CN106842961A - The symmetrical hysteresis control method of supersonic motor servo-control system based on Stop operators - Google Patents

Abstract

The present invention relates to a kind of symmetrical hysteresis control method of the supersonic motor servo-control system based on Stop operators：One pedestal and the supersonic motor on pedestal are provided, supersonic motor side output shaft is connected with photoelectric encoder, opposite side output shaft is connected with flywheel inertia load, the output shaft of the flywheel inertia load is connected through shaft coupling with torque sensor, and the signal output part of the photoelectric encoder, the signal output part of the torque sensor are respectively connected to a control system；The control system is set up on the basis of stop operator compensating controllers, and the stop operators compensating controller is with minimum its Tuning function of Identification Errors, so as to obtain more preferable input and output controlled efficiency.Not only control accuracy is high for the present invention, and simple structure, compact, and using effect is good.

Description

The symmetrical hysteresis control method of supersonic motor servo-control system based on Stop operators

Technical field

The present invention relates to a kind of symmetrical hysteresis control method of the supersonic motor servo-control system based on Stop operators.

Background technology

When being in proper states due to torque output in the design of existing supersonic motor servo-control system, its hysteresis With symmetry, there is certain error when controlling cycle repeating signal.In order to improve trace performance, we devise and are based on The supersonic motor servo-control system of stop operators symmetrical compensation control.From implementation result, it has been found that the torque of system Length velocity relation is substantially linear, therefore can effectively promote the controlled efficiency of system based on the control of stop operators symmetrical compensation, goes forward side by side one Step reduction system can obtain preferable dynamic characteristic for probabilistic influence degree, torque and the speed control of motor.

The content of the invention

In view of this, it is an object of the invention to provide a kind of supersonic motor servo-control system based on Stop operators Symmetrical hysteresis control method, not only control accuracy is high, and simple structure, compact, and using effect is good.

To achieve the above object, the present invention is adopted the following technical scheme that：A kind of supersonic motor based on Stop operators is watched Take the symmetrical hysteresis control method of control system, it is characterised in that comprise the following steps：

Step S1：One pedestal and the supersonic motor on pedestal, supersonic motor side output shaft are provided It is connected with photoelectric encoder, opposite side output shaft is connected with flywheel inertia load, the output shaft of the flywheel inertia load Be connected with torque sensor through shaft coupling, the signal output part of the photoelectric encoder, the torque sensor signal it is defeated Go out end and be respectively connected to a control system；

Step S2：The control system is set up on the basis of stop operator compensating controllers, the stop operators compensation Controller is with minimum its Tuning function of Identification Errors, so as to obtain more preferable input and output controlled efficiency；The control system Dynamical equation be：

Wherein A_p=-B/J, B_P=J/K_t＞ 0, C_P=-1/J；B is damped coefficient, and J is rotary inertia, K_tIt is current factor, T_fV () is frictional resistance torque, T_LIt is loading moment, U (t) is the output torque of motor, θ_rT () is to be surveyed by photoelectric encoder The position signalling for measuring.

Further, in the step S1, the control system includes supersonic motor drive control circuit, the ultrasound Ripple motor driving controling circuit include control chip circuit and driving chip circuit, the signal output part of the photoelectric encoder with The respective input of the control chip circuit is connected, the output end of the control chip circuit and the driving chip circuit Respective input be connected, to drive the driving chip circuit, the driving frequency Regulate signal of the driving chip circuit The respective input of output end and driving half-bridge circuit Regulate signal output end respectively with the supersonic motor is connected, described Stop operator compensating controllers are arranged in the control chip circuit.

Further, in the step S1, the shaft coupling is yielding coupling.

Further, in the step S1, the supersonic motor, photoelectric encoder, torque sensor are respectively through ultrasound Ripple motor fixed rack, photoelectric encoder fixed support, torque sensor fixed support are fixed on the pedestal.

