CN105843043A - Electro hydraulic load simulator self-adaptive robust force control method - Google Patents

Abstract

The invention discloses an electro hydraulic load simulator self-adaptive robust force control method. For the system characteristics of an electro hydraulic load simulator, the friction characteristic of the system is analyzed through friction identification. A system nonlinear mathematical model which comprises a continuous differentiable friction model is established. Based on a traditional self-adaptive control method, a nonlinear robust control law is designed. The parameter estimation of the system is not affected in the presence of parametric uncertainties and uncertain nonlinearities, and an asymptotic tracking performance is acquired. According to the electro hydraulic load simulator self-adaptive robust force control method provided by the invention, the robustness of uncertain nonlinearities of traditional external load disturbance self-adaptive control and the like is enhanced to acquire a better tracking performance.

Description

A kind of electro-hydraulic load simulator ADAPTIVE ROBUST force control method

Technical field

The invention belongs to electro-hydraulic servo control field, be specifically related to a kind of electro-hydraulic load simulator ADAPTIVE ROBUST power control Method processed.

Background technology

The performance of precision strike weapon is the most important factor determining modern war victory or defeat, the attitude of weapon, track and side To control be crucial, this process is to receive cable system by the inertia device on weapon or guide to experience target location, then by Centre controls computer and calculates control instruction, then controls output.All these controls output will implement to final servo In executing agency's (steering wheel).Therefore, the property relationship of executing agency is to the full side of the national defense industry such as aviation, aerospace, naval vessel, cannon Position development, the application in civilian industry simultaneously is paid attention to the most widely.Which determine the modern whole big control of precision guided weapon The structure of system processed, layout, the especially key factor of weapon control dynamic characteristic.

The spy such as non-linear, uncertain that electro-hydraulic load simulator on the one hand has that general electrohydraulic servo system had Property, on the other hand the most again by being loaded the strong jamming of object motion so that system architecture is increasingly complex, therefore its network analysis with Controller design is increasingly difficult compared with general electrohydraulic servo system.The development of electrohydraulic servo-controlling system is it may be said that and control reason The development of opinion is complementary, and on the one hand, as the application of control system, control is managed by the development of electrohydraulic servo-controlling system The achievement of opinion is committed to application；On the other hand, due to the exclusive complex characteristics of electrohydraulic servo system and the highest performance Index request, the development of its control system has also promoted the development of control theory.

Currently for the Advanced Control Strategies of electrohydraulic servo system, there is the control method such as feedback linearization, sliding formwork.Feedback line Property control method not only design simply, and can ensure that the high-performance of system, but it require the systematic mathematical set up Model must be very accurate, and this is difficult to be guaranteed in actual applications；As a kind of robust control method, classical sliding formwork controls The modeling that can effectively process any bounded is uncertain, and obtains the steady-state behaviour of asymptotic tracking.But classical sliding formwork control Discontinuous controller designed by system easily causes the Flutter Problem of sliding-mode surface, thus deteriorates the tracking performance of system.To this end, Classical sliding formwork is controlled to be improved by many researchs, as used smooth continuous print hyperbolic tangent function to substitute discontinuous standard Sign function.But the most just lose the steady-state behaviour of asymptotic tracking, the tracking error of bounded can only be obtained.Sum up Say: traditional control method is difficult to meet the tracking accuracy requirement of Uncertain nonlinear；And control strategy design advanced in recent years The most more complicated, it is not easy to Project Realization.

Summary of the invention

It is an object of the invention to provide a kind of electro-hydraulic load simulator ADAPTIVE ROBUST force control method, solve existing Have in electro-hydraulic load simulator and there is uncared-for model uncertainty, based on designed by the control method of traditional sliding formwork The problems such as controller is discontinuous, based on traditional self-adaptation control method, make by designing nonlinear robust control rule dexterously System exist concurrently with parameter uncertainty and uncertainty nonlinear in the case of parameter Estimation unaffected and obtain The performance of asymptotic tracking, enhances the uncertain nonlinear robustness such as the tradition external load disturbance of Self Adaptive Control, it is thus achieved that More preferable tracking performance.

The present invention solves that the problems referred to above adopt the technical scheme that: a kind of electro-hydraulic load simulator ADAPTIVE ROBUST power Control method, comprises the following steps:

Step 1, based on continuously differentiable friction model, set up the Mathematical Modeling of electro-hydraulic load simulator；

Step 2, for arbitrary power track following, propose three reasonable assumptions, according to described reasonable assumption, design electro-hydraulic Load simulating device ADAPTIVE ROBUST force controller；

Step 3, the performance evaluation of ADAPTIVE ROBUST force controller.

In step 1, based on continuously differentiable friction model, set up the Mathematical Modeling of electro-hydraulic load simulator, concrete grammar As follows:

Step 1-1, foundation continuously differentiable friction model based on tanh approximation

$F_f\left(\stackrel{\·}{y}\right)=a_1\[\tanh \left(c_1\stackrel{\·}{y}\right)-\tanh \left(c_2\stackrel{\·}{y}\right)\]+a_2\tanh \left(c_3\stackrel{\·}{y}\right)+a_3\stackrel{\·}{y}---\left(1\right)$

In formula (1), a₁,a₂,a₃Represent the amplification level of differentiated friction characteristic, c respectively₁,c₂,c₃It is sign friction spy The form factor of property,Characterize movement velocity；Tanh represents hyperbolic tangent function.

Step 1-2, set up the kinetics equation of electro-hydraulic load simulator:

$\begin{array}{c}F={\mathrm{AP}}_L-F(t,y,\stackrel{\·}{y})\\ F(t,y,\stackrel{\·}{y})=F_f\left(\stackrel{\·}{y}\right)+f(t,y,\stackrel{\·}{y})\end{array}---\left(2\right)$

In formula (2), F is output torque, and A is the discharge capacity of load hydraulic cylinder, hydraulic cylinder load pressure P_L=P₁-P₂, P₁For The pressure of hydraulic cylinder oil suction chamber, P₂Go out the pressure of oil pocket for hydraulic cylinder, y is the position output that steering wheel produces,For the most true Determine nonlinear terms,For non-linear friction,For Unmarried pregnancy and outer interference.

Therefore formula (2) can be write as:

$F={\mathrm{AP}}_L-a_1\[\tanh \left(c_1\stackrel{\·}{y}\right)-\tanh \left(c_2\stackrel{\·}{y}\right)\]-a_2\tanh \left(c_3\stackrel{\·}{y}\right)-a_3\stackrel{\·}{y}-f(t,y,\stackrel{\·}{y})---\left(3\right)$

OrderThen have:

$F={\mathrm{AP}}_L-a_1{S}_f\left(\stackrel{\·}{y}\right)-a_2{P}_f\left(\stackrel{\·}{y}\right)-a_3\stackrel{\·}{y}-f(t,y,\stackrel{\·}{y})---\left(4\right)$

Step 1-3, set up hydraulic cylinder oil suction chamber and go out the Pressure behaviour equation of oil pocket:

$\left\{\begin{array}{c}{\stackrel{\·}{P}}_1=\frac{{\mathrm{\β}}_e}{V_1}(-A\stackrel{\·}{y}-C_t{P}_L+Q_1)\\ {\stackrel{\·}{P}}_2=\frac{{\mathrm{\β}}_e}{V_2}(A\stackrel{\·}{y}+C_t{P}_L-Q_2)\end{array}\right.---\left(5\right)$

In formula (5), β_eFor the effective bulk modulus of hydraulic oil, the control volume V of oil suction chamber₁=V₀₁+ Ay, V₀₁For oil-feed The original volume in chamber, goes out the control volume V of oil pocket₂=V₀₂-Ay, V₀₂For going out the original volume of oil pocket, C_tFor letting out in hydraulic cylinder Dew coefficient, Q₁For the flow of oil suction chamber, Q₂Flow for oil back chamber.

