CN111781836A - Self-adaptive asymptotic control method for hydraulic pressure preset performance - Google Patents

Self-adaptive asymptotic control method for hydraulic pressure preset performance Download PDF

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CN111781836A
CN111781836A CN202010710755.6A CN202010710755A CN111781836A CN 111781836 A CN111781836 A CN 111781836A CN 202010710755 A CN202010710755 A CN 202010710755A CN 111781836 A CN111781836 A CN 111781836A
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CN111781836B (en
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徐张宝
刘庆运
郭永存
涂德浴
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Anhui University of Technology AHUT
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    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
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Abstract

The invention discloses a hydraulic pressure preset performance self-adaptive asymptotic control method in the technical field of hydraulic control systems, which comprises the following steps: establishing a double-rod hydraulic cylinder servo system model; designing a hydraulic pressure preset performance self-adaptive asymptotic controller; adjusting the parameters in the steps to enable the system to meet the control performance index, suppressing the system interference through a robust error sign integral function, and simultaneously approaching the system parameters by using a self-adaptive controller to greatly weaken the influence of uncertainty on the system; in addition, in consideration of the problem of transient control precision of the system, the control precision of the system is constrained through the design of a preset performance function, the asymptotic tracking of the system is theoretically realized, and the output force of the double-rod hydraulic cylinder servo system can accurately track an expected force instruction; the invention simplifies the design of the controller and is more beneficial to the application in engineering practice.

Description

Self-adaptive asymptotic control method for hydraulic pressure preset performance
Technical Field
The invention relates to the technical field of hydraulic control systems, in particular to a hydraulic pressure preset performance self-adaptive asymptotic control method.
Background
The electro-hydraulic servo system has the advantages of high output power, flexible signal processing and the like, is easy to realize feedback of various parameters, is most suitable for occasions with high power and quality, and has been applied to various fields of national defense and industry, such as control of steering engines of airplanes and ships, control of radars and artillery, position control of machine tool working tables, plate thickness control of plate and strip rolling mills, electrode position control of electric furnace smelting, pressure control of material testing machines and other experimental machines, and the like. However, the uncertainties prevalent in electro-hydraulic servo systems increase the design difficulty of the control system and may severely degrade the control performance that can be achieved, resulting in low control accuracy, limit ring oscillations, and even instability. In order to improve the control performance of the electro-hydraulic system, designers have studied many advanced nonlinear controllers, such as robust adaptive control, Adaptive Robust Control (ARC), sliding mode control, and so on. Although these controllers can ensure good steady-state performance, transient control performance is not concerned, and based on this, the invention designs a hydraulic pressure preset performance adaptive asymptotic control method to solve the above problems.
Disclosure of Invention
The present invention is directed to a hydraulic pressure preset performance adaptive asymptotic control method, so as to solve the problems in the background art.
In order to achieve the purpose, the invention provides the following technical scheme: a hydraulic pressure preset performance self-adaptive asymptotic control method comprises the following steps:
s1: establishing a double-rod hydraulic cylinder servo system model;
s2: designing a hydraulic pressure preset performance self-adaptive asymptotic controller;
s3: and adjusting the parameters in the step S2 to enable the system to meet the control performance index.
Further, the step S1 is specifically: according to Newton's second law, the dynamic model equation of the inertial load of the double-rod hydraulic cylinder is as follows:
Figure BDA0002596465850000021
in the formula: y is load displacement, T represents driving force, PL＝P1-P2Is the load driving pressure, P1And P2Are respectively liquidThe method comprises the following steps that two cavity pressures of a hydraulic cylinder are measured, A is the effective working area of a piston rod, b represents a viscous friction coefficient, f is other unmodeled interferences including nonlinear friction, external interference and unmodeled dynamics, and the load pressure dynamic equation of the hydraulic cylinder is as follows:
Figure BDA0002596465850000022
in the formula: vtIs the total effective volume of two cavities of the hydraulic cylinder, CtIs the leakage coefficient, Q, of the hydraulic cylinderL＝(Q1+Q2) Per 2 is the load flow, Q1For supply of oil to the cylinder inlet chamber, Q2The oil return flow of an oil return cavity of the hydraulic cylinder is q (t), and modeling errors and unmodeled dynamics are q (t);
QLfor spool displacement x of servo valvevFunction of (c):
Figure BDA0002596465850000023
in the formula:
Figure BDA0002596465850000024
is the gain factor of the flow servovalve, CdIs the flow coefficient of the servo valve, and w is the area gradient of the servo valve; rho is the density of the hydraulic oil, PsSign (x) for supply pressurev) Comprises the following steps:
Figure BDA0002596465850000025
assuming that the servo valve spool displacement is proportional to the control input u, i.e., xv＝kiu, wherein ki>0 is the scaling factor and u is the control input voltage, therefore equation (3) can be translated into
Figure BDA0002596465850000026
In the formula: k is a radical oft＝kqkiRepresenting the total flow gain.
