CN105700352B - A kind of electrohydraulic load simulator error symbol integral robust control method - Google Patents

A kind of electrohydraulic load simulator error symbol integral robust control method Download PDF

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CN105700352B
CN105700352B CN201610053083.XA CN201610053083A CN105700352B CN 105700352 B CN105700352 B CN 105700352B CN 201610053083 A CN201610053083 A CN 201610053083A CN 105700352 B CN105700352 B CN 105700352B
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load simulator
electrohydraulic load
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error
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CN105700352A (en
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岳欣
姚建勇
仇杨
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Nanjing University of Science and Technology
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Nanjing University of Science and Technology
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    • GPHYSICS
    • G05CONTROLLING; REGULATING
    • G05BCONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators

Abstract

The invention discloses a kind of electrohydraulic load simulator error symbols to integrate robust control method, for the system features of electrohydraulic load simulator, the frictional behavior of the system is analyzed by Friction identification, establish continuously differentiable friction model, by parameter error, modeling error, Unmarried pregnancy and it is outer interference be included into systematic uncertainty it is non-linear in, designed error symbol integral robust controller also achieves the asymptotically stability performance of system under conditions of not using High Gain Feedback, with good robust effect, and it can guarantee that the output torque of electrohydraulic load simulator accurately tracks the torque instruction being arbitrarily designated;This invention simplifies controller designs, and the control voltage of controller is continuous, are conducive to apply in practice in engineering.

Description

A kind of electrohydraulic load simulator error symbol integral robust control method
Technical field
The invention belongs to electro-hydraulic servo control field, especially a kind of electrohydraulic load simulator error symbol integrates robust control Method processed is based on continuously differentiable friction model.
Background technique
The performance of precision strike weapon is to determine the most important factor of modern war victory or defeat, posture, track and the side of weapon To control be it is crucial, this process is that the inertia device or guiding on weapon receive cable system and experience target position, then in Centre control computer calculates control instruction, then control output.All these control outputs will implement to final servo In executing agency's (steering engine).Therefore, full side of the property relationship of executing agency to national defense industry such as aviation, aerospace, naval vessel, cannons Position development, while also widely being paid attention in the application of civilian industry.It determines that modern precision guided weapon is entirely controlled greatly Structure, the layout, the even more key factor of weapon control dynamic characteristic of system processed.
The typical case of load simulator is loaded to the steering engine position servomechanism of aircraft, in ground simulation rudder Face aerodynamic load suffered in flight course, to constitute the HWIL simulation of flight control system.Simulation computer is embedded According to the aircraft six degrees of freedom model that wind tunnel data and Aviate equation are established, according to flying height, speed, rudder face corner and The related physical quantity such as atmosphere data calculates the Aerodynamic binge moment load in entire flight course in real time, forms torque and carries Lotus spectrum.The main function of load simulator is exactly the load instruction of real-time reception simulation computer, and it is accurately applied to In steering engine servo mechanism.Load simulator, can be to week product life-cycle of steering engine development as a kind of test and emulator Phase all plays an important role, it has run through optimization design, performance test and the calibration and fault diagnosis of steering engine.So load mould The design requirement of quasi- device is usually very high, especially precision and dynamic characteristic.
It is directed to the Advanced Control Strategies of electrohydraulic servo system at present, there is feedback linearization, sliding formwork and ADAPTIVE ROBUST etc. Control method.Modified feedback linearization control method not only designs simply, but also can guarantee the high-performance of system, but it requires institute The system mathematic model of foundation must be very accurate, this is difficult to be guaranteed in practical applications.Sliding-mode control is simply real There is certain robustness with and to the outer interference etc. of system, but the method based on general sliding formwork control can cause trembling for sliding-mode surface It is dynamic, keep designed controller discontinuous, to make the penalty of system, is unfavorable for applying in practice in engineering.Adaptively Robust control is but easy to be interfered by the noise in system mode, and the precision of its parameter Estimation is also not achieved in certain occasions It is required that the output tracking performance of indirect self-adaptive is not although this can be solved by using the method for indirect self-adaptive It is ideal.In summary: traditional control method is difficult to meet the tracking accuracy requirement of Uncertain nonlinear;And advanced control in recent years Strategy controller design processed is more complicated, is not easy to Project Realization.
