Research on Anti-Fluctuation Control of Winding Tension System Based on Feedforward Compensation

1  Introduction

In the fiber winding process, one of the most critical factors affecting the winding quality is the adjustment of fiber tension [1]. The guide wire device in the winding machine drives the fiber to wind along the surface of the wound part according to a particular trajectory, in which the fiber ensures the stability of the winding tension under the action of the adjusting device [2,3]. Suppose there is no appropriate controller to control the tension. In that case, the fiber may be unable to retract and release the yarn in time, resulting in fiber tension fluctuation and output tension instability, which in turn affects the performance of the winding product. In order to overcome the above difficulties, the traditional fiber winding machine adjustment device [4] is usually used to adjust the fiber tension, such as swing rod, mechanical spring and lag brake. However, when the winding speed is accelerated, the regulating device cannot respond quickly and cannot ensure that the winding tension is constant near the set value. At this time, fiber jitter and tension fluctuation will be caused. In order to solve the above problems, this paper is devoted to using the active tension control method to analyze the tension variation law and the composition of the fluctuation, primarily focusing on the tension estimation combined with the observer and the control of the tension system by the fractional-order PID controller to improve the performance of the fiber winding machine.

Researchers have conducted many investigations on constant tension control [5,6]. Chen et al. [7] proposed a nonlinear model predictive control method to eliminate tension disturbance and improve tension control accuracy. Chen et al. [8] designed a robust linear parameter varying model predictive control scheme to improve the tension tracking performance. Imamura et al. [9] used system identification to model the winding tension and designed a 2-DOF PID controller based on a genetic algorithm to realize the real-time control of winding tension. In addition, many modern control methods have been introduced into tension control, including PID control [10], fuzzy control [11], and fractional order control [12]. Among them, the fractional-order PID controller has been successfully applied in the stable control of tension with its excellent control performance [13]. Meng designed a control strategy based on fractional-order PID to achieve the control goal of constant tension in the winding system [14]. However, the control parameters of the PID controller are independent of each other and lack the ability to coordinate control [15]. In order to eliminate the shortcomings of the PID controller, scientists have carried out further research. Zeng et al. [16] proposed a parameter coupling strategy and established the relationship between the control parameters of the proportional, integral, and differential links in the integer-order PID controller. This provides a new idea for the tuning of fractional-order PID controller parameters.

However, the framework of existing control methods cannot effectively suppress tension fluctuation by using the main characteristics of tension fluctuation, which is conservative. Because of multi-factor disturbance for speed in the winding process (the non-circularity, coulomb friction, and vibration of the guide roller) and strong coupling effect between fiber speed and tension, resulting in periodic tension fluctuation [17]. In order to achieve a satisfactory control effect, the rejection of periodic signals should be considered in the design of signal control schemes. In most studies, various methods have proved effective for uncertain periodic systems [18–20]. In [21], the adaptive frequency estimation scheme combined with the fractional-order controller to suppress the offset sinusoidal disturbance of the unknown parameter state and an adaptive orthogonal signal generator based on the third-order generalized integrator to estimate the disturbance. However, in the adaptive algorithm, the control and observation parameters are solved separately, which leads to a significant increase in the calculation of the parameters and affects the convergence speed of the estimation. In addition, the disturbance observer (DOBC) [22,23] realizes the compensation and rejection of periodic fluctuations [24,25] by estimating the critical fluctuation frequency information [26,27]. In [28], a two-layer observer is proposed to reconstruct the unknown frequency disturbance, which can eliminate the parameter uncertainty condition on the system uncertainty. However, the tension system always encounters disturbances with different characteristics [29], such as friction and noise. Due to the lack of robust control, the performance of disturbance compensation may deteriorate.