Q₁、Q₂With valve core of servo valve displacement x_vThere is a following relation:

$\left\{\begin{array}{c}{Q}_1=k_q{x}_v\[s\left(x_v\right)\sqrt{P_s-P_1}+s(-x_v)\sqrt{P_1-P_r}\]\\ Q_2=k_q{x}_v\[s\left(x_v\right)\sqrt{P_2-P_r}+s(-x_v)\sqrt{P_s-P_2}\]\end{array}\right.---\left(6\right)$

In formula (6), valve parameterC_dFor servo valve discharge coefficient for orifices, w₀For servo valve throttle orifice Area gradient, P_sFor electro-hydraulic load simulator charge oil pressure, P_rFor electro-hydraulic load simulator return pressure, ρ is hydraulic oil Density, x_vFor spool displacement, s (x_v) it is sign function, and described sign function is defined as:

$s\left(x_v\right)=\left\{\begin{array}{cc}1& x_v\&GreaterEqual;0\\ 0& x_v<0\end{array}\right.---\left(7\right)$

Ignore the dynamic of valve core of servo valve, it is assumed that act on the control input u and spool displacement x of spool_vProportional relation, I.e. meet x_v=k_lU, wherein k_lFor voltage-spool displacement gain coefficient, u is input voltage.

Therefore, formula (6) is written as

$\left\{\begin{array}{c}{Q}_1=gu\&lsqb;s\left(x_v\right)\sqrt{P_s-P_1}+s(-x_v)\sqrt{P_1-P_r}\&rsqb;\\ Q_2=gu\&lsqb;s\left(x_v\right)\sqrt{P_2-P_r}+s(-x_v)\sqrt{P_s-P_2}\&rsqb;\end{array}\right.---\left(8\right)$

The most total servo valve gain coefficient g=k_qk_l。

Based on formula (4), (5), (8), the power output dynamical equation of electro-hydraulic load simulator, i.e. electro-hydraulic load simulator Mathematical Modeling be:

$\begin{array}{c}\stackrel{\&CenterDot;}{F}=(\frac{R_1}{V_1}+\frac{R_2}{V_2}){\mathrm{A\&beta;}}_{e}gu-(\frac{1}{V_1}+\frac{1}{V_2}){\mathrm{\&beta;}}_e{A}^2\stackrel{\&CenterDot;}{y}-(\frac{1}{V_1}+\frac{1}{V_2}){\mathrm{A\&beta;}}_e{C}_t{P}_L\\ -a_1{\stackrel{\&CenterDot;}{S}}_f\left(\stackrel{\&CenterDot;}{y}\right)-a_2{\stackrel{\&CenterDot;}{P}}_f\left(\stackrel{\&CenterDot;}{y}\right)-a_3\stackrel{\&CenterDot;\&CenterDot;}{y}-d(t,y,\stackrel{\&CenterDot;}{y})\end{array}---\left(9\right)$

(9) in formula, the model uncertainty of electro-hydraulic load simulatorR₁And R₂Definition such as Under:

$\left\{\begin{array}{c}{R}_1=s\left(u\right)\sqrt{P_s-P_1}+s(-u)\sqrt{P_1-P_r}\\ R_2=s\left(u\right)\sqrt{P_2-P_r}+s(-u)\sqrt{P_s-P_2}\end{array}\right.---\left(10\right)$

R is understood by formula (10)₁＞ 0, R₂＞ 0, R₁And R₂It is intermediate variable.

For arbitrary power track following in step 2, three reasonable assumptions are proposed, according to described reasonable assumption, design electricity Hydraulic load simulator ADAPTIVE ROBUST force controller, specifically comprises the following steps that

Step 2-1, for ease of electro-hydraulic load simulator error symbol integration robust Controller Design, turn for arbitrary Square track following, has following 3 reasonable assumptions:

Assume 1: actual hydraulic pressure electro-hydraulic load simulator works in normal conditions, due to P_rAnd P_sImpact, P₁With P₂Meet condition: 0≤P_r＜ P₁＜ P_s, 0≤P_r＜ P₂＜ P_s, i.e. P₁And P₂It is all bounded；

Assume 2: desired torque instruction F_dT () is that single order is continuously differentiable, and instruct F_dT () and first derivative thereof are all It is bounded, motion artifacts y,It is the most all bounded.

Assume 3: parameter uncertainty and Uncertain nonlinear meet following condition:

$\left\{\begin{array}{c}\mathrm{\&theta;}\&Element;{\mathrm{\&Omega;}}_{\mathrm{\&theta;}}=\{\mathrm{\&theta;}:{\mathrm{\&theta;}}_{\min }\&le;\mathrm{\&theta;}\&le;{\mathrm{\&theta;}}_{\max }\}\\ \left|d(t,y,\stackrel{\&CenterDot;}{y})\right|\&le;{\mathrm{\&delta;}}_d(t,y,\stackrel{\&CenterDot;}{y})\end{array}\right.---\left(11\right)$

In formula (11), θ_min=[θ_1min,…,θ_6min]^T, θ_max=[θ_1max,…,θ_6max]^T, Ω_θFor the boundary of parameter θ, δ_dFor The interference function of one bounded.

Step 2-2, for simplifying electro-hydraulic load simulator dynamical equation, it is simple to the design of controller, the unknown constant value of definition Parameter vector θ=[θ₁,θ₂,θ₃,θ₄,θ₅,θ₆]^T, wherein θ₁=β_eG, θ₂=β_e, θ₃=β_eC_t, θ₄=a₁, θ₅=a₂, θ₆=a₃, because of This dynamical equation (9) is write as:

$\stackrel{\&CenterDot;}{F}={\mathrm{\&theta;}}_1{f}_{1}u-{\mathrm{\&theta;}}_2{f}_2-{\mathrm{\&theta;}}_3{f}_3-{\mathrm{\&theta;}}_4{\stackrel{\&CenterDot;}{S}}_f\left(\stackrel{\&CenterDot;}{y}\right)-{\mathrm{\&theta;}}_5{\stackrel{\&CenterDot;}{P}}_f\left(\stackrel{\&CenterDot;}{y}\right)-{\mathrm{\&theta;}}_6\stackrel{\&CenterDot;\&CenterDot;}{y}-d(t,y,\stackrel{\&CenterDot;}{y})---\left(12\right)$

Nonlinear function f in formula (12)₁,f₂,f₃It is defined as follows:

$\left\{\begin{array}{c}{f}_1(P_1,P_2,y)=A(\frac{R_1}{V_1}+\frac{R_2}{V_2})\\ f_2(y,\stackrel{\&CenterDot;}{y})=A^2\stackrel{\&CenterDot;}{y}(\frac{1}{V_1}+\frac{1}{V_2})\\ f_3(P_1,P_2,y)={\mathrm{AP}}_L(\frac{1}{V_1}+\frac{1}{V_2})\end{array}\right.---\left(13\right)$

Step 2-3, design electro-hydraulic load simulator ADAPTIVE ROBUST force controller, step is as follows:

The discontinuous parameter mapping that parameter adaptive is used first was given: make before the design being controlled deviceRepresent Estimation to system unknown parameter θ,For parameter estimating error, i.e.For guaranteeing the stability of adaptive control laws, Parameter uncertainty based on system is bounded, i.e. assumes 3, and the parameter adaptive being defined as follows discontinuously maps

${\mathrm{Proj}}_{\widehat{\mathrm{\&theta;}}}\left({\mathrm{\&tau;}}_i\right)=\left\{\begin{array}{cc}0& if{\widehat{\mathrm{\&theta;}}}_i={\widehat{\mathrm{\&theta;}}}_{i\max }{\mathrm{and\&tau;}}_i>0\\ 0& if{\widehat{\mathrm{\&theta;}}}_i={\widehat{\mathrm{\&theta;}}}_{i\min }{\mathrm{and\&tau;}}_i<0\\ {\mathrm{\&tau;}}_i& otherwise\end{array}\right.---\left(14\right)$

