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CN108181818B - Robust self-adaptive control method for electro-hydraulic position servo system containing unmodeled friction dynamics - Google Patents

Robust self-adaptive control method for electro-hydraulic position servo system containing unmodeled friction dynamics Download PDF

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CN108181818B
CN108181818B CN201810158842.8A CN201810158842A CN108181818B CN 108181818 B CN108181818 B CN 108181818B CN 201810158842 A CN201810158842 A CN 201810158842A CN 108181818 B CN108181818 B CN 108181818B
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姚建勇
郑靖重
徐缙恒
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Nanjing University of Science and Technology
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Abstract

The invention discloses a robust self-adaptive control method of an electro-hydraulic position servo system containing unmodeled friction dynamics, which belongs to the field of electromechanical hydraulic servo control, and comprises the steps of firstly establishing a mathematical model of the electro-hydraulic position servo system; then designing a robust adaptive controller; and finally, analyzing the performance and stability of the robust adaptive controller. The method selects the electro-hydraulic position servo system as a research object, establishes a nonlinear model of the system, and simultaneously considers the parameter uncertainty of the system and the interference of unmodeled friction; aiming at the parameter uncertainty of the system, a self-adaptive control method is fused, and the designed parameter self-adaptive law can effectively estimate the unknown constant parameter and make the estimated value converge; the controller designed based on the dynamic standard signal method has a good compensation effect on unmodeled friction interference existing in the system.

Description

Robust self-adaptive control method for electro-hydraulic position servo system containing unmodeled friction dynamics
Technical Field
The invention relates to the technical field of electromechanical servo control, in particular to a robust self-adaptive control method of an electro-hydraulic position servo system containing unmodeled friction dynamics.
Background
The electro-hydraulic servo system has the outstanding advantages of large power-weight ratio, quick dynamic response, good pressure and flow controllability, flexible power transmission and the like, and is widely applied to the fields of aviation, aerospace, automobiles, ships, engineering machinery and the like. With the development of these fields and the continuous progress of the technical level, a high-performance electro-hydraulic servo system is urgently needed as a support, and the control performance obtained by the traditional linearization-based method can not meet the system requirement gradually. The nonlinearity of the electro-hydraulic servo system, such as pressure dynamic nonlinearity, servo valve pressure flow nonlinearity, friction nonlinearity, etc., gradually becomes a bottleneck factor limiting the performance improvement of the electro-hydraulic servo system. Besides, the electro-hydraulic servo system has parameter uncertainties such as load inertia, leakage coefficients, hydraulic oil elastic modulus and the like and uncertainty nonlinearity such as unmodeled friction dynamics, external interference and the like. The existence of these uncertainties has been a major obstacle to the development of advanced control strategies.
In order to solve the problem of nonlinear control of the electro-hydraulic servo system, a plurality of methods are sequentially proposed. As with feedback linearization control based on inverse design, which assumes that the system model is precisely known, the basic idea of the design is to dynamically linearize the error by precisely compensating for the nonlinear function in the controller. Although perfect asymptotic tracking performance can be obtained theoretically, the model of the actual system cannot be accurately known, modeling uncertainty always exists, and therefore the tracking performance obtained by theoretical analysis is certainly deteriorated. The self-adaptive control method is an effective method for processing the problem of uncertainty of parameters, and can obtain the steady-state performance of asymptotic tracking. However, the adaptive controller is designed on the premise that uncertainty nonlinearity such as external load interference does not exist in the system, and theoretically, the system parameter estimation can be guaranteed to converge to a true value when the system expected instruction meets the continuous excitation condition, and the system obtains the performance of gradual tracking. However, the system may be unstable when it is subjected to large unmodeled disturbances. The friction dynamics which is not modeled widely exists in the electro-hydraulic servo system, and has important influence on the system performance, particularly the low-speed servo performance. In order to describe more accurately the macro-microscopic behavior of friction, many researchers have proposed dynamic friction models, such as Dahl models, bristle models, LuGre models, and the like. The compensation based on the friction model has certain friction prediction capability, so that the effect of the friction compensation is more obvious, however, the accurate friction model is not easy to obtain, and the complicated friction model also has the problems that the identification of model parameters is extremely difficult and the like. When dealing with these unmodeled tribodynamic components, if they are simply ignored, the controller is ideally designed so that the actual control thereof often fails to achieve the desired effect.
Disclosure of Invention
The invention aims to provide a robust self-adaptive control method for an electro-hydraulic position servo system containing unmodeled friction dynamics.
The technical scheme for realizing the purpose of the invention is as follows: a robust self-adaptive control method for an electro-hydraulic position servo system containing unmodeled friction dynamics comprises the following steps:
step 1, establishing a mathematical model of an electro-hydraulic position servo system;
step 2, designing a robust adaptive controller;
and 3, analyzing the performance and stability of the robust adaptive controller.