In order to reduce the tension fluctuation of fiber winding, the purpose of this technical description is to propose a tension fluctuation rejection strategy based on feedforward compensation. A composite control scheme of tension fluctuation observer and time-varying parameter fractional PID controller is developed using DOBC [30] control structure and PID parameters coupling [31] control rules. It solves the tension fluctuation control problem of the winding system with an unknown state periodic signal and bounded noise. In the design process of the proposed tension fluctuation observer, the tension periodic signal is regarded as an exogenous system [32]. Considering the friction between the fiber and the device, the uncertain periodic signal is a bias harmonic. The fluctuation characteristics are excited by designing the nominal model and the inverse filter of the controlled system. Then, the frequency generator is designed to estimate the critical frequency parameter information of the bias harmonic periodic signal, and a third-order auxiliary filter is constructed to give the equivalent form of the uncertain bias harmonic compensation signal. Finally, the tension signal is reconstructed in the fluctuation reconstructor, and the actual system is compensated. The asymptotic convergence and uniform ultimate boundedness of the reconstructed tension fluctuation under a multi-factor disturbance environment are analyzed by using the Lyapunov theorem. The coupling between harmonic parameters and the solution of complex differential equations can be avoided. Combined with the designed time-varying fractional-order PID controller, the bounded noise in tension fluctuation is attenuated. The simulation data show that when the winding speed disturbs the system, the proposed control strategy can control the tension fluctuation deviation in a small range. It is verified that the proposed scheme improves the tracking ability and robustness of the system in response to tension fluctuation.

The structure of this paper is as follows: Section 2 gives a description of the tension fluctuation problem. In Section 3, a tension fluctuation observer is designed for biased harmonic signals, and the stability of the composite system is analyzed. Section 4 gives the design of a time-varying parameter fractional-order PID controller. In Section 5, various simulation examples are given to illustrate the effectiveness. Section 6 gives the conclusion of this paper and the future research work.

The main contributions and innovations of this paper can be summarized as follows:

(1) The tension fluctuation generated by the actual tension system is analyzed, and the compensation signal of the uncertain disturbance is reconstructed by its characteristics. The equivalent form of the tension fluctuation of the uncertain bias harmonic compensation signal is given.

(2) A third-order auxiliary filter is constructed. The disturbance signal with unknown frequency is introduced, and the relationship between the disturbance frequency and the compensation coefficient is given. A tension fluctuation observer (TFO) is proposed to realize the accuracy and robustness of tension fluctuation estimation.

(3) By introducing the dynamic parameters into the fractional order PID controller, a time-varying parameters fractional-order PID controller (TPFOPID) is established, which can realize the efficient control of the tension system with complex disturbance.

(4) The proposed TFO observer and TPFOPID controller are independent of each other and remove the necessity of estimating the system state to complete the control target, making the system stability easier to analyze.

2  Problem Formulation

In this section, we consider the control model of fiber winding tension regulator. Additionally, based on actual tension data, the main types of disturbances causing tension fluctuations were identified. This serves to further inform the design of control strategies.

2.1 Tension Control Model

According to Xu et al. [33], the kinematics control model of tension adjusting rod can be expressed as:

RJθ⋅⋅+(Rβ+KeKf)θ⋅=KfKb(u+d)(1)

where R is the equivalent resistance, J is the equivalent rotational inertia, β is the equivalent friction coefficient, Ke is the electromotive force coefficient, Kf is the motor torque coefficient, Kb is the pendulum coefficient, θ is the angle of the swing rod, u is the control input, d is the disturbance signal.

The transfer function for the tension regulating device can be expressed as:

Gk(s)=KfRJs+Rβ+KeKf⋅Kbs(2)

Define the state vector x=[x1x2]T=[θw]T.

The state space control model describing the motion of the winding tension regulating device can be expressed as:

{x⋅(t)=Ax(t)+C(u+d∗+Δd)y=Fx(t)(3)

A=[010−a1a2],C=[0b0],F=[1a20](4)

a2=RJ,a1=Rβ+KeKf,b0=KfKb.(5)

In the research problem shown in Fig. 1, the movement of the fiber driven by the guide tube during winding is a repetitive process in the form of harmonics, and there is friction between the fiber and the device. Due to the strong coupling relationship between the winding speed and the winding tension, the tension fluctuation of the fiber is present in the form of a biased harmonic signal.

Figure 1: A multi-bundle fiber winding device with tension adjustment

2.2 Tension Fluctuation Description

We consider the tension experimental data from the actual winding system and fit the equivalent tension output curve that characterizes the tension fluctuation, as shown in Fig. 2.

