CN104847715B - Control method for asymmetric electric-hydraulic proportional system based on model transferring - Google Patents

Abstract

The invention discloses a control method for an asymmetric electric-hydraulic proportional system based on model transferring. By converting a transfer function, an asymmetric mathematical model of the asymmetric electric-hydraulic proportional system is converted into a symmetric system model, the symmetry of a load flow model is achieved, and when a unified controller is adopted, responses of forward and reverse motions are identical; by means of signal conversion, the load flow model of the asymmetric system is established, and the purposes of transfer function converting and model converting are achieved, and a new established system model serves as a symmetric load flow model. The achieving method comprises the steps that a reference signal symmetrical to a proportional valve is given, after a signal is converted, a new input signal of the proportional valve is obtained, and proportional valve is controlled to work, so that load flows of the asymmetric electric-hydraulic proportional system in a proportional valve linear region in the forward motion and the reverse motion is identical; by means of the method, an asymmetric effect is eliminated by an asymmetric factor of the system through mathematical conversion, and the identical response of the asymmetric electric-hydraulic proportional system is achieved simply and intelligently.

Description

A kind of control method of the asymmetric electro-hydraulic proportional system based on model conversion

Technical field

The invention belongs to machinery and electronic technology field, it is related to a kind of asymmetric electro-hydraulic proportional system based on model conversion Control method.

Background technology

125mn extruder hydraulic control system uses the control form of " pilot valve combines choke valve ", is substantially One valve-controlled cylinder hydraulic system, when the speed of moved cross beam is excessive, reduces the Lift of choke valve, when speed becomes hour, Increase the opening degree of throttle valve core, systematic schematic diagram is as shown in Figure 1.

Live applicable cases show: throttle valve core is different in the characteristic opening and closing both direction, and valve element is opened Process speed is very slow, and closing velocity is very fast, and dynamic characteristic is asymmetric, is mainly manifested in stabilization time, overshoot etc..Using Single controller can not meet the rapidity opening and closing both direction and stability requirement simultaneously.Analyze its reason, except itself Unsymmetric structure outside, opening and closing two directions, external applied load also exist significantly different it is believed that in extruder work process The middle state change that there is external applied load and this body structure of system.Specific problem is, the asymmetrical characteristic of proportioning valve is not right Claim to become readily apparent under load effect, can not solve the problems, such as that two direction controlling characteristics are consistent using unified control strategy.

The positive and negative speed characteristics agreement of existing electrohydraulic system is typically carried out in terms of structure and control theory two, and Achieve great successes.Propose in structure and control asymmetrical cylinder using double spool, Asymmetry Valve etc., but due to valve In the case of processed complex, and high frequency divertical motion and load change frequency height, not right with move for eliminating pressure jump Title property effect is poor, and industrial application is not very wide in addition it is also necessary to further investigate.In terms of control theory, traditional solution It is based on unified model, by detecting system state parameter, using adaptive compensation technique it is proposed that a lot of compensation sides Method.Although these methods can effectively solve the problem that asymmetry problem in theory.But it is more complicated that their common feature is theory, ginseng Number regulation is more, and design and maintenance difficulties are big, and engineers and technicians are had high demands.And it is because control program is more complicated, real When property is poor, is difficult to be used widely in industry spot.

Analyzed according to above, in manufacturing cost, complex working condition, stability aspect cannot expire existing asymmetrical control strategy Sufficient practical.Therefore propose industry spot and can have important engineering significance by practical asymmetrical control strategy.

Content of the invention

The invention provides a kind of control method of the asymmetric electro-hydraulic proportional system based on model conversion, its purpose exists In overcoming in prior art and respond conforming defect for realizing general asymmetric electrohydraulic system, using conversion transmission Function, by asymmetrical for asymmetric electro-hydraulic proportional system mathematical model, is transformed into symmetrical system model, realizes load flow mould The symmetry transformation of type is so that consistent using forward and reverse motion response during unified controller.

