CN108415252A - Electrohydraulic servo system modeling forecast Control Algorithm based on extended state observer - Google Patents
Electrohydraulic servo system modeling forecast Control Algorithm based on extended state observer Download PDFInfo
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Abstract
The electrohydraulic servo system modeling forecast Control Algorithm based on extended state observer that the invention discloses a kind of initially setting up the dynamics mathematical model of hydraulic system and uses single order Euler method by its discretization；Being then based on Design of Mathematical Model has the model predictive controller of state constraint；It is finally based on Design of Mathematical Model Discrete Extended State Observer.The present invention has merged expansion state observation thought on the basis of conventional model predictive controller, by cleverly design a model predictive equation and using the Interference Estimation of extended state observer as control output compensation, so that when system exists concurrently with state constraint and matching interference, control performance is unaffected and still keeps higher steady state controling precision；Invention enhances conventional model PREDICTIVE CONTROLs to that can not survey the inhibiting effect interfered outside, can handle state constraint simultaneously and inhibit that influence of the outer interference to system control performance can not be surveyed, and obtain good tracking performance.
Description
Technical field
The present invention relates to electro-hydraulic servo control technical field, relates generally to a kind of electro-hydraulic based on extended state observer and watch
Dress system model predictive control method.
Background technology
The application of electrohydraulic servo system has the history of last 100 years, since it is with light-weight, size is small, it is fast to be swift in response
The advantages that big with load stiffness, so being widely used in defence equipment, civilian industry, such as：In flat pad, human body sense
The adjusting etc. of device, the constant frequency of engine and constant speed is answered all to use hydraulic control.Electrohydraulic servo system is one typical non-
Linear system, it has many nonlinear characteristics and model uncertainty；Nonlinear characteristic mainly has non-linear in tribology, pressure current
Measure non-linear etc., model uncertainty can be divided into parameter uncertainty and Uncertain nonlinear, and wherein parameter uncertainty is main
There are the viscosity friction coefficient of actuator, the elasticity modulus of hydraulic oil, leakage coefficient etc., Uncertain nonlinear, which mainly has not model, to rub
Wipe dynamic, outer interference, system high-order dynamic etc..These nonlinear characteristics are present in all hydraulic systems and affect electro-hydraulic watch
Dress system develops towards high-precision, high frequency sound direction.The control of electrohydraulic servo system can also be influenced by being performed simultaneously the presence of device saturation
Performance processed.With the development of society, industrial quarters requires more the control performance of electrohydraulic servo system to want high, traditional classical control
Make theory be difficult meet demand, therefore be directed to electrohydraulic servo system in nonlinear characteristic work out it is more advanced non-thread
Property control theory is compeled as pressing as a fire singeing one's eyebrows.
For the nonlinear Control problem of electrohydraulic servo system, proposed in succession there are many method.It is wherein adaptive
Control method is very effective method for processing parameter uncertain problem, can obtain the steady-state behaviour of asymptotic tracking.
But for but seeming barely satisfactory when uncertain non-linear and state constraint problem, and the object of actual electrohydraulic servo system
Rationality can all be limited and all non-linear in the presence of uncertainty, therefore self-adaptation control method can not obtain in practical applications
Obtain high-precision control performance；As a kind of robust control method, it is uncertain that classical sliding formwork control can effectively handle model
And external disturbance, and the steady-state behaviour of asymptotic tracking is obtained, but the discontinuous controller designed by classical sliding formwork control holds
The Flutter Problem for easily leading to sliding-mode surface, to influence the control performance of system.In order to solve parameter uncertainty and not true simultaneously
The problem of qualitative non-linear, adaptive robust control method are suggested, which can be in Parameter uncertainties and outer interference
In the case of existing simultaneously, system is made to obtain preferable transient state and steady-state behaviour.But this method when response state constrains still
It is so helpless；Later researcher combines obstacle Li Yapu loves function and adaptive robust control, solve state constraint and
Model uncertain problem, although this method can constrain state variable, this method proposes higher to the initial value of system
It is required that.