Figure 2: The actual measured curve and simulation fitting curve of wire tension

The composite disturbance causing tension fluctuation through fitting and comparison analysis has the following characteristics: random noise, unknown frequency simple harmonic wave disturbance, and bias disturbance. In order to facilitate the verification of the performance of the controller and observer proposed in this paper, in the following research, these disturbance factors are reasonably amplified and described. Therefore, the tension fluctuation d(t) in our system can be described as:

d(t)=d∗(t)+Δd(t)=ψ0+ψsin⁡(ωct+ϕ)+Δd(t)(6)

where, ψ is the amplitude of the bias harmonic signal, ωc is the fluctuation frequency, ϕ is the phase, ψ0 is the offset, and the bias harmonic signal is in the parameter unknown state.

Δd(t)=η(x,t) satisfies:

Δd(t)=‖η(x,t)‖≤κ(7)

The bias harmonic signal can be equivalent to a third-order external system, d∗(t) can be formulated by external linear system [34]:

{ω⋅(t)=Mω(t)d(t)=Qω(t)(8)

where ω(t)∈R3×1, Q∈R1×3, M∈R3×3, and M, Q can be expressed as:

M=[0100010−ϖ0],Q=[100](9)

Most of the current work needs to estimate the system state and signal parameters information in Eq. (8) simultaneously, which will increase the operation and reduce the convergence speed of the estimation. On the contrary, in the proposed tension fluctuation observation structure, the reconstruction d∗(t) only needs to estimate one parameter, which also dramatically improves the compensation efficiency of the system disturbance.

3  Design of Fluctuation Reconstructor

This section describes the nominal model of the controlled system for cases where the frequency deviation harmonic signal is unknown. An inverse filter is then designed to stimulate signal features, and a frequency generator is used to estimate tension fluctuation frequency. Further, the fluctuation signal is reconstructed using a third-order auxiliary filter based on the estimated frequency information. Finally, feedforward compensation is implemented for practical. Fig. 3 gives the overall structure of proposed method. The design of the TPFOPID controller is described in the next section.

Figure 3: The proposed control architecture block diagram

The tension fluctuation observer consists of three parts: inverse filter, third-order auxiliary filter and frequency generator. The control input is defined as u=uTP−d^∗, where uTP is the output of the TPFOPID controller and d^∗ is the estimated bias harmonic signal; y∼ is the difference between the controlled system tension output y and the nominal model output yn. r is system reference input.

We can get the error equation transfer function from d to y∼:

Gdy∼(s)=y∼d=Gk(s)1+L0⋅Gk(s)=b0a2s2+a1s+b0⋅L0(10)

(1) Compared with other disturbance rejection methods, it only needs to use the disturbance characteristics without estimating the disturbance state, which avoids the conservatism of the design.

(2) The tension fluctuation observer can be well combined with the designed time-varying parameter fractional order PID controller to control the controlled system, which can realize the independence of the observer and the controller.

(3) It can be applied to fractional order control systems to solve the problem of disturbance estimation and compensation in such nonlinear systems. At the same time, the rejection of a class of biased sinusoidal disturbances with unknown frequencies in the tension system is realized, and the large-scale asymptotic stability of the system is analyzed by using uniform ultimate boundedness.

3.1 Reconstruction Signal

The inverse filter is designed to excite the tension fluctuation characteristics. The equivalent input tension compensation form is derived by introducing the reconstructed fluctuation signal.

The inverse filter is constructed by using G0−1(s) and n first-order inertial links in series. The inverse filter can be described as:

H(s)=a2s2+a1s+a0+b0⋅L0b0⋅∏i=1n(s+mi)(11)

where n is the minimum integer order such that the H(s) function holds.

According to Eq. (11), ηi(s),(1≤i≤n) are defined as:

η1(s)=η2(s)s+m1,ηi(s)=ηi+1(s)s+mi,ηn(s)=d(s)s+mn(12)

From Eq. (12), we can obtain the relationship in time domain:

η⋅1(t)=−m1η1(t)+η2(t)η⋅i(t)=−miηi(t)+ηi+1(t)η⋅n(t)=−mnηn(t)+d(t)(13)

where 1≤i≤n, ηi(t) are the inverse Laplace transform of ηi(s). mi is the undetermined parameter to be designed; ηi(t),(1≤i≤n) is a set of cascade relationship of unmeasurable signal. ηn(t) will converge to the harmonic signal gradually with frequency ωc. For simplicity, it can be expressed as:

ηi(t)=di∗(t)+Δdi(t)+εi(t)(14)

The noise signal Δdi(t)(1≤i≤n) is equivalent bounded disturbance. According to Fig. 2 and Eq. (12), we can see that Δdi(1≤i≤n) and Δd satisfy the relationship of Eq. (15):

Δdi(s)=Δdi+1(s)s+mi,(i=1,2,…,n−1)Δdn(s)=Δd(s)s+mn(15)

The decay term obeys

ε⋅n(t)=−mnεn(t)(16)

According to the above analysis, the error signal y∼ through the inverse filter contains signal characteristics. Therefore, ηi,(2≤i≤n) can be decomposed into bias harmonic signal, equivalent bounded noise and decay term.