Defining a state variable x1T, then the entire system can be written in the following state space form:
Figure BDA0002596465850000031
defining an unknown parameter set θ ═ θ1,θ2,θ3,θ4]TWherein
Figure BDA0002596465850000032
Figure BDA0002596465850000033
θ4＝B，
Figure BDA0002596465850000034
Assume that 1: the derivatives of d (x, t) and d (x, t) are bounded, i.e.
Figure BDA0002596465850000035
In the formula: zeta1、ζ2Is a known constant.
Further, the step S2 includes the following steps:
s2 a: constructing a projection self-adaptive law with rate limitation;
s2 b: designing a robust controller;
s2 c: and verifying the stability of the system of the controller.
Further, the step S2a is specifically: order to
Figure BDA0002596465850000036
The estimate of the value of theta is represented,
Figure BDA0002596465850000037
error in the estimate of theta, i.e.
Figure BDA0002596465850000038
A projection function is established as follows：
Figure BDA0002596465850000039
In the formula, zeta ∈ R2,(t)∈R2×2Is a positive definite symmetric matrix that varies with time,
Figure BDA00025964658500000310
and
Figure BDA00025964658500000311
each represents omegaθThe inner portion and the boundary of (a),
Figure BDA00025964658500000312
to represent
Figure BDA00025964658500000313
An outer unit normal vector of time;
for the projection function (8), the preset adaptive limiting speed is used in the control parameter estimation process, and thus, a saturation function is established as follows:
Figure BDA00025964658500000314
in the formula:
Figure BDA00025964658500000315
the method is characterized in that the method is a preset limiting rate, and the structural characteristics of a parameter estimation algorithm to be used by the system are summarized through the following quotation;
introduction 1: assume that the following projection-type adaptation law and preset adaptive rate limit are used
Figure BDA0002596465850000041
Updating an estimated parameter
Figure BDA0002596465850000042
Figure BDA0002596465850000043
In the formula: tau is an adaptive function, and (t) > 0 is a continuous slightly positive symmetrical adaptive rate matrix, so that the following ideal characteristics can be obtained by the adaptive law:
p1) parameter estimate is always at a known bounded ΩθIn-set, i.e. for any t, there is always
Figure BDA0002596465850000044
Thus, from hypothesis 1
Figure BDA0002596465850000045
P2)
Figure BDA0002596465850000046
P3) the law of parameter variation is consistently bounded, i.e. it is determined that the parameter variation is uniformly bounded
Figure BDA0002596465850000047
Further, the step S2b is specifically: defining the control error e ═ x of the hydraulic cylinder1-x1dIt is assumed that it needs to meet the following performance criteria:
Figure BDA0002596465850000048
in the formula:l，uthe parameters to be designed are used for assisting in restraining the upper limit and the lower limit of the control error; ρ (t) is a positive strictly increasing smoothing function, as shown by:
Figure BDA0002596465850000049
in the formula: rho0、ρ∞And k are both positive designable parameters;
in formula (11) -lρ0Anduρ0the maximum undershoot and overshoot of the output force control error e (t) are constrained respectively,the parameter k constrains the convergence speed, ρ, of the error e (t)∞The steady state bound of the error is constrained, so that equation (11) gives a specific plan for the performance of the output force control error by selecting an appropriate parameter ρ0、ρ∞、k、lAnduthe transient and stable performance of the output force control error can be planned in advance by the aid of the parameters, and the transient performance can be improved according to actual requirements of the system;
the following increasing function is established:
Figure BDA0002596465850000051
in the formula: z is a radical of1(t) is a conversion error variable corresponding to the control error e (t), and it is easy to analyze that equation (13) is equivalent to e (t) ═ ρ (t) S (z)1(t)), and z1(t) when the interface is bounded, the preset performance characteristic formula (8) is always satisfied;
