CN106026832A - Improved ADRC control algorithm-based permanent magnet synchronous linear motor control method - Google Patents

Abstract

The invention discloses an improved ADRC control algorithm-based permanent magnet synchronous linear motor control method. A speed controller output signal is input to a TD1 tracking differentiator. A noised feedback signal is input to a TD2 tracking differentiator with a differential pre-compensation factor. The output subtraction error e of the TD1 and the TD2 is input into a PD module. An ESO1 state observer tracks the input of a system wherein the total interference value z of the system is a reduction factor 1/b. the output of the PD module is subtracted by the output of the reduction factor 1/b to obtain an input with a net controlled variable u1 to be an amplification factor b, and the output thereof is adopted as the input of the ESO1 state observer together with the output controlled variable of the TD2. According to the technical scheme of the invention, the rapid and non-overshoot speed control is realized, and the algorithm does not depend on motor parameters. The curve of output net controlled variables is smooth and free of tremble, and the parameter adjustment is simplified. Moreover, the accurate compensation is achieved by adopting an ESO1 to estimate the total interference value. Therefore, a quick response is made to the sudden interference of the current, so that the progressive stability of the current can be ensured.

Description

Based on the permanent magnetic linear synchronous motor control method improving ADRC control algolithm

Technical field

The present invention relates to the control method of permanent magnetic linear synchronous motor, particularly relate to a kind of based on improving ADRC control calculation The permanent magnetic linear synchronous motor control method of method.

Background technology

As in figure 2 it is shown, the NLSEF nonlinearity erron state feedback law in tradition ADRC control algolithm, due to NLSEF state In equation, fal () function is operated in linear interval at (-δ, δ), in other scope fal () function works between inelastic region, and fal () function is rough at-δ and 2 position function characteristic curves of δ, and this rough characteristic easily causes the clean control of ADRC output Amount processed occurs that jitter, NLSEF state equation need to regulate k₁,k₂,δ₃,δ₄Parameter, δ affects the non-linear spy of ADRC Property, if value is too big, ADRC is only operated in linear zone, and value is the least makes function curve the most smooth, increases parameter and adjusts Joint difficulty.

As in figure 2 it is shown, the state observer (being called for short ESO) in tradition ADRC control algolithm, ESO output z1 and TD exports x₁ Relatively obtaining error amount, ESO exports z₂X is exported with TD₂Relatively obtain error amount, the least by closed loop control error amount be When zero, ESO exports z₁Perfect tracking TD exports x₁, ESO exports z₂Perfect tracking TD exports x₂, NLSEF output clean controlled quentity controlled variable and Described Fdb value of feedback is as the input of ESO, z₁Follow the tracks of Fdb feedback signal, z₂Follow the tracks of Fdb feedback signal differential value, ESO state side When the journey margin of error is gradually decreased as zero, z₁And z₂Also can perfect tracking Fdb feedback signal, such feedback signal Fdb perfect tracking ADRC give input signal while perfect tracking total interference value z₃But, the ESO state equation rough spy of fal () function curve Point will result in z₁And z₂Can not perfect tracking x₁And x₂, due to z in ESO state equation₁And z₂The error followed the tracks of will influence whether z3 The tracking of total interference value, and then have influence on the degree of accuracy of whole ADRC control algolithm.

In permanent-magnetism linear motor control, dynamo-electric noise jamming is big especially on system dynamics response impact, feedback channel On often exist deviation actual value outlier, feedback signal Fdb at feedback channel as shown in Figure 2 also exists a lot of outlier, this A little outlier are the most eventful from the higher hamonic wave produced during drive and control of electric machine, the existence of outlier have a strong impact on velocity control accuracy and System rejection to disturbance ability, at present great majority use karr to fill the air and wait wave filter removal outlier.

Summary of the invention

It is an object of the invention to provide a kind of based on the permanent magnetic linear synchronous motor controlling party improving ADRC control algolithm Method, solve the curve of output that causes of the NLSEF rough characteristic of nonlinearity erron Feedback Control Laws tremble, parameter regulation quantity many and ESO is followed the tracks of the impact of total interference value by the outlier that there is deviation actual value on feedback channel.