Similarly, we can see that η1(t) also tracks the bias sine curve with frequency ωc. Next, a third-order auxiliary filter is introduced to estimate the key parameter information ϖ=ωc2 by using η1(t).

3.2 Tension Fluctuation Observer Design

Under the framework of the tension fluctuation observer, the output signal of the inverse filter η1(t) is reconstructed n times to obtain the estimated value of the bias harmonic signal.

The structure of the third-order auxiliary filter is shown in Fig. 4. For the convenience of analysis, d^i+1∗(1≤i≤n) in Fig. 4 is defined as the estimated value of the bias harmonic reconstruction signal, d^n+1∗ is the estimated value of the input tension fluctuation, g2i,g1i(1≤i≤n) is an optional positive scalar, Ki1Ki2Ki3(1≤i≤n) is the undetermined coefficient, μi1(s),μi2(s),μi3(s)(1≤i≤n) is the internal signal of the tension fluctuation observer, which can be described as:

Figure 4: Third-order auxiliary filter structure diagram

{μi1(s)=1s2+g2is+g1iηi(s)μi2(s)=ss2+g2is+g1iηi(s)μi3(s)=s2s2+g2is+g1iηi(s)(17)

From Eq. (17):

μ⋅⋅i1(t)=−g2iμ⋅i1(t)−g1iμi1(t)+ηi(t)μ⋅⋅i2(t)=−g2iμ⋅i2(t)−g1iμi2(t)+η⋅i(t)μ⋅⋅i3(t)=−g2iμ⋅i3(t)−g1iμi3(t)+η⋅⋅i(t)μ⋅i1(t)=μi2(t)μ⋅i2(t)=μi3(t)(18)

where μi1(t),μi2(t),μi3(t) are Laplace transformed into μi1(s),μi2(s),μi3(s). Combined with Eq. (13), we can get:

μ⋅i3(t)=μ⋅⋅i2(t)=−g2iμ⋅i2(t)−g1iμi2(t)+η⋅i(t)=−g2i(−g2iμ⋅i1(t)−g1iμi1(t)+ηi(t))−g1iμi2(t)+η⋅i(t)=g1ig2iμi1(t)+(g2i2−g1i)μi2(t)−g2iηi(t)+η⋅i(t)=g1ig2iμi1(t)+(g2i2−g1i)μi2(t)−g2iηi(t)−miηi(t)+ηi+1(t)=g1ig2iμi1(t)+(g2i2−g1i)μi2(t)−(g2i+mi)(μ⋅⋅i1(t)+g2iμ⋅i1(t)+giμi1(t))+ηi+1(t)=−mig1iμi1(t)−(g1i+mig2i)μi2(t)−(mi+g2i)μi3(t)+ηi+1(t)(19)

Theorem 1. If there exists dynamics μi1(t),μi2(t),μi3(t)(1≤i≤n) satisfying Eq. (19). The bias harmonic reconstruction signal di+1∗(t)(1≤i≤n−1) can be described as:

di+1∗(t)=ζiTμi(t)+ζiTδi(t)(20)

ζ1=[K11K12K13]=[m1g1g1+m1g2−ϖm1+g2],μi(t)=[μi1(t)μi2(t)μi3(t)](21)

And ϖ=ω2, and δi(t)(1≤i≤n) obeys

δ⋅i(t)=Biδi(t)−D(εi+1+Δdi+1)(22)

Bi=[010001−m1g1−g1−m1g2−m1−g2],D=[001](23)

Proof. According to Theorem 1 in [28], there exists vector μ∨i(t)∈R3×1 satisfying:

μ∨⋅(t)=Bμ∨(t)+Dd∗(t)(24)

The bias harmonic signal di+1∗(t) can be denoted as:

di+1∗(t)=ζiTμ∨i(t)+ζiTδ∨i(t)(25)

where δ∨(t) satisfies:

δ∨⋅(t)=Bδ∨(t)(26)

Thus, the bias harmonic reconstruction signal is expressed as follows:

di+1∗(t)=μ∨(t)Tζ+ζTδ∨(t)=(μ∨(t)−μ(t)+μ(t))Tζ+ζTδ∨(t)=μ(t)Tζ+ζTδ∨(t)+ζT(μ∨(t)−μ(t))(27)

where,

μ(t)=[μi1(t)μi2(t)μi3(t)]T(28)

According to Eqs. (14), (18), (19), (24) and (26) it follows that:

{μ∨⋅(t)−μ⋅(t)=B(μ∨(t)−μ(t))−Dεi+1−DΔdi+1μ∨⋅(t)−μ⋅(t)+δ∨⋅(t)=B(μ∨(t)−μ(t)+δ∨(t))−Dεi+1−DΔdi+1(29)

Combining with Eqs. (23) and (28), it can be seen that Eq. (27) upholds by introducing δ(t)=μ∨(t)−μ(t)+δ∨(t) into Eq. (20).

3.3 Tension Fluctuation Information Estimation

This section constructed a frequency generator to track the unknown constant scalar ϖ and estimate the frequency of the biased harmonic signal. Combining Eqs. (14), (18), (19) and (20), we have:

{μ⋅11(t)=μ12(t)μ⋅12(t)=μ13(t)μ⋅13(t)=−ϖμ12(t)+ζ1Tδ1(t)+Δd2(t)+ε2(t)(30)

Theorem 2. The reconstruction signal di+1∗(t) is obtained according to ϖ. Correspondingly, consider the designed observer:

{ϖ∧(t)=z(t)+p(t)z⋅(t)=−αμ11(t)μ122(t)ϖ∧(t)+αμ122(t)μ13(t)+αμ11(t)μ132(t)p(t)=−αμ11(t)μ12(t)μ13(t)(31)

where α>0, is a given constant value.

From Eqs. (30) and (31), the estimation error:

ϖ∼⋅(t)=−αμ11(t)μ122(t)ϖ~(t)+αμ11(t)μ122(t)(ζiTδi(t)+Δdn(t)+εn(t))

We could see that if there is no bounded noise Δd(t), the estimation error ϖ~(t) is asymptotically stable. So μ11(t),μ12(t),μ13(t) are essentially stable and converges to a bias harmonic curve with frequency ϖ.

According to Eq. (20), the estimation forms of reconstruction signal dn∗∧(t) can be given by:

dn∗∧(t)=μ(n−1)1(t)mg1+μ(n−1)2(t)(g1+mg2−ϖ)+μ(n−1)3(t)(m+g2)(32)

After obtaining the estimated reconstruction signal di+1∗∧(t), with Eqs. (13) and (14) obtained the estimated reconstruction d∗(t) is carried out and described in the next section.

3.4 Fluctuation Reconstructor and Stability

In the existing methods of signal estimation, due to the uncertain tension fluctuation characteristics, the equivalent state of d∗(t) needs to be constructed. However, this could be simplified in proposed method of this paper.

As shown in Fig. 3, the tension fluctuation reconstructor provides an alternative form of equivalent input signal based on reconstructed tension fluctuation. From the Eqs. (13), (14), (15) and (16), we can see

d∗(t)=η⋅n(t)+mnηn(t)=mn(dn∗(t)+Δdn(t)+εn(t))+dn∗⋅(t)+Δd⋅n(t)+ε⋅n(t)=mndn∗(t)+dn∗⋅(t)(33)

The next task of the tension fluctuation observer is to use the estimator dn∗(t) instead of d∗(t). From Eqs. (18), (19) and (20), we can directly derive the derivative of dn∗(t) as:

dn∗⋅(t)=ζTMμ(t)+ζTδ⋅(t)+ε⋅n(t)=ζTMμ(t)+ζTBδ(t)−mnεn(t)(34)

Furthermore, the input signal d∗(t) can be redescribed as:

d∗(t)=η⋅n(t)+mnηn(t)=ζnTMμn(t)+ζnTBδ(t)−mnεn(t)+mndn∗(t)(35)

We obtain the alternative form of d∗(t), which depends on the unknown parameters. Applying Eqs. (32) to (35), we can reconstruct the compensation signal of tension fluctuation, as shown in Theorem 3.

