CN103233946B - A kind of Pneumatic Position Servo System backstepping control method - Google Patents

Abstract

The invention discloses a kind of Pneumatic Position Servo System backstepping control method, the method is specifically implemented according to following steps: step 1, set up the model of controlled Pneumatic Position Servo System, and each above-mentioned variable parameter detects in real time respectively by respective detecting instrument and obtains; Negligible friction, obtain three rank linear models of pneumatic system, control objectives is the desired output made required by Model output tracking; Step 2, set up the Backstepping Controller model of Pneumatic Position Servo System; Step 3, estimate Pneumatic Position Servo System unknown parameters ' value, by the numerical value input computer estimating to obtain, for the parameter of real-time update Backstepping Controller model, the signal of computer control amplifier exports, the displacement amount of real-time regulating piston.The inventive method does not need to increase Pressure testing hardware or algorithm, can obtain the control accuracy of better tracking effect and Geng Gao.

Description

A kind of Pneumatic Position Servo System backstepping control method

Technical field

The invention belongs to pneumatic system high precision position tracking control technology field, relate to a kind of Pneumatic Position Servo System backstepping control method.

Background technique

Pneumatic system take pressurized air as working medium, fire prevention, explosion-proof, anti-electromagnetic interference, not by the impact of radioactive rays and noise, and to vibrating and impacting also insensitive, and there is the advantages such as life-span long, safety, clean, high power/weight ratio, pneumatics has been widely used every field.

But because the compressibility of gas, gas are relatively large by the Complex Flows characteristic of valve port, frictional force between cylinder and slide block, it is very difficult that these factors make the high precision tracking of Pneumatic Position Servo System control.

Summary of the invention

The object of the present invention is to provide a kind of Pneumatic Position Servo System backstepping control method, solve prior art to the not high enough problem of the tracing control precision of Pneumatic Position Servo System.

The technical solution used in the present invention is, a kind of Pneumatic Position Servo System backstepping control method, and the method is specifically implemented according to following steps:

Step 1, set up the model of controlled Pneumatic Position Servo System

The mathematical model of proportional valve control Pneumatic Position Servo System is as shown in the formula (1):

$\left\{\begin{array}{c}{\stackrel{\·}{m}}_a=f_a(u,p_a)\\ {\stackrel{\·}{m}}_b=f_b(u,p_b)\\ \mathrm{KRT}{\stackrel{\·}{m}}_a=K{p}_a{A}_a\stackrel{\·}{y}+A_a(y_0+y){\stackrel{\·}{p}}_a\\ \mathrm{KRT}{\stackrel{\·}{m}}_b=K{p}_b{A}_b\stackrel{\·}{y}+A_b(y_0-y){\stackrel{\·}{p}}_b\\ M\stackrel{\·\·}{y}=p_a{A}_a-p_b{A}_b-F_f\end{array}\right.,---\left(1\right)$

Wherein with be respectively the gas mass flow flowing into cylinder A side and B side, u is control signal, p _aand p _bbe respectively cylinder A side and B side pressure, A _aand A _bbe respectively cylinder A side and B side piston cross-section to amass, y is piston displacement, y ₀for piston-initial-position, M is slide block quality, F _ffor frictional force, f _a() and f _b() is respectively the nonlinear function relevant with B side external and internal pressure with cylinder A side, and K, R and T are dependent constant, and above-mentioned piston displacement y is obtained by displacement detecting instrument;

Negligible friction, and linearization is carried out to the nonlinear function of formula (1), obtain three rank linear models of pneumatic system as shown in the formula (2):

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=x_3\\ {\stackrel{\·}{x}}_3=a_1{x}_1+a_2{x}_2+a_3{x}_3+\mathrm{bu}\\ y=x_1\end{array}\right.,---\left(2\right)$

Wherein x ₁, x ₂, x ₃for system mode, u is control inputs, a ₁, a ₂, a ₃, b is unknown model parameters, and control objectives makes the desired output y required by model output y tracking _d;

Step 2, set up the Backstepping Controller model of Pneumatic Position Servo System