In order to reach foregoing invention purpose, the step of the technical solution used in the present invention is as follows:

1) the Ref speed control output signal input as TD1 Nonlinear Tracking Differentiator of ADRC control algolithm is improved；

2) noisy feedback signal Fdb of permanent magnetic linear synchronous motor pre-compensates for the tracking differential of the factor as TD2 with differential The input of device, TD2 is carried out with the output of TD1 Nonlinear Tracking Differentiator with output one tunnel of Nonlinear Tracking Differentiator of the differential precompensation factor Relatively obtain error amount e, export the input as ESO1 state observer of another road；

3) output of the Nonlinear Tracking Differentiator that the output of TD1 Nonlinear Tracking Differentiator pre-compensates for the factor with TD2 with differential is compared Error amount e is as the input of PD module；

4) the output z of ESO1 state observer is as the input of coefficient of reduction 1/b；

5) output of PD module exports as clean controlled quentity controlled variable with the output fiducial value u1 of coefficient of reduction 1/b；

6) output of amplification coefficient b is as the input of ESO1 state observer；

7) output fiducial value u1 mono-tunnel of coefficient of reduction 1/b is as the input of amplification coefficient b, and another road is as clean controlled quentity controlled variable Output drives permanent-magnetic synchronous linear motor driver input；

8) permanent magnetic linear synchronous motor exports noisy fed-back current signals Fdb as TD2 with the differential precompensation factor Nonlinear Tracking Differentiator inputs, and TD2 compares with the output of TD1 Nonlinear Tracking Differentiator with the Nonlinear Tracking Differentiator output of the differential precompensation factor Error control amount e arrived, by closed loop control be gradually reduced e be zero time, the noisy feedback signal of permanent magnetic linear synchronous motor follow the tracks of The input signal of TD1 differentiator, reaches to control effect.

The state equation of described TD1 Nonlinear Tracking Differentiator:

x_1(k+1)=x_1(k)+hx_2(k),x_2(k+1)=x_2(k)+hfst(x_1(k)-r_(k),x_2(k),λ,h₀), in formula: x_1(k)For following the tracks of r_(k)Signal, x_2(k)For input r_(k)Differential signal, x_1(k+1)It is x_1(k)Signal subsequent time is straight, x_2(k+1)It is x_2(k)Signal next Moment is straight, and h is numerical integration step-length, and λ is to determine the tracking velocity speed factor, h₀It is noise filtering factor, r_(k)For input quantity. Fst () is discrete domain time-optimal control comprehensive function, fst (x₁,x₂,λ,h₀)=λ sign (a) (| a | ＞ d)；fst(x₁,x₂,λ,h₀) =λ a/d (| a |≤d)；

D=λ h₀；d₀=dh₀；Y=x₁+h₀x₂；a₀=(d²+8λ|y|)^1/2；

A=x₂+(a₀-d) sign (y)/2 (| y | ＞ d₀)；A=x₂+y/h(|y|≤d₀)；Sign (y)=1 (y ＞ 0), sign (y)=0 (y ＜ 0)；

The state equation of described ESO1 state observer:

E (k)=z_1(k)-y_(k),z_1(k+1)=z_1(k)+h[z_2(k)-β₁e(k)]

$z_{2(k+1)}=z_{2\left(k\right)}+h\[z_{3\left(k\right)}-{\mathrm{\β}}_{2}fal\left(e\right(k),{\∂}_1,{\mathrm{\δ}}_1)\]+bu\left(t\right),z_{3(k+1)}=z_{3\left(k\right)}-{\mathrm{h\β}}_{3}fal\left(e\right(k),{\∂}_2,{\mathrm{\δ}}_2)$

Fal (e, α, δ)=e δ^1-α(|e|≤δ)；Fal (e, α, δ)=| e |^αSign (e) (| e | ＞ δ)

In formula: z_1(k)It is that ESO1 follows the tracks of input y_(k)Signal value, z_2(k)It is that ESO1 follows the tracks of input y_(k)Differential value, z_3(k)It is ESO1 follows the tracks of total interference, z_1(k+1)It is z_1(k)Subsequent time value, z_2(k+1)It is z_2(k)Subsequent time value, z_3(k+1)It is z_3(k)Subsequent time Value, β₁,β₂,β₃For fal function coefficients, y_(k)For ESO1 input signal, δ₁,δ₂Affect the nonlinear characteristic of described ADRC, For power parameter, e is error amount, and b is amplification coefficient.