I=1 in formula (14) ..., 6, τ is parameter adaptive function, and it is concrete to provide it in follow-up controller design Form: be given below parameter adaptive rate:

$\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}={\mathrm{Proj}}_{\widehat{\mathrm{\&theta;}}}\left(\mathrm{\&Gamma;}\mathrm{\&tau;}\right){\mathrm{with\&theta;}}_{\min }\&le;\widehat{\mathrm{\&theta;}}\left(0\right)\&le;{\mathrm{\&theta;}}_{max},---\left(15\right)$

Г in formula > 0 is positive definite diagonal matrix；For arbitrary auto-adaptive function τ, discontinuous map (14) have following property Matter:

$\begin{array}{cc}\left(P1\right)& \widehat{\mathrm{\&theta;}}\&Element;{\mathrm{\&Omega;}}_{\mathrm{\&theta;}}=\{\widehat{\mathrm{\&theta;}}:{\mathrm{\&theta;}}_{\min }\&le;\widehat{\mathrm{\&theta;}}\&le;{\mathrm{\&theta;}}_{max}\}\end{array}---\left(16\right)$

$\begin{array}{ccc}\left(P2\right)& {\widetilde{\mathrm{\&theta;}}}^T\left({\mathrm{\&Gamma;}}^{-1}{\mathrm{Proj}}_{\widehat{\mathrm{\&theta;}}}\right(\mathrm{\&Gamma;}\mathrm{\&tau;})-\mathrm{\&tau;})\&le;0& \&ForAll;\mathrm{\&tau;}\end{array}---\left(17\right)$

Definition z=F-F_dFor the tracking error of system, the derivative of its time can be write as:

$\stackrel{\&CenterDot;}{z}=\stackrel{\&CenterDot;}{F}-{\stackrel{\&CenterDot;}{F}}_d={\mathrm{\&theta;}}_1{f}_{1}u-{\mathrm{\&theta;}}_2{f}_2-{\mathrm{\&theta;}}_3{f}_3-{\mathrm{\&theta;}}_4{\stackrel{\&CenterDot;}{S}}_f\left(\stackrel{\&CenterDot;}{y}\right)-{\mathrm{\&theta;}}_5{\stackrel{\&CenterDot;}{P}}_f\left(\stackrel{\&CenterDot;}{y}\right)-{\mathrm{\&theta;}}_6\stackrel{\&CenterDot;\&CenterDot;}{y}-d(t,y,\stackrel{\&CenterDot;}{y})-{\stackrel{\&CenterDot;}{F}}_d---\left(18\right)$

According to formula (18), System design based on model device u may be designed as:

U=u_m+u_r

$u_m=\frac{1}{{\widehat{\mathrm{\&theta;}}}_1{f}_1}\&lsqb;{\widehat{\mathrm{\&theta;}}}_2{f}_2+{\widehat{\mathrm{\&theta;}}}_3{f}_3+{\widehat{\mathrm{\&theta;}}}_4{\stackrel{\&CenterDot;}{S}}_f+{\widehat{\mathrm{\&theta;}}}_5{\stackrel{\&CenterDot;}{P}}_f+{\widehat{\mathrm{\&theta;}}}_6\stackrel{\&CenterDot;\&CenterDot;}{y}+{\stackrel{\&CenterDot;}{F}}_d\&rsqb;$

u_r=(-kz+u_s)/f₁ (19)

U in formula_mIt it is the adaptive model compensation term of the on-line parameter adaptive law be given by formula (15)；K is positive anti- Feedforward gain；u_rFor Robust Control Law, u_sIt is that non linear robust item is for overcoming the model uncertainty impact on tracking performance.

The Lyapunov function being defined as follows:

$V\left(t\right)=\frac{1}{2}{z}^2---\left(20\right)$

Based on controller (19), function V time differential is:

Formula returns deviceIt is defined as:

For robust item u_sDesign, need to meet following condition:

eu_s≤0 (23)

In formula, ξ represents given robust precision, and be can be arbitrarily small positive design parameter.

Function h is made to meet following condition:

θ in formula (24)_M=θ_max-θ_min, u_sCan be chosen for:

$u_s=-k_{m}z=-\frac{h}{2{\mathrm{\&theta;}}_{1\min }\mathrm{\&xi;}}z---\left(25\right)$

K in formula (25)_mFor positive non-linear gain, now u_sMeet condition (23).

The performance evaluation of ADAPTIVE ROBUST force controller described in step 3, specific as follows:

Controller performance: use and discontinuously map adaptive law (15) and auto-adaptive functionThe self adaptation proposed Robust Control Law (19) can ensure following performance:

A. in closed signal, all signals are all bounded, and positive definite integral form V meets such as lower inequality:

$V\left(t\right)\&le;\exp (-\mathrm{\&lambda;}t)V\left(0\right)+\frac{\mathrm{\&xi;}}{\mathrm{\&lambda;}}\&lsqb;1-\exp (-\mathrm{\&lambda;}t)\&rsqb;---\left(26\right)$

λ=2 θ in formula (26)_1minK is exponential convergence rate.

If the most at a time t₀Afterwards, system only exists parameter uncertainty, i.e. d=0, then except conclusion A it Outward, the performance of asymptotic tracking also can be obtained, i.e. as t → ∞, e → 0；

Stability analysis: choose following liapunov function, uses Lyapunov stability theory to carry out stable Property analyze:

$V_s=\frac{1}{2}{z}^2+\frac{1}{2}{\widetilde{\mathrm{\&theta;}}}^T{\mathrm{\&Gamma;}}^{-1}\widetilde{\mathrm{\&theta;}}---\left(27\right)$

And using Barbalat lemma can obtain the globally asymptotically stable result of system, therefore regulation parameter k, ξ and Γ can Make the tracking error of system the time tend to infinite under conditions of go to zero.

Compared with prior art, its remarkable advantage is the present invention:

(1) for the system features of electro-hydraulic load simulator, analyzed the frictional behavior of this system by Friction identification, build Stood more accurate new type of continuous can micro tribology model, lay the foundation for promoting the stability of this system.

(2) based on traditional self-adaptation control method, by design nonlinear robust control rule, system is deposited at the same time Parameter Estimation in the case of parameter uncertainty and uncertainty are nonlinear unaffected and obtain asymptotic tracking performance.

(3) the nonlinear adaptive robust controller designed by the present invention is simple and its control output is smooth continuously, more It is beneficial to apply in engineering reality.Its validity of simulation results show.

Accompanying drawing explanation

Fig. 1 is the electro-hydraulic load simulation dress of a kind of electro-hydraulic load simulator ADAPTIVE ROBUST force control method of the present invention Put schematic diagram.

Fig. 2 is the control strategy figure of a kind of electro-hydraulic load simulator ADAPTIVE ROBUST force control method of the present invention.

Fig. 3 is embodiment middle controller u time history plot, and controller input voltage meets-10V's～+10V Input range, meets actual application.

Fig. 4 is systematic parameter θ under the electro-hydraulic load simulator ADAPTIVE ROBUST force controller effect designed by the present invention₁ The time dependent exemplary curve of estimate.

Fig. 5 is systematic parameter θ under the electro-hydraulic load simulator ADAPTIVE ROBUST force controller effect designed by the present invention₂ The time dependent exemplary curve of estimate.

Fig. 6 is systematic parameter θ under the electro-hydraulic load simulator ADAPTIVE ROBUST force controller effect designed by the present invention₃ The time dependent exemplary curve of estimate.

Fig. 7 is systematic parameter θ under the electro-hydraulic load simulator ADAPTIVE ROBUST force controller effect designed by the present invention₄ The time dependent exemplary curve of estimate.

Fig. 8 is systematic parameter θ under the electro-hydraulic load simulator ADAPTIVE ROBUST force controller effect designed by the present invention₅ The time dependent exemplary curve of estimate.

