Speed Tracking Performance for a Coreless Linear Motor Servo System Based on a Fitted Adaptive Fuzzy Controller

Abstract

Fuzzy control is widely used in linear motor servo systems. However, simple fuzzy rules reduce the control accuracy of the servo system, while complex fuzzy rules reduce the speed of its decision making. This paper proposes a fitted adaptive fuzzy controller (FAFC) to improve the speed tracking performance of a coreless linear motor servo system. The FAFC took a planned curve as a motion target. The planned curve is compounded by multiple performance curves of fuzzy control under the same given input. These multiple performance curves cover the variation range of motor parameters, which are offline-obtained. The performance of the planned curve is ensured by the multiple offline performance curves. The FAFC only needs simple fuzzy rules to fit the planned curve, and achieves high control accuracy without affecting the decision speed. The experimental results verified the feasibility of an FAFC. This research shows that an FAFC can effectively shorten the online calculation time of complex algorithms and keep the consistency of performance.

1. Introduction

Tracking performance is an important index of linear motor servo system, which can directly determine the quality of a processed product, such as laser processing, contour scanning, or artificial intelligence. The stiffness of laser welding products is seriously affected by the speed tracking performance of linear motion platforms. The speed tracking performance of linear motion platforms also determines the accuracy of the scanning system and the cooperation of multiple components of artificial intelligence. Therefore, the study of the tracking performance of linear motor servo systems is signal.

Fuzzy control is widely used in linear motor servo systems, due to the fact that it is based on rules with no need for motor control models. There are many studies on using fuzzy control algorithms to improve reliability, antijamming capability, and tracking performance of nonlinear systems. Double-loop recurrent feature selection fuzzy neural networks can compensate for approximation deviation and suppress the chattering phenomenon of permanent magnet linear synchronous motor servo systems [1], which takes position tracking error and filtered velocity error as the input to output dynamic compensation gain. Dynamic compensation gain is designed by the dynamic neural network to refine the adaptive feedback. Universal adaptive fuzzy control has the advantage of resisting external disturbances [2], which is achieved through global asymptotic model-free trajectory-independent tracking [3]. The universal fuzzy state observer is the core of control strategy, which is used to estimate unknown dynamics and dominant unknown residuals. Type-2 fuzzy control is better for uncertain systems. Compare to conventional fuzzy control, Type-2 fuzzy control broadens the membership function and changes the exact membership function into footprint of uncertainty. In [4], an interval type-2 sampled-data fuzzy-model-based output-feedback is adopted to protect against the uncertainties of the nonlinear plant, which realized better robustness of the fuzzy controller. A new fuzzy relaxed matrix technique was developed [5], which solved the periodic tracking control problem in nonlinear systems. The fuzzy relaxed matrix technique was based on the fuzzy observer-based controller and the fuzzy repetitive controller. The fuzzy sliding mode control has the characteristics of fast response, strong anti-interference ability, no steady-state error, and insensitivity to uncertain disturbance. A sliding mode control method based on fuzzy switching gain adjustment improved the dynamic performance of the linear motor system and weakened chattering [6]. Fractional order system has the characteristics of slow energy transfer and gradual convergence, which is conducive to weaken the amplitude and frequency of buffeting of sliding mode control. Furthermore, an adaptive fuzzy fractional-order sliding-mode control strategy was studied, which performs the precise tracking response for motor drive systems against parameter variations and external disturbances [7]. To enhance the reliability of the motor drive system, the fuzzy logic approach [8] was applied to process the fault symptom variables and obtain the faulty information of power switches. This fuzzy-based fault diagnosis method is available in detecting not only the intermittent power switch faults, but also the multiple kinds of open-circuit faults in power switches, which can improve the warning ability of the permanent magnet synchronous motor drive system. Fuzzy neural network combines the advantages of neural network system and fuzzy system. It has great advantages in dealing with nonlinear and fuzzy problems, and has great potential in intelligent information processing. In [9], a high-performance position servo drive system was constructed through intelligent backstepping controls using recurrent feature selection fuzzy neural networks. A fuzzy proportional-integral (PI) controller is often used to enhance the tracking performance of motion systems, and a quadcopter with a fuzzy-PI controller can better track a moving target [10]. The mentioned Fuzzy-PI controller had its own rule base which can either stabilize the quadcopter over a non-moving target or track a moving target under varying speed. The fuzzy PI controller in [11] had a simple structure and a clear physical meaning, which greatly improved tracking performance by using backstepping method with fuzzy logic systems. An adaptive fuzzy PI controller with robust updating rules in [12] reduced the impacts of disturbances, which helped improve speed tracing performance. With a fuzzy tuner, the gains of PI controller were regulated online, and the scaling factors of the fuzzy tuner were online optimized as initial information of the dynamic non-linear system was not enough for designing fixed parameters fuzzy tuner in advance.