The Pantograph Equation of described PD module:P is ratio control, and D is that differential differential controls, k₁ For error amount coefficient, k₂For error differential value coefficient, e is error,For differential error.

Described TD2 is with the state equation of the Nonlinear Tracking Differentiator of the compensation prediction factor:

x₁=x₁+hx₂,x₂=x₂+hfst(x₁-y,x₂,λ,h₀),x₃=x₁+ηhx₂, x in formula₃For the letter after phase compensation Number.

The invention have the advantages that:

Present invention effect in terms of speed regulating control and anti-Parameter Perturbation is obvious, it is possible to realize quick non-overshoot speed regulating control, Control algolithm is independent of the parameter of electric machine, exports clean controlled quentity controlled variable line smoothing, it is to avoid tremble, and simplifies the regulation of parameter, utilizes ESO1 estimates that total interference value reaches accurately to compensate, and the interference that electric current is unexpected can be made response rapidly, it is ensured that electric current is progressive Stablize.

Accompanying drawing explanation

Fig. 1 is based on improving ADRC algorithm structure schematic diagram.

Fig. 2 is based on tradition ADRC algorithm structure schematic diagram.

Fig. 3 is to improve the clean controlled quentity controlled variable of PD output in ADRC control algolithm to export clean with NLSEF in tradition ADRC control algolithm Controlled quentity controlled variable contrast schematic diagram.

Fig. 4 is to improve ADRC control algolithm and tradition ADRC control algolithm output controlled quentity controlled variable contrast schematic diagram.

Fig. 5 is to improve ESO state observer in ADRC control algolithm to follow the tracks of interference value schematic diagram.

Fig. 6 is to improve after ADRC control algolithm feedback channel TD adds differential precompensation factor filtering and former pollution signal pair Compare schematic diagram.

Fig. 7 is by extraneous unexpected increase load and to alleviate suddenly load at permanent-magnetism linear motor based on ADRC control algolithm Time output speed and estimate total interference value schematic diagram.

In figure: 301, TD1 Nonlinear Tracking Differentiator, 302, PD module, 303, ESO1 state observer, 304, TD2 follow the tracks of differential Device, 305, speed control output signal, 306, noisy feedback signal, 307, the output of clean controlled quentity controlled variable, 308, follow the tracks of total interference value, 309, coefficient of reduction 1/b, 310, amplification coefficient b.

Detailed description of the invention

The invention will be further described with embodiment below in conjunction with the accompanying drawings.

As it is shown in figure 1, be based on improving ADRC algorithm structure schematic diagram.Speed control output signal 305 input TD1 with Track differentiator 301, noisy feedback signal 306 is as the input of the TD2 Nonlinear Tracking Differentiator 304 with the differential precompensation factor, TD1 The error e that Nonlinear Tracking Differentiator 301 output and TD2 Nonlinear Tracking Differentiator 304 output are subtracted each other is as the input of PD module 302, ESO1 shape System total interference value z 308 followed the tracks of by state observer 303, follows the tracks of the input that total interference value 308 is coefficient of reduction 1/b 309, PD mould The output of the output of block 302 and coefficient of reduction 1/b 309 is subtracted each other and is obtained clean controlled quentity controlled variable output u1 307, clean controlled quentity controlled variable output u1 307 is the input of amplification coefficient b 310, and the output controlled quentity controlled variable of amplification coefficient b 310 and TD2 Nonlinear Tracking Differentiator 304 output control Measure the input as ESO1 state observer 303.

The step of the method is as follows:

1) the Ref speed control output signal input as TD1 Nonlinear Tracking Differentiator of ADRC control algolithm is improved；

2) noisy feedback signal Fdb of permanent magnetic linear synchronous motor pre-compensates for the tracking differential of the factor as TD2 with differential The input of device, TD2 is carried out with the output of TD1 Nonlinear Tracking Differentiator with output one tunnel of Nonlinear Tracking Differentiator of the differential precompensation factor Relatively obtain error amount e, export the input as ESO1 state observer of another road；

3) output of the Nonlinear Tracking Differentiator that the output of TD1 Nonlinear Tracking Differentiator pre-compensates for the factor with TD2 with differential is compared Error amount e is as the input of PD module；

4) the output z of ESO1 state observer is as the input of coefficient of reduction 1/b；

5) output of PD module exports as clean controlled quentity controlled variable with the output fiducial value u1 of coefficient of reduction 1/b；