Choose Backstepping Controller to the Pneumatic Position Servo System modular form (2) that upper step obtains, the Controlling model of Backstepping Controller is as shown in the formula (3):

$u^{\′}=\frac{1}{\hat{b}}(-z_2-{\hat{a}}_1{z}_1-{\hat{a}}_1{y}_d-{\hat{a}}_2{z}_2-{\hat{a}}_2{\mathrm{\α}}_1-{\hat{a}}_3{\mathrm{\α}}_2+{\stackrel{\·}{\mathrm{\α}}}_2),---\left(3\right)$

Wherein $\left\{\begin{array}{c}{z}_1=x_1-y_d\\ z_2=x_2-{\mathrm{\α}}_1\\ z_3=x_3-{\mathrm{\α}}_2\end{array}\right.,{\mathrm{\α}}_1={\stackrel{\·}{y}}_d-c_1{z}_1,{\mathrm{\α}}_2=-z_1+{\stackrel{\·}{\mathrm{\α}}}_1-c_2{z}_2,{\hat{a}}_1,{\hat{a}}_2,{\hat{a}}_3,\hat{b}$ Be the estimated value of system unknown parameter, y _dfor desired output, the derivative of variable is asked for by Euler's formula, and concrete form is as shown in the formula (4):

$\stackrel{\·}{x}\left(\mathrm{k\ΔT}\right)=\frac{x\left((k+1)\mathrm{\ΔT}\right)-x\left(\mathrm{k\ΔT}\right)}{\mathrm{\ΔT}},---\left(4\right)$

Δ T is wherein the sampling time, represent in the value of k sampling instant;

Actual export controlled quentity controlled variable carry out amplitude limit such as formula (5):

$u=\left\{\begin{array}{cc}{U}_{\max }& u^{\&prime;}>U_{\max }\\ u^{\&prime;}& -U_{\max }\&le;u^{\&prime;}\&le;U_{\max }\\ -U_{\max }& u^{\&prime;}<-U_{\max }\end{array};---\left(5\right)\right.$

Step 3, Pneumatic Position Servo System unknown parameters ' value to be estimated

Method of estimation is with reference to following formula (6):

$\left\{\begin{array}{c}\frac{1}{\hat{b}}=\&Integral;(-\mathrm{\&lambda;})z_1{y}_d\mathrm{dt}\\ {\stackrel{\&CenterDot;}{\hat{a}}}_1=-{\mathrm{\&beta;}}_1{z}_1{x}_1\\ {\stackrel{\&CenterDot;}{\hat{a}}}_2=-{\mathrm{\&beta;}}_2{z}_1{x}_2\\ {\stackrel{\&CenterDot;}{\hat{a}}}_3=-{\mathrm{\&beta;}}_3{z}_1{x}_3\end{array}\right.,---\left(6\right)$

λ > 0, β wherein _i> 0, i=1,2,3 is adaptive gain, and will estimate that the numerical value obtained is used for the parameter of real-time update Backstepping Controller modular form (3), computer is exported by the signal of D/A control amplifier, the displacement amount of the piston of real-time adjustment Rodless cylinder.

The beneficial effect of the inventive method is: 1) do not need to increase Pressure testing hardware or algorithm; 2) do not need the priori of object, just can implement effective control; 3) compared with more existing controlling methods, the control accuracy of better tracking effect and Geng Gao can be obtained.

Accompanying drawing explanation

Fig. 1 is the structural representation of the control object (proportional valve control Rodless cylinder) of the inventive method;

Fig. 2 is the Steady Experimental result adopting the inventive method to follow the tracks of S curve;

Fig. 3 is the Steady Experimental result adopting the inventive method to follow the tracks of sinusoidal signal;

Fig. 4 is the Steady Experimental result adopting the inventive method to follow the tracks of multifrequency sine signal;

Fig. 5 is the Steady Experimental result adopting sliding moding structure 1 method to follow the tracks of S curve;

Fig. 6 is the Steady Experimental result adopting sliding moding structure 1 method to follow the tracks of sinusoidal signal;

Fig. 7 is the Steady Experimental result adopting sliding moding structure 1 method to follow the tracks of multifrequency sine signal;

Fig. 8 is the Steady Experimental result adopting sliding moding structure 2 method to follow the tracks of S curve;

Fig. 9 is the Steady Experimental result adopting sliding moding structure 2 method to follow the tracks of sinusoidal signal;

Figure 10 is the Steady Experimental result adopting sliding moding structure 2 method to follow the tracks of multifrequency sine signal.