Further, in the step S2, the hysteresis of motor torque-speed characteristics has symmetry, shows to reduce this As the influence that causes while reducing operand, compensated using the symmetrical hysteresis of stop operators and controlled it：Stop operation symbol Output is the function of its critical value s and input v (t), and the stop operators output of input v (t) ∈ C [0, T] can be expressed as：

E_s[v] (0)=e_s(v(0))

E_s[v] (t)=e_s(v(t)-v(t_i)+E_s[v](t_i))

For t_i＜ t ＜ t_i+1And 0≤i≤N-1,

e_s=min (s, max (- s, v))

Wherein 0=t₀<t₁<......<t_l=T is the subregion of T [0, T] so that function v (t) ∈ C [0, T], C [0, T] table Show the space of the continuous function on [0, T], in each subinterval [t_i, t_i+1] on be dull；

Under different threshold value s, stop operators are output as：

Above formula discretization, output are described by n stop operator, 0=s₀<s₁<......<s_n=S, that is,：

Wherein w_sThe weight of density function is represented, i.e.,

Due to Lipschitz continuitys, stop operators E_sFor density function, therefore the model pair based on stop operators can be accumulated It is Lipschitz continuous in given input v (t) ∈ C [0, T], it is monotonic operation symbol, weight that can further obtain this model Function integrable and for just；

When the system is operating, input signal v (t) first passes through inverse system ψ, and its output enters balanced system as control signal Φ, the mapping expected between input v (t) and output u (t) is obtained using feedforward compensation：

U (t)=Φ [ψ [v]] (t)

Initial load curve and given threshold value r based on PPI models_iWith corresponding weight p_i, obtain the two of stop operators Individual parameter：Threshold value s_iWith weight w_i；

Assuming that the original upload curve of PPI models is expressed as：

Wherein r ∈ [r₀, r_np] and r₀=0, n_pIt is the number of operator p, p is the operator of PPI models, r_iIt is the threshold of PPI models Value；Function phi_p：R^+→R⁺It is convex function and increasing function, in order to obtain the parameter of compensator, the original upload based on stop operators Curve φ_sIt is defined as：

Wherein φ_s：：R^+→R⁺It is concave function and increasing function, n_sIt is the number of operator, s ∈ [0, s₀], s₀It is set to big positive reality Number, meets s₀>Max (v (t)), it is ensured that the Rigid Monotony of stop Operator Models；

In order to obtain the weight and threshold value of stop Operator Models, the threshold value and initial load curve of stop Operator Models expire Foot：

s_i=φ_r(r_i) (3.16)

φ_s(s_i)=r_i (3.17)

Any point B (r on initial load curve according to PPI_k, φ_r) meeting equation (3.16) and (3.17), it is always Corresponding points C (s can be found on the initial load curve of stop operators_k, φ_s)；The threshold value of stop Operator Models can be with following Mode is related to the threshold value of PPI models：

s₁=r₁p₀

s₂=(r₂-r₁)p₁+r₂p₀

s₃=(r₃-r₁)p₁+(r₃-r₂)p₂+r₃p₀

s_n=(r_n-r₁)p₁+(r_n-r₂)p₂+...+r_np₀ (3.18)

Stop Operator Models w_iCan be calculated as according to (3.17)：

Equation group (3.19) includes (n+1) individual known variables, and the quantity of equation is n, in order to solve equation (3.19) and Obtain weight w_n, should first solve weight w₀；

Using annex point as (s on the initial load curve of SPI models_n+ 1, φ_s(s_n+ 1)), by makingWhereinIt is arithmetic number, ξ=φ can be expressed as_s(s_n+1)；

According to (3.16) and (3.17)：

Formula (3.19) and (3.21)：

Equation (3.20) can be expressed as：

W is drawn from (3.22) and (3.23)₀：

The weight w of stop Operator Models is readily available finally by solution equation (3.19)_i。

The present invention has the advantages that compared with prior art：The present invention is using the control of stop operators symmetrical compensation Supersonic motor servo controller, system has significant improvement, this symmetrical compensation control energy on torque velocities tracking effect The effective controlled efficiency for promoting system, and system is further reduced for probabilistic influence degree, improve control Accuracy, can obtain preferable dynamic characteristic.Additionally, the device is reasonable in design, simple structure, compact, low cost of manufacture, tool There is very strong practicality and wide application prospect.

Brief description of the drawings

Fig. 1 is the structural representation of one embodiment of the invention.

Fig. 2 is the control circuit theory diagrams of one embodiment of the invention.

Fig. 3 is open-loop control system figure of the invention.

