CN104698844B - Uncertainty compensatory sliding-mode control method of hydraulic position servo system - Google Patents

Uncertainty compensatory sliding-mode control method of hydraulic position servo system Download PDF

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CN104698844B
CN104698844B CN201510066478.9A CN201510066478A CN104698844B CN 104698844 B CN104698844 B CN 104698844B CN 201510066478 A CN201510066478 A CN 201510066478A CN 104698844 B CN104698844 B CN 104698844B
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姚建勇
刘龙
邓文翔
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Nanjing University of Science and Technology
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Abstract

The invention discloses an uncertainty compensatory sliding-mode control method of a hydraulic position servo system. The uncertainty compensatory sliding-mode control method comprises the steps of firstly, establishing a mathematic model of the hydraulic position servo system, then respectively designing a mismatching disturbance observer and a matching disturbance observer, designing a sliding-mode controller based on the mismatching disturbance observer and the matching disturbance observer, and finally proving the global asymptotic stability of the system according to a Lyapunov stability principle. By means of the uncertainty compensatory sliding-mode control method, the gain of discontinuous terms of a sliding mode is remarkably reduced, and meanwhile accelerated speed information of the system is not used, so that matching uncertainty nonlinearity and mismatching uncertainty nonlinearity simultaneously exist in the system, asymptotic tracking steady-state performance is obtained, the capacity of resisting to matching and mismatching uncertainty and nonlinearity of the uncertainty compensatory sliding-mode control method applied in the hydraulic position servo system is improved, and good tracking performance is obtained.

Description

The sliding-mode control of the uncertain compensation of Hydraulic Position Servo
Technical field
The invention belongs to electro-hydraulic servo control technical field, the uncertain benefit of particularly a kind of Hydraulic Position Servo The sliding-mode control repaid.
Background technology
Hydrauservo System has that power to weight ratio is big, response is fast and the rigid outstanding advantages such as strong of anti-loading, in numerous important necks Find broad application in domain.Electrohydraulic servo system is a typical nonlinear system, comprising many nonlinear characteristics and modeling It is uncertain.Nonlinear characteristic has servo valve flow pressure non-linear, non-linear in tribology etc..Modeling is uncertain to include parameter not Certainty and uncertainty is non-linear, and wherein parameter uncertainty mainly has load quality, the viscosity friction coefficient of executor, lets out The coefficient of leakage, servo valve flow gam, hydraulic oil elastic modelling quantity etc., it is uncertain it is non-linear mainly have do not model friction dynamic, System high-order dynamic, outer interference and do not model leakage etc..When electrohydraulic servo system is to high accuracy, high frequency sound development, system is presented Impact of the nonlinear characteristic to systematic function it is more notable, and model probabilistic presence and can make to set with system nominal plant model The controller of meter is unstable or depression of order, therefore electrohydraulic servo system nonlinear characteristic and modeling uncertainty are to limit systematic function The key factor of lifting.With industry and the continuous progress of national defence technical merit, conventional linear Design Theory was based in the past Controller can not gradually meet the high performance demands of system, it is therefore necessary to for the nonlinear characteristic in electrohydraulic servo system The more advanced non-linear control strategy of research.
Matching and mismatched uncertainty and nonlinear Control problem for electrohydraulic servo system, many methods quilt in succession Propose.In the design of Hydraulic Position Servo controller, for electrohydraulic servo system exist matching and it is unmatched not Definitiveness, the basic thought of back stepping control design is by accurately being compensated nonlinear function so that error in the controller Dynamical linearization.Although can obtain perfect asymptotic tracking performance in theory, the model of real system is impossible essence Really known, always presence modeling is uncertain, therefore can deteriorate the tracking performance of theory analysis acquisition.Self-adaptation control method Although can the probabilistic problem of processing parameter, be could to obtain gradually in the case where hypothesis system does not have outer interference The steady-state behaviour of close tracking.Due to there is mismatched uncertainty in system, the basic ideas of traditional sliding-mode control are Mismatched with matching uncertainties so as to reach sliding-mode surface to overcome by the robustness for increasing controller, but, even if reaching After sliding-mode surface, systematically tracking error cannot be still zero under the interference of mismatching uncertain, can only obtain one and not With a related uniformly bounded steady track error of uncertain supremum.Also, by increasing discontinuous term gain Method is likely to activating system high frequency dynamic in practice, makes system unstability increasing the robustness of controller.Thus pass The sliding-mode control of system has very big engineering limitations.