In figure, 1. piston, 2. load, 3. Rodless cylinder, 4. displacement detecting instrument, 5. Proportional valve, 6. amplifier, 7. computer, 8. reduction valve, 9. air pump.

Embodiment

Below in conjunction with the drawings and specific embodiments, the present invention is described in detail.

Pneumatic Position Servo System backstepping control method of the present invention, specifically implement according to following three steps:

Step 1, set up the model of controlled Pneumatic Position Servo System

With reference to Fig. 1, the structure of the controlled Pneumatic Position Servo System that the inventive method relies on is, the piston 1 of Rodless cylinder 3 is fixedly connected with load 2, and piston 1 also contacts with displacement detecting instrument 4 is corresponding simultaneously, and the output signal of location detector 4 sends into computer 7 by A/D conversion; Proportional valve 5 is five-way valve, the air cavity A side of Rodless cylinder 3 and air cavity B side respectively with two outlet sides (two po hold) the corresponding UNICOM of Proportional valve 5, Proportional valve 5 inlet end (pu end) is by reduction valve 8 and air pump 9 UNICOM, the valve element position of Proportional valve 5 is connected with the controller of amplifier 6, and amplifier 6 is connected by signaling line with computer 7.

Suppose that pneumatic system meets following condition: the working medium (air) that 1) system uses is perfect gas; 2) flowing state of gas flow when valve port or other restriction is constant entropy adiabatic process; 3) in same cavity volume, gas pressure and temperature are equal everywhere; 4) leakage do not considered is ignored; 5), during piston movement, the change procedure of two intracavity gas is adiabatic process; 6) bleed pressure and atmospheric pressure constant; 7) compared with system dynamic characteristic, the inertia of Proportional valve can be ignored.Obtain the mathematical model of proportional valve control Pneumatic Position Servo System of the present invention accordingly as shown in the formula (1):

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{m}}_a=f_a(u,p_a)\\ {\stackrel{\&CenterDot;}{m}}_b=f_b(u,p_b)\\ \mathrm{KRT}{\stackrel{\&CenterDot;}{m}}_a=K{p}_a{A}_a\stackrel{\&CenterDot;}{y}+A_a(y_0+y){\stackrel{\&CenterDot;}{p}}_a\\ \mathrm{KRT}{\stackrel{\&CenterDot;}{m}}_b=K{p}_b{A}_b\stackrel{\&CenterDot;}{y}+A_b(y_0-y){\stackrel{\&CenterDot;}{p}}_b\\ M\stackrel{\&CenterDot;\&CenterDot;}{y}=p_a{A}_a-p_b{A}_b-F_f\end{array}\right.,---\left(1\right)$

Wherein with be respectively the gas mass flow flowing into cylinder A side (the air cavity A in Fig. 1) and B side (the air cavity B in Fig. 1), u is control signal, p _aand p _bbe respectively cylinder A side and B side pressure, A _aand A _bbe respectively cylinder A side and B side piston cross-section amasss (equal for both the Rodless cylinders in native system), y is piston displacement, y ₀for piston-initial-position, M is slide block quality, F _ffor frictional force, f _a() and f _b() is respectively the nonlinear function relevant with B side external and internal pressure with cylinder A side, and K, R and T are dependent constant, and above-mentioned piston displacement y is obtained by displacement detecting instrument 4;

Negligible friction, and linearization is carried out to the nonlinear function of formula (1), obtain three rank linear models of pneumatic system as shown in the formula (2):

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=x_3\\ {\stackrel{\&CenterDot;}{x}}_3=a_1{x}_1+a_2{x}_2+a_3{x}_3+\mathrm{bu}\\ y=x_1\end{array}\right.,---\left(2\right)$

Wherein x ₁, x ₂, x ₃for system mode, u is control inputs, a ₁, a ₂, a ₃, b is unknown model parameters, and control objectives makes the desired output y required by model output y tracking _d.