an increasing function S (z) satisfying the characteristic formula (13)1) The following can be selected:
Figure BDA0002596465850000052
the inverse function of equation (14) is found:
Figure BDA0002596465850000053
for the conversion error z1Designing a controller;
the establishment function is as follows:
Figure BDA0002596465850000054
in the formula: k is a radical of1Is the feedback gain;
the controller is designed as follows:
Figure BDA0002596465850000061
in the formula: k is a radical of2Is the feedback gain;
the controller (17) can be substituted into the formula (16):
Figure BDA0002596465850000062
the adaptive law is designed as follows:
Figure BDA0002596465850000063
in the formula:
Figure BDA0002596465850000064
then, it is possible to obtain:
Figure BDA0002596465850000065
design robust controller us2The following were used:
Figure BDA0002596465850000066
in the formula β1Are parameters to be designed.
Further, the step S2c is specifically: by performance theorem 1: selecting initial conditions for system controllρ(0)<e(0)<uρ (0) -l<λ(0)<uSimultaneous parameter β1Satisfies the following inequality:
Figure BDA0002596465850000067
while designing a sufficiently large parameter k1And k2So that the following matrix Λ is a positive definite matrix
Figure BDA0002596465850000071
Then canEnsuring that the control error of the output force is bounded all the time, realizing better instruction tracking of the output force and adjusting rho0、ρ∞、k、lAnduthe parameters are equal, so that the control error can meet the preset performance requirement designed by the formula (11);
the following was demonstrated:
the following Lyapunov function is defined:
Figure BDA0002596465850000072
further, by further deriving V and substituting the following equations (16) and (20):
Figure BDA0002596465850000073
in the formula: z ═ Z1,z2]TThe matrix Λ is defined as equation (23) if the reasonable design parameter k is passed1And k2Making the matrix Λ positive definite makes the following satisfied:
Figure BDA0002596465850000081
in the formula: lambda [ alpha ]min(Λ) represents the minimum eigenvalue of the matrix Λ, and the analytic expression (26) shows that the Lyapunov function is bounded, and the W integral is bounded, and further shows that the conversion error amount z is bounded1And z2All are bounded, and in combination with equations (8), (16) and (17), all signals in the system are bounded, so that the derivative of W is bounded, as can be seen from the Barbalt theorem, as time approaches infinity, W approaches zero, i.e., the transformation error amount z1And approaches zero, so that the control error e (t) is always bounded in conjunction with equation (11), thereby proving that the controller is convergent and the system is stable.
Further, the third step is specifically to adjust a parameter k of the control law u1、k2、ρ0、ρ∞、k、l、u、β1And the system meets the control performance index.
Compared with the prior art, the invention has the beneficial effects that: aiming at the characteristics of a hydraulic servo system, a hydraulic servo system model and a designed hydraulic system output force preset performance self-adaptive asymptotic controller are established, unmodeled interference is estimated through a robust error sign integral function and feedforward compensation is carried out, meanwhile, system parameters are estimated through the self-adaptive controller, the problems of uncertain nonlinearity and uncertain parameters of a motor servo system can be effectively solved, the preset performance controller is designed based on the preset performance function, the overall stability of the system is finally proved through Lyapunov, the parameter convergence is good under the existing interference condition, and the transient and steady state control accuracy of the system meets the performance indexes; meanwhile, the design of the controller is simplified, and the final simulation result shows that the method has effectiveness and is more beneficial to application in engineering practice.