The content of the invention
It is an object of the invention to provide a kind of sliding formwork control side of the uncertain compensation of liquid Hydraulic Position Servo Method.
The technical solution for realizing the object of the invention is:A kind of cunning of the uncertain compensation of Hydraulic Position Servo Mould control method, comprises the following steps:
Step 1, the mathematical model for setting up Hydraulic Position Servo;
Step 2, separately design mismatch and match interference observer;
Step 3, design are based on the sliding mode controller for mismatching and matching interference observer;
Step 4, according to Liapunov stability principle proof system asymptotically stable in the large.
Compared with prior art, its remarkable advantage is the present invention:(1) dexterously design terminal sliding formwork disturbance-observer of the invention The matching of device observation Hydraulic Position Servo and mismatched uncertainty, and will matching and not in design sliding mode controller Fall with uncertain compensation, significantly cut down sliding formwork discontinuous term gain;(2) present invention does not use system acceleration information, So that system exists concurrently with matching and the non-linear performance for obtaining asymptotic tracking of mismatched uncertainty, sliding formwork control is enhanced Approach application resists matching and mismatched uncertainty and nonlinear ability in Hydraulic Position Servo;(3) present invention Solve the problems, such as that sliding-mode control tracking error in the case where system has mismatched uncertainty cannot converge to zero, And obtain the tracking performance that steady track error is zero.
Description of the drawings
Fig. 1 is the sliding-mode control flow chart of the uncertain compensation of the Hydraulic Position Servo of the present invention.
Fig. 2 is the schematic diagram of the Hydraulic Position Servo of the present invention.
Fig. 3 is the sliding-mode control principle schematic of the uncertain compensation of the Hydraulic Position Servo of the present invention.
Fig. 4 is the lower system output of the sliding mode controller effect of the matching of the present invention and mismatched uncertainty compensation to expecting The tracking process of instruction.
Fig. 5 is the position tracking of the lower system of the sliding mode controller effect of the matching of the present invention and mismatched uncertainty compensation The time dependent curve of error.
Fig. 6 be the mismatch of the lower system of the sliding mode controller effect of the matching of the present invention and mismatched uncertainty compensation not Definitiveness observation curve.
Fig. 7 be the mismatch of the lower system of the sliding mode controller effect of the matching of the present invention and mismatched uncertainty compensation not Definitiveness observation error changes over curve.
Fig. 8 is that the matching of the lower system of the sliding mode controller effect of the matching of the present invention and mismatched uncertainty compensation is not true Qualitative observation curve.
Fig. 9 is that the matching of the lower system of the sliding mode controller effect of the matching of the present invention and mismatched uncertainty compensation is not true Qualitative observation error changes over curve.
Figure 10 is that the sliding mode controller effect of the matching of the present invention and mismatched uncertainty compensation is lower and without mismatching not Position tracking error under the sliding mode controller effect of definitiveness compensation changes over curve.
Figure 11 is that the control of the lower system of the sliding mode controller effect of the matching of the present invention and mismatched uncertainty compensation is defeated Enter time dependent curve.
Figure 12 be the lower system of sliding mode controller effect without mismatched uncertainty compensation of the present invention control input with The curve of time change.
Specific embodiment
Below in conjunction with the accompanying drawings and specific embodiment is described in further detail to the present invention.