Further, the establishing of the mathematical model of the electro-hydraulic position servo system in the step 1 is as follows:
step 1-1, for an electro-hydraulic position servo system, controlling a hydraulic cylinder to drive an inertial load through a servo valve, wherein the motion equation of the inertial load is as follows:
Figure BDA0001582295380000021
in the formula (1), m is the inertial load mass, y is the system output displacement, and PL＝P1-P2Is the pressure difference between two cavities of the hydraulic cylinder, P1And P2The hydraulic pressure of the left cavity and the right cavity of the hydraulic cylinder is respectively, A is the effective piston area of the left cavity and the right cavity of the hydraulic cylinder, delta (t) is friction interference, and t is a time variable.
Neglecting the external leakage of the hydraulic cylinder, the pressure dynamic equation of the two cavities of the hydraulic cylinder is as follows:
Figure BDA0001582295380000022
in the formula (2), V01And V02Respectively the initial volumes of the two chambers of the cylinder, betaeIs effective oil elastic modulus, CtIs the internal leakage coefficient, Q1And Q2The hydraulic flow into/out of the left/right chambers of the hydraulic cylinder from the servo valve is respectively. Wherein Q is1And Q2And servo valve displacement xvThe relationship of (1) is:
Figure BDA0001582295380000023
in the formula (3)
Figure BDA0001582295380000024
And defining a function s (×) as:
Figure BDA0001582295380000031
wherein, CdIs the flow coefficient, w is the spool area gradient, ρ is the oil density, PsFor supply pressure, PrIs the return oil pressure.
By adopting a servo valve with high response, the displacement of the valve core and the control input are approximately proportional links, namely xv＝kiu, so equation (3) can be written as:
Figure BDA0001582295380000032
formula (5) wherein g ═ kqkiRepresents the total flow gain, and
Figure BDA0001582295380000033
in order to accurately describe the macro-micro behavior of friction, the LuGre friction model is widely used, which describes the friction force as:
Figure BDA0001582295380000034
σ in formula (7)0，σ1，σ2The bristle stiffness, damping and system viscous damping coefficients are respectively expressed, omega is the relative motion speed of the friction surface, z is the average deformation of the bristle between two contact surfaces, and the average deformation behavior can be expressed as:
Figure BDA0001582295380000035
the mathematical model of the nonlinear function g (ω) in equation (8) is:
Figure BDA0001582295380000036
in the formula (9) fcAnd fsRespectively characterize macroscopic coulomb friction and static friction, omegasIs the Stribeck speed.
Step 1-2, defining state variables:
Figure BDA0001582295380000037
then the formula (1) is usedThe equation of motion is converted into an equation of state:
Figure BDA0001582295380000041
due to the parameters g, beta of the systeme,CtThere may be a large variation to subject the system to parameter uncertainty, and therefore, in order to simplify equation (10), an uncertainty parameter set θ is defined as [ θ ═ θ1,θ2,θ3]TWherein theta1＝gβe，θ2＝βe，θ3＝βeCt。
The state space equation (10) can be rewritten as:
Figure BDA0001582295380000042
in the formula (11)
Figure BDA0001582295380000043
The design goals of the system controller are: given system reference signal yd(t)＝x1d(t) designing a bounded control input u such that the system output y is x1The reference signal of the system is tracked as much as possible.
For the controller design, assume the following:
assume that 1: system reference command signal yd(t) is three-order continuous and the system expects that the position command, the velocity command, the acceleration command, and the jerk command are bounded; the electro-hydraulic position servo system works under the common working condition, namely the pressure of two cavities of the hydraulic cylinder meets 0<Pr<P1<Ps，0<Pr<P2<Ps。
Further, the robust adaptive controller designed in step 2 includes the following steps:
step 2-1, definition of e1＝x1-ydIs the heel of the systemTracking error, taking into account the second equation of equation (11), selecting α1Is x2Virtual control of alpha1And the true state x2Error e of2＝x2-α1Taking the Lyapunov function
Figure BDA0001582295380000051
To V1The derivation can be:
Figure BDA0001582295380000052
design of virtual control law α1The following were used:
Figure BDA0001582295380000053
in the formula k1For positive feedback gain, alpha1aFor improving the feedforward control rate of the model compensation, α1sIs a linear robust feedback term. Substituting formula (15) into formula (14):
Figure BDA0001582295380000054
step 2-2, selecting a Lyapunov function Vz(z)＝z2To V pairz(z) the derivation yields:
Figure BDA0001582295380000055
based on the LuGre model where the signal z is bounded, there is β1(|z|)＝0.5z2，β2(|z|)＝2z2，γ0(|x2|)＝x2 2E.k ∞ class function and appropriate constant c0>0，d0Function V of >0z(z) satisfies:
β1(|z|)≤Vz(z)≤β2(|z|) (18)
Figure BDA0001582295380000056
without loss of generality, assume γ within some neighborhood of the origin0(s)＝O(s2) Derived from hypothesis 1
Figure BDA0001582295380000061
(d1Is a proper constant which is constant in the above-mentioned manner,
Figure BDA0001582295380000062
is a suitable function), combined with (19) having
Figure BDA0001582295380000063
The dynamic standard signal is designed as follows:
Figure BDA0001582295380000064
wherein
Figure BDA0001582295380000065
Constant d is greater than or equal to d0+d1Then the following properties hold:
(P1) has a finite time of existence
Figure BDA0001582295380000066
When T is more than or equal to T, D (T) is 0; when t is greater than or equal to 0, there are
Vz(z)≤r(t)+D(t) (22)
Wherein D (t) ≧ 0 is defined at t ≧ 0, and r (0) > 0.