The mentioned coreless linear motor servo system in this paper is a part of laser process equipment. It needs to cooperate with other linear moving parts to complete the arc motion under the same speed with rectilinear motion. The tracking performance of velocity in transient state is an important index of the linear servo system. Complex fuzzy control is difficult to achieve desired control performance with because of its long decision time, while simple fuzzy control is difficult to maintain the original performance with, due to the change in the motor control model. As more factors are considered, the control algorithm becomes more complex. In the starting stage of the motor servo system, the output control parameter of the complex algorithm is less because of the large acceleration. Even if these control parameters are more accurate, the larger output interval of control parameter will reduce the performance of the servo system, especially in transient state. In this paper, an FAFC is proposed to fit the synthesized performance curves with uncomplicated fuzzy rules, which has precise control parameters but less output interval. These synthesized performance curves in an FAFC are obtained offline through the analysis of tracking performance of complex fuzzy control within the variation range of motor parameters under the same fuzzy rules. Fine-turned fuzzy controller of FAFC has simple fuzzy rules and shorter output interval of control parameter, which is used to track the synthesized performance curves online.

The paper is organized as follows: Section 2 elaborates on the principle of fitted adaptive fuzzy controllers (FAFC); The target performance curve is synthesized by the offline performance curves in Section 3; A fine-turned fuzzy controller is designed in Section 4; and Section 5 provides the experimental tests and discussion.

2. Principle of Fitted Adaptive Fuzzy Controllers

A fitted fuzzy adaptive controller consists of a fitted data generator and a fine-tuned fuzzy controller, which is shown in Figure 1. v*(t), v_t(t), and v(t) correspond to speed curve, target speed curve, and actual speed response curve of linear motion system, respectively. e_v(t) is the difference between v*(t) and v_t(t), e_vc(t) is the derivative of e_v(t). E_v(t) is the difference between v_t(t) and v(t), and E_vc(t) is the derivative of E_v(t). i(t) is the armature current and x(t) is the position of the mover. A general fuzzy controller takes e_v(t) and e_vc(t) as input to output control parameters, while a fitted fuzzy controller takes E_v(t) and E_vc(t) as input. The speed controller in Figure 1 is a PI controller; its transfer function W_s(s) can be expressed as W_s(s) = [K_p(t)s + K_i(t)]/s. K_p(t) and K_i(t) are proportional magnification factor and integral factor respectively, and

$$\left\{\begin{array}{c}{\mathit{K}}_{\mathrm{p}}{\left(\mathit{t}\right)=\mathit{K}}_{\mathrm{p}0}{+\mathit{K}}_{\mathrm{pf}}{\left(\mathit{t}\right)+\mathsf{\Delta }\mathit{K}}_{\mathrm{p}}\left(\mathit{t}\right)\\ {\mathit{K}}_{\mathrm{i}}{\left(\mathit{t}\right)=\mathit{K}}_{\mathrm{i}0}{+\mathit{K}}_{\mathrm{if}}{\left(\mathit{t}\right)+\mathsf{\Delta }\mathit{K}}_{\mathrm{i}}\left(\mathit{t}\right) \end{array},\right.$$

(1)

where K_p0 and K_i0 are inherent attributes of the speed controller. K_pf(t) and K_if(t) are proportional magnification factor and integral factor of the output of the fitted data generator, which brings v(t) closer to v_t(t). ΔK_p(t) and ΔK_i(t) are factors of the output of the fine-tuned fuzzy controller, which guarantees that E_v(t) is within a certain range.