Traditional model prediction (MPC) control method can effectively handle restricted problem present in controlled system.MPC
Early stage is in chemical field and slow time-varying industrial circle using relatively broad；With the development of science and technology the raising of computer performance, is opened
Beginning is gradually used in the fields such as motor, robot, unmanned；MPC is based primarily upon state equation and is derived from model prediction
Equation；Then consider input constraint or state variable constraint solving quadratic form equation, reach to the look-ahead of future output and
Output planning.Processing restricted problem is a big advantage place of model prediction (MPC) control；But traditional Model Predictive Control
Not can solve in system control can not survey time-varying interference, thus when in face of severe external environment, control effect
Fruit is barely satisfactory.
Invention content
The purpose of the present invention is to provide a kind of moulds of the high electrohydraulic servo system of strong robustness, anti-saturation, tracking performance
Type forecast Control Algorithm, and extended state observer is cleverly combined to solve the state of electrohydraulic servo system in practical applications not
It can survey and interference problem, to realize the high-precision control to electrohydraulic servo system.
To realize that said program, the technical solution adopted in the present invention are as follows：
A kind of electrohydraulic servo system modeling forecast Control Algorithm based on extended state observer, includes the following steps：
Step 1, it establishes the dynamics mathematical model of hydraulic system and uses single order Euler method by its discretization；
Step 2, there is the model predictive controller of state constraint based on Design of Mathematical Model；
Step 3, it is based on Design of Mathematical Model Discrete Extended State Observer.
Further, step 1 establishes the dynamics mathematical model of hydraulic system and with single order Euler method by its discretization,
It is specific as follows：
Step 1-1, for typical electrohydraulic servo system, wherein driving inertia negative by the hydraulic actuator of a valve control
It carries；Therefore, according to Newton's second law, the equation of motion of the system is：
In formula (1)：M is inertia load quality；PLFor the pressure difference of two chamber of hydraulic pressure, B is viscosity friction coefficient；A is hydraulic cylinder
Effective piston area；F (t) is that other do not model friction and interference；Y is the displacement of inertia load；T is time variable；Ignore liquid
The Pressure behaviour equation of the outward leakage of pressure system, two chamber of hydraulic actuator is：
In formula (2)：VtFor the sum of two cavity volume of actuator；βeFor the effective modulus of elasticity of hydraulic oil；CtIt is for interior leakage
Number；QLFor the load flow of system；Q (t) is model error；Due to the use of the servo valve of high response, it is assumed here that control is defeated
Enter relationship proportional to the spool displacement of servo valve, i.e. xv=kiu；Therefore QLThere can be following equation to calculate, i.e.,：
In formula (3)：ktFor total flow gain；PsFor system charge oil pressure；PrFor system oil return pressure；CdFor flow system
Number；ω is spool area gradient；ρ is the density of oil；kiFor proportionality coefficient；Wherein sign (u) is defined as：
(2.2) definition status variable：Then formula (1) equation of motion is converted into state side
Journey：
In formula (5):
d
(t) it is the total interference of system and model error；It can be obtained using single order Euler discrete method：
In formula (6)：TsFor sampling time, Ad=I3×3+TsA, Bud=TsBu, Bdn=TsBd, Cd=C；I3×3For the list of three ranks
Bit vector；
For ease of controller design, it is assumed that as follows：
Assuming that 1：Total interference in system, can be estimated by observer, and wherein the evaluated error of observer is in controller
Design in do not consider, i.e.,：
In formula (7)：It is interfered for the estimation of observer；W (k) is observer Interference Estimation error；
Assuming that 2：According to the principle of model prediction, need with newest measured value initial value as input, by a certain
Moment predicts future time instance；Therefore prediction time domain is set as Np, and the control time domain of system is Nc and Nc≤Np；For controller
Subsequent design need to assume：
Further, the Model Predictive Control based on Design of Mathematical Model with state constraint designed by abovementioned steps 2
Device, steps are as follows：
Definition：Δ x (k)=x (k)-x (k-1) is the increment of two moment states, similarly Δ u (k)=u (k)-u (k-1)
