CN106019981A - Stability control method for time-delay asymmetric double-teleoperation system - Google Patents

Abstract

The invention discloses a stability control method for a time-delay asymmetric double-teleoperation system. The method is characterized by, to begin with, establishing a model of the asymmetric double-teleoperation system; then, designing wave variables; and finally, establishing a time delay model. According to the method, speed and force information is converted into the wave variables and then, are transmitted; according to the passivity theory, passivity of channels under any fixed time delay condition can be ensured, and passivity of the double-teleoperation system under any fixed time delay condition can be ensured; and wave transform can be realized only by carrying out simple arithmetic operation on the speed and force information, and the method is easy to realize. Finally, the method can ensure the passivity of the channels under any fixed time delay condition; and master ends, a slave end and an environmental model are passive, so that passivity and stability of the whole system under any fixed time delay condition can be ensured.

Description

A kind of stable control method for the asymmetric double remote control system of time delay

[technical field]

The invention belongs to remote operating control field, be specifically related to a kind of steady for the asymmetric double remote control system of time delay Locking control method.

[background technology]

Teleoperation plays the heaviest at numerous areas such as such as robot for space, high accuracy assembling, operations The role wanted.In remote control system, operator is by interacting with distal environment from the mechanical arm of end.Teleoperation one Aspect can provide the environment of a safety to operator, on the other hand can provide the strongest telepresenc to operator.Tradition The teleoperation of single unit operator is required higher, and concertedness is strong, has been used for complex task and has had office Sex-limited.So, many people multimachine teleoperation arises at the historic moment, and the especially double remote control system of unsymmetric structure has height The plurality of advantages such as motility, operability.But time delay strong influence is to the stability of remote operating, and time delay is widely present In communication channel, so the stable control method of time delay remote control system should be focused on the stability ensureing channel.Ripple becomes It is theoretical that metering method is derived from Passive Shape Control, can well solve the remote operating stability problem brought by time delay.

[summary of the invention]

It is an object of the invention to solve the stability problem that the time delay in asymmetric double remote control system is brought, it is provided that A kind of stable control method for the asymmetric double remote control system of time delay.

For reaching above-mentioned purpose, the present invention is achieved by the following technical solutions:

A kind of stable control method for the asymmetric double remote control system of time delay, comprises the following steps:

1) model of asymmetric double remote control system is set up；

Set up two main sides and from the kinetic model held:

$Z_{m_1}{V}_{h_1}=F_{{\mathrm{cm}}_1}+F_{h_1}---(1-1)$

$Z_{m_2}{V}_{h_2}=F_{{\mathrm{cm}}_2}+F_{h_2}---(1-2)$

Z_sV_e=F_cs-F_e (1-3)

Wherein:

$F_{{\mathrm{cm}}_1}=-C_{m_1}{V}_{h_1}-C_{4m_1}{V}_{h_{1}d}+C_{6m_1}{F}_{h_1}-C_{2m_1}{F}_{h_{1}d}---(1-4)$

$F_{{\mathrm{cm}}_2}=-C_{m_2}{V}_{h_2}-C_{4m_2}{V}_{h_{2}d}+C_{6m_2}{F}_{h_1}-C_{2m_2}{F}_{h_{2}d}---(1-5)$

F_cs=-C_sV_e+C₁V_ed-C₅F_e+C₃F_ed (1-6)

And V_eRepresent main side 1, main side 2 and the speed from end respectively,And Z_sRepresent main side 1, main side 2 and from end linear kinetic model,And F_csRepresent main side 1, main side 2 and the control power from end respectively,WithRepresent that operator is applied to main side 1 and the power of main side 2, F respectively_eRepresent the active force of environment；And V_edTable respectively Show main side 1, main side 2 and the desired speed from end,And F_edRepresent that main side 1, main side 2 and the expectation from end are made respectively Firmly； C_s、C₁、C₅And C₃It is respectively different control Device parameter；

The desired speed of main and slave terminal is respectively:

$\left\{\begin{array}{c}{V}_{h_{1}d}\left(t\right)={\mathrm{\α}}_1{V}_e^{d_1}\left(t\right)+(1-{\mathrm{\α}}_1)V_{h_2}^d\left(t\right)\\ V_{h_{2}d}\left(t\right)=(1-{\mathrm{\α}}_2)V_e^{d_2}\left(t\right)+{\mathrm{\α}}_2{V}_{h_1}^d\left(t\right)\\ V_{ed}\left(t\right)={\mathrm{\α}}_3{V}_{h_1}^{d_1}\left(t\right)+(1-{\mathrm{\α}}_3)V_{h_2}^{d_2}\left(t\right)\end{array}\right.---(1-7)$

The expected force of main and slave terminal is respectively:

$\left\{\begin{array}{c}{F}_{h_{1}d}\left(t\right)={\mathrm{\α}}_1{F}_e^{d_1}\left(t\right)+(1-{\mathrm{\α}}_1)F_{h_2}^d\left(t\right)\\ F_{h_{2}d}\left(t\right)=(1-{\mathrm{\α}}_2)F_e^{d_2}\left(t\right)+{\mathrm{\α}}_2{F}_{h_1}^d\left(t\right)\\ F_{ed}\left(t\right)={\mathrm{\α}}_3{F}_{h_1}^{d_1}\left(t\right)+(1-{\mathrm{\α}}_3)F_{h_2}^{d_2}\left(t\right)\end{array}\right.---(1-8)$

Wherein, α₁、α₂And α₃Represent the dominant factor of asymmetric double remote control system respectively；Subscript d₁、d₂, d respectively table Show main side 1 and between end, main side 2 and between end, time delay size between main side 1 and main side 2；The speed of main side 1 And powerThe information from end that is transferred to becomesWithThe information being transferred to main side 2 becomesWithMain side 2 SpeedAnd powerThe information from end that is transferred to becomesWithThe information being transferred to main side 1 becomesWithSpeed V from end_e(t) and F_eT () is transferred to the information of main side 1 and becomesWithIt is transferred to the information of main side 2 BecomeWith

2) design wave variables

Owing to co-existing in three channels, speed and force information are transferred to the other side by each channel, on each channel profit The stability of channel is ensured by wave variables method；Comprise the following steps that；

For each channel design wave variables, time domain variable is become wave variables and is transmitted；

$u_m\left(t\right)=\frac{bv\left(t\right)+F^{\′}\left(t\right)}{\sqrt{2b}}---(1-9)$

$v_m\left(t\right)=\frac{bv\left(t\right)-F^{\′}\left(t\right)}{\sqrt{2b}}---(1-10)$

$u_s\left(t\right)=\frac{{\mathrm{bv}}^{\′}\left(t\right)+F\left(t\right)}{\sqrt{2b}}---(1-11)$

$v_s\left(t\right)=\frac{{\mathrm{bv}}^{\′}\left(t\right)-F\left(t\right)}{\sqrt{2b}}---(1-12)$

T represents the time delay size of transmission, u_mT () represents the wave variables that left end is propagated to the right, v (t) represents left end Speed, F ' (t) represents that right-hand member is transferred to the expected force of left end, v_mT () represents the wave variables that left end is propagated to the left, b represents wave resistance Anti-, u_sT () represents the wave variables that right-hand member is propagated to the right, v_sT () represents the wave variables that right-hand member is propagated to the left, v ' (t) represents that left end passes Being passed to the desired speed of right-hand member, F (t) represents the power of right-hand member；The introducing of wave variables, can make v ' at the conditionality ensureing passivity T () follows the tracks of v (t), F ' (t) follows the tracks of F (t)；

$u_{m_2}\left(t\right)=\frac{{\mathrm{bF}}_2\left(t\right)+v_2^{\′}\left(t\right)}{\sqrt{2b}}---(1-13)$

$v_{m_2}\left(t\right)=\frac{{\mathrm{bF}}_2\left(t\right)-v_2^{\′}\left(t\right)}{\sqrt{2b}}---(1-14)$