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In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the description of the embodiments will be briefly introduced below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art that other drawings can be obtained according to the drawings without creative efforts.
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a schematic view of a dual-out-rod hydraulic cylinder system of the present invention;
FIG. 3 is a schematic diagram of the control error preset capability of the present invention;
FIG. 4 shows the function S (z) according to the invention1) A schematic diagram;
FIG. 5 is a graph illustrating parameter estimation according to an embodiment of the present invention;
FIG. 6 is a graph comparing control errors of two controllers according to an embodiment of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Referring to fig. 1-2, the present invention provides a technical solution: a hydraulic pressure preset performance self-adaptive asymptotic control method comprises the following steps:
s1: establishing a double-rod hydraulic cylinder servo system model;
s2: designing a hydraulic pressure preset performance self-adaptive asymptotic controller;
s3: and adjusting the parameters in the step S2 to enable the system to meet the control performance index.
Wherein, step S1 specifically includes: according to Newton's second law, the dynamic model equation of the inertial load of the double-rod hydraulic cylinder is as follows:
Figure BDA0002596465850000091
in the formula: y is load displacement, T represents driving force, PL＝P1-P2Is the load driving pressure, P1And P2The method is characterized in that the pressure of two cavities of the hydraulic cylinder is respectively, A is the effective working area of a piston rod, b represents a viscous friction coefficient, f is other unmodeled interferences including nonlinear friction, external interference and unmodeled dynamics, and the dynamic equation of the load pressure of the hydraulic cylinder is as follows:
Figure BDA0002596465850000092
in the formula: vtIs the total effective volume of two cavities of the hydraulic cylinder, CtIs the leakage coefficient, Q, of the hydraulic cylinderL＝(Q1+Q2) Per 2 is the load flow, Q1For supply of oil to the cylinder inlet chamber, Q2The oil return flow of an oil return cavity of the hydraulic cylinder is q (t), and modeling errors and unmodeled dynamics are q (t);
QLfor spool displacement x of servo valvevFunction of (c):
Figure BDA0002596465850000101
in the formula:
Figure BDA0002596465850000102
is the gain factor of the flow servovalve, CdIs the flow coefficient of the servo valve, and w is the area gradient of the servo valve; rho is the density of the hydraulic oil, PsSign (x) for supply pressurev) Comprises the following steps:
Figure BDA0002596465850000103
assuming that the servo valve spool displacement is proportional to the control input u, i.e., xv＝kiu, wherein ki>0 is the scaling factor and u is the control input voltage, therefore equation (3) can be translated into
Figure BDA0002596465850000104
In the formula: k is a radical oft＝kqkiRepresenting the total flow gain.
Defining a state variable x1T, then the entire system can be written in the following state space form:
Figure BDA0002596465850000105
defining an unknown parameter set θ ═ θ1,θ2,θ3,θ4]TWherein
Figure BDA0002596465850000106
Figure BDA0002596465850000107
θ4＝B，
Figure BDA0002596465850000108
Assume that 1: the derivatives of d (x, t) and d (x, t) are bounded, i.e.
Figure BDA0002596465850000109
In the formula: zeta1、ζ2Is a known constant.
Step S2 includes the following steps:
s2 a: constructing a projection self-adaptive law with rate limitation;
s2 b: designing a robust controller;
s2 c: and verifying the stability of the system of the controller.
Step S2a specifically includes: order to
Figure BDA0002596465850000111
The estimate of the value of theta is represented,
Figure BDA0002596465850000112
error in the estimate of theta, i.e.