Step 2-3, considering the third equation of formula (11), selecting alpha2Is x3Virtual control of alpha2And the true state x3Error e of3＝x3-α2Taking the Lyapunov function
Figure BDA0001582295380000067
In the formula (23), λ is a design parameter, for V2The derivation can be:
Figure BDA0001582295380000068
presence of non-negative smooth function
Figure BDA0001582295380000069
(d 20 or more is an appropriate constant),
Figure BDA00015822953800000610
and unknown normality p1And p2Satisfy the requirement of
Figure BDA00015822953800000611
Thus, it can be obtained from formula (25):
Figure BDA00015822953800000612
to handle the two terms on the right side of the above equation, we refer to the hyperbolic tangent function property: for any >0, 0 ≦ x | -xtanh (x /) ≦ 0.2785, which may result in:
Figure BDA0001582295380000071
wherein,1is an arbitrary normal number which is a constant,
Figure BDA0001582295380000072
is a smooth function.
Further, the following equations (18), (22) and (27) can be given:
Figure BDA0001582295380000073
wherein,2is an arbitrary normal number which is a constant,
Figure BDA0001582295380000074
and
Figure BDA0001582295380000075
formula (27) and formula (28) are substituted in formula (25):
Figure BDA0001582295380000076
definition of
Figure BDA0001582295380000077
Representing an estimation of the unknown parameter theta, let
Figure BDA0001582295380000078
An estimation error for the parameter theta, i.e.
Figure BDA0001582295380000079
Design of virtual control law α2The following were used:
Figure BDA00015822953800000710
k in formula (31)2For positive feedback gain, alpha2aFor improving the feedforward control rate of the model compensation, α2sIs a linear robust feedback term. Substituting formula (30) into formula (29):
Figure BDA00015822953800000711
step 2-4, defining unknown parameter set p ═ p1,p2]TConsidering the fourth equation of equation (11), design practiceTaking the Lyapunov function as the control input u
Figure BDA0001582295380000081
To V3The derivation can be:
Figure BDA0001582295380000082
in the formula>0，γ>0 is a design parameter, #1＝[g3u,-f31,-f32]T。
From formula (26):
Figure BDA0001582295380000083
to handle the two terms on the right side of the above equation, we refer to the hyperbolic tangent function property: for any >0, 0 ≦ x | -xtanh (x /) ≦ 0.2785, which may result in:
Figure BDA0001582295380000084
wherein,3is an arbitrary normal number which is a constant,
Figure BDA0001582295380000085
is a smooth function.
Further, the following equations (18), (22) and (35) can be used:
Figure BDA0001582295380000086
wherein,4is an arbitrary normal number which is a constant,
Figure BDA0001582295380000087
formula (35) and formula (36) are substituted in formula (33):
Figure BDA0001582295380000091
in the formula sigmaθ>0，
Figure BDA0001582295380000092
σp>0，
Figure BDA0001582295380000093
Are all design constants, and
Figure BDA0001582295380000094
the actual controller u is designed as follows:
Figure BDA0001582295380000095
in the formula (39), k3For positive feedback gain, uaFor improving the feedforward control rate of model compensation, usIs a linear robust feedback term. Substituting formula (39) into formula (37) to obtain:
Figure BDA0001582295380000096
in the formula (40)
Figure BDA0001582295380000097
Phi in formula (41)4＝[g3ua,-f31,-f32]T。
Further, the performance and stability analysis of the robust adaptive controller in step 3 is specifically as follows:
if the electro-hydraulic position servo system (11) satisfies the assumption 1, then for bounded initial conditions, the closed loop system obtained by applying the controller has the following properties:
(P2) all signals of the closed loop system are globally consistent and ultimately bounded;
(P3) by adjusting the design parameters appropriately, the tracking error e can be made1Adjust to an arbitrarily small neighborhood of the origin.
Compared with the prior art, the invention has the following remarkable effects: (1) the method selects the electro-hydraulic position servo system as a research object, establishes a nonlinear model of the system, and simultaneously considers the parameter uncertainty of the system and the interference of unmodeled friction; aiming at the parameter uncertainty of the system, a self-adaptive control method is fused, and the designed parameter self-adaptive law can effectively estimate the unknown constant parameter and make the estimated value converge; (2) the controller designed based on the dynamic standard signal method has a good compensation effect on unmodeled friction interference existing in the system; (3) the robust self-adaptive controller of the electro-hydraulic position servo system with unmodeled friction dynamics is a full-state feedback controller, and the position output of the electro-hydraulic servo system has globally consistent and finally bounded tracking performance; (4) the controller designed by the invention has continuous control voltage and is more beneficial to application in engineering practice.