$u_{s_2}\left(t\right)=\frac{{\mathrm{bF}}_2^{\′}\left(t\right)+v_2\left(t\right)}{\sqrt{2b}}---(1-15)$

$v_{s_2}\left(t\right)=\frac{{\mathrm{bF}}_2^{\′}\left(t\right)-v_2\left(t\right)}{\sqrt{2b}}---(1-16)$

Represent the wave variables that left end is propagated to the right, F₂T () represents the power of left end, v '₂T () represents that right-hand member is transferred to a left side The desired speed of end,Represent the wave variables that left end is propagated to the left,Represent the wave variables that right-hand member is propagated to the right,Table Show the wave variables that right-hand member is propagated to the left, F '₂T () represents that left end is transferred to the expected force of right-hand member, v₂T () represents the speed of right-hand member；Ripple The introducing of variable, can make F ' at the conditionality ensureing stability₂T () follows the tracks of F₂(t), v '₂T () follows the tracks of v₂(t)；

For the channel between main side 1 and main side 2, when the wave variables of this channel designs, T is between main side 1 and main side 2 Time delay d；V (t) isV ' (t) isF (t) isF ' (t) isF₂(t) beF′₂(t) bev₂(t) bev′₂(t) be

For main side 1 and from end between channel, this channel wave variables design time, T be main side 1 and from end between time Prolong d₁；V (t) isV ' (t) isF (t) is F_eT (), F ' (t) isF₂(t) beF′₂(t) bev₂T () is V_e(t), v '₂(t) be

For main side 2 and from end between channel, this channel wave variables design time, T be main side 2 and from end between time Prolong d₂；V (t) isV ' (t) isF (t) is F_eT (), F ' (t) isF₂(t) beF′₂(t) bev₂T () is V_e(t), v '₂(t) be

3) setup delay model

When wave variables is propagated in the channel that time delay is T, the Time Delay Model according to following:

u_s(t)=u_m(t-T) (1-17)

v_m(t)=v_s(t-T) (1-18)

$u_{s_2}\left(t\right)=u_{m_2}(t-T)---(1-19)$

$v_{m_2}\left(t\right)=v_{s_2}(t-T)---(1-20).$

Compared with prior art, the method have the advantages that

The present invention is transmitted after speed and force information are converted into wave variables again, according to Passivity Theory, it is ensured that Channel passivity under the conditions of any fixed response time, it is ensured that double remote control system stability under the conditions of fixed response time；This Bright have only to that speed and force information are carried out simple arithmetic operator and can realize wave conversion, it is easy to accomplish.Finally, the present invention can To ensure channel passivity under the conditions of any fixed response time, and main side, from end and environmental model be all passive, so can To ensure whole system passivity under any fixed response time and stability.

[accompanying drawing explanation]

Fig. 1 represents Time Delay Model；

Fig. 2 represents wave variables computational methods.

[detailed description of the invention]

Below in conjunction with the accompanying drawings the present invention is described in further detail:

Seeing Fig. 1 and Fig. 2, Fig. 1 and specifically describe main side 1, main side 2 and the transmission situation of the information between end, the present invention uses In the stable control method of the asymmetric double remote control system of time delay, comprise the following steps:

Set up two main sides and from the kinetic model held:

$Z_{m_1}{V}_{h_1}=F_{{\mathrm{cm}}_1}+F_{h_1}---(1-21)$

$Z_{m_2}{V}_{h_2}=F_{{\mathrm{cm}}_2}+F_{h_2}---(1-22)$

Z_sV_e=F_cs-F_e (1-23)

Wherein:

$F_{{\mathrm{cm}}_1}=-C_{m_1}{V}_{h_1}-C_{4m_1}{V}_{h_{1}d}+C_{6m_1}{F}_{h_1}-C_{2m_1}{F}_{h_{1}d}---(1-24)$

$F_{{\mathrm{cm}}_2}=-C_{m_2}{V}_{h_2}-C_{4m_2}{V}_{h_{2}d}+C_{6m_2}{F}_{h_1}-C_{2m_2}{F}_{h_{2}d}---(1-25)$

F_cs=-C_sV_e+C₁V_ed-C₅F_e+C₃F_ed (1-26)

In above formulaZ_s=0.1s,C₁=0.1s+2+10/s, S represents Laplace operator.