It is respectively the two moment increments for inputting and interfering with Δ d (k)=d (k)-d (k-1)；It can be obtained by the discretization model of formula (6)
It arrives：
Definition：Δ x (k+1 | k) indicates in k moment Δ x (k) prediction Δ x (k+1), and similarly y (k+1 | k) was indicated at the k moment
Δ x (k) predicts y (k+1), therefore can obtain following formula：
By the way that formula (10) is substituted into formula (9), the predicted value of output y can be obtained, i.e.,：
Definition：
Ye (k+1 | k)=[y (k+1 | k), y (k+2 | k) ..., y (k+Np | k)]T
Δ U (k)=[Δ u (k), Δ u (k+1) ..., Δ u (k+Nc-1)]T
Therefore status predication equation is：
Ye(k+1 | k)=HxΔx(k)+HIy(k)+HdnΔd(k)+HuΔU(k) (12)
In formula (12)：HI、Hx、HuAnd HdnIt can be obtained according to formula (11)；Due to system mode x2And x3Pass through succeeding state
Observer is estimated and is obtained, therefore there are evaluated errors；Definition： For the state increment of state estimation,
Estimated state incremental error, definition：Regard evaluated error approximation as interference, wherein
γ is gain term；Following formula can be obtained by (12) formula：
In order to track target instruction target word xd, objective function does to reflect system control performance according to object function
Go out the input of next step control forecasting；Objective function of the present invention is as follows：
In formula (14)：R and Q is respectively NpRank tracking error and NcThe diagonal weight matrix of rank controlling increment is weighed by adjusting
Value determines expectation target；In addition, Xd(k+1) it is reference instruction target.In order to solve the quadratic form of formula (13), need formula
(13) be updated in formula (14), by arrange by its abbreviation be standard type, i.e.,：
J=Δ U (k)THΔU(k)-G(k+1|k)TΔU(k) (15)
In formula (15)：
H=Hu TRTRHu+QTQ
G (k+1 | k)=2Hu TRTREp(k+1|k)
Consider state constraint：
Xmin≤Xi≤XmaxI=1,2,3
Wherein：
Xi=[xij,xij,..,xij]TJ=1,2 ..., Np
Xmin=[xmin,xmin,..,xmin]T
Xmax=[xmax,xmax,..,xmax]T
Xd(k+1)=[xd,xd,...,xd]T
It can be obtained according to the inference method of model prediction：
In formula (16)：Hix,Hidn,HiuIt equally can be by Ad、BudAnd BdnIt shows, being substituted into above-mentioned state constraint can
To obtain：
It is available by arranging：
EΔU(k)≤F (17)
In formula (17)：
Consider state constraint by optimization aim of J by being based on Hildreth quadratic forms method for solving, solves Δ U controls
Increment sequence is：
In formula (18)：hijRepresenting matrix E (2H)-1ETIn i-th and j elements；N is matrix E (2H)-1ETColumns；kiFor
Vector (F+EH-1G i-th of element)；N is natural number；In formula (19)：U (k) is current time control law.
Further, Design of Mathematical Model Discrete Extended State Observer is based on described in abovementioned steps 3, it is specific as follows：
It can be obtained by formula (5), specific design is as follows：
H (t) is the derivative for interfering d (t), definition in formula (20)： X=[x1,x2,x3,x4]T；Therefore it can be obtained by formula (20)：
Can obtain discretization model by single order Euler method is：
X (k+1)=AodX(k)+G(u,x)du(k)+Δd(k) (22)
In formula (22)：
Therefore design observer is as follows：
In formula (23)I=1,2,3,4 be design parameter, and Ho is the gain of observer, ω o>0 is observer bandwidth；It is fixed
Justice：Formula (23) is subtracted by formula (22), following formula can be obtained：
Definition：It indicates evaluated error, following formula can be obtained by formula (24)：
In formula (25)：
SelectionI=1,2,3,4 makes AooMeet Hull dimension hereby matrix, so certainly existing matrix P meets Lyapunov equatioies, i.e.,：
In formula (26):I4×4For 4 rank unit matrixs.
Compared with prior art, the present invention its remarkable advantage is：Electricity proposed by the present invention based on extended state observer
Fluid servo system model predictive control method (ESOMPC), the fusion expansion shape on traditional model predictive control method (MPC)
The thought of state observer (ESO), by design a model predictive equation and using the Interference Estimation of extended state observer as control
System output compensation so that when system exists concurrently with state constraint and matching interference, control performance is unaffected and still
Higher steady state controling precision is kept, it is outer to that can not survey to enhance conventional model PREDICTIVE CONTROL using the said program of the present invention
The inhibiting effect of interference can handle state constraint and inhibit that influence of the outer interference to system control performance can not be surveyed simultaneously, and
Obtain good tracking performance.
Description of the drawings
Fig. 1 is the schematic diagram of electrohydraulic servo-controlling system.