A kind of control method of the asymmetric electro-hydraulic proportional system based on model conversion, will be non-right using conversion transmission function Claim the asymmetrical mathematical model of electro-hydraulic proportional system, be transformed into symmetrical system model, realize the symmetrical change of load flow model Change so that using unified controller when forward and reverse motion response consistent；

Load flow model after model conversion is described symmetrical system model, as follows:

$t\left(s\right)=m\left(s\right)g\left(s\right)=\mathrm{\α}l\sqrt{\frac{(1+\mathrm{\η})(p_s-p_r)}{2(1+{\mathrm{\η}}^3)}}{e}^{-sz}$

$t^{\′}\left(s\right)=m^{\′}\left(s\right)g^{\′}\left(s\right)=\mathrm{\α}l\sqrt{\frac{(1+\mathrm{\η})(p_s-p_r)}{2(1+{\mathrm{\η}}^3)}}{e}^{sz}$

Wherein, t (s) is positive load flow model transfer function after conversion, t'(s) it is reverse load flow mould after conversion Type transmission function；

Wherein u₁And u₂It is respectively the left position of proportioning valve and right position dead zone signals value；

Described conversion transmission function is as follows:

$m\left(s\right)=\frac{1}{m_i}{e}^{s\frac{u_1+u_2}{2}},m^{\′}\left(s\right)=\frac{1}{{m_i}^{\′}}{e}^{s\frac{u_1+u_2}{2}}$

M (s) is positive load flow model conversion transmission function, m'(s) convert transmission letter for reverse load discharge model Number；

$m_i=\frac{q_{l_i}}{q_{l_0}}=\sqrt{\frac{p_s-{\mathrm{\ηp}}_r-p_{l_i}}{\frac{1+\mathrm{\η}}{2}(p_s-r_r)}},{m_i}^{\′}=\frac{{q_{l_i}}^{\′}}{{q_{l_0}}^{\′}}=\sqrt{\frac{{\mathrm{\ηp}}_s-p_r+{p_{l_i}}^{\′}}{\frac{1+\mathrm{\η}}{2}(p_s-r_r)}};$

Positive load flow ratio isWherein, i represents different load pressure states, and i=0 is basic Load pressure state, i=0,1,2...n, n are positive integer；Reverse load flow ratio is

Load flow is the flow flowing in or out rodless cavity, and the load flow flowing into rodless cavity during positive movement is During adverse movement, the load flow of outflow rodless cavity isFor negative value；

Before g (s) is Asymmetric Electric liquid proportional system changeover, in the proportioning valve range of linearity, positive load flow model passes Delivery function:

$g\left(s\right)=\mathrm{\α}l\sqrt{\frac{(p_s-{\mathrm{\ηp}}_r-p_{l_i})}{1+{\mathrm{\η}}^3}}{e}^{-{\mathrm{su}}_1}$

G'(s, before) being Asymmetric Electric liquid proportional system changeover, in the proportioning valve range of linearity, reverse load discharge model passes Delivery function:

$g^{\′}\left(s\right)=\mathrm{\α}l\sqrt{\frac{{\mathrm{\ηp}}_s-p_r+{p_{l_i}}^{\′}}{1+{\mathrm{\η}}^3}}{e}^{-{\mathrm{su}}_2}$

α is intermediate variable, whereinc_dFor discharge coefficient, value is between 0.68～0.8, and ω is ratio Valve aperture area gradient, ρ is Media density；

η is the ratio of two cavity areas of asymmetrical cylinder, η=a₂/a₁, p_sFor the pressure of the fuel feeding of system, p_rPressure for oil return Power, p₁, p₂The pressure of rodless cavity and rod chamber respectively, load pressureDuring positive movement, load pressure isInstead To load pressure during motion it isL is maximum spool displacement.

Described conversion transmission function is that signal transformation for mula is by signal mapping mode Equivalent realization:

$\begin{array}{c}\left[\begin{array}{c}{u}_i\\ {u_i}^{\′}\end{array}\right]=\left[\begin{array}{cc}\frac{1}{m_i}& 0\\ 0& \frac{1}{{m_i}^{\′}}\end{array}\right]\left[\begin{array}{c}u\\ u^{\′}\end{array}\right]+\left[\begin{array}{c}(1-\frac{1}{m_i})u_1\\ (1-\frac{1}{{m_i}^{\′}})u_2\end{array}\right]+\left[\begin{array}{c}\frac{1}{m_i}\left(\frac{u_1+u_2}{2}\right)\\ \frac{1}{{m_i}^{\′}}\left(\frac{u_1+u_2}{2}\right)\end{array}\right]\\ =\left[\begin{array}{c}\frac{1}{m_i}u+(1-\frac{1}{m_i})u_1+\frac{1}{m_i}\frac{u_1+u_2}{2}\\ \frac{1}{{m_i}^{\′}}{u}^{\′}+(1-\frac{1}{{m_i}^{\′}})u_2+\frac{1}{{m_i}^{\′}}\frac{u_1+u_2}{2}\end{array}\right]\end{array}$

Wherein, (u, u') is forward and reverse reference signal of given proportioning valve, (u_i,u_i') be converted by signal after obtain Forward and reverse signal of inputting to proportioning valve of reality.