With reference to Fig. 1~3, the sliding-mode control of the uncertain compensation of the Hydraulic Position Servo of the present invention, including Following steps:
Step 1, the mathematical model for setting up Hydraulic Position Servo;
Step 1-1, Hydraulic Position Servo are the system of the fluid motor-driven inertia load by servo valve control; According to Newton's second law, the equation of motion of inertia load is:
In formula (1), m is inertia load parameter;PLFor two chamber pressure reduction of hydraulic motor;Discharge capacities of the A for hydraulic motor;B is viscosity Coefficient of friction;For modeling error, including m, PL, B nominal value and actual value between deviation and outer load it is dry Disturb;Displacements of the y for inertia load;For the speed of inertia load,For the acceleration of inertia load;T is time variable;
Ignore the external leakage of hydraulic motor, then the Pressure behaviour equation in two chamber of hydraulic motor is:
P in formula (2)1And P2The respectively pressure in two chamber of hydraulic motor,WithRespectively P1And P2Derivative;V1=V01+ Ay,V2=V02- Ay, V1And V2The control volume in two chamber of hydraulic motor is represented respectively;V01And V02Respectively hydraulic motor two chamber Original volume;βeFor effective fluid elastic modelling quantity;CtFor internal leakage coefficient;q1(t) and q2T () is respectively P1And P2Dynamical equation Modeling error;Q1And Q2The respectively oil suction chamber flow and oil back chamber flow of hydraulic motor;Q1And Q2With servo valve displacement xvPass It is to be:
In formula (3)s(xv) definition be:
Wherein, kqFor flow gain, CdDischarge coefficient;ω is valve element area gradient;ρ is fluid density;PsIt is to supply oil pressure Power, PrFor return pressure;Two cavity pressure of hydraulic motor meets 0 < Pr< P1< Ps, 0 < Pr< P2< Ps,|PL| < < PS
Due to considering that servo valve dynamic needs to install extra displacement transducer to obtain the displacement of valve core of servo valve, and For tracking performance only has small lifting;Therefore the dynamic of servo valve is all ignored in a large amount of related researchs, it is assumed that use The servo valve of high response, spool displacement are approximately proportional component i.e. x with control inputv=kiU, therefore formula (3) can be write as
K in formula (5)t=kqkiRepresent total flow gain, kiFor servo valve gain, u is that Hydraulic Position Servo is defeated Enter,
Step 1-2, definition status variable:Then the state equation of system is:
System physical parameter m, B, β in formula (6)e、kt、V01、V02And CtIt is name in the design of observer and controller Value, its deviation between actual value are concentrated and are placed in non-modeled terms, be d (x, t), be q in third channel in leading to second (t);Wherein, d (x, t) is the mismatched uncertainty of system, including outer load disturbance, does not model friction and Unmarried pregnancy;q T () is the uncertainty of the modeling error of Pressure behaviour, i.e. system matches;Wherein:
Due to hydraulic system parameters m, B, βe,ktAnd CtAffected to change by various factors (such as temperature, component wear degree etc.) It is very big, therefore for simplified system state equation, definition:
d1(x, t)=d (x, t)
Because | PL| < < PS, so as to g (x) ≠ 0;Second channel mismatched uncertainty d1(x, t) and third channel are matched Uncertain d2(x, t) is all bounded, i.e.,:
|d1(x,t)|≤D1,|d2(x,t)|≤D2
D in formula1、D2Respectively | d1(x, t) | and | d2(x, t) | the upper bound, is all known positive number, and d1(x, t) single order is led Number is present;Then Hydraulic Position Servo model is
Step 2, separately designs mismatch and matches interference observer, and step is as follows:
Step 2-1, design mismatch interference observer:
Definition mismatches interference observer sliding-mode surface s1For:
s1=z1-x2 (10)
Wherein, z1To mismatch dynamic in interference observer;
In formula (11), k1、β1、ε1、p1And q1It is mismatch interference observer coefficient;p1< q1, and positive odd number is, k1、 β1、ε1It is positive number, β1≥D1