Summary of the invention
The purpose of the present invention is to provide a kind of electrohydraulic load simulator error symbols to integrate robust control method, solves There are designed by ignored model uncertainty, the control method based on traditional sliding formwork in existing electrohydraulic load simulator Controller is discontinuous, there are problems that High Gain Feedback phenomenon based on general adaptive robust control method.
The technical solution adopted by the present invention to solve the above problem is as follows: a kind of electrohydraulic load simulator error symbol integrates Shandong Stick control method, comprising the following steps:
Step 1 is based on continuously differentiable friction model, establishes the mathematical model of electrohydraulic load simulator;
Step 1-1, it establishes and is based on the approximate continuously differentiable friction model of tanh
In formula (1), a1,a2Respectively indicate the amplification level of differentiated friction characteristic, c1,c2,c3For characterization frictional behavior Form factor, w characterize movement velocity;Tanh indicates hyperbolic tangent function.
Step 1-2, the output torque dynamical equation of electrohydraulic load simulator is established:
In formula (2), T is output torque, and A is negative the discharge capacity of carrier fluid pressure motor, hydraulic motor load pressure PL=P1-P2, P1For the pressure of motor oil suction chamber, P2For the pressure of motor oil outlet chamber, B is total viscous damping coefficient, y electrohydraulic load simulator Position,For the speed of electrohydraulic load simulator,The w being equivalent in continuously differentiable friction model;For external disturbance ?;For static friction;For Unmarried pregnancy.
Therefore formula (1) is write as:
It enablesThen have:
Step 1-3, the Pressure behaviour equation of motor oil suction chamber and oil outlet chamber is established:
In formula (5), βeFor the effective bulk modulus of hydraulic oil, the control volume V of oil suction chamber1=V01+ Ay, V01For oil inlet The original volume of chamber, the control volume V of oil outlet chamber2=V02- Ay, V02For the original volume of oil outlet chamber, CtFor the interior leakage of motor Coefficient, Q1For the flow of oil suction chamber, Q2For the flow of oil back chamber.Q1、Q2With valve core of servo valve displacement xvThere is following relationship:
In formula (6), valve parameterCdFor servo valve discharge coefficient for orifices, w0For servo valve throttle orifice Area gradient, PsFor electrohydraulic load simulator charge oil pressure, PrFor electrohydraulic load simulator return pressure, ρ is the close of hydraulic oil Degree, xvFor spool displacement, s (xv) it is sign function, and the sign function is defined as:
Ignore the dynamic of valve core of servo valve, it is assumed that act on the control input u and spool displacement x of spoolvProportional relationship, Meet xv=klU, wherein klFor voltage-spool displacement gain coefficient, u is input voltage.Therefore, formula (6) is written as
Wherein total servo valve gain coefficient g=kqkl
Based on formula (4), (5), (8), output torque controls the dynamical equation of electrohydraulic load simulator, i.e., electro-hydraulic load simulation The mathematical model of device are as follows:
(9) in formula, the model uncertainty of electrohydraulic load simulatorR1And R2Definition such as Under:
The R known to formula (10)1> 0, R2> 0, R1And R2It is intermediate variable.
Step 2, for arbitrary torque track following, propose three reasonable assumptions, according to the reasonable assumption, design electricity Liquid load simulator error symbol integrates robust controller, and steps are as follows:
Step 2-1, to integrate robust Controller Design convenient for electrohydraulic load simulator error symbol, for arbitrary torque Track following has following 3 reasonable assumptions:
Assuming that 1: actual hydraulic electrohydraulic load simulator works in normal conditions, due to PrAnd PsInfluence, P1And P2 Meet condition: 0≤Pr<P1<Ps, 0≤Pr<P2<Ps, i.e. P1And P2It is all bounded.
Assuming that 2: desired torque instruction TdIt (t) is that single order is continuously differentiable, and instructs Td(t) and its first derivative all It is bounded, motion artifacts y,It also is all bounded.
Assuming that 3: uncertain non-linearThere are 2 order derivatives, and 1 rank, the equal bounded of 2 order derivatives, that is, have following formula at It is vertical:
In formula (11), σ1, σ2It is all known constant.