Figure BDA0002596465850000113
A projection function is established as follows:
Figure BDA0002596465850000114
in the formula, zeta ∈ R2,(t)∈R2×2Is a positive definite symmetric matrix that varies with time,
Figure BDA0002596465850000115
and
Figure BDA0002596465850000116
each represents omegaθThe inner portion and the boundary of (a),
Figure BDA00025964658500001116
to represent
Figure BDA0002596465850000117
An outer unit normal vector of time;
for the projection function (8), the preset adaptive limiting speed is used in the control parameter estimation process, and thus, a saturation function is established as follows:
Figure BDA0002596465850000118
in the formula:
Figure BDA0002596465850000119
the method is characterized in that the method is a preset limiting rate, and the structural characteristics of a parameter estimation algorithm to be used by the system are summarized through the following quotation;
introduction 1: assume that the following projection-type adaptation law and preset adaptive rate limit are used
Figure BDA00025964658500001110
Updating an estimated parameter
Figure BDA00025964658500001111
Figure BDA00025964658500001112
In the formula: tau is an adaptive function, and (t) > 0 is a continuous slightly positive symmetrical adaptive rate matrix, so that the following ideal characteristics can be obtained by the adaptive law:
p1) parameter estimate is always at a known bounded ΩθIn-set, i.e. for any t, there is always
Figure BDA00025964658500001113
Thus, from hypothesis 1
Figure BDA00025964658500001114
P2)
Figure BDA00025964658500001115
P3) the law of parameter variation is consistently bounded, i.e. it is determined that the parameter variation is uniformly bounded
Figure BDA0002596465850000121
Step S2b specifically includes: defining the control error e ═ x of the hydraulic cylinder1-x1dIt is assumed that it needs to meet the following performance criteria:
Figure BDA0002596465850000122
in the formula:l，uthe parameters to be designed are used for assisting in restraining the upper limit and the lower limit of the control error; ρ (t) is a positive strictly increasing smoothing function, as shown by:
Figure BDA0002596465850000123
in the formula: rho0、ρ∞And k are both positive designable parameters; the rough curve of the performance index inequality (11) is shown in fig. 3;
in formula (11) -lρ0Anduρ0respectively constraining the maximum downward impulse and the maximum overshoot of the output force control error e (t), and constraining the convergence rate, rho, of the error e (t) by the parameter k∞The steady state bound of the error is constrained, so that equation (11) gives a specific plan for the performance of the output force control error by selecting an appropriate parameter ρ0、ρ∞、k、lAnduthe transient and stable performance of the output force control error can be planned in advance by the aid of the parameters, and the transient performance can be improved according to actual requirements of the system;
the following increasing function is established:
Figure BDA0002596465850000124
in the formula: z is a radical of1(t) is a conversion error variable corresponding to the control error e (t), and it is easy to analyze that equation (13) is equivalent to e (t) ═ ρ (t) S (z)1(t)), and z1(t) when the interface is bounded, the preset performance characteristic formula (8) is always satisfied;
an increasing function S (z) satisfying the characteristic formula (13)1) The following can be selected:
Figure BDA0002596465850000131
increasing function S (z)1) The curve of (a) is shown in fig. 4;
the inverse function of equation (14) is found:
Figure BDA0002596465850000132
for the conversion error z1Designing a controller;
the establishment function is as follows:
Figure BDA0002596465850000133
in the formula: k is a radical of1Is the feedback gain;
the controller is designed as follows:
Figure BDA0002596465850000134
in the formula: k is a radical of2Is the feedback gain;
the controller (17) can be substituted into the formula (16):
Figure BDA0002596465850000135
the adaptive law is designed as follows:
Figure BDA0002596465850000136
in the formula:
Figure BDA0002596465850000141
then, it is possible to obtain:
Figure BDA0002596465850000142
design robust controller us2The following were used:
Figure BDA0002596465850000143
in the formula β1Are parameters to be designed.