[asymmetric electrohydraulic system is by pump, proportioning valve, asymmetrical cylinder, load source and its necessary Hydraulic Elements composition, non-right Title factor includes that cylinder structure is asymmetric, and proportioning valve dead band is asymmetric, and the load suffered by system is asymmetric；】

Using signal conversion, system model is entered line translation, the model obtaining is as follows:

$t\left(s\right)=\frac{l\[q_{l_i}(\frac{1}{m_i}u+(1-\frac{1}{m_i})u_1+\frac{1}{m_i}\frac{u_1+u_2}{2})\]}{u\left(s\right)}=\mathrm{\α}l\sqrt{\frac{(1+\mathrm{\η})(p_s-p_r)}{2(1+{\mathrm{\η}}^3)}}{e}^{-sz}$

$t^{\′}\left(s\right)=\frac{l\[{q_{l_i}}^{\′}(\frac{1}{{m_i}^{\′}}{u}^{\′}+(1-\frac{1}{{m_i}^{\′}})u_2+\frac{1}{{m_i}^{\′}}\frac{u_1+u_2}{2})\]}{u^{\′}\left(s\right)}=\mathrm{\α}l\sqrt{\frac{(1+\mathrm{\η})(p_s-p_r)}{2(1+{\mathrm{\η}}^3)}}{e}^{sz}$

From formula above can be seen that and adds asymmetric system to control described signal conversion, it is non-right to be capable of Claim model symmetry so that using unified controller when forward and reverse motion response consistent；And this implementation method simple, feasible, Effectively.

Beneficial effect

The invention provides a kind of control method of the asymmetric electro-hydraulic proportional system based on model conversion, passed using conversion Delivery function, by asymmetrical for asymmetric electro-hydraulic proportional system mathematical model, is transformed into symmetrical system model, realizes load flow The symmetry of model is so that consistent using forward and reverse motion response during unified controller；Converted by signal and build asymmetric system The load flow model of system reaches the purpose of conversion translation of transfer function model, and the new system model building is symmetrical load Discharge model.Implementation is: given proportioning valve symmetrical reference signal during both forward and reverse directions motion, obtains after being converted by signal The input signal of new proportioning valve, controls proportioning valve to work so that asymmetric electro-hydraulic proportional system is in the proportioning valve range of linearity Positive and heterodromous load flow is consistent；The asymmetric factor of system is passed through mathematic(al) manipulation by the method, eliminates asymmetric The impact of factor, thus system Asymmetric Model is transformed into symmetry model；By the pressure state value of acquisition system, simple intelligence Can realize electric-hydraulic proportion asymmetric system response consistent, greatly reduce the debugging work load at scene, and solve load The response consensus of asymmetric system during change.This control method is simply effective for solving the problems, such as asymmetrical response, is suitable for In industry spot, low cost, reliability is high.

Brief description

Fig. 1 is the 125mn throttle system schematic diagram of the present invention；

Fig. 2 is the cut-away view of asymmetric electro-hydraulic proportional system；

Fig. 3 is the model conversion implementation figure of the present invention；

Fig. 4 is asymmetric electro-hydraulic proportional system laboratory table schematic diagram；

Fig. 5 is the plc control schematic diagram of the present invention；

Fig. 6 is when load condition changes, and applies the controlling stream to asymmetric electro-hydraulic proportional system for the method for the invention Cheng Tu；

Fig. 7 be the present invention varying duty under asymmetric system control move forward and backward response diagram；

Fig. 8 is shown in the pressure state change procedure shown in Fig. 7, and asymmetric system is provided without controlling and adopts the present invention The dynamic respond curve of both forward and reverse directions after described control method.

Specific embodiment

Below in conjunction with drawings and Examples, the present invention is described further.

A kind of control method of the asymmetric electro-hydraulic proportional system based on model conversion, will be non-right using conversion transmission function Claim the asymmetrical mathematical model of electro-hydraulic proportional system, be transformed into symmetrical system model, realize the symmetrical change of load flow model Change so that using unified controller when forward and reverse motion response consistent；

The cut-away view of asymmetric electro-hydraulic proportional system, as shown in Figure 2.