The desired speed of main and slave terminal is respectively:

$\left\{\begin{array}{c}{V}_{h_{1}d}\left(t\right)=0.5V_e^{d_1}\left(t\right)+0.5V_{h_2}^d\left(t\right)\\ V_{h_{2}d}\left(t\right)=0.8V_e^{d_2}\left(t\right)+0.2V_{h_1}^d\left(t\right)\\ V_{ed}\left(t\right)=0.3V_{h_1}^{d_1}\left(t\right)+0.7V_{h_2}^{d_2}\left(t\right)\end{array}\right.---(1-27)$

The expected force of main and slave terminal is respectively:

$\left\{\begin{array}{c}{F}_{h_{1}d}\left(t\right)=0.5F_e^{d_1}\left(t\right)+0.5F_{h_2}^d\left(t\right)\\ F_{h_{2}d}\left(t\right)=0.8F_e^{d_2}\left(t\right)+0.2F_{h_1}^d\left(t\right)\\ F_{ed}\left(t\right)=0.3F_{h_1}^{d_1}\left(t\right)+0.7F_{h_2}^{d_2}\left(t\right)\end{array}\right.---(1-28)$

For the channel between main side 1 and main side 2, make b=2.5

$\left\{\begin{array}{c}{u}_m\left(t\right)=\frac{2.5V_{h_1}\left(t\right)+F_{h_2}^d\left(t\right)}{\sqrt{5}}\\ v_m\left(t\right)=\frac{2.5V_{h_1}\left(t\right)-F_{h_2}^d\left(t\right)}{\sqrt{5}}\\ u_s\left(t\right)=\frac{2.5V_{h_1}^d\left(t\right)-F_{h_2}\left(t\right)}{\sqrt{5}}\\ v_s\left(t\right)=\frac{2.5V_{h_1}^d\left(t\right)-F_{h_2}\left(t\right)}{\sqrt{5}}\end{array}\right.---(1-29)$

$\left\{\begin{array}{c}{u}_{m_2}\left(t\right)=\frac{2.5F_{h_1}\left(t\right)+V_{h_2}^d\left(t\right)}{\sqrt{5}}\\ v_{m_2}\left(t\right)=\frac{2.5F_{h_1}\left(t\right)-V_{h_2}^d\left(t\right)}{\sqrt{5}}\\ u_{s_2}\left(t\right)=\frac{2.5F_{h_1}^d\left(t\right)+V_{h_2}\left(t\right)}{\sqrt{5}}\\ v_{s_2}\left(t\right)=\frac{2.5F_{h_1}^d\left(t\right)-V_{h_2}\left(t\right)}{\sqrt{5}}\end{array}\right.---(1-30)$

For main side 1 and the channel between end, make b=2.5

$\left\{\begin{array}{c}{u}_m^{\′}\left(t\right)=\frac{2.5V_{h_1}\left(t\right)+F_e^{d_1}\left(t\right)}{\sqrt{5}}\\ v_m^{\′}\left(t\right)=\frac{2.5V_{h_1}\left(t\right)-F_e^{d_1}\left(t\right)}{\sqrt{5}}\\ u_s^{\′}\left(t\right)=\frac{2.5V_{h_1}^{d_1}\left(t\right)+F_e\left(t\right)}{\sqrt{5}}\\ v_s^{\′}\left(t\right)=\frac{2.5V_{h_1}^{d_1}\left(t\right)-F_e\left(t\right)}{\sqrt{5}}\end{array}\right.---(1-31)$