Step 2-2, to simplify electrohydraulic load simulator equation, convenient for the design of controller, unknown constant parameter arrow is defined Measure θ=[θ123456]T, wherein θ1eG, θ2e, θ3eCt, θ4=B, θ5=a1, θ6=a2, therefore dynamic Equation (9) is write as
U=θ in formula (12)1f1U, parametric function f1,f2,f3It is defined as follows:
According to formula (13), actual control inputs u=U/ θ1f1, therefore, only need design error symbolic integration robust control Device U carrys out processing parameter uncertainty and uncertainty is non-linear.
Step 2-3, design electrohydraulic load simulator error symbol integrates robust controller:
It is defined as follows error variance:
In formula: TdFor torque trace command;z1For electrohydraulic load simulator tracking error;R is the auxiliary margin of error;k1It is positive Feedback oscillator.
From (14) formula:
It is as follows to design electrohydraulic load simulator error symbol integral robust controller:
UaIndicate model compensation controller;krThe feedback oscillator being positive;Us1It is linear feedback item;Us2It is non linear robust , for overcoming influence of the model uncertainty to tracking performance.
(16) formula is substituted into (15) Shi Ke get:
(18) in formula, robust gain beta > 0, sign (z1) it is about z1Standard signum function.
It is transferred to step 3.
Step 3 integrates robust to designed electrohydraulic load simulator error symbol with Lyapunov stability theory Controller, which carries out stability, to be proved, it was demonstrated that process is as follows:
To above-mentioned (17) formula derivation, obtain:
Before the performance that designed controller is presented, following lemma is provided:
Lemma 1: defined variable L (t) and auxiliary function P (t) are as follows:
If robust gain beta meets such as lower inequality:
It is positive value that then auxiliary function P (t) is permanent;
By lemma 1 it is found that the differential of auxiliary function P (t) are as follows:
Theorem 1: for the mathematical model of electrohydraulic load simulator, if the error symbol as shown in formula (16) integrates Shandong The robust gain beta of stick controller meets inequality (22), and its feedback oscillator k1,krIt is sufficiently large, so that such as undefined matrix Λ For positive definite matrix:
Then all equal boundeds of signal in closed loop electrohydraulic load simulator, and controller obtain Asymptotic Stability, i.e., when t → When ∞, z → 0, wherein error vector z is defined as z=[z1,r];
It proves: being defined as follows non-negative function V
Its time diffusion are as follows:
Wushu (14), (19), (23) are brought into (26) and can obtain:
Z=[z in formula1,r]T, Λ such as formula (24) is the matrix of a positive definite;
Therefore, there is following formula establishment:
λ in formulamin(Λ) is the minimal eigenvalue of matrix Λ, it can thus be appreciated that V bounded, therefore z bounded, therefore closed loop controller All equal boundeds of signal;The W ∈ L known to formula (28) again2, from error dynamics equationTherefore variable W is unanimously to connect Continuous, by Barbalat lemma, thus W → 0 as t → ∞ demonstrates theorem 1, which can guarantee that tracking error is progressive Converge on zero.
The beneficial effects of the present invention are:
(1) system features of electrohydraulic load simulator are directed to, the frictional behavior of the system is analyzed by Friction identification, are established More accurate new type of continuous can micro tribology model, lay the foundation to promote the stability of the system.
(2) by parameter error, modeling error, Unmarried pregnancy and it is outer interference be included into systematic uncertainty it is non-linear in, Designed error symbol integral robust controller is also realized under conditions of not using High Gain Feedback (Sign function) The asymptotically stability performance of system has good robust effect, and can guarantee that the output torque of electrohydraulic load simulator is accurate Track the torque instruction being arbitrarily designated.
(3) controller design is simplified, and the control voltage of controller is continuous, is conducive to apply in practice in engineering, The contrast simulation result verification validity of controller.
Detailed description of the invention
Fig. 1 is the electro-hydraulic load simulation that a kind of electrohydraulic load simulator error symbol of the invention integrates robust control method Device schematic diagram.
Fig. 2 is the control strategy figure that a kind of electrohydraulic load simulator error symbol of the invention integrates robust control method.