Asymmetrical cylinder is system positive movement in y positive movement, otherwise moves for system reverse；Two Cavity surfaces of asymmetrical cylinder Long-pending ratio η=a₂/a₁, when η=1, it is asymmetric cylinder；p_sFor the pressure of the fuel feeding of system, p_rFor the pressure of oil return, p₁, p₂ The pressure of rodless cavity and rod chamber respectively, load pressureIf load pressure is during positive movementReversely transport When dynamic, load pressure isForward and reverse signal u (%) of proportioning valve, u'(%), meetThe ratio of negative opening The dead band of the left position of valve and right position is-ε₁, ε₂, the corresponding left position of proportioning valve, right position dead zone signals value is u₁, u₂,

Load flow is the flow flowing in or out rodless cavity, and the load flow flowing into rodless cavity during positive movement is q_l, The load flow flowing out rodless cavity during adverse movement is q_l', q_l' it is negative value.

The basic status setting asymmetric system is state during balanced system as this system or asymmetric factor is mutual The state offset, sets the load in basic status for the asymmetric system as basic load pressure state.Other inconsistent information Can passing ratio and the basic status offseting the system that is transformed into.

The load pressure of basic load pressure stateComputing formula is:

$p_{l_0}=\frac{(1-\mathrm{\η})(p_s+p_r)}{2}$

Defining positive load flow ratio isWherein i (i=0,1,2...) represents different load pressures State, i=0 is basic load pressure state, and reverse load flow ratio is

When proportioning valve opening degree is identical, the load flow ratio of the load flow of different conditions and basic load pressure state m_i, m_i' computing formula:

$\left\{\begin{array}{c}{m}_i=\frac{q_{l_i}}{q_{l_0}}=\sqrt{\frac{p_s-{\mathrm{\ηp}}_r-p_{l_i}}{\frac{1+\mathrm{\η}}{2}(p_s-r_r)}}\\ {m_i}^{\′}=\frac{{q_{l_i}}^{\′}}{{q_{l_0}}^{\′}}=\sqrt{\frac{{\mathrm{\ηp}}_s-p_r+{p_{l_i}}^{\′}}{\frac{1+\mathrm{\η}}{2}(p_s-r_r)}}\end{array}\right.$

G (s) is positive load flow model transfer function before conversion, g'(s) pass for reverse load discharge model before conversion Delivery function, in the proportioning valve range of linearity before asymmetric system conversion, the transmission function of forward and reverse load flow is:

$g\left(s\right)=\frac{q_{l_i}\left(s\right)}{u\left(s\right)}=\mathrm{\α}l\sqrt{\frac{(p_s-{\mathrm{\ηp}}_r-p_{l_i})}{1+{\mathrm{\η}}^3}}{e}^{-{\mathrm{su}}_1}$

$g^{\′}\left(s\right)=\frac{{q_{l_i}}^{\′}\left(s\right)}{u^{\′}\left(s\right)}=\mathrm{\α}l\sqrt{\frac{{\mathrm{\ηp}}_s-p_r+{p_{l_i}}^{\′}}{1+{\mathrm{\η}}^3}}{e}^{-{\mathrm{su}}_2}$

IfT (s) is positive load flow model transfer function after conversion, t'(s) it is reverse load after conversion Discharge model transmission function, positive negative side in the proportioning valve range of linearity, after ratio and skew conversion for the load flow model To model it is:

$t\left(s\right)=\frac{l\[q_{l_i}(\frac{1}{m_i}u+(1-\frac{1}{m_i})u_1+\frac{1}{m_i}\frac{u_1+u_2}{2})\]}{u\left(s\right)}=\mathrm{\α}l\sqrt{\frac{(1+\mathrm{\η})(p_s-p_r)}{2(1+{\mathrm{\η}}^3)}}{e}^{-sz}$

$t^{\′}\left(s\right)=\frac{l\[{q_{l_i}}^{\′}(\frac{1}{{m_i}^{\′}}{u}^{\′}+(1-\frac{1}{{m_i}^{\′}})u_2+\frac{1}{{m_i}^{\′}}\frac{u_1+u_2}{2})\]}{u^{\′}\left(s\right)}=\mathrm{\α}l\sqrt{\frac{(1+\mathrm{\η})(p_s-p_r)}{2(1+{\mathrm{\η}}^3)}}{e}^{sz}$

Load flow model after conversion is basic status load flow model, and that is, forward and reverse load flow model is symmetrical. Model conversion process is as shown in Figure 3.