Step 2, set up the Backstepping Controller model of Pneumatic Position Servo System

Choose contragradience (self adaption) controller to the Pneumatic Position Servo System modular form (2) that upper step obtains, the Controlling model of Backstepping Controller is as shown in the formula (3):

$u^{\&prime;}=\frac{1}{\hat{b}}(-z_2-{\hat{a}}_1{z}_1-{\hat{a}}_1{y}_d-{\hat{a}}_2{z}_2-{\hat{a}}_2{\mathrm{\&alpha;}}_1-{\hat{a}}_3{\mathrm{\&alpha;}}_2+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_2),---\left(3\right)$

Wherein $\left\{\begin{array}{c}{z}_1=x_1-y_d\\ z_2=x_2-{\mathrm{\&alpha;}}_1\\ z_3=x_3-{\mathrm{\&alpha;}}_2\end{array}\right.,{\mathrm{\&alpha;}}_1={\stackrel{\&CenterDot;}{y}}_d-c_1{z}_1,{\mathrm{\&alpha;}}_2=-z_1+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1-c_2{z}_2,{\hat{a}}_1,{\hat{a}}_2,{\hat{a}}_3,\hat{b}$ Be the estimated value of system unknown parameter, y _dfor desired output, the derivative of variable is asked for by Euler's formula, and concrete form is as shown in the formula (4):

$\stackrel{\&CenterDot;}{x}\left(\mathrm{k\&Delta;T}\right)=\frac{x\left((k+1)\mathrm{\&Delta;T}\right)-x\left(\mathrm{k\&Delta;T}\right)}{\mathrm{\&Delta;T}},---\left(4\right)$

Δ T is wherein the sampling time, represent in the value of k sampling instant, such as, in formula (3) then have ${\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_2\left(\mathrm{k\&Delta;T}\right)=\frac{{\mathrm{\&alpha;}}_2\left((k+1)\mathrm{\&Delta;T}\right)-{\mathrm{\&alpha;}}_2\left(\mathrm{k\&Delta;T}\right)}{\mathrm{\&Delta;T}};$

Actual export controlled quentity controlled variable carry out amplitude limit such as formula (5):

$u=\left\{\begin{array}{cc}{U}_{\max }& u^{\&prime;}>U_{\max }\\ u^{\&prime;}& -U_{\max }\&le;u^{\&prime;}\&le;U_{\max }\\ -U_{\max }& u^{\&prime;}<-U_{\max }\end{array};---\left(5\right)\right.$

Step 3, Pneumatic Position Servo System unknown parameters ' value to be estimated

Method of estimation is with reference to following formula (6):

$\left\{\begin{array}{c}\frac{1}{\hat{b}}=\&Integral;(-\mathrm{\&lambda;})z_1{y}_d\mathrm{dt}\\ {\stackrel{\&CenterDot;}{\hat{a}}}_1=-{\mathrm{\&beta;}}_1{z}_1{x}_1\\ {\stackrel{\&CenterDot;}{\hat{a}}}_2=-{\mathrm{\&beta;}}_2{z}_1{x}_2\\ {\stackrel{\&CenterDot;}{\hat{a}}}_3=-{\mathrm{\&beta;}}_3{z}_1{x}_3\end{array}\right.,---\left(6\right)$

λ > 0, β wherein _i> 0, i=1,2,3 is adaptive gain, and will estimate that the numerical value obtained is used for the parameter of real-time update Backstepping Controller modular form (3), computer is exported by the signal of D/A control amplifier 6, the displacement amount of the piston 1 of real-time adjustment Rodless cylinder 3.