Then d1The estimation of (x, t)For:
Had by formula (10), (11):
Definition mismatches interference observer Lyapunov Equation:
Again because of β1≥D1, then:
If there is a positive definite integral form V0T () is met with lower inequality:
Then, V0T () is in time tsEquilibrium point is converged to inside, wherein
Wherein, 0,0 < γ < 1 of α > 0, λ >;
Therefore, V1T () will be in Finite-time convergence to equilibrium point, i.e. s1To be zero in finite time, nowTo also receive Zero is held back, and because of d1(x, t) estimation difference
Then probabilistic estimation differenceAlso will be 0 in finite time;I.e. after finite time
Obtain mismatching interference observer:
Step 2-2, design matching interference observer:
Definition matching interference observer sliding-mode surface s2For:
s2=z2-x3 (20)
Wherein, z2To match dynamic in interference observer;
In formula (21), k2、β2、ε2、p2And q2It is matching interference observer coefficient;Wherein p2< q2, and positive odd number is, k2、β2、ε2It is positive number, β2≥D2;Then d2The estimation of (x, t)For:
Had by formula (21), (22):
Definition matching interference observer Lyapunov Equation:
Again because of β2> D2, then have,
Therefore, V2T () will be zero in finite time, i.e. s2To be zero in finite time, nowZero will also be converged to again Because of d2(x, t) estimation difference
The estimation difference then disturbedAlso will be zero in finite time, i.e., after finite time Obtain matching interference observer:
Step 3, design are based on the sliding mode controller for mismatching and matching interference observer, specific as follows:
Define Hydrauservo System position tracking error e0, speed Tracking error e1, acceleration tracking error e2, plus accelerate Degree tracking error e3
e0(t)=x1-xd(t) (27)
Wherein, xdT () is system reference signal, xdT () is that three ranks are continuous, and system reference position signalling xd(t), it is System reference speed signalSystem reference acceleration signalAnd system reference acceleration signalAll it is bounded;
Define sliding mode controller sliding-mode surface s:
Wherein c1、c2、c3Sliding mode controller parameter is, and is all higher than zero, and cause expression formula It is Hurwitz, then has:
Obtaining sliding mode controller u is:
Wherein k is sign (s) gains, and k > 0.
Step 4, stability is carried out with Lyapunov stability theory prove, obtain the globally asymptotically stable of system As a result;It is specific as follows:
Step 4-1, by formula (33) substitute into formula (32) have:
Define sliding mode controller Lyapunov Equation:
Then have:
Then sliding mode controller sliding-mode surface s will be zero in finite time;Now have:
Again because of s1、s2And finite time in be 0, if t1For the moment that s is zero, t2For s1It is zero moment, t3For s2For Zero moment, then there is t4=max { t1,t2,t3, through t4Have after moment:
And because:
Then:
e2(t)+c1e1(t)+c2e0(t)+c3∫e0(t) dt=d1(t) (40)
I.e.:
Step 4-2, whenWhen:
Then have, when t → ∞ has:
Therefore e0(t)=x1-xd(t) the time tend to it is infinite under conditions of go to zero;
WhenWhen δ is a certain normal number:
Then have, when t → ∞ has:
Therefore e0(t)=x1-xd(t) the time tend to it is infinite under conditions of converge in Uniformly stable circle.
In summary, for Hydraulic Position Servo design matching and the sliding formwork control of mismatched uncertainty compensation Device can make system obtain globally asymptotically stable result.Adjust observer coefficient k1、β1、ε1、p1、q1、k2、β2、ε2、p2、q2Can So that the tracking error of observer goes to zero in finite time, gain c is adjusted1、c2、c3, k can make the tracking error of system The time tend to it is infinite under conditions of go to zero.The sliding formwork control that Hydraulic Position Servo is matched and mismatched uncertainty is compensated Device principle schematic processed is as shown in Figure 3.
With reference to specific embodiment, the present invention will be further described.