$\left\{\begin{array}{c}{u}_{m_2}^{\′}\left(t\right)=\frac{2.5F_{h_1}\left(t\right)+V_e^{d_1}\left(t\right)}{\sqrt{5}}\\ v_{m_2}^{\′}\left(t\right)=\frac{2.5F_{h_1}\left(t\right)-V_e^{d_1}\left(t\right)}{\sqrt{5}}\\ u_{s_2}^{\′}\left(t\right)=\frac{2.5F_{h_1}^{d_1}\left(t\right)+V_e\left(t\right)}{\sqrt{5}}\\ v_{s_2}^{\′}\left(t\right)=\frac{2.5F_{h_1}^{d_1}\left(t\right)-V_e\left(t\right)}{\sqrt{5}}\end{array}\right.---(1-32)$

For main side 2 and the channel from end, make b=2.5

$\left\{\begin{array}{c}{u}_m^{\′\′}\left(t\right)=\frac{2.5V_{h_2}\left(t\right)+F_e^{d_2}\left(t\right)}{\sqrt{5}}\\ v_m^{\′\′}\left(t\right)=\frac{2.5V_{h_2}\left(t\right)-F_e^{d_2}\left(t\right)}{\sqrt{5}}\\ u_s^{\′\′}\left(t\right)=\frac{2.5V_{h_2}^{d_2}\left(t\right)+F_e\left(t\right)}{\sqrt{5}}\\ v_s^{\′\′}\left(t\right)=\frac{2.5V_{h_2}^{d_2}\left(t\right)-F_e\left(t\right)}{\sqrt{5}}\end{array}\right.---(1-33)$

$\left\{\begin{array}{c}{u}_{m_2}^{\′\′}\left(t\right)=\frac{2.5F_{h_2}\left(t\right)+V_e^{d_2}\left(t\right)}{\sqrt{5}}\\ v_{m_2}^{\′\′}\left(t\right)=\frac{2.5F_{h_2}\left(t\right)-V_e^{d_2}\left(t\right)}{\sqrt{5}}\\ u_{s_2}^{\′\′}\left(t\right)=\frac{2.5F_{h_2}^{d_2}\left(t\right)+V_e\left(t\right)}{\sqrt{5}}\\ v_{s_2}^{\′\′}\left(t\right)=\frac{2.5F_{h_2}^{d_2}\left(t\right)-V_e\left(t\right)}{\sqrt{5}}\end{array}\right.---(1-34)$

Setup delay model, d₁=0.1s, d₂=0.2s, d=0.05s, s represents the second herein:

$\left\{\begin{array}{c}{u}_s\left(t\right)=u_m(t-0.05)\\ v_m\left(t\right)=v_s(t-0.05)\\ u_{s_2}\left(t\right)=u_{m_2}(t-0.05)\\ v_{m_2}\left(t\right)=v_{s_2}(t-0.05)\end{array}\right.---(1-35)$

$\left\{\begin{array}{c}{u}_s^{\′}\left(t\right)=u_m^{\′}(t-0.01)\\ v_m^{\′}\left(t\right)=v_s^{\′}(t-0.01)\\ u_{s_2}^{\′}\left(t\right)=u_{m_2}^{\′}(t-0.01)\\ v_{m_2}^{\′}\left(t\right)=v_{s_2}^{\′}(t-0.01)\end{array}\right.---(1-36)$

$\left\{\begin{array}{c}{u}_s^{\′\′}\left(t\right)=u_m^{\′\′}(t-0.2)\\ v_m^{\′\′}\left(t\right)=v_s^{\′\′}(t-0.2)\\ u_{s_2}^{\′\′}\left(t\right)=u_{m_2}^{\′\′}(t-0.2)\\ v_{m_2}^{\′\′}\left(t\right)=v_{s_2}^{\′\′}(t-0.2)\end{array}\right.---(1-37)$

Above content is only the technological thought that the present invention is described, it is impossible to limit protection scope of the present invention with this, every presses The technological thought proposed according to the present invention, any change done on the basis of technical scheme, each fall within claims of the present invention Protection domain within.

