cello 5/60), and a TMS320F28377 digital signal processor (DSP; Texas Instruments) [7]. The real-time control software developed in the DSP comprises one main program and one interrupt service routine (ISR). In the main program, parameters and I/O initializations are initially established, and the interrupt interval for the ISR is set. When the interrupt is enabled, the ISR, with 1 ms execution frequency, calculates the mover position from the encoder interfaces and then determines the control signals through the designed PID, FFPID, and VOFFPID control systems. After that, the control signals are sent to the servo drivers via the 14-bit resolution digital-to-analog converters (DACs) of the DSP. Afterward, the servo drivers convert the control signals to drive currents so that the VCMs can produce the required thrust force for high-precision contour tracking. In this study, a flower contour and a window contour are designed for the reference nonlinear contour commands as shown in Figures 7 and 8, respectively.

(a)

(b)

(a)

(b)

The maximum, average, and standard deviation of the contour tracking error T_m, T_A, and T_S are measured as follows to compare the different positioning performance levels of the PID, FFPID, and VOFFPID control systems [7]:where I is the current iteration number and N_T is the total number of iterations. Moreover, two conditions are tested in this study: nominal (Case 1) and payload (Case 2) cases. In Case 2, one payload with a 5 kg weight is added to the mover.

5.2. Discretization of FO Integral and Differential

As seen from the FO definitions shown in (17)–(20), the Laplace transform of the FO differential and integral of function f(t) can be represented by s^λF(s), where s = jω is the Laplace transform operator. Hence, the Tustin method is used to obtain the coefﬁcients and the form of the direct discretization of s^λ. To simplify the presentation, only the recursive formula for a positive λ is considered. Thus, the continuous Laplace operator can be replaced by a generating function as follows [18, 25]:where z is the shifting operator and T is the sampling period:

Consequently, the Laplace operator can be approximated to derive the FO integral and differential based on any given order of approximation n, as follows:

Thus, the FO operations can be realized via the digital implementation. Table 3 lists the expressions of for n = 0, 1, 3, and 5.

5.3. Experimental Results

In the experiment, the control parameters of the PID controller were chosen as = 25, K_Ix = 110, K_Dx = 10, = 25, K_Iy = 110, and K_Dy = 12, respectively. Moreover, the control parameters of the FFPID controller are selected as = 25, K_Ix = 100, K_Dx = 9, λ_x = 1, a_x = 0.6, b_x = 0.5, = 25, K_Iy = 100, K_Dy = 11, λ_y = 1, a_y = 0.5, and b_y = 0.5, respectively. In addition, a third-order approximation was used for the FO digital realization; that is, n = 3. In this study, the control parameters were selected on the basis of several trials to achieve the favorable transient responses, considering the requirement of steady-state stability. However, designing an optimal set for all the control parameters is difficult because of the occurrence of uncertainties. Additionally, the PID and FFPID controllers cannot maintain ideal positioning performances by adopting the constant control parameters.

5.3.1. Flower Contour Tracking Results

The experimental results, including the tracking errors and drive currents of the VCMs-driven dual-axis positioning stage controlled by the PID, FFPID, and VOFFPID control systems due to the flower contour tracking in Cases 1 and 2, are shown in Figures 9 and 10, respectively. As can be seen from Figures 9(a) and 10(a), the mover of the stage can be successfully controlled by all the controllers to track the reference nonlinear contour shown in Figure 7. Furthermore, the drive currents in Case 2 are larger than those in Case 1, so a higher thrust force for the additional payload can be generated. The maximum tracking errors obtained in Case 1 for the PID, FFPID, and VOFFPID control systems were 0.2807 mm, 0.2363 mm, and 0.1752 mm, respectiely, whereas those obtained in Case 2 were 0.3973 mm, 0.2986 mm, and 0.2731 mm, respectively.

(a)

(b)

(a)

(b)

The tracking errors of the PID control system were unfavorable because of the large tracking errors. Although selecting larger control gains can diminish the amplutide of tracking errors, the excessive aggressive control gains may result in the oscillation of control responses.

As seen in Figures 9 and 10, the FFPID with two well-designed variables a and b and FLS can derive more effective and smooth control signals to restrain the contouring errors related to the specified dynamics of VCMs and the possible occurrence of uncertainties during the flower contour tracking. The corresponding tracking errors were reduced compared with those of the PID controller. Moreover, the FFPID controller has a good ability to diminish the effect of the disturbance, as shown in Figures 9(a) and 10(a). Although the control parameters of the FFPID controller were selected with several trials, the maximum and average tracking errors in the nominal and payload conditions are obviously reduced by the self-tuned strategy.

5.3.2. Window Contour Tracking Results

The experimental results due to the window contour tracking in Cases 1 and 2 are shown in Figures 11 and 12, respectively. Similar behaviors on the tracking responses of flower contour tracking can be observed. As seen from Figures 11(a) and 12(a), the maximum tracking errors obtained in Case 1 for the PID, FFPID, and VOFFPID control systems were 0.4877 mm, 0.2843 mm, and 0.2344 mm, respectively, whereas those obtained in Case 2 were 0.5944 mm, 0.3512 mm, and 0.3098 mm, respectively. From the comparison in Figures 11(a) and 12(a), the tracking performances of the PID are evidently deteriorated when the contour command changes instantaneously. In contrast, the proposed VOFFPID demonstrates its robustness in the tracking performance during both test conditions. On the other hand, the control oscillations in the PID control system as shown in Figures 11(b) and 12(b) are evident due to its inefficient tracking ability. As opposed to the PID, more effective and smooth control signal was derived by the proposed VOFFPID to carry out the best control performance.

(a)

(b)

(a)

(b)

The experimental results and observations reveal that the optimized control parameters can improve the tracking performance in practical control applications. In Figures 9–12, the best control performance of the VOFFPID controller due to the flower and window contours under the nominal and payload conditions can be clearly observed. The improvement of the proposed VOFFPID controller in terms of the contour tracking accuracy is significant compared with that given by traditional PID and FFPID controllers.

The contour tracking performance measures of the PID, FFPID, and VOFFPID control systems for the tracking of the flower and window reference nonlinear contours are shown in Tables 4 and 5, respectively. They indicate that the FFPID controller with the integration of the PID control, FO operation, and FLS outperforms the conventional PID controller. Moreover, the proposed VOFFPID controller further improves the tracking performance of the FFPID controller because all the control parameters were globally and dynamically optimized by the CSA algorithm. The VOFFPID controller apparently exhibits a high-precision contour tracking performance by effectively handling the payload and uncertainty during control processes.

6. Conclusions

In this study, a VOFFPID control system is successfully developed and applied to control the mover position of a new VCMs-driven dual-axis positioning stage for tracking nonlinear reference contours. First, the structural and operating principles of the stage are introduced. Then, the CSA with the adaptive strategy for the optimization of control parameters is described. Subsequently, the theoretical bases of the PID, FFPID, and VOFFPID control systems are given in detail. With an additional degree of freedom to the control parameters and FLS operation, the FFPID controller can upgrade the contouring performances of the PID controller. Moreover, in the proposed VOFFPID controller, the CSA with the adaptive strategy can enhance the robustness of the FFPID controller by tuning the control parameters online. The experimental results subjected to different performance measures are given to verify the effectiveness of the proposed VOFFPID controller.

Data Availability

The experimental data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors acknowledge the financial support of the Ministry of Science and Technology in Taiwan, R.O.C., under Grant no. MOST 108-2221-E-003-022-MY2.

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Copyright © 2021 Syuan-Yi Chen and Meng-Chen Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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