Step S2c specifically includes: by performance theorem 1: selecting initial conditions for system controllρ(0)<e(0)<uρ (0) -l<λ(0)<uSimultaneous parameter β1Satisfies the following inequality:
Figure BDA0002596465850000144
while designing a sufficiently large parameter k1And k2So that the following matrix Λ is a positive definite matrix
Figure BDA0002596465850000145
The control error of the output force can be guaranteed to be bounded all the time, better instruction tracking can be realized by the output force, and rho is adjusted0、ρ∞、k、lAnduthe parameters are equal, so that the control error can meet the preset performance requirement designed by the formula (11);
the following was demonstrated:
the following Lyapunov function is defined:
Figure BDA0002596465850000151
further, by further deriving V and substituting the following equations (16) and (20):
Figure BDA0002596465850000152
in the formula: z ═ 2-z1,z2]TThe matrix Λ is defined as equation (23) if the reasonable design parameter k is passed1And k2Making the matrix Λ positive definite makes the following satisfied:
Figure BDA0002596465850000153
in the formula: lambda [ alpha ]min(Λ) represents the minimum eigenvalue of the matrix Λ, and the analytic expression (26) shows that the Lyapunov function is bounded, and the W integral is bounded, and further shows that the conversion error amount z is bounded1And z2All are bounded, and in combination with equations (8), (16) and (17), all signals in the system are bounded, so that the derivative of W is bounded, as can be seen from the Barbalt theorem, as time approaches infinity, W approaches zero, i.e., the transformation error amount z1And approaches zero, so that the control error e (t) is always bounded in conjunction with equation (11), thereby proving that the controller is convergent and the system is stable.
Step three is specifically to adjust the parameter k of the control law u1、k2、ρ0、ρ∞、k、l、u、β1And the system meets the control performance index.
The first embodiment is as follows:
the system was modeled in the simulation using the following parameters, m 40kg, a 2 × 10-4m2，B＝80N·s/m，βe＝200Mpa，V01＝1×10-3m3,V02＝1×10-3m3,Ct＝9×10-12m5/Ns,
Figure BDA0002596465850000161
θmin＝[0.01,0.1,1000,20]T,θmax＝[10,30,500000，500]T,
Figure BDA0002596465850000162
Г ═ diag {6,10,390,50}, selected
Figure BDA0002596465850000163
Far from the true value of the parameter to evaluate the effect of the adaptive control law, force input signal
Figure BDA0002596465850000164
The unit KN; the interference applied by the system is f-30 sin (2 π t) KN.
The invention provides a hydraulic pressure self-adaptive preset performance controller (APFRISE) based on robust error symbolic integral, and relevant parameters of a controller design parameter controller are selected as follows: k is a radical of1＝200,k2＝300,β＝100,ρ0＝800,ρ∞＝500,k＝0.0001,l＝u1 is ═ 1; PID controller parameter is kp＝500,ki＝100,kd＝0；
The control law effect is shown in fig. 5 and 6, and fig. 5 is a parameter estimation curve; FIG. 6 is a control error versus curve for the designed controller (APFRISE) and the PID controller.
The figure shows that the algorithm provided by the invention can accurately estimate the system parameters in a simulation environment, and compared with a PID (proportion integration differentiation) controller, the controller designed by the invention can obtain good control precision and can ensure the preset transient and steady-state control precision requirements of the system.
In the description herein, references to the description of "one embodiment," "an example," "a specific example" or the like are intended to mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, the schematic representations of the terms used above do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
The preferred embodiments of the invention disclosed above are intended to be illustrative only. The preferred embodiments are not intended to be exhaustive or to limit the invention to the precise embodiments disclosed. Obviously, many modifications and variations are possible in light of the above teaching. The embodiments were chosen and described in order to best explain the principles of the invention and the practical application, to thereby enable others skilled in the art to best utilize the invention. The invention is limited only by the claims and their full scope and equivalents.

Claims (7)

1. A hydraulic pressure preset performance self-adaptive asymptotic control method is characterized by comprising the following steps:
s1: establishing a double-rod hydraulic cylinder servo system model;
s2: designing a hydraulic pressure preset performance self-adaptive asymptotic controller;
s3: and adjusting the parameters in the step S2 to enable the system to meet the control performance index.