Fig. 9 is systematic parameter θ under the electro-hydraulic load simulator ADAPTIVE ROBUST force controller effect designed by the present invention₆ The time dependent exemplary curve of estimate.

Figure 10 is system output under the electro-hydraulic load simulator ADAPTIVE ROBUST force controller effect designed by the present invention With desired output time history plot.

Figure 11 be under the electro-hydraulic load simulator ADAPTIVE ROBUST force controller effect designed by the present invention system with Track error time history plot.

Detailed description of the invention:

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

In conjunction with Fig. 2, a kind of electro-hydraulic load simulator ADAPTIVE ROBUST force control method, its electro-hydraulic load simulator is tied Structure principle is as it is shown in figure 1, comprise the following steps:

A kind of electro-hydraulic load simulator ADAPTIVE ROBUST force control method, comprises the following steps:

Step 1, based on continuously differentiable friction model, set up the Mathematical Modeling of electro-hydraulic load simulator, concrete grammar is such as Under:

Step 1-1, foundation continuously differentiable friction model based on tanh approximation

$F_f\left(\stackrel{\&CenterDot;}{y}\right)=a_1\&lsqb;\tanh \left(c_1\stackrel{\&CenterDot;}{y}\right)-\tanh \left(c_2\stackrel{\&CenterDot;}{y}\right)\&rsqb;+a_2\tanh \left(c_3\stackrel{\&CenterDot;}{y}\right)+a_3\stackrel{\&CenterDot;}{y}---\left(1\right)$

In formula (1), a₁,a₂,a₃Represent the amplification level of differentiated friction characteristic, c respectively₁,c₂,c₃It is sign friction spy The form factor of property,Characterize movement velocity；Tanh represents hyperbolic tangent function.

Step 1-2, set up the kinetics equation of electro-hydraulic load simulator:

$\begin{array}{c}F={\mathrm{AP}}_L-F(t,y,\stackrel{\&CenterDot;}{y})\\ F(t,y,\stackrel{\&CenterDot;}{y})=F_f\left(\stackrel{\&CenterDot;}{y}\right)+f(t,y,\stackrel{\&CenterDot;}{y})\end{array}---\left(2\right)$

In formula (2), F is output torque, and A is the discharge capacity of load hydraulic cylinder, hydraulic cylinder load pressure P_L=P₁-P₂, P₁For The pressure of hydraulic cylinder oil suction chamber, P₂Go out the pressure of oil pocket for hydraulic cylinder, y is the position output that steering wheel produces,For the most true Determine nonlinear terms,For non-linear friction,For Unmarried pregnancy and outer interference.

Therefore formula (2) can be write as:

$F={\mathrm{AP}}_L-a_1\&lsqb;\tanh \left(c_1\stackrel{\&CenterDot;}{y}\right)-\tanh \left(c_2\stackrel{\&CenterDot;}{y}\right)\&rsqb;-a_2\tanh \left(c_3\stackrel{\&CenterDot;}{y}\right)-a_3\stackrel{\&CenterDot;}{y}-f(t,y,\stackrel{\&CenterDot;}{y})---\left(3\right)$

OrderThen have:

$F={\mathrm{AP}}_L-a_1{S}_f\left(\stackrel{\&CenterDot;}{y}\right)-a_2{P}_f\left(\stackrel{\&CenterDot;}{y}\right)-a_3\stackrel{\&CenterDot;}{y}-f(t,y,\stackrel{\&CenterDot;}{y})---\left(4\right)$

Step 1-3, set up hydraulic cylinder oil suction chamber and go out the Pressure behaviour equation of oil pocket:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{P}}_1=\frac{{\mathrm{\&beta;}}_e}{V_1}(-A\stackrel{\&CenterDot;}{y}-C_t{P}_L+Q_1)\\ {\stackrel{\&CenterDot;}{P}}_2=\frac{{\mathrm{\&beta;}}_e}{V_2}(A\stackrel{\&CenterDot;}{y}+C_t{P}_L-Q_2)\end{array}\right.---\left(5\right)$

In formula (5), β_eFor the effective bulk modulus of hydraulic oil, the control volume V of oil suction chamber₁=V₀₁+ Ay, V₀₁For oil-feed The original volume in chamber, goes out the control volume V of oil pocket₂=V₀₂-Ay, V₀₂For going out the original volume of oil pocket, C_tFor letting out in hydraulic cylinder Dew coefficient, Q₁For the flow of oil suction chamber, Q₂Flow for oil back chamber.Q₁、Q₂With valve core of servo valve displacement x_vThere is a following relation:

$\left\{\begin{array}{c}{Q}_1=k_q{x}_v\&lsqb;s\left(x_v\right)\sqrt{P_s-P_1}+s(-x_v)\sqrt{P_1-P_r}\&rsqb;\\ Q_2=k_q{x}_v\&lsqb;s\left(x_v\right)\sqrt{P_2-P_r}+s(-x_v)\sqrt{P_s-P_2}\&rsqb;\end{array}\right.---\left(8\right)$

In formula (6), valve parameterC_dFor servo valve discharge coefficient for orifices, w₀For servo valve throttle orifice Area gradient, P_sFor electro-hydraulic load simulator charge oil pressure, P_rFor electro-hydraulic load simulator return pressure, ρ is hydraulic oil Density, x_vFor spool displacement, s (x_v) it is sign function, and described sign function is defined as:

$s\left(x_v\right)=\left\{\begin{array}{cc}1& x_v\&GreaterEqual;0\\ 0& x_v<0\end{array}\right.---\left(7\right)$

Ignore the dynamic of valve core of servo valve, it is assumed that act on the control input u and spool displacement x of spool_vProportional relation, I.e. meet x_v=k_lU, wherein k_lFor voltage-spool displacement gain coefficient, u is input voltage.

Therefore, formula (6) is written as

$\left\{\begin{array}{c}{Q}_1=gu\&lsqb;s\left(x_v\right)\sqrt{P_s-P_1}+s(-x_v)\sqrt{P_1-P_r}\&rsqb;\\ Q_2=gu\&lsqb;s\left(x_v\right)\sqrt{P_2-P_r}+s(-x_v)\sqrt{P_s-P_2}\&rsqb;\end{array}\right.---\left(8\right)$

The most total servo valve gain coefficient g=k_qk_l。

Based on formula (4), (5), (8), the power output dynamical equation of electro-hydraulic load simulator, i.e. electro-hydraulic load simulator Mathematical Modeling be:

$\begin{array}{c}\stackrel{\&CenterDot;}{F}=(\frac{R_1}{V_1}+\frac{R_2}{V_2}){\mathrm{A\&beta;}}_{e}gu-(\frac{1}{V_1}+\frac{1}{V_2}){\mathrm{\&beta;}}_e{A}^2\stackrel{\&CenterDot;}{y}-(\frac{1}{V_1}+\frac{1}{V_2}){\mathrm{A\&beta;}}_e{C}_t{P}_L\\ -a_1{\stackrel{\&CenterDot;}{S}}_f\left(\stackrel{\&CenterDot;}{y}\right)-a_2{\stackrel{\&CenterDot;}{P}}_f\left(\stackrel{\&CenterDot;}{y}\right)-a_3\stackrel{\&CenterDot;\&CenterDot;}{y}-d(t,y,\stackrel{\&CenterDot;}{y})\end{array}---\left(9\right)$

(9) in formula, the model uncertainty of electro-hydraulic load simulatorR₁And R₂Definition such as Under:

$\left\{\begin{array}{c}{R}_1=s\left(u\right)\sqrt{P_s-P_1}+s(-u)\sqrt{P_1-P_r}\\ R_2=s\left(u\right)\sqrt{P_2-P_r}+s(-u)\sqrt{P_s-P_2}\end{array}\right.---\left(10\right)$

R is understood by formula (10)₁＞ 0, R₂＞ 0, R₁And R₂It is intermediate variable.