Fig. 3 is controller u time history plot in embodiment, controller input voltage satisfaction -10V~+10V Input range meets practical application.
Fig. 4 is the curve graph that torque (i.e. command torque and output torque) is tracked in embodiment.
Fig. 5 is tracking error in embodiment (i.e. command torque and output torque error) curve graph.
Specific embodiment:
Present invention is further described in detail with reference to the accompanying drawing.
In conjunction with Fig. 2, a kind of electrohydraulic load simulator error symbol integral robust control method, electrohydraulic load simulator knot Structure principle is as shown in Figure 1, comprising the following steps:
Step 1 is based on continuously differentiable friction model, establishes the mathematical model of electrohydraulic load simulator:
Step 1-1, it establishes and is based on the approximate continuously differentiable friction model of tanh
In formula (1), a1,a2Respectively indicate the amplification level of differentiated friction characteristic, c1,c2,c3For characterization frictional behavior Form factor, w characterize movement velocity;Tanh indicates hyperbolic tangent function;
Step 1-2, the output torque dynamical equation of electrohydraulic load simulator is established:
In formula (2), T is output torque, and A is negative the discharge capacity of carrier fluid pressure motor, hydraulic motor load pressure PL=P1-P2, P1For the pressure of motor oil suction chamber, P2For the pressure of motor oil outlet chamber, B is total viscous damping coefficient, y electrohydraulic load simulator Position,For the speed of electrohydraulic load simulator,The w being equivalent in continuously differentiable friction model;For external disturbance ?;For static friction;For Unmarried pregnancy.
Therefore formula (1) can be write as:
It enablesThen have:
Step 1-3, the Pressure behaviour equation of motor oil suction chamber and oil outlet chamber is established:
In formula (5), βeFor the effective bulk modulus of hydraulic oil, the control volume V of oil suction chamber1=V01+ Ay, V01For oil inlet The original volume of chamber, the control volume V of oil outlet chamber2=V02- Ay, V02For the original volume of oil outlet chamber, CtFor the interior leakage of motor Coefficient, Q1For the flow of oil suction chamber, Q2For the flow of oil back chamber, Q1、Q2With valve core of servo valve displacement xvThere is following relationship:
In formula (6), valve parameterCdFor servo valve discharge coefficient for orifices, w0For servo valve throttle orifice Area gradient, PsFor electrohydraulic load simulator charge oil pressure, PrFor electrohydraulic load simulator return pressure, ρ is the close of hydraulic oil Degree, xvFor spool displacement, s (xv) it is sign function, and the sign function is defined as:
Due to ignoring the dynamic of valve core of servo valve, it is assumed that act on spool using high performance servo valve Control input u and spool displacement xvProportional relationship, that is, meet xv=klU, wherein klFor voltage-spool displacement gain coefficient, u For input voltage.
Therefore, formula (6) can be written as
Wherein total servo valve gain coefficient g=kqkl
Based on formula (4), (5), (8), output torque controls the dynamical equation of electrohydraulic load simulator, i.e., electro-hydraulic load simulation The mathematical model of device are as follows:
(9) in formula, the model uncertainty of electrohydraulic load simulatorR1And R2Definition such as Under:
The R known to formula (10)1> 0, R2> 0, R1And R2It is intermediate variable.
Step 2, for arbitrary torque track following, propose three reasonable assumptions, according to the reasonable assumption, design electricity Liquid load simulator error symbol integrates robust controller, and steps are as follows:
Step 2-1, to integrate robust Controller Design convenient for electrohydraulic load simulator error symbol, for arbitrary torque Track following has following 3 reasonable assumptions:
Assuming that 1: actual hydraulic electrohydraulic load simulator works in normal conditions, due to PrAnd PsInfluence, P1And P2 Meet condition: 0≤Pr<P1<Ps, 0≤Pr<P2<Ps, i.e. P1And P2It is all bounded.
Assuming that 2: desired torque instruction TdIt (t) is that single order is continuously differentiable, and instructs Td(t) and its first derivative all It is bounded, motion artifacts y,It also is all bounded.
Assuming that 3: uncertain non-linearThere are 2 order derivatives, and 1 rank, the equal bounded of 2 order derivatives, that is, have following formula at It is vertical:
In formula (11), σ1, σ2It is all known constant.