The fitted data generator is the core of proposed method, and its internal structure is shown in Figure 2. The target speed curve generator is the important part of the fitted data generator, which outputs v_t(t). v*(t), v(t), T_w(t₀), T_A(t₀), T_p(t₀), x(t₀), and i(t₁) are the input parameters of the target speed curve generator. T_w(t₀), T_A(t₀), and T_p(t₀) are the temperatures of winding, ambient, and permanent magnet (PM) at t = t₀, respectively, which can be detected directly or deduced [13,14]. t_{0 has} the start time as v*(t). The winding temperature influences the value of the armature resistance, the ambient temperature influences the accuracy of the grating ruler, and the PM temperature influences the value of the armature inductance and the magnitude of the thrust/back-EMF coefficient. Therefore, the target speed curve generator takes T_w(t₀), T_A(t₀), and T_p(t₀) instead of variable parameters of linear motor as input. Force of friction f is an important parameter of linear servo motion, and f(t₀) is the friction at t = t₀, which is another input of the fitted data generator. f(t₀) is variable and difficult to detect directly, which can be inferred by

$${\mathit{f}(\mathit{t}}_0)=\mathrm{g}\left[{\mathit{x}(\mathit{t}}_0{),\mathit{T}}_{\mathrm{p}}{(\mathit{t}}_0{),\mathit{i}(\mathit{t}}_1)\right]{=\mathit{k}}_{\mathrm{f}}\left[{\mathit{x}(\mathit{t}}_0{),\mathit{T}}_{\mathrm{p}}{(\mathit{t}}_0)\right]{\mathit{i}(\mathit{t}}_1),$$

(2)

where i(t₁) is the armature current at t = t₁, and t₁ is the first time of v(t) ≠ 0. k_f[x(t₀), T_p(t₀)] is the thrust coefficient of linear motor at t = t₀, which is related with T_p(t₀) and x(t₀). x(t₀) is the position of the mover at t = t₀. T_p(t₀) has influence on B_r (remanent flux density). For the coreless motor, B_r is proportional to B_δ (gap flux density). B_δ is directly related to k_f[x(t₀), T_p(t₀)], therefore k_f[x(t₀), T_p(t₀)] can be deduced by Figure 3.

When the whole curve of v*(t) is detected and v(t) ≠ 0 at the first time, the proportional magnification factor K_p1(t), integral factor K_i1(t), and v_t(t) can be output by the target speed curve generator depending on the offline speed performance curves and the values of f(t₀), T_w(t₀), T_A(t₀), and T_p(t₀). When the whole curve of v*(t) is detected and v(t) = 0, the proportional magnification factor K_p3(t) and integral factor K_i3(t) are the initial output of the fitted data generator, which are used to ensure that the terminal voltage of the armature winding is maximum.

The fuzzy controller is another part of the fitted data generator, which is used to enhance the robustness of the motion system. If the motion system has a strong distraction, it is difficult for v(t) to re-fit v_t(t) through the fine-turned fuzzy controller. The fuzzy controller in the fitted data generator takes e_v(t) and e_vc(t) as input to output control parameters. When e_vt(t) is more than e_x, the standby fuzzy controller will output an off signal to the fine-turned fuzzy controller, and replace the target speed curve generator to output the proportional magnification factor K_p2(t) and integral factor K_i2(t) for the speed controller until v*(t) and v(t) are completed. e_vt(t) is the difference between v_t(t) and v(t), and e_x is a threshold value. This standby fuzzy controller is conventional and its principle will not be elaborated in this paper.

The fine-turned fuzzy controller takes E_v(t) and E_vc(t) as input, which makes v(t) closer to v_t(t). This is detailed in later sections.