Drawings
FIG. 1 is a schematic diagram of an electro-hydraulic position servo system of the present invention.
FIG. 2 is a schematic diagram of a robust adaptive control method of an electro-hydraulic position servo system with unmodeled friction dynamics.
FIG. 3 is a schematic diagram of a process for tracking the expected command output by the system under the control of a controller.
FIG. 4 is a graph of tracking error versus tracking error for a system under the influence of a designed controller and a conventional PID controller.
FIG. 5 is a graph of estimated values of parameters over time under the control of a designed controller.
Fig. 6 is a graph of the actual control input u of the system over time under the control of a designed controller.
Detailed Description
The invention is described in further detail below with reference to the figures and the embodiments.
With reference to fig. 1-2, a robust adaptive control method for an electro-hydraulic position servo system with unmodeled friction dynamics comprises the following steps:
step 1, establishing a mathematical model of an electro-hydraulic position servo system;
(1.1) FIG. 1 is a schematic diagram of a typical electro-hydraulic position servo system in which an inertial load is driven by a servo-valve controlled hydraulic cylinder. Therefore, according to newton's second law, the equation of motion for an inertial load is:
Figure BDA0001582295380000101
in the formula (1), m is the inertial load mass, y is the system output displacement, and PL＝P1-P2Is the pressure difference between two cavities of the hydraulic cylinder, P1And P2The hydraulic pressure of the left cavity and the right cavity of the hydraulic cylinder is respectively, A is the effective piston area of the left cavity and the right cavity of the hydraulic cylinder, delta (t) is friction interference, and t is a time variable.
Neglecting the external leakage of the hydraulic cylinder, the pressure dynamic equation of the two cavities of the hydraulic cylinder is as follows:
Figure BDA0001582295380000111
in the formula (2), V01And V02Respectively the initial volumes of the two chambers of the cylinder, betaeIs effective oil elastic modulus, CtIs the internal leakage coefficient, Q1And Q2The hydraulic flow into/out of the left/right chambers of the hydraulic cylinder from the servo valve is respectively. Wherein Q is1And Q2And servo valve displacement xvThe relationship of (1) is:
Figure BDA0001582295380000112
in the formula (3)
Figure BDA0001582295380000113
And defining a function s (×) as:
Figure BDA0001582295380000114
wherein, CdIs the flow coefficient, w is the spool area gradient, ρ is the oil density, PsFor supply pressure, PrIs the return oil pressure.
Because the servo valve dynamic state is considered, an additional displacement sensor is required to be installed to obtain the displacement of the servo valve spool, and the tracking performance is only slightly improved. Thus, in the present embodiment, assuming a highly responsive servo valve is used, the spool displacement and control input are approximately proportional, i.e., xv＝kiu, so equation (3) can be written as:
Figure BDA0001582295380000115
formula (5) wherein g ═ kqkiRepresents the total flow gain, and
Figure BDA0001582295380000116
in order to accurately describe the macro-micro behavior of friction, the LuGre friction model is widely used, which describes the friction force as:
Figure BDA0001582295380000121
σ in formula (7)0，σ1，σ2The bristle stiffness, damping and system viscous damping coefficients are respectively expressed, omega is the relative motion speed of the friction surface, z is the average deformation of the bristle between two contact surfaces, and the average deformation behavior can be expressed as:
Figure BDA0001582295380000122
the mathematical model of the nonlinear function g (ω) in equation (8) is:
Figure BDA0001582295380000123
in the formula (9) fcAnd fsRespectively characterize macroscopic coulomb friction and static friction, omegasIs the Stribeck speed.
(1.2) defining state variables:
Figure BDA0001582295380000124
equation of motion (1) is converted to an equation of state:
Figure BDA0001582295380000125
due to the parameters g, beta of the systeme,CtThere may be a large variation to subject the system to parameter uncertainty, and therefore, in order to simplify equation (10), an uncertainty parameter set θ is defined as [ θ ═ θ1,θ2,θ3]TWherein theta1＝gβe，θ2＝βe，θ3＝βeCt。
The state space equation (10) can be rewritten as:
Figure BDA0001582295380000131
in the formula (11)
Figure BDA0001582295380000132
The design goals of the system controller are: given system reference signal yd(t)＝x1d(t) designing a bounded control input u such that the system output y is x1The reference signal of the system is tracked as much as possible.