Fig. 2 is the electrohydraulic servo system modeling forecast Control Algorithm principle schematic in extended state observer.
Fig. 3 is that system interference is d (t)=sin (2t) [1-exp (- 0.01t3)] when, the present invention designed by ESOMPC control
Tracking process schematic of the lower system output of device effect processed to expectation instruction.
Fig. 4 is that system interference is d (t)=sin (2t) [1-exp (- 0.01t3)] when, the present invention designed by ESOMPC control
The contrast curve that the tracking error of device processed and the lower system of PID control effect changes over time.
Fig. 5 is that system interference is d (t)=sin (2t) [1-exp (- 0.01t3)] when, the present invention designed by ESOMPC control
To the estimation condition figure of state and interference under device effect processed.
Fig. 6 is system speed state constraint and system interference is d (t)=sin (2t) [1-exp (- 0.01t3)] when.This hair
The state variable x of system under bright designed ESOMPC controller actions2Time history plot.
Fig. 7 is system speed state constraint and system interference is d (t)=sin (2t) [1-exp (- 0.01t3)] Shi Benfa
The control of system exports time history plot under bright designed ESOMPC controller actions.
Fig. 8 is system speed state constraint and system interference is d (t)=sin (2t) [1-exp (- 0.01t3)] Shi Benfa
The contrast curve that the tracking error of bright designed ESOMPC controllers and the lower system of PID control effect changes over time.
Fig. 9 is that system interference is d (t)=t4sin(4t)[1-exp(-0.01t3)] when ESOMPC controller actions under system
Tracking error time history plot.
Figure 10 system interferences are d (t)=t4sin(4t)[1-exp(-0.01t3)] when the present invention designed by ESOMPC control
Device, MPC controller and PID controller processed act on the tracking error contrast curve of lower system.
Specific implementation mode
In conjunction with Fig. 1~2, the present invention is based on the electrohydraulic servo system modeling forecast Control Algorithm of extended state observer, packets
Include following steps：
Step 1, it establishes the dynamics mathematical model of hydraulic system and uses single order Euler method by its discretization；
(1.1) it is that inertia is driven by the hydraulic actuator of a valve control if Fig. 1 is a typical electrohydraulic servo system
Load；Therefore, according to Newton's second law, the equation of motion of the hydraulic system is：
In formula (1)：M is inertia load parameter；PLFor the pressure difference of two chamber of hydraulic pressure, B is viscosity friction coefficient；A is effectively living
Fill in area；F (t) is that other do not model friction and interference；Y is the displacement of inertia load；T is time variable.
Ignore the outward leakage of hydraulic actuator, then the Pressure behaviour equation of two chambers of hydraulic actuator is：
In formula (2)：VtFor the sum of two cavity volume of actuator；βeFor the effective modulus of elasticity of hydraulic oil；CtIt is for interior leakage
Number；QLFor the load flow of system；Q (t) is model error；Due to the use of the servo valve of high response, it is assumed here that control is defeated
Enter relationship proportional to the spool displacement of servo valve, i.e. xv=kiu；Therefore, QLThere can be following equation to calculate, i.e.,：
In formula (3)：ktFor total flow gain；PsFor system charge oil pressure；PrFor system oil return pressure；CdFor flow system
Number；ω is spool area gradient；ρ is the density of oil；kiFor proportionality coefficient；Wherein sign (u) is defined as：
(1.2) definition status variable：Then formula (1) equation of motion is converted into state equation
For：
In formula (5)：
d
(t) it is the total interference of system, including outer load disturbance, does not model friction, Unmarried pregnancy, system actual parameter and modeling parameters
Deviation caused by interfere；It can be obtained using single order Euler discrete method：
In formula (6)：TsFor sampling time, Ad=I3×3+TsA, Bud=TsBu, Bdn=TsBd, Cd=C；I3×3For the list of three ranks
Bit vector；
For the ease of controller design, it is assumed that as follows：
Assuming that 1：Total interference in system, can be estimated by observer, and wherein the evaluated error of observer is in controller
Design in do not consider, i.e.,：
In formula (7)：It is interfered for the estimation of observer；W (k) is observer Interference Estimation error；
Assuming that 2：According to the principle of model prediction, need with newest measured value initial value as input, by a certain