Step 2, for arbitrary power track following, propose three reasonable assumptions, according to described reasonable assumption, design electro-hydraulic Load simulating device ADAPTIVE ROBUST force controller, specifically comprises the following steps that

Step 2-1, for ease of electro-hydraulic load simulator error symbol integration robust Controller Design, turn for arbitrary Square track following, has following 3 reasonable assumptions:

Assume 1: actual hydraulic pressure electro-hydraulic load simulator works in normal conditions, due to P_rAnd P_sImpact, P₁With P₂Meet condition: 0≤P_r＜ P₁＜ P_s, 0≤P_r＜ P₂＜ P_s, i.e. P₁And P₂It is all bounded；

Assume 2: desired torque instruction F_dT () is that single order is continuously differentiable, and instruct F_dT () and first derivative thereof are all It is bounded, motion artifacts y,It is the most all bounded.

Assume 3: parameter uncertainty and Uncertain nonlinear meet following condition:

$\left\{\begin{array}{c}\mathrm{\&theta;}\&Element;{\mathrm{\&Omega;}}_{\mathrm{\&theta;}}=\{\mathrm{\&theta;}:{\mathrm{\&theta;}}_{\min }\&le;\mathrm{\&theta;}\&le;{\mathrm{\&theta;}}_{\max }\}\\ \left|d(t,y,\stackrel{\&CenterDot;}{y})\right|\&le;{\mathrm{\&delta;}}_d(t,y,\stackrel{\&CenterDot;}{y})\end{array}\right.---\left(11\right)$

In formula (11), θ_min=[θ_1min,…,θ_6min]^T, θ_max=[θ_1max,…,θ_6max]^T, Ω_θFor the boundary of parameter θ, δ_dFor The interference function of one bounded.

Step 2-2, for simplifying electro-hydraulic load simulator dynamical equation, it is simple to the design of controller, the unknown constant value of definition Parameter vector θ=[θ₁,θ₂,θ₃,θ₄,θ₅,θ₆]^T, wherein θ₁=β_eG, θ₂=β_e, θ₃=β_eC_t, θ₄=a₁, θ₅=a₂, θ₆=a₃, because of This dynamical equation (9) is write as:

$\stackrel{\&CenterDot;}{F}={\mathrm{\&theta;}}_1{f}_{1}u-{\mathrm{\&theta;}}_2{f}_2-{\mathrm{\&theta;}}_3{f}_3-{\mathrm{\&theta;}}_4{\stackrel{\&CenterDot;}{S}}_f\left(\stackrel{\&CenterDot;}{y}\right)-{\mathrm{\&theta;}}_5{\stackrel{\&CenterDot;}{P}}_f\left(\stackrel{\&CenterDot;}{y}\right)-{\mathrm{\&theta;}}_6\stackrel{\&CenterDot;\&CenterDot;}{y}-d(t,y,\stackrel{\&CenterDot;}{y})---\left(12\right)$

Nonlinear function f in formula (12)₁,f₂,f₃It is defined as follows:

$\left\{\begin{array}{c}{f}_1(P_1,P_2,y)=A(\frac{R_1}{V_1}+\frac{R_2}{V_2})\\ f_2(y,\stackrel{\&CenterDot;}{y})=A^2\stackrel{\&CenterDot;}{y}(\frac{1}{V_1}+\frac{1}{V_2})\\ f_3(P_1,P_2,y)={\mathrm{AP}}_L(\frac{1}{V_1}+\frac{1}{V_2})\end{array}\right.---\left(13\right)$

Step 2-3, design electro-hydraulic load simulator ADAPTIVE ROBUST force controller, step is as follows:

The discontinuous parameter mapping that parameter adaptive is used first was given: make before the design being controlled deviceRepresent Estimation to system unknown parameter θ,For parameter estimating error, i.e.For guaranteeing the stability of adaptive control laws, Parameter uncertainty based on system is bounded, i.e. assumes 3, and the parameter adaptive being defined as follows discontinuously maps

${\mathrm{Proj}}_{\widehat{\mathrm{\&theta;}}}\left({\mathrm{\&tau;}}_i\right)=\left\{\begin{array}{cc}0& if{\widehat{\mathrm{\&theta;}}}_i={\widehat{\mathrm{\&theta;}}}_{i\max }{\mathrm{and\&tau;}}_i>0\\ 0& if{\widehat{\mathrm{\&theta;}}}_i={\widehat{\mathrm{\&theta;}}}_{i\min }{\mathrm{and\&tau;}}_i<0\\ {\mathrm{\&tau;}}_i& otherwise\end{array}\right.---\left(14\right)$

I=1 in formula (14) ..., 6, τ is parameter adaptive function, and it is concrete to provide it in follow-up controller design Form: be given below parameter adaptive rate:

$\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}={\mathrm{Proj}}_{\widehat{\mathrm{\&theta;}}}\left(\mathrm{\&Gamma;}\mathrm{\&tau;}\right){\mathrm{with\&theta;}}_{\min }\&le;\widehat{\mathrm{\&theta;}}\left(0\right)\&le;{\mathrm{\&theta;}}_{max},---\left(15\right)$

Г in formula > 0 is positive definite diagonal matrix；For arbitrary auto-adaptive function τ, discontinuous map (14) have following property Matter:

$\begin{array}{cc}\left(P1\right)& \widehat{\mathrm{\&theta;}}\&Element;{\mathrm{\&Omega;}}_{\mathrm{\&theta;}}=\{\widehat{\mathrm{\&theta;}}:{\mathrm{\&theta;}}_{\min }\&le;\widehat{\mathrm{\&theta;}}\&le;{\mathrm{\&theta;}}_{max}\}\end{array}---\left(16\right)$

$\begin{array}{ccc}\left(P2\right)& {\widetilde{\mathrm{\&theta;}}}^T\left({\mathrm{\&Gamma;}}^{-1}{\mathrm{Proj}}_{\widehat{\mathrm{\&theta;}}}\right(\mathrm{\&Gamma;}\mathrm{\&tau;})-\mathrm{\&tau;})\&le;0& \&ForAll;\mathrm{\&tau;}\end{array}---\left(17\right)$

Definition z=F-F_dFor the tracking error of system, the derivative of its time can be write as:

$\stackrel{\&CenterDot;}{z}=\stackrel{\&CenterDot;}{F}-{\stackrel{\&CenterDot;}{F}}_d={\mathrm{\&theta;}}_1{f}_{1}u-{\mathrm{\&theta;}}_2{f}_2-{\mathrm{\&theta;}}_3{f}_3-{\mathrm{\&theta;}}_4{\stackrel{\&CenterDot;}{S}}_f\left(\stackrel{\&CenterDot;}{y}\right)-{\mathrm{\&theta;}}_5{\stackrel{\&CenterDot;}{P}}_f\left(\stackrel{\&CenterDot;}{y}\right)-{\mathrm{\&theta;}}_6\stackrel{\&CenterDot;\&CenterDot;}{y}-d(t,y,\stackrel{\&CenterDot;}{y})-{\stackrel{\&CenterDot;}{F}}_d---\left(18\right)$

According to formula (18), System design based on model device u may be designed as:

U=u_m+u_r

$u_m=\frac{1}{{\widehat{\mathrm{\&theta;}}}_1{f}_1}\&lsqb;{\widehat{\mathrm{\&theta;}}}_2{f}_2+{\widehat{\mathrm{\&theta;}}}_3{f}_3+{\widehat{\mathrm{\&theta;}}}_4{\stackrel{\&CenterDot;}{S}}_f+{\widehat{\mathrm{\&theta;}}}_5{\stackrel{\&CenterDot;}{P}}_f+{\widehat{\mathrm{\&theta;}}}_6\stackrel{\&CenterDot;\&CenterDot;}{y}+{\stackrel{\&CenterDot;}{F}}_d\&rsqb;$

u_r=(-kz+u_s)/f₁ (19)

U in formula_mIt it is the adaptive model compensation term of the on-line parameter adaptive law be given by formula (15)；K is positive anti- Feedforward gain, u_rFor Robust Control Law, u_sIt is that non linear robust item is for overcoming the model uncertainty impact on tracking performance.