Theorem 3. Applying the estimation rate of ϖ, for parameters γ1,2>0 and α>0, if there exists P>0, ρ0>0 satisfying the following inequality, the estimation error of the bias harmonic reconstruction signal is uniformly bounded:

Γ=[−2ααζ¯Tααζ¯γ2(BTP+PB)−γ2PDα−γ2DTPT−2m2γ1]<0(36)

where ζ¯ is upper bound of ζ, then reconstruct input signal d∗(t) as:

dn+1∗∧(t)=mnmn−1g1μ(n−1)1(t)+mn(g1+mn−1g2−ϖ∧(t))μ(n−1)2(t)+mn(g2+mn−1)μ(n−1)3(t)+mn−1g1μ(n−1)2(t)+(g1+mn−1g2−ϖ∧(t))μ(n−1)3(t)−ϖ∧(t)(g2+mn−1)μ(n−1)2(t)(37)

Further, when time t→+∞, the error dynamic d∗(t)−dn+1∗∧(t) converges to zero.

Proof. Denoting bias harmonic signal estimation error:

d∗~(t)=d∗(t)−dn+1∗∧(t)=−mnϖ~(t)μ(n−1)2(t)−ϖ~(t)μ(n−1)3(t)−(g2+mn−1)ϖ~(t)μ(n−1)2(t)+ζTBδ(t)−mnεn(t)+ζTδ(t)(38)

From Eqs. (31) and (32), we can get:

ϖ∧⋅(t)=αμ11(t)μ122(t)ϖ~(t)−αμ11(t)μ122(t)(ζiTδi(t)+Δdn(t)+εn(t))(39)

then the estimation error:

ϖ∼⋅(t)=−αμ11(t)μ122(t)ϖ~(t)+αμ11(t)μ122(t)(ζiTδi(t)+Δdn(t)+εn(t))(40)

Define the following Lyapunov function:

V(t)=ϖT~(t)ϖ~(t)+γ1εnT(t)εn(t)+γ2δT(t)Pδ(t)(41)

where, P,γ1,2>0, ρ0>0, then from Eqs. (16), (22) and (40):

V⋅(t)=−2α‖μ110.5(t)μ12(t)ϖ~(t)‖2+2αεn(t)μ11(t)μ122(t)ϖ~(t)+2αδT(t)ζμ11(t)μ122(t)ϖ~(t)+γ2δT(t)(BTP+PB)δ(t)−2γ2δT(t)PDεn(t)−2m2γ1‖εn(t)‖2−γ2Δdi+1DTPδ(t)−γ2δT(t)PDΔdi+1=[μ110.5(t)μ12(t)ϖ~(t)δ(t)εn(t)]TΓ[μ110.5(t)μ12(t)ϖ~(t)δ(t)εn(t)]−2γ2δT(t)PDΔdi+1≤−ρ0‖μ110.5(t)μ12(t)ϖ~(t)δ(t)εn(t)‖2+2γ2μ‖δTPD‖(42)

where, Γ=[−2ααζTααζγ2(BTP+PB)−γ2PDα−γ2DTPT−2m2γ1]<0, γ2ρ0≤1μ⋅‖εn‖‖P‖⋅‖D‖

In consideration that ζ¯ is upper bound of ζ, Γ<0 can be guaranteed by Eq. (36). The above analysis can prove that ϖ~(t), εn(t) and δ(t) are bounded, and we can test that V⋅⋅(t) is also bounded. According to the Eq. (38), d∗~(t) can converge to zero, which shows that Theorem 4 is supported.

With the above design, the influence of the unknown frequency bias harmonic curve d∗(t) can be directly eliminated, and the attenuation of the remaining bounded disturbance Δd(t) can be realized by the controller design in the next section.

For the uncertainty of the biased harmonic signal, we derive the derivative of ϖ is bounded to prove that our method is effective. It is easy to see from the simulation that the frequency information has good estimation ability. According to Eq. (42), it can be obtained that the uniform ultimate boundedness of the biased harmonic signal d∗~(t) can be guaranteed under the condition of Eq. (36). At the same time, we give the relationship between parameters α, m1, m2, g1, and the convergence performance of the ϖ~, d∗~. By reasonably selecting α and B, it can avoid the initial value of the frequency parameter estimation value being too large to obtain faster convergence.