Claims (1)

1. the stable control method for the asymmetric double remote control system of time delay, it is characterised in that comprise the following steps:

1) model of asymmetric double remote control system is set up；

Set up two main sides and from the kinetic model held:

$Z_{m_1}{V}_{h_1}=F_{{\mathrm{cm}}_1}+F_{h_1}---(1-1)$

$Z_{m_2}{V}_{h_2}=F_{{\mathrm{cm}}_2}+F_{h_2}---(1-2)$

Z_sV_e=F_cs-F_e (1-3)

Wherein:

$F_{{\mathrm{cm}}_1}=-C_{m_1}{V}_{h_1}-C_{4m_1}{V}_{h_{1}d}+C_{6m_1}{F}_{h_1}-C_{2m_1}{F}_{h_{1}d}---(1-4)$

$F_{{\mathrm{cm}}_2}=-C_{m_2}{V}_{h_2}-C_{4m_2}{V}_{h_{2}d}+C_{6m_2}{F}_{h_1}-C_{2m_2}{F}_{h_{2}d}---(1-5)$

F_cs=-C_sV_e+C₁V_ed-C₅F_e+C₃F_ed (1-6)

And V_eRepresent main side 1, main side 2 and the speed from end respectively,And Z_sRepresent main side 1, main side 2 and from The linear kinetic model of end,And F_csRepresent main side 1, main side 2 and the control power from end respectively,WithRespectively Represent that operator is applied to main side 1 and the power of main side 2, F_eRepresent the active force of environment；And V_edRepresent main side respectively 1, main side 2 and the desired speed from end,And F_edRepresent main side 1, main side 2 and the expectation function power from end respectively； C_s、C₁、C₅And C₃It is respectively different controller ginsengs Number；

The desired speed of main and slave terminal is respectively:

$\left\{\begin{array}{c}{V}_{h_{1}d}\left(t\right)={\mathrm{\α}}_1{V}_e^{d_1}\left(t\right)+(1-{\mathrm{\α}}_1)V_{h_2}^d\left(t\right)\\ V_{h_{2}d}\left(t\right)=(1-{\mathrm{\α}}_2)V_e^{d_2}\left(t\right)+{\mathrm{\α}}_2{V}_{h_1}^d\left(t\right)\\ V_{ed}\left(t\right)={\mathrm{\α}}_3{V}_{h_1}^{d_1}\left(t\right)+(1-{\mathrm{\α}}_3)V_{h_2}^{d_2}\left(t\right)\end{array}\right.---\left(1-7\right)$

The expected force of main and slave terminal is respectively:

$\left\{\begin{array}{c}{F}_{h_{1}d}\left(t\right)={\mathrm{\α}}_1{F}_e^{d_1}\left(t\right)+(1-{\mathrm{\α}}_1)F_{h_2}^d\left(t\right)\\ F_{h_{2}d}\left(t\right)=(1-{\mathrm{\α}}_2)F_e^{d_2}\left(t\right)+{\mathrm{\α}}_2{F}_{h_1}^d\left(t\right)\\ F_{ed}\left(t\right)={\mathrm{\α}}_3{F}_{h_1}^{d_1}\left(t\right)+(1-{\mathrm{\α}}_3)F_{h_2}^{d_2}\left(t\right)\end{array}\right.---(1-8)$

Wherein, α₁、α₂And α₃Represent the dominant factor of asymmetric double remote control system respectively；Subscript d₁、d₂, d represent main respectively End 1 and between end, main side 2 and between end, time delay size between main side 1 and main side 2；The speed of main side 1And powerThe information from end that is transferred to becomesWithThe information being transferred to main side 2 becomesWithThe speed of main side 2 DegreeAnd powerThe information from end that is transferred to becomesWithThe information being transferred to main side 1 becomesWithSpeed V from end_e(t) and F_eT () is transferred to the information of main side 1 and becomesWithIt is transferred to the information of main side 2 BecomeWith

2) design wave variables

Owing to co-existing in three channels, speed and force information are transferred to the other side by each channel, utilize ripple on each channel Variable method ensures the stability of channel；Comprise the following steps that；