Embodiment

All parts in Pneumatic Position Servo System structure is selected respectively: the Rodless cylinder (model of employing is DGPL-25-450-PPV-A-B-KF-GK-SV) of FESTO company; Five-way Proportional valve (model of employing is MPYE-5-1/8-HF-010-B); Swept resistance formula linear displacement detecting instrument (model of employing is MLO-POT-450-TLF, position detection accuracy 0.15mm after coordinating with capture card); Computer (model of employing is CPU is P21.2GHz); Universal data collection card (model of employing is PCI2306); Other element such as air pump forms Pneumatic Position Servo System.The control software design of built-in computer adopts VB establishment, by the change curve of correlated variables in On Screen Display control procedure.

Control objectives is set to respectively

Reference signal 1:S curve

y _d＝-(A/ω ²)sin(ωt)+(A/ω)t， （7）

The value of A is the value of 55.825, w is 0.5 π.

Reference signal 2: single frequency sinusoidal signal

y _d＝111.65sin0.5πt， （8）

Reference signal 3: multifrequency sine signal

y _d＝167.475sinπt+167.475sin0.5πt+167.475sin(2πt/7)， （9）

+167.475sin(πt/6)+167.475sin(2πt/17)

The contragradience adaptive controller of employing formula (3)-Shi (6) carries out Control release, the controling parameters c in formula (3)-Shi (6) ₁, c ₂, λ, β ₁, β ₂, β ₃value can repeatedly test examination and gather.

Be set to 50,50,1,1,1,1 in the present embodiment, control amplitude limit U _max=1.56V, when following the tracks of expectation target and being respectively formula (7)-Shi (9), steady track curve such as Fig. 2, Fig. 3, Fig. 4 provide.

Fig. 5-Figure 10 gives control effects when two kinds of modes (i.e. sliding moding structure 1 and sliding moding structure 2) tracking equating expections adopting prior art exports, and visible by contrasting, the inventive method tracking accuracy is higher.

Method reference literature [the Gary M.Bone of sliding moding structure 1, Shu Ning.Experimental Comparison of Position Tracking Control Algorithms for Pneumatic Cylinder Actuators [J] .IEEE/ASME Transactions on Mechatronics, 2007,12 (5): 557-561], its variable-structure controller is expressed as form:

$S=\stackrel{\&CenterDot;\&CenterDot;}{y}-{\stackrel{\&CenterDot;\&CenterDot;}{y}}_d+2\mathrm{\&lambda;}(\stackrel{\&CenterDot;}{y}-{\stackrel{\&CenterDot;}{y}}_d)+{\mathrm{\&lambda;}}^2(y-y_d)---\left(10\right)$

$u_{\mathrm{eq}}=\frac{1}{m_0}[{\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{y}}_d+n_2\stackrel{\&CenterDot;\&CenterDot;}{y}+n_1\stackrel{\&CenterDot;}{y}+n_{0}y-2\mathrm{\&lambda;}(\stackrel{\&CenterDot;\&CenterDot;}{y}-{\stackrel{\&CenterDot;\&CenterDot;}{y}}_d)-{\mathrm{\&lambda;}}^2(\stackrel{\&CenterDot;}{y}-{\stackrel{\&CenterDot;}{y}}_d)]---\left(11\right)$

u _s=-k _s1sat(S/φ) （12）

u′=u _eq+u _s（13）

Actual control is provided by formula (12), and the amplitude limit controlling to export provides identical with the inventive method such as formula (5), wherein model nominal parameters n ₂=29.5544, n ₁=218.436, n ₀=0, m ₀=5531.3305, controller parameter λ=50, k _s1=2.44 × 10 ⁴, φ=0.05, controls result curve with reference to shown in Fig. 5-Fig. 7.