Embodiment 1
With reference to Fig. 1~Fig. 3, it is the designed controller performance of examination, takes following parameter in simulations and electro-hydraulic position is watched Dress system is modeled:
Load rotating inertia m=40kgm2, motor displacement A=2 × 10-4m3/ rad, viscosity friction coefficient B=80Nm S/rad, charge oil pressure Ps=7MPa, return pressure Pr=0, fluid elastic modelling quantity βe=2 × 108Pa, two chamber original volume V of motor01 =V02=1 × 10-3m3, leadage coefficient Ct=9 × 10-12m3/s/Pa, overall throughput gainPressure Power dynamic modeling error q1(t)=q2(t)=6 × 10-6m3Rad/s, outer load disturbance f (t)=200 [1-exp (- 0.1t3)] (N·m)。
The expectation instruction of given system is x1d=sin (t) [1-exp (- 0.01t3)](rad)。
Take following controller to compare:
Matching and the sliding mode controller of mismatched uncertainty compensation:Take mismatch interference observer parameter k1=1000, β1 =0.002, ε1=0.05, p1=3, q1=5, match interference observer parameter k2=1000, β2=0.006, ε2=0.05, p2= 5, q2=7, controller parameter c1=24, c2=192, c3=512, k=2.
The sliding mode controller of matching uncertainties compensation:Matching interference observer parameter k2=1000, β2=0.006, ε2= 0.05, p2=5, q2=7, controller parameter c1=24, c2=192, c3=512, k=200.
The lower system of sliding mode controller effect of matching and mismatched uncertainty compensation exports tracking, device to expectation instruction Tracking error is as shown in Figure 4, Figure 5;In Fig. 4, expectation instruction is almost overlapped with system curve of output;From Fig. 4 and Fig. 5, With under the sliding mode controller effect compensated with mismatched uncertainty, the position of Hydrauservo System exports the tracking essence to instructing Degree is very high, the amplitude about 1 × 10 of steady track error-6(rad)。
Fig. 6, Fig. 7 are the mismatching uncertains of the lower system of the sliding mode controller effect of matching and mismatched uncertainty compensation Property observation curve and observation error change over curve, and in Fig. 6, mismatched uncertainty estimated value is truly mismatched with system Uncertain curve is essentially coincided;Designed terminal sliding mode interference observer be can be seen that from Fig. 6, Fig. 7 to watch hydraulic position The mismatched uncertainty of dress system estimates that very accurately the observation error of mismatched uncertainty is after the very short time Converge to rapidly zero;
Fig. 8, Fig. 9 are the matching uncertainties of the lower system of the sliding mode controller effect of matching and mismatched uncertainty compensation Observation curve and observation error change over curve, and in Fig. 8, matching uncertainties estimated value is not known with system true match Linearity curve is essentially coincided;As can be seen that designed terminal sliding mode interference observer is to hydraulic position servo system from Fig. 8, Fig. 9 The matching uncertainties of system estimate that very accurately the observation error of matching uncertainties is received also rapid after the very short time Hold back zero;
Figure 10, Figure 11 are the sliding mode controller of matching and mismatched uncertainty compensation and compensate without mismatched uncertainty The tracking error of sliding mode controller change over curve and control input changes over curve;From fig. 10 it can be seen that The tracking accuracy of the sliding mode controller with mismatching uncertain compensation is very high, steady-state error about 1 × 10-6(rad), and Under sliding mode controller effect without mismatched uncertainty compensation, steady track error about 3 × 10-5(rad), meanwhile, by scheming Understand, the transient state tracking error of the lower system of sliding mode controller effect without mismatched uncertainty compensation is also mismatched not than having Definitiveness compensating controller it is big.
From Figure 11, Figure 12, the discontinuous term ratio of gains of the sliding mode controller with mismatching uncertain compensation is invariably The sliding mode controller of matching uncertainties compensation is much smaller, and with less steady-state error.Therefore understand, with mismatch The sliding mode controller of uncertain compensation largely reducing the shake of controller, and ensure that more accurate tracking accuracy.