The Lyapunov function being defined as follows:

$V\left(t\right)=\frac{1}{2}{z}^2---\left(20\right)$

Based on controller (19), function V time differential is:

Formula returns deviceIt is defined as:

For robust item u_sDesign, need to meet following condition:

eu_s≤0 (23)

In formula, ξ represents given robust precision, and be can be arbitrarily small positive design parameter.

Function h is made to meet following condition:

θ in formula (24)_M=θ_max-θ_min, u_sCan be chosen for:

$u_s=-k_{m}z=-\frac{h}{2{\mathrm{\&theta;}}_{1\min }\mathrm{\&xi;}}z---\left(25\right)$

K in formula (25)_mFor positive non-linear gain, now u_sMeet condition (23).

Step 3, the performance evaluation of ADAPTIVE ROBUST force controller, specific as follows:

Controller performance: use and discontinuously map adaptive law (15) and auto-adaptive functionThe self adaptation proposed Robust Control Law (19) can ensure following performance:

A. in closed signal, all signals are all bounded, and positive definite integral form V meets such as lower inequality:

$V\left(t\right)\&le;\exp (-\mathrm{\&lambda;}t)V\left(0\right)+\frac{\mathrm{\&xi;}}{\mathrm{\&lambda;}}\&lsqb;1-\exp (-\mathrm{\&lambda;}t)\&rsqb;---\left(26\right)$

λ=2 θ in formula (26)_1minK is exponential convergence rate.

If the most at a time t₀Afterwards, system only exists parameter uncertainty, i.e. d=0, then except conclusion A it Outward, the performance of asymptotic tracking also can be obtained, i.e. as t → ∞, e → 0；

Stability analysis: choose following liapunov function, uses Lyapunov stability theory to carry out stable Property analyze:

$V_s=\frac{1}{2}{z}^2+\frac{1}{2}{\widetilde{\mathrm{\&theta;}}}^T{\mathrm{\&Gamma;}}^{-1}\widetilde{\mathrm{\&theta;}}---\left(27\right)$

Owing to unknown parameter θ is constant value, so that

$\stackrel{\&CenterDot;}{\widetilde{\mathrm{\&theta;}}}=\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}-\stackrel{\&CenterDot;}{\mathrm{\&theta;}}=\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}---\left(28\right)$

Therefore function V_sTo the derivative of time it is:

${\stackrel{\&CenterDot;}{V}}_s=z\stackrel{\&CenterDot;}{z}+{\widetilde{\mathrm{\&theta;}}}^T{\mathrm{\&Gamma;}}^{-1}\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}---\left(29\right)$

Can be obtained by formula (21) and condition (23):

In conjunction with the Property P 2 in the definition of τ and formula (17), can obtain:

${\stackrel{\&CenterDot;}{V}}_s\&le;-{\mathrm{\&theta;}}_1{\mathrm{kz}}^2=-W---\left(31\right)$

This means Vs≤Vs (0).Therefore W ∈ L2 and Vs ∈ L ∞, due to all signals all bounded, easy according to formula (18) Know that W is bounded, therefore W congruous continuity.Using Barbalat lemma to understand, as t → ∞, W → 0, this implies conclusion B.

Embodiment:

Hydraulic cylinder power controls load simulator parameter:

A=2 × 10^-4m³/rad,β_e=2 × 10⁸Pa,C_t=9 × 10^-12m⁵/ (N s), P_s=21 × 10⁶Pa, P_r=0Pa, V₀₁=V₀₂=1.7 × 10^-4m³, a₃=80N m s/rad, J=0.32kg m²,a₁=5 ×10^-4,a₂=3.5 × 10^-4,c₁=15, c₂=1.5, c₃=900

Controller parameter is chosen for: feedback oscillator K=k+k_m=100, adaptive gain Г=diag{7.3 × 10^-5,1× 10¹¹,3×10^-11,5×10^-4,2×10^-4, 30}, the sampling time of emulation is 0.2ms.Outside system time-varying, interference is chosen for d= 200sint, movement locus isThe torque command that system expectation is followed the tracks of is curve

Control law action effect:

Fig. 3 is that under embodiment middle controller effect, system controls input u time history plot, can from figure Going out, obtained controls the signal that input is low frequency and continuous, is more conducive to execution in actual applications.

Fig. 4～Fig. 9 is system under the electro-hydraulic load simulator ADAPTIVE ROBUST force controller effect designed by the present invention The time dependent exemplary curve of estimates of parameters, it can be seen that the partial parameters of system is estimated under controller action True value can preferably be restrained by meter.

Figure 10 is system output under the electro-hydraulic load simulator ADAPTIVE ROBUST force controller effect designed by the present invention With desired output time history plot.

Figure 11 be under the electro-hydraulic load simulator ADAPTIVE ROBUST force controller effect designed by the present invention system with Track error time history plot.

In conjunction with Figure 10 and Figure 11, it can be seen that tracking error is boundedly convergent, and this boundary is relative to expectation instruction Amplitude for be the least.From upper figure, it is uncertain that the algorithm that the present invention proposes can process model under simulated environment Property, compared to traditional PID control, the controller of present invention design can greatly improve and there is parameter uncertainty and uncertain The control accuracy of property nonlinear system.Result of study shows, under the influence of Uncertain nonlinear and parameter uncertainty, to carry herein The method gone out disclosure satisfy that performance indications.

Claims (4)

1. an electro-hydraulic load simulator ADAPTIVE ROBUST force control method, it is characterised in that comprise the following steps:

Step 1, based on continuously differentiable friction model, set up the Mathematical Modeling of electro-hydraulic load simulator；

Step 2, for arbitrary power track following, propose three reasonable assumptions, according to described reasonable assumption, design electro-hydraulic load Analogue means ADAPTIVE ROBUST force controller；

Step 3, the performance evaluation of ADAPTIVE ROBUST force controller.

Electro-hydraulic load simulator ADAPTIVE ROBUST force control method the most according to claim 1, it is characterised in that step In 1, based on continuously differentiable friction model, setting up the Mathematical Modeling of electro-hydraulic load simulator, concrete grammar is as follows:

Step 1-1, foundation continuously differentiable friction model based on tanh approximation

$F_f\left(\stackrel{\&CenterDot;}{y}\right)=a_1\&lsqb;\tanh \left(c_1\stackrel{\&CenterDot;}{y}\right)-\tanh \left(c_2\stackrel{\&CenterDot;}{y}\right)\&rsqb;+a_2\tanh \left(c_3\stackrel{\&CenterDot;}{y}\right)+a_3\stackrel{\&CenterDot;}{y}---\left(1\right)$

In formula (1), a₁,a₂,a₃Represent the amplification level of differentiated friction characteristic, c respectively₁,c₂,c₃It is and characterizes frictional behavior Form factor,Characterize movement velocity；Tanh represents hyperbolic tangent function；

Step 1-2, set up the kinetics equation of electro-hydraulic load simulator:

$\begin{array}{c}F={\mathrm{AP}}_L-F(t,y,\stackrel{\&CenterDot;}{y})\\ F(t,y,\stackrel{\&CenterDot;}{y})=F_f\left(\stackrel{\&CenterDot;}{y}\right)+f(t,y,\stackrel{\&CenterDot;}{y})\end{array}---\left(2\right)$

In formula (2), F is output torque, and A is the discharge capacity of load hydraulic cylinder, hydraulic cylinder load pressure P_L=P₁-P₂, P₁For hydraulic pressure The pressure of cylinder oil suction chamber, P₂Go out the pressure of oil pocket for hydraulic cylinder, y is the position output that steering wheel produces,For uncertain non- Linear term,For non-linear friction,For Unmarried pregnancy and outer interference；

Therefore formula (2) can be write as:

$F={\mathrm{AP}}_L-a_1\&lsqb;\tanh \left(c_1\stackrel{\&CenterDot;}{y}\right)-\tanh \left(c_2\stackrel{\&CenterDot;}{y}\right)\&rsqb;-a_2\tanh \left(c_3\stackrel{\&CenterDot;}{y}\right)-a_3\stackrel{\&CenterDot;}{y}-f(t,y,\stackrel{\&CenterDot;}{y})---\left(3\right)$

OrderThen have:

$F={\mathrm{AP}}_L-a_1{S}_f\left(\stackrel{\&CenterDot;}{y}\right)-a_2{P}_f\left(\stackrel{\&CenterDot;}{y}\right)-a_3\stackrel{\&CenterDot;}{y}-f(t,y,\stackrel{\&CenterDot;}{y})---\left(4\right)$

Step 1-3, set up hydraulic cylinder oil suction chamber and go out the Pressure behaviour equation of oil pocket:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{P}}_1=\frac{{\mathrm{\&beta;}}_e}{V_1}(-A\stackrel{\&CenterDot;}{y}-C_t{P}_L+Q_1)\\ {\stackrel{\&CenterDot;}{P}}_2=\frac{{\mathrm{\&beta;}}_e}{V_2}(A\stackrel{\&CenterDot;}{y}+C_t{P}_L-Q_2)\end{array}\right.---\left(5\right)$

In formula (5), β_eFor the effective bulk modulus of hydraulic oil, the control volume V of oil suction chamber₁=V₀₁+ Ay, V₀₁For oil suction chamber Original volume, goes out the control volume V of oil pocket₂=V₀₂-Ay, V₀₂For going out the original volume of oil pocket, C_tInterior leakage system for hydraulic cylinder Number, Q₁For the flow of oil suction chamber, Q₂Flow for oil back chamber；

Q₁、Q₂With valve core of servo valve displacement x_vThere is a following relation:

$\left\{\begin{array}{c}{Q}_1=k_q{x}_v\&lsqb;s\left(x_v\right)\sqrt{P_s-P_1}+s(-x_v)\sqrt{P_1-P_r}\&rsqb;\\ Q_2=k_q{x}_v\&lsqb;s\left(x_v\right)\sqrt{P_2-P_r}+s(-x_v)\sqrt{P_s-P_2}\&rsqb;\end{array}\right.---\left(6\right)$

In formula (6), valve parameterC_dFor servo valve discharge coefficient for orifices, w₀For servo valve throttle hole area Gradient, P_sFor electro-hydraulic load simulator charge oil pressure, P_rFor electro-hydraulic load simulator return pressure, ρ is the close of hydraulic oil Degree, x_vFor spool displacement, s (x_v) it is sign function, and described sign function is defined as:

$s\left(x_v\right)=\left\{\begin{array}{cc}1& x_v\&GreaterEqual;0\\ 0& x_v<0\end{array}\right.---\left(7\right)$

Ignore the dynamic of valve core of servo valve, it is assumed that act on the control input u and spool displacement x of spool_vProportional relation, the fullest Foot x_v=k_lU, wherein k_lFor voltage-spool displacement gain coefficient, u is input voltage；

Therefore, formula (6) is written as

$\left\{\begin{array}{c}{Q}_1=gu\&lsqb;s\left(x_v\right)\sqrt{P_s-P_1}+s(-x_v)\sqrt{P_1-P_r}\&rsqb;\\ Q_2=gu\&lsqb;s\left(x_v\right)\sqrt{P_2-P_r}+s(-x_v)\sqrt{P_s-P_2}\&rsqb;\end{array}\right.---\left(8\right)$

The most total servo valve gain coefficient g=k_qk_l；

Based on formula (4), (5), (8), the power output dynamical equation of electro-hydraulic load simulator, the i.e. number of electro-hydraulic load simulator Model is:

$\begin{array}{c}\stackrel{\&CenterDot;}{F}=(\frac{R_1}{V_1}+\frac{R_2}{V_2}){\mathrm{A\&beta;}}_{e}gu-(\frac{1}{V_1}+\frac{1}{V_2}){\mathrm{\&beta;}}_e{A}^2\stackrel{\&CenterDot;}{y}-(\frac{1}{V_1}+\frac{1}{V_2}){\mathrm{A\&beta;}}_e{C}_t{P}_L\\ -a_1{\stackrel{\&CenterDot;}{S}}_f\left(\stackrel{\&CenterDot;}{y}\right)-a_2{\stackrel{\&CenterDot;}{P}}_f\left(\stackrel{\&CenterDot;}{y}\right)-a_3\stackrel{\&CenterDot;\&CenterDot;}{y}-d(t,y,\stackrel{\&CenterDot;}{y})\end{array}---\left(9\right)$

(9) in formula, the model uncertainty of electro-hydraulic load simulatorR₁And R₂It is defined as follows:

$\left\{\begin{array}{c}{R}_1=s\left(u\right)\sqrt{P_s-P_1}+s(-u)\sqrt{P_1-P_r}\\ R_2=s\left(u\right)\sqrt{P_2-P_r}+s(-u)\sqrt{P_s-P_2}\end{array}\right.---\left(10\right)$

R is understood by formula (10)₁＞ 0, R₂＞ 0, R₁And R₂It is intermediate variable.

Electro-hydraulic load simulator ADAPTIVE ROBUST force control method the most according to claim 1, it is characterised in that step For arbitrary power track following in 2, propose three reasonable assumptions, according to described reasonable assumption, design electro-hydraulic load simulation dress Put ADAPTIVE ROBUST force controller, specifically comprise the following steps that

Step 2-1, for ease of electro-hydraulic load simulator error symbol integration robust Controller Design, for arbitrary torque rail Mark is followed the tracks of, and has following 3 reasonable assumptions:

Assume 1: actual hydraulic pressure electro-hydraulic load simulator works in normal conditions, due to P_rAnd P_sImpact, P₁And P₂Full Foot condition: 0≤P_r＜ P₁＜ P_s, 0≤P_r＜ P₂＜ P_s, i.e. P₁And P₂It is all bounded；

Assume 2: desired torque instruction F_dT () is that single order is continuously differentiable, and instruct F_dT () and first derivative thereof all have Boundary, motion artifacts y,It is the most all bounded；

Assume 3: parameter uncertainty and Uncertain nonlinear meet following condition:

$\left\{\begin{array}{c}\mathrm{\&theta;}\&Element;{\mathrm{\&Omega;}}_{\mathrm{\&theta;}}=\{\mathrm{\&theta;}:{\mathrm{\&theta;}}_{\min }\&le;\mathrm{\&theta;}\&le;{\mathrm{\&theta;}}_{\max }\}\\ \left|d(t,y,\stackrel{\&CenterDot;}{y})\right|\&le;{\mathrm{\&delta;}}_d(t,y,\stackrel{\&CenterDot;}{y})\end{array}\right.---\left(11\right)$

In formula (11), θ_min=[θ_1min,…,θ_6min]^T, θ_max=[θ_1max,…,θ_6max]^T, Ω_θFor the boundary of parameter θ, δ_dIt is one to have The interference function on boundary；

Step 2-2, for simplifying electro-hydraulic load simulator dynamical equation, it is simple to the design of controller, the unknown constant parameter of definition Vector theta=[θ₁,θ₂,θ₃,θ₄,θ₅,θ₆]^T, wherein θ₁=β_eG, θ₂=β_e, θ₃=β_eC_t, θ₄=a₁, θ₅=a₂, θ₆=a₃, therefore move State equation (9) is write as

$\stackrel{\&CenterDot;}{F}={\mathrm{\&theta;}}_1{f}_{1}u-{\mathrm{\&theta;}}_2{f}_2-{\mathrm{\&theta;}}_3{f}_3-{\mathrm{\&theta;}}_4{\stackrel{\&CenterDot;}{S}}_f\left(\stackrel{\&CenterDot;}{y}\right)-{\mathrm{\&theta;}}_5{\stackrel{\&CenterDot;}{P}}_f\left(\stackrel{\&CenterDot;}{y}\right)-{\mathrm{\&theta;}}_6\stackrel{\&CenterDot;\&CenterDot;}{y}-d(t,y,\stackrel{\&CenterDot;}{y})---\left(12\right)$