Method reference literature [the T.Nguyen of sliding moding structure 2, J.Leavitt, F.Jabbari, J.E.Bobrow.Accurate Slide-Mode Control of Pneumatic Systems Using Low-Cost Solenoid Valves [J] .IEEE/ASME Transactions on Mechatronics, 2007,12 (2): 216-219], its variable structure control method as shown in the formula:

$S=\stackrel{\&CenterDot;\&CenterDot;}{y}-{\stackrel{\&CenterDot;\&CenterDot;}{y}}_d+2\mathrm{\&zeta;\&omega;}(\stackrel{\&CenterDot;}{y}-{\stackrel{\&CenterDot;}{y}}_d)+{\mathrm{\&omega;}}^2(y-y_d)---\left(14\right)$

u=-k _s2sgn(S) （15）

This existing method is for switch valve control cylinder, and actual control is provided by formula (15), k _s2the corresponding valve of=1, u=1 is opened, and the corresponding valve of u=-1 closes.

In the present invention, adoption rate valve, therefore gets k _s2=1.56(lies prostrate) the aperture amplitude of control ratio valve, controller parameter ξ=1, ω=50, realize controlling, and control result curve with reference to shown in Fig. 8-Figure 10.

Table 1 the inventive method and the error of existing controlling method when tracking type (7) outputs signal contrast

Table 2 the inventive method and the error of existing controlling method when tracking type (8) outputs signal contrast

In order to the control effects of the inventive method is described more intuitively, calculate tracking error quantitatively when following the tracks of different expectation target, error is defined as definition root-mean-square error and is:

$\mathrm{RMSE}=\sqrt{\frac{1}{N_2-N_1}\underset{k=N_1}{\overset{N_2}{\mathrm{\&Sigma;}}}{e}_k^2},---\left(5.7\right)$

Wherein N ₁for comparing start time, N ₂for comparing finish time, e _k=y (k Δ T)-y _d(k Δ T), Δ T is sampling time interval.For avoiding the impact of the different initial value of self adaptive control and random disturbances, having carried out test of many times, provide the experimental result of wherein five times to the tracking of often kind of input signal, its result such as table 1-table 3 provides.

In contrast table, result is visible, and when various expectation target, the average tracking error of the inventive method is all less than existing method.

Table 3 the inventive method and the error of existing controlling method when tracking type (9) outputs signal contrast

Claims (2)

1. a Pneumatic Position Servo System backstepping control method, is characterized in that, the method is specifically implemented according to following steps:

Step 1, set up the model of controlled Pneumatic Position Servo System

The mathematical model of proportional valve control Pneumatic Position Servo System is as shown in the formula (1):

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{m}}_a=f_a(u,p_a)\\ {\stackrel{\&CenterDot;}{m}}_b=f_b(u,p_b)\\ {\mathrm{kRT}\stackrel{\&CenterDot;}{m}}_a={\mathrm{Kp}}_a{A}_a\stackrel{\&CenterDot;}{y}+A_a(y_0+y){\stackrel{\&CenterDot;}{p}}_a\\ {\mathrm{KPT}\stackrel{\&CenterDot;}{m}}_b={\mathrm{Kp}}_b{a}_b\stackrel{\&CenterDot;}{y}+A_b(y_0-y){\stackrel{\&CenterDot;}{p}}_b\\ M\stackrel{\&CenterDot;\&CenterDot;}{y}=p_a{A}_a-p_b{A}_b-F_f\end{array}\right.,---\left(1\right)$

Wherein, with be respectively the gas mass flow flowing into air cavity A side and air cavity B side, and p _bbe respectively air cavity A side and air cavity B side pressure, with be respectively and p _bfirst derivative, for the nonlinear function relevant with air cavity A side external and internal pressure, f _b(u, p _b) be the nonlinear function relevant with air cavity B side external and internal pressure, and A _bbe respectively air cavity A side and air cavity B side piston cross-section to amass, y ₀for piston-initial-position, y is piston displacement, with be respectively first derivative and the second dervative of y, u is control signal, and M is slide block quality, F _ffor frictional force, K, R and T are dependent constant, and above-mentioned piston displacement y is obtained by displacement detecting instrument (4);

Negligible friction, and linearization is carried out to the nonlinear function of formula (1), the Pneumatic Position Servo System model after linearization process is such as formula (2):

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=x_3\\ {\stackrel{\&CenterDot;}{x}}_3=a_1{x}_1+a_3{x}_3+a_3{x}_3+\mathrm{bu}\\ y=x_1\end{array}\right.,---\left(2\right)$