For each channel design wave variables, time domain variable is become wave variables and is transmitted；

$u_m\left(t\right)=\frac{bv\left(t\right)+F^{\′}\left(t\right)}{\sqrt{2b}}---(1-9)$

$v_m\left(t\right)=\frac{bv\left(t\right)-F^{\′}\left(t\right)}{\sqrt{2b}}---(1-10)$

$u_s\left(t\right)=\frac{{\mathrm{bv}}^{\′}\left(t\right)+F\left(t\right)}{\sqrt{2b}}---(1-11)$

$v_s\left(t\right)=\frac{{\mathrm{bv}}^{\′}\left(t\right)-F\left(t\right)}{\sqrt{2b}}---(1-12)$

T represents the time delay size of transmission, u_mT () represents the wave variables that left end is propagated to the right, v (t) represents the speed of left end, F ' (t) represents that right-hand member is transferred to the expected force of left end, v_mT () represents the wave variables that left end is propagated to the left, b represents natural impedance, u_s T () represents the wave variables that right-hand member is propagated to the right, v_sT () represents the wave variables that right-hand member is propagated to the left, v ' (t) represents that left end is transferred to The desired speed of right-hand member, F (t) represents the power of right-hand member；The introducing of wave variables, can make v ' (t) at the conditionality ensureing passivity Following the tracks of v (t), F ' (t) follows the tracks of F (t)；

$u_{m_2}\left(t\right)=\frac{{\mathrm{bF}}_2\left(t\right)+v_2^{\′}\left(t\right)}{\sqrt{2b}}---(1-13)$

$v_{m_2}\left(t\right)=\frac{{\mathrm{bF}}_2\left(t\right)-v_2^{\′}\left(t\right)}{\sqrt{2b}}---(1-14)$

$u_{s_2}\left(t\right)=\frac{{\mathrm{bF}}_2^{\′}\left(t\right)+v_2\left(t\right)}{\sqrt{2b}}---(1-15)$

$v_{s_2}\left(t\right)=\frac{{\mathrm{bF}}_2^{\′}\left(t\right)-v_2\left(t\right)}{\sqrt{2b}}---(1-16)$

Represent the wave variables that left end is propagated to the right, F₂T () represents the power of left end, v '₂T () represents that right-hand member is transferred to left end Desired speed,Represent the wave variables that left end is propagated to the left,Represent the wave variables that right-hand member is propagated to the right,Represent the right side The wave variables that end is propagated to the left, F '₂T () represents that left end is transferred to the expected force of right-hand member, v₂T () represents the speed of right-hand member；Wave variables Introducing, can ensure stability conditionality make F '₂T () follows the tracks of F₂(t), v '₂T () follows the tracks of v₂(t)；

For the channel between main side 1 and main side 2, when the wave variables of this channel designs, T is the time delay between main side 1 and main side 2 d；V (t) isV ' (t) isF (t) isF ' (t) isF₂(t) beF′₂(t) bev₂ (t) bev′₂(t) be

For main side 1 and the channel between end, when the wave variables of this channel designs, T is main side 1 and time delay d between end₁； V (t) isV ' (t) isF (t) is F_eT (), F ' (t) isF₂(t) beF′₂(t) bev₂(t) For V_e(t), v '₂(t) be

For main side 2 and the channel between end, when the wave variables of this channel designs, T is main side 2 and time delay d between end₂； V (t) isV ' (t) isF (t) is F_eT (), F ' (t) isF₂(t) beF′₂(t) bev₂ T () is V_e(t), v '₂(t) be

3) setup delay model

When wave variables is propagated in the channel that time delay is T, the Time Delay Model according to following:

u_s(t)=u_m(t-T) (1-17)

v_m(t)=v_s(t-T) (1-18)

$u_{s_2}\left(t\right)=u_{m_2}(t-T)---(1-19)$

$v_{m_2}\left(t\right)=v_{s_2}(t-T)---(1-20).$