3. The Offline Performance Curves and Output v_t(t)

It has been introduced that T_A(t), T_p(t), f(t), and T_w(t) are closely related to the control model of linear motors. Under the fuzzy controller, the change in the motor control model can be equivalent to the change in T_A(t), T_p(t), f(t), and T_w(t). On the basis of the variation range of T_A(t), T_p(t), f(t), and T_w(t), they are equally divided and numbered, which is shown in

Table 1. Divided and numbered T_A(t), T_p(t), f(t), and T_w(t).

Parameter	Variation Range	Interval	Unit	Code
TA(t)	0~40	0.1	degrees Celsius	i = 0, 1, …, 400
Tp(t)	0~60	0.2	degrees Celsius	j = 0, 1, …, 300
f(t)	3.5~4.5	0.02	N	m = 0, 1, …, 50
Tw(t)	0~70	1	degrees Celsius	n = 0, 1, …, 70

. The variation ranges of these parameters are based on working conditions and surroundings. The value of the interval depends on the accuracy of the feedback element and the degree of influence of the parameters. T_A(t) is the fundamental parameter, which determines the values of other parameters.

Based on divided and numbered T_A(t), T_p(t), f(t), and T_w(t), the motor control models are numbered as Model(i,j,m,n). For example, Model(250,150,25,35) respects the motor control model with T_A(t) = 25 °C, T_p(t) = 30 °C, f = 4 N, and T_w(t) = 35 °C. In order to ensure the performance of the fuzzy controller as the motor models are changed, an exclusive fuzzy controller is designed for each numbered motor control model. The core of the fuzzy controller is to output control parameters, and the design of fuzzy rules, domain, and membership function are all methods of obtaining accurate control parameters. Therefore, each exclusive fuzzy controller corresponds to a numbered motor control model, which is shown in Figure 4.

Fuzzy controller (i,j,m,n) is exclusive for Model(i,j,m,n). The exclusive fuzzy controllers can have different fuzzy rules, domain, or membership function, which guarantees that their output control parameters can achieve better control performance. K_p(t)_(i,j,m,n) and K_i(t)_(i,j,m,n) are the output proportional magnification factor and integral factor of fuzzy controller (i,j,m,n). With the input of v*(t), v(t)_(i,j,m,n) is the speed-response curve of Model_(i,j,m,n) under K_p(t)_(i,j,m,n) and K_i(t)_(i,j,m,n). With the design of exclusive fuzzy controllers for all numbered models, the offline speed performance curves and control parameters are obtained.

When T_A(t₀), T_p(t₀), f(t₀), and T_w(t₀) are detected by the target speed curve generator, they are numbered as i₀, j₀, m₀, and n₀ respectively. Refer to Table 1, α, β, λ, and σ.

$${\mathit{i}}_0{=\mathit{T}}_{\mathrm{A}}{(\mathit{t}}_0)/0{.1,\mathit{j}}_0{=\mathit{T}}_{\mathrm{p}}{(\mathit{t}}_0)/0{.2,\mathit{m}}_0{=\mathit{f}(\mathit{t}}_0)/0{.02-150,\mathit{n}}_0{=\mathit{T}}_{\mathrm{w}}{(\mathit{t}}_0),$$

(3)

i₀, j₀, m₀, and n₀ must belong to [α (α+1)], [β (β+1)], [λ (λ+1)], and [σ (σ+1)], respectively. α, β, λ, and σ are positive integers less than 400, 300, 50, and 70, respectively, which indicates the current motor control model is strongly related to Model(α, β, λ, σ) and Model(α+1, β+1, λ+1, σ+1). There is less difference between the two Models, and their speed performance curves are similar. Then, K_p1(t) and K_i1(t) can be expressed as