Wherein x ₁, x ₂, x ₃be system mode, correspond to x respectively ₁, x ₂, x ₃first derivative, for unknown model parameters, control objectives makes the desired output y required by piston displacement y tracking _d;

Step 2, set up the Backstepping Controller model of Pneumatic Position Servo System

Choose Backstepping Controller to the Pneumatic Position Servo System modular form (2) that upper step obtains, Backstepping Controller model is as shown in the formula (3):

$u^{\&prime;}=\frac{1}{\hat{b}}({-z}_2-{\hat{a}}_1{z}_1-{\hat{a}}_1{y}_d-{\hat{a}}_2{z}_2-{\hat{a}}_2{\mathrm{\&alpha;}}_1-{\hat{a}}_3{\mathrm{\&alpha;}}_2+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_2),---\left(3\right)$

Wherein ${\mathrm{\&alpha;}}_1={\stackrel{\&CenterDot;}{y}}_d-c_1{z}_1,{\mathrm{\&alpha;}}_2=-z_1+{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1-c_2{z}_2,$ Z ₁=x ₁-y _d, u ' is the control signal before amplitude limit, for desired output y _dfirst derivative, z ₁, z ₂for intermediate variable, c ₁and c ₂for design parameter, for first derivative, for first derivative, with be unknown model parameters estimated value,

Amplitude limit is carried out to control signal u, such as formula (4):

$u=\left\{\begin{array}{cc}{U}_{\max }& u^{\&prime;}>U_{\max }\\ u^{\&prime;}& {-U}_{\max }\&le;u^{\&prime;}{U}_{\max }\\ {-U}_{\max }& u^{\&prime;}<-U_{\max }\end{array}\right.,---\left(4\right)$

Wherein, for the amplitude limit value of control signal u;

The estimated value of step 3, calculating unknown model parameters

The estimated value computational methods of four unknown model parameters are with reference to following formula (5):

$\left\{\begin{array}{c}\frac{1}{\hat{b}}=\&Integral;(-\mathrm{\&lambda;})z_1{y}_d\mathrm{dt}\\ {\stackrel{\&CenterDot;}{\hat{a}}}_1=-{\mathrm{\&beta;}}_1{z}_1{x}_1\\ {\stackrel{\&CenterDot;}{\hat{a}}}_2={-\mathrm{\&beta;}}_2{z}_1{x}_2\\ {\stackrel{\&CenterDot;}{\hat{a}}}_3=-{\mathrm{\&beta;}}_3{z}_1{x}_3\end{array}\right.,---\left(5\right)$

Wherein with be respectively with derivative, λ, β ₁, β ₂, β ₃for the adaptive gain of corresponding four unknown model parameters estimated values, and λ > 0, β ₁> 0, β ₂> 0, β ₃> 0, will estimate to obtain by formula (5) with numerical value be used for the parameter of real-time update Backstepping Controller modular form (3), computer is exported by the signal of D/A control amplifier (6), the displacement amount of the piston (1) of real-time adjustment Rodless cylinder (3).

2. Pneumatic Position Servo System backstepping control method according to claim 1, it is characterized in that: in described step 1, the structure of controlled Pneumatic Position Servo System is, five-way valve selected by Proportional valve (5), the piston (1) of Rodless cylinder (3) is fixedly connected with load (2), and the piston (1) of Rodless cylinder (3) is also arranged with corresponding contact of displacement detecting instrument (4) simultaneously; The air cavity A side of Rodless cylinder (3) and air cavity B side respectively with two corresponding UNICOMs of Stress control end of Proportional valve (5), the double piston-rod tail end of Proportional valve (5) is connected with the controller of amplifier (6), and in the middle part of the double piston-rod of Proportional valve (5), inlet end is by reduction valve (8) and air pump (9) UNICOM; The double piston-rod two ends pressure releasing chamber of Proportional valve (5) is provided with outlet side; Amplifier (6) is connected by signaling line with computer (7), and computer (7) is connected with displacement detecting instrument (4) by other signaling line.