2. The hydraulic pressure preset performance adaptive asymptotic control method according to claim 1, characterized in that: the step S1 specifically includes: according to Newton's second law, the dynamic model equation of the inertial load of the double-rod hydraulic cylinder is as follows:
Figure FDA0002596465840000011
in the formula: y is load displacement, T represents driving force, PL＝P1-P2Is the load driving pressure, P1And P2The method is characterized in that the pressure of two cavities of the hydraulic cylinder is respectively, A is the effective working area of a piston rod, b represents a viscous friction coefficient, f is other unmodeled interferences including nonlinear friction, external interference and unmodeled dynamics, and the dynamic equation of the load pressure of the hydraulic cylinder is as follows:
Figure FDA0002596465840000012
in the formula: vtIs the total effective volume of two cavities of the hydraulic cylinder, CtIs the leakage coefficient, Q, of the hydraulic cylinderL＝(Q1+Q2) Per 2 is the load flow, Q1For supply of oil to the cylinder inlet chamber, Q2For the return oil flow of the return oil cavity of the hydraulic cylinder, q (t) for modeling error and unmodeled motionState;
QLfor spool displacement x of servo valvevFunction of (c):
Figure FDA0002596465840000013
in the formula:
Figure FDA0002596465840000014
is the gain factor of the flow servovalve, CdIs the flow coefficient of the servo valve, and w is the area gradient of the servo valve; rho is the density of the hydraulic oil, PsSign (x) for supply pressurev) Comprises the following steps:
Figure FDA0002596465840000015
assuming that the servo valve spool displacement is proportional to the control input u, i.e., xv＝kiu, wherein ki>0 is the scaling factor and u is the control input voltage, therefore equation (3) can be translated into
Figure FDA0002596465840000021
In the formula: k is a radical oft＝kqkiRepresenting the total flow gain.
Defining a state variable x1T, then the entire system can be written in the following state space form:
Figure FDA0002596465840000022
defining an unknown parameter set θ ═ θ1,θ2,θ3,θ4]TWherein
Figure FDA0002596465840000023
Figure FDA0002596465840000024
θ4＝B，
Figure FDA0002596465840000025
Assume that 1: the derivatives of d (x, t) and d (x, t) are bounded, i.e.
Figure FDA0002596465840000026
In the formula: zeta1、ζ2Is a known constant.
3. The hydraulic pressure preset performance adaptive asymptotic control method according to claim 1, characterized in that: the step S2 includes the following steps:
s2 a: constructing a projection self-adaptive law with rate limitation;
s2 b: designing a robust controller;
s2 c: and verifying the stability of the system of the controller.
4. The hydraulic pressure preset performance adaptive asymptotic control method according to claim 3, characterized in that: the step S2a specifically includes: order to
Figure FDA0002596465840000027
The estimate of the value of theta is represented,
Figure FDA0002596465840000028
error in the estimate of theta, i.e.
Figure FDA0002596465840000029
A projection function is established as follows:
Figure FDA00025964658400000210
in the formula, zeta ∈ R2,(t)∈R2×2Is a positive definite symmetry varying with timeThe matrix is a matrix of a plurality of matrices,
Figure FDA0002596465840000031
and
Figure FDA0002596465840000032
each represents omegaθThe inner portion and the boundary of (a),
Figure FDA0002596465840000033
to represent
Figure FDA0002596465840000034
An outer unit normal vector of time;
for the projection function (8), the preset adaptive limiting speed is used in the control parameter estimation process, and thus, a saturation function is established as follows:
Figure FDA0002596465840000035
in the formula:
Figure FDA0002596465840000036
the method is characterized in that the method is a preset limiting rate, and the structural characteristics of a parameter estimation algorithm to be used by the system are summarized through the following quotation;
introduction 1: assume that the following projection-type adaptation law and preset adaptive rate limit are used
Figure FDA0002596465840000037
Updating an estimated parameter
Figure FDA0002596465840000038
Figure FDA0002596465840000039
In the formula: tau is an adaptive function, and (t) > 0 is a continuous slightly positive symmetrical adaptive rate matrix, so that the following ideal characteristics can be obtained by the adaptive law:
p1) parameter estimate is always at a known bounded ΩθIn-set, i.e. for any t, there is always
Figure FDA00025964658400000310
Thus, from hypothesis 1
Figure FDA00025964658400000311
P2)
Figure FDA00025964658400000312
P3) the law of parameter variation is consistently bounded, i.e. it is determined that the parameter variation is uniformly bounded
Figure FDA00025964658400000313