Fig. 4 is the asymmetric hydraulic system laboratory table schematic diagram in the embodiment of the present invention, and laboratory table includes asymmetric system Hydraulic circuit, provides load hydraulic circuit, to workbench of taking blame for others.Asymmetric system hydraulic circuit is the control object of the present invention, There is dead band in the proportioning valve in loop, hydraulic cylinder is asymmetrical cylinder, and provides asymmetrically placed load by load circuit.

Fig. 5 is the detecting and controlling system in the embodiment of the present invention, and detecting and controlling system system is by detecting element (pressure sensing Device, displacement transducer), plc control system and host computer are constituted.The pressure of the fuel feeding of detecting element sensed system parameter system p_s, the pressure p of oil return_r, the pressure p of rodless cavity and rod chamber₁, p₂, and the dynamic respond of hydraulic cylinder.

Fig. 6 is the plc control flow chart in inventive embodiments, plc control system acquisition system state parameter, by judging The forward and reverse motion of system, selects forward and reverse load flow ratio to carry out signal conversion, the signal output after conversion is to proportioning valve.

(u, u') is forward and reverse reference signal of given proportioning valve, signal transformation calculations formula:

$\begin{array}{c}\left[\begin{array}{c}{u}_i\\ {u_i}^{\′}\end{array}\right]=\left[\begin{array}{cc}\frac{1}{m_i}& 0\\ 0& \frac{1}{{m_i}^{\′}}\end{array}\right]\left[\begin{array}{c}u\\ u^{\′}\end{array}\right]+\left[\begin{array}{c}(1-\frac{1}{m_i})u_1\\ (1-\frac{1}{{m_i}^{\′}})u_2\end{array}\right]+\left[\begin{array}{c}\frac{1}{m_i}\left(\frac{u_1+u_2}{2}\right)\\ \frac{1}{{m_i}^{\′}}\left(\frac{u_1+u_2}{2}\right)\end{array}\right]\\ =\left[\begin{array}{c}\frac{1}{m_i}u+(1-\frac{1}{m_i})u_1+\frac{1}{m_i}\frac{u_1+u_2}{2}\\ \frac{1}{{m_i}^{\′}}{u}^{\′}+(1-\frac{1}{{m_i}^{\′}})u_2+\frac{1}{{m_i}^{\′}}\frac{u_1+u_2}{2}\end{array}\right]\end{array}$

Upper computer software be used for asymmetric system parameter setting (such as displacement of targets, displacement control parameters, proportioning valve dead Zones values etc.), system state monitoring (pressure, displacement, speed monitoring etc.).

Fig. 7, Fig. 8 are the control effect figure after present invention enforcement.In Fig. 7 display embodiment of the present invention, laboratory table passes through load Control loop, the load pressure of system is with the change of displacement.Fig. 8 is shown in the pressure state change procedure shown in Fig. 7, non- Balanced system is provided without controlling and the dynamic respond curve using both forward and reverse directions after control method of the present invention, can from figure Know system when not using control method, the forward and reverse motion response of system differs greatly, more positive between adverse movement adjustment time Motion is short.After using control method of the present invention, forward and reverse motion response of system has preferable concordance, stable Time reaches basically identical.

Claims (2)

1. a kind of control method of the asymmetric electro-hydraulic proportional system based on model conversion is it is characterised in that transmitted using conversion Function, by asymmetrical for asymmetric electro-hydraulic proportional system mathematical model, is transformed into symmetrical system model, realizes load flow mould The symmetry transformation of type is so that consistent using forward and reverse motion response during unified controller；

Load flow model after model conversion is described symmetrical system model, as follows:

$t\left(s\right)=m\left(s\right)g\left(s\right)=\mathrm{\αl}\sqrt{\frac{(1+\mathrm{\η})(p_s-p_r)}{2(1+{\mathrm{\η}}^3)}}{e}^{-\mathrm{sz}}$

$t^{\′}\left(s\right)=m^{\′}\left(s\right)g^{\′}\left(s\right)=\mathrm{\αl}\sqrt{\frac{(1+\mathrm{\η})(p_s-p_r)}{2(1+{\mathrm{\η}}^3)}}{e}^{\mathrm{sz}}$