6) output of amplification coefficient b is as the input of ESO1 state observer；

7) output fiducial value u1 mono-tunnel of coefficient of reduction 1/b is as the input of amplification coefficient b, and another road is as clean controlled quentity controlled variable Output drives permanent-magnetic synchronous linear motor driver input；

8) permanent magnetic linear synchronous motor exports noisy fed-back current signals Fdb as TD2 with the differential precompensation factor Nonlinear Tracking Differentiator inputs, and TD2 compares with the output of TD1 Nonlinear Tracking Differentiator with the Nonlinear Tracking Differentiator output of the differential precompensation factor Error control amount e arrived, by closed loop control be gradually reduced e be zero time, the noisy feedback signal of permanent magnetic linear synchronous motor follow the tracks of The Setting signal of TD1 differentiator, reaches to control effect.

The state equation of described TD1 Nonlinear Tracking Differentiator:

x_1(k+1)=x_1(k)+hx_2(k),x_2(k+1)=x_2(k)+hfst(x_1(k)-r_(k),x_2(k),λ,h₀), in formula: x_1(k)For following the tracks of r_(k)Signal, x_2(k)For input r_(k)Differential signal, x_1(k+1)It is x_1(k)Signal subsequent time is straight, x_2(k+1)It is x_2(k)Signal next Moment is straight, and h is numerical integration step-length, and λ is to determine the tracking velocity speed factor, h₀It is noise filtering factor, r_(k)For input quantity. Fst () is discrete domain time-optimal control comprehensive function, fst (x₁,x₂,λ,h₀)=λ sign (a) (| a | ＞ d)；fst(x₁,x₂,λ,h₀) =λ a/d (| a |≤d)；

D=λ h₀；d₀=dh₀；Y=x₁+h₀x₂；a₀=(d²+8λ|y|)^1/2；

A=x₂+(a₀-d) sign (y)/2 (| y | ＞ d₀)；A=x₂+y/h(|y|≤d₀)；Sign (y)=1 (y ＞ 0), sign (y)=0 (y ＜ 0)；

The state equation of described ESO1 state observer:

E (k)=z_1(k)-y_(k),z_1(k+1)=z_1(k)+h[z_2(k)-β₁e(k)]

$z_{2(k+1)}=z_{2\left(k\right)}+h\[z_{3\left(k\right)}-{\mathrm{\β}}_{2}fal\left(e\right(k),{\∂}_1,{\mathrm{\δ}}_1)\]+bu\left(t\right),z_{3(k+1)}=z_{3\left(k\right)}-{\mathrm{h\β}}_{3}fal\left(e\right(k),{\∂}_2,{\mathrm{\δ}}_2)$

Fal (e, α, δ)=e δ^1-α(|e|≤δ)；Fal (e, α, δ)=| e |^αSign (e) (| e | ＞ δ)

In formula: z_1(k)It is that ESO1 follows the tracks of input y_(k)Signal value, z_2(k)It is that ESO1 follows the tracks of input y_(k)Differential value, z_3(k)It is ESO1 follows the tracks of total interference, z_1(k+1)It is z_1(k)Subsequent time value, z_2(k+1)It is z_2(k)Subsequent time value, z_3(k+1)It is z_3(k)Subsequent time Value, β₁,β₂,β₃For fal function coefficients, y_(k)For ESO1 input signal, δ₁,δ₂Affect the nonlinear characteristic of described ADRC,For Power parameter, e is error amount, and b is amplification coefficient.

The Pantograph Equation of described PD module:P is ratio control, and D is that differential differential controls, k₁ For error amount coefficient, k₂For error differential value coefficient, e is error,For differential error.

Described TD2 is with the state equation of the Nonlinear Tracking Differentiator of the compensation prediction factor:

x₁=x₁+hx₂,x₂=x₂+hfst(x₁-y,x₂,λ,h₀),x₃=x₁+ηhx₂, x in formula₃For the letter after phase compensation Number.

The characteristic curve of fal () function rough in NLSEF nonlinearity erron Feedback Control Laws, this rough spy Property easily cause jitter, need to regulate parameter many, PD proportion differential device can not only rapid drop error, extraneous to system Suddenly interference make response rapidly, and curve of output is smooth, jitter easily occurs, needs to regulate parameter few.