For the controller design, assume the following:
assume that 1: system reference command signal yd(t) is three-order continuous and the system expects that the position command, the velocity command, the acceleration command, and the jerk command are bounded; the electro-hydraulic position servo system works under the common working condition, namely the pressure of two cavities of the hydraulic cylinder meets 0<Pr<P1<Ps，0<Pr<P2<Ps。
Step 2, designing a robust adaptive controller, comprising the following steps:
(2.1) definition of e1＝x1-ydSelecting alpha for the tracking error of the system by considering the second equation of equation (11)1Is x2Virtual control of alpha1And the true state x2Error e of2＝x2-α1Taking the Lyapunov function
Figure BDA0001582295380000133
To V1The derivation can be:
Figure BDA0001582295380000134
design of virtual control law α1The following were used:
Figure BDA0001582295380000141
in the formula k1For positive feedback gain, alpha1aFor improving the feedforward control rate of the model compensation, α1sIs a linear robust feedback term. Substituting formula (15) into formula (14):
Figure BDA0001582295380000142
(2.2) selecting a Lyapunov function Vz(z)＝z2To V pairz(z) the derivation yields:
Figure BDA0001582295380000143
based on the LuGre model where the signal z is bounded, there is β1(|z|)＝0.5z2，β2(|z|)＝2z2，γ0(|x2|)＝x2 2E.k ∞ class function and appropriate constant c0>0，d0Function V of >0z(z) satisfies:
β1(|z|)≤Vz(z)≤β2(|z|) (18)
Figure BDA0001582295380000144
without loss of generality, assume γ within some neighborhood of the origin0(s)＝O(s2) Derived from hypothesis 1
Figure BDA0001582295380000145
(d1Is a proper constant which is constant in the above-mentioned manner,
Figure BDA0001582295380000146
is a suitable function), combined with (19) having
Figure BDA0001582295380000147
The dynamic standard signal is designed as follows:
Figure BDA0001582295380000148
wherein
Figure BDA0001582295380000149
Constant d is greater than or equal to d0+d1Then the following properties hold:
(P1) has a finite time of existence
Figure BDA00015822953800001410
When T is more than or equal to T, D (T) is 0; when t is greater than or equal to 0, there are
Vz(z)≤r(t)+D(t) (22)
Wherein D (t) ≧ 0 is defined at t ≧ 0, and r (0) > 0.
Consider the demonstration of property (P1). Combining equations (20) and (21) and the gram-wale inequality can be obtained:
Figure BDA0001582295380000151
definition of
Figure BDA0001582295380000152
And due to
Figure BDA0001582295380000153
And r (0)>0, so property (P1) holds.
(2.3) selecting α in consideration of the third equation of the formula (11)2Is x3Virtual control of alpha2And the true state x3Error e of3＝x3-α2Taking the Lyapunov function
Figure BDA0001582295380000154
In the formula (24), λ is a design parameter, for V2The derivation can be:
Figure BDA0001582295380000155
although the mathematical model of Δ in the formula is known, it is due to σ0，σ1And sigma2Are unknown and difficult to identify, so friction compensation based on the LuGre model is not favorable for direct use in controller design. However, there is a non-negative smoothing function
Figure BDA0001582295380000156
(d 20 or more is an appropriate constant),
Figure BDA0001582295380000157
and unknown normality p1And p2Satisfy the requirement of
Figure BDA0001582295380000158
Thus, from equation (26):
Figure BDA0001582295380000159
to handle the two terms on the right side of the above equation, we refer to the hyperbolic tangent function property: for any >0, 0 ≦ x | -xtanh (x /) ≦ 0.2785, which may result in:
Figure BDA00015822953800001510
wherein,1is an arbitrary normal number which is a constant,
Figure BDA0001582295380000161
is a smooth function.
Further, the following equations (18), (22) and (28) can be used:
Figure BDA0001582295380000162
wherein,2is an arbitrary normal number which is a constant,
Figure BDA0001582295380000163
and
Figure BDA0001582295380000164
formula (28) and formula (29) are substituted in formula (25):
Figure BDA0001582295380000165
definition of
Figure BDA0001582295380000166
Representing an estimation of the unknown parameter theta, let
Figure BDA0001582295380000167
An estimation error for the parameter theta, i.e.
Figure BDA0001582295380000168
Design of virtual control law α2The following were used:
Figure BDA0001582295380000169
k in formula (31)2For positive feedback gain, alpha2aFor improving the feedforward control rate of the model compensation, α2sIs a linear robust feedback term. Substituting formula (31) into formula (30) to obtain:
Figure BDA00015822953800001610
(2.4) define unknown parameter set p ═ p1,p2]TConsidering the fourth equation of equation (11), the actual control input u is designed, and the Lyapunov function is taken
Figure BDA00015822953800001611
To V3The derivation can be:
Figure BDA0001582295380000171
in the formula>0，γ>0 is the design gain, #1＝[g3u,-f31,-f32]T。
From formula (26):
Figure BDA0001582295380000172
to handle the two terms on the right side of the above equation, we refer to the hyperbolic tangent function property: for any >0, 0 ≦ x | -xtanh (x /) ≦ 0.2785, which may result in:
Figure BDA0001582295380000173
wherein,3is an arbitrary normal number which is a constant,
Figure BDA0001582295380000174
is a smooth function.