In figure：1- photoelectric encoders, 2- photoelectric encoder fixed supports, 3- supersonic motor output shafts, 4- ultrasonic waves electricity Machine, 5- supersonic motor fixed supports, 6- supersonic motor output shafts, 7- flywheel inertia loads, the output of 8- flywheels inertia load Axle, 9- yielding couplings, 10- torque sensors, 11- torque sensor fixed supports, 12- pedestals, 13- control chip circuits, 14- driving chip circuits, 15,16,17- photoelectric encoders output A, B, Z phase signals, 18,19,20,21- driving chip circuits The driving frequency Regulate signal of generation, 22- driving chips circuit produce driving half-bridge circuit Regulate signal, 23,24,25,26, 27th, the signal of the driving chip circuit that 28- control chips circuit is produced, 29- supersonic motor drive control circuits.

Specific embodiment

Below in conjunction with the accompanying drawings and embodiment the present invention will be further described.

The present invention provides a kind of symmetrical hysteresis control method of supersonic motor servo-control system based on Stop operators, its It is characterised by, comprises the following steps：

Step S1：Refer to Fig. 1, there is provided a pedestal 12 and the supersonic motor on pedestal 12 4, the ultrasonic wave electricity The side output shaft 3 of machine 4 is connected with photoelectric encoder 1, and opposite side output shaft 6 is connected with flywheel inertia load 7, the flywheel The output shaft 8 of inertia load 7 is connected through yielding coupling 9 with torque sensor 10, the signal output of the photoelectric encoder 1 End, the signal output part of the torque sensor 10 are respectively connected to control system.

Above-mentioned supersonic motor 4, photoelectric encoder 1, torque sensor 10 are respectively through supersonic motor fixed support 5, light Photoelectric coder fixed support 2, torque sensor fixed support 11 are fixed on the pedestal 12.

As shown in Fig. 2 above-mentioned control system includes supersonic motor drive control circuit 29, the supersonic motor drives Control circuit 29 includes control chip circuit 13 and driving chip circuit 14, the signal output part of the photoelectric encoder 1 and institute The respective input for stating control chip circuit 13 is connected, output end and the driving chip electricity of the control chip circuit 13 The respective input on road 14 is connected, to drive the driving chip circuit 14, the driving frequency of the driving chip circuit 14 Regulate signal output end and drive half-bridge circuit Regulate signal output end respectively with the respective input phase of the supersonic motor 4 Connection.The driving chip circuit 14 produces driving frequency Regulate signal and drives half-bridge circuit Regulate signal, to ultrasonic wave electricity Frequency, phase and the break-make that machine exports A, B two phase PWM are controlled.Ultrasound is controlled by opening and turning off the output of PWM ripples The startup of ripple motor and out of service；By adjust output PWM ripples frequency and two-phase phase difference come regulation motor most Good running status.

Step S2：The control system is set up on the basis of stop operator compensating controllers, the stop operators compensation Controller is arranged in the control chip circuit；The stop operators compensating controller is with minimum its adjustment letter of Identification Errors Number, so as to obtain more preferable input and output controlled efficiency；The dynamical equation of the control system is：

Wherein A_p=-B/J, B_P=J/K_t＞ 0, C_P=-1/J；B is damped coefficient, and J is rotary inertia, K_tIt is current factor, T_fV () is frictional resistance torque, T_LIt is loading moment, U (t) is the output torque of motor, θ_rT () is to be surveyed by photoelectric encoder The position signalling for measuring.

When the loading moment of motor is moderate, the hysteresis of motor torque-speed characteristics has symmetry, shows to reduce this As the influence for causing is while reduce operand, we are compensated using the symmetrical hysteresis of stop operators and controlled it：Stop operation The output of symbol is the function of its critical value s and input v (t), and the stop operators output of input v (t) ∈ C [0, T] can be expressed as：

E_s[v] (0)=e_s(v(0))

E_s[v] (t)=e_s(v(t)-v(t_i)+E_s[v](t_i))

For t_i＜ t ＜ t_i+1And 0≤i≤N-1,

e_s=min (s, max (- s, v))

Wherein 0=t₀<t₁<......<t_l=T is the subregion of T [0, T] so that function v (t) ∈ C [0, T], C [0, T] table Show the space of the continuous function on [0, T], in each subinterval [t_i, t_i+1] on be dull；

Under different threshold value s, stop operators are output as：

Above formula discretization, output are described by n stop operator, 0=s₀<s₁<......<s_n=S, that is,：