As in figure 2 it is shown, the tradition ADRC control algolithm output of output z1 and the TD Nonlinear Tracking Differentiator of ESO state observer X1 compares and obtains error amount e1, and the output x2 of output z2 and the TD Nonlinear Tracking Differentiator of ESO state observer compares and obtains error amount E2, using error amount e1 and error amount e2 as the input of NLSEF, the output valve of described NLSEF deducts described ESO total estimates z3 Obtaining clean controlled quentity controlled variable u, described clean controlled quentity controlled variable u and feedback signal value are described ESO input value, the ADRC control algolithm of described improvement Directly exporting with the TD2 containing the compensation prediction factor compares as error control amount e with the output of TD1, by described error control Amount e is as the input of PD proportion differential device, and the output valve of described PD deducts ESO state observer total interference value z3 and only controlled Amount u1, eliminates and follows the tracks of, from ESO state observer, the error that x1, x2 bring, simplify algorithm structure.

Add on feedback channel can not only solve unruly-value rejecting problem containing compensation prediction factor TD2 tracker, also The phase place of loss can be carried out forecast to revise, further increase algorithm controls precision.

Based on permanent-magnetism linear motor mathematical model, in Simulink emulates, Setting signal is 1 for following the tracks of amplitude, and frequency is The sine wave of 0.2HZ, additional amplitude is 1, and the cycle is the interference of 6.25s square wave, tradition ADRC control algolithm clean output emulation effect As shown by dotted lines in figure 3, in tradition ADRC control algolithm, the clean controlled quentity controlled variable of NLSEF nonlinear Feedback Control rule output is locally lying in fruit Certain trembles, and described improvement ADRC control algolithm clean output simulated effect as shown in figure 3 by the solid lines, improves ADRC control algolithm Described in PD proportion differential device 302 to export clean controlled quentity controlled variable curve more smooth.

ADRC control algolithm follows the tracks of original signal effect as shown in Figure 4, and tradition ADRC control algolithm follows the tracks of original signal output width Value is about 1.3, and the ADRC controller algorithm of described improvement follows the tracks of original signal output more to approach primary signal, curve of output Amplitude is about 1.1.

In described improvement ADRC control algolithm, ESO1 state observer 303 follows the tracks of interference value as it is shown in figure 5, described ESO1 shape State observer 303 exports z3 can perfect tracking interference value.

Effective controlled quentity controlled variable is extracted in simulation from noise, adds, at feedback channel, the random white noise interference letter that amplitude is 0.5 Number, described effective controlled quentity controlled variable such as Fig. 6 that can extract from signals and associated noises with compensation prediction factor TD2 Nonlinear Tracking Differentiator 304 Shown in.Simulation parameter is as follows:

λ=100 h₀=0.01 P=100, D=10 λ=100, h₀=0.01, η=30

k₁=100, k₂=10,σ=0.01, b=1

β₁=100, β₂=65, β₃=80,σ=0.01, b=1

When permanent magnetic linear synchronous motor gives constant load 5N, increased suddenly by 30 loads at 0.16 second and dashed forward at 0.32 second When so reducing by 30 load, ADRC speed control ESO can estimate total interference value accurately, can during speed regulates More quickly precisely compensating for reaching stable, simulated effect is as shown in Figure 7.

Claims (5)

1. a permanent magnetic linear synchronous motor control method based on improvement ADRC control algolithm, it is characterised in that the method Step is as follows:

1) the Ref speed control output signal input as TD1 Nonlinear Tracking Differentiator of ADRC control algolithm is improved；

2) noisy feedback signal Fdb of permanent magnetic linear synchronous motor pre-compensates for the Nonlinear Tracking Differentiator of the factor as TD2 with differential Input, TD2 compares with the output of TD1 Nonlinear Tracking Differentiator with output one tunnel of Nonlinear Tracking Differentiator of the differential precompensation factor Obtain error amount e, export the input as ESO1 state observer of another road；

3) error that the output of the Nonlinear Tracking Differentiator that the output of TD1 Nonlinear Tracking Differentiator pre-compensates for the factor with TD2 with differential is compared Value e is as the input of PD module；

4) the output z of ESO1 state observer is as the input of coefficient of reduction 1/b；

5) output of PD module exports as clean controlled quentity controlled variable with the output fiducial value u1 of coefficient of reduction 1/b；

