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 PDFInfo
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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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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 nonlinear 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
Technical field
The invention belongs to electrohydraulic 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 realtime reception simulation computer, and it is accurately applied to
In steering engine servo mechanism.Load simulator, can be to week product lifecycle 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 highperformance 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.Slidingmode 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 slidingmode 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 selfadaptive is not although this can be solved by using the method for indirect selfadaptive
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 11, it establishes and is based on the approximate continuously differentiable friction model of tanh
In formula (1), a_{1},a_{2}Respectively indicate the amplification level of differentiated friction characteristic, c_{1},c_{2},c_{3}For characterization frictional behavior
Form factor, w characterize movement velocity；Tanh indicates hyperbolic tangent function.
Step 12, 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 P_{L}=P_{1}P_{2},
P_{1}For the pressure of motor oil suction chamber, P_{2}For 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 13, the Pressure behaviour equation of motor oil suction chamber and oil outlet chamber is established:
In formula (5), β_{e}For the effective bulk modulus of hydraulic oil, the control volume V of oil suction chamber_{1}=V_{01}+ Ay, V_{01}For oil inlet
The original volume of chamber, the control volume V of oil outlet chamber_{2}=V_{02} Ay, V_{02}For the original volume of oil outlet chamber, C_{t}For the interior leakage of motor
Coefficient, Q_{1}For the flow of oil suction chamber, Q_{2}For the flow of oil back chamber.Q_{1}、Q_{2}With valve core of servo valve displacement x_{v}There is following relationship:
In formula (6), valve parameterC_{d}For servo valve discharge coefficient for orifices, w_{0}For servo valve throttle orifice
Area gradient, P_{s}For electrohydraulic load simulator charge oil pressure, P_{r}For electrohydraulic load simulator return pressure, ρ is the close of hydraulic oil
Degree, x_{v}For spool displacement, s (x_{v}) 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 spool_{v}Proportional relationship,
Meet x_{v}=k_{l}U, wherein k_{l}For voltagespool displacement gain coefficient, u is input voltage.Therefore, formula (6) is written as
Wherein total servo valve gain coefficient g=k_{q}k_{l}。
Based on formula (4), (5), (8), output torque controls the dynamical equation of electrohydraulic load simulator, i.e., electrohydraulic load simulation
The mathematical model of device are as follows:
(9) in formula, the model uncertainty of electrohydraulic load simulatorR_{1}And R_{2}Definition such as
Under:
The R known to formula (10)_{1}> 0, R_{2}> 0, R_{1}And R_{2}It 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 21, 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 P_{r}And P_{s}Influence, P_{1}And P_{2}
Meet condition: 0≤P_{r}<P_{1}<P_{s}, 0≤P_{r}<P_{2}<P_{s}, i.e. P_{1}And P_{2}It is all bounded.
Assuming that 2: desired torque instruction T_{d}It (t) is that single order is continuously differentiable, and instructs T_{d}(t) and its first derivative all
It is bounded, motion artifacts y,It also is all bounded.
Assuming that 3: uncertain nonlinearThere 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), σ_{1}, σ_{2}It is all known constant.
Step 22, to simplify electrohydraulic load simulator equation, convenient for the design of controller, unknown constant parameter arrow is defined
Measure θ=[θ_{1},θ_{2},θ_{3},θ_{4},θ_{5},θ_{6}]^{T}, wherein θ_{1}=β_{e}G, θ_{2}=β_{e}, θ_{3}=β_{e}C_{t}, θ_{4}=B, θ_{5}=a_{1}, θ_{6}=a_{2}, therefore dynamic
Equation (9) is write as
U=θ in formula (12)_{1}f_{1}U, parametric function f_{1},f_{2},f_{3}It is defined as follows:
According to formula (13), actual control inputs u=U/ θ_{1}f_{1}, therefore, only need design error symbolic integration robust control
Device U carrys out processing parameter uncertainty and uncertainty is nonlinear.