Moment predicts future time instance；Therefore prediction time domain is set as Np, and the control time domain of system is Nc and Nc≤Np；For controller
Subsequent design need to assume：
Step 2, there is the model predictive controller of state constraint based on Design of Mathematical Model, steps are as follows：
Definition：Δ x (k)=x (k)-x (k-1) is the increment of two moment states, similarly Δ u (k)=u (k)-u (k-1)
It is respectively the two moment increments for inputting and interfering with Δ d (k)=d (k)-d (k-1)；It can be obtained by the discretization model of formula (6)
It arrives：
It defines Δ x (k+1 | k) to indicate in k moment Δ x (k) prediction Δ x (k+1), similarly y (k+1 | k) is indicated in k moment Δs
X (k) predicts y (k+1)；Therefore it is as follows recurrence equation can be obtained：
By the way that formula (10) is substituted into formula (9), the predicted value of output y can be obtained, i.e.,：
Definition：
Ye (k+1 | k)=[y (k+1 | k), y (k+2 | k) ..., y (k+Np | k)]T
Δ U (k)=[Δ u (k), Δ u (k+1) ..., Δ u (k+Nc-1)]T
Therefore status predication equation is：
Ye(k+1 | k)=HxΔx(k)+HIy(k)+HdnΔd(k)+HuΔU(k) (12)
In formula (12)：HI、Hx、HuAnd HdnIt can be obtained according to formula (11)；Due to system mode x2And x3Pass through succeeding state
Observer is estimated and is obtained, therefore there are evaluated errors；Definition： For the state increment of state estimation,
Estimated state incremental error, definition：Regard evaluated error approximation as interference；Wherein
γ is gain term；Following formula can be obtained by (12) formula：
In order to track target instruction target word xd, objective function does to reflect system control performance according to object function
Go out the input of next step control forecasting；Objective function of the present invention is as follows：
In formula (14)：R and Q is respectively NpRank tracking error and NcThe diagonal weight matrix of rank controlling increment is weighed by adjusting
Value determines desired target；In addition Xd(k+1) it is reference instruction target.In order to solve the quadratic form of formula (14), need formula
(13) be updated in formula (14), by arrange by its abbreviation be standard type, i.e.,：
J=Δ U (k)THΔU(k)-G(k+1|k)TΔU(k) (15)
In formula (15)：
H=Hu TRTRHu+QTQ
G (k+1 | k)=2Hu TRTREp(k+1|k)
Consider state constraint：
Xmin≤Xi≤XmaxI=1,2,3
Wherein：
Xi=[xij,xij,..,xij]TJ=1,2 ..., Np
Xmin=[xmin,xmin,..,xmin]T
Xmax=[xmax,xmax,..,xmax]T
Xd(k+1)=[xd,xd,...,xd]T
According to the inference method of status predication equation, can obtain：
In formula (16)：Hix,Hidn,HiuIt equally can be by Ad、BudAnd BdnIt shows, being substituted into above-mentioned constraint can obtain
It arrives：
It is available by arranging：
EΔU(k)≤F (17)
In formula (17)：
By using Hildreth quadratic form method for solving, that is, consider that the state constraint situation of solution using J as optimization aim, is asked
Solving Δ U controlling increment sequences is, i.e.,
In formula (18)：hijRepresenting matrix E (2H)-1ETIn ith row and jth column element, n be matrix E (2H)-1ETRow
Number, kiFor vector (F+EH-1G i-th of element), N are natural number；In formula (19)：U (k) is current time control law.
Step 3, it is based on Design of Mathematical Model Discrete Extended State Observer, it is specific as follows：
Due to system can not be surveyed there may be state and it is outer interference be unable to measure situation, it is therefore desirable in conjunction with system object mould
Type designs expansion state observation (ESO), and the output by the way that Interference Estimation to be used as to controller output compensates, to inhibit outer interference pair
The influence of control accuracy；According to formula (5), specific design is as follows：
H (t) is the derivative for interfering d (t), definition in formula (20)： X=[x1,x2,x3,x4]T；Therefore, it can be obtained by formula (20)：
It can be obtained by single order Euler's discrete method by formula (21)：
X (k+1)=AodX(k)+G(u,x)du(k)+Δd(k) (22)
In formula (22)：
Therefore design observer is as follows：
In formula (22)I=1,2,3,4 be design parameter, and Ho is the gain of observer, ω o>0 is observer bandwidth；It is fixed
Justice：I=1 ... 4, formula (23) is subtracted by formula (22), following formula can be obtained：
Definition：It indicates evaluated error, can be obtained by formula (24)：
In formula (25)：
SelectionI=1,2,3,4 makes AooMeet Hull dimension hereby matrix, so certainly existing matrix P meets Lyapunov equatioies, i.e.,：
In formula (26)：I4×4For quadravalence unit matrix.