The present invention has merged the thought of disturbance-observer and compensation based on traditional sliding Mode Robust Control (SMC) method, for Hydraulic Position Servo designs the sliding formwork control of the matching based on terminal sliding mode interference observer and mismatched uncertainty compensation Method processed;The method dexterously observe the matching of Hydraulic Position Servo and mismatch not by design terminal sliding formwork interference observer Definitiveness, and matching and mismatched uncertainty compensation are fallen in design sliding mode controller, significantly cut down sliding formwork and do not connect Continuous item gain, while not using system acceleration information so that system exists concurrently with matching and mismatched uncertainty non-thread Property obtain the steady-state behaviour of asymptotic tracking, enhance sliding-mode control be used in Hydraulic Position Servo opposing matching and Mismatched uncertainty and nonlinear ability, and obtain good tracking performance.

Claims (4)

1. the sliding-mode control that a kind of uncertainty of Hydraulic Position Servo is compensated, it is characterised in that including following step Suddenly:
Step 1, the mathematical model for setting up Hydraulic Position Servo;
Step 2, separately design mismatch and match interference observer;
Step 3, design are based on the sliding mode controller for mismatching and matching interference observer;
Step 4, according to Liapunov stability principle proof system asymptotically stable in the large;
Wherein, the mathematical model for setting up Hydraulic Position Servo described in step 1, specific as follows:
Step 1-1, Hydraulic Position Servo are the system of the fluid motor-driven inertia load by servo valve control;According to Newton's second law, the equation of motion of inertia load is:
In formula (1), m is inertia load parameter;PLFor two chamber pressure reduction of hydraulic motor;Discharge capacities of the A for hydraulic motor;B is viscous friction Coefficient;For modeling error, including m, PL, B nominal value and actual value between deviation and outer load disturbance;Y is The displacement of inertia load;For the speed of inertia load,For the acceleration of inertia load;T is time variable;
Ignore the external leakage of hydraulic motor, the Pressure behaviour equation in two chamber of hydraulic motor is:
P in formula (2)1And P2The respectively pressure in two chamber of hydraulic motor,WithRespectively P1And P2Derivative;V1=V01+Ay,V2 =V02- Ay, V1And V2The control volume in two chamber of hydraulic motor is represented respectively;V01And V02The respectively initial appearance in two chamber of hydraulic motor Product;βeFor effective fluid elastic modelling quantity;CtFor internal leakage coefficient;q1(t) and q2T () is respectively P1And P2The modeling of dynamical equation is missed Difference;Q1And Q2The respectively oil suction chamber flow and oil back chamber flow of hydraulic motor;Q1And Q2With servo valve displacement xvRelation be:
In formula (3)s(xv) definition be:
Wherein, kqFor flow gain, CdDischarge coefficient;ω is valve element area gradient;ρ is fluid density;PsFor charge oil pressure, PrFor Return pressure;Two cavity pressure of hydraulic motor meets 0 < Pr< P1< Ps, 0 < Pr< P2< Ps,|PL| < < PS
Servo valve of the Hydraulic Position Servo using altofrequency response, spool displacement are approximately proportional component i.e. with control input xv=kiU, therefore formula (3) can be write as
K in formula (5)t=kqkiRepresent total flow gain, kiFor servo valve gain, u is Hydraulic Position Servo input,
Step 1-2, definition status variable:Then the state equation of system is:
System physical parameter m, B, β in formula (6)e、kt、V01、V02And CtIt is nominal value in the design of observer and controller, its Deviation between actual value is concentrated and is placed in non-modeled terms, is d (x, t) in the second channel, is q in third channel (t);Wherein, d (x, t) is the mismatched uncertainty of system, including outer load disturbance, does not model friction and Unmarried pregnancy;q T () is the uncertainty of the modeling error of Pressure behaviour, i.e. system matches;Wherein:
For simplified system state equation, definition:
Because | PL| < < PS, then g (x) ≠ 0;Second channel mismatched uncertainty d1The matching of (x, t) and third channel is not true Qualitative d2(x, t) is all bounded, i.e.,:
|d1(x,t)|≤D1,|d2(x,t)|≤D2
D in formula1、D2Respectively | d1(x, t) | and | d2(x, t) | the upper bound, is all known positive number, and d1(x, t) first derivative is deposited ;Then Hydraulic Position Servo model is
2. the sliding-mode control that the uncertainty of Hydraulic Position Servo according to claim 1 is compensated, its feature It is that mismatch is separately designed described in step 2 and interference observer is matched, step is as follows:
Step 2-1, design mismatch interference observer:
Definition mismatches interference observer sliding-mode surface s1For:
s1=z1-x2 (10)
Wherein, z1To mismatch dynamic in interference observer;
In formula (11), k1、β1、ε1、p1And q1It is mismatch interference observer coefficient;p1< q1, and positive odd number is, k1、β1、 ε1It is positive number, β1≥D1
Then d1The estimation of (x, t)For:
Had by formula (10), (11):
Definition mismatches interference observer Lyapunov Equation:
Again because of β1≥D1, then:
If there is a positive definite integral form V0T () is met with lower inequality:
Then, V0T () is in time tsEquilibrium point is converged to inside, wherein
Wherein, 0,0 < γ < 1 of α > 0, λ >;
Therefore, V1T () will be in Finite-time convergence to equilibrium point, i.e. s1To be zero in finite time, nowAlso will convergence To zero, and because of d1(x, t) estimation difference
Then probabilistic estimation differenceAlso will be 0 in finite time;I.e. after finite time
Obtain mismatching interference observer:
Step 2-2, design matching interference observer:
Definition matching interference observer sliding-mode surface s2For:
s2=z2-x3 (20)
Wherein, z2To match dynamic in interference observer;
In formula (21), k2、β2、ε2、p2And q2It is matching interference observer coefficient;Wherein p2< q2, and positive odd number is, k2、 β2、ε2It is positive number, β2≥D2;Then d2The estimation of (x, t)For:
Had by formula (21), (22):
Definition matching interference observer Lyapunov Equation:
Again because of β2> D2, then have,
Therefore, V2T () will be zero in finite time, i.e. s2To be zero in finite time, nowZero will also be converged to again because of d2 (x, t) estimation difference
The estimation difference then disturbedAlso will be zero in finite time, i.e., after finite time
Obtain matching interference observer:
3. the sliding-mode control that the uncertainty of Hydraulic Position Servo according to claim 2 is compensated, its feature It is that the design described in step 3 is based on the sliding mode controller for mismatching and matching interference observer, specific as follows:
Define Hydrauservo System position tracking error e0, speed Tracking error e1, acceleration tracking error e2, acceleration with Track error e3
e0(t)=x1-xd(t) (27)
Wherein, xdT () is system reference signal, xdT () is that three ranks are continuous, and system reference position signalling xdT (), system are joined Examine rate signalSystem reference acceleration signalAnd system reference acceleration signalAll it is bounded;
Define sliding mode controller sliding-mode surface s:
Wherein c1、c2、c3Sliding mode controller parameter is, and is all higher than zero, and cause expression formulaIt is Hurwitz's, then have:
Obtaining sliding mode controller u is:
Wherein k is sign (s) gains, and k > 0.
4. the sliding-mode control that the uncertainty of Hydraulic Position Servo according to claim 3 is compensated, its feature It is, it is according to Liapunov stability principle proof system asymptotically stable in the large described in step 4, specific as follows:
Step 4-1, by formula (33) substitute into formula (32) have:
Define sliding mode controller Lyapunov Equation:
Then have:
Then sliding mode controller sliding-mode surface s will be zero in finite time;Now have:
Again because of s1、s2And finite time in be 0, if t1For the moment that s is zero, t2For s1It is zero moment, t3For s2It is zero Moment, then there is t4=max { t1,t2,t3, through t4Have after moment:
And because:
Then:
e2(t)+c1e1(t)+c2e0(t)+c3∫e0(t) dt=d1(t) (40)
I.e.:
Step 4-2, whenWhen:
Then have, when t → ∞ has:
Therefore e0(t)=x1-xd(t) the time tend to it is infinite under conditions of go to zero;
WhenWhen δ is positive number:
Then have, when t → ∞ has:
Therefore e0(t)=x1-xd(t) the time tend to it is infinite under conditions of converge in Uniformly stable circle.
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