Step 23, design electrohydraulic load simulator error symbol integrates robust controller:
It is defined as follows error variance:
In formula: T_{d}For torque trace command；z_{1}For electrohydraulic load simulator tracking error；R is the auxiliary margin of error；k_{1}It is positive
Feedback oscillator.
From (14) formula:
It is as follows to design electrohydraulic load simulator error symbol integral robust controller:
U_{a}Indicate model compensation controller；k_{r}The feedback oscillator being positive；U_{s1}It is linear feedback item；U_{s2}It 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 (z_{1}) it is about z_{1}Standard 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 abovementioned (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 k_{1},k_{r}It 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=[z_{1},r]；
It proves: being defined as follows nonnegative function V
Its time diffusion are as follows:
Wushu (14), (19), (23) are brought into (26) and can obtain:
Z=[z in formula_{1},r]^{T}, Λ such as formula (24) is the matrix of a positive definite；
Therefore, there is following formula establishment:
λ in formula_{min}(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) again_{2}, 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 nonlinear 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 electrohydraulic 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 11, it establishes and is based on the approximate continuously differentiable friction model of tanh
In formula (1), a_{1},a_{2}Respectively indicate the amplification level of differentiated friction characteristic, c_{1},c_{2},c_{3}For characterization frictional behavior
Form factor, w characterize movement velocity；Tanh indicates hyperbolic tangent function；
Step 12, 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 P_{L}=P_{1}P_{2},
P_{1}For the pressure of motor oil suction chamber, P_{2}For 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 13, the Pressure behaviour equation of motor oil suction chamber and oil outlet chamber is established:
In formula (5), β_{e}For the effective bulk modulus of hydraulic oil, the control volume V of oil suction chamber_{1}=V_{01}+ Ay, V_{01}For oil inlet
The original volume of chamber, the control volume V of oil outlet chamber_{2}=V_{02} Ay, V_{02}For the original volume of oil outlet chamber, C_{t}For the interior leakage of motor
Coefficient, Q_{1}For the flow of oil suction chamber, Q_{2}For the flow of oil back chamber, Q_{1}、Q_{2}With valve core of servo valve displacement x_{v}There is following relationship:
In formula (6), valve parameterC_{d}For servo valve discharge coefficient for orifices, w_{0}For servo valve throttle orifice
Area gradient, P_{s}For electrohydraulic load simulator charge oil pressure, P_{r}For electrohydraulic load simulator return pressure, ρ is the close of hydraulic oil
Degree, x_{v}For spool displacement, s (x_{v}) 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 x_{v}Proportional relationship, that is, meet x_{v}=k_{l}U, wherein k_{l}For voltagespool displacement gain coefficient, u
For input voltage.
Therefore, formula (6) can be written as
Wherein total servo valve gain coefficient g=k_{q}k_{l}；
Based on formula (4), (5), (8), output torque controls the dynamical equation of electrohydraulic load simulator, i.e., electrohydraulic load simulation
The mathematical model of device are as follows:
(9) in formula, the model uncertainty of electrohydraulic load simulatorR_{1}And R_{2}Definition such as
Under:
The R known to formula (10)_{1}> 0, R_{2}> 0, R_{1}And R_{2}It 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 21, 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 P_{r}And P_{s}Influence, P_{1}And P_{2}
Meet condition: 0≤P_{r}<P_{1}<P_{s}, 0≤P_{r}<P_{2}<P_{s}, i.e. P_{1}And P_{2}It is all bounded.
Assuming that 2: desired torque instruction T_{d}It (t) is that single order is continuously differentiable, and instructs T_{d}(t) and its first derivative all
It is bounded, motion artifacts y,It also is all bounded.
Assuming that 3: uncertain nonlinearThere 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), σ_{1}, σ_{2}It is all known constant.