Further, the following equations (18), (22) and (36) can be used:
Figure BDA0001582295380000175
wherein,4is an arbitrary normal number which is a constant,
Figure BDA0001582295380000176
formula (36) and formula (37) are substituted in formula (34):
Figure BDA0001582295380000177
in the formula sigmaθ>0，
Figure BDA0001582295380000178
σp>0，
Figure BDA0001582295380000179
Are all design constants, and
Figure BDA0001582295380000181
the actual controller u is designed as follows:
Figure BDA0001582295380000182
k in formula (40)3For positive feedback gain, uaFor improving the feedforward control rate of model compensation, usIs a linear robust feedback term. Substituting formula (40) into formula (38) to obtain:
Figure BDA0001582295380000183
in the formula (41)
Figure BDA0001582295380000184
In formula (42) < CHEM >4＝[g3ua,-f31,-f32]T。
Step 3, analyzing the performance and stability of the robust adaptive controller, specifically as follows:
if the electro-hydraulic position servo system (11) satisfies the assumption 1, then for bounded initial conditions, the closed loop system obtained by applying the controller has the following properties:
(P2) all signals of the closed loop system are globally consistent and ultimately bounded;
(P3) by adjusting the design parameters appropriately, the tracking error e can be made1Adjust to an arbitrarily small neighborhood of the origin.
Consider the demonstration of property (P2). In order to prove the stability of a closed-loop system, a Lyapunov function form is selected as follows: v is V3Defining:
Figure BDA0001582295380000191
selection of adaptive law
Figure BDA0001582295380000192
The formula (40) can be rewritten as
Figure BDA0001582295380000193
Obviously, when t ≧ 0, d3(t) is not less than 0; when T is more than or equal to T, d3(t) is 0. Thus, it is possible to obtain:
Figure BDA0001582295380000194
combination of formula (45) and formula (46) is readily demonstrated
Figure BDA0001582295380000195
Thus, based on the definition of equation (33), signal ei，r，
Figure BDA0001582295380000196
And xi(i ═ 1,2,3) is globally consistent and finally bounded, as evidenced by property (P2).
Consider the demonstration of property (P3). From formula (47) to
Figure BDA0001582295380000197
Existence time T '(T is less than or equal to T'<+ ∞) such that when t is>At T' | e1(t)|<. By adjusting the design parameter k appropriatelyi，j，λ，，γ，σθ，σp(j ═ 1,2,3,4) may be such that
Figure BDA0001582295380000198
Arbitrarily small, and thus arbitrarily small, properties (P3) are warranted.
Examples
In order to assess the performance of the designed controller, the following parameters are taken in simulation to model the electro-hydraulic position servo system:
pressure P of fuel supplys＝12×106Pa, oil return pressure Pr0Pa, initial volume V of two chambers of the hydraulic cylinder01＝V02＝3.981×10-5m3Effective piston area a of the left and right chambers of the hydraulic cylinder is 9.0478 × 10-4m2Mass m of inertial load is 30kg, and internal leakage coefficient Ct＝3×10-12m3(s/Pa), total flow gain
Figure BDA0001582295380000199
Effective oil elastic modulus betae＝7×108Pa, bristle stiffness σ0＝1×105N/m, damping
Figure BDA0001582295380000201
Coefficient of system viscous damping σ280N · s/m, coulomb friction fc30N, static friction fs42N, Stribeck speed ωs＝0.01m/s。
The expected instruction for a given system is x1d＝0.02sin(2t)[1-exp(-0.01t3)](m), the parameters of the controller designed by the invention are selected as follows: k is a radical of1＝1000，k2＝600，k3＝100，
Figure BDA0001582295380000202
Figure BDA0001582295380000203
1＝2＝3＝4＝0.5，λ＝1，＝diag{1×10-5,5.4×109,1×10-9}，Υ＝diag{1,5×10-3}，σθ＝1×10-16，σp＝0.5，θ°＝0，p°＝1，
Figure BDA0001582295380000204
d＝0.09，d2＝0；PID, selecting control parameters as follows: k is a radical ofp＝1600，ki＝1000，kd＝0。
Comparing simulation results:
fig. 3 is a tracking process of the system output to the expected instruction under the action of the designed controller, fig. 4 is a curve of the position tracking error of the system under the action of the controller designed by the present invention and the conventional PID controller respectively, which changes with time, and it can be seen from fig. 4 that the tracking error of the system under the action of the controller designed by the present invention is significantly smaller than the tracking error of the system under the action of the PID controller, so that the tracking performance is greatly improved.
FIG. 5 is a real value of a parameter of an electro-hydraulic position servo system and a time-varying curve of an estimated value thereof, and it can be seen from the curve that an adaptive law is designed to make the estimated value of the parameter of the system converge to the real value thereof, so that an unknown constant parameter of the system can be accurately estimated.