With reference to embodiment and attached drawing, the present invention is described in detail.
Embodiment
For the controller performance for examining designed, following parameter is taken to build electrohydraulic servo system in Matlab emulation
Mould：
Inertia load parameter M=30kg；Viscosity friction coefficient B=4000Nms/rad；Effective piston area A=
9.0478×10-4m2；The sum of two cavity volumes Vt=7.962 × 10-5；Charge oil pressure Ps=12Mpa；Return pressure Pr=0Mpa；The effective modulus of elasticity β of hydraulic oile=7 × 108Pa；Reveal coefficient Ct=4 × 10- 11m3/s/Pa.The expectation instruction of given system is x1d=10sin (2t) [1-exp (- 0.01t3)]mm。
According to three kinds of different system conditions, simulation process is divided into three parts：Following controller is taken to compare：
PID controller：The controller is common system controller in industry, main proportional item, integral term and differential
Item composition；The selecting step of PID controller parameter is：Proportional is being adjusted first, is allowing its system to stablize, then adjusts integral term
Its control accuracy is improved, finally comprehensive fine tuning parameters make system obtain best tracking performance.The controller parameter of selection
For：kP=-5000, kI=1000, kD=0.
MPC controller：In the slow time-varyings industrial circle such as chemical industry using relatively broad, which can propose output
Preceding prediction, the system control problem in the case of can handling with state constraint, controller parameter are：Predict time domain Np=5；Control
Time domain N processedc=2；Weight coefficient is：R=diag { 100,100,500,200,100 } Q=diag { 0.1,0.1 }.
ESOMPC controllers:I.e. designed controller, compares and traditional controller, can take into account state constraint and when
Become matching interference.Take the parameter of controller：Predict time domain Np=5；Control time domain Nc=2；Interference feedback gain term γ=200；It sees
The parameter for surveying device is as follows：H=[0.8,64,800,4 × 107] weight coefficient is：R=diag { 100,100,500,200,100 },
Q=diag { 0.1,0.1 }.
1. time-varying interferes d (t)=sin (2t) [1-exp (- 0.01t to operating mode3)], no restraint condition；
System output is to the tracking of expectation instruction, ESOMPC controllers and PID controller under ESOMPC controller actions
System tracking error correlation curve, ESOMPC controllers are to the estimation situation of system mode and interference respectively such as Fig. 3, Fig. 4 and Fig. 5
It is shown.As can be seen from figs. 3 and 4 system is under the control of ESOMPC controllers, steady track error about within 0.05mm,
And the steady-state error of traditional PID control control accuracy is about within 0.1mm；In comparison, designed controller tracking performance
It is better than PID controller；Designed ESOMPC controllers accurate estimating system state and can not be surveyed dry as can be seen from Figure 5
It disturbs.
2. time-varying interferes d (t)=sin (2t) [1-exp (- 0.01t to operating mode3)], x2The state of ∈ [- 19,19] mm/s is about
Beam；
This is to consider the electricity in the case of state constraint the case where meeting to encounter in practical electro-hydraulic servo control
The characteristics of liquid SERVO CONTROL problem, ESOMPC controllers is exactly that can be very good processing state constraint and input constraint problem, this
It is also that Model Predictive Control (MPC) is introduced into the reason in electro-hydraulic servo control, it can be with look-ahead system using ESOMPC
Output and planning control list entries, can preferably have electrohydraulic servo system in the case that encountering state constraint still
The control of effect.
Fig. 6 is the output speed versus time curve under the control of ESOMPC controllers of system, and Fig. 7 is under the operating mode
The control of controller exports situation；Fig. 8 is ESOMPC and PID controller position tracking error correlation curve under the operating mode；By scheming
In 6 as can be seen that ESOMPC controllers can in the case where ensureing precision, accurately by constraint of velocity prescribed limit it
It is interior；Fig. 7 in order to control device control output；As shown in Figure 8, under conditions of identical constraint, the steady state controling precision of ESOMPC is wanted
Higher than PID controller.