4  Design of the Proposed TPFOPID Controller

Given the above observer design, the new controller will be further introduced. In this section, an improved fraction-order PID controller is incorporated into the framework of tension fluctuation observer. A time-varying parameters fractional-order PID controller (TPFOPID) is realized to improve the system control performance and attenuate the bounded noise.

4.1 TPFOPID Controller Design

The control strategy is defined as:

uTP(t)=Kp⋅e1(t)+Ki⋅D−λe0(t)+Kd⋅Dμe2(t)(43)

TPFOPID reference the parameter self-coupling control strategy method [31], the parametersKp, Ki and Kd can be defined as:

{Kp=a∨⋅f2(t)Ki=b∨⋅f3(t)Kd=c∨⋅f(t)(44)

where a∨,b∨,c∨ are adjustable parameters, b∨∈[0,1],a∨∈[1,10],c∨∈[1,100].

According to [16], an adaptive regulator f(t) is defined:

f(t)=τ⋅(1−e−t)(45)

The parameter τ is determined by the time characteristics of the controlled system. The general value range is 1<τ<100, and the faster the system responds, the value is larger. By assigning the gain parameters to time-varying characteristics, the system can have a large setting margin to achieve large-scale stability. However, when the adaptive regulator in the variable parameters is too large, it is greatly affected by the proportional integral control force. While improving the response speed and anti-disturbance ability of the system, it will also cause overshoot, which makes the tension system vibrate obviously and not conducive to the operation of the actual actuator. When the adaptive regulator is too small, it will affect the response speed and anti-disturbance ability of the system, resulting in unstable tension output.

Theorem 4. When the adjustable parameters in the system controller satisfy a∨c∨−a2b∨>0, the closed-loop system is robustly stable in a large range.

Proof. When the saturation of the controller integrator is limited, 0<b∨≤1 can be set, and the control law is:

uTP(t)=a∨⋅f2(t)⋅e1(t)+b∨⋅f3(t)⋅e0(t)+c∨⋅f(t)⋅e2(t)(46)

e1(t) is the error of system input and output, e0(t) is the error integral, e2(t) is the error differential.

For the tension control system Eq. (3):

{x1⋅=x2x2⋅=−a0a2x1−a1a2x2+1a2(u+d∗~+Δd)y=b0x1(47)

The controlled error system can be established by bringing Eqs. (46) into (47):

{e0⋅=e1e1⋅=e2e2⋅=a1a2y⋅+a0a2y−1a2(a∨⋅f2(t)⋅e1+b∨⋅f3(t)⋅e0+c∨⋅f(t)⋅e2)(48)

The Laplace change of Eq. (48) can be obtained:

E(s)=a1s2+a0sa2s3+c∨⋅f(t)⋅s2+a∨⋅f2(t)⋅s+b∨⋅f3(t)Y(s)(49)

The characteristic polynomial of the transfer function of the closed-loop system under the error state can be defined as follows:

A¯(s)=a2s3+c∨⋅f(t)⋅s2+a∨⋅f2(t)⋅s+b∨⋅f3(t)(50)

If to ensure that A¯(s) is a Hurwitz polynomial if and only if the first column elements of the Routh matrix of A¯(s) are all positive definite, and the first column elements are:

{A1=a2A2=c∨f(t)A3=c∨f(t)×a∨f2(t)−a2×b∨f3(t)c∨f(t)=1c∨(a∨c∨−a2b∨)f2(t)A4=b∨f3(t)(51)

It can be seen that the necessary and sufficient condition for the Eq. (50) to be a Hurwitz polynomial. The coefficients of the closed-loop characteristic equation of the system are positive, and the relationship in Eq. (52) is satisfied.

a2>0, a∨c∨−a2b∨>0(52)

When the integral control force is under the condition of 0<b∨≤1 limiting, the necessary and sufficient conditions of Hurwitz can be satisfied, which ensures that the poles of the system characteristic polynomial are in the left half plane of the S domain. The closed-loop system is robust and stable in a wide range under the controller. The designed TPFOPID controller aims to improve the system’s convergence and tracking performance and can effectively reduce the bounded noise in the tension fluctuation signal.