FIG. 6 is a time-varying control input curve of the electro-hydraulic position servo system, and it can be seen that the control input signals obtained by the present invention are continuous, which is advantageous for practical engineering applications.

Claims (3)

1. A robust self-adaptive control method for an electro-hydraulic position servo system containing unmodeled friction dynamics is characterized by comprising the following steps of:
step 1, establishing a mathematical model of an electro-hydraulic position servo system; the method comprises the following specific steps:
step 1-1, for an electro-hydraulic position servo system, controlling a hydraulic cylinder to drive an inertial load through a servo valve, wherein the motion equation of the inertial load is as follows:
Figure FDA0002720495150000015
in the formula (1), m is the inertial load mass, y is the system output displacement, and PL＝P1-P2Is the pressure difference between two cavities of the hydraulic cylinder, P1And P2Respectively hydraulic pressure in left and right chambers of the hydraulic cylinder, A is the hydraulic cylinder with the left and right chambersEffective piston area, delta (t) is frictional interference, and t is time variable;
neglecting the external leakage of the hydraulic cylinder, the pressure dynamic equation of the two cavities of the hydraulic cylinder is as follows:
Figure FDA0002720495150000011
in the formula (2), Va＝V01+Ay、Vb＝V02Ay is the total volume of the two chambers, V01And V02Respectively the initial volumes of the two chambers of the cylinder, betaeIs effective oil elastic modulus, CtIs the internal leakage coefficient, Q1And Q2Hydraulic flow rates entering/exiting from the left/right chambers of the hydraulic cylinder through the servo valve respectively; wherein Q is1And Q2And servo valve displacement xvThe relationship of (1) is:
Figure FDA0002720495150000012
in the formula (3)
Figure FDA0002720495150000013
And defining a function s (×) as:
Figure FDA0002720495150000014
wherein, CdIs the flow coefficient, w is the spool area gradient, ρ is the oil density, PsFor supply pressure, PrIs the return oil pressure;
the valve core displacement and the control input are approximately proportional links, namely xv＝kiu, so equation (3) can be written as:
Figure FDA0002720495150000021
formula (5) wherein g ═ kqkiRepresents the total flow gain, and
Figure FDA0002720495150000022
the LuGre friction model describes the friction force as:
Figure FDA0002720495150000023
σ in formula (7)0，σ1，σ2The bristle stiffness, damping and system viscous damping coefficients are respectively expressed, omega is the relative motion speed of the friction surface, z is the average deformation of the bristle between two contact surfaces, and the average deformation behavior can be expressed as:
Figure FDA0002720495150000024
the mathematical model of the nonlinear function g (ω) in equation (8) is:
Figure FDA0002720495150000025
in the formula (9) fcAnd fsRespectively characterize macroscopic coulomb friction and static friction, omegasIs the Stribeck speed;
step 1-2, defining state variables:
Figure FDA0002720495150000026
equation of motion (1) is converted to an equation of state:
Figure FDA0002720495150000027
defining an uncertainty parameter set θ ═ θ1,θ2,θ3]TWherein theta1＝gβe，θ2＝βe，θ3＝βeCt；
The state space equation (10) can be rewritten as:
Figure FDA0002720495150000031
in the formula (11)
Figure FDA0002720495150000032
The design goals of the system controller are: given system reference signal yd(t)＝x1d(t) designing a bounded control input u such that the system output y is x1Tracking the reference signal of the system as much as possible;
for the controller design, assume the following:
assume that 1: system reference command signal yd(t) is three-order continuous and the system expects that the position command, the velocity command, the acceleration command, and the jerk command are bounded; the electro-hydraulic position servo system works under the common working condition that the pressure of two cavities of the hydraulic cylinder meets the condition that P is more than 0r＜P1＜Ps，0＜Pr＜P2＜Ps；
Step 2, designing a robust adaptive controller;
and 3, analyzing the performance and stability of the robust adaptive controller.