Wherein w_sThe weight of density function is represented, i.e.,

Due to Lipschitz continuitys, stop operators E_sFor density function, therefore the model pair based on stop operators can be accumulated It is Lipschitz continuous in given input v (t) ∈ C [0, T], it is monotonic operation symbol, weight that can further obtain this model Function integrable and for just；

As shown in figure 3, when the system is operating, input signal v (t) first passes through inverse system ψ, its output is entered as control signal Enter balanced system Φ, we obtain the mapping expected between input v (t) and output u (t) using feedforward compensation：

U (t)=Φ [ψ [v]] (t)

Initial load curve and given threshold value r based on PPI models_iWith corresponding weight p_i, stop operators can be obtained Two parameters：Threshold value s_iWith weight w_i；

Assuming that the original upload curve of PPI models is expressed as：

Wherein r ∈ [r₀, r_np] and r₀=0, n_pIt is the number of operator p, p is the operator of PPI models, r_iIt is the threshold of PPI models Value；Function phi_p：R^+→R⁺It is convex function and increasing function, in order to obtain the parameter of compensator, the original upload based on stop operators Curve φ_sIt is defined as：

Wherein φ_s：：R^+→R⁺It is concave function and increasing function, n_sIt is the number of operator, s ∈ [0, s₀], s₀It is set to big positive reality Number, meets s₀>Max (v (t)), it is ensured that the Rigid Monotony of stop Operator Models；

In order to obtain the weight and threshold value of stop Operator Models, the threshold value and initial load curve of stop Operator Models expire Foot：

s_i=φ_r(r_i) (3.16)

φ_s(s_i)=r_i (3.17)

Any point B (r on initial load curve according to PPI_k, φ_r) meeting equation (3.16) and (3.17), it is always Corresponding points C (s can be found on the initial load curve of stop operators_k, φ_s)；The threshold value of stop Operator Models can be with following Mode is related to the threshold value of PPI models：

s₁=r₁p₀

s₂=(r₂-r₁)p₁+r₂p₀

s₃=(r₃-r₁)p₁+(r₃-r₂)p₂+r₃p₀

s_n=(r_n-r₁)p₁+(r_n-r₂)p₂+...+r_np₀ (3.18)

Stop Operator Models w_iCan be calculated as according to (3.17)：

Equation group (3.19) includes (n+1) individual known variables, and the quantity of equation is n, in order to solve equation (3.19) and Obtain weight w_n, should first solve weight w₀；

Therefore, using annex point as (s on the initial load curve of SPI models_n+ 1, φ_s(s_n+ 1)), by makingWhereinIt is arithmetic number, ξ=φ can be expressed as_s(s_n+1)；

According to (3.16) and (3.17)：

Formula (3.19) and (3.21)：

Equation (3.20) can be expressed as：

W is drawn from (3.22) and (3.23)₀：

The weight w of stop Operator Models is readily available finally by solution equation (3.19)_i。

The foregoing is only presently preferred embodiments of the present invention, all impartial changes done according to scope of the present invention patent with Modification, should all belong to covering scope of the invention.

Claims (5)

1. a kind of symmetrical hysteresis control method of supersonic motor servo-control system based on Stop operators, it is characterised in that bag Include following steps：

Step S1：One pedestal and the supersonic motor on pedestal, supersonic motor side output shaft and light are provided Photoelectric coder is connected, and opposite side output shaft is connected with flywheel inertia load, and the output shaft of the flywheel inertia load is through connection Axle device is connected with torque sensor, the signal output part of the photoelectric encoder, the signal output part of the torque sensor It is respectively connected to a control system；

Step S2：The control system is set up on the basis of stop operator compensating controllers, the stop operators compensation control Device is with minimum its Tuning function of Identification Errors, so as to obtain more preferable input and output controlled efficiency；The control system it is dynamic State equation is：

$\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}\left(t\right)=A_p{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r\left(t\right)+\frac{1}{B_P}U\left(t\right)+C_P(T_L+T_f(v\left)\right)---\left(1\right)$

Wherein A_p=-B/J, B_P=J/K_t＞ 0, C_P=-1/J；B is damped coefficient, and J is rotary inertia, K_tIt is current factor, T_f V () is frictional resistance torque, T_LIt is loading moment, U (t) is the output torque of motor, θ_rT () is to be measured by photoelectric encoder The position signalling for obtaining.