3. time-varying interferes d (t)=t to operating mode4sin(4t)[1-exp(-0.01t3)], no restraint condition；
This is a kind of extreme operating condition, if if designed control method can be well adapted for this extreme operating condition, it can
Show that designed control method is widely portable to all kinds of operating modes of engineering in practice.It can be with according to the expression formula of time-varying interference
Find out, interference value and interference are why to consider this extreme feelings as the time is constantly increased to the first derivative values of time
Condition be because fully to show the validity that designed controller controls electrohydraulic servo system in the case where coping with extreme situation,
Additionally fully prominent ESOMPC is compared to advantages and typical PID of traditional MPC when reply can not survey time-varying interference
Controller compares, and fully demonstrates advantages of the ESOMPC in terms of coping with system time-varying interference problem.
Fig. 9 is ESOMPC controllers position tracking error curve under the operating mode；As the time is increasing, the amplitude of interference
It is consequently increased, but system, under ESOMPC controls, control accuracy remains to be maintained at 0.1mm；Figure 10 be system at three kinds not
Tracking error comparison under being controlled with controller, as can be seen from the figure PID controller can not inhibit time-varying to interfere at all, system
Tracking error increase as time increases；And under the control of MPC, the control accuracy of system all maintains always
Within 0.2mm；Compared to more traditional MPC controller, improved in the control accuracy of ESOMPC controllers twice.
Claims (4)
1. a kind of electrohydraulic servo system modeling forecast Control Algorithm based on extended state observer, which is characterized in that including with
Lower step：
Step 1, the dynamics mathematical model for establishing hydraulic system simultaneously use single order Euler method by its discretization；
Step 2, the model predictive controller based on Design of Mathematical Model with state constraint；
Step 3 is based on Design of Mathematical Model Discrete Extended State Observer.
2. the electrohydraulic servo system modeling forecast Control Algorithm according to claim 1 based on extended state observer,
It is characterized in that, the dynamics mathematical model of hydraulic system is established described in step 1 and with single order Euler method by its discretization, specifically
It is as follows：
Step 1-1 drives inertia load for electrohydraulic servo system by the hydraulic actuator of a valve control；Therefore, according to
The equation of motion of Newton's second law, the system is：
In formula (1)：M is inertia load quality；PLFor the pressure difference of two chamber of hydraulic cylinder；B is viscosity friction coefficient；A is that hydraulic cylinder is effective
Piston area；F (t) is that other do not model friction and interference；Y is the displacement of inertia load；T is time variable；Ignore hydraulic pressure horse
The outward leakage reached, then the Pressure behaviour equation of hydraulic actuator be：
In formula (2)：VtFor the sum of two cavity volume of actuator；βeFor the effective modulus of elasticity of hydraulic oil；CtFor interior leakage coefficient；QLFor
Load flow；Q (t) is model error；Due to the use of the servo valve of high response, it is assumed here that the valve of control input and servo valve
The proportional relationship of core displacement, i.e. xv=kiu；Therefore QLIt is the relationship controlled between inputting：
In formula (3)：ktFor total flow gain；PsFor system charge oil pressure；PrFor system oil return pressure；CdFor discharge coefficient；ω
For spool area gradient；ρ is the density of oil；kiFor proportionality coefficient；Wherein sign (u) is defined as：
Step 1-2, definition status variable：Then the state equation of system is：
In formula (5)：
d(t)
It is the total interference of system；It can be obtained using single order Euler discrete method：
In formula (6)：TsFor sampling time, Ad=I3×3+TsA, Bud=TsBu, Bdn=TsBd, Cd=C；I3×3For three ranks unit to
Amount；
Assuming that as follows：
Assuming that 1：Total interference in system, can be estimated by observer, wherein the evaluated error of observer setting in controller
It is not considered in meter, i.e.,：
In formula (7)：It is interfered for the estimation of observer；W (k) is observer Interference Estimation error；