2. The robust adaptive control method for the electro-hydraulic position servo system with unmodeled friction dynamics as claimed in claim 1, wherein the step 2 of designing the robust adaptive controller comprises the following steps:
step 2-1, definition of e1＝x1-ydSelecting alpha for the tracking error of the system by considering the second equation of equation (11)1Is x2Is virtualizedControl of alpha1And the true state x2Error e of2＝x2-α1Taking the Lyapunov function
Figure FDA0002720495150000033
To V1The derivation can be:
Figure FDA0002720495150000034
design of virtual control law α1The following were used:
Figure FDA0002720495150000041
in the formula k1For positive feedback gain, alpha1aFor improving the feedforward control rate of the model compensation, α1sIs a linear robust feedback term; substituting formula (15) into formula (14):
Figure FDA0002720495150000042
step 2-2, selecting a Lyapunov function Vz(z)＝z2To V pairz(z) the derivation yields:
Figure FDA0002720495150000043
based on the LuGre model where the signal z is bounded, there is β1(|z|)＝0.5z2，β2(|z|)＝2z2，γ0(|x2|)＝x2 2Function of class e K and constant c0＞0，d0Function V of >0z(z) satisfies:
β1(|z|)≤Vz(z)≤β2(|z|) (18)
Figure FDA0002720495150000044
let y be assumed to be within a certain neighborhood of the origin0(s)＝O(s2) Derived from hypothesis 1
Figure FDA0002720495150000045
d1Is a constant number of times that the number of the first,
Figure FDA0002720495150000046
is a function, combined with (19) having
Figure FDA0002720495150000047
The dynamic standard signal is designed as follows:
Figure FDA0002720495150000048
wherein
Figure FDA0002720495150000049
Constant d is greater than or equal to d0+d1Then the following properties hold:
(P1) has a finite time of existence
Figure FDA0002720495150000051
When T is more than or equal to T, D (T) is 0; when t is greater than or equal to 0, there are
Vz(z)≤r(t)+D(t) (22)
Wherein D (t) is not less than 0, t is not less than 0, and r (0) > 0;
step 2-3, considering the third equation of formula (11), selecting alpha2Is x3Virtual control of alpha2And the true state x3Error e of3＝x3-α2Taking the Lyapunov function
Figure FDA0002720495150000052
In the formula (23), λ is a design parameter, for V2The derivation can be:
Figure FDA0002720495150000053
presence of non-negative smooth function
Figure FDA0002720495150000054
And unknown normality p1And p2Satisfy the requirement of
Figure FDA0002720495150000055
d2More than or equal to 0 is a constant;
thus, it can be obtained from formula (25):
Figure FDA0002720495150000056
to handle the two terms on the right side of the above equation, we refer to the hyperbolic tangent function property: for any >0, 0 ≦ x | -x tanh (x /) for 0.2785, yielding:
Figure FDA0002720495150000057
wherein,1is an arbitrary normal number which is a constant,
Figure FDA0002720495150000058
is a smooth function;
further, the following equations (18), (22) and (27) can be given:
Figure FDA0002720495150000061
wherein,2is an arbitrary normal number which is a constant,
Figure FDA0002720495150000062
and
Figure FDA0002720495150000063
formula (27) and formula (28) are substituted in formula (25):
Figure FDA0002720495150000064
definition of
Figure FDA0002720495150000065
Representing an estimation of the unknown parameter theta, let
Figure FDA0002720495150000066
An estimation error for the parameter theta, i.e.
Figure FDA0002720495150000067
Design of virtual control law α2The following were used:
Figure FDA0002720495150000068
k in formula (31)2For positive feedback gain, alpha2aFor improving the feedforward control rate of the model compensation, α2sIs a linear robust feedback term; substituting formula (30) into formula (29):
Figure FDA0002720495150000069
step 2-4, defining unknown parameter set p ═ p1,p2]TConsidering the fourth equation of equation (11), the actual control input u is designed, and the Lyapunov function is taken
Figure FDA00027204951500000610
To V3The derivation can be:
Figure FDA0002720495150000071
wherein >0, upsilon >0 is design gain, psi1＝[g3u,-f1,-f2]T；
From formula (26):
Figure FDA0002720495150000072
to handle the two terms on the right side of the above equation, we refer to the hyperbolic tangent function property: for any >0, 0 ≦ x | -x tanh (x /) for 0.2785, yielding:
Figure FDA0002720495150000073
wherein,3is an arbitrary normal number which is a constant,
Figure FDA0002720495150000074
is a smooth function;
further, the following equations (18), (22) and (35) can be used:
Figure FDA0002720495150000075
wherein,4is an arbitrary normal number which is a constant,
Figure FDA0002720495150000076
formula (35) and formula (36) are substituted in formula (33):
Figure FDA0002720495150000077
in the formula sigmaθ＞0，
Figure FDA0002720495150000078
σp＞0，
Figure FDA0002720495150000079
Are all design constants, and
Figure FDA0002720495150000081
the actual controller u is designed as follows:
Figure FDA0002720495150000082
in the formula (39), k3For positive feedback gain, uaFor improving the feedforward control rate of model compensation, usIs a linear robust feedback term; substituting formula (39) into formula (37) to obtain:
Figure FDA0002720495150000083
in the formula (40)
Figure FDA0002720495150000084
Phi in formula (41)4＝[g3ua,-f1,-f2]T。
3. The robust adaptive control method for the electro-hydraulic position servo system containing unmodeled friction dynamics as recited in claim 2, wherein the performance and stability analysis of the robust adaptive controller in step 3 is as follows:
if the electro-hydraulic position servo system meets the assumption 1, the closed-loop system obtained by applying the controller has the following properties for the bounded initial condition:
(P2) all signals of the closed loop system are globally consistent and ultimately bounded;
(P3) adjusting the design parameters to make the tracking error e1Adjust to an arbitrarily small neighborhood of the origin.
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