6) output of amplification coefficient b is as the input of ESO1 state observer；

7) output fiducial value u1 mono-tunnel of coefficient of reduction 1/b is as the input of amplification coefficient b, and another road exports as clean controlled quentity controlled variable Driving permanent-magnetic synchronous linear motor driver inputs；

8) permanent magnetic linear synchronous motor exports noisy fed-back current signals Fdb and pre-compensates for the tracking of the factor as TD2 with differential The differentiator introduction, TD2 exports to compare with TD1 Nonlinear Tracking Differentiator with the Nonlinear Tracking Differentiator output of the differential precompensation factor and obtains Error control amount e, by closed loop control be gradually reduced e be zero time, permanent magnetic linear synchronous motor noisy feedback signal perfect tracking The Setting signal of TD1 differentiator, reaches to control effect.

A kind of permanent magnetic linear synchronous motor control method based on improvement ADRC control algolithm the most according to claim 1, It is characterized in that: the state equation of described TD1 Nonlinear Tracking Differentiator:

x_1(k+1)=x_1(k)+hx_2(k),x_2(k+1)=x_2(k)+hfst(x_1(k)-r_(k),x_2(k),λ,h₀), in formula: x_1(k)For following the tracks of r_(k)Letter Number, x_2(k)For input r_(k)Differential signal, x_1(k+1)It is x_1(k)Signal subsequent time is straight, x_2(k+1)It is x_2(k)Signal subsequent time Directly, h is numerical integration step-length, and λ is to determine the tracking velocity speed factor, h₀It is noise filtering factor, r_(k)For input quantity.fst() For discrete domain time-optimal control comprehensive function, fst (x₁,x₂,λ,h₀)=λ sign (a) (| a | ＞ d)；fst(x₁,x₂,λ,h₀)=λ a/ d(|a|≤d)；

D=λ h₀；d₀=dh₀；Y=x₁+h₀x₂；a₀=(d²+8λ|y|)^1/2；

A=x₂+(a₀-d) sign (y)/2 (| y | ＞ d₀)；A=x₂+y/h(|y|≤d₀)；Sign (y)=1 (y ＞ 0), sign (y) =0 (y ＜ 0)；

A kind of permanent magnetic linear synchronous motor control algolithm based on improvement ADRC control algolithm the most according to claim 1, It is characterized in that: the state equation of described ESO1 state observer:

E (k)=z_1(k)-y_(k),z_1(k+1)=z_1(k)+h[z_2(k)-β₁e(k)]

$z_{2(k+1)}=z_{2\left(k\right)}+h\[z_{3\left(k\right)}-{\mathrm{\β}}_{2}fal\left(e\right(k),{\∂}_1,{\mathrm{\δ}}_1)\]+bu\left(t\right),z_{3(k+1)}=z_{3\left(k\right)}-{\mathrm{h\β}}_{3}fal\left(e\right(k),{\∂}_2,{\mathrm{\δ}}_2)$

Fal (e, α, δ)=e δ^1-α(|e|≤δ)；Fal (e, α, δ)=| e |^αSign (e) (| e | ＞ δ)

In formula: z_1(k)It is that ESO1 follows the tracks of input y_(k)Signal value, z_2(k)It is that ESO1 follows the tracks of input y_(k)Differential value, z_3(k)ESO1 with Track always disturbs, z_1(k+1)It is z_1(k)Subsequent time value, z_2(k+1)It is z_2(k)Subsequent time value, z_3(k+1)It is z_3(k)Subsequent time value, β₁, β₂,β₃For fal function coefficients, y_(k)For ESO1 input signal, δ₁,δ₂Affect the nonlinear characteristic of described ADRC,Join for power Number, e is error amount, and b is amplification coefficient.

A kind of permanent magnetic linear synchronous motor control algolithm based on improvement ADRC control algolithm the most according to claim 1, It is characterized in that: the Pantograph Equation of described PD module:P is ratio control, and D is differential control, k₁For Error amount coefficient, k₂For error differential value coefficient, e is error,For differential error.

A kind of permanent magnetic linear synchronous motor control algolithm based on improvement ADRC control algolithm the most according to claim 1, It is characterized in that: described TD2 is with the state equation of the Nonlinear Tracking Differentiator of the compensation prediction factor:

x₁=x₁+hx₂,x₂=x₂+hfst(x₁-y,x₂,λ,h₀),x₃=x₁+ηhx₂, x in formula₃For the signal after phase compensation.