$$\left\{\begin{array}{c}{\mathit{K}}_{\mathrm{p}1}{\left(\mathit{t}\right)=(\mathit{i}}_0{-\mathit{i}}_1{)\mathit{K}}_{\mathrm{p}}{\left(\mathit{t}\right)}_{\left(\mathit{\alpha }+1,\mathit{\beta }+1,\mathit{\lambda }+1,\mathit{\sigma }+1\right)}{+(1+\mathit{i}}_1{-\mathit{i}}_0{)\mathit{K}}_{\mathrm{p}}{\left(\mathit{t}\right)}_{\left(\mathit{\alpha },\mathit{\beta },\mathit{\lambda },\mathit{\sigma }\right)}\\ {\mathit{K}}_{\mathrm{i}1}\left(\mathit{t}\right)= {(\mathit{i}}_0{-\mathit{i}}_1{)\mathit{K}}_{\mathrm{i}}{\left(\mathit{t}\right)}_{\left(\mathit{\alpha }+1,\mathit{\beta }+1,\mathit{\lambda }+1,\mathit{\sigma }+1\right)}{+(1+\mathit{i}}_1{-\mathit{i}}_0{)\mathit{K}}_{\mathrm{i}}{\left(\mathit{t}\right)}_{\left(\mathit{\alpha },\mathit{\beta },\mathit{\lambda },\mathit{\sigma }\right)} \end{array}\right.,$$

(4)

Considering v(t) _(i,j,m,n) is not consistent and v(t) will be disturbed by noise errors when it is at steady state, v_t(t) can be expressed as

$${\mathit{v}}_{\mathit{t}}\left(\mathit{t}\right)=\left\{\begin{array}{c}\frac{{(\mathit{i}}_0{-\mathit{i}}_1){{\displaystyle \sum }}_{\mathit{h}=1,2,\dots }^{\mathit{N}}{\mathit{v}}_{\mathit{h}}{\left(\mathit{t}\right)}_{\left(\mathit{\alpha }+1,\mathit{\beta }+1,\mathit{\lambda }+1,\mathit{\sigma }+1\right)}}{\mathit{N}}+\frac{{(1+\mathit{i}}_1{-\mathit{i}}_0){{\displaystyle \sum }}_{\mathit{h}=1,2,\dots }^{\mathit{N}}{\mathit{v}}_{\mathit{h}}{\left(\mathit{t}\right)}_{\left(\mathit{\alpha },\mathit{\beta },\mathit{\lambda },\mathit{\sigma }\right)}}{\mathit{N}},{\mathit{t}<\mathit{t}}_2\\ \mathit{v}*\left(\mathit{t}\right),\mathit{t}\ge {\mathit{t}}_2\end{array}\right.,$$

(5)

where v_h(t) is hth v(t) _(i,j,m,n) of Model_(i,j,m,n) under the same K_p(t)_(i,j,m,n) and K_i(t)_(i,j,m,n), N is the number of samples, and t₂ is the time that both v*(t) and v(t) are at steady state.

4. Design of the Fine-Turned Fuzzy Controller

Because the exclusive fuzzy controllers are designed for different motor control models, v_t(t) is a curve that can be fitted by v(t) under a fine-turned fuzzy controller. Referring to [12], the real inputs are fuzzed to fuzzy variables, α₁E_v(t) and β₁E_vc(t). λ₁ΔK_p(t) and σ₁ΔK_i(t) are output fuzzy variables. The fuzzy domains of α₁E_v(t), β₁E_vc(t), λ₁ΔK_p(t), and σ₁ΔK_i(t) are all [−3 3] and invariable. α₁, β₁, λ₁, and σ₁ are scaling factors of E_v(t), E_vc(t), ΔK_p(t), and ΔK_i(t), respectively, which are variable. The values of α₁ and β₁ depend on v*(t) and the accuracy of speed feedback. λ₁ and σ₁ are related to the proportional magnification factor and integral factor of approximate Models, which can be expressed as