For giving desired torque instruction Td(t), the design object of controller is that the control input u of design bounded makes Output torque T tracks T in the case where uncertain despite the presence of various modelings as much as possibled(t)。
Step 2-2, to simplify electrohydraulic load simulator equation, convenient for the design of controller, unknown constant parameter arrow is defined Measure θ=[θ123456]T, wherein θ1eG, θ2e, θ3eCt, θ4=B, θ5=a1, θ6=a2, therefore dynamic Equation (9) can be write as
U=θ in formula (12)1f1U, parametric function f1,f2,f3It is defined as follows:
A new variable U has been incorporated herein to replace the control of system to input u, because of θ1f1It can calculate in real time, If having obtained the expression formula of new control input U, actual control inputs u=U/ θ1f1.Therefore, design error symbol is only needed Integral robust controller U carrys out processing parameter uncertainty and uncertainty is non-linear.
Step 2-3, design electrohydraulic load simulator error symbol integrates robust controller:
It is defined as follows error variance:
In formula: TdFor torque trace command;z1For electrohydraulic load simulator tracking error;R is the auxiliary margin of error;k1It is positive Feedback oscillator.
From (14) formula:
It is as follows to design electrohydraulic load simulator error symbol integral robust controller:
UaIndicate model compensation controller;krThe feedback oscillator being positive;Us1It is linear feedback item;Us2It is non linear robust , for overcoming influence of the model uncertainty to tracking performance;
(16) formula is substituted into (15) Shi Ke get:
If model uncertainty is not present in system, i.e., it can be seen from (17) formulaNon linear robust Item Us2It need not use, linear feedback item Us1Ensure that nominal systemRealize the property of asymptotically stability Energy.Obviously, engineering is in practice and there is no such systemsTherefore, it is necessary to introduce non linear robust item Us2 To handle the influence of model uncertainty.This design of control method error symbol integrates robust item us2It is as follows:
(18) β > 0 is robust gain in formula.sign(z1) it is about z1Standard signum function, this design of control method Us2Purpose be desirable to by selecting suitable control gain k1,kr, β carrys out the model uncertainty in compensation system, passes through conjunction The stability analysis of reason makes the performance of system realization asymptotically stability.
From formula (16) and (18): in controller and without containing the error symbol r assisted, include is measurable letter Number, therefore controller can execute.In addition, in view of UaUsed in be continuous friction model, sign function is in Us2In It is worked in the form of integral, it can thus be appreciated that final control input is continuously, (to be based on compared to some discontinuous controllers Discontinuous friction model control algolithm and traditional synovial membrane control algolithm) for, the method be more advantageous to engineering in practice Using.
Step 3 integrates robust to designed electrohydraulic load simulator error symbol with Lyapunov stability theory Controller, which carries out stability, to be proved;
To above-mentioned (17) formula derivation, obtain:
Before the performance that designed controller is presented, following lemma is provided:
Lemma 1: defined variable L (t) and auxiliary function P (t) are as follows:
If robust gain beta meets such as lower inequality:
It is positive value that then auxiliary function P (t) is permanent;
By lemma 1 it is found that the differential of auxiliary function P (t) are as follows:
Theorem 1: for nonlinear system (9), if the robust gain beta of error symbol integral robust controller (16) meets Inequality (22) and its feedback oscillator k1,krIt is wide enough so that undefined matrix Λ such as is positive definite matrix:
Then all equal boundeds of signal in closed-loop system, and controller can get Asymptotic Stability, i.e., as t → ∞, z → 0, wherein z is defined as z=[z1,r];
It proves: being defined as follows non-negative function
Its time diffusion are as follows:
Wushu (14), (19), (23) are brought into (26) and can obtain:
Z=[z in formula1,r]T, Λ such as formula (24) is the matrix of a positive definite.
Therefore, there is following formula establishment:
λ in formulamin(Λ) is the minimal eigenvalue of matrix Λ, it can thus be appreciated that V bounded, therefore z bounded, therefore closed loop controller All equal boundeds of signal.The W ∈ L known to formula (28) again2, from error dynamics equationTherefore W is congruous continuity , by Barbalat lemma, thus W → 0 as t → ∞ demonstrates theorem 1, which can guarantee the progressive receipts of tracking error It holds back in zero.