Assuming that 2：According to the principle of model prediction, need with newest measured value initial value as input, to pass through a certain moment
To predict future time instance；Therefore prediction time domain is set as Np, the control time domain of system is NcAnd Nc≤Np；After controller
Continuous design needs to assume：
。
3. the electrohydraulic servo system modeling forecast Control Algorithm according to claim 2 based on extended state observer,
It is characterized in that, design described in step 2 has the model predictive controller of state constraint based on Design of Mathematical Model, and steps are as follows：
Definition：△ x (k)=x (k)-x (k-1) is the increment of two moment states, similarly △ u (k)=u (k)-u (k-1) and △ d
(k)=d (k)-d (k-1) is respectively the two moment increments for inputting and interfering；It can be increased by the discretization model of formula (6)
Amount formula state equation is：
It defines △ x (k+1 | k) to indicate in k moment △ x (k) prediction △ x (k+1), similarly y (k+1 | k) is indicated in k moment △ x (k)
Predict y (k+1)；Therefore it is as follows recurrence equation can be obtained：
By the way that formula (10) is substituted into formula (9), the predicted value of output y can be obtained, i.e.,：
Definition：
Ye(k+1 | k)=[y (k+1 | k), y (k+2 | k) ..., y (k+Np|k)]T
△ U (k)=[△ u (k), △ u (k+1) ..., △ u (k+Nc-1)]T
Therefore status predication equation is：
Ye(k+1 | k)=Hx△x(k)+HIy(k)+Hdn△d(k)+Hu△U(k) (12)
In formula (12)：HI、Hx、HuAnd HdnIt can be obtained according to formula (11)；Due to system mode x2And x3Pass through succeeding state observer
Estimate and obtain, therefore there are evaluated errors；Definition： For the state increment of state estimation,Estimate shape
State incremental error, definition：Regard evaluated error approximation as interference, wherein γ is to increase
Beneficial item；Following formula can be obtained by (12) formula：
In order to track target instruction target word xd, objective function reflects system control performance, and makes according to object function next
Walk control forecasting input；Objective function is as follows：
In formula (14)：R and Q is respectively NpRank tracking error and NcThe diagonal weight matrix of rank controlling increment, by adjust weights come
Determine desired target；Xd(k+1) it is reference instruction target.In order to solve the quadratic form of formula (14), formula (13) is updated to formula
(14) in, by arrange by its abbreviation be standard type, i.e.,：
J=△ U (k)TH△U(k)-G(k+1|k)T△U(k) (15)
In formula (15)：
H=Hu TRTRHu+QTQ
G (k+1 | k)=2Hu TRTREp(k+1|k)
Consider state constraint：
Xmin≤Xi≤XmaxI=1,2,3
Wherein：
Xi=[xij,xij,..,xij]TJ=1,2 ..., Np
Xmin=[xmin,xmin,..,xmin]T
Xmax=[xmax,xmax,..,xmax]T
Xd(k+1)=[xd,xd,...,xd]T
According to the inference method of status predication equation, can obtain：
In formula (16)：Hix,Hidn,HiuIt equally can be by Ad、BudAnd BdnIt shows, being substituted into above-mentioned state constraint can obtain
It arrives：
It is available by arranging：
E△U(k)≤F (17)
In formula (17)：
By using Hildreth quadratic form method for solving, that is, consider that the state constraint situation of solution using J as optimization aim, solves
△ U controlling increment sequences, i.e.,
In formula (18)：hijRepresenting matrix E (2H)-1ETIn i-th and j elements；N is matrix E (2H)-1ETColumns；kiFor vector
(F+EH-1G i-th of element)；N is natural number；In formula (19)：U (k) is current time control law.
4. the electrohydraulic servo system modeling forecast Control Algorithm according to claim 3 based on extended state observer,
It is characterized in that, step 3 is based on Design of Mathematical Model Discrete Extended State Observer, specific as follows：
According to formula (5), specific design is as follows：
H (t) is the derivative for interfering d (t), definition in formula (20)： X=[x1,x2,x3,x4]T；Therefore, it can be obtained by formula (20)：
It can be obtained by the way that single order Euler method is discrete：
X (k+1)=AodX(k)+G(u,x)du(k)+△d(k) (22)
In formula (22)：
It is as follows to design observer：
In formula (23)I=1,2,3,4 be design parameter, HoFor the gain of observer；ωo>0 is observer bandwidth, definition：I=1 ... 4, formula (23) is subtracted by formula (22), following formula can be obtained：
Definition：I=1 ..., 4 indicates evaluated error, and following formula can be obtained by formula (24)：
In formula (25)：SelectionI=1,2,3,4 makes AooMeet Hull dimension hereby matrix, so certainly existing matrix P meets Lyapunov equatioies, i.e.,：
I in formula (26)4×4For 4 rank unit matrixs.
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