5. The hydraulic pressure preset performance adaptive asymptotic control method according to claim 4, characterized in that: the step S2b specifically includes: defining the control error e ═ x of the hydraulic cylinder1-x1dIt is assumed that it needs to meet the following performance criteria:
Figure FDA00025964658400000314
in the formula:l，uthe parameters to be designed are used for assisting in restraining the upper limit and the lower limit of the control error; ρ (t) is a positive strictly increasing smoothing function, as shown by:
Figure FDA0002596465840000041
in the formula: rho0、ρ∞And k are both positive designable parameters;
in formula (11) -lρ0Anduρ0respectively restrain the control error of the output forceMaximum undershoot and overshoot of e (t), and parameter k constrains the convergence rate, ρ, of error e (t)∞The steady state bound of the error is constrained, so that equation (11) gives a specific plan for the performance of the output force control error by selecting an appropriate parameter ρ0、ρ∞、k、lAnduthe transient and stable performance of the output force control error can be planned in advance by the aid of the parameters, and the transient performance can be improved according to actual requirements of the system;
the following increasing function is established:
Figure FDA0002596465840000042
in the formula: z is a radical of1(t) is a conversion error variable corresponding to the control error e (t), and it is easy to analyze that equation (13) is equivalent to e (t) ═ ρ (t) S (z)1(t)), and z1(t) when the interface is bounded, the preset performance characteristic formula (8) is always satisfied;
an increasing function S (z) satisfying the characteristic formula (13)1) The following can be selected:
Figure FDA0002596465840000043
the inverse function of equation (14) is found:
Figure FDA0002596465840000044
for the conversion error z1Designing a controller;
the establishment function is as follows:
Figure FDA0002596465840000051
in the formula: k is a radical of1Is the feedback gain;
the controller is designed as follows:
Figure FDA0002596465840000052
in the formula: k is a radical of2Is the feedback gain;
the controller (17) can be substituted into the formula (16):
Figure FDA0002596465840000053
the adaptive law is designed as follows:
Figure FDA0002596465840000054
in the formula:
Figure FDA0002596465840000055
then, it is possible to obtain:
Figure FDA0002596465840000056
design robust controller us2The following were used:
Figure FDA0002596465840000057
in the formula β1Are parameters to be designed.
6. The hydraulic pressure preset performance adaptive asymptotic control method according to claim 5, characterized in that: the step S2c specifically includes: by performance theorem 1: selecting initial conditions for system controllρ(0)<e(0)<uρ (0) -l<λ(0)<uSimultaneous parameter β1Satisfies the following inequality:
Figure FDA0002596465840000061
while designing sufficiently large parametersk1And k2So that the following matrix Λ is a positive definite matrix
Figure FDA0002596465840000062
The control error of the output force can be guaranteed to be bounded all the time, better instruction tracking can be realized by the output force, and rho is adjusted0、ρ∞、k、lAnduthe parameters are equal, so that the control error can meet the preset performance requirement designed by the formula (11);
the following was demonstrated:
the following Lyapunov function is defined:
Figure FDA0002596465840000063
further, by further deriving V and substituting the following equations (16) and (20):
Figure FDA0002596465840000071
in the formula: z ═ Z1,z2]TThe matrix Λ is defined as equation (23) if the reasonable design parameter k is passed1And k2Making the matrix Λ positive definite makes the following satisfied:
Figure FDA0002596465840000072
in the formula: lambda [ alpha ]min(Λ) represents the minimum eigenvalue of the matrix Λ, and the analytic expression (26) shows that the Lyapunov function is bounded, and the W integral is bounded, and further shows that the conversion error amount z is bounded1And z2All are bounded, and in combination with equations (8), (16) and (17), all signals in the system are bounded, so that the derivative of W is bounded, as can be seen from the Barbalt theorem, as time approaches infinity, W approaches zero, i.e., the transformation error amount z1Approaches zero, so that the combination (11) knows that the control error e (t) is always bounded, thus proving that the controller is convergentThe system is stable.
7. The hydraulic pressure preset performance adaptive asymptotic control method according to claim 6, characterized in that: step three is specifically to adjust the parameter k of the control law u1、k2、ρ0、ρ∞、k、l、u、β1And the system meets the control performance index.
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