Nonlinear function f in formula (12)₁,f₂,f₃It is defined as follows:

$\left\{\begin{array}{c}{f}_1(P_1,P_2,y)=A(\frac{R_1}{V_1}+\frac{R_2}{V_2})\\ f_2(y,\stackrel{\&CenterDot;}{y})=A^2\stackrel{\&CenterDot;}{y}(\frac{1}{V_1}+\frac{1}{V_2})\\ f_3(P_1,P_2,y)={\mathrm{AP}}_L(\frac{1}{V_1}+\frac{1}{V_2})\end{array}\right.---\left(13\right)$

Step 2-3, design electro-hydraulic load simulator ADAPTIVE ROBUST force controller, step is as follows:

The discontinuous parameter mapping that parameter adaptive is used first was given: make before the design being controlled deviceRepresent being The estimation of system unknown parameter θ,For parameter estimating error, i.e.For guaranteeing the stability of adaptive control laws, based on The parameter uncertainty of system is bounded, i.e. assumes 3, and the parameter adaptive being defined as follows discontinuously maps

${\mathrm{Proj}}_{\widehat{\mathrm{\&theta;}}}\left({\mathrm{\&tau;}}_i\right)=\left\{\begin{array}{cc}0& if{\widehat{\mathrm{\&theta;}}}_i={\widehat{\mathrm{\&theta;}}}_{i\max }{\mathrm{and\&tau;}}_i>0\\ 0& if{\widehat{\mathrm{\&theta;}}}_i={\widehat{\mathrm{\&theta;}}}_{i\min }{\mathrm{and\&tau;}}_i<0\\ {\mathrm{\&tau;}}_i& otherwise\end{array}\right.---\left(14\right)$

I=1 in formula (14) ..., 6, τ is parameter adaptive function, and provides its concrete shape in follow-up controller design Formula: be given below parameter adaptive rate:

$\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}={\mathrm{Proj}}_{\widehat{\mathrm{\&theta;}}}\left(\mathrm{\&Gamma;}\mathrm{\&tau;}\right){\mathrm{with\&theta;}}_{\min }\&le;\widehat{\mathrm{\&theta;}}\left(0\right)\&le;{\mathrm{\&theta;}}_{max},---\left(15\right)$

Г in formula > 0 is positive definite diagonal matrix；For arbitrary auto-adaptive function τ, discontinuous map (14) have the property that

$\begin{array}{cc}\left(P1\right)& \widehat{\mathrm{\&theta;}}\&Element;{\mathrm{\&Omega;}}_{\mathrm{\&theta;}}=\{\widehat{\mathrm{\&theta;}}:{\mathrm{\&theta;}}_{\min }\&le;\widehat{\mathrm{\&theta;}}\&le;{\mathrm{\&theta;}}_{max}\}\end{array}---\left(16\right)$

$\begin{array}{ccc}\left(P2\right)& {\widetilde{\mathrm{\&theta;}}}^T\left({\mathrm{\&Gamma;}}^{-1}{\mathrm{Proj}}_{\widehat{\mathrm{\&theta;}}}\right(\mathrm{\&Gamma;}\mathrm{\&tau;})-\mathrm{\&tau;})\&le;0& \&ForAll;\mathrm{\&tau;}\end{array}---\left(17\right)$

Definition z=F-F_dFor the tracking error of system, the derivative of its time can be write as:

$\stackrel{\&CenterDot;}{z}=\stackrel{\&CenterDot;}{F}-{\stackrel{\&CenterDot;}{F}}_d={\mathrm{\&theta;}}_1{f}_{1}u-{\mathrm{\&theta;}}_2{f}_2-{\mathrm{\&theta;}}_3{f}_3-{\mathrm{\&theta;}}_4{\stackrel{\&CenterDot;}{S}}_f\left(\stackrel{\&CenterDot;}{y}\right)-{\mathrm{\&theta;}}_5{\stackrel{\&CenterDot;}{P}}_f\left(\stackrel{\&CenterDot;}{y}\right)-{\mathrm{\&theta;}}_6\stackrel{\&CenterDot;\&CenterDot;}{y}-d(t,y,\stackrel{\&CenterDot;}{y})-{\stackrel{\&CenterDot;}{F}}_d---\left(18\right)$

According to formula (18), System design based on model device u may be designed as:

U=u_m+u_r

$u_m=\frac{1}{{\widehat{\mathrm{\&theta;}}}_1{f}_1}\&lsqb;{\widehat{\mathrm{\&theta;}}}_2{f}_2+{\widehat{\mathrm{\&theta;}}}_3{f}_3+{\widehat{\mathrm{\&theta;}}}_4{\stackrel{\&CenterDot;}{S}}_f+{\widehat{\mathrm{\&theta;}}}_5{\stackrel{\&CenterDot;}{P}}_f+{\widehat{\mathrm{\&theta;}}}_6\stackrel{\&CenterDot;\&CenterDot;}{y}+{\stackrel{\&CenterDot;}{F}}_d\&rsqb;$

u_r=(-kz+u_s)/f₁ (19)

U in formula_mIt it is the adaptive model compensation term of the on-line parameter adaptive law be given by formula (15)；K is that positive feedback increases Benefit, u_rFor Robust Control Law, u_sIt is that non linear robust item is for overcoming the model uncertainty impact on tracking performance；

The Lyapunov function being defined as follows:

$V\left(t\right)=\frac{1}{2}{z}^2---\left(20\right)$

Based on controller (19), function V time differential is:

Formula returns deviceIt is defined as:For robust item u_sDesign, Need to meet following condition:

eu_s≤0 (23)

In formula, ξ represents given robust precision, and be can be arbitrarily small positive design parameter；

Function h is made to meet following condition:

θ in formula (24)_M=θ_max-θ_min, u_sCan be chosen for:

$u_s=-k_{m}z=-\frac{h}{2{\mathrm{\&theta;}}_{1\min }\mathrm{\&xi;}}z---\left(25\right)$

K in formula (25)_mFor positive non-linear gain, now u_sMeet condition (23).

Described electro-hydraulic load simulator ADAPTIVE ROBUST force control method the most according to claim 3, its feature exists In, the performance evaluation of ADAPTIVE ROBUST force controller described in step 3, specific as follows:

Controller performance: use and discontinuously map adaptive law (15) and auto-adaptive functionProposed is adaptive

Answer Robust Control Law (19) can ensure following performance:

A. in closed signal, all signals are all bounded, and positive definite integral form V (t) meets such as lower inequality:

$V\left(t\right)\&le;\exp (-\mathrm{\&lambda;}t)V\left(0\right)+\frac{\mathrm{\&xi;}}{\mathrm{\&lambda;}}\&lsqb;1-\exp (-\mathrm{\&lambda;}t)\&rsqb;---\left(26\right)$

λ=2 θ in formula (26)_1minK is exponential convergence rate；

If the most at a time t₀Afterwards, system only exists parameter uncertainty, i.e. d=0, then remove

Outside conclusion A, also can obtain the performance of asymptotic tracking, i.e. as t → ∞, e → 0；

Stability analysis: choose following liapunov function, uses Lyapunov stability theory to carry out analysis of stability Analysis:

$V_s=\frac{1}{2}{z}^2+\frac{1}{2}{\widetilde{\mathrm{\&theta;}}}^T{\mathrm{\&Gamma;}}^{-1}\widetilde{\mathrm{\&theta;}}---\left(27\right)$

And use Barbalat lemma can obtain the globally asymptotically stable result of system, therefore regulation parameter k, ξ and Γ can make be System tracking error the time tend to infinite under conditions of go to zero.