Embodiment:
Twayblade hydraulic motor power controls load simulator parameter are as follows:
A=2 × 10-4m3/ rad, B=80Nms/rad, βe=2 × 108Pa,Ct=9 × 10-12m5/ (Ns),Ps=21 × 106Pa, Pr=0Pa, V01=V02=1.7 × 10-4m3, J=0.32kgm2, a1=3.5 × 10-4,a2=5 × 10-4,c1=700, c2=15, c3=1.5
The controller parameter of design is chosen are as follows: k1=200, kr=0.01, β=0.1.It is interfered outside system time-varying and is chosen for d= 200sint, motion profile areThe torque command of system expectation tracking is curve
Control law function and effect:
Fig. 3 is interference suppressor input voltage u-curve, and control input signal is continuous, and input voltage satisfaction -10V~+ The input range of 10V, meets practical application.
In conjunction with Fig. 4 and Fig. 5, it can be seen that command signal and tracking error curve can be seen that tracking error is bounded convergence , and this boundary is very little for the amplitude of instruction.By Fig. 4 and Fig. 5 it is found that algorithm proposed by the present invention is imitative It is capable of handling model uncertainty under true environment, compared to traditional PID control, the controller that the present invention designs can greatly be mentioned There are the control precision of parameter uncertainty and Uncertain nonlinear systems for height.Result of study show in Uncertain nonlinear and Under the influence of parameter uncertainty, method proposed in this paper can satisfy performance indicator.

Claims (1)

1. a kind of electrohydraulic load simulator error symbol integrates robust control method, which comprises the following steps:
Step 1 is based on continuously differentiable friction model, establishes the mathematical model of electrohydraulic load simulator, method particularly includes:
Step 1-1, it establishes and is based on the approximate continuously differentiable friction model of tanh
In formula (1), a1,a2Respectively indicate the amplification level of differentiated friction characteristic, c1,c2,c3It is the shape for characterizing frictional behavior Shape coefficient, w characterize movement velocity;Tanh indicates hyperbolic tangent function;
Step 1-2, the output torque dynamical equation of electrohydraulic load simulator is established:
In formula (2), T is output torque, and A is negative the discharge capacity of carrier fluid pressure motor, hydraulic motor load pressure PL=P1-P2, P1For The pressure of motor oil suction chamber, P2For the pressure of motor oil outlet chamber, B is total viscous damping coefficient, the position of y electrohydraulic load simulator It sets,For the speed of electrohydraulic load simulator,The w being equivalent in continuously differentiable friction model;For external disturbance ?;For static friction;For Unmarried pregnancy;
Therefore formula (1) is write as:
It enablesThen have:
Step 1-3, the Pressure behaviour equation of motor oil suction chamber and oil outlet chamber is established:
In formula (5), βeFor the effective bulk modulus of hydraulic oil, the control volume V of oil suction chamber1=V01+ Ay, V01For oil suction chamber Original volume, the control volume V of oil outlet chamber2=V02- Ay, V02For the original volume of oil outlet chamber, CtFor the interior leakage coefficient of motor, Q1For the flow of oil suction chamber, Q2For the flow of oil back chamber;
Q1、Q2With valve core of servo valve displacement xvThere is following relationship:
In formula (6), valve parameterCdFor servo valve discharge coefficient for orifices, w0For servo valve throttle hole area Gradient, PsFor electrohydraulic load simulator charge oil pressure, PrFor electrohydraulic load simulator return pressure, ρ is the density of hydraulic oil, xv For spool displacement, s (xv) it is sign function, and the sign function is defined as:
Ignore the dynamic of valve core of servo valve, it is assumed that act on the control input u and spool displacement x of spoolvProportional relationship, i.e., it is full Sufficient xv=klU, wherein klFor voltage-spool displacement gain coefficient, u is input voltage;
Therefore, formula (6) is written as
Wherein total servo valve gain coefficient g=kqkl
Based on formula (4), (5), (8), output torque controls the dynamical equation of electrohydraulic load simulator, i.e. electrohydraulic load simulator Mathematical model are as follows:
(9) in formula, the model uncertainty of electrohydraulic load simulatorR1And R2It is defined as follows:
The R known to formula (10)1> 0, R2> 0, R1And R2It is intermediate variable;
Step 2, for arbitrary torque track following, propose three reasonable assumptions, according to the reasonable assumption, design electro-hydraulic negative It carries simulator error symbol and integrates robust controller, the specific steps are as follows:
Step 2-1, to integrate robust Controller Design convenient for electrohydraulic load simulator error symbol, for arbitrary torque track Tracking, there is following 3 reasonable assumptions:
Assuming that 1: actual hydraulic electrohydraulic load simulator works in normal conditions, due to PrAnd PsInfluence, P1And P2Meet Condition: 0≤Pr< P1< Ps, 0≤Pr< P2< Ps, i.e. P1And P2It is all bounded;
Assuming that 2: desired torque instruction TdIt (t) is that single order is continuously differentiable, and instructs Td(t) and its first derivative all has Boundary, motion artifacts y,It also is all bounded;
Assuming that 3: uncertain non-linearThere are 2 order derivatives, and 1 rank, the equal bounded of 2 order derivatives, that is, there is following formula establishment:
In formula (11), σ1, σ2It is all known constant;
Step 2-2, to simplify electrohydraulic load simulator equation, convenient for the design of controller, define unknown constant parameter vector theta= [θ123456]T, wherein θ1eG, θ2e, θ3eCt, θ4=B, θ5=a1, θ6=a2, therefore dynamical equation (9) it is write as
U=θ in formula (12)1f1U, parametric function f1,f2,f3It is defined as follows:
According to formula (13), actual control inputs u=U/ θ1f1, therefore, only need design error symbolic integration robust controller U Carry out processing parameter uncertainty and uncertainty is non-linear;
Step 2-3, design electrohydraulic load simulator error symbol integrates robust controller:
It is defined as follows error variance:
In formula: TdFor torque trace command;z1For electrohydraulic load simulator tracking error;R is the auxiliary margin of error;k1What is be positive is anti- Feedforward gain;
From (14) formula:
It is as follows to design electrohydraulic load simulator error symbol integral robust controller:
UaIndicate model compensation controller;krThe feedback oscillator being positive;Us1It is linear feedback item;Us2It is non linear robust item, uses In overcoming influence of the model uncertainty to tracking performance;
(16) formula is substituted into (15) Shi Ke get:
(18) in formula, robust gain beta > 0, sign (z1) it is about z1Standard signum function;
It is transferred to step 3;
Step 3, with Lyapunov stability theory, robust control is integrated to designed electrohydraulic load simulator error symbol Device, which carries out stability, to be proved, it was demonstrated that process is as follows:
To above-mentioned (17) formula derivation, obtain:
Before the performance that designed controller is presented, following lemma is provided:
Lemma 1: defined variable L (t) and auxiliary function P (t) are as follows:
If robust gain beta meets such as lower inequality:
It is positive value that then auxiliary function P (t) is permanent;
By lemma 1 it is found that the differential of auxiliary function P (t) are as follows:
Theorem 1: for the mathematical model of electrohydraulic load simulator, if the error symbol as shown in formula (16) integrates robust control The robust gain beta of device processed meets inequality (22), and its feedback oscillator k1,krIt is sufficiently large, so that as undefined matrix Λ is positive Set matrix:
Then all equal boundeds of signal in closed loop electrohydraulic load simulator, and controller obtains Asymptotic Stability, i.e., as t → ∞ When, z → 0, wherein error vector z is defined as z=[z1,r];
It proves: being defined as follows non-negative function V
Its time diffusion are as follows:
Wushu (14), (19), (23) are brought into (26) and can obtain:
Z=[z in formula1,r]T, Λ such as formula (24) is the matrix of a positive definite;
Therefore, there is following formula establishment:
λ in formulamin(Λ) is the minimal eigenvalue of matrix Λ, it can thus be appreciated that V bounded, therefore z bounded, therefore closed loop controller is all The equal bounded of signal;The W ∈ L known to formula (28) again2, from error dynamics equationTherefore variable W is congruous continuity , by Barbalat lemma, thus W → 0 as t → ∞ demonstrates theorem 1, which can guarantee the progressive receipts of tracking error It holds back in zero.
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