2. the symmetrical hysteresis controlling party of the supersonic motor servo-control system based on Stop operators according to claim 1 Method, it is characterised in that：In the step S1, the control system includes supersonic motor drive control circuit, the ultrasonic wave Motor driving controling circuit includes control chip circuit and driving chip circuit, the signal output part of the photoelectric encoder and institute The respective input for stating control chip circuit is connected, the output end of the control chip circuit and the driving chip circuit Respective input is connected, and to drive the driving chip circuit, the driving frequency Regulate signal of the driving chip circuit is defeated Go out end and drive respective input of the half-bridge circuit Regulate signal output end respectively with the supersonic motor to be connected, it is described Stop operator compensating controllers are arranged in the control chip circuit.

3. the symmetrical hysteresis controlling party of the supersonic motor servo-control system based on Stop operators according to claim 1 Method, it is characterised in that：In the step S1, the shaft coupling is yielding coupling.

4. the symmetrical hysteresis controlling party of the supersonic motor servo-control system based on Stop operators according to claim 1 Method, it is characterised in that：In the step S1, the supersonic motor, photoelectric encoder, torque sensor are respectively through ultrasonic wave electricity Machine fixed support, photoelectric encoder fixed support, torque sensor fixed support are fixed on the pedestal.

5. the symmetrical hysteresis controlling party of the supersonic motor servo-control system based on Stop operators according to claim 1 Method, it is characterised in that：In the step S2, the hysteresis of motor torque-speed characteristics has symmetry, is made to reduce this phenomenon Into influence reduce operand simultaneously, compensated using the symmetrical hysteresis of stop operators and controlled it：Stop operation the output of symbol It is the function of its critical value s and input v (t), the stop operators output of input v (t) ∈ C [0, T] can be expressed as：

E_s[v] (0)=e_s(v(0))

E_s[v] (t)=e_s(v(t)-v(t_i)+E_s[v](t_i))

For t_i＜ t ＜ t_i+1And 0≤i≤N-1,

e_s=min (s, max (- s, v))

Wherein 0=t₀<t₁<......<t_l=T is the subregion of T [0, T] so that function v (t) ∈ C [0, T], C [0, T] expressions [0, T] on continuous function space, in each subinterval [t_i, t_i+1] on be dull；

Under different threshold value s, stop operators are output as：

$\stackrel{\&OverBar;}{\mathrm{\&Psi;}}\&lsqb;v\&rsqb;\left(t\right)=\underset{0}{\overset{s}{\&Integral;}}{\stackrel{\&OverBar;}{w}}_s\left(s\right)E_s\&lsqb;v\&rsqb;\left(t\right)dr$

Above formula discretization, output are described by n stop operator, 0=s₀<s₁<......<s_n=S, that is,：

$\mathrm{\&Psi;}\&lsqb;v\&rsqb;\left(t\right)=\underset{i=1}{\overset{n_s}{\&Sigma;}}{w}_s\left(s_i\right)E_{s_i}\&lsqb;v\&rsqb;\left(t\right)$

Wherein w_sThe weight of density function is represented, i.e.,

$w_s\left(s_i\right)={\stackrel{\&OverBar;}{w}}_s\left(s_i\right)(s_{i+1}-s_i)$

Due to Lipschitz continuitys, stop operators E_sFor density function, therefore the model based on stop operators can be accumulated for giving Surely input v (t) ∈ C [0, T] is Lipschitz continuous, and it is monotonic operation symbol, weighting function that can further obtain this model Integrable and for just；

When the system is operating, input signal v (t) first passes through inverse system ψ, and its output enters balanced system Φ as control signal, The mapping expected between input v (t) and output u (t) is obtained using feedforward compensation：

U (t)=Φ [ψ [v]] (t)

Initial load curve and given threshold value r based on PPI models_iWith corresponding weight p_i, obtain two ginsengs of stop operators Number：Threshold value s_iWith weight w_i；

Assuming that the original upload curve of PPI models is expressed as：

Wherein r ∈ [r₀, r_np] and r₀=0, n_pIt is the number of operator p, p is the operator of PPI models, r_iIt is the threshold value of PPI models； Function phi_p：R^+→R⁺It is convex function and increasing function, in order to obtain the parameter of compensator, the original upload based on stop operators is bent Line φ_sIt is defined as：