$$\left\{\begin{array}{c}{\mathit{\lambda }}_1=\max \left\{{\mathit{K}}_{\mathrm{p}}{\left(\mathit{t}\right)}_{(\mathit{\alpha }+1,\mathit{\beta }+1,\mathit{\lambda }+1,\mathit{\sigma }+1)}{,\mathit{K}}_{\mathrm{p}}{\left(\mathit{t}\right)}_{(\mathit{\alpha },\mathit{\beta },\mathit{\lambda },\mathit{\sigma })}\right\}/3\\ {\mathit{\sigma }}_1=\max \left\{{\mathit{K}}_{\mathrm{i}}{\left(\mathit{t}\right)}_{(\mathit{\alpha }+1,\mathit{\beta }+1,\mathit{\lambda }+1,\mathit{\sigma }+1)}{,\mathit{K}}_{\mathrm{i}}{\left(\mathit{t}\right)}_{(\mathit{\alpha },\mathit{\beta },\mathit{\lambda },\mathit{\sigma })}\right\}/3\end{array}\right.,$$

(6)

The fuzzy word set of α₁E_v(t), β₁E_vc(t), λ₁ΔK_p(t), and σ₁ΔK_i(t) are {PB, NM, NS, ZO, PS, PM, PB}, which corresponds, respectively, to {−3, −2, −1, 0, 1, 2, 3}. Their membership functions are all triangular shape, which is shown in Figure 5.

The fuzzy rule of fine-turned fuzzy controllers is relatively simple. When the acceleration of the linear motor changes greatly, α₁E_v(t) is bigger, which needs a large ΔK_p(t) and minimized ΔK_i(t). As the change in acceleration of the mover decreases, α₁E_v(t) varies around zero, and it is more appropriate to take the moderate values of ΔK_p(t) and ΔK_i(t). When the change in acceleration of the mover approaches zero, increasing ΔK_p(t) and ΔK_i(t), it is conducive to reduce the overshoot of v(t). Based on the above principles, the fuzzy rules of ΔK_p(t) and ΔK_i(t) are shown in

Table 2. Fuzzy rule of λ₁ΔK_p(t).

α1Ev(t)	β1Evc(t)
	NB	NM	NS	ZO	PS	PM	PB
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	PS	ZO	NS
NS	PM	PM	PM	PS	ZO	NS	NS
ZO	PM	PM	PS	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

and

Table 3. Fuzzy rule of σ₁ΔK_i(t).

α1Ev(t)	β1Evc(t)
	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NB	NM	NM	NS	ZO	ZO
NM	NB	NB	NM	NS	NS	ZO	ZO
NS	NB	NM	NS	ZO	ZO	PS	PS
ZO	NM	NM	NS	PS	PS	PS	PM
PS	NM	NS	ZO	PS	PS	PM	PB
PM	ZO	ZO	PS	PM	PM	PM	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

, respectively.

5. Experimental Test and Discussion

To verify the speed tracking performance of the FAFC, a test platform of coreless linear motion systems is presented, which is shown in Figure 6, and its main parameters are shown in

Table 4. Main parameters of motion system.

Item	Value	Unit
Effective travel	±50	mm
Maximum speed	0.15	m/s
Phase inductance	5.5 ± 10%	Ω
Phase resistance	1.24 ± 20%	mH
Force constant	11.06 ± 10%	N/A
Frictional force	3.83 ± 5%	N

.

The verification of speed tracking performance of an FAFC is based on two cases. When the coreless linear motor has been working for a little time, there is not a lot of heat, where T_A(t) = 20 °C, T_p(t) = 21 °C, f = 3.88 N, and T_w(t) = 25 °C, which is named Case (200, 105, 194, 25). When the coreless linear motor has been working for some time, there is a lot of heat, where T_A(t) = 20 °C, T_p(t) = 35 °C, f = 3.82 N, and T_w(t) = 50 °C, which is named Case (200, 175, 191, 50).

Figure 7a shows the given input speed-curve v*(t). Figure 7b,c show the target speed curve v_t(t) in Case (200, 105, 194, 25) and Case (200, 175, 191, 50) respectively. e_t(t) is the difference between v*(t) and v_t(t). The e_t(t) of Case (200, 105, 194, 25) and Case (200, 175, 191, 50) are shown in Figure 7d. The maximum error and overshoot of Case (200, 105, 194, 25) are 1.1 mm/s and 0.074 mm/s, respectively, while for Case (200, 175, 191, 50) they are 1.158 mm/s and 0.097 mm/s. Compared to Case (200, 105, 194, 25), the speed tracking performance of Case (200, 175, 191, 50) decreases lightly, because of the change in heat.