Wherein, t (s) is positive load flow model transfer function after conversion, t'(s) pass for reverse load discharge model after conversion Delivery function；

Wherein u₁And u₂It is respectively the left position of proportioning valve and right position dead zone signals value；

Described conversion transmission function is as follows:

$m\left(s\right)=\frac{1}{m_i}{e}^{s\frac{u_1+u_2}{2}},m^{\′}\left(s\right)=\frac{1}{{m_i}^{\′}}{e}^{s\frac{u_1+u_2}{2}}$

M (s) is positive load flow model conversion transmission function, m'(s) convert transmission function for reverse load discharge model；

$m_i=\frac{q_{l_i}}{q_{l_0}}=\sqrt{\frac{p_s-{\mathrm{\ηp}}_r-p_{l_i}}{\frac{1+\mathrm{\η}}{2}(p_s-p_r)}},{m_i}^{\′}=\frac{{q_{l_i}}^{\′}}{{q_{l_0}}^{\′}}=\begin{array}{}\end{array}\sqrt{\frac{{\mathrm{\ηp}}_s-p_r+{p_{l_i}}^{\′}}{\frac{1+\mathrm{\η}}{2}(p_s-p_r)}};$

Positive load flow ratio isWherein, i represents different load pressure states, and i=0 is basic load Pressure state, i=0,1,2...n, n are positive integer；Reverse load flow ratio is

Load flow is the flow flowing in or out rodless cavity, and the load flow flowing into rodless cavity during positive movement isReversely During motion, the load flow of outflow rodless cavity isFor negative value；

Before g (s) is Asymmetric Electric liquid proportional system changeover, in the proportioning valve range of linearity, positive load flow Model Transfer letter Number:

$g\left(s\right)=\mathrm{\αl}\sqrt{\frac{(p_s-{\mathrm{\ηp}}_r-p_{l_i})}{1+{\mathrm{\η}}^3}}{e}^{-{\mathrm{su}}_1}$

G'(s, before) being Asymmetric Electric liquid proportional system changeover, in the proportioning valve range of linearity, reverse load discharge model transmits letter Number:

$g^{\′}\left(s\right)=\mathrm{\αl}\sqrt{\frac{{\mathrm{\ηp}}_s-p_r+{p_{l_i}}^{\′}}{1+{\mathrm{\η}}^3}}{e}^{{-\mathrm{su}}_2}$

α is intermediate variable, whereinc_dFor discharge coefficient, value is that between 0.68～0.8, ω opens for proportioning valve Open area gradient, ρ is Media density；

η is the ratio of two cavity areas of asymmetrical cylinder, η=a₂/a₁, p_sFor the pressure of the fuel feeding of system, p_rFor the pressure of oil return, p₁, p₂The pressure of rodless cavity and rod chamber respectively, load pressure p_li=p₁-ηp₂, during positive movement, load pressure isAdverse movement When load pressure beL is maximum spool displacement.

2. the control method of a kind of asymmetric electro-hydraulic proportional system based on model conversion according to claim 1, it is special Levy and be, described conversion transmission function is that signal transformation for mula is by signal mapping mode Equivalent realization:

$\left[\begin{array}{c}{u}_i\\ {u_i}^{\′}\end{array}\right]=\left[\begin{array}{cc}\frac{1}{m_i}& 0\\ 0& \frac{1}{{m_i}^{\′}}\end{array}\right]\left[\begin{array}{c}u\\ u^{\′}\end{array}\right]+\left[\begin{array}{c}(1-\frac{1}{m_i})u_1\\ (1-\frac{1}{{m_i}^{\′}})u_2\end{array}\right]+\left[\begin{array}{c}\frac{1}{m_i}\left(\frac{u_1+u_2}{2}\right)\\ \frac{1}{{m_i}^{\′}}\left(\frac{u_1+u_2}{2}\right)\end{array}\right]$

$=\left[\begin{array}{c}\frac{1}{m_i}u+(1-\frac{1}{m_i})u_1+\frac{1}{m_i}\frac{u_1+u_2}{2}\\ \frac{1}{{m_i}^{\′}}{u}^{\′}+(1-\frac{1}{{m_i}^{\′}})u_2+\frac{1}{{m_i}^{\′}}\frac{u_1+u_2}{2}\end{array}\right]$

Wherein, (u, u') is forward and reverse reference signal of given proportioning valve, (u_i,u_i') be converted by signal after obtain reality Forward and reverse signal that border inputs to proportioning valve.