For giving desired torque instruction T_{d}(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 possible_{d}(t)。
Step 22, to simplify electrohydraulic load simulator equation, convenient for the design of controller, unknown constant parameter arrow is defined
Measure θ=[θ_{1},θ_{2},θ_{3},θ_{4},θ_{5},θ_{6}]^{T}, wherein θ_{1}=β_{e}G, θ_{2}=β_{e}, θ_{3}=β_{e}C_{t}, θ_{4}=B, θ_{5}=a_{1}, θ_{6}=a_{2}, therefore dynamic
Equation (9) can be write as
U=θ in formula (12)_{1}f_{1}U, parametric function f_{1},f_{2},f_{3}It is defined as follows:
A new variable U has been incorporated herein to replace the control of system to input u, because of θ_{1}f_{1}It can calculate in real time,
If having obtained the expression formula of new control input U, actual control inputs u=U/ θ_{1}f_{1}.Therefore, design error symbol is only needed
Integral robust controller U carrys out processing parameter uncertainty and uncertainty is nonlinear.
Step 23, design electrohydraulic load simulator error symbol integrates robust controller:
It is defined as follows error variance:
In formula: T_{d}For torque trace command；z_{1}For electrohydraulic load simulator tracking error；R is the auxiliary margin of error；k_{1}It is positive
Feedback oscillator.
From (14) formula:
It is as follows to design electrohydraulic load simulator error symbol integral robust controller:
U_{a}Indicate model compensation controller；k_{r}The feedback oscillator being positive；U_{s1}It is linear feedback item；U_{s2}It 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 U_{s2}It need not use, linear feedback item U_{s1}Ensure 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 U_{s2}
To handle the influence of model uncertainty.This design of control method error symbol integrates robust item u_{s2}It is as follows:
(18) β > 0 is robust gain in formula.sign(z_{1}) it is about z_{1}Standard signum function, this design of control method
U_{s2}Purpose be desirable to by selecting suitable control gain k_{1},k_{r}, 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 U_{a}Used in be continuous friction model, sign function is in U_{s2}In
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 abovementioned (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 k_{1},k_{r}It is wide enough so that undefined matrix Λ such as is positive definite matrix:
Then all equal boundeds of signal in closedloop system, and controller can get Asymptotic Stability, i.e., as t → ∞, z →
0, wherein z is defined as z=[z_{1},r]；
It proves: being defined as follows nonnegative function
Its time diffusion are as follows:
Wushu (14), (19), (23) are brought into (26) and can obtain:
Z=[z in formula_{1},r]^{T}, Λ such as formula (24) is the matrix of a positive definite.
Therefore, there is following formula establishment:
λ in formula_{min}(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) again_{2}, 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^{4}m^{3}/ rad, B=80Nms/rad, β_{e}=2 × 10^{8}Pa,C_{t}=9 × 10^{12}m^{5}/ (Ns),P_{s}=21 × 10^{6}Pa, P_{r}=0Pa, V_{01}=V_{02}=1.7 × 10^{4}m^{3}, J=0.32kgm^{2},
a_{1}=3.5 × 10^{4},a_{2}=5 × 10^{4},c_{1}=700, c_{2}=15, c_{3}=1.5
The controller parameter of design is chosen are as follows: k_{1}=200, k_{r}=0.01, β=0.1.It is interfered outside system timevarying 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 ucurve, 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 11, it establishes and is based on the approximate continuously differentiable friction model of tanh
In formula (1), a_{1},a_{2}Respectively indicate the amplification level of differentiated friction characteristic, c_{1},c_{2},c_{3}It is the shape for characterizing frictional behavior
Shape coefficient, w characterize movement velocity；Tanh indicates hyperbolic tangent function；
Step 12, 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 P_{L}=P_{1}P_{2}, P_{1}For
The pressure of motor oil suction chamber, P_{2}For 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 13, the Pressure behaviour equation of motor oil suction chamber and oil outlet chamber is established:
In formula (5), β_{e}For the effective bulk modulus of hydraulic oil, the control volume V of oil suction chamber_{1}=V_{01}+ Ay, V_{01}For oil suction chamber
Original volume, the control volume V of oil outlet chamber_{2}=V_{02} Ay, V_{02}For the original volume of oil outlet chamber, C_{t}For the interior leakage coefficient of motor,
Q_{1}For the flow of oil suction chamber, Q_{2}For the flow of oil back chamber；
Q_{1}、Q_{2}With valve core of servo valve displacement x_{v}There is following relationship:
In formula (6), valve parameterC_{d}For servo valve discharge coefficient for orifices, w_{0}For servo valve throttle hole area
Gradient, P_{s}For electrohydraulic load simulator charge oil pressure, P_{r}For electrohydraulic load simulator return pressure, ρ is the density of hydraulic oil, x_{v}
For spool displacement, s (x_{v}) 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 spool_{v}Proportional relationship, i.e., it is full
Sufficient x_{v}=k_{l}U, wherein k_{l}For voltagespool displacement gain coefficient, u is input voltage；
Therefore, formula (6) is written as
Wherein total servo valve gain coefficient g=k_{q}k_{l}；
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 simulatorR_{1}And R_{2}It is defined as follows:
The R known to formula (10)_{1}> 0, R_{2}> 0, R_{1}And R_{2}It is intermediate variable；
Step 2, for arbitrary torque track following, propose three reasonable assumptions, according to the reasonable assumption, design electrohydraulic negative
It carries simulator error symbol and integrates robust controller, the specific steps are as follows:
Step 21, 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 P_{r}And P_{s}Influence, P_{1}And P_{2}Meet
Condition: 0≤P_{r}< P_{1}< P_{s}, 0≤P_{r}< P_{2}< P_{s}, i.e. P_{1}And P_{2}It is all bounded；
Assuming that 2: desired torque instruction T_{d}It (t) is that single order is continuously differentiable, and instructs T_{d}(t) and its first derivative all has
Boundary, motion artifacts y,It also is all bounded；
Assuming that 3: uncertain nonlinearThere are 2 order derivatives, and 1 rank, the equal bounded of 2 order derivatives, that is, there is following formula establishment:
In formula (11), σ_{1}, σ_{2}It is all known constant；
Step 22, to simplify electrohydraulic load simulator equation, convenient for the design of controller, define unknown constant parameter vector theta=
[θ_{1},θ_{2},θ_{3},θ_{4},θ_{5},θ_{6}]^{T}, wherein θ_{1}=β_{e}G, θ_{2}=β_{e}, θ_{3}=β_{e}C_{t}, θ_{4}=B, θ_{5}=a_{1}, θ_{6}=a_{2}, therefore dynamical equation
(9) it is write as
U=θ in formula (12)_{1}f_{1}U, parametric function f_{1},f_{2},f_{3}It is defined as follows:
According to formula (13), actual control inputs u=U/ θ_{1}f_{1}, therefore, only need design error symbolic integration robust controller U
Carry out processing parameter uncertainty and uncertainty is nonlinear；
Step 23, design electrohydraulic load simulator error symbol integrates robust controller:
It is defined as follows error variance:
In formula: T_{d}For torque trace command；z_{1}For electrohydraulic load simulator tracking error；R is the auxiliary margin of error；k_{1}What is be positive is anti
Feedforward gain；
From (14) formula:
It is as follows to design electrohydraulic load simulator error symbol integral robust controller:
U_{a}Indicate model compensation controller；k_{r}The feedback oscillator being positive；U_{s1}It is linear feedback item；U_{s2}It 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 (z_{1}) it is about z_{1}Standard 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 abovementioned (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 k_{1},k_{r}It 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=[z_{1},r]；
It proves: being defined as follows nonnegative function V
Its time diffusion are as follows:
Wushu (14), (19), (23) are brought into (26) and can obtain:
Z=[z in formula_{1},r]^{T}, Λ such as formula (24) is the matrix of a positive definite；
Therefore, there is following formula establishment:
λ in formula_{min}(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) again_{2}, 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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