Wherein φ_s：：R^+→R⁺It is concave function and increasing function, n_sIt is the number of operator, s ∈ [0, s₀], s₀Big arithmetic number is set to, it is full Sufficient s₀>Max (v (t)), it is ensured that the Rigid Monotony of stop Operator Models；

In order to obtain the weight and threshold value of stop Operator Models, the threshold value and initial load curve of stop Operator Models meet：

s_i=φ_r(r_i) (3.16)

φ_s(s_i)=r_i (3.17)

Any point B (r on initial load curve according to PPI_k, φ_r) meeting equation (3.16) and (3.17), it can be always Corresponding points C (s are found on the initial load curve of stop operators_k, φ_s)；The threshold value of stop Operator Models can be with the following methods Threshold value to PPI models is related：

s₁=r₁p₀

s₂=(r₂-r₁)p₁+r₂p₀

s₃=(r₃-r₁)p₁+(r₃-r₂)p₂+r₃p₀

s_n=(r_n-r₁)p₁+(r_n-r₂)p₂+...+r_np₀ (3.18)

Stop Operator Models w_iCan be calculated as according to (3.17)：

$\begin{array}{c}{s}_1{w}_0+s_1{w}_1+s_1{w}_2+s_1{w}_3+\mathrm{...}+s_1{w}_k+\mathrm{...}+s_1{w}_n=r_1\\ s_2{w}_0+s_1{w}_1+s_2{w}_2+s_2{w}_3+\mathrm{...}+s_2{w}_k+\mathrm{...}+s_2{w}_n=r_2\\ s_3{w}_0+s_1{w}_1+s_2{w}_2+s_3{w}_3+\mathrm{...}+s_3{w}_k+\mathrm{...}+s_3{w}_n=r_3\\ .\\ .\\ .\\ s_k{w}_0+s_1{w}_1+s_2{w}_2+s_3{w}_3+\mathrm{...}+s_k{w}_k+\mathrm{...}+s_k{w}_n=r_k\\ .\\ .\\ .\\ s_n{w}_0+s_1{w}_1+s_2{w}_2+s_3{w}_3+\mathrm{...}+s_k{w}_k+\mathrm{...}+s_n{w}_n=r_n\end{array}---\left(3.19\right)$

Equation group (3.19) includes (n+1) individual known variables, and the quantity of equation is n, in order to solve equation (3.19) and obtain Weight w_n, should first solve weight w₀；

Using annex point as (s on the initial load curve of SPI models_n+ 1, φ_s(s_n+ 1)), by makingWhereinIt is arithmetic number, ξ=φ can be expressed as_s(s_n+1)；

According to (3.16) and (3.17)：

${\mathrm{\&xi;p}}_0+(\mathrm{\&xi;}-r_1)p_1+...+(\mathrm{\&xi;}-r_k)p_k+...+(\mathrm{\&xi;}-r_n)p_n=s_n+\stackrel{\&OverBar;}{\mathrm{\&rho;}}---\left(3.20\right)$

$(s_n+\stackrel{\&OverBar;}{\mathrm{\&rho;}})w_0+s_1{w}_1+...+s_k{w}_k+...+s_n{w}_n=\mathrm{\&xi;}---\left(3.21\right)$

Formula (3.19) and (3.21)：

$\stackrel{\&OverBar;}{\mathrm{\&rho;}}{w}_0=\mathrm{\&xi;}-r_n---\left(3.22\right)$

Equation (3.20) can be expressed as：

$(\mathrm{\&xi;}-r_n)p_0+r_n{p}_0+(\mathrm{\&xi;}-r_n)p_1+(r_n-r_1)p_1+...+(\mathrm{\&xi;}-r_n)p_k+(r_n-r_k)p_k+(\mathrm{\&xi;}-r_n)p_n=s_n+\stackrel{\&OverBar;}{\mathrm{\&rho;}}---\left(3.23\right)$

W is drawn from (3.22) and (3.23)₀：

$w_0=\frac{1}{p_0+p_1+...+p_k+...p_n}---\left(3.24\right)$

The weight w of stop Operator Models is readily available finally by solution equation (3.19)_i。