Figure 8a shows the speed tracking error e_v(t) of CFC, AFC, and FAFC with the input speed curve of Figure 7a under Case (200, 105, 194, 25), and Figure 8b is that of Case (200, 175, 191, 50).

Table 5. Performance of CFC, AFC, and FAFC under Case (200, 105, 194, 25).

Item	Maximum Error (mm/s)	Overshoot (mm/s)	RMS of Error (mm/s)	Average Computation Cost (ms)
CFC	1.171	0.084	0.317	0.126
AFC	1.125	0.076	0.306	0.208
FAFC	1.097	0.072	0.289	0.141

and

Table 6. Performance of CFC, AFC, and FAFC under Case (200, 175, 191, 50).

Item	Maximum Error (mm/s)	Overshoot (mm/s)	RMS of Error (mm/s)	Average Computation Cost (ms)
CFC	1.683	0.239	0.474	0.128
AFC	1.216	0.135	0.350	0.219
FAFC	1.146	0.117	0.305	0.144

indicate the performance of the three control algorithms under the two Cases respectively. CFC here is the acronym for conventional fuzzy control, which has a fixed domain and membership function and simple fuzzy rules. AFC here is the acronym for adaptive fuzzy control, which can choose the appropriate domain, membership function, and complex fuzzy rules according to the current criteria. Under the two Cases, FAFC has the best speed tracking performance within the three control algorithms, because the target speed curve of FAFC is generated offline by complex fuzzy rules and the fine-turned fuzzy controller makes the actual speed curve close to the target speed curve online. The fine-turned fuzzy controller has simple fuzzy rules. The average computation cost of an FAFC is slightly longer than that of CFC, but FAFC has more complex fuzzy rules. Complex fuzzy rules need more calculation time than simple fuzzy rules online, so the FAFC has less average computation cost than AFC, as they have the same complex fuzzy rules.

When the control model is changed, the performance of CFC is clearly declined, even though it has less computation time. AFC can act against the change in control model, but the long computation time limits its performance. FAFC combines the advantages of CFC and AFC, which confers it better performance.

The standard deviation of speed tracking error σ_e(t) of FAFC and AFC under 20 sampling times are shown in Figure 9a and Figure 9b, respectively, which can reflect their consistency of speed tracking performance. The mean value of σ_e(t) of FAFC is 0.033 mm/s, and the mean value of σ_e(t) of AFC is 0.036 mm/s. AFC can deal with the variation in motor control model by the change in fuzzy rules or domain, but its consistency of speed tracking performance is less than that of an FAFC. Because the output v_t(t) of FAFC is synthesized based on the average of multiple outputs of the Model, the random errors in motion process are effectively neutralized.

6. Conclusions

The FAFC presented in this paper not only has a complex fuzzy algorithm, but also does not cost a lot of online time. The FAFC can achieve better speed tracking performance, and also better consistency. In this paper, the variational parameters involved for FAFC are T_A(t), T_p(t), f(t), and T_w(t). In the future, as more variational parameters are analyzed, FAFC will further improve the tracking performance of the motion system.

The major contributions of this paper are as follows.

A proposed FAFC takes online complex algorithms offline and turns them into performance curves that can be fitted with simple algorithms, which indirectly reduces the online computation time of complex algorithms.

The generated performance curves are synthesized from the results of multiple measurements, which can effectively avoid random error and improve the consistency of performance.

The more complex the fuzzy rules, the greater advantage of an FAFC over AFC.

With the exception of a few particularly good algorithms, the accuracy and complexity of algorithms are often proportional. More complex algorithms require longer computing time, which will limit the control performance of algorithms. This paper provides an effective method to shorten the online calculation time of complex algorithms.