4.2 Noisy Scenario

It can be seen from the previous derivation that the tension fluctuation observer effectively compensates and rejects the harmonic disturbance in the tension system, and the variance of the feedback σyTHOb2 and the feedforward σyTHOf2 obtained under ideal conditions approaches zero. However, the actual non-optimal environment additional output variance σy02.

The output variance of the actual tension system:

σy2=σyTHOb2+σyTHOf2+σy02+σΔd2(53)

We consider the scenario where random noise disturbance destroys the system. It is assumed that the bounded noise in the system obeys the Gaussian distribution and satisfies Δd∼N(0, ρ2). yd represents the output under noise disturbance, the actual system output y=yd−Δd. yyTP denotes the output of the controller. The residual noise of the controller after noise attenuation, that is, the error χ between yyTP and y, can be expressed as: χ=yyTP−y.

y∧ represents the predicted output after passing through the tension system. It is assumed that the error between the predicted output y∧ based on the tension system and the actual output y of the system obeys Δdc∼N(0, ρ1). And y=y∧−Δdc.

The likelihood functions of y based on y∧ and yd are:

L(y|y∧)=12πρ1exp⁡[−12ρ12(y∧−y)2](54)

L(y|yd)=12πρ2exp⁡[−12ρ22(yd−y)2](55)

Using the Bayes formula, the posterior distribution of y is expressed as:

p(y|y∧,yd)∝L(y|y∧)L(y|yd)p(y)=12πρ1ρ2exp⁡{−12[(yd−y)2ρ22+(y∧−y)2ρ12]}(56)

Finally, we use the maximum likelihood method to obtain the estimated value of the system output:

yyTP=y∧+(1+ρ22ρ1−2)−1(yd−y∧)(57)

From Eq. (56), we obtain:

E(χ2)=(ρ1−2+ρ2−2)−1<min(ρ12,ρ22)(58)

The output variance under the controller:

σyTP2=σyTHOb2+σyTHOf2+σy02+σχ2(59)

According to Eq. (58), σχ2<σΔd2, so we get σyTP2<σy2. Further analysis shows that the TPFOPIFD controller reduces the output variance of the tension system and attenuates the bounded noise signal to a certain extent. We know that random noise can not be completely eliminated, and the degree of noise attenuation is the need for further improvement in our future work.

5  Numerical Simulation

Ultimately, various simulations are carried out to prove the effectiveness of the proposed algorithm. The simulation is carried out on the MATLAB platform using SIMULINK and S function editor.

5.1 Simulation for Tension Fluctuation

To verify the effectiveness of proposed method to deal with tension fluctuation, simulation considering different control target angles. The tension control system of Eq. (53) is taken as:

RJθ⋅⋅+(Rβ+KeKf)θ⋅=KfKb(u+d)(60)

where J=0.00105,Kb=8.33,Kf=1.43,Ke=0.8256,β=0.207,R=11.95. The Eq. (53) is converted into the second-order system form of Eq. (1), and the coefficient matrix is obtained:

A=[010−291],C=[0949.34],F=[10](61)

According to Eq. (10), the gain L0=19.18.

The transfer function from d to y~ is:

Gdy∼(s)=Gk(s)1+H⋅Gk(s)=949.34s2+291s+18227(62)

Therefore, according to Eq. (11), the low-pass filter can be designed as follows:

H(s)=s2+291s+18227949.34⋅(s+m1)(s+m2)(63)

Case 1: d(t)=d∗(t)=200+100sin⁡(10t−π4)

Next, the disturbance observer parameters are selected as: m1=m2=10, g11=g12=g1=100, g21=g22=g2=20, α=5000; Considering the low-frequency disturbance rejection, the parameters in the matrix in Eq. (23) are selected as:

B=[010001−1000−300−30](64)

Case 1 uses the traditional PID controller as the nominal controller, and its parameters are: [KpKiKd]=[590.1].

Fig. 5 shows the estimation performance using the proposed observer. The estimation error and disturbance error of ϖ can converge to zero and the error is very small.

Figure 5: System performance using proposed algorithm (d∗(t)=200+100sin⁡(10t−π4))

Case 2: d(t)=d∗(t)=500+1000sin⁡(20t−π4)+Δd

The random noise disturbance Δd amplitude is 100, and then the disturbance observer parameters are selected as: m1=m2=10, g11=g12=g1=100, g